M. Beneke/Physics Reports 317 (1999) 1}142
RENORMALONS
M. BENEKE Theory Division, CERN, CH-1211 Geneva 23, Switzerland
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Physics Reports 317 (1999) 1}142
Renormalons M. Beneke Theory Division, CERN, CH-1211 Geneva 23, Switzerland Received July 1998: editor: R. Petronzio Contents 1. Introduction 2. Basic concepts 2.1. Divergent series 2.2. Renormalons 2.3. Factorization and operator product expansions 2.4. The Borel plane 3. Renormalons from Feynman diagrams 3.1. The #avour expansion 3.2. Ultraviolet renormalons 3.3. Infrared renormalons 3.4. Renormalization scheme dependence 3.5. Calculating &bubble' diagrams 4. Renormalons and non-perturbative e!ects 4.1. The O(N) p-model 4.2. IR renormalons and power corrections
3 6 6 9 14 18 21 21 24 40 46 49 55 55 63
5. Phenomenological applications of renormalon divergence 5.1. Directions 5.2. Hard QCD processes I 5.3. Hard QCD processes II 5.4. Heavy quarks 6. Connections with lattice "eld theory 6.1. KM and the quark mass from HQET 6.2. The gluon condensate 7. Conclusion Acknowledgements References
70 71 76 91 112 129 129 131 135 137 137
Abstract A certain pattern of divergence of perturbative expansions in quantum "eld theories, related to their small and large momentum behaviour, is known as renormalons. We review formal and phenomenological aspects of renormalon divergence. We "rst summarize what is known about ultraviolet and infrared renormalons from an analysis of Feynman diagrams. Because infrared renormalons probe large distances, they are closely connected with non-perturbative power corrections in asymptotically free theories such as QCD. We discuss this aspect of the renormalon phenomenon in various contexts, and in particular the successes and failures of renormalon-inspired models of power corrections to hard processes in QCD. 1999 Elsevier Science B.V. All rights reserved. PACS: 03.70.#k; 11.10.Gh; 12.38.!t
0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 3 0 - 6
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1. Introduction Quantum "eld theories seem to be well understood when the interactions between elementary degrees of freedom are weak. The rules of "eld theory and renormalization allow us to express observables R as series R" r aL (1.1) L L in the (renormalized) interaction strength a. Almost invariably, however, these series are divergent for any a, r L & KaLn! n@ , L
(1.2)
and it is not at all obvious how the equality sign in Eq. (1.1) should be interpreted. In this report we will be concerned with a particular source of divergence that has become known as renormalon divergence. Originally discovered in the 1970s (Gross and Neveu, 1974; Lautrup, 1977; 't Hooft, 1977), it has continued to receive attention in a much more phenomenological context since about 1992. Indeed, the divergent behaviour of perturbative expansions is more than a mathematical curiosity. It often indicates profound physics such as a non-trivial, non-perturbative structure of the vacuum and its excitations. Many of the early studies of large-order behaviour in perturbation theory, starting from the work of Dyson (1952) and others (Hurst, 1952; Thirring, 1953; Peterman, 1953), have hence focused on the question of whether a quantum "eld theory can be constructed non-perturbatively from the perturbative expansions and analyticity properties of their Green functions. This turns out not to be the case for quantum "eld theories of phenomenological relevance. The renaissance period of large-order behaviour, and renormalons in particular, dating from Brown and Ya!e (1992), Zakharov (1992) and Mueller (1992), addresses di!erent questions. From the 1970s to 1992 quantum chromodynamics (QCD) had been growing from a qualitative to a quantitative theory of strong interaction phenomena. The "rst third-order perturbative calculations had just become available for e>e\ annihilation (Gorishny et al., 1991; Surguladze and Samuel, 1991) and deepinelastic scattering (Larin et al., 1991; Larin and Vermaseren, 1991), and experiments were reaching a precision that had to be matched by theoretical accuracy. It was therefore natural to ask how much could be learned about the parameters that enter the asymptotic formula (1.2) and whether asymptotic estimates could have anything to do with exact multi-loop results, that is, whether they could be extrapolated to n&2}3. If so, one could estimate yet higher orders and improve the theoretical precision. Another aspect has drawn more attention later. As will be discussed at length, renormalon divergence is a direct consequence of the short- and long-distance behaviour of "eld theories. The long-distance behaviour is especially interesting in theories like QCD, whose coupling a grows with distance and eventually eludes a perturbative treatment. Sensitivity to Q non-perturbative long-distance/large-time behaviour is inevitable to some degree in any measurement, that refers to asymptotic states, even if the fundamental scattering process occurs at small distances such as in high-energy electron-positron annihilation or deep inelastic lepton-nucleon scattering at large momentum transfer. Perturbative factorization allows us to separate the
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short-distance part, characterized by a large momentum scale Q, from the long-distance part, characterized by a small momentum scale K&1 GeV, up to power corrections. Schematically, R(Q,K)"C(Q,k) 1O2(k,K)#power corrections (K/Q)N ,
(1.3)
with k a factorization scale. But perturbative factorization tells us little about the form of power corrections. Power corrections can be large at intermediate energies, sometimes up to M & 8 90 GeV, or they are important to ascertain the parametric accuracy that could at best be achieved perturbatively. Most of the interest in renormalons derives from the fact that the (infrared) renormalon behaviour of C(Q,k) is related to power corrections. Strictly speaking, only the scaling behaviour (in Q) of the power correction can be inferred through renormalon divergence. However, it is also interesting to take one step further and to construct models that quantify the absolute magnitude of power corrections. Models of this kind, inspired by renormalons, pro"t from being consistent with the short-distance behaviour of QCD, but su!er from being somewhat unspeci"c as far as non-perturbative properties of hadrons are concerned. Not all expectations at some time connected with the subject have been ful"lled. It may be fair to say that the conceptual progress remained little compared to the pioneering work of 't Hooft (1977), Parisi (1978), Parisi (1979), David (1984) and Mueller (1985). On the other hand, while the early discussions of renormalons refer almost exclusively to the two-point function of electromagnetic currents and its operator product expansion, the generality of the phenomenon, and its usefulness for observables that do not admit an operator product expansion, has been appreciated only recently. This development has reached the point where it has inspired new experimental QCD studies. This report re#ects this development in that it puts emphasis on results with potential phenomenological implications. It is divided roughly in two parts. The "rst part is more theoretical and collects what is known about renormalon divergence from a general point of view. The second part addresses applications to speci"c processes. The report is not intended to be comprehensive in details regarding this second part. Rather, the idea is that it summarizes, for each topic, the principal ideas and results, and that it could serve as a guide to the original literature. In Section 2 we begin with basic concepts and terminology related to divergent series and renormalons. We embark on an introductory tour through the Borel plane and treat an example of ultraviolet (UV) and infrared (IR) renormalon divergence. We then explain the connection of renormalons with operator production expansions and, more generally, perturbative factorization. This connection is crucial. In fact, many of the results on power corrections summarized in the phenomenology part could have equally been obtained from extending perturbative factorization without ever using the concept of renormalons. This section could be read as a basic introduction to the subject, summarizing the status prior to and around 1993. In Section 3 we deal with renormalons from an entirely diagrammatic point of view. Since it is the asymptotic behaviour of perturbative coe$cients in large orders which is under discussion, one should, after all, be able to extract it from Feynman graphs. Treating separately UV and IR
The long-distance part at leading power, 1O2(k,K), may vanish altogether in e>e\ annihilation observables, for example event shape variables.
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renormalons, we discuss how the values of a and b in Eq. (1.2) are computed and why K cannot be computed. The starting point is an expansion in the number of #avours in QED and QCD, which allows us to check our expectations for &real QCD'. The perturbative coe$cients r depend on L renormalization conventions to de"ne the coupling a (and, possibly, other relevant parameters) and are arbitrary to a large extent. Section 3 concludes with a discussion of how scheme dependence is re#ected in the large-order behaviour of the r and an overview of methods to calculate &bubble L graphs', which play a prominent role in applications of renormalons. In Section 4 we ask what the divergence of perturbative series tells us about non-perturbative e!ects and explain the relation of IR renormalons and power corrections. This is "rst studied in "rst orders of the 1/N expansion of the two-dimensional O(N) p-model, which, contrary to #avour expansions in QED and QCD, provides a non-perturbative set-up for the problem. We shall learn that the existence of IR renormalons is speci"c to performing infrared factorization in dimensional regularization: they are indirect manifestations of power-like factorization scale dependence, which is otherwise absent in dimensional renormalization. As a consequence, IR renormalons are related to the UV renormalization properties, power divergences, to be precise, of operators that parametrize power corrections, if such can be identi"ed. This interpretation of IR renormalons in terms of operator mixing between operators of di!erent dimension also clari"es that without additional assumptions IR renormalons can tell us little about the matrix elements of these operators. We exemplify the matching between IR renormalons and UV behaviour of power corrections for twist-four corrections to deep inelastic scattering, using the #avour expansion as a toy model. Section 5 constitutes the second part in its entirety; it reviews applications of ideas based on or related to renormalon behaviour to processes of phenomenological interest. We identify three main strains of applications: related to the size and estimation of perturbative coe$cients, related to the scaling behaviour of power corrections, and related to modelling the absolute magnitude of power corrections. Because several of these aspects can be interesting for any given process, the section is divided by processes. The "rst set of processes consists of those where the large, perturbative momentum scale is given by a large momentum transfer. Inclusive observables in high-energy e>e\ annihilation and q decay, structure functions in deep-inelastic scattering, and hadronic reactions such as Drell-Yan production belong to this class. Power corrections of order 1/Q to event shape observables in e>e\ annihilation are reviewed in some detail because of their considerable experimental interest. For the second set of observables the large scale is given by the mass of a heavy quark, of a bottom quark in practice. Beginning with the quark mass parameter itself, we then consider exclusive and inclusive heavy quark decays and, "nally, systems of two heavy quarks, described by non-relativistic QCD. The problem of power UV divergences mentioned above is even more acute in lattice computations of power-suppressed e!ects. Renormalons enter here mainly to remind us that power divergences have to be subtracted non-perturbatively. Section 6 gives a brief account of activities in this direction. In Section 7 we summarize and collect open questions.
At least in its conventional usage, that is, if one does not subtract poles in dimensions other than n for theories in n dimensions.
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2. Basic concepts In this section we brie#y introduce some concepts that appear in connection with renormalons. We begin with the notions of divergent/asymptotic series and the Borel transform. We then compute as an elementary example the leading IR and UV renormalon singularity of the vector current}current correlation function in the bubble chain approximation. This approximation is already su$cient to work out the main aspects of renormalons, with generalizations and re"nements being delegated to later sections. Because the concepts of factorization and the operator product expansion (OPE) are crucial in this context and will lead as a red thread through this review, a separate subsection expands on the relation between the OPE and renormalons. We then return to the current}current correlation function and discuss its singularities in the Borel plane. This section may be read as a "rst overview of basic ideas, which will recur in more general treatments or further examples later. The section is fairly self-contained on an elementary level, but points to later sections for more details. A more detailed and formal discussion of the divergent series problem in the context of renormalons can be found in Fischer (1997). The reprint volume (Le Guillou and Zinn-Justin, 1990) collects many of the early papers on divergent series in quantum "eld theories, with emphasis on instanton-induced divergence, and provides an introduction to the subject. 2.1. Divergent series Divergent series are common in applied mathematics and there is nothing &wrong' with them. However, given the divergent series expansion R& r aL of R, the following questions arise: LL 1. How does one assign a numerical value (&sum') to the series? 2. How is the series or its sum related to the original (&exact') function R(a)? Is the sum of the series identical to R? There is little to say about the second question for series expansions that occur in renormalizable "eld theories realized in nature, because we do not know how to de"ne R non-perturbatively. In order that a divergent series be useful as an approximation to R, it should be asymptotic to R in a region C of the complex a-plane. Then there exist numbers K such that ,
, R(a)! r aL (K a,> (2.1) L ,> L for all a in C and the truncation error at order N is uniformly bounded to be of order a,>. If r L & KaLn! n@ L
(2.2)
Lattice regularization provides the exception. In this case, one has to deal with the continuum and in"nite volume limit. We adhere to continuum de"nitions at this point.
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with constants K, a, b, one often "nds that also K Ja,N! N@. The truncation error follows the , same pattern as the terms of the series themselves. It "rst decreases until N夹&1/"a"a ,
(2.3)
beyond which the approximation of R does not improve through the inclusion of further terms in the series. If N夹<1, the approximation is good up to terms of order (2.4) K 夹a,夹&e\ ? ? . , Provided r &K , the best approximation is achieved when the series is truncated at its minimal L L term and the truncation error is roughly given by the minimal term of the series. Since there is no rigorous non-perturbative de"nition of R in theories such as QED and QCD, we cannot even ask whether series expansions are asymptotic. It is usually assumed that they are. The justi"cation is that if QED (QCD) is the theory of electromagnetic (strong) interactions, non-perturbative results are provided by (ideal) measurements. The fact that independent determinations of the coupling constant a are consistent with each other indicates that the series which enter these determinations are not entirely arbitrary. It is also usually assumed that r &K . L L Note that if nothing is known of R but its series expansion, there is actually no di!erence between a divergent and convergent series regarding the second question above. The sum of a convergent series may still di!er from R by exponentially small terms exp(!1/a). In turn, while a divergent series implies that R is non-analytic at a"0, non-analyticity does not imply divergence. The answer to the second question is trivial only if R is analytic in a"0. To improve over the best approximation (2.4), the divergent series has to be summed. There may be many ways of doing this. For factorially divergent series, Borel summation is most useful. We "rst de"ne the Borel transform as tL . (2.5) R& r aL>NB[R](t)" r L L n! L L If B[R](t) has no singularities for real positive t and does not increase too rapidly at positive in"nity, we can de"ne the Borel integral (a positive) as
RI "
dt e\R ? B[R](t) , (2.6) which has the same series expansion as R. The integral RI , if it exists, gives the Borel sum of the original divergent series. To determine whether the Borel sum equals R non-perturbatively requires that we know more about R than its formal series expansion. The Watson-Nevanlinna-Sokal theorem (Sokal, 1980) guarantees this equality, provided R meets certain analyticity requirements in addition to satisfying asymptotic estimates of the form (2.1). These requirements are too strong for renormalizable theories ('t Hooft, 1977). It is convenient to denote by r the coe$cient of aL> rather than aL. Without loss of generality, we can assume that L R has no constant term or we can treat the constant term separately.
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Returning to the Borel transform, assume that r "KaLC(n#1#b) L exactly. Unless b is a negative integer, the Borel transform of the series is given by B[R](t)"KC(1#b)/(1!at)>@ .
(2.7)
(2.8)
For b"!m a negative integer (in which case the "rst few r are discarded), it follows from (2.5) L that B[R](t)"((!1)K/C(m)) (1!at)K\ ln(1!at)# polynomial in t .
(2.9)
Hence, non-sign-alternating series (a'0), which as we shall see are expected in QED and QCD, yield singularities at positive t. It follows that already the Borel integral does not exist. Nevertheless, the Borel transform and Borel integral are useful concepts. The Borel transform can be considered as a generating function for the series coe$cients r . As seen from Eqs. (2.7) and L (2.8) the divergent behaviour of the original series is encoded in the singularities of its Borel transform. Hence, divergent behaviour is often referred to through poles/singularities in the Borel plane. This language is particularly convenient for subleading divergent behaviour. Note that larger a, i.e. faster divergence, leads to singularities closer to the origin t"0 of the Borel plane. When there are singularities at positive t, the Borel integral may still be de"ned by moving the contour above or below the singularities. For the series (2.7) with a'0, the so-de"ned Borel integral acquires an imaginary part Im RI (a)"G(pK/a) e\ ?? (aa)\@ ,
(2.10)
where the sign depends on whether the integration is taken in the upper or lower complex plane. The di!erence between the two de"nitions is often called &ambiguity of the Borel integral'. It is exponentially small in the expansion parameter a and in this sense non-perturbative. It is also parametrically of the same order as the minimal term (2.4) of the series. (We did not keep track of pre-exponential factors in Eq. (2.4).) It is customary to take these ambiguities in the Borel integral as an indication that exponentially small terms of the same form as Eq. (2.10) must be added explicitly to the series expansion, after which ambiguities in de"ning the sum of the perturbative series cancel and an improved approximation to the exact function is obtained. As a simplistic example of how this is supposed to work, let us assume that the &exact' result is given by W(n) , (2.11) R(a), (!1)L n!aL L which de"nes an analytic function in the entire complex plane except for a"0. (W is the logarithmic derivative of the C-function.) Its complete asymptotic expansion, for a'0, is given by
Because the coupling a (Q) depends logarithmically on Q, exponentially small terms (in a (Q)) are referred to as power Q Q corrections (in Q) in QCD applications.
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a divergent series and an exponentially small term: R(a)"! n!aL># e\ ? (!ln aGin) . L
(2.12)
If the divergent sum is understood as the Borel integral in the upper complex plane (upper sign) or lower plane (lower sign), Eq. (2.12) is exactly equal to Eq. (2.11) and the ambiguity in the Borel integral of the divergent series is indeed cancelled by the twofold ambiguity in the exponential term. Without more knowledge of the exact function than what is usually available in "eld theories, this is a heuristic line of thought. It also assigns a privileged role to Borel summation, as signalternating series (a(0) are then believed not to require adding exponentially small terms, while from the point of view of Eq. (2.4) there is no di!erence between sign-alternating and "xed-sign series. As will be seen later, the chain fixed-sign factorial ambiguity of the N Naddition of exponentially small terms divergence Borel integral
(2.13)
is supported by physics arguments and calculations in toy models. However, it is important to bear in mind that it is not rigorous. 2.2. Renormalons This section provides a "rst, non-technical introduction to renormalon divergence. We begin with a short and classic calculation and interpret it afterwards. Consider the correlation functions of two vector currents j "q c q of massless quarks I I
(!i) dx e\GOV 10"¹ ( j (x) j (0))"02"(q q !qg ) P(Q) I J I J IJ
(2.14)
with Q"!q. We now compute the contribution of the fermion bubble diagrams shown in Fig. 1 to the Adler function D(Q)"4p dP(Q)/dQ .
(2.15)
The set of selected diagrams is gauge-invariant, but it is not the only set of diagrams that contributes to renormalon divergence. It is selected here for illustration and a systematic investigation is postponed to Section 3. Renormalons were originally found in bubble diagrams (Gross and Neveu, 1974; Lautrup, 1977; 't Hooft, 1977), and these diagrams still feature so prominent in discussions of renormalons that sometimes they are even identi"ed with them. The Adler function requires no additional subtractions beyond those contained in the renormalized QCD Lagrangian. Therefore, no regularization is needed, provided the fermion loop insertions are renormalized. The renormalized fermion loop is given by !b a [ln(!k/k)#C] D Q
(2.16)
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with a scheme-dependent constant C and b "N ¹/(3p) the fermion contribution to the one-loop D D b-function. In the MS scheme C"!5/3. Proceeding with the diagrams of Fig. 1, we integrate over the loop momentum of the &large' fermion loop and the angles of the gluon momentum k. De"ning kK "!k/Q, we obtain
dkK Qe\ L D" a . (2.19) F(kK ) b a ln kK Q D Q kK k L The exact expression for F can be found in Neubert (1995b), but we do not need it for our present purpose. Rather than calculating the "nal integral exactly, we evaluate it approximately for n<1. Provided the renormalization scale k is kept "xed with order of perturbation theory and is taken of order Q, the dominant contributions to the integral come from k
(2.20)
(2.21)
Note that UV and IR "niteness of the Adler function implies that F must have a power-like approach to zero for both large and small kK . The integrand of Eq. (2.19) is shown in Fig. 2 for n"0 and n"2. It is clearly seen how the integrand is dominated by loop momentum of order Q for n"0, but peaks at large and small kK for n as small as 2. Splitting the integral (2.19) at kK "k/(Qe\ ) and inserting Eq. (2.20) for the small-kK interval and Eq. (2.21) for the large-kK interval, one obtains
3 Q \ L 1 Q b 11 C e\ ! D n!# e\ bL n! n# , (2.22) D" $ aL> Q D 4 k 3 k 6 2 p L where the "rst term comes from small kK and the second from large kK . Accordingly, the factorial divergence exhibited by the two series components is called infrared (IR) renormalon and ultraviolet
Unless otherwise stated, a denotes the strong coupling renormalized in the modi"ed minimal subtraction (MS) Q scheme (Bardeen et al., 1978) at the subtraction point k. We use the following convention for the b-function: b(a )"k*a /*k"b a#b a#2 . (2.17) Q Q Q Q The b-function is scheme-dependent, but the "rst two coe$cients are scheme-independent in the class of massless subtraction schemes. We will often need
1 11C 4N ¹ ! D b "b #b "! , , D 4p 3 3
(2.18)
where C "N "3, ¹"1/2 and N the number of massless quark #avours. For future use we recall that A D C "(N!1)/(2N )"4/3. $ A A The function F(kK )/(4nkK ) is called w( in Neubert (1995b). "
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Fig. 1. The simplest set of &bubble' diagrams for the Adler function consists of all diagrams with any number of fermion loops inserted into a single gluon line.
Fig. 2. The integrand of Eq. (2.19) for n"0 and n"2 as function of kK . The vertical scale is arbitrary.
(UV) renormalon. Eq. (2.22) is accurate up to relative corrections of order n (2/3)L from the infrared and (1/2)L from the ultraviolet region. The corresponding singularities in the Borel plane lie at t"!2/b (IR renormalon) and t"1/b (UV renormalon). Using Eqs. (2.7) and (2.8), the Borel D D transform obtained from Eq. (2.22) reads
\ 1 3C Q e\ ("rst IR renormalon) B[D](u)" $ 2!u 2p k
C Q 5 1 1 # $ e\ # 3p k (1#u) 6 1#u
("rst UV renormalon) ,
(2.23)
where we de"ned u"!b t. The large-order behaviour of the Adler function is dominated by the D UV renormalon. The UV renormalon singularity is a double pole (Beneke, 1993a), which is equivalent to the additional factor of n in Eq. (2.22) and can be traced back to the logarithm of kK in Eq. (2.21). Eq. (2.23) provides us with the singularities closest to the origin of the Borel plane. The exact Borel transform of the set of diagrams of Fig. 1 is known (Beneke, 1993a; Broadhurst, 1993) Some etymology: the word &renormalon' "rst appeared in 't Hooft (1977). Apparently, it was chosen, because the only other known source of divergent behaviour, related to instantons, had been called &instanton divergence'. The divergent behaviour discussed here was then novel and is characteristic of renormalizable "eld theories.
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and we return to it in Section 5.2.1. One "nds an in"nite sequence of IR (UV) renormalon poles at positive (negative) integer u with the exception of u"1. In the following, we de"ne the term &renormalon' as a singularity of the Borel transform related to large or small loop momentum behaviour. The set of bubble graphs provides an approximation to renormalon singularities. We have seen how renormalon divergence arises technically. Let us now collect some observations on the calculation, which are essential to its understanding: 1. The Adler function is UV and IR "nite and hence depends only on one scale, Q. Hence we expect that the loop integrals should be dominated by k&Q. Renormalon divergence is related to the fact that this is not the case when the number of loops becomes large. The leading contributions to Eq. (2.19) arise from k &k e e\L , (2.24) '0 k &k e eL . (2.25) 34 Hence, each logarithm of kK counts as a factor of n. The presence of two very di!erent scales and &large logarithms' suggests a renormalization group treatment. In contrast with more familiar applications of renormalization group methods, the hierarchy of scales is not "xed by external parameters, but generated by the loop diagrams themselves. All results on renormalon divergence that are independent of special classes of diagrams follow, in one way or another, from renormalization group methods or simply from the fact that there exist two di!erent scales. 2. To compute the leading divergent behaviour, only the expansion at small or large kK of the integrand of the skeleton diagrams (Fig. 1 without the fermion loop insertions) was needed. One can turn this statement around and say that the fermion loop insertions (and hence renormalon divergence) probe the large and small momentum tails of F(kK ), which would otherwise give a small contribution to the integral of F, see the case n"0 in Fig. 2. The possibility to use IR renormalons to keep track of IR sensitivity of Feynman integrals will be essential in the analysis of power corrections in QCD. In this respect the absence of a kK -term in Eq. (2.20) (and, hence, the absence of a singularity at u"1 in Eq. (2.23)) has signi"cance and corresponds to the absence of a dimension2 operator in the operator product expansion of the current-current correlation function as we discuss later in this section. For UV renormalons we observe a similarity to ordinary UV renormalization, for quadratic (logarithmic) UV divergences would be in correspondence with a kK (ln kK ) term in the large momentum expansion (2.21) of F. Hence, the suggestion of Parisi (1978) that the leading UV renormalon at u"!1 can be compensated by dimension-6 counterterms. This will be explained in detail in Section 3.2. 3. Renormalons are often associated with the notion of the &running coupling'. Interchanging the sum over n and the integration in Eq. (2.19), one obtains
D"
dkK F(kK ) a (ke\ ) , Q kK
(2.26)
Note that the recent literature is not always precise on this point. For example, &renormalon' can be found used as a synonym for &power correction', especially in the context of QCD applications.
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where a (k) 1 Q a (k)" , (2.27) Q 1!a (k)b ln(k/k) !b ln k/K Q is the familiar one-loop running coupling which follows from Eq. (2.17). Hence, the set of diagrams with a single chain of fermion loops can be obtained by integrating the skeleton diagram with the one-loop running coupling at the vertices. In writing the previous two equations, we have in fact taken the "rst step beyond the set of bubble graphs. It is evident that in QCD the fermion bubble graphs give (2.27) with the fermion contribution b to the b-function only. We may add the gluon and ghost bubbles, but the resulting D coe$cient would be gauge-dependent. The integral over the running coupling (2.26) with (2.27) literally implicitly incorporates some contributions from vertex diagrams. The substitution of b by b has profound consequences, because it changes the location of D renormalon singularities. Since the signs of b and b are di!erent, UV renormalons move to the D negative real axis in the Borel plane (implying sign-alternating factorial divergence), while IR renormalons move to the positive real axis and obstruct (naive) Borel summation. According to the discussion in Section 2.1, this implies that in QCD IR renormalons indicate that non-perturbative corrections should be added to de"ne the theory unambiguously, while the same is true for UV renormalons in QED. This is of course exactly what one expects, because the coupling becomes strong in the infrared (ultraviolet) in QCD (QED). Nevertheless, the extrapolation to the full non-abelian b at this stage seems to be ad hoc and has often been shrouded in mystery. We will argue in Section 3 that the substitution of b by b can be D justi"ed diagrammatically, so that indeed renormalon singularities are located at multiples of 1/b in QCD. We have already seen that renormalon divergence is related to the counting of logarithms of loop momentum. Since for an observable like the Adler function, the b-function (broken scale invariance) is the only source of logarithms, it seems clear that one must end up with b also in the non-abelian theory (QCD). Eventually we will see that the location of renormalon singularities is "xed by renormalization group arguments alone (Parisi, 1978; Mueller, 1985) once UV and IR factorization is established for the loop momentum regions from which renormalons arise. For the further discussion we will therefore assume that the location of renormalon singularities is dictated by the "rst coe$cient of the b-function also in QCD. 4. In spite of what has been said, the running coupling is of minor importance once one is interested in IR renormalons as probes of power corrections. This point is often not well understood. The physics of power corrections resides in the small-momentum behaviour of the skeleton diagram, see Eq. (2.20), and the running coupling is unrelated to it. The running coupling turns the small momentum behaviour into factorial divergence and makes it visible in the perturbative expansion. From the formulae of Section 2.1, we "nd, using Eq. (2.27), that the "rst IR renormalon pole in Eq. (2.23) yields an ambiguity in the de"nition of the Adler function that scales as dD(Q)Je @?Q/&(K/Q) ,
(2.28)
Hence, in QCD, u"!b t is understood in Eq. (2.23). Then, in QCD, u is positive when t is positive and this is the reason for the minus sign in the de"nition of u.
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where K is the QCD scale parameter. The power behaviour follows from Eq. (2.20). If F(k)&k? at small k, an ambiguity of order (K/Q)? would have followed, together with a leading IR renormalon singularity at u"a/2. There is a simple way to understand this: the minimal term of the series associated with the IR renormalon occurs at n such that k &K in Eq. (2.24). Hence if '0 F(k)&k? the contribution from such k scales as K?. 5. The interchange of summation and integration that led to Eq. (2.26) is actually not justi"ed, because in QCD (QED) the one-loop running coupling has a Landau pole in the infrared (ultraviolet) region. The problem this causes in de"ning the integral (2.26) is technically equivalent to the problem of de"ning the sum of the divergent series expansion of the integral. However, it is important to note that the renormalon and Landau pole phenomenon are logically disconnected in general. Whether a Landau pole exists or not is a strong-coupling problem and it depends on higher coe$cients b , etc., of the b-function and on power corrections to the running of the coupling. On the other hand, renormalons always exist as seen from the fact that the location of renormalon singularities does not depend on higher coe$cients of the b-function. (It is a simple exercise to convert Eq. (2.26) with two-loop running coupling into an expression for the Borel transform by a change of variables and to check what happens whether or not the b-function has a "xed point.) More details on this point are found in Grunberg (1996), Dokshitzer and Uraltsev (1996) and Peris and de Rafael (1996). 2.3. Factorization and operator product expansions We have already alluded to the fact that the ideas of factorization, the operator product expansion (OPE) and the renormalization group could be applied to renormalons, because there exist two very di!erent scales in the problem. Mathematically, OPEs amount to constructing an expansion in powers and logarithms of the small ratio of the two scales; so it seems that this could (almost) always be done. But there is more to factorization and OPEs, because the quantity under consideration should be broken into di!erent pieces each of which depends on only one of the two scales. The simplest and earliest example of factorization is renormalization itself. To de"ne QCD or any other renormalizable "eld theory, one has to introduce an ultraviolet cut-o! K . Renor34 malizability guarantees that all cut-o! dependence can be absorbed into universal renormalization constants. These constants being universal, i.e. independent of external momenta of Green functions, they disappear from relations of physical quantities, thus rendering them cut-o! insensitive up to terms that scale with inverse powers of the cut-o!. The residual cut-o! dependence could be further reduced by adding higher-dimension operators to the Lagrangian together with their respective set of renormalization/coupling constants. Ultraviolet renormalons, which originate from loop momentum larger than external momenta, can be understood entirely in terms of such renormalization theory methods. This will be explained in detail in Section 3.2. In QCD, which is strongly coupled in the infrared, the concept of infrared factorization is crucial. In this case, once factorization is achieved, the short-distance contributions can be computed and
The scale parameter is scheme-dependent. Without quali"cation we have in mind a scale of order 0.5-1 GeV.
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the long-distance contributions parametrized. Since the latter do not depend on the short-distance scale, they drop out in relations of physical quantities which di!er only in their short-distance set-up. Infrared factorization was "rst applied in QCD to deep-inelastic scattering (Christ et al., 1972), based on the OPE of Wilson (1969). The OPE is a powerful method, but it applies to a restricted class of observables. Most of QCD phenomenology, from jet physics to hadron}hadron collisions, relies on perturbative factorization, developed from the late 1970s on and reviewed in Collins et al. (1989). The idea of factorization is the same as in the OPE, but the approach is di!erent in that one inspects the factorization properties of Feynman diagrams. It is more di$cult in this approach to go beyond the leading power in the ratio of the two disparate scales and it has rarely been done (Ellis et al., 1982; Balitsky and Braun, 1991; Qiu and Sterman, 1991). In every case, the procedure is to identify and isolate the IR-sensitive regions in Feynman integrals and then to substitute them by non-perturbative and process-independent parameters. (As an example one may have in mind how parton densities are introduced in the perturbative factorization approach to deep inelastic scattering and compare this with the OPE treatment of deep inelastic scattering.) IR renormalons are a useful addition to this strategy. As mentioned above, IR renormalons cause ambiguities/prescription dependences in summing the associated divergent series and we expect them to be cancelled only after exponentially small terms in a have been added, or, according to Eq. (2.27), power corrections in Q. Q Let us return to the Adler function to illustrate how IR renormalons lead us to non-perturbative parameters for power corrections. First, the sequence of IR renormalons is related to terms in the small-momentum expansion in the gluon momentum. The only scale Q can be factored out and hence the IR parameter must be the matrix element of a local operator. Since there are no external hadrons, one needs a vacuum matrix element. It is a single gluon line that is soft in Fig. 1 which requires the operator to be bilinear in the gluon "elds. The Adler function is a Lorentz scalar, and gauge invariance excludes A A I, where A denotes the gluon "eld. This leaves covariant I I derivatives acting on the product of two "eld strength tensors with all Lorentz indices contracted. Thus, starting with the operator of lowest dimension (four), one is uniquely led to introduce the gluon condensate 10"G G IJ"02 (2.29) IJ as a parameter for the leading infrared contributions to the Adler function. (The argument that leads to this conclusion is worked out more thoroughly in Mueller (1985).) The gluon condensate adds to the Adler function a non-perturbative contribution of order (K/Q), in coincidence with Eq. (2.28). We also see that the potential IR renormalon at u"1 can be excluded because we would not be able to write down any operator matrix element of dimension two for it. The gluon condensate contribution to current}current correlation functions could have been discovered in this way. Historically, Shifman et al. (1979) were led to introduce it when they considered the OPE of the correlation function. The connection with IR renormalons was noted soon after by Parisi (1979). The OPE for the current}current correlation function reads 1 D(Q)"C (Q/k)# [C (Q/k) 10"G G IJ"02(k)#C (Q/k) m 10"q q"02(k)] O IJ OO Q %% #O(1/Q) ,
(2.30)
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where we assumed that the fermion in the large fermion loop in Fig. 1 has mass m ;Q. Starting O from Eq. (2.30), we conclude this section with a few general remarks regarding the relation of IR renormalons and parameters for power corrections. Most of these remarks are taken up again in Section 4 in a more concrete context. There we will compute explicit examples, non-perturbatively for the non-linear p model, and perturbatively for twist-4 corrections to deep-inelastic scattering. In constructing the OPE one introduces a factorization scale k. This is often controversially discussed in the context of renormalons, although the problem seems to be one of semantics. The loop momentum region k&Q
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themselves cannot. Except that one refers to power-like Q-dependence, the situation with IR renormalons is just the same. We have kept the quark mass in Eq. (2.30) to make the following important point: while IR renormalons lead one to introduce non-perturbative parameters for power corrections, the gluon condensate (and higher-dimension gluonic operators with derivatives) in case of Eq. (2.30), one cannot be sure that one obtains all of them. In Eq. (2.30) one would obviously miss the quark condensate, because it is the order parameter for chiral symmetry breaking, which does not occur to any ("nite) order in perturbation theory. In general, those operators will be missed that are protected from mixing with lower-dimensional ones, which usually means that their matrix elements are unambiguous and physical. In particular, there is the possibility that power corrections parametrically larger than those found through IR renormalons are missed. However, since operators do mix unless there is a particular reason that they should not (such as a symmetry), such cases can often be identi"ed. Still, it requires some understanding of the form of operators, which one does not have in all applications considered to date. IR renormalons (and condensates) evidently refer to power corrections that originate from long distances. The OPE, which factorizes long and short distances, does not exclude power corrections/non-perturbative contributions from short distances, which are logically part of the coe$cient functions (contrary to IR renormalon contributions, there is no ambiguity in this assignment). Very little is known about such contributions and the only known source of such contributions is small-size instantons. While the power-suppressed terms discussed in this report are typically of order 1/Q\, small-size instantons give rise to terms of order (1/Q)\p@ or smaller, which are strongly suppressed in comparison. For this reason, we will ignore them altogether. For the current}current correlation function the IR renormalon phenomenon reinforces that the notion of perturbative and non-perturbative e!ects is ambiguous and requires a prescription. On the other hand, one does not learn from IR renormalons anything new about power corrections beyond the content of the OPE treatment of Shifman et al. (1979). The situation is very di!erent for observables that do not admit an operator product expansion, even though they may be treated at leading power with standard perturbative methods, for instance fragmentation processes in e>e\ annihilation and the related event shape variables. Power corrections to these processes do not lend themselves easily to an operator interpretation, and IR renormalons turned out to be very useful in taking the step beyond leading power. In these cases renormalon-based methods are conceptually connected to an extension of perturbative factorization techniques beyond the leading power. Many of the questions concerning the large-order behaviour of the series expansion in a can also Q be asked about the operator product expansion, i.e. the expansion in K/Q. But much less is known for the latter. There is good reason to believe that the OPE is also divergent (Shifman, 1994), but the precise behaviour is not known, not even whether the divergence is sign-alternating or not. It is not known whether the OPE is asymptotic and whether exponentially small terms in K/Q have to be added to recover the exact result. If the OPE is asymptotic the important question arises in what
If m "0, one can instead "nd dimension-6 four-fermion operators protected by chiral symmetry. O In the case of the quark condensate, m 10"q q"02 is physical, as follows for example from the Gell}Mann} O Oakes}Renner relation.
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region in the complex Q plane it is asymptotic. For example, the expansion might be asymptotic in the euclidian region (Q real and positive), but the bound on the remainder may not be analytically continued to the cuts at negative Q or may degrade as the domain of validity in the complex plane increases. In this case further calculation of power-suppressed terms would not improve the approximation of Minkowskian quantities. The fact that the OPE may not provide an asymptotic expansion for Minkowskian quantities provides a mathematical de"nition of what is usually referred to as &violations of parton-hadron duality', although the terminology is not homogeneous in the literature. The question has so far been addressed only in models (Chibisov et al., 1997; Grinstein and Lebed, 1998; Blok et al., 1998; Bigi et al., 1998). Alternatively, one can demonstrate a certain behaviour under analytic continuation, which is independent of the dynamics of a particular theory, provided certain conditions are met by the exact result (Fischer, 1997). In this report we will not pursue this very interesting but still uncertain subject. Finally, we emphasize that renormalons can be discussed only in the context of processes for which a hard scale, say Q
Fig. 3. Singularities in the Borel plane of P(Q), the current}current correlation function in QCD. Shown are the singular points, but not the cuts attached to each of them. Recall that b (0 according to Eq. (2.18).
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Zakharov, 1992) shows that it is of order dD &(QK/k); logarithms . (2.31) 34 However, since UV renormalons produce sign-alternating factorial divergence in QCD, we do not take them as an indication that extra terms should be added to the perturbative expansion. Eq. (2.31) supports this interpretation: since the coupling renormalization scale k is arbitrary, one can make the minimal term small by increasing k. In this way, one systematically cancels (approximately) factorially large constants against powers of ln(Q/k). Note that dD is polynomial in Q (up 34 to logarithms) and therefore cannot be confused with an infrared 1/Q power correction. For the current}current correlation function all UV renormalons are double poles, if one restricts oneself to the set of bubble graphs in Fig. 1. Beyond this approximation, only the "rst singularity at u"!1 has been analysed in detail (Beneke et al., 1997a). This analysis uses renormalization group methods suggested by Parisi (1978) and developed further in Vainshtein and Zakharov (1994), Di Cecio and Pa!uti (1995), Beneke (1995) and Beneke and Smirnov (1996). These will be the subject of Section 3.2. The result is a complicated branch point structure attached to the point u"!1. UV renormalons are theory-speci"c, but process-independent. In theories with four-dimensional rotational invariance, UV renormalons are always located at positive integer multiples of 1/b , provided the theory contains no power divergences. If it does, the semi-in"nite series of UV renormalons begins at some negative integer multiple of 1/b . If O(4) invariance is broken, UV renormalons can also occur at half-integer u. An example of this kind is heavy quark e!ective theory, because it contains the heavy quark velocity four-vector (Beneke and Braun, 1994). IR renormalons are located at t"!m/b , with m"2,3,2 , i.e. u"2,3,2 . As discussed in Section 2.3 the minimal term associated with the subseries due to the "rst IR renormalon is of order (K/Q). Contrary to the situation for UV renormalons, the minimal term is k-independent and cannot be decreased (Beneke and Zakharov, 1992). (We are using dimensional regularization, see the remarks in Section 2.3.) This suggests that the ambiguities caused by IR renormalons have physical signi"cance. For current}current correlation functions one can associate them with condensates. The singularity at u"1 is absent, because there is no dimension-2 condensate in the OPE (Parisi, 1979). The set of diagrams of Fig. 1 leads to double poles for all IR renormalons expect for u"2, which is a single pole (Beneke, 1993a). Beyond this approximation, only the "rst singularity has been analysed in detail (Mueller, 1985; Zakharov, 1992), making use of the renormalization properties of the gluon condensate. This will be discussed in Section 3.3. The result is that the simple pole is turned into a branch cut, but the structure is simpler than for the "rst UV renormalon. IR renormalons are process-dependent and the absence of an IR renormalon at u"1 is speci"c to processes without identi"ed hadrons in the initial and "nal state, for which vacuum matrix elements are relevant. For example, there exists a leading singularity at u"1 in deep inelastic scattering, that is naturally connected with 1/Q twist-4 corrections (Mueller, 1993). In general, observables that can be related to o!-shell Green functions (non-exceptional external momentum
Read: The process dependence factorizes and is calculable, see Section 3.2.
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con"gurations) have IR renormalons at positive integer u. For time-like processes and on-shell Green functions, singularities at half-integer u are quite common, often beginning at u"1/2, which leads to power corrections suppressed only as K/Q (Korchemsky and Sterman, 1995a; Dokshitzer and Webber, 1995; Akhoury and Zakharov, 1995). For time-like processes one can also construct physical quantities, which are IR "nite, but arbitrarily IR sensitive (Manohar and Wise, 1995; Beneke et al., 1997b). Such quantities have IR renormalon poles at u"c with c positive and arbitrarily close to zero. Note that theories without self-interactions of massless particles such as &real' QED with massive leptons are not expected to have IR renormalons. In addition to renormalon singularities, instantons are known to produce factorially divergent series (Lipatov, 1977). In QCD instantons carry topological charge and hence they cannot be related to the perturbative expansion. However, con"gurations of n instantons and n antiinstantons with topological charge zero produce singularities at t"4pn (Bogomolny and Fateyev, 1977), the position of the singularity being related to the action of the "eld con"guration. Instanton singularities are not associated with either large or small momenta, but with the number of diagrams, which increases rapidly with order of perturbation theory. Because of their semiclassical origin, instanton singularities are under better control than renormalon singularities. For example, not only the form of the singularity, but also the residue can be calculated. For the current}current correlation function this calculation is carried out in Balitsky (1991). However, in QCD, and in fact most other interesting renormalizable theories, instanton singularities are far away from the origin of the Borel plane. Hence, we do not expect them to play a role in the large-order behaviour of perturbative expansions in QCD. Nor do they represent a dominant source of power corrections. Instanton-induced factorial divergence is reviewed in Le Guillou and Zinn-Justin (1990). What do we really know? What we have said appears to be compelling on physics grounds and is (probably) correct, but mathematical proofs are rare. Although the rules are set by specifying the Lagrangian, results on global properties of series expansions are di$cult to obtain in renormalizable "eld theories. For example, in the above discussion we have implicitly assumed that one has not applied arbitrary subtractions in de"ning the coupling. Otherwise any singularity could be obtained. Provided that only minimal subtractions are applied, it was shown in Beneke and Smirnov (1996) for o!-shell Green functions that to any "nite order in an expansion in 1/N of D (massless) QED and QCD, where N is the number of #avours, the Borel transform is analytic, D except for UV and IR renormalon singularities at the expected positions. But this may tell us more about de"ciencies of the 1/N expansion than anything else: instanton singularities are absent, D because they are exponentially small e!ects in 1/N . To the knowledge of the author, the strongest D result has been obtained by David et al. (1988), although for the scalar U theory. There it was shown that the Borel transform is analytic in a disc around the origin of the Borel plane of radius at least as large as the distance of the "rst UV renormalon from the origin. The existence of the "rst UV renormalon singularity was almost established and could be avoided only through improbable cancellations. 't Hooft (1977) has shown that, even if the Borel transform of Green functions in QCD had no singularities on the positive real axis, it could not reconstruct the Green function non-perturbatively, because the analyticity domain in a of the Borel sum would be in con#ict with the horn-shaped Q analyticity region that follows from the (assumed) non-perturbative analyticity properties of Green
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functions in momentum space. This is often interpreted to the e!ect that the Borel integral must diverge at positive in"nity. However, once we give up the idea that Green functions should be reconstructible from their Borel integrals, this conclusion does not follow. IR renormalons signal that further contributions should be added to perturbative expansions. In fact only after one sums not only perturbative series, but the entire OPE, can one hope to recover the correct analyticity properties. As far as the Borel transform de"ned by Eq. (2.5) is concerned, one usually "nds that the Borel integral (de"ned as principal value) converges, provided that all kinematic invariants (Q for the Adler function) are larger than cK, where K is the QCD scale and c a constant of order 1.
3. Renormalons from Feynman diagrams This section deals solely with properties of perturbative expansions, and phenomenological applications do not concern us here. We will try to learn as much as possible about renormalons from Feynman diagrams. Our basic tool to look at diagrams is an expansion in the number of massless fermions, although, of course, we are mainly interested in statements that are true beyond this expansion. After setting up the rules of the 1/N expansion, we consider UV renormalons in D Section 3.2, "rst to next-to-leading order in 1/N . The purpose of this exercise is to motivate the D subsequent, general, renormalization group analysis. In Section 3.3 we discuss IR renormalons. Our treatment will be more qualitative for these, mainly because a general process independent factorization theorem for IR renormalons does not hold. The subsequent two subsections address the question of scheme dependence of large-order behaviour and methods to calculate or represent bubble diagrams, which we will need in Section 5. More precisely, let us anticipate that the asymptotic behaviour due to UV and IR renormalons takes the form
c (3.1) r " K (a b )L n! n@G 1# G#2 . L G G n G We will try to calculate the parameters K , a , etc., and to understand why b enters. We emphasize G G the diagrammatic point of view, although we shall then see that every positive result can be obtained more elegantly by solving renormalization group equations. However, we believe that the diagrammatic analysis is useful to understand why some quantities in Eq. (3.1) can be calculated and others cannot. 3.1. The yavour expansion We begin the analysis with the set-up of the yavour expansion. That is, we consider QCD (or any SU(N ) gauge theory) or QED with N massless fermion #avours and we expand in 1/N . Because A D D we are interested in properties of (classes of) Feynman diagrams, we use the #avour expansion also for QCD, even though one loses asymptotic freedom and everything that is crucial for the QCD By non-perturbative we mean that the existence of resonances is crucial. It is not enough that the cut in the Adler function is generated by perturbative logarithms ln(!q/k). See Khuri (1981) for a discussion of this point.
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vacuum. In the #avour expansion (the large-N limit) the gluon self-couplings are perturbations, D generally speaking. Even if the #avour expansion could be summed, it would not reproduce QCD. We have already mentioned instanton e!ects as exponentially small, and hence non-perturbative e!ects in 1/N . D Besides there is evidence from lattice QCD (Iwasaki et al., 1997) and supersymmetric QCD (Seiberg, 1994) for a phase structure in N , so that the large-N (IR free) region and small-N D D D (asymptotically free) region are not analytically connected. This being said, we shall nevertheless see that the #avour expansion is quite instructive also in QCD. The #avour expansion is obtained in the limit N PR, where N is the number of massless D D fermion #avours in QED or QCD, keeping a "!b a JN a "xed. In this limit, fermion loops Q D Q D Q with two gluon legs are special, because they count as N a Ja "O(1). In leading order one is led D Q Q to the set of diagrams with a single chain of fermion bubbles, such as in Fig. 1 for the current}current two-point functions. The #avour expansion as an organizing principle is implicit in the works of Lautrup (1977) and 't Hooft (1977). It was used in Coquereaux (1981), Espriu et al. (1982), Palanques-Mestre and Pascual (1984) and Kawai et al. (1991) to obtain renormalization group functions in QED in the MS and on-shell schemes and then in Beneke (1993a) and Broadhurst (1993) for the photon propagator. Since then it has been applied to a variety of processes in QCD, for which we refer to Section 5. More precisely, we call a gluon propagator with any number of fermion bubbles inserted and summed over a chain. The e!ective propagator for a chain in covariant gauge is
kk kk 1 (!i) g ! I J #(!i) m I J , D (k)" IJ IJ k 1#P (k) k k
(3.2)
where P is given by Eq. (2.16) and m is the gauge-"xing parameter. The counterterms for the fermion loops are included and we have taken the limit e"(4!d)/2P0 in dimensional regularization. In Landau gauge, m"0, the propagator is particularly simple. When Feynman diagrams are written in terms of chains, all other interactions are suppressed by powers of N . Let c be D a diagram consisting of n chains, f fermion loops with more than two gluon legs (i.e. fermion loops A other than those absorbed into chains), and v three-gluon (four-gluon) vertices. Then the diagram contributes to the #avour expansion at order N\BA, where D d(c)"n !f!v !v . A
(3.3)
Examples of diagrams at leading and next-to-leading order to pair creation from an external current are shown in Fig. 4. The chain propagator becomes particularly useful after applying Borel transformation (Beneke, 1993a). Using the de"nition (2.5), one has
kk (!i) g ! I J B[a D ](u)" IJ Q IJ k k
S kk k ! e\! #(!i) m I J , k k
(3.4)
For gauge-invariant quantities, and in QED, one can neglect all k k -terms, as long as one is interested only in I J large-order behaviour.
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Fig. 4. Pair creation of quarks by an external current: (a) leading order in the #avour expansion; (b) representatives at next-to-leading order. Chains are displayed as dashed lines.
where u"!b t. Hence Borel transformation of a chain results in an analytically regularized D gluon propagator, except for the gauge parameter dependent piece. Note that inserting the renormalized chain propagator after taking eP0 is correct only if the diagram into which it is inserted does not require further subtractions. For the moment, we postpone the issue of subtractions. The Borel transform of diagrams with one chain is obtained by Borel-transforming the chain as in the previous paragraph. If the number of chains n '1, one uses the fact that the Borel transform A of a product of series is a convolution. Suppressing the Lorentz indices, the relevant identity is
1 LA LA S LA LA du d u! u B[a D(k )](u ) . (3.5) B a D(k ) (u)" Q H H H Q H H A (!b )L \ D H H H H If both ends of a chain attach to fermion or ghost lines, one obtains D (k) always in conjunction IJ with a factor of a . On the other hand, if a chain attaches to a three-gluon or four-gluon vertex, Q factors of 1/a will be left over after use of Eq. (3.5). These factors can be dealt with by applying an Q appropriate number of derivatives in u at the end. Thus, the Borel transform can be obtained by "rst calculating the skeleton diagram with all gluon propagators analytically regularized with regularization parameters u . Then for a given value of the Borel parameter u one integrates over all H regularization parameters u with the delta-function constraint of Eq. (3.5). This completes the H set-up of the #avour expansion. Our goal is to "nd the factorially divergent contributions from diagrams with an arbitrary number of chains, i.e. the singularities in u of their Borel transforms. As far as singularity structure is concerned, the second step above } integrating over the u } is trivial, once the singularities in the H space of variables u are known. Finally, we will see in the #avour expansion regularities that allow H us to sum partial contributions to all orders in 1/N . D To prepare the subsequent discussion, consider expanding Eq. (3.1) in 1/N . For simpli"cation, D we assume that there is only one component (no sum over i). Write
b !b D , (3.6) ab "ab 1# D b D the second term being O(1/N ). Furthermore, we expand K"K (1#K /N #2) and likewise D D for b and c . We assume that c "0. This is always true, if the leading order contribution to the
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#avour expansion is a one-loop skeleton diagram, because one-loop diagrams can result only in a simple pole in the Borel transform. Then
1 c
1 b !b D n#b ln n#K # #O N r "K (ab )L n! n@ 1# D b L D N n n D D
#O
1 N D
.
(3.7)
In the following we will identify the origin of the various terms in this equation. 3.2. Ultraviolet renormalons In this section, we discuss in detail the "rst UV renormalon, located at u"!1, in QED and QCD. 3.2.1. QED Most explicit calculations of renormalon behaviour have foucssed on diagrams with one chain. Large-order behaviour due to UV renormalons from diagrams with two chains (other than chains inserted into chains) was "rst considered in Vainshtein and Zakharov (1994). Further work is due to Beneke and Smirnov (1996) and Peris and de Rafael (1997). The characterization of singularities of the Borel transform for an arbitrary number of chains below follows Beneke and Smirnov (1996). Sometimes, instead of being general we take pair creation of quarks from a vector current and the two-point function of two vector currents as illustrative examples. Let C represent a diagram with n chains. Such a diagram is expressed as a series in a , whose A Q Borel transform is denoted by BC(u). Let GC(u),GC(u ,2,u A) be the Feynman integral that is L obtained by replacing each chain/dressed gluon by its Borel transform of form 1/(k)>SG (cf. Eq. G (3.2)), where k is the momentum of the ith dressed gluon line. The two latter quantities are related G by
S LA 1 LA BC(u)" du d u! u GC(u) . (3.8) G G (!b )LA\ G D G The singularity structure of GC(u) follows straightforwardly from earlier results on analytic regularization (Speer, 1968; Pohlmeyer, 1974; Breitenlohner and Maison, 1977) in the context of renormalization of "eld theories. Consider one-particle irreducible (1PI) subgraphs c of C and let u(c) be the (naive) degree of UV divergence of c obtained in the standard way from UV power counting of lines and vertices in c. For a given point u "(u ,2,u A) in the space of (complex) regularization parameters u de"ne G L the modi"ed degree of divergence uu(c)"u(c)!2 Re(u (c)), (3.9) where u (c)" u is the sum over the analytic regularization parameters of all lines of c. With JZA J this de"nition the subgraph has no overall UV divergence if uu(c)(0. One then "nds that GC(u) has poles of ultraviolet origin at those points u for which there exists a 1PI subgraph c of C such that u (c) is integer and uu(c)50.
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For example, the vertex graph in Fig. 4a has u(c)"0 and hence leads to singularities at u "0,!1,!2,2 , where u is the single regularization parameter. The box subgraph in the two-chain vertex graph in Fig. 4b is ultraviolet convergent, u(c)"!2 and leads to singularities at u #u "!1,!2,2 , where u are the two analytic regularization parameters for the two gluon propagators. A forest is a set of non-overlapping subgraphs. In the present context we can restrict these subgraphs to be 1PI. Let F be a maximal forest, i.e. a forest such that for any c not in F the union F6c is no longer a forest. Then the singularities of GC(u) in the vicinity of u are characterized by gF(u) GC(u)" , (3.10) u (c)!u(c) F AZF: uuAY SA where the functions gF are analytic in a vicinity of the point u , the sum extends over all maximal forests, and u(c) is de"ned analogously to u (c). Barring cancellations between di!erent forests, Eq. (3.10) allows us to obtain the nature of UV renormalon singularities for any diagram in the #avour expansion. Note that a maximal forest of an n-loop skeleton diagram can have at most n elements. Hence, an n-loop skeleton diagram can have at most n singular factors in Eq. (3.10). Let us illustrate Eq. (3.10) by examples: One chain. The single regularization parameter u coincides with the Borel parameter. The diagram of Fig. 4a gives rise to simple poles at u"0,!1,2 . Any single chain one-loop diagram must result in simple poles. The pole at u"0 corresponds to an explicit logarithmic ultraviolet divergence. It is cancelled by the self-energy diagrams, so that the pair creation amplitude is UV "nite. The pole at u"!1 gives rise to the "rst UV renormalon singularity at lowest order in the #avour expansion. Its residue gives K in Eq. (3.7). Furthermore, b "0 and, since there is no subleading singularity at leading order in 1/N , c "0, as assumed for Eq. (3.7). The explicit D expression is e! 1 (qc !q. q )#2 , BC?(u)" I I 6pk1#u
(3.11)
where C comes from the fermion loop (2.16) and the dots denote terms that vanish when the external &quarks' are on-shell. The residue of the pole at u"!1 follows from the coe$cient of the dk/k-term in the expansion of the Feynman integrand for k
S, and it is seen that the pole occurs when u is such that the integral is logarithmic by power counting. The fact that the pole follows from the expansion of the Feynman integrand is very important, because it implies that the residue is polynomial in the external momentum q. On dimensional grounds alone, the residue of a pole at u"!n can be written as the insertion of an operator of dimension 4#2n. From this point of view there is not
Note that we consider ultraviolet renormalon poles only.
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M. Beneke / Physics Reports 317 (1999) 1}142
much di!erence between ordinary UV divergences and UV renormalon singularities. The former produce poles at u"0 in the Borel transform. They can be compensated by counterterms, that is, insertions of operators of dimension 4 with appropriately chosen coe$cients. The latter can be compensated by insertions of higher-dimension operators (Parisi, 1978). In particular, the leading UV renormalon at u"!1 leads to considering dimension-6 operators. From the structure qc !q. q in Eq. (3.11) it can be deduced that the "rst UV renormalon is proportional to the I I zero-momentum insertion of the operator (Vainshtein and Zakharov, 1994; Di Cecio and Pa!uti 1995) O "(1/g) (tM c t) * FIJ (3.12) Q I J into the three-point function. At this order, the three-point function with insertion of O is needed only at tree level and the coe$cient of O is adjusted to reproduce the normalization of Eq. (3.11). In Eq. (3.12) FIJ is the "eld strength of an external abelian gauge "eld that relates to the external vector current through * FIJ"jJ . The factor 1/g is convention. I 4 Q Note that there is only a limited number of 1PI one-loop graphs C, which can have a UV renormalon pole at u"!1. The condition is u(C)5!2. This generalizes to all loops: a diagram C with u(C)(!2 can have a singularity at u"!1 only from subgraphs with a larger degree of divergence. For the vector current two-point function (see Fig. 5) a maximal forest contains two elements, for example the left one-loop vertex subgraph and the two-loop (skeleton) diagram itself. Each of the two gives one singular factor 1/(1#u). This explains why all UV renormalons in the Adler function turned out to be double poles as discussed in Section 2. The double pole arises from the loop momentum region, where both loop momenta are large but ordered: k, cf. Eq. (2.22). In the upper diagram Q of Fig. 5, after contraction of the vertex subgraph, one can have p
M. Beneke / Physics Reports 317 (1999) 1}142
27
Fig. 5. Leading order contribution in the #avour expansion to the two point function of vector currents j (left) and its 4 reduced diagrams with operator insertions (right). The momentum p is the loop momentum for the fermion loop.
of O is the product of the coe$cient of O and an entry of the mixing matrix. In the second case, p&q, the integration over p does not produce further singularities in u and the net result is 1/(1#u) from the insertion of O . The residue of this pole is determined by the coe$cient of O times the value of the one-loop p-integral. Because p&q, the residue is non-polynomial in q. (It contains a logarithm of q.) To summarize, the singularity at u"!1 of the two-point function of two currents at leading order in the #avour expansion is described by two universal constants, one from the one-loop vertex subgraph and the other from the region k&p
(3.14) (3.15)
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M. Beneke / Physics Reports 317 (1999) 1}142
Fig. 6. A two-chain vertex integral and the contributions to its UV renormalon singularity. The straight arrows indicate the contractions that lead to insertions of dimension-6 operators. The arrows to the right indicate contractions that correspond to logarithmic operator mixing among the dimension-6 operators.
The arrows indicate the resulting singularity of the Borel transform after integration over u according to Eq. (3.8). The second forest, containing the box subgraph, results in a double pole, to be compared with the single pole at leading order in the #avour expansion (Fig. 4a). This translates into an enhancement of the large-order behaviour of perturbative coe$cients by a factor of n, which was "rst noted in Vainshtein and Zakharov (1994). This enhancement can also be obtained by counting logarithms of loop momentum ln k. It should also be taken into account that there are of the order of n ways to distribute n fermion loops over the two photon lines of the box subgraph. Viewed this way, the enhancement is combinatorial in origin. Let us analyse again in more detail the relation between singular terms near u"!1 and loop momentum regions. A pictorial representation of this relation is shown in Fig. 6. We begin with the forest F . When k
After summation of all diagrams, the anomalous dimension of O is found to vanish.
M. Beneke / Physics Reports 317 (1999) 1}142
29
singular factor and we end up with ln(1#u) in the Borel transform. Using Eq. (2.9), this corresponds to a 1/n-suppression in large orders relative to leading order in the #avour expansion. It can be obtained from the order-a correction to the vertex function with one insertion of O . Q Hence (part of ) c follows from a "rst-order perturbative calculation. Finally, when k &k
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non-cancelling renormalization parts of the electromagnetic vertex function reside in the photon vacuum polarization. They "rst appear in the second diagram of Fig. 4b. Call the Borel parameter of the chain in the bubble u and those of the chains that connect to the bubble u . The singularities from the two-loop/one-chain vacuum polarization have already been discussed in part. From u P!1, one obtains 1/((1#u) (1#u )), which after integration over the u results in ln(1#u)/(1#u). In the sum of all diagrams, this contribution is always cancelled G (Beneke and Smirnov, 1996). But the vacuum polarization is also UV divergent and this results in a single pole 1/u for the two-loop vacuum polarization subdiagram with coe$cient proportional to b "N /(4p), the two-loop coe$cient of the QED b-function. After adding the diagram with D the charge renormalization counterterm, the singularity structure is
b b K
K
! ! "nite terms , (3.18) 1#u #u u 1#u #u #u u where the second term comes from the counterterm and K is the residue of the simple pole in the one-chain vertex graph (Fig. 4a). Note that in the counterterm the "rst factor has no u . This can be seen as follows: let k be the momentum of the two photon lines with indices u that join to the vacuum polarization insertion. On dimensional grounds the vacuum polarization insertion is proportional to (!k/k)S; this factor combines with the other two chain propagators to u #u #u . On the other hand the counterterm insertion has no momentum dependence and the two chain propagators combine with index u #u only. The UV divergence at u "0 cancels in the di!erence (3.18) and one obtains, after integration over the u , H !(K /(1#u)) b ln(1#u) (3.19) for the most singular term. It yields a ln n enhancement in the large-order behaviour relative to the leading order vertex in the #avour-expansion and gives another contribution to b in Eq. (3.7). The "nite terms in Eq. (3.18) are renormalization scheme dependent. If we assume that the subtractions do not themselves introduce factorial divergence } as is true in MS-like schemes } the subtraction dependent singular terms are ln(1#u), and hence are suppressed by one power of n relative to the leading order vertex in the #avour expansion. This scheme-dependence a!ects only c of Eq. (3.7). The ln n-enhancement from inserting a chain into a chain has been noted in Zakharov (1992). This paper also demonstrates diagrammatically how these logarithms exponentiate to n@ @, when one iterates the process of inserting chains into chains. We will see later how this exponentiation follows from renormalization group equations. In addition to the leading singularity (3.19), one also obtains 1/(1#u) with a residue that does not factorize into a one chain residue and an anomalous dimension (such as b ). It is to be interpreted as a 1/N correction to the coe$cient function of O . This correction was noted by D Grunberg (1993) and Beneke and Zakharov (1993) and provided the "rst diagrammatic evidence that the over-all renormalization of renormalon divergence cannot be computed without resorting to the #avour expansion. Its value was calculated in Beneke (1995) and can also be inferred from Broadhurst (1993). For the following discussion we imply that the self-energy type contributions are added inside the vacuum polarization insertion in Fig. 4b.
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31
To summarize: the parameter b in the asymptotic estimate of Eq. (3.1) follows from the anomalous dimension matrix of dimension-6 operators and the b-function. A rather straightforward extension of the above analysis leads to the conclusion that only one-loop anomalous dimensions and the two-loop b-function are required. Higher coe$cients contribute to preasymptotic corrections parametrized by c , etc. These pre-asymptotic corrections are also access ible through calculations involving a xnite number of loops. In particular, in addition to two-loop anomalous dimensions and the three-loop b-function, the one-loop corrections to Green functions with one zero-momentum insertion of a dimension-6 operator are required. Only the normalization K cannot be computed in a "nite number of loops. But its scheme-dependence is trivial and arises only through the counterterm for the simple fermion loop (C in Eq. (2.16)). All other schemedependence is 1/n-suppressed, see Section 3.4 for a further discussion of scheme dependence. Finally, we mention that the leading large-n behaviour can also be found by counting logarithms of loop momentum. (Recall the remarks in Section 2.) The logarithms arise from the running coupling and logarithmically divergent loop integrals. For diagrams with two chains one has to take care of the correct argument of the coupling and the hierarchy of loop momenta. The contribution from the forest F is treated by this method in Vainshtein and Zakharov (1994). 3.2.2. QCD There exists no complete analysis of non-abelian diagrams at the time of writing. The analysis of multi-chain diagrams in QED showed that higher-order corrections in 1/N do not modify the D location of UV renormalon singularities, but only their &strength', speci"ed by b in Eq. (3.1). Not so in QCD, where one expects that higher-order contributions move the singularity from m/b to D m/b after a partial resummation of the #avour expansion. As seen from Eq. (3.7) this shift should be visible in the #avour expansion as systematically enhanced corrections of the form
b !b I db I D nI, nI b b D D
(3.20)
at order 1/NI . The goal of this section is to identify the new elements in the non-abelian theory that D lead to precisely this factor for k"1. The general pattern for arbitrary k should then be transparent also. Consider again pair creation from an external abelian vector current and the "rst UV renormalon singularity at u"!1. The additional non-abelian diagrams are shown in Fig. 7a}e. In QCD gauge cancellations are more complicated than in QED and, in a general covariant gauge, the longitudinal piece in the chain propagator (3.2) has to be kept. It is convenient to perform the analysis in Landau gauge, m"0. If one chooses another gauge, one encounters UV divergent subgraphs, which are not regulated by one of the regularization parameters u , because the G longitudinal part of the chain propagator carries no u . It is then necessary to choose another G intermediate regularization for these subgraphs. For a complete treatment, this is also necessary for fermion loops with three and four gluon legs. According to Eq. (3.20) we should "nd contributions to the large-order behaviour which are enhanced by one power of n. Some of the contributions of this type are clearly not related to the b-function, for example the contribution from the box subgraph/four-fermion operator insertions to the diagram of Fig. 6. It is not di$cult to keep these contributions apart from those relevant to
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M. Beneke / Physics Reports 317 (1999) 1}142
Fig. 7. Some non-abelian vertex diagrams (a}e) at next-to-leading order in the #avour expansion. The long-dashed circle denotes a ghost loop.
restoring the non-abelian b-function. Consider the non-abelian vertex subgraph c of Fig. 7d. It is logarithmically UV divergent. When the loop momentum of c is large compared to all other momenta, the subgraph can be contracted to a point. The two leading contributions in its UV behaviour correspond to a dimension-4 counterterm (u (c)"0 in Eq. (3.10)) and a dimension-6 counterterm (u (c)"!1 in Eq. (3.10)). The second contribution, where one picks up the dk/k term from the "rst loop, is of the same type as discussed for QED. The "rst contribution, however, has an obvious connection with the b-function and we follow only this type of contribution in this section. The "rst coe$cient of the b-function follows from the one-loop pole part of the charge renormalization constant Z "Z Z\ Z\ , (3.21) E where Z is the quark wave function renormalization constant, Z the gluon wave function renormalization constant, and Z the renormalization constant for the quark-gluon vertex. Note that at the one-loop order a pole in 1/e in dimensional regularization is in one-to-one correspondence with a pole of the form 1/u(c) in a logarithmically UV divergent subgraph c. In QED, Z "Z and the only non-cancelling logarithmically divergent subgraphs occur in the photon vacuum polarization. We have already seen that these subgraphs give rise to a logarithmically enhanced contribution to the large-order behaviour proportional to the second coe$cient of the b-function. In QCD one has to keep track of all other logarithmically UV divergent subgraphs and their counterterms. There is a potential di$culty in QCD, because the gluon self-energy is quadratically divergent by power counting. A quadratic divergence gives rise to a UV renormalon singularity at u"#1 on top of an IR renormalon singularity at the same position. Consider for example the tadpole diagram in Fig. 7c. It contains dBk , (3.22) (k)>S
M. Beneke / Physics Reports 317 (1999) 1}142
33
which is zero in dimensional regularization, but should be interpreted as a UV and IR renormalon pole at u "1 with opposite signs. A UV renormalon at u"1 would complicate the discussion, because a singularity at u"!1 could in principle be obtained by inserting a dimension-2 counterterm "rst and then a dimension-8 operator. However, gauge invariance requires the gluon self-energy to have the tensor structure g !k k /k in the external gluon momentum k, while IJ I J a non-cancelling quadratic divergence would have the tensor structure of the metric tensor g . IJ Consequently, the pole at u"1 should be interpreted as purely infrared. Another way to say this is that there is no gauge-invariant dimension-2 operator in QCD that could serve as a counterterm. Consider the diagrams of Fig. 7 explicitly. Fig. 7a is obtained by substituting one fermion loop by a ghost loop, b [ln(!k/k)#C]P!(N /48p)[ln(!k/k)#C ] , (3.23) D A where N is the number of colours. The enhancement of Fig. 7a by a factor of n in the large-order A behaviour is combinatorial: at order n#1 in perturbation theory the chain in the leading order diagram of Fig. 4 has n loops and there are n ways to replace one fermion loop by a ghost loop. In terms of the Borel transform, since one factor of n is equivalent to one factor 1/(1#u), the ghost loop diagram results in
1 N K
A ! . 48pb 1#u 1#u D For the gluon loop (Fig. 7b) the same argument leads to a contribution
(3.24)
K
1 25N ! A . (3.25) 1#u 48pb 1#u D The full contribution from the gluon loop is more involved, because the gluon loop itself consists of chains. This results in further singular terms at u"!1, but they are not related to the b-function. Turning to Fig. 7d, we pick up the logarithmic UV divergence of the vertex subgraph as discussed above. Together with the counterterm, the relevant singularity structure is
1 1 1 ! , (3.26) 1#u u #u 1#u #u #u where u is the parameter of the lower chain in Fig. 7d and u are the parameters in the vertex subgraph. Integrating over the u one obtains 1/(1#u) as the leading singularity at u"!1. G However, when using Eq. (3.5) for the diagram with three chains, we assumed three powers of a from the vertices while Fig. 7d has only two. To compensate for the factor 1/a , one has to take Q Q one derivative in u. The result, putting in the correct constants and taking into account that there is an identical contribution from a symmetric diagram, reads
3N 1 K
A ! . (3.27) 8pb 1#u 1#u D The factor 1/b arises from the prefactor in Eq. (3.5). The tadpole Fig. 7c vanishes. The only D logarithmically divergent subgraph of Fig. 7e is the fermion loop. However, since the fermion loop
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itself produces no singularities in any u , this region can be considered as an order-a renormalizG Q ation of the three-gluon vertex. Hence, for the present discussion, this diagram can be considered 1/n-suppressed relative to Fig. 7d. The Fig. 7f has to be reconsidered, because its colour factor in the non-abelian case is C (C !N /2). The C-part of the logarithmic UV divergence cancels with $ $ A $ a self-energy insertion as in the abelian case. The non-abelian part contributes to Z /Z in general. But in Landau gauge the vertex subgraph is in fact UV "nite and Fig. 7f does not contribute to Z /Z . Adding together the three non-vanishing contributions (3.24),(3.25) and (3.27), one obtains
11N 1 K
A ! 12pb 1#u 1#u D
(3.28)
or, since db "!(11N )/(12p), A db n . r &K bL n! (3.29) L D b D Comparison with Eq. (3.7) shows that this is exactly what is needed at sub-leading order in the #avour expansion to restore the non-abelian b-function. As already mentioned, Eq. (3.28) represents only a fraction of all contributions to the singularity at u"!1 from the non-abelian diagrams. The ones not discussed should be associated with insertions of dimension-6 operators for some of the subgraphs of the diagrams and their interpretation parallels the QED case. A complete analysis of these contributions remains to be done. Expanding Eq. (3.1) to yet higher order in 1/N , one obtains terms of the form (db ), db b , D (b ) enhanced by n, n ln n and ln n, respectively. The origin of these terms is roughly as follows: consider a forest of nested subgraphs c Lc LC of a three-loop diagram at next-to-next-to leading order in the #avour expansion. Assume that c are both logarithmically UV divergent. Then Eq. (3.10) permits three contributions to the singularity at u"!1 from such a forest:
u (c )"0, u (c )"0, u (C)"1 , (ii) u (c )"0, u (c )"1, u (C)"1 , (3.30) (iii) u (c )"1, u (c )"1, u (C)"1 . The "rst line amounts to picking up the logarithmic UV divergences in the "rst two subgraphs. This is a contribution to the terms of the form K (db ). The third line amounts to picking up the dk/k term in c . The subsequent two contractions can then be associated with logarithmic operator mixing among dimension-6 operators. This is a contribution to terms of the form (b ). The second line represents the obvious intermediate case. This discussion has been crudely simpli"ed in that we ignored again that four-fermion operators also lead to an enhancement by a factor of n. However, this enhancement occurs only once. The e!ect of the anomalous dimension of these operators is then ln n for every loop in the #avour expansion. (i)
The required enhancement by n is most easily seen in the contribution from two ghost loops in one chain. In this particular case it can be obtained again from a simple counting argument.
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Despite the somewhat sketchy treatment of the QCD case, the general pattern that leads to the restoration of the non-abelian b seems to be simple. It con"rms the heuristic argument that b has to appear, because this coe$cient is tied to the leading ultraviolet logarithms. It would be nice to recover the full QCD b already from vacuum polarization subgraphs in order to preserve the association of renormalons with the running coupling at each vertex, which is suggested by abelian theories. This can indeed be done (Watson, 1997), at least at one loop, by a diagrammatic rearrangement (the &pinch technique') that absorbs parts of the vertex graphs into an &e!ective charge'. As far as large-order behaviour is concerned, one then has to demonstrate that after this rearrangement no contributions enhanced by a factor of n (and not related to dimension-6 insertions) are left over. 3.2.3. Renormalization group analysis We have treated the diagrammatic approach at length in order to familiarize the reader with the idea that UV factorization can be applied to the problem of UV renormalons. Diagrammatically in the #avour expansion a recursive construction of operator insertions emerges, which is completely analogous to the recursive structure of renormalization in an expansion in the coupling, except that higher-dimension operators are implied in the case of renormalons. This paves the ground to introducing the renormalization group treatment, originally suggested by Parisi (1978) and exempli"ed in the scalar -theory. The idea was worked out for QCD in Beneke et al. (1997a), on which this section is based. The renormalization group equations are formulated most easily for ambiguities or, equivalently, imaginary parts of Borel-type integrals introduced in Section 2.1. In QCD UV renormalons lie on the negative Borel axis and do not lead to ambiguities. It is technically convenient to consider the integral
\RA> C
2 1 (3.31) dt e\R ?QB[R](t), ! 't '! '0 , A b b > C given a series expansion R and its Borel transform as de"ned in Eq. (2.5). The integral is complex and its imaginary part is unambiguously related to the "rst UV renormalon singularity at t"1/b (u"!1) or large-order behaviour (compare Eqs. (2.7) and (2.10)). The statement of factorization is that the imaginary part of I[R] can be represented as I[R](a )" Q
1 (3.32) Im I[R](a , p )" C (a )ROG(a , p ) . G Q Q I Q I k G In this equation, O denote dimension-6 operators and ROG the Green function from which R is G derived with a single zero-momentum insertion of O . C (a ) are the coe$cient functions, which are G G Q independent of any external momentum p of R and in fact independent of the quantity R. They I play the same role as the universal renormalization constants in ordinary renormalization. The coe$cient function being universal, the dependence of the UV renormalon divergence on the observable R is contained in the factors ROG. These factors can be computed order by order in a by Q conventional methods. The dimension-6 operators may be thought of as an additional term, *L"!(i/k) C (a )O , G Q G G
(3.33)
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in the QCD Lagrangian with coe$cients such that for any R the imaginary part of I[R] is compensated by the additional contribution to R from *L. From the requirement that *L be independent of the renormalization scale k or from a comparison of the renormalization group equations satis"ed by I[R] and ROG it can be derived that
d 1 b(a ) !1 d ! c (a ) C (a )"0 , (3.34) Q da GH 2 GH Q H Q Q where c(a ) is the anomalous dimension matrix of the dimension-6 operators de"ned such that the Q renormalized operators satisfy (d k(d/dk)#c )O "0 . (3.35) GH GH H The unusual &!1' in Eq. (3.34) originates from the factor 1/k in Eq. (3.32). The solution to the di!erential equation (3.34) can be written as C (a )"e\ @?Qa\@ @F(a )E (a ) , Q Q G Q G Q where
(3.36)
b 1 1 ! ! (3.37) b x bx b(x) has a regular series expansion in a and incorporates the e!ect of terms of higher order than b in Q the b-function and ?Q
F(a )"exp Q
E (a )"exp G Q
dx
c2 (x) dx GH CK 2b(x) H
?Q
(3.38) ? takes into account the anomalous dimension matrix. Thus, the coe$cient functions are determined up to a -independent integration constants CK . Because the a -dependence in Eq. (3.31) translates Q G Q into n-dependence of large-order behaviour, we deduce that this n-dependence is completely determined. Only overall normalization factors related to the integration constants do not follow from the renormalization group equation. However, these integration constants are processindependent numbers; they depend only on the Lagrangian that speci"es the theory. It is in this precise sense that ultraviolet renormalon divergence is universal. When Eq. (3.32), together with the solution for the coe$cient functions, is translated into large-order behaviour and expanded formally in 1/N , one can verify that it is consistent with the D diagrammatic analysis. The unspeci"ed integration constants are related to the over-all normalization of renormalon singularities. We have already seen that its calculation requires more input than renormalization group properties. Since in fact we concluded that it cannot be calculated at "nite N , it follows that the renormalization group treatment already gives everything one can D hope to obtain for UV renormalons without approximations. To proceed we specify a basis of dimension-6 operators. In general, one is also interested in processes induced by external currents. For simplicity, we consider only vector and axial-vector The lower limit a in Eq. (3.38) is arbitrary. A change of a can be compensated by adjusting the integration constants.
M. Beneke / Physics Reports 317 (1999) 1}142
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currents and we let them be #avour singlets. Thus, in expressions like (tM Mt), a sum over #avour, colour and spinor indices is implied, and M is a matrix in colour and spinor space, but unity in #avour space. The generalization to broken #avour symmetry will be indicated below. To account for the external currents, two (abelian) background "elds v and a , which couple to the vector and I I axial-vector current, are introduced. Their "elds strengths F "R v !R v and H "R a !R a IJ I J J I IJ I J J I satisfy R FIJ"jJ and R HIJ"jJ . A basis of dimension-6 operators is then given by I 4 I O "(tM c t)(tM cIt), I
O "(tM c c t)(tM cIc t) , I
O "(tM c ¹ t)(tM cI¹ t), I
O "(tM c c ¹ t)(tM cIc ¹ t) , I
O "(1/g ) f G GJ GMI! , Q ! IJ M O "(1/g)(tM c t)R FJI, Q I J O "(1/g)R FJIRMF , Q J MI
O "(1/g)(tM c c t)R HJI , Q I J O "(1/g)R HJIRMH , Q J MI
(3.39)
where the overall factors 1/gI have been inserted for convenience. We neglected gauge-variant Q operators and operators that vanish by the equations of motion. We also assume that all N quarks D are massless. Chirality then allows us to omit four-fermion operators of scalar, pseudo-scalar or tensor type. Diagrammatically, they cannot be generated in massless QCD, because the number of Dirac matrices on any fermion line that connects to an external fermion in a four-point function is always odd. The coe$cients C corresponding to these operators therefore vanish exactly. G The leading-order anomalous dimension matrix is easily obtained. The mixing of four-fermion operators was obtained in Shifman et al. (1979) and the mixing of O into itself can be inferred from Narison and Tarrach (1983) and Morozov (1984). Writing c"ca /(4p)#2 and Q
A 0
c" 0 0
B
c 0 , 0 C
(3.40)
the mixing of four-fermion operators is described by
0
0
8 3
12
0
0
44 3
0
0
6C $ N A
6C $ N A
0
A"
9N#4 8N ! A # D 3N 3 A 3(N!4) 4 A ! N 3N A A
3(N!4) A N A !3N A
,
(3.41)
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with C "(N!1)/(2N ), N the number of colours. The non-zero entries of the 4;4 sub-matrices $ A A A B,C are B "B "8(2N N #1)/3 , A D B "B "8/3 , B "B "B "B "8C /3 , $ C "C "!2b , C "C "!4b , C "C "8N N /3 . A D
(3.42)
The mixing of O into itself is given by c "!8(N !N )/3. Note that due to a cancellation of A D di!erent diagrams the entry c vanishes. As a consequence O decouples from the mixing at leading order (Narison and Tarrach, 1983). To solve Eq. (3.38) with c(a ) and b(a ) evaluated at leading order, let b"!4pb , and let Q Q 2bj , i"1,2,4, be the eigenvalues of A and j "c /(2b). Let ; be the matrix that diagonalizes A. G Since the integration constants CK cannot be calculated and can be considered as non-perturbative, G we do not keep track of factors multiplying these constants in the following, unless they are exactly zero. Thus, we only note that no element of ; vanishes for values of N of interest. Since C is D triangular, one obtains E (a )" C a\HI, i"1,2,4, G Q GI Q I E (a )"C a\H , Q Q E (a )"C a # C a\HI, i"6, 7, G Q G Q GI Q I E (a )"C a #C a# C a\HI, G Q G Q G Q GI Q I
i"8, 9
(3.43)
with a -independent non-vanishing constants C J that depend on the nine integration constants Q CK and the elements of c. The exponents j are reported in Table 1. At leading order it is G I consistent to set F(a )"1. This completes the evaluation of the coe$cient functions in Eq. (3.36). Q As an example, we consider the Adler function de"ned in Eq. (2.15). In addition to the coe$cient functions we need the ROG , the current}current correlation function with a single insertion of O . G Since we do not follow overall constants, it is su$cient to know that ROG(a ,q)Ja, i"1,2,4, Q Q RO(a ,q)Ja , ROG(a ,q)Ja\, i"6,7, and ROG(a ,q)Ja\ for i"8,9. Having determined the Q Q Q Q Q Q a -dependence of Eq. (3.32), we use Eqs. (2.7) and (2.10), and "nd Q
r L " bL n!n@ @ K n>HG#K n\>H#K #K n (1#O(1/n)) L G G
(3.44)
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Table 1 Numerical values of j (N "3) G A N D
j
j
j
j
j
3 4 5 6
0.379 0.487 0.630 0.817
0.126 0.140 0.155 0.172
!0.332 !0.302 !0.275 !0.254
!0.753 !0.791 !0.843 !0.910
0 4/25 8/23 4/7
for the coe$cient at order aL>. Because we consider vector currents, the operators O are not Q needed. The normalization constants K are undetermined. The leading asymptotic behaviour is G r L " K bL n!n>@ @>H"K bL n!n+ , , L
(3.45)
for N "+3,4,5,. Note that the leading-order result in the #avour expansion corresponds to the D term K n in Eq. (3.44) because in the large-N limit K is suppressed by one power of N compared D D to K . For the Adler function, UV renormalons dominate the large-order behaviour and hence Eq. (3.45) represents the strongest divergent behaviour at large n. We assumed that the external vector current is #avour-symmetric. In reality, the current is j "tM c Qt, with Q "diag(e ,e ,2) a matrix in #avour space and #avour indices are summed I I GH S B over. Since #avour symmetry is broken only by the external current (all quarks are still considered as massless), the &QCD operators' O remain unaltered. The basis of ¤t operators' O has \ \ to be modi"ed to include the operators (tr Q)tM cItR FJI and tM cIQtR FJI instead of O . This ensures J J that mixing of four-fermion operators into the current operators contributes proportionally to tr Q" e and (tr Q)"( e ), as required by the existence of &#avour non-singlet' and D D D D &light-by-light scattering' terms. The matrices B and C in Eq. (3.40) change, but their pattern of non-zero entries does not. Thus, as we are not interested in over-all constants, Eq. (3.44) carries over to the present case. Eq. (3.44) holds when the series is expressed in terms of the MS renormalized coupling a . If Q a di!erent coupling is employed that is related to the MS coupling by a factorially divergent series, the coe$cients r change accordingly and Eq. (3.44) may not be valid. We return to the problem of L scheme dependence in Section 3.4. It is interesting to note that sub-leading corrections to the asymptotic behaviour can be computed without introducing further &non-perturbative' parameters in addition to the constants CK already present at leading order. As a rule, to obtain the coe$cient of the 1/nI correction, one G needs the b-function coe$cients b ,2,b , the (k#1)th loop anomalous dimension matrix and I> the k-loop correction to Green functions with operator insertions. For simplicity, suppose there is only a single operator O and RO"1#e a #2 Then, using Eqs. (3.36)}(3.38), one "nds Q Im I[R](a ,p )"const ) e\ @?Q(!b a )\@ @>A @(1#s a #2) , Q Q Q I
(3.46)
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where c c b b b s "e # ! ! # . 2b 2b b b The corresponding large-order behaviour is
(3.47)
1 b c 1 s #O . (3.48) r L " KbL C n#1# ! 1# ! L n b 2b b n The extension to higher terms in 1/n is straightforward. The renormalization group treatment can in principle be extended to the next singularity in the Borel plane at u"!2. One has to consider single insertions of dimension-8 operators and double insertions of dimension-6 operators. In practice, this is probably already too complicated to be useful. Note that the idea of compensating UV renormalons (to be precise, the imaginary part of the Borel integral due to UV renormalons) by adding higher-dimension operators has much in common with the idea of reducing the cut-o! dependence of lattice actions by adding higherdimension operators, known as Symanzik improvement (Symanzik, 1983). This analogy has been taken up by Berge`re and David (1984). The fact that the normalization of UV renormalons cannot be calculated is re#ected in the statement that the coe$cients of higher-dimension operators in Symanzik-improved actions have to be tuned non-perturbatively in order that a certain power behaviour in the lattice spacing is eliminated completely. 3.3. Infrared renormalons Infrared renormalons are more interesting than ultraviolet renormalons from the phenomenological point of view. Despite this fact, there has been less work on diagrammatic aspects beyond diagrams with a single chain. A general classi"cation of IR renormalon singularities for an arbitrary Green function comparable to the classi"cation of UV renormalons presented above is not known at this time. This is probably due to the fact that IR properties of Green functions depend crucially on external momentum con"gurations, while UV properties depend on external momenta trivially, through diagrams with counterterm insertions. The structure of UV renormalization is also simpler than IR factorization, which deals with collinear and soft divergences on a process-by-process basis. The same increase in complexity may be expected when dealing with IR renormalons. Nevertheless, this is an area where progress can be made and should be expected in the nearer future. In the following we restrict ourselves to a qualitative discussion of diagrammatic aspects of IR renormalons. This discussion divides into o!-shell and on-shell processes. More details on the connection of IR renormalons and non-perturbative power corrections can be found in Section 4 and many explicit cases will be reviewed in Section 5. 3.3.1. Ow-shell processes In QCD o!-shell, Euclidian Green functions of external (electromagnetic or weak) currents are of interest. They are related to physical processes such as the total cross section in e>e\Phadrons or moments of deep inelastic scattering structure functions through dispersion relations.
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In the #avour expansion the Borel transform of a diagram with chains is represented by the integral (3.8). We suppose that there are no power-like infrared divergences. Then for o!-shell Green functions at euclidian momenta it follows from properties of analytic regularization that the Borel transform has IR renormalon singularities at non-negative integer u. However, the structure of the singularity in terms of subgraphs is di!erent from Eq. (3.10) as di!erent notions of irreducibility apply to ultraviolet and infrared properties. The methods used in Beneke and Smirnov (1996) could be extended to this situation. Consider the two-point function of two quark currents, de"ned in Eq. (2.14), with external momentum q. IR renormalons arise from regions of small loop momentum k;q, where the integrand becomes IR sensitive. For massless, o!-shell Green functions, the IR sensitive points are those where a collection of internal lines has zero momentum. There has to be a connected path of large external momentum from one external vertex to the other. Hence, a general graph can be divided into a sum of contributions of the form shown in Fig. 8a: A &hard' subgraph to which both external vertices connect and a &soft' subgraph of small momentum lines, which connects to the hard subgraph through an arbitrary number of soft lines. In terms of the operator expansion (OPE), the soft subgraph corresponds to the matrix element of an operator and the hard part to the coe$cient function. An analysis of the leading IR renormalon contribution (t"!2/b ) to the current}current correlation function based on factorization of hard and soft subgraphs can be found in Mueller (1985). In Section 2.2 we considered the leading-order diagrams of Fig. 1 in the loop momentum region, where the soft part consisted of a single gluon line (or chain). The general classi"cation would also allow a quark line or more than one line in the soft part. These parts are associated with condensates in the OPE containing quark "elds. For the analysis of IR renormalons soft quark lines alone play no role, because they cannot be &dressed' with bubbles, which is necessary in order to turn IR sensitivity in a skeleton diagram into a factorially divergent series expansion. An immediate consequence of the factorization expressed by Fig. 8a is that in order for the diagram to contribute to an IR renormalon at t"!m/b , the soft part must connect to the hard part by not more than 2m gluon lines. This follows from the fact that each additional such line adds one hard propagator to the hard part, which counts as 1/q. On dimensional grounds this factor must be compensated by a power of one of the small momenta k . Such factors result in G a suppression of the large-order behaviour which is related to integrals that generalize
O
dk kK\[b ln(k/q)]L&(!2b /m)Ln! . (3.49) In general, the location of IR renormalons and the possible contributions to a singularity at a particular point follow from such IR power counting arguments. The leading-order diagrams in the #avour expansion (Fig. 1) result in dk/k for small k. This leads to a singularity at t"!1/b for each diagram, which can be associated with the operator A AI . Gauge invariance of the current}current two-point function requires that these leading I contributions cancel in the sum of diagrams. After this cancellation the leading term is dk, Recall that in the #avour expansion u"!b t, where t is the Borel parameter. In QCD u"!b t, so that in both D cases, QED and QCD, IR renormalons are located at positive u.
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associated with a singularity at t"!2/b and the operator G GIJ as discussed in Section 2. IJ Consider now the diagram with two chains shown in Fig. 9a. If both gluon momenta are small, power counting gives dk /kdk /k which can contribute to the singularity at t"!2/b . This contribution must be associated with the (A ) term in the operator G GIJ and it is hence related I IJ to the leading order in the #avour expansion by gauge invariance. Except for this trivial contribution, the region when both gluon momenta are small contributes only to subleading renormalon singularities at t'!2/b . When one of the gluon lines is hard and only one is soft, a contribution to the order a correction of the coe$cient function of G GIJ is obtained. Because one loses one Q IJ power of a , this contribution is 1/n-suppressed in large orders relative to the leading order in the Q #avour expansion. We conclude that the leading IR renormalon at u"2 is determined by diagrams with only a single soft chain, up to contributions constrained by gauge invariance and up to a calculable multiplicative factor that follows from the coe$cient function of G GIJ . These IJ diagrams are shown in Fig. 9b, where the shaded circle denotes an arbitrary collection of soft lines. Note the di!erence with the corresponding analysis for UV renormalon singularities, in which case Fig. 9a was found to be enhanced relative to the leading order in the #avour expansion rather than suppressed. The diagrams of Fig. 9b have been considered further in Zakharov (1992), Grunberg
Fig. 8. Infrared regions that give rise to infrared renormalons. (a) For a current}current two-point function at euclidian momentum. The external currents are shown as dashed lines. (b) For an event-shape variable in e>e\ annihilation near the two-jet limit. Wavy lines represent collections of soft lines.
Fig. 9. Two diagrams at higher order in the #avour expansion.
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(1993) and Beneke and Zakharov (1993). It was found that the residue of the IR renormalon singularity receives contributions from arbitrarily complicated graphs in the shaded circle and remains uncalculable (Grunberg, 1993; Beneke and Zakharov, 1993) despite the simpler overall diagram structure compared to the UV renormalon case. A graph-by-graph comparison of some contributions to the "rst IR and "rst UV renormalon is summarized in Table 2. A complete characterization of IR renormalon singularities must account not only for powers of small momenta but also for logarithms of k/q. The soft subgraphs contains renormalization parts, when some soft momenta are larger than others: k ;k . These renormalization parts lead to logarithms whose coe$cients are given by renormalization group functions and introduce the e!ect of higher order coe$cients in the b-function and operator anomalous dimensions into the large-order behaviour. Technically, in the #avour expansion, this occurs in a way similar to the UV renormalon case. In particular, there is no di!erence between UV and IR renormalons as far as the mechanism that restores the non-abelian b-function coe$cient b is concerned (see Section 3.2.2). Once factorization is established, the most elegant characterization of IR renormalon singularities follows from "rst identifying the &operator content' of the soft subgraph and then from deriving an evolution (renormalization group) equation for it. Consider a physical quantity such as the Adler function (2.15) or its discontinuity and its series expansion r aL>(Q) in a normalized at Q. L Q Q The IR renormalon behaviour of the coe$cients r leads to an ambiguity in the Borel integral with L a certain scaling behaviour in Q. This scaling behaviour must be matched exactly by higherdimension terms in the OPE. For simplicity, we assume that there is only one operator O of dimension d with anomalous dimension c as de"ned in Eq. (3.35) and coe$cient function C(1,a (Q))"c #c a (Q)#2. The scaling behaviour is given by Q Q 1 C(Q/k,a )10"O"02(k)"const eB @?Q/(!b a (Q))B@ @ Q Q QB
;F(a (Q))B exp ! Q
?Q/ c(x) dx C(1,a (Q)) , Q 2b(x) ?
(3.50)
where F is de"ned in Eq. (3.37). Using Eqs. (2.7) and (2.10), the large-order behaviour r L " K L
2b L db c C n#1! # d 2b 2b
1 d s #O 1# ! n 2b n
(3.51)
Table 2 Comparison of contributions of various diagrams to the leading UV and IR renormalon behaviour. For the UV renormalon the displayed factor multiplies bL n!, for the IR renormalon (!b /2)Ln!. In the case of Fig. 9b we refer to the diagram with a chain inserted into a chain analogous to Fig. 4b Diagram
Fig. 1
Fig. 9a
Fig. 9b
UV IR
n 1
n 1/n
n ln n ln n
Ignoring O(1) "xed by gauge invariance.
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with c c b db db c (3.52) s " ! # # ! c 2b 2b 2b 2b follows. Note the di!erent signs of the anomalous dimension terms compared to Eq. (3.48). (Otherwise the "rst UV renormalon can formally be obtained from setting d"!2.) The global normalization K is not determined. This equation is valid provided the renormalization counterterms do not absorb factorial divergence into the de"nition of renormalized parameters (Mueller, 1985; Beneke, 1993b); see also Section 3.4. For current}current correlation functions the leading IR renormalon corresponds to d"4 and O"a G GIJ . Taking into account that for this operator c "0 and c "2b , one reproduces the Q IJ leading asymptotic behaviour and the 1/n correction, obtained in Mueller (1985) and Beneke (1993b), respectively. The 1/n correction could be computed also, if the two-loop correction to the coe$cient function of the gluon condensate were known. An important point is that the unknown constant K is a universal property of the soft part in Fig. 8a, that is a property of the operator O. Hence for correlation functions with di!erent currents, which di!er only in their hard part, the diwerence in the leading IR renormalon behaviour is calculable. We refer to this property as universality of the leading IR renormalon or 1/Q power correction. Note, however, that universality is more restricted for IR renormalons than for UV renormalons, because it refers to a speci"c class of processes, in the present case given by various current}current correlation functions. Let us also note that for certain operators K can be exactly zero. These are operators like q q, which are protected from perturbative contributions to all orders in perturbation theory. Our discussion has focused on the current}current correlation functions. The generalization to other o!-shell quantities is straightforward. 3.3.2. On-shell processes For on-shell, Minkowskian processes the classi"cation of IR sensitive regions of a Feynman integral is more complicated than for o!-shell quantities. As is well known, in addition to soft, zero-momentum lines, collinear con"gurations of massless lines (&jets') have to be considered. Furthermore, on-shell propagators do not give power suppression, even if the line momentum is of order q. As a consequence, soft subgraphs, which connect to on-shell propagators, cannot be parametrized by local operators. As an example, the infrared regions that contribute to power corrections to two-jet-like observables in e>e\ annihilation are shown in Fig. 8b. Non-local operators that parametrize power corrections to a class of jet observables were "rst analysed in Korchemsky and Sterman (1995a). It is characteristic of o!-shell processes that IR renormalons occur only at positive integer u, which implies power corrections as powers of 1/Q and not powers of 1/Q, where Q is the &hard' scale of the process. For on-shell quantities the generic situation leads to IR renormalons at positive half-integers and integers and a series of power corrections in 1/Q. To illustrate this point, we consider a simpler case than Fig. 8b, a system with one heavy quark. More precisely, we consider the mass shift dm"m!m (m ), the di!erence between the pole mass and the MS mass +1 +1 of a heavy quark. This is in fact the quantity where IR renormalons leading to linear suppression in the hard scale, here m, have been found "rst (Beneke and Braun 1994; Bigi et al., 1994b).
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Fig. 10. Infrared regions that contribute to the "rst IR renormalon in the mass shift dm.
It is a trivial consequence of IR power counting to see that the IR contribution to the mass shift is suppressed only linearly in m. The one-loop contribution to the heavy quark self-energy R(p) evaluated at p"m is
R(m)Jm
dk & dk . k(2p ) k#k)
(3.53)
When the one-loop diagram is dressed with vacuum polarization (&bubble') insertions one obtains (!2b )Ln!aL> in large orders, i.e. an IR renormalon singularity at u"1/2. The IR sensitive Q regions in an arbitrary diagram are shown in Fig. 10. The important di!erence to Fig. 8a is that one obtains a contribution to the singularity at u"1/2 for an arbitrary number of gluon couplings to the heavy quark line, because the heavy quark propagators are nearly on-shell. The IR renormalon singularity cannot be associated with a local operator as in the case of o!-shell correlation functions. The situation is still simple, though. As far as the leading IR renormalon is concerned, the numerator of the heavy quark propagator can be approximated by mv. #m, where p"mv is the heavy quark momentum and v"1. Hence, using also the on-shell condition, gluons couple only through the combination v ) A. In a temporal axial gauge with v ) A"0, they decouple and the leading IR renormalon can be seen to correspond to an operator bilinear in the quark "eld with "elds at non-coincident positions. In a general gauge a phase factor E QT Q Pe !
(3.54)
accounts for the non-vanishing temporal soft gluon couplings (Bigi et al., 1994b) and makes the non-local operator gauge-invariant. In high-energy processes involving massless quarks there are in addition collinear-sensitive regions such as &J' in Fig. 8b. However, it seems that power corrections from hard-collinear regions (energy u much larger than transverse momentum k ) are always suppressed by powers of , Q rather than Q. There is no proof to all orders of this statement yet, but the following heuristic argument may illustrate the point: let p be the momentum of a fast on-shell particle, p&Q, after emission of a hard-collinear on-shell particle with u&Q, k ;Q, where k is the transverse , , momentum relative to p. Then the propagator 1 1 " (3.55) (p#k) p(u!(u#k ) , is expanded in k /u&k /Q and Q enters only quadratically. Since the same is true of the , , hard-collinear phase space, it may be argued that the transverse momentum, and hence Q, always enters quadratically as long as energies are large.
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As a consequence, if 1/Q power corrections exist and if one is interested only in those, the diagram of Fig. 8b can be somewhat simpli"ed. The jet parts J can be replaced by Wilson line operators to which soft gluons couple through a phase factor. In general, the leading-order eikonal approximation may not be su$cient and the "rst-order correction to it must be kept. However, many hadronic event shape observables, which are particularly interesting with respect to 1/Q power corrections, have a linear suppression of soft regions built into their de"nitions. For such observables, the analysis simpli"es further, since the conventional eikonal approximation can be used. Systematic investigations of power corrections to such quantities beyond one gluon emission have been started in Korchemsky et al. (1997) and Dokshitzer et al. (1998a). We will return to this topic in Section 5.3.2 in connection with the phenomenology of power corrections to hadronic event shape variables. 3.4. Renormalization scheme dependence The answer to the following question is overdue: Since the perturbative coe$cients can be altered arbitrarily by changing the renormalization convention, which convention has been implicit in the derivation of large-order behaviour? The short answer is that a renormalization prescription must be used in which the subtraction constants are not factorially divergent. This ensures that bare and renormalized parameters are related by convergent series (although every coe$cient of the series diverges when the the cut-o! is removed) and that no factorial divergence is &hidden' in the formal de"nition of the renormalized parameters. Such schemes have been called regular in Beneke (1993b). Before addressing scheme transformations in general, let us consider the issue of subtractions in the #avour expansion. Suppose R is a renormalization scheme-invariant quantity, which depends only on the strong coupling, for example the Adler function (2.15). We calculate its Borel transform in leading order in the #avour expansion in four dimensions by inserting renormalized fermion loops (2.16) into a gluon line. Since by assumption the quantity is scheme-invariant, no further subtractions, except for C in Eq. (2.16) are needed and the calculation in four dimensions is justi"ed. The result has the form
B[R](u)"
Q \S F(u) . e! k
(3.56)
The function F is scheme and scale independent, but the Borel transform is not, because it is de"ned as a Borel transform with respect to the scheme and scale-dependent coupling a . The prefactor in Q Eq. (3.56) can be combined with the exponent in the (formal) Borel integral
1 Q !b ln #C F(u) (3.57) D k a Q such that the exponent is manifestly scheme and scale invariant at leading order in the #avour expansion. The de"nition of the Borel transform can be modi"ed in such a way as to preserve dt exp !t
A detailed analysis of (the cancellation of) K/m power corrections at two loops for inclusive heavy quark decay can be found in Sinkovics et al. (1998).
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manifest scale and scheme independence beyond the leading order of the #avour expansion (Beneke, 1993a; Grunberg, 1993), but the de"nition in terms of perturbative coe$cients becomes complicated. Suppose now that R is not in itself physical, but requires additional subtractions beyond the renormalization of the fermion loops in the chain. For example, R may be the gluon/photon vacuum polarization or the quark mass shift discussed in Section 3.3. In this case the result of the calculation takes the form B[R](u)"((Q/k)e!)\SF(u)!S(u),
(3.58)
where F(u) has a pole at u"0. This pole is cancelled by the scheme-dependent but momentumindependent subtraction function S(u), which is arbitrary otherwise. UV and IR power counting relates the UV and IR renormalon poles of F(u) to the behaviour of loop diagrams at large and small momentum. In order that these relations remain valid, the function S(u) must not introduce singularities in u other than at u"0. At this leading order in the #avour expansion, the subtraction function can be expressed in terms of renormalization group functions (Espriu et al., 1982; Palanques-Mestre and Pascual, 1984; Beneke and Braun, 1994), the b-function in the case of the vacuum polarization, and the anomalous dimension of the quark mass in the case of the mass shift. The requirement that S(u) be analytic except at u"0 results in the requirement that the renormalization group functions have convergent series expansions in a , or at least they should not diverge Q as fast as factorials. This is indeed true, at least to leading order in the #avour expansion, for the MS de"nition of the coupling and the quark mass, but it is obviously not true for &physical' de"nitions of the coupling, because the perturbative expansions of physical quantities do have renormalons. Of course, once the large-order behaviour of two physical quantities expressed as series in a coupling, de"ned in a regular scheme, is known, the two physical quantities can always be related directly to each other, and the large-order behaviour of this relation can be found. Once S(u) is speci"ed at leading order in the #avour expansion, it appears as a counterterm in higher orders, for example as a vacuum polarization insertion in the second diagram of Fig. 4b. In this case S(u)!b /u" s uI appears as &"nite terms' in Eq. (3.18) and contributes to the I I singularity at u"1 as (K /(1#u #u ))s uI Ps (1#u)Iln(1#u) . (3.59) I I Since s is proportional to the b in the large-N limit, it follows that the scheme-dependent I I> D b-function coe$cient b (k'0) enters as a 1/nI> correction to the large-order behaviour, I> provided S(u) is analytic in the complex plane with the origin removed. This is in accordance with the general results (3.47), (3.48) and (3.51), (3.52). We emphasize that these general results are valid only in regular renormalization schemes. It is reasonable to conjecture (and true to leading order in the #avour expansion) that the MS scheme is regular, but since the MS scheme is de"ned only order by order in a or 1/N , there is no proof of this conjecture. Above and below when we state(d) that Q D a certain large-order behaviour is valid in the MS scheme, it is (has been) always tacitly assumed that the MS anomalous dimensions are convergent series in a or, at least, do not diverge as fast as Q factorials. In this context it is interesting to note that the series expansion of the b-function up to the highest order known today (van Ritbergen et al., 1997) is indeed much better behaved than physical quantities, which are expected to have divergent series expansions.
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The transformation properties of the large-order behaviour under changes of the series expansion parameter a are as follows: suppose R" r aL>, with the large-order behaviour Q L L Q 1 L c C(n#1#b) 1#S #2 r "K (3.60) L S n#b
and suppose that a is related to a , the coupling in the new scheme, by Q Q a "a #d a#d a#2 . (3.61) Q Q Q Q Then the parameters of the expression analogous to Eq. (3.60) in the new scheme are given by KM "KeB1 ,
(3.62)
SM "S,
(3.63)
bM "b,
(3.64)
c "c !S(d!d )!bd . (3.65) For these relations to be valid one can allow that the couplings are related by divergent series, provided the divergence is slower than for the r . The easiest way to obtain these transformation L properties is to examine the transformation of the ambiguity of the Borel integral or the variant (3.31). Recall that b and c are calculable, but K is not. However, the scheme dependence of the normalization is known and involves only the relation of the couplings at one loop. This is analogous to the transformation property of the QCD scale parameter K. The case, where the scheme is "xed, but the renormalization scale of a is changed, is covered as Q a special case of Eq. (3.61). With a "a (k ) and d "!b ln(k /k), this leads to a trivial scale Q Q dependence of K, K(k )"(k /k)\@1K(k) .
(3.66)
For UV renormalons in QCD (!b S) is a negative integer and the overall normalization decreases when the renormalization scale is increased (Beneke and Zakharov, 1992). For IR renormalons it is exactly opposite. The transformation properties can be generalized to the case, where Eq. (3.61) is allowed to be arbitrary. In this case, the large-order behaviour of R may end up being dominated by the large-order behaviour of Eq. (3.61). From the point of view of analysing power corrections (IR and UV behaviour) to R via renormalons, expressing R through a non-regular coupling seems unnatural, since the coupling parameter &imports' power corrections not related to the physical process R itself. As in the case of low orders in perturbation theory (Stevenson, 1981), one can "nd certain scheme independent combinations of the parameters that characterize the large-order behaviour (Beneke, 1993b). Restricting attention to physical quantities that depend on only one scale (&e!ective charges'), these parameters can be read o! from the large-order behaviour of the e!ective charge The subsequent equations are valid not only for the dominant large-order behaviour but also for subleading components from the second UV or IR renormalon, etc. Keeping b in the denominator of the 1/n correction term in Eq. (3.60) proves convenient, when one goes to yet higher order in 1/n.
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b-functions just as in low orders of perturbation theory (Grunberg, 1980). One "nds that S, b and K "b Ke\P1,
(3.67)
c "c !((b#2)/S)#br #S(r!r )#b /b
(3.68)
are scheme and scale independent, provided the relation (3.61) does not diverge too fast. One may also wonder about the situation when a quark has intermediate mass m or b ,D determines the factorial growth of perturbative coe$cients. The answer depends on whether one considers UV or IR renormalons. For UV renormalons, b ,D> is relevant. For IR renormalons, the typical loop momentum falls below m beyond a certain order, in which case the massive quark e!ectively decouples. In large orders the perturbative coe$cients become close to those of the N #avour theory even though Q is much larger than m, provided the D coupling constants in the N #1 and N #avour theory are matched as usual. The decoupling of D D intermediate mass quarks has been studied in Ball et al. (1995a). 3.5. Calculating &bubble' diagrams Many of the applications reviewed in Section 5 are based on the analysis of diagrams with a single chain of fermion loops. In this section we summarize various methods to represent or calculate this class of diagrams and the relations between these methods. We begin with some de"nitions. We consider observables R and subtract the tree contribution. The radiative corrections take the form r aL>. We assume that R is gauge-invariant and does L Q not involve external gluon legs at tree L level, so that the "rst-order correction r comes from diagrams with a single gluon line. The coe$cients r are polynomials in N : L D r "r #r N #2#r NL . L L L D LL D
(3.69)
The set of fermion loop diagrams (&bubble diagrams') is gauge-invariant and gives the coe$cient r with the largest power of N , the number of light #avours. In the following we do not consider LL D the other terms in Eq. (3.69). In general, the "rst-order correction to R may be the sum of a one-loop virtual and a one-gluon real emission contribution. The fermion bubble corrections are (Fig. 11): Fermion loops inserted into the virtual gluon line [cut (a)] or fermion loops inserted into the &real' gluon line, which can be either part of the "nal state [cut (b)] or split into a fermion pair (&cut bubble') [cut (c)]. In case (c), the gluon is not real anymore. In case (b) the fermion loops are scaleless integrals, which vanish in dimensional regularization. The virtual corrections of type (a) can be represented as
a 1 a (k exp[C/2]) Q " dk F (k,Q) Q , R " dk F (k,Q) k 1#P(k) k
(3.70)
where P(k) is given by Eq. (2.16), k is the momentum of the gluon line, and Q stands collectively for external momenta. The fermion loop insertions are summed to all orders into 1/(1#P(k)). The
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Fig. 11. The three di!erent types of cuts relevant for bubble graphs, here for e>e\P hadrons. The cuts may be weighted to give an event shape variable.
real corrections (c) can be represented as
b a 1 D Q , R " dk F (k,Q) k "1#P(k)"
(3.71)
where the virtuality of the gluon line, k, is now the invariant mass of the fermion pair into which the gluon splits. In writing Eq. (3.71) we have separated the two-particle phase space over k for the cut bubble by introducing a factor dk d(k!k !k ). Note that all dependence on a in Eqs. Q (3.70) and (3.71) is either explicit or in P(k). If R requires subtractions in addition to those for the fermion loops, the above integrals have divergences. Even if R is "nite after coupling renormalization, the integrals are ill-de"ned, because the Landau pole lies in the integration domain. However, their perturbative expansions are de"ned (but divergent). The integral (3.71), understood as an expansion in a , does not include the "rst-order correction with no gluon splitting, as seen Q from the fact that its expansion starts at order a. It turns out that in the summed expression (3.71) Q } appropriately de"ned } the "rst-order real correction is contained as an &end-point' contribution of order 1/a from the lower limit k"0 and that (3.71) gives the correct result for (b) and (c) Q together. 3.5.1. The Borel transform method The Borel transform B[R](u)" r /n! (!b )\LuL can be used as a generating function for the LL D perturbative coe$cients: r "(!b )L(dL/duL)B[R](u) S . L D The Borel transform of bubble graphs is obtained using the relations
B
a \S k Q , " ! e! 1#P(k) k
B
b a \S sin(pu) k D Q , "! ! e! "1#P(k)" p k
(3.72)
(3.73)
(3.74)
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on Eqs. (3.70) and (3.71) to obtain B[R ](u) and B[R ](u). The integrals for B[R ](u) then look like those that appear in evaluating the lowest-order correction r , except that the gluon propaga tor is raised to the power 1#u (Beneke, 1993a). However, the integral over k obtained for B[R ](u) does not converge in the vicinity of u"0 and cannot be used in Eq. (3.72). Constructing the analytic continuation of the integral in the usual way by integrating by parts and de"ning m"!k/k e!, we obtain
dm m\S F (m,Q/k) , (3.75) m sin(pu) K d (3.76) B[R ](u)"! dm m\S F (m,Q/k) , pu dm with a kinematic upper limit m . The virtual and real corrections have infrared divergences
separately. These result in singularities at u"0, which cancel in the sum of virtual and real corrections. With this pole subtracted B[R ](u) approaches a constant at u"0 and hence gives rise to a contribution to r , see Eq. (3.72). It can be shown that this contribution is exactly the order a contribution from real gluon emission despite the fact that this contribution belongs to the cuts Q (b) in Fig. 11 while Eq. (3.71) followed from the cuts (c). The resolution to the paradox lies in the unconventional IR regularization implied in calculating the Borel transforms (Beneke and Braun 1995b). If we keep dimensional regularization, the cuts (b) vanish, except for the one with no fermion loop. However, we also have to take into account the counterterms for the fermion loops that do not lead to vanishing scaleless integrals. The Borel transform of the one-gluon emission together with the counterterm contributions is proportional to exp(!u/e), which should be set to zero in the limit eP0. Thus the one-gluon emission contribution disappears together with all other contributions of type (b). It reappears as part of (c) in Eq. (3.76). If R requires ultraviolet renormalization in addition to coupling constant renormalization, Eq. (3.75) has to be amended by a subtraction function as discussed in Section 3.4. The calculation of the subtraction function is described in detail in Espriu et al. (1982), Palanques-Mestre and Pascual (1984), Beneke and Braun (1994) and Ball et al. (1995a). If R needs infrared subtractions and receives only virtual corrections, the procedure is essentially identical. The case when R requires IR subtractions and receives real and virtual corrections has not been worked out in detail so far. B[R ](u)"
3.5.2. The dispersive method The bubble diagrams can also be calculated by using the dispersion relation
1 Im P(j) 1 j 1 1 * dj dj " # d(j!j) * k!j "1#P(j)" k!j (!b a ) 1#P(k) p \ D Q in Eq. (3.70) (Beneke and Braun, 1995a). Here
(3.77)
j"!kexp[!1/(!b a )!C] (3.78) * D Q is the position of the Landau pole. This leads to a very intuitive characterization of IR renormalon singularities (Beneke et al., 1994; Beneke and Braun, 1995a; Ball et al., 1995a; Dokshitzer et al., 1996). Note that, since Im P(j)"pb a , the "rst term on the right-hand side has the same D Q
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a dependence as the real term (3.71). Moreover, the integral over k left after inserting Eq. (3.77) in Q Eq. (3.70),
1 , r (j,Q)" dk F (k,Q) k!j
(3.79)
coincides with the "rst-order virtual correction calculated with a massive gluon. Because the a dependence for virtual and real corrections is the same after application of the dispersive Q representation (3.77), the Borel transform can be represented in the particularly simple form (Beneke and Braun 1995a; Ball et al., 1995a)
d sin(pu) (3.80) dm m\S ¹(m,Q/k) , B[R](u)"! dm pu ¹(m,Q/k)"r (m,Q/k)#F (m,Q/k) h(m !m) , (3.81)
where we set m"j/k e! in the virtual contribution. If the observable R is su$ciently inclusive, one "nds that F (m,Q/k)"r (m,Q/k) , (3.82) where r denotes the correction from emission of a virtual gluon with mass j. That is, the set of bubble diagrams can be evaluated by taking an integral over the "rst-order virtual and real correction evaluated with a "nite gluon mass. &Su$ciently inclusive' means that the cuts (c) in Fig. 11 are not weighted. Total cross sections and total decay widths are su$ciently inclusive, but event shape observables in e>e\ annihilation are not (Nason and Seymour, 1995; Beneke and Braun, 1995b). It follows from Eqs. (3.72) and (3.80) that the coe$cients r can be computed in L terms of logarithmic moments of the function ¹(m,Q/k). The series given by the bubble graphs are divergent because of IR and UV renormalons. One may still dexne the sum of the series by de"ning the Borel integral (2.6) as a principal value or in the upper/lower complex plane. Let a "!b a and u"!b t. Then Q Q >GC dt e\R ?Q B[R](t) R, d " dm U(m) ¹(m,Q/k)#[¹(m !ie,Q/k)!¹(0,Q/k)] , (3.83) * dm where the ewective coupling U is given by
ap 1 Q U(m)"! arctan !h(!m !m) * 1#a ln(m) p Q
(3.84)
Since we assumed that the observable is gauge-invariant and does not involve the three-gluon coupling in the order a correction, this identi"cation is meaningful. Because of this the present method is often called the &massive gluon' Q method. In general, the identi"cation holds only for virtual corrections. For the set of bubble graphs b "b "N ¹/(3p). In order that a be positive for positive a , we formally consider D D Q Q negative N . In practical applications of the following equation one usually departs from the literal evaluation of fermion D bubble graphs and uses the full QCD b . Since it is negative, a is then positive. Q
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and m "!exp(!1/a ) is related to the position of the Landau pole, cf. Eq. (3.78). The derivation * Q of Eqs. (3.83) and (3.84) requires some care and can be found in Beneke and Braun (1995a) and Ball et al. (1995a). Despite the h-function the e!ective coupling is continuous at m"!m and * approaches a "nite value as mP0. The attractiveness of the dispersive method results from the fact that renormalon properties follow directly from the distribution function ¹(m) (we omit the second argument for brevity) without the integration over m having to be done. R, de"ned by Eq. (3.83), has an imaginary part due to the term ¹(m !ie). This imaginary part persists as eP0, because m (0 and ¹(m) has a cut * * for m(0. The imaginary part of the Borel integral is directly related to renormalon singularities, cf. Eqs. (2.8) and (2.10) in Section 2.1. Because m "!exp(1/a );1, one can expand * Q (3.85) ¹(m)" c ((m)I lnJ m , IJ I J where we anticipated that the expansion goes in powers of (m and logarithms of m. Since only the imaginary part for negative m is related to IR renormalons, it follows that IR renormalon singularities are characterized by non-analytic terms in the small-m expansion of the distribution ¹ (Beneke et al., 1994). Taking into account the value of m , the following correspondences are * found between non-analytic terms in m, renormalon singularities and power corrections (n, m non-negative integer): mL lnK> m
K L 1 lnK(K/Q) , Q (n!u)K
m >L lnK m
1 K >L lnK(K/Q) . (1/2#n!u)K Q
(3.86) (3.87)
These relations provide a direct implementation of the correspondence between perturbative infrared behaviour and power corrections. It is clear that analytic terms in Eq. (3.85) are not related to IR renormalons, because analytic terms arise from large and small momenta. Note, however, that analytic terms in ¹(m !ie) in * Eq. (3.83) are important for the real part of Eq. (3.83) to coincide with the principal value of the Borel integral. Although the relevance of the principal value is far from obvious, the term ¹(m !ie)!¹(0), which is exponentially small in a (&non-perturbative'), should still be kept for the * Q following reason. One would like the sum of the bubble diagrams to equal roughly the sum of the
The representation (3.83) and (3.84) of bubble graphs has been derived in a slightly di!erent way by Dokshitzer et al. (1996). There, the e!ective coupling U is called a and the distribution function ¹ &characteristic function', denoted by F. Dokshitzer et al. (1996) do not include the Landau pole contribution in the dispersion relation (3.77), because they have in mind a physical coupling rather than the MS coupling. As a consequence the term ¹(m !ie,Q/k)!¹(0,Q/k) is absent * from their result. This di!erence is irrelevant for the study of power corrections induced by IR renormalons, because one needs to know only the function ¹(m) for this purpose, and not the Borel integral. As shown in Ball et al. (1995a) leaving out the Landau pole contribution in Eq. (3.77) implies a rede"nition of the strong coupling, which di!ers from the standard one by K/Q power corrections not related to renormalons and infrared properties. The possible implications of such additional power corrections are also discussed in Grunberg (1997) and Akhoury and Zakharov (1997a). For large k the propagator 1/(k!j) can be Taylor-expanded and gives rise to (only) analytic terms in j.
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perturbative series truncated at its minimal term. There are cases (Ball et al., 1995a) for which the real part of ¹(m !ie)!¹(0) is parametrically larger in Q than the minimal term. In these cases, * Eq. (3.83) without the Landau pole contribution comes nowhere close to the sum of the perturbative expansion truncated at its minimal term. There may of course be non-perturbative corrections parametrically larger than the minimal term. However, without any positive evidence for them, one would like to avoid introducing them by hand. If one takes a negative, ambiguities in the Borel integral arise from ultraviolet renormalons. In Q this case one "nds a correspondence between UV renormalon singularities and non-analytic terms in the expansion of the distribution function ¹(m) at large m. If R requires renormalization beyond coupling renormalization, this manifests itself as ¹(m)&ln m at large m. Then the integral over m in Eq. (3.83) does not converge. The renormalized R includes subtractions, after which the integral becomes convergent. The modi"cations of Eqs. (3.80) and (3.83) relevant to quantities requiring additional renormalization can be found in Ball et al. (1995a). The subtraction function analogous to S(u) in Eq. (3.58) can in fact be determined entirely from the asymptotic behaviour of the "rst-order virtual corrections in the limit of large gluon mass. In the MS scheme, the subtractions do not introduce factorial divergence. As a consequence the non-analytic terms in the small-m expansion of ¹(m) remain una!ected. 3.5.3. The loop momentum distribution function The fact that for Euclidian quantities renormalons can be characterized in terms of the loop momentum distribution function F (k/Q) of Eq. (3.70) in a transparent way has been empha sized by Neubert (1995b). We have already exploited in Section 2.2 the fact that the small and large momentum expansion of F (k/Q) su$ces for this purpose. In addition to this, the loop momentum distribution function provides an easily visualized answer to the question of which momentum scales contribute most to a given perturbative coe$cient. From this perspective, the summation of bubble graphs can be considered as the extension of Brodsky}Lepage}Mackenzie scale-setting (Brodsky et al., 1983) envisaged in Lepage and Mackenzie (1993). Thus extended, the BLM scale QH is given by r a (QH)"Eq. (3.83) . (3.88) Q Note that the BLM scale is small compared to Q if the cumulative e!ect of higher-order perturbative corrections is large. But a small BLM scale need not be indicative of a large intrinsic perturbative uncertainty, as renormalon ambiguities can still be small. For minkowskian quantities a loop momentum distribution function that generalizes Eq. (3.70) does not exist (Neubert, 1995c) and the distribution function ¹(m,Q/k) is more useful. For Euclidian quantities the relation between the loop momentum distribution function and the distribution function ¹(m,Q/k) is given by Ball et al. (1995b) and Neubert (1995c)
¹(m,1)"
ds
s F (s) , s#m
(3.89)
If one uses MS subtractions for infrared divergences, one cancels a ln m term in the small-m expansion of the distribution ¹ of the corresponding hard scattering coe$cient, but all other non-analytic terms remain unmodi"ed.
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where ¹(m,1)"r (m) is only from virtual corrections. In turn it follows from Eq. (3.75), that the loop momentum distribution function can be obtained from the Borel transform by an inverse Mellin transformation. 4. Renormalons and non-perturbative e4ects In the previous section we have emphasised on the diagrammatic analysis of renormalon divergence. In QCD, IR renormalons are taken as an indication that a perturbative treatment is not complete and that further terms in a power expansion in K/Q, where K is the QCD scale and Q is a &hard' scale, should be added. The perturbative expansion itself is ambiguous to the accuracy of such terms unless it is given a de"nite summation prescription. In this section we address the question in what sense IR renormalons are related to non-perturbative, power-like corrections and how perturbative and non-perturbative contributions combine to an unambiguous result. In order to examine non-perturbative corrections, one has to resort to a solvable model. In Section 4.1 we consider the non-linear O(N) p-model in the 1/N expansion as a toy model. After general remarks regarding QCD, we consider explicitly the matching of IR contributions to twist-2 coe$cient functions in deep-inelastic scattering and UV contributions to the matrix elements of twist-4 operators. 4.1. The O(N)p-model The Euclidian action of the non-linear O(N) p-model is given by
1 S" dBx R p?R p? , I I 2
(4.1)
where d"2!e and the "elds are subject to the constraint p?p?"N/g. The index &a' is summed from 1 to N. The &length' of the p "eld is chosen such that a 1/N expansion can be obtained. Solving the constraint locally for p,, an interacting theory for the remaining N!1 components is obtained, which can be treated perturbatively in g. Perturbation theory is rather complicated in this theory, because the p "eld is dimensionless and the Lagrangian contains an in"nite number of interaction vertices after elimination of p,. In perturbation theory the "elds p?, a"1,2,N!1, are massless and the perturbative expansion is plagued by severe IR divergences. Despite this fact, O(N) invariant Green functions are IR "nite (Elitzur, 1983; David, 1981) and a sensible perturbation expansion is obtained for them. The non-linear O(N)p-model can be solved non-perturbatively (in g) in an expansion in 1/N (Bardeen et al., 1976). The 1/N expansion follows from introducing a Lagrange multiplier "eld a(x), which makes the generating functional
Z[J]" D[p]D[a] exp !S[p,a]# dBx J?(x)p?(x) , with
1 a N S[p,a]" dBx R p?R p?# p?p?! I I 2 g (N
(4.2)
(4.3)
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quadratic in the p "eld. One then integrates over p and performs a saddle point expansion of the a integral. There is a non-trivial saddle point at
e C a "(N g kCC (4p)C\ , 2
(4.4)
where g denotes the bare coupling and k is the renormalization scale of dimensional regulariz ation. De"ning the renormalized coupling g(k) by g\"Zg\ with e (4p)C\ , (4.5) Z"1#g(k)C 2
the saddle point approaches a,(Nm"(Nk e\p EI
(4.6)
as eP0. As a consequence the p "eld acquires a mass m, which is non-perturbative in g. Furthermore, at leading order in the 1/N expansion, b(g)"kRg/Rk"b g, b "!1/4p (4.7) is exact and the model is asymptotically free. The Feynman diagrams of the 1/N expansion are constructed from the p propagator d?@/(p#m), the propagator for a!a,
(p#4m#(p \ D (p)"4p(p(p#4m) ln , ? (p#4m!(p
(4.8)
and the pa vertex d?@/(N. By de"nition bubble graphs of p "elds are already summed into the a propagator and are to be omitted. The non-linear O(N) p-model has often been used as a toy "eld theory, because it has some interesting features in common with QCD. It has only massless particles in perturbation theory, but exhibits dynamical mass generation non-perturbatively and a mass gap in the spectrum. It is asymptotically free, as is QCD, and m is the analogue of the QCD scale K. In the following we consider the structure of the short-distance/operator product expansion (OPE) of Euclidian correlation functions in the p-model as a toy model for the OPE in QCD. The p-model has been analysed from this perspective in the papers (David, 1982, 1984; Novikov et al., 1984, 1985; Terent'ev, 1987; Beneke et al., 1998), on which this section is based. Because the p "eld is dimensionless, there exist an in"nite number of operators of any given dimension that can appear in the OPE. In leading order of the 1/N expansion, the matrix elements factorize and, using the constraint p?p?"N/g, the number of independent matrix elements is greatly reduced. In the following it will be su$cient to consider the (vacuum) matrix elements of the operators O "1, O "gR p?R p?, O "gR p?R p?R p@R p@ I I I I J J to illustrate the point. Note that the equations of motion yield a"!(g/(N)R p?R p? , I I
(4.9)
(4.10)
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so that 1O 2"!m at leading order in 1/N. Because of factorization one has 1O 2"m at this order. One can consider as examples the OPE of the amputated two-point function C(p) of the p "eld and of the two-point function of the a "eld. Because of Eq. (4.10) the second quantity can also be interpreted as the two-point correlation function of the scale invariant current j"(!g)/(N R p?R p?. Introducing a factorization scale k satisfying m;k;p, the OPE of C(p) I I reads (4.11) C(p)" CC(p,k)1O 2(k,m)"p#m#O(1/N) L L L and realizes an expansion in m/p. From the second equality one deduces CC"p and CC"!1. All other coe$cient functions vanish in leading order in 1/N. The OPE of the current-current correlation function reads
S(p,m),i dBx e NV10"¹( j(x) j(0))"02" C1(p,k)1O 2(k,m) L L "(2p)d(p)1a2#D (p)#O(1/N) . (4.12) ? In the following we drop the disconnected term proportional to 1a2. At leading order in 1/N, the expansion m m 1 D (p) ? "g( (p)# (2g( (p)!2g( (p))# (!2g( (p)!g( (p)#4g( (p))#2 p p 4p p
(4.13)
follows from Eq. (4.8). (We introduced g( (k),!b g(k)"1/ln(k/m).) Each power correction is multiplied by a "nite series in g(p). At leading order in 1/N there are no renormalons and there is no factorization scale dependence. The power corrections in m/p follow from the factorizable part of matrix elements of p "elds (Novikov et al., 1984). Note that the truncated expansion in m/p and g( (p) has a Landau pole at p"m due to the IR behaviour of g( (p). The correct analyticity properties of S(p,m) are restored only after the OPE (the expansion in m/p) is summed. To see the interplay of IR renormalons and operator matrix elements, one has to go to the "rst subleading order in 1/N. The relevant Feynman diagrams in the 1/N expansion are shown in Fig. 12. At this order one has to specify a factorization prescription in the OPE. If one uses dimensional regularization (David, 1982, 1984) one is led to the usual situation that the coe$cient functions have IR renormalons and to the problem how the corresponding ambiguities are cancelled. One can also use an explicit factorization scale in loop momentum integrals (Novikov et al., 1984, 1985). In this case the coe$cient functions contain only integrations over loop momenta k'k and therefore have no IR renormalon divergence. The IR renormalon divergence appears as a perturbative contribution to the vacuum expectation values, if one attempts to separate such a perturbative part from the whole. It is somewhat easier to begin with cut-o! factorization, since it su$ces to calculate the operator matrix elements. The leading non-factorizable contributions to the matrix elements of O and O are shown in Fig. 12e and f, respectively. The OPE of the self-energy diagram, Fig. 12b, is trivially given by the "rst correction to 1O 2. The non-factorizable contribution to 1O 2 appears as
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Fig. 12. (a,b) p-self-energy diagrams at order 1/N. (c, d) Connected contributions to the a propagator at order 1/N. (e) Non-factorizable contribution to the vacuum expectation value of O Ja. (f ) Non-factorizable contribution to the vacuum expectation value of O Ja. The solid lines represent the p propagator, the wavy lines the leading order a propagator (4.8).
part of Fig. 12a, c and d, when the a line is soft and the p lines are hard. The contribution to the vacuum expectation value of the operator a (which is proportional to O ) from Fig. 12f is given by dp D (p) . (4.14) 1a2(k,m)" (2p) ? N I Note that the restriction p(k dexnes the otherwise singular operator product a. The integral can be evaluated (Novikov et al., 1984) with the result
1a2(k,m)"m[Ei(ln A)#Ei(!ln A)!ln ln A!ln(!ln A)!2c ] , (4.15) # where c "0.57722 is Euler's constant, Ei(!x)"! dt e\R/t the exponential integral function # V and
A"
k k . 1# # 4m 4m
(4.16)
Note that F(x)"Ei(!x)!ln x has an essential singularity at x"0 but no discontinuity. By assumption k<m, hence ln AP2/g( (k)<1. To expand Eq. (4.15) in this limit, up to terms that vanish as kPR, one needs the asymptotic expansion of F(x) at large x. For positive argument the asymptotic expansion is n! F(x)"!ln x#e\V (!1)L> . xL> L
(4.17)
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If the divergent series is understood as its Borel sum, the right-hand side equals F. For negative, real argument, one obtains the asymptotic expansion F(!x)"eV
n! !e\V(ln xGip) . xL> L
(4.18)
Note the &ambiguous' imaginary part in the exponentially small term. The interpretation of Eq. (4.18) is as follows (compare the discussion at the end of Section 2.1): the upper (lower) sign is to be taken, if the (non-Borel-summable!) divergent series is interpreted as the Borel integral in the upper (lower) complex plane. With this interpretation, Eq. (4.18) is exact and unambiguous. Inserting these expansions, the condensate is given by
g( (k) L> n!#2g( (k) km 1a2(k,m)"k 2 L #m !2ln
m 2 g( (k) $ip!2c !4g( (k)# #O . # k g( (k) 2
(4.19)
The expansion for large k has quartic and quadratic terms in k, parametrically larger than the &natural magnitude' of the condensate of order m. The power terms in k arise from the quartic and quadratic divergence of the Feynman integral (4.14), i.e. from loop momentum p&k. The k dependence cancels with the k-dependence of the coe$cient functions in the OPE. In particular, the k-term cancels with the coe$cient function of the unit operator. The important point to note is that the condensate is unambiguous, but separating the &perturbative part' of order k is not, since the asymptotic expansion for k/m<1 leads to divergent, non-sign-alternating series expansions, which require a summation prescription. The &non-perturbative part' of order m depends on this prescription (via $ip in Eq. (4.19)). In a purely perturbative calculation, one would only obtain the divergent series expansion. The infrared renormalon ambiguity of this expansion would lead us to correctly infer the existence of a non-perturbative power correction of order m. However, it does not allow us to say much about the magnitude of the power correction which is determined by other terms, such as ln(2/g( ) in Eq. (4.19). In dimensional regularization power dependence on the factorization scale k is absent and IR renormalon divergence is part of the coe$cient function. If the power terms in k in Eq. (4.19) are deleted, it seems that the remainder has a twofold ambiguity. This should be taken as an indication that the de"nition of a renormalized condensate in dimensional regularization requires some care, because the summation prescription for the coe$cient functions depends on it. This point has been studied in detail by David (1982, 1984).
At "rst sight there seems to be a problem with the argument, because of the term proportional to km. However, this is a pure cut-o! term, which cancels in physical quantities when the condensates are combined with coe$cient functions. In dimensional regularization such terms are absent. The fact that coe$cient functions depend on the de"nition of the condensates is of course true in any factorization scheme. However, in some schemes the subtleties in handling divergent series expansions may be avoided.
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Consider as in Eq. (4.14) the condensate of a, but de"ned in dimensional regularization instead of a momentum cut-o!:
m/(4p) dBp m \C p p \C D (p,d)" d D (p,d) . (4.20) ? C(1!e/2) 4pk (2p)B ? m m Since the integral contains no scale other than m, it must be proportional to m. D (p,d) denotes ? the a propagator (4.8) before the limit eP0 is taken. However, for the following short-cut of the detailed analysis of David (1982, 1984) it is su$cient to set d"2 in the a propagator. From the treatment of the integral in cut-o! regularization we learn that we should focus on the UV behaviour of the integral. Hence expanding (cf. Eq. (4.13)) 1a2(k,m)"kC
c 1 D (p)"4pmu IJ ? uI lnJ u I J with u"p/m, we obtain
(4.21)
m \C u\C\I m c du , (4.22) 1a2(k,m)" IJ lnJu C(1!e/2) 4pk S I J with an (arbitrary and irrelevant) IR cut-o! u '1. Now write 1 (4.23) " dv vJ\uT . lnJ u The u-integration leads to UV poles of the form 1/(!2#e#k#v). Keeping only those,
m m \C vJ\ 1a2(k,m)& (4.24) c dv IJ C(1!e/2) 4pk !2#e#k#v I J follows. To de"ne the v-integral it is necessary to take complex e. (We also take Re(e)'0, because the p-model is super-renormalizable in d(2.) For k"2, and only for k"2, one obtains a pole in e which can be subtracted as usual. This pole arises from a logarithmically UV divergent integral. The terms with k"0 (k"1) correspond to the quartically (quadratically) divergent terms in Eq. (4.20). For these terms the limit eP0 is "nite, but depends on whether it is taken from the upper or the lower complex plane, because of the pole at v"2!e!k in Eq. (4.24). The di!erence between the two de"nitions of the dimensionally renormalized condensate is
lim ! lim 1a2(k,m)"2pi m c (2!k)J\ . IJ C >G C \G I J
(4.25)
From Eq. (4.13) only c "1, c "!c "2 are non-zero for k(2 and the result is
lim ! lim 1a2(k,m)"2pi m . C >G C \G
(4.26)
The approximations made do not allow us to calculate the condensate itself. However, comparison of Eq. (4.26) with Eq. (4.19) demonstrates that the di!erence between the two limits coincides with
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the di!erence in the m terms in Eq. (4.19), when the perturbative parts are subtracted. It is interesting to note that although power divergences do not give rise to counterterms in dimensional regularization, they have not completely disappeared in the limit eP0 in the sense that they render the limit non-unique. A more precise analysis also demonstrates that the summation prescription for the divergent series expansions of the coe$cient functions depends on how the limit eP0 is taken. The OPE of Green functions is unambiguous, if the limit is taken in the same way as for the condensates. To this end the works of David (1982, 1984) begin with an analysis of the OPE of bare Green functions in d dimensions. The OPE exists also in the regularized theory (e "nite), (4.27) C(p,e)" CC(p,e)1O 2(m,e) , L L L taking the self-energy as an example. In the regularized theory the separation in coe$cient functions and matrix elements is unique and well-de"ned without further prescriptions. However, the analytic structure in e of the individual terms on the right-hand side is di!erent from that of the unexpanded self-energy. The latter has a straightforward limit as eP0 (we assume that counterterms have been included), but a condensate of dimension d on the right-hand side has poles at e"2k/l (k(d, l positive integer) related to the power divergences of the operator. The poles accumulate at e"0 and hence the limit eP0 has to be taken from complex e. But the limit eP0 has to be taken in the same way for all terms in the OPE and this is how the de"nitions of renormalized condensates and coe$cient functions are related to each other. At "nite e there are no renormalon singularities in the coe$cient functions and the Borel integrals are de"ned. When e approaches zero, singularities develop in the Borel transform but the limit also entails a prescription of how the contour is to be chosen in the Borel integral to avoid the singularities. To see this in more detail, it may be helpful to consider the integrals
H H (4.28) gL> dk bL lnL kP dk(k)@R . L To the right of the arrow the Borel transform of the series is indicated, which has a single IR renormalon pole at b t"!1. In the regularized theory, the corresponding series is H b (e) L (k)\LC , (4.29) gL> dk ! e L where b (e) is a function that approaches b as eP0. These integrals do not lead to divergent series, contrary to the logarithmic integrals for vanishing e. For any given n, e can be made small enough for the integral to converge. However, for any given e, there always exists an n beyond which the integrals diverge and have to be de"ned in the sense of an analytic continuation in n. This is the reason why the limit eP0 and the large-order behaviour nPR do not commute. Taking the integrals, one "nds accumulating poles at e"1/n in the sum of the series. The Borel transform with respect to g is given by
H
b (e)t dk exp ! [(k)\C!1] . e
(4.30)
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The &!1' in the exponent takes into account the coupling renormalization counterterms. As eP0 one recovers the Borel transform in Eq. (4.28). But for any "nite e the behaviour of the k integrals at small k is very di!erent. In fact, for e'0, the integral diverges, so we de"ne the integral as the analytic continuation from negative e. As a result one "nds that the pole at b t"!1, which arises in the limit eP0, should be interpreted as 1/(1$i0#b t) depending on whether the limit is taken from the upper or lower right half plane. The corresponding di!erence in the Borel integrals cancels exactly the di!erence in the condensates. The OPE of the self-energy can be obtained exactly at order 1/N (Beneke et al., 1998), and the result con"rms what one would expect from the above discussion. The expansion in m/p of diagram (a) of Fig. 12 can be expressed in the form
m L 1 p e\R EN FL[t] #GL[t] !HL[t] . (4.31) dt ! R(p)" p g(p) N L The explicit expressions for the functions FL[t], GL[t], HL[t] can be found in Beneke et al. (1998), but only the structure of the result is of importance. The two terms that are multiplied by e\R EN can be interpreted as Borel transforms of perturbative expansions of coe$cient functions. The function HL[t] originates from the loop momentum region, where the momentum of the a propagator in diagram (a) is of order m, and hence probes its long-distance behaviour. This term corresponds to condensates of a. The functions FL[t], GL[t], HL[t] have singularities in the complex t plane at integer values t"$k, k"1,2,2 These are just the UV and IR renormalon singularities. All IR renormalon singularities at positive values of t cancel in the integrand, so that the integral, and hence the OPE, is well de"ned. The cancellation of a particular singularity at t"t occurs between GL[t] and HL>R[t] and thus involves a cancellation between a short-distance coe$cient and an operator matrix element over di!erent orders in the power expansion. As a consequence of the singularities in individual terms of the sum over n, the summation and the integration over t cannot be interchanged, unless the integration contour is shifted slightly above (or below) the real axis. This amounts to a simultaneous prescription for summing the divergent series expansions of coe$cient functions as well as a de"nition of the renormalized condensates. Only after such a de"nition can the OPE be truncated at a given order in m/p. In Section 2.1 we asked why the Borel integral should play a privileged role in de"ning divergent series and whether the association of IR renormalons with power corrections does not rely too much on this idea. The O(N) p model in the 1/N expansion provides an example which con"rms the picture assumed there. The Borel integral emerges as the natural way to de"ne the divergent series that arise in the limit eP0. In particular, the Borel representation (4.31) emerges naturally in the exact OPE of the self-energy. The p model is still special because the leading, factorizable contributions in 1/N to the condensates are unambiguous or factorization scale independent. As a consequence the power-like ambiguities in de"ning perturbative expansions are parametrically smaller in 1/N than the actual
The self-energy in Beneke et al. (1998) is subtracted at zero momentum, in which case diagram (b) is subtracted completely.
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condensates. This tells us that some caution is necessary in identifying the magnitude of the &renormalon ambiguity' with the magnitude of power corrections. It is probably more appropriate to say that power corrections are expected to be at least as large as perturbative ambiguities. However, a similar parametric suppression of perturbative ambiguities does not seem to take place in neither the large-N nor the large-N limit of QCD. A D 4.2. IR renormalons and power corrections We have shown above how IR renormalons arise in asymptotically free theories, when one performs an asymptotic expansion in K/Q, where K is the intrinsic scale of the theory and Q a large external scale. In the following we summarize the conclusions from the p-model with respect to applications in QCD, recollecting in part the remarks of Section 2.3. The tacit assumption is that the structure of short-distance expansions in QCD is as in the p-model. We then check perturbatively in a QCD example that the power divergences of matrix elements match with IR renormalons. 4.2.1. Summary First, let us emphasize that IR renormalon ambiguities are not a problem of QCD, but a problem of doing perturbative calculations in QCD, which implicitly or explicitly require some kind of factorization, and an expansion in a ratio like K/Q. If we could do non-perturbative calculations, IR renormalons would just be artefacts that appear in the expansion of the exact (and well-de"ned) result. Let us imagine that an observable R, which depends on at least the two scales Q and K, can be written as an expansion C (Q,k) 11O 22(k,K) . (4.32) R(Q,K)" L L QBL L The product may be a normal product or a convolution and the operator O of dimension d may L L be local or non-local. The matrix element may be a vacuum matrix element or a matrix element between hadron states. (We use the double bracket to indicate that the external state may be complicated.) We assume that the C (Q/k) can be calculated as a series in a . It is not obvious that L Q such an expansion in powers and logarithms of K/Q always exists. Or one may know only the form of the "rst term, but not the form of power corrections to it. This is the most interesting situation from the point of view of IR renormalons. We assume that factorization is done in dimensional regularization. If one uses another factorization scheme, the wording of the following changes but the conclusions do not. In dimensional regularization the coe$cient functions C (Q,k) have IR renormalons from integrating L Feynman integrals over loop momenta much smaller than Q. With regard to power corrections we note: (i) Renormalon ambiguities in C (Q,k) are power-suppressed. Non-perturbatively they are L cancelled by ambiguities in de"ning the (renormalized) matrix elements 11O 22 with d 'd . K K L We assume that R has been made dimensionless.
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Contrary to the p-model one cannot trace this cancellation non-perturbatively in QCD. However, if QCD is a consistent theory and if R is physical, this cancellation must occur. In this way, IR renormalons in C lead us to introduce parameters for power corrections with a dependence on L Q (given by C /QBK11O 22) that matches the scaling behaviour of the renormalon ambiguity. This K K is the minimalistic, but also most rigorous and most universally applicable use of IR renormalons. (ii) The analysis of Feynman diagrams gives some information on the form of the operator O . K But IR renormalons provide no information on the magnitude of 11O 22. It is natural to think of K 11O 22 as at least as large as the renormalon ambiguity. The p-model is an example where the K matrix elements are parametrically larger than their ambiguities, both at large N and at g;1. (iii) The IR renormalon approach to power corrections does not provide a &non-perturbative method'. Viewed from the low-energy side, IR renormalons are related to ultraviolet properties of operators and not to matrix elements. The analysis of the p-model shows that the IR renormalon in C is related to a power divergence of degree d !d of O . In a cut-o! factorization scheme with L K L K factorization scale k, divergent series appear in the expansion of matrix elements for k/K<1, and the same statement holds. In Section 4.2.2 we demonstrate this for deep inelastic scattering in QCD by evaluating the ultraviolet contributions to twist-4 operator matrix elements perturbatively. As a consequence of being UV with respect to the scale K, IR renormalons do not distinguish matrix elements of the same operator but taken between di!erent states. (iv) Renormalon factorial divergence is closely connected with logarithms of loop momentum, which in turn are related to the running coupling. This leads to the universal appearance of b , b , etc., in the large-order behaviour. On the other hand, power corrections inferred from IR renormalons and power corrections in general have nothing to do with the low-energy properties of the running coupling. They are process-dependent and, generally speaking, non-universal. IR renormalons can be universal for a restricted set of observables, if the same operator appears in their short-distance expansion. However, universality of IR renormalons does not imply universality of non-perturbative e!ects. This is true only if the operator is not only the same, but is also taken between the same external states. (v) If this strong form of universality holds for a set of observables, one can relate power corrections to them on the basis of knowing only the IR renormalon behaviour of coe$cient functions. In particular, one can relate the leading power correction on the basis of the perturbative expansion at leading power. For simplicity, consider two observables R"C #(C /QB)11O22 , (4.33) (4.34) R"C #(C /QB)11O22 , and denote by dC the renormalon ambiguity in C of order 1/QB, which is related directly R\B @ to the large-order behaviour. Then it follows that "C /C , (4.35) dC /dC " R\B @ and this ratio can be expanded in a (Q). In particular, the ratio of the uncalculable normalizations Q of IR renormalon behaviour is given by the ratio of the coe$cient functions C , C evaluated to If the operator is non-local and is multiplied with the coe$cient function in the sense of a convolution, the situation is more complicated, because one also has to unfold the convolution.
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lowest order in a . Conversely, knowing the left-hand side of Eq. (4.35), the relative magnitude of Q 1/QB power corrections of the two observables can be predicted systematically as an expansion in a (Q). One observable has to be used to "x the absolute normalization, i.e. to determine C 11O22 Q from data. The procedure described parallels the phenomenological use of the OPE in standard situations such as QCD sum rules or deep inelastic scattering. Note that in practice, in connection with renormalons, universality often takes the status of an assumption. This is so, because to establish universality, one needs to know enough of the operator structure of power corrections that it may be possible to compute C and C directly, thus by-passing (4.35) and the IR renormalon argument. (vi) There is the problem of consistently combining (divergent) perturbative expansions in dimensional renormalization with phenomenological parametrizations of power corrections. For the purpose of discussion, let us consider the simpli"ed structure of (4.33) with only one parameter and a single, corresponding IR renormalon singularity in C at t"!d/(2b ). If we knew the singularity exactly, we could subtract it from the series. Recalling that the k-dependence of the IR renormalon singularity is an overall factor (k/Q)B (up to logarithms), we write Eq. (4.33) as
k B 1 # [C kB#C 11O22] , R" C !C Q QB
(4.36)
where K(k(Q and C denotes the exact asymptotic behaviour. Both square brackets are now separately well-de"ned. Note that this rewriting results in exactly the same representation as would be obtained with cut-o! factorization. In reality, the subtraction can be carried out at best approximately. Moreover, C is known only in the "rst few orders. Suppose we choose k as close to K as possible for a (k) to be perturbative. In this case the Q subtraction is e!ective only as one gets close to the minimal term of the series expansion of C . It may turn out that the phenomenological determination of the power suppressed term is large compared to the last term kept in the expansion of C . In this case C kB is small and IR renormalons are not an issue. It is sometimes argued (Novikov et al., 1985) that such a numerical fact is at the basis of the success of QCD sum rules. It may also turn out that the phenomenological determination of the power suppressed term is not large, but of the order of the last known term in the truncated series. In this case the phenomenological power correction may parametrize the e!ect of higher order perturbative corrections rather than a truly non-perturbative e!ect. It is still reasonable to use such an e!ective parametrization, because, as illustrated by Eq. (4.36), the dominant contribution to perturbative coe$cients in su$ciently large orders can be combined with 11O22. Moreover, if the minimal term in C is reached at not very high orders, the sum of higher order corrections parametrized in this way, may indeed scale approximately like a power correction. The important conclusion is that combining power corrections with truncated perturbative series is meaningful in the sense that the error incurred is never larger and most likely smaller than the error one would obtain without using information on power corrections. The improvement comes from the fact that the error is now determined by the degree to which the perturbative correction is non-universal in intermediate orders rather than by the size of the perturbative correction itself. For a related discussion, see David (1984) and Martinelli and Sachrajda (1996).
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4.2.2. Example: DIS structure functions In this section we demonstrate the cancellation of IR renormalons in coe$cient functions with ultraviolet contributions to matrix elements at the one-loop order and to twist-4 accuracy for the longitudinal structure function in deep inelastic scattering. This example serves to illustrate the operator interpretation of IR renormalons in a more involved situation than the OPE of current}current correlation functions discussed in Section 2.3. The motivation for choosing this more complicated example is that it is of interest in context with the phenomenological modelling of twist-4 corrections discussed in Section 5.2.4. We begin with some notation. The (spin-averaged) deep-inelastic scattering cross section of a virtual photon with momentum q from a nucleon with momentum p is obtained from the hadronic tensor
1 = " dz e OX1N(p,p)" j (z)j (0)"N(p,p)2 I J IJ 4p N qq F q p q #p q F J I , pp! I J " g ! I J *! g # IJ IJ I J 2x (p ) q) q 2x p)q
(4.37)
where x"Q/(2p ) q) and Q"!q and j is the electromagnetic current tM c t. In the following, I I the spin average over p is always implicitly understood. At leading order in the expansion in 1/Q, the longitudinal structure function can be written as a convolution
dm C (m,a (Q),Q/k)F(x/m,k)#gluon contribution . m * Q V Here F is the usual quark distribution, de"ned through the matrix element F (x,Q) " * \
1N(p)"tM (y)y. t(!y)"N(p)2(k)"2p ) y
(4.38)
dm e KN WF(m,k) , (4.39) \ where y is the light-like projection of z, y "z !(zp )/(2p ) z) for p"0. The quark "elds at I I I positions y and !y are joined by a path-ordered exponential that makes the operator product gauge-invariant. We do not write out the path-ordered exponential explicitly. We will check the matching of IR renormalons and UV contributions to twist-4 operators only to leading order in the #avour expansion. The N massless quarks are assumed to have identical electric charges and in D (4.39) a sum over the N quark #avours is assumed. The leading order in the #avour expansion is D equivalent to the analysis of IR regions of the one-loop diagrams (see Fig. 13a) with an important exception: there is also a gluon contribution to F at one loop (not shown in the "gure), but it does * not have an internal gluon line. Consequently, there are no contributions of order aL> (with n'0) Q to the gluon matrix elements in leading order of the #avour expansion, and hence we will not consider them here. However, it is important to keep in mind that there are 1/Q power corrections from gluon matrix elements as well and that they are not suppressed in any way. The #avour This section is based on unpublished notes worked out in collaboration with V.M. Braun and L. Magnea. It is somewhat more technical and can be omitted for "rst reading. The reader may return to it in connection with Section 5.2.4, where we discuss the renormalon model of twist-4 corrections to deep-inelastic scattering structure functions.
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Fig. 13. (a) Diagrams that contribute to the twist-2 coe$cient function in the operator product expansion of the hadronic tensor. (Wave function renormalization on the external quark legs is not shown.) The wavy lines denote the external current with momentum q. When the gluons are dressed with fermion loops, these diagrams contribute at leading order in the #avour expansion. (b) Diagrams that give contributions to the matrix elements of twist-4 operators to leading order in the #avour expansion. The third diagram is scaleless and vanishes.
expansion does not treat soft quark lines and renormalons appear in the gluon matrix elements only at next-to-leading order in the #avour expansion. In leading order of the #avour expansion there is a contribution from diagrams where a quark (or anti-quark) in a cut quark loop connects to the external hadron state, which we do not consider here. This contribution is not relevant for pure non-singlet quantities. With these restrictions in mind, we continue to analyse the non-singlet contribution to the quark matrix elements as shown in Fig. 13. The coe$cient function C vanishes at order a. To obtain it at leading order in the #avour * Q expansion, it is therefore su$cient to evaluate Eq. (4.39) between quark states at tree level, which gives F(m)"d(1!m). One then "nds C from the quark deep-inelastic scattering cross section * according to Eq. (4.38). The hadronic tensor at leading order in the #avour expansion requires the one-loop diagrams of Fig. 13a dressed with fermion loops. It has been calculated in Beneke and Braun (1995b), Dokshitzer et al. (1996) Stein et al. (1996) and Dasgupta and Webber (1996). For the Borel transform of the longitudinal structure function close to the leading IR renormalon pole at u"1, we obtain B
F K * (u,x)" +!8m#4d(1!m), * F(x/m) , 2x Q
(4.40)
where &*' denotes the convolution product as in Eq. (4.38) and C ke\! . K" $ 4p 1!u
(4.41)
(C is the subtraction constant for the fermion loop, C"!5/3 in the MS scheme.) The IR renormalon pole at u"1 corresponds to a twist-4 1/Q power correction to Eq. (4.38). In the
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remainder of this section, we reproduce the leading IR renormalon in the twist-2 coe$cient function C from the analysis of twist-4 matrix elements. * A complete analysis of twist-4 operators and their coe$cient functions has been performed in Ja!e and Soldate (1981), Ellis et al. (1982) and Ja!e (1983). Here we follow the treatment of Balitsky and Braun (1988/89), who work directly with non-local twist-4 operators rather than their expansion into local operators. The twist-4 contributions to the longitudinal structure function can be written as
1 (4.42) F (x,Q) " d+m, CG (x,+m,,a (Q),Q/k)¹ (+m,,k) * Q G * U Q G with multi-parton correlations ¹ de"ned below. At tree level the relevant part of the light-cone G expansion of the current product is
4g 1 IJ i¹( j (z) j (!z)) " dq q(1#ln q)Q (qz)#2 , I J U 128p z!i0 where
(4.43)
dv[4O (v,y)!2i(1!v)O (v,y)#2]#(y !y) (4.44) \ and the three-particle operators O are de"ned as (4.45) O (v,y)"e tM (y)y?c@c g GAB(vy)t(!y) , Q ?@AB O (v,y)"tM (y)y. y@D?g G (vy)t(!y) . (4.46) Q ?@ Path-ordered exponentials that connect "elds at di!erent points are again understood. The dots denote contributions that can be found in Balitsky and Braun (1988/89), but which are needed neither for the longitudinal part of the structure function nor in leading order of the #avour expansion. Two-gluon operators and four-fermion operators not related to a tM Gt operator through the equation of motion are not relevant at leading order. For the two-gluon operators, this follows from the fact that the third diagram in Fig. 13b vanishes because it contains no scale. The nucleon matrix elements of the three-particle operators O are parametrized as Q (y)"
1N(p)"O (v,y)"N(p)2(k)"2p ) y dm dm e N W K\T>K>T ¹ (m ,m ,k) ,
1N(p)"O (v,y)"N(p)2(k)"2(p ) y) dm dm e N W K\T>K>T ¹ (m ,m ,k) .
(4.47) (4.48)
The dependence on the renormalization scale will be suppressed in the following. It is now straightforward to take the Fourier transform and discontinuity of Eq. (4.43) to obtain the longitudinal structure function in the form of Eq. (4.42):
1 F (x,Q) * " dm dm [C (x,m ,m )¹ (m ,m )#C (x,m ,m )¹ (m ,m )] , U * * Q 2x
(4.49)
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where
4x x x C (x,m ,m )" 1#ln h(m !x)!(m m ) , (4.50) * m !m m m x 2 x 1 4x 1#ln # ln h(m !x)#(m m ) . C (x,m ,m )"! * m m (m !m ) m (m !m ) m (4.51)
Since we are interested in UV contributions to the matrix elements of multi-parton operators, the coe$cient functions at tree level as quoted su$ce. Up to this point we have been rather general. Let us now consider the UV renormalization of the three-particle operators O . There are logarithmic UV divergences, which lead to logarithmic scaling violations. However, power counting tells us that the operators also have quadratic divergences, which can appear in quark matrix elements through the diagrams shown in Fig. 13b. Since the quadratic divergences depend on the factorization scheme, one has to compute the quark matrix elements of O in the same way as the twist-2 coe$cient function, which means that we consider their Borel transform in leading order of the #avour expansion. Then the (Borel transform of the) matrix element of O between quark states of momentum p is given by dk ky !(k ) y)k I I 1p"O (v,y)"p2"(!i)4pC (!ke\!)\Se N W $ (2p) (k)>S(p!k)
;u (p)+e\ I W>Ty. (p. !k. )c #e\ I W\Tc (p. !k. )y. ,u(p) (4.52) I I and a similar result holds for O . Strictly speaking, the integral vanishes for p"0, because it does not contain a scale. This is the usual fact that matrix elements vanish perturbatively in factorization schemes that do not introduce an explicit factorization scale. One can isolate the quadratic divergence by keeping pO0, since the quadratic divergence is independent of p. Power divergences lead to non-Borel summable UV renormalon singularities in QCD and the quadratic divergence is seen as a pole at u"1 in the integral above. The integral can be done exactly. It is crucial for the singularity structure in u that y is exactly light-like. Close to u"1, we "nd for the Borel transforms
da(2!a)+tM (y)y. t(y[av!a])#tM (y[av#a])y. t(!y), , (4.53) (4.54) O (v,y) "2p ) yK da a+tM (y)y. t(y[av!a])#tM (y[av#a])y. t(!y), , where a"1!a and K is as de"ned in Eq. (4.41). Note that K is proportional to k, so these equations take the form expected for a quadratic divergence. The quadratic divergence is independent of the external states and Eqs. (4.53) and (4.54) are written as operator relations. The power divergent part takes the form of an integral over the leading-twist operator (4.39). This is exactly O (v,y) "(!K)
To avoid cumbersome notation, we do not write B[2] in what follows, but the Borel transform is understood.
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what one needs to match the IR renormalon singularity in the coe$cient function at leading twist. Taking the nucleon matrix elements, the quadratically divergent part of ¹ is expressed in terms of the twist-2 quark distribution as follows:
1 m ¹ (m ,m ) "(!K) 1# F(m )h(m !m )#(m m ) h(m )h(m ) , (4.55) m m m ¹ (m ,m ) "2K F(m )h(m !m )#(m m ) h(m )h(m ) . (4.56) m Inserting these expressions into Eq. (4.49), using Eqs. (4.50) and(4.51), and taking the remaining integrals except for one convolution, one "nds that the result takes the following form:
B
F K * (u,x) " +G (m)#G (m), * F(x/m) , 2x U Q
(4.57)
where
G (m)"4m 1#2(1#ln m) ln
1!m #lnm#2Li (1!m) m
G (m)"4m 1!2(1#ln m) ln
1!m !lnm!2Li (1!m)!d(1!m) m
(4.58)
(4.59)
with Li the dilogarithm function. There is a remarkable cancellation (for which we do not have an explanation) between the two contributions from the two operators and the sum G (m)#G (m)"8m!4d(1!m) (4.60) leads to the coincidence of Eqs. (4.57) and (4.40), except for the overall sign. Hence, we have shown that the "rst IR renormalon singularity in C cancels the UV renormalon singularity at the same * position in a perturbative evaluation of UV contributions to twist-4 matrix elements. The matching of IR renormalons in coe$cient functions and UV contributions to matrix elements exhibited here and in the p-model is a general feature of perturbative factorization and short-distance expansions, or asymptotic expansions in ratios of mass scales in general, in quantum "eld theories. QCD has a mass gap and is supposed to be well de"ned in the infrared. The complicated structure of the short-distance expansion, including renormalons, re#ects the fact that quantum #uctuations are distributed over all distance scales. However, if care is taken of de"ning all terms in the expansion consistently, the unambiguous expansion that is obtained may be hoped to be asymptotic to the exact, non-perturbative result.
5. Phenomenological applications of renormalon divergence In this section we turn to manifestations of renormalon divergence in particular physical processes. During the past few years the number of processes considered has been rapidly expanding as has been the number of next-to-leading and next-to-next-to-leading perturbative calculations. The interest in renormalons stems from the fact that they provide a link between
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perturbative and non-perturbative physics, because, on the one hand, renormalons account for a large part of the higher-order perturbative coe$cients and, on the other hand, in still higher orders they merge with the treatment of non-perturbative power corrections. Following this general idea, three main strains of applications with more or less emphasis on the perturbative or power correction aspect have developed. We brie#y summarize the questions, methods and problems associated with each of the three in Section 5.1 before turning to the details of process-speci"c applications. These applications deal exclusively with QCD processes. In order that there be renormalons, there must be a perturbative expansion. Hence the processes analysed in what follows satisfy two requirements: they contain at least one scale, which is large compared with K, the scale where QCD becomes non-perturbative, and one can isolate a part of the process that depends only on large scales, such that it can be expanded perturbatively in a . The Q large scale may be provided by large energy transfer in the high-energy collision of massless particles or by the mass of a quark much heavier than K. Applications of renormalons to hard reactions of massless particles are reviewed in Sections 5.2 and 5.3. The "rst section concentrates on processes that admit an operator product expansion (OPE) or are related to an OPE by dispersion relations. The second section deals with genuinely time-like processes. Finally, observables involving heavy quarks are discussed in Section 5.4. 5.1. Directions We summarize the main uses of renormalons. The starting point is series expansions in a "a (k), Q Q R(+q,,a ,k)" r (+q,,k)aL> , (5.1) Q L Q L where +q, denotes a set of kinematic variables which must all be large compared to K, and k denotes the renormalization and, if present, factorization scale. R may be either a physical quantity or a short-distance coe$cient in a factorization formula for a physical quantity. Without loss of generality the series starts at order a . Q 5.1.1. Large perturbative corrections Since renormalons dominate the large-order behaviour of the perturbative coe$cients r , the L question of whether they can be used to improve truncated perturbative series suggests itself. In an ideal situation we would compute the asymptotic behaviour and combine it with exact results in low orders so as to approximate the Borel integral, as was done using the instanton-induced divergence for improving perturbative calculations of critical exponents (Le Guillou and ZinnJustin, 1977). Even if the series were not Borel-summable, we would be able to improve the perturbative prediction to an accuracy limited only by the leading power correction. The large-order behaviour due to renormalons cannot be used in this way, because the overall normalization cannot be computed. As a consequence only ratios of coe$cients can be computed. If
c c #2 , r "K(ab )LC(n#1#b) 1# # L n#b (n#b)(n#b#1)
(5.2)
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the ratio of consecutive coe$cients is given by r
c c!2c L "ab (n#b) 1! #2 # (5.3) r (n#b)(n#b!1) (n#b) L\ and the parameters a,b,c are calculable as discussed in Section 3. An attempt to use this G observation for the cross section in e>e\ into hadrons was made in Beneke (1993b). There exist a few observables for which the "rst correction term in brackets is known and one } the di!erence between the pole mass and the MS mass of a heavy quark } for which even the 1/n correction can be obtained (see Section 5.4.1). In practice it is often di$cult to carry out this idea, because the large-order behaviour is not as simple as Eq. (5.2). There may be several components with the same value of a, but with di!erent normalization constants. This is the case with ultraviolet (UV) renormalon divergence as discussed in Section 3.2.3. Then the ratio of asymptotic coe$cients depends on ratios of normalization constants. For UV renormalons these ratios are processindependent and in principle one may think of determining them from a set of observables. In practice, this does not seem feasible, given in particular that the available exact series are not very long and reach n"2 at best. The application of the strategy outlined here is therefore restricted to observables whose large-order behaviour is dominated by infrared (IR) renormalons and which in addition exhibit a relatively simple IR renormalon structure. For all these reasons one resorts to either qualitative or less rigorous approaches. There are indeed interesting patterns in low order perturbative coe$cients. Referring to Table 3, which compares the perturbative expansions of three observables, we observe that the series in brackets (i) all have positive coe$cients and (ii) the larger the coe$cients are the larger the leading power correction is. All this may well be accidental. But remembering that larger power corrections are associated with faster growth of perturbative coe$cients due to IR renormalons, one may also speculate whether the observed pattern may be a manifestation of IR renormalon behaviour down to very low orders. This raises obvious questions: Why then is there no trace of sign-alternating UV renormalon behaviour, which should dominate the asymptotic behaviour for the Adler function ("rst line) and perhaps also deep inelastic scattering sum rules (second line)? The coe$cients in the table are scheme-dependent, and the comparison may look completely di!erent in another scheme. Why should the MS scheme be special? There is a simple &approximation' to the perturbative coe$cients that allows us to study part of these questions. Write r "r #r N #2#r NL "r [d (!b )L#d ] , (5.4) L L L D LL D L L where d "(!6p)Lr /r , b "!(11!2N /3)/(4p) and N is the number of massless quarks. We L LL D D then obtain the coe$cients d from a calculation of fermion bubble graphs (see Section 3.5) and L neglect the remainder d . The &model' for the series constructed in this way has UV and IR L renormalons at the correct positions, although the nature of the singularity and the overall
We recall that the leading UV renormalon leads to a minimal term of the series of order 1/Q and hence dominates all observables with IR power corrections smaller than 1/Q.
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Table 3 Comparison of perturbative series in a "a (Q)/p (MS scheme) and leading power corrections (LPC). Results are taken Q Q from Gorishny et al. (1991) and Surguladze and Samuel (1991) (Adler function of vector currents, 1st line, N "3), Larin D and Vermaseren (1991) (Gross}Llewellyn}Smith sum rule, 2nd line, N "3) and Kunszt et al. (1989) (average 1!¹, 3rd D line, N "5) D Observable
Series
LPC
dP 4pQ dQ 1 dxF (x,Q) 6 11!¹2(Q)
1#a (1#1.6a #6.4a#2) Q Q Q
1/Q
1!a (1#3.6a #19a#2) Q Q Q
1/Q
1.05a (1#9.6a #2) Q Q
1/Q
normalization are not reproduced correctly. It has been suggested in Beneke and Braun (1995a), Neubert (1995b) and Ball et al. (1995a) to use this approximation quantitatively and the procedure is often referred to as &Naive Non-Abelianization' (NNA). The term was coined by Broadhurst (Broadhurst and Grozin, 1995) who observed empirically that the remainder d at second-order a is typically rather small compared to d (!b ) in the MS scheme. This empirical Q fact, together with the fact that the method can be viewed as an extension of Brodsky}Lepage}Mackenzie scale setting (Brodsky et al., 1983), still provides the principal motivation for considering fermion loop diagrams. An important point is that one should expect the NNA or large-b approximation to work quantitatively only, if the contribution associated with the (one-loop) running coupling is large in higher orders. If it turns out to be small, there is no reason to expect that the NNA approximation is a good approximation to the exact higher order coe$cient. There are very few calculations that go beyond the calculation of fermion loop diagrams, and much of what follows relies on this class of diagrams. An interesting observation in the context of fermion bubble calculations is that in the MS scheme the (scheme-dependent, see Section 3.4) normalization of UV renormalons is suppressed compared to the normalization of IR renormalons, and hence the onset of UV renormalon behaviour is delayed (Beneke, 1993b). This suggests that UV renormalons are irrelevant to intermediate orders in perturbation theory in that scheme; it also suggests an explanation for why the series in Table 3 exhibit a "xed-sign pattern. We will return to estimates of perturbative coe$cients from NNA in Sections 5.2 and 5.4. 5.1.2. The power of power corrections Aside from their obvious connection with perturbation theory, renormalons are primarily discussed in connection with power corrections. If a(0 in Eq. (5.2), the attempt to sum the series
Note that this is not a systematic approximation, because it has no tunable small parameter. Formally, however, it can be obtained as a &large-b ' limit or a large (and negative!) N limit. Instead of &Naive non-abelianization' we will refer D to the approximation as the &large-b approximation' or &large-b limit'.
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M. Beneke / Physics Reports 317 (1999) 1}142
with the help of Borel summation leads to ambiguities of order
dR&
K ? ;logarithms of q/K q
(5.5)
in de"ning the perturbative contribution. In QCD these ambiguities arise from long distances and are interpreted as the ambiguity in de"ning what one means by &perturbative' and &non-perturbative'. As a consequence one identi"es the scaling with q of some power corrections through the value of a. Additional logarithmic variations of Eq. (5.5) can also be determined, but not the absolute magnitude of the power suppressed contribution. Note the analogy with the standard formalism for deep-inelastic scattering: scaling violations, logarithmic only at leading power, can be computed in perturbation theory, but not the parton densities. Early phenomenologically oriented discussions of IR renormalons concentrated on the question of whether or not there could be a 1/Q power correction to current}current correlation functions (Brown and Ya!e, 1992; Zakharov, 1992; Beneke, 1993a) which would imply larger non-perturbative corrections than the 1/Q correction incorporated through the OPE. From the present perspective this discussion appears historical. If there is an OPE the IR renormalon structure is consistent with it by construction. At the same time one has to be aware of the fact that IR renormalons imply power corrections, but that the converse is not true. There may be power corrections larger than those indicated by renormalons, especially for time-like processes, and next to nothing is known theoretically about them. These may be power corrections to coe$cient functions from short distances, power corrections from long-distances that do not &mix' with perturbation theory, or, for time-like processes, power corrections related to violations of parton-hadron duality, i.e. the possibility that the power expansion is not an asymptotic one after continuation to the time-like region. Our attitude towards this problem is that if power corrections indicated by renormalons are large, there is a good chance that one has found the dominant ones. Identifying power corrections through IR renormalons is especially interesting for processes that do not admit an OPE and for which the result is not obvious. Along this line the heavy quark mass was considered in Beneke and Braun (1994) and Bigi et al. (1994b). The "rst investigations of hard QCD processes, in particular event shape observables in e>e\ annihilation into hadrons, from this perspective appeared in Contopanagos and Sterman (1994), Manohar and Wise (1995) and Webber (1994a). Most often power corrections to these observables are large, being suppressed only by one power of the large momentum scale. For event shape observables and other hadronic quantities the understanding of even the leading power corrections is still not complete, although signi"cant progress has been made over the past four years. At this point it seems that the analysis of IR renormalons would merge with the general problem of classifying IR-sensitive regions in Feynman integrals beyond leading power. 5.1.3. Models The absolute magnitude of power corrections cannot be calculated with perturbative methods. Additional assumptions are needed, which may be di$cult to justify. The result is a model for power corrections. Such models have the advantage that they are consistent with short-distance properties of QCD } exactly the point that is most problematic for other models of low-energy
M. Beneke / Physics Reports 317 (1999) 1}142
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QCD } although they cannot be derived from QCD. Two models, for di!erent purposes, have been developed. In the Dokshitzer}Webber}Akhoury}Zakharov (DWAZ) model for event shape variables (Dokshitzer and Webber, 1995; Akhoury and Zakharov, 1995) it is assumed that 1/Q (where Q is the centre-of-mass energy) power corrections in the fragmentation of quarks and gluons in e>e\P hadrons can be accounted for by one parameter only. Hence, for all (averaged) event shape variables we may write schematically S "K 1k 2/Q . / 1
(5.6)
It follows from the universality assumption that the relative magnitudes of 1/Q power corrections to di!erent observables are predicted (see the discussion in Section 4.2.1). The simplicity of the model is appealing and has led to numerous comparisons with experimental data on the energy dependence of averaged event shapes. We follow this in more detail in Section 5.3.2. The second model, proposed in Dokshitzer et al. (1996) and Stein et al. (1996), concerns the dependence of twist-4 corrections to deep-inelastic scattering cross sections on the scaling variable x. The OPE constrains these to be of the form of Eq. (4.42) with unknown multi-parton correlations. The model assumes that the x-dependence can be approximated by the x-dependence of the IR renormalon contribution to the twist-2 coe$cient function folded with the ordinary parton densities. The structure functions are then expressed as
dm K f (x/m,k) CG (m,Q,k)#AG (m) #2 , F (x,Q)/(2x)" . . Q . m G V G
(5.7)
with calculable functions AG (m). Usually, only quarks are taken into account in the sum over i. It is . clear that the target dependence at twist-4 is the same as at twist-2 in this model and a prerequisite for it to work is that the genuine target dependence at twist-4 is small compared to the twist-4 correction as a whole. In Dokshitzer et al. (1996) the model has been motivated by the assumption that the bulk of the twist-4 correction can be accounted for as an integral over a universal, IR-"nite coupling constant. In Beneke et al. (1997b) it was argued that the model could be justi"ed, if the twist-4 matrix elements normalized at k were dominated by their UV contributions from K(k(k rather than by #uctuations with k&K. This interpretation follows indeed directly from the matching calculation performed in Section 4.2.2. Since the UV contributions to twist-4 matrix elements are equivalent to IR contributions to twist-2 coe$cient functions, the &ultraviolet dominance' suggestion amounts to stating that the model provides an e!ective parametrization of perturbative contributions not taken into account in the truncated series expansion of CG . We . discuss this model further in Section 5.2.4 and, for fragmentation functions in e>e\ annihilation, in Section 5.3.1. The advantage of both models introduced here is simplicity. In both cases, success or failure in comparison with data leads to interesting hints on the nature of power corrections.
The models proposed by Dokshitzer and Webber (1995) and Akhoury and Zakharov (1995) do not coincide exactly. They share however the crucial assumption that power corrections are universal. See Section 5.3.2 for a more discriminative discussion.
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5.2. Hard QCD processes I In this section we summarize results on renormalons for inclusive hadronic observables in e>e\ annihilation, q decay and deep inelastic scattering (DIS). Since the scaling of power corrections is known from OPEs, the emphasis is on potentially large higher-order perturbative corrections and, in Section 5.2.4, on modelling the x-dependence of twist-4 corrections to DIS structure functions. 5.2.1. Inclusive hadroproduction in e>e\ annihilation The inclusive cross section e>e\PcHPhadrons is given by the vector current spectral function:
p>\ (5.8) R > \(q)" C C "12p e Im P(q#i0) , O C C p>\ >\ C C I I O where the vacuum polarization P(q) is de"ned by Eq. (2.14). The Adler function (see Eq. (2.15)) is expanded as a dP(Q) "1# Q aL[d (!b )L#d ] , (5.9) Q L L p dQ L see Eq. (5.4). The normalization is such that d "1, d "0. As mentioned after Eq. (5.4) the coe$cients d are computed in terms of fermion bubble diagrams, in the present case the diagrams L shown in Fig. 1. The exact result for these diagrams was obtained in Beneke (1993a) and Broadhurst (1993) in the context of QED. Adjusting the colour factors and overall normalization, the Borel transform is found to be D(Q)"4pQ
(!1)Ik \S u d 32 Q e! . (5.10) B[D](u)" L uL" (k!(1!u)) 1!(1!u) n! 3 k I L The representation in terms of a single sum is due to Broadhurst (1993). In the MS scheme C"!5/3. The coe$cients d are presented in Table 4 for k"Q. With reference to Eq. (5.9), we L call the approximation of neglecting the d the &large-b ' approximation. For comparison we show L d obtained from the exact perturbative coe$cients (Gorishny et al., 1991; Surguladze and Samuel, 1991) and d obtained from the estimate of Kataev and Starchenko (1995). The &large-b ' approximation is quite good at order a but overestimates the coe$cient at order a considerQ Q ably. It should be noted that the comparison depends on the choice k"Q and the approximation cannot be expected to work well for arbitrary choices of scale or scheme (Beneke and Braun, 1995a; Ball et al., 1995a). This has been a point of criticism of the &large-b ' approximation (Chyla, 1995). We discuss this point further in the context of q decay below. The renormalon singularities of the Adler function have already been discussed in Section 2.4. The UV renormalon poles at u"!1,!2,2 are double poles. The IR renormalon poles at u"2,3,2 are also double poles, with the exception of u"2. In the large-N limit one expects an D A di!erent quark electric charge factor is understood for the &light-by-light' contributions. However, in the following &light-by-light' terms do not play an important role.
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Table 4 Perturbative corrections to the Adler function in the MS scheme: the &large-b limit' in comparison with the remainder, d , to the exact result and an estimate thereof for d . Results for N "3 D n
d L
d (!b )L L
d L
0 1 2 3 4 5 6 7 8
1 0.6918 3.1035 2.1800 30.740 !34.534 759.74 !3691.4 42251
1 0.4955 1.5919 0.8009 * * * * *
0 0.0265 !0.9464 0.0860 * * * * *
IR renormalon pole at n to take the form 1/(n!u)>A @D, where c is the N -part of the one-loop D anomalous dimension of an operator of dimension 2n, see Eq. (3.51). It follows that the singularity at n"2 has to be a simple pole, because the operator a GG has no anomalous dimension in the Q large-N limit. It has been checked by Beneke (1993c) that there is a dimension-6 operator with D c "2b , which leads to a double pole at n"3. Since there is no operator of dimension 2 in the D OPE of the Adler function, there is no IR renormalon pole at u"1. The Borel transform of the vacuum polarization is obtained by dividing B[D](u) by (!u). One then notes (Beneke, 1993a; Lovett-Turner and Maxwell, 1994) the symmetry B[P](1#u)" B[P](1!u), which interchanges UV and IR renormalon poles. This symmetry implies that the small and large momentum behaviours of the diagrams of Fig. 1 are related (Beneke, 1993c). Note that this symmetry relates the IR renormalon pole at u"2 which corresponds to the gluon operator a GG to the pole at u"0, which corresponds to (external) charge renormalization. Q Likewise the IR renormalon pole at u"3 and the UV renormalon pole at u"!1 are related, and both are described in terms of dimension-6 operators. It is not known whether this symmetry persists in higher orders of the #avour expansion. It is interesting to break down the d into contributions from the leading renormalon pole in L order to check how fast the asymptotic regime is reached. To this end we decompose B[D](u) into the sum of the leading poles according to
B[D](u)"e\
4 1 10 1 2 # #e 9 (1#u) 9 1#u 2!u
2 1 1 1 #e\ ! ! #2 . 9 (2#u) 2 2#u
(5.11)
This breakdown is given in Table 5. One can see that the asymptotic behaviour sets in late and the low-order coe$cients n&1}5 are not dominated by a single renormalon pole. The irregularities in low orders are due to cancellations between IR and UV renormalons in every second order. The sum over contributions from IR renormalon poles does not converge, because of the overall factors
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Table 5 Breakdown of d into contributions from the leading IR and UV renormalon poles. The integer in brackets denotes the L position of the pole n
d L
UV(!1)
IR(2)
UV(!2)
IR(3)
IR(4)
0 1 2 3 4 5 6 7 8
1 0.6918 3.1035 2.1800 30.740 !34.534 759.74 !3691.4 42251
0.294 !0.378 0.923 !3.27 15.1 !85.6 574 !4442 38923
28.03 14.02 14.02 21.02 42.05 105.1 315.4 1104 4415
!0.011 0.006 !0.007 0.013 !0.028 0.078 !0.256 0.975 !4.214
!11.0 !11.0 !12.2 !17.1 !29.3 !59.7 !141 !380 !1149
!50.9 !7.28 !0.91 1.36 3.41 6.82 14.1 31.3 76.1
eL for an IR renormalon pole at u"n. If one chooses the scheme with C"0, the asymptotic regime sets in earlier. In this case the series is dominated by sign-alternating behaviour from UV renormalons starting at low order. In Beneke (1993b) a result for the ratio of asymptotic coe$cients due to the "rst IR renormalon was obtained that does not rely on the large-b limit. This uses the known anomalous dimension of the operator a GG and the second-order Wilson coe$cient (Chetyrkin et al., 1985; Surguladze and Q Tkachov, 1990) to obtain b and c in Eq. (5.3). The result can only be useful in intermediate orders, before the asymptotically dominant UV renormalon behaviour takes over. However, Table 5 suggests that higher IR renormalons are very important at low orders because of their enhanced overall normalization in the MS scheme. Hence, the method outlined in Section 5.1.1 is not expected to be useful for the Adler function, at least in the MS scheme. Taking the large-b approximation as a model for the entire series, we can also estimate the ambiguity in summing the series. We estimate this by dividing the absolute value of the imaginary part of the Borel integral (2.10) by p, an estimate that comes close to the minimal term of the series. Restricting the attention to the "rst IR renormalon pole, we "nd 2 e K 0.06 GeV +1+ . dD(Q)" ! p Q b Q
(5.12)
This should be compared with the contribution from the gluon condensate
1 0.08 GeV 2p a Q GG + , Q Q 3 p
(5.13)
The factor 1/p comes from the 1/p in Eq. (5.9). We determine K from a (m )"0.33, using the one-loop relation (to Q O +1 be consistent with the large-b approximation) and N "3. This gives K "215 MeV. D +1
M. Beneke / Physics Reports 317 (1999) 1}142
79
which is marginally larger than the perturbative ambiguity. (The present estimate agrees with Neubert (1995b).) Note that the phenomenological value of the gluon condensate (Shifman et al., 1979) may in part parametrize higher-order perturbative corrections, because it is extracted from comparison of data with a theoretical prediction that includes only a "rst-order radiative correction. In the large-b approximation there is a simple relation between the Borel transform of the Adler function and that of the inclusive cross section e>e\P hadrons, because the b-function has exactly one term b a. Writing Q
a R > \"N 1# Q aL [d0(!b )L#d0] , A Q L L C C p L
(5.14)
and neglecting d0, we have (Brown and Ya!e, 1992) L d0 sin(pu) B[R](u)" L uL" B[D](u) . n! pu L
(5.15)
This follows directly from the fact that the Q-dependence factorizes in Eq. (5.10) in the large-b approximation (Beneke, 1993a). The sin attenuates the renormalon singularities. In particular, the "rst IR renormalon pole at u"2 is eliminated. This is an artefact of the large-b approximation. Beyond this approximation the renormalon singularities are branch cuts, which are suppressed but not eliminated by analytic continuation to Minkowski space. In large orders, d /d0&n. L L More on numerical aspects of the Adler function in the large-b approximation can be found in Neubert (1995b), Ball et al. (1995a) and Lovett-Turner and Maxwell (1995). The distribution function ¹(m) that enters the integral representation (3.83) of the (principal value) Borel integral is given in Ball et al. (1995a) (for R > \) and Neubert (1995c) (for D). C C 5.2.2. Inclusive q decay into hadrons The inclusive q decay rate into hadrons yields one of the most accurate determinations of the strong coupling a . Subsequent to the detailed analysis of (Braaten et al., 1992) in the framework of Q the OPE (Shifman et al., 1979), a lot of e!ort has gone into controlling and understanding the uncertainties in the perturbative series that enters the prediction and into the question of whether there could be other non-perturbative corrections than those incorporated in the OPE, in particular power corrections suppressed only by K/m. The latter question touches also the issue O of parton-hadron duality, although from the point of view of duality there is no reason that violations of it should scale as 1/m. Since renormalons have nothing to say about this and since O experimental evidence does not support &non-standard' non-perturbative corrections (such as small-size instanton corrections (Nason and Porrati, 1994; Balitsky et al., 1993; Nason and Palassini, 1995)), we focus on the accuracy of the perturbative prediction in this section. Its renormalon structure was analysed in Beneke (1993c). Numerical investigations of the large-b limit were performed by Ball et al. (1995a) and Neubert (1995c) and by Lovett-Turner and Maxwell (1995) and Maxwell and Tonge (1996) for the total decay width and for weighted spectral functions by Neubert (1996). Altarelli et al. (1995) investigated the uncertainties due to UV renormalons speci"cally.
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The total hadronic width is very well known experimentally, and we quote the result from the ALEPH Collaboration (Barate et al., 1998) C(q\Pl # hadrons) O "3.647$0.014 . (5.16) R" O C(q\Pl e\l ) O C The error in a (m ) obtained from this measurement is largely theoretical. The theoretical prediction Q O follows from the correlation functions of the charged vector and axial-vector currents, which are decomposed as PIJ (q)"(q q !g q)P (q)#q q P (q) . (5.17) 4 I J IJ 4 I J 4 Making use of the exact, non-perturbative analyticity properties of the correlation functions, one obtains
ds s s 1! 1#2 P(s)#P(s) , (5.18) m m m O Q K O O O where the integral extends over a circle of radius m in the s"q plane and PG(s)" O PG(s)#PG(s). This equation includes decays into strange quarks. Small electroweak corrections 4 have to be applied. Eq. (5.18) has a meaningful perturbative expansion, because the smallest scale involved is m . O We treat quark mass terms as power corrections in m /m and refer to the perturbative B Q O expansion of R in a in the limit m "0 as the perturbative contribution. As before, we write O Q B Q a (5.19) R "N ("< "#"< ") 1# Q aL [dO (!b )L#dO ] , Q L L O A SB SQ p L and obtain an exact result in the approximation where the remainders dO are neglected. The Borel L transform follows from inserting Eq. (5.10) into Eq. (5.18). Taking advantage of the factorized dependence on s"!Q in Eq. (5.10), the result is (Beneke, 1993c) R "6pi O
dO 1 2 2 1 # ! # . (5.20) B[R ](u)" L uL"B[D](u) sin(pu) O n! pu p(1!u) p(3!u) p(4!u) L The sin attenuates all renormalon poles except those at u"3,4. The point u"1 is regular, but we note that if a power correction of order K/m to D existed, it would not be suppressed by a factor O of a after taking the integral in Eq. (5.18). Q In Table 6 we show the coe$cients dO in the MS scheme and in the scheme with C"0, with L k"m in both cases. In the present approximation the second scheme coincides with the < scheme, O where the coupling is de"ned through the static heavy quark potential. The table also shows the partial sums , M (a )"1# d (!b a )L , (5.21) , Q L Q L which quantify how much the "rst-order radiative correction is modi"ed by higher order corrections. Compared to the Adler function (see Table 4) the onset of the sign-alternating UV
M. Beneke / Physics Reports 317 (1999) 1}142
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Table 6 Perturbative corrections to R in the MS and < scheme. For the partial sums we take a (m )"0.32 in the MS scheme. The O Q O last three columns compare the &large-b limit' with the remainder, dO , to the exact result and an estimate thereof for dO . MO +1 gives partial sums with dO taken into account L L n
dO +1 L
dO 4 L
MO +1 L
dO +1(!b )L L
dO L
MO +1 L
0 1 2 3 4 5 6 7 8
1 2.2751 5.6848 13.754 35.147 84.407 248.83 525.38 3036.0
1 0.6084 0.8788 !0.3395 3.7796 !14.680 99.483 !664.00 5400.1
1 1.521 1.819 1.984 2.081 2.134 2.170 2.187 2.210
1 1.629 2.916 5.053 ! ! ! ! !
0 0.027 !0.245 !1.650 ! ! ! ! !
1 1.530 1.803 1.915 ! ! ! ! !
renormalon divergence is delayed, because the integration in Eq. (5.18) enhances the over-all normalization of IR renormalons relative to UV renormalons. (This e!ect holds beyond the large-b limit.) In the MS scheme the low orders are dominated by "xed-sign behaviour and the series can be summed to a parametric accuracy of order K/m without interference of UV O renormalons. The situation is di!erent in the < scheme, where UV renormalon residues are larger and IR renormalon residues are smaller. Comparison with exact results shows that the large-b approximation is very good at order a , but seems to overestimate the next order, if we trust the Q estimate of Kataev and Starchenko (1995) more than the large-b estimate. In Table 7 we show the contributions to dO +1 from the leading renormalon poles, to be compared with Table 5 for the L Adler function. The relevant decomposition of the Borel transform is now
1 8 1 2 8 1 2 1 #e\ #e ! #2 , B[R ]"e\ O 3 (3!u) 9 3!u 135 2#u 15 1#u
(5.22)
which shows explicitly the suppression of residues of the leading UV renormalon poles. However, Table 7 illustrates that the coe$cients dO +1 are only approximately dominated by the IR renorL malon pole at u"3. On the other hand, in the < scheme (not shown in the table) the leading UV renormalon pole describes the coe$cients well for n'5. The a correction (n"2) adds about 0.3 to the partial sums in Table 6. If we truncate the series at Q its minimal term (n"7) the cumulative e!ect of higher-order corrections amounts to 0.4, slightly larger than the third-order correction. This amounts to a reduction of a (m ) needed to reproduce Q O the data. To make this more precise (Ball et al., 1995a) (see also Neubert, 1995c; Lovett-Turner and One may object that the large-b approximation overestimates this number, because it may overestimate already the coe$cient for n"3. However, if the actual growth of coe$cients were slower than in the large-b approximation, we would be able to add more terms.
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Table 7 Breakdown of dO +1 into contributions from the leading IR and UV renormalon poles. The integer in parantheses denotes L the position of the pole n
dO +1 L
UV(!1)
IR(3)
IR(4)
0 1 2 3 4 5 6 7 8
1 2.2751 5.6848 13.754 35.147 84.407 248.83 525.38 3036.0
0.025 !0.025 0.050 !0.151 0.604 !3.022 18.13 !126.9 1015
0 14.66 19.54 29.32 52.12 108.6 260.6 709.4 2162
!87.31 !27.28 !16.37 !14.32 !16.37 !23.02 !38.37 !73.86 !161.1
Maxwell, 1995) computed the principal value of the Borel integral as a function of a . For Q a (m )"0.32, they "nd MO "2.23, close to the value MO "2.19 that would have been obtained Q O from truncating the series expansion (see Table 6). Note that MO is scheme-dependent, but a MO is Q not, provided schemes are consistently related in the large-b approximation (Beneke and Braun, 1995a). Accounting for electroweak and power corrections, R is given by O R "3 ("< "#"< ") S +1#d#d #d ,. (5.23) O SB SQ #5 #5 Making use of the analysis of power corrections in Braaten et al. (1992) and their approximate a -independence, the experimental measurement quoted above translates into Q d"0.211$0.005. (5.24) The error is purely experimental and no theoretical error has been assigned to d . (The analysis of power corrections by the ALEPH Collaboration (Barate et al., 1998) leads to d"0.20.) The theoretical prediction, based on the series in the large-b approximation, is a (m ) (5.25) d" Q O [MO (a (m ))#dO a (m )#dO a (m )] , Q O Q O Q O p where the terms in the series known exactly are taken into account. This result for d is shown as curve &i' in Fig. 14. Compared to perturbation theory truncated at order a (curve &ii'), the value of Q a (m ) is reduced by 15% from about 0.35 to 0.31. This is somewhat less than the reduction caused Q O by adding the a correction (compare curves &ii' and &iv'). Q How reliable is the large-b approximation for the unknown higher order perturbative contribu tions? Clearly, there is no answer to this question. If we knew, we could do better. It seems safe to conclude that higher order corrections add positively and reduce a . As a consequence, we may Q
However, the corrections to the large-b approximation may be di!erent in di!erent schemes.
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Fig. 14. d as a function of a (m ) (MS scheme) for various truncations/partial resummations of the perturbative Q O expansion: (i) Large-b resummation according to Eq. (5.25). (ii) Fixed-order perturbation theory up to (including) a. (iii) Q Resummation of running coupling e!ects from the contour integral only (see text for discussion). (iv) Fixed-order perturbation theory up to a. The shaded bar gives the experimental measurement with experimental errors only. The Q "gure is an update from Ball et al. (1995a).
argue that the theoretical error should not be taken symmetric around the "xed order a result, but Q rather as the variation between curves &i' and &ii'. This understanding of the &systematics' of higher order corrections is taken into account in (Barate et al., 1998), where the error of "xed-order perturbation theory is computed from a variation around an assumed positive value for the a correction. An important point is that incorporating systematic shifts due to higher order Q perturbative corrections in q decay may bring us closer to the &true' value of a , but need not Q improve the consistency with other measurements, if similar systematic e!ects exist there and are not taken into account. In the above discussion, renormalon ambiguities in the perturbative prediction play no role, because they are very small, re#ecting the fact that the minimal term is attained at rather large n. In principle, there is an error of order K/m that arises when the series is truncated at the onset of UV O renormalon divergence. The large-b approximation suggests that the numerical coe$cient of this term is very small, so that this uncertainty is insigni"cant in the MS scheme. (Recall that the magnitude of this term is scheme-dependent, see Section 3.4 and Beneke and Zakharov (1992).) Related to this is the observation made above that the coe$cients do not show sign-alternation up to relatively high orders, see Table 6. Altarelli et al. (1995) have investigated UV renormalons in q decay in great detail, using conformal mappings to eliminate this uncertainty. They found rather sizeable variations of $0.05 in a (m ), depending on the precise implementation of the mapping Q O procedure. There is a problem in applying these mappings to series that do not yet show sign-alternation, because the mapping then produces ampli"cations of coe$cients rather than cancellations. We therefore feel that the conclusion of Altarelli et al. (1995) may be too pessimistic.
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In curve &iii' of Fig. 14 we show the result for the perturbative contribution to R based on the O implementation of a partial resummation of running coupling e!ects suggested by Le Diberder and Pich (1992). This resummation takes into account a series of &p-terms' that arise when integrals of powers of a (!(s) are taken according to Eq. (5.18). Because the largest e!ect comes from b , this Q resummation is included in the large-b approximation which takes into account running coupling e!ects not only in the contour integral (5.18) but also in the spectral functions. Comparison of &i' and &iii' with &ii' shows that the e!ect of the two resummations tends into di!erent directions relative to the "xed-order result. The explanation suggested in Ball et al. (1995a) reads that the convergence of the partial resummation of Le Diberder and Pich (1992) is limited by the UV renormalon behaviour of the Adler function. As seen from Table 4 this limitation is more serious for D than it is for R . O The large-b approximation is scheme and scale dependent in the sense that the terms dropped (the remainders d ) are of di!erent size in di!erent schemes. Such scheme dependence is expected for L partial resummations and the real question is in which schemes the approximation works best. The requirement of scheme-independence emphasized by Chyla (1995) and Maxwell and Tonge (1996) misses this point. Since empirically the approximation seems to work well in the MS scheme, one cannot expect it to work well in schemes that di!er from MS by large parameter rede"nitions that are formally of sub-leading order. Maxwell and Tonge (1996) proposed to implement the large-b limit for the e!ective charge b-function that corresponds to R . In Fig. 14 this implementation falls O below the "xed order result &ii'. This resummation scheme implies that the correction to be added to the third order result in the MS scheme is negative despite the regular "xed-sign behaviour observed in the exact coe$cients up to order a. Q As the spectral functions in q decay are well measured, additional information can be obtained from their moments. Neubert (1996) has analysed in detail the leading-b resummations for the moments. Finally, we mention that when Eq. (3.83) is used to compute the principal value Borel integral M , the &Landau pole contribution' in square brackets is very important. Although formally of order K/m, leaving this term out results in a very small value for M . The omission of this term is O equivalent to a rede"nition of the coupling constant which is related to the MS coupling by large 1/Q corrections not related to renormalons. This point is discussed in detail in Ball et al. (1995a). 5.2.3. Deep-inelastic scattering: sum rules Consider the Gross}Llewellyn}Smith (GLS) and polarized Bjorken (Bj) sum rules,
a dx FJN>J N(x,Q)"6 1! Q aL[d%*1(!b )L#d%*1] , Q L L p L
1g a dx gCN\CL(x,Q)" 1! Q aL[d (!b )L#d ] . Q L L 3g p 4 L
The distribution function required for q decay can be found in Ball et al. (1995a).
(5.26)
(5.27)
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The nucleon structure functions F and g are de"ned in the standard way. In both cases the twist-4 K/Q corrections are given by the matrix element of a single operator (Ja!e and Soldate, 1981; Shuryak and Vainshtein, 1982; Ellis et al., 1982). The perturbative corrections are known exactly to order a (Larin and Vermaseren, 1991). The normalization is such that d "1, d "0. Q The IR renormalon singularity at t"!1/b (u"1) that corresponds to the twist-4 operator was "rst discussed by Mueller (1993). The strength of the leading UV renormalon at t"1/b is determined in Beneke et al. (1997a). Combining both pieces of information, we "nd (Beneke et al., 1997a) C (a ) L " (!b )L n! [K34 (!1)L n>@ @>H# K'0 n\@ @\ @,A\ ,A] aL> , (5.28) %*1 Q %*1 Q %*1 L where C (a ) denotes the perturbative contribution to the GLS sum rule and the anomalous %*1 Q dimension of the twist-4 operator calculated by Shuryak and Vainshtein (1982) has been used. In this equation b are the "rst two coe$cients of the b-function, b"!4pb , and j is related to the anomalous dimension matrix of four-fermion operators, see Table 1. For N '2, the UV D renormalon behaviour dominates the asymptotic behaviour at very large n because of its larger power of n. However, the overall normalizations are not known. Since the MS scheme favours large residues of IR renormalons, one expects "xed-sign IR renormalon behaviour in intermediate orders. The "rst three terms in the series known exactly are indeed of the same sign in the MS scheme. The large-b approximation to the perturbative part of the sum rules has been investigated in Ji (1995a) and Lovett-Turner and Maxwell (1995). The large-b approximations to the GLS and Bj sum rules coincide, because the perturbative contributions to the sum rules di!er only by &light-by-light' contributions starting at order a. These contributions are subleading in the Q large-b approximation. The Borel transform that is relevant in the large-b approximation can be inferred from Broadhurst and Kataev (1993) and is given by
4 5 1 d%*1
Q \S 1 8 uL" e! # ! ! . (5.29) B[GLS/Bj](u)" L n! k 9 1!u 1#u 2!u 2#u L It is much simpler than the Borel transform for the Adler function (5.10), because the a correction Q comes from one-loop diagrams in DIS and from two-loop diagrams for the Adler function. In particular, there are only four renormalon poles, all other being suppressed at leading order. But since the leading singularities at u"$1,$2 are present, we may still try a numerical analysis. The coe$cients d%*1"d are displayed in Table 8 and compared with the exact result and an L L estimate of the a correction from Kataev and Starchenko (1995). We note that while the large-b Q approximation gives the higher-order corrections with the correct sign, it generally overestimates them, a tendency already observed for the Adler function and q decay. Taken at face value, the large-b approximation implies that the minimal term of the series is reached at order a at Q Q"3 GeV, a momentum transfer relevant to the CCFR experiment. Hence it is not clear whether at Q"3 GeV the perturbative prediction could be improved by further exact calculations of higher-order corrections. Further improvement would then require the inclusion of twist-4 contributions, and in particular a practically realizable procedure to combine them consistently with the perturbative series.
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Table 8 Perturbative corrections to the GLS (Bj) sum rules in the large-b limit. All results in the MS scheme and for N "3. To D compute the partial sums we take a (Q"3 GeV)"0.33. The last three columns compare the large-b limit with the Q remainder, d%*1, to the exact result and an estimate thereof for d%*1. M%*1 gives partial sums with d%*1 taken into L L account n
d%*1 L
M%*1 L
d%*1(!b )L L
d%*1 L
M%*1 L
0 1 2 3 4 5 6 7 8
1 2 6.389 22.41 103.7 525.9 3362 22990 1.92 ) 10
1 1.473 1.830 2.125 2.449 2.837 3.423 ! !
1 1.432 3.277 8.233 ! ! ! ! !
0 !0.291 !1.354 !4.040 ! ! ! ! !
1 1.376 1.586 1.737 ! ! ! ! !
In this context it is interesting to note that the integral over loop momentum is dominated by k&450 MeV at order a and k&330 MeV at order a. As for the Adler function, we estimate the Q Q ambiguity in summing the perturbative expansion by the imaginary part of the Borel integral (2.10) (divided by p) from the "rst IR renormalon pole alone. This gives (K "215 MeV as above) +1 1 1 8e K 0.10 GeV +1+ dGLS(Q)" ! . (5.30) 6 9p Q b Q This should be compared to the twist-4 contribution to the same quantity estimated by QCD sum rules (Braun and Kolesnichenko, 1987),
(5.31) ! 11O 22/Q+!0.1 GeV/Q , where 11O 22 is the reduced nucleon matrix element of a certain local twist-4 operator. The two are comparable, which suggests that the treatment of perturbative corrections beyond those known exactly is as important for a determination of a as the twist-4 correction. Q Stein et al. (1996) and Mankiewicz et al. (1997) have considered moments of the longitudinal structure function F and the non-singlet contribution to F , respectively, in the large-b approxi* mation. The second case is more di$cult, because it requires collinear factorization to be carried out in the large-b limit, while this is not necessary for F in leading order. The approximation is * found to be quite good for larger moments (N'4), typically overestimating the exact result by some amount, but fails completely for the lower moments of F . This may be due to the fact that smaller moments are more sensitive to the small-x region in which other e!ects not incorporated in the large-b limit are important (Stein et al., 1996). These estimates can be obtained from converting the Borel transform into the loop momentum distribution (Neubert, 1995b), see Section 3.5.3.
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5.2.4. Twist-4 corrections to DIS structure functions In this section we discuss applications of the &renormalon model' for twist-4 corrections to deep-inelastic scattering (DIS) quantities suggested in Dokshitzer et al. (1996) and Stein et al. (1996). The basic aspects of the model, its virtues and limitations, have already been outlined in Section 5.1.3, see Eq. (5.7). To make the idea more explicit, we consider the structure functions F and F as examples. One * "rst computes the dependence of the "rst IR renormalon residue (related to twist-4 operators, see Section 4.2.2) on the scaling variable x"!q/(2p ) q). At present all such calculations have been done only for one-loop diagrams dressed by vacuum polarization insertions, i.e. in the formal large-b limit. It is usually most convenient to extract the residue from the expansion of the distribution function ¹(m) introduced in Section 3.5.2. The result is (Beneke and Braun, 1995b; Dokshitzer et al., 1996; Stein et al., 1996; Dasgupta and Webber, 1996) A(x)"8x!4d(1!x) , *
(5.32)
A(x)"!(4/[1!x] )#4#2x#12x!9d(1!x)!d (1!x) >
(5.33)
for F /(2x) and F /(2x). The &#' prescription is de"ned as usual by dx [ f (x)] t(x)" * > dx f (x) (t(x)!t(1)) for test functions t(x). The result is then represented as
D (x,Q) #O(1/Q) , F (x,Q)"FU(x,Q) 1# . . . Q
(5.34)
where FU(x,Q) is the leading-twist result for the structure function F and . .
dm 1 f (x/m,k) KA G(m) D (x,Q)" G . . m G FU(x,Q) . G V
(5.35)
is the model parametrization of the (relative) twist-4 correction. Here f (x/m,k) are standard G (leading-twist) parton densities, i sums over quarks and gluons, and K are scales of order K which G provide the overall normalization. We recall (Section 4.2.2) that twist-4 corrections take the form Eq. (5.35) if the twist-4 matrix elements are substituted by their power divergence (Beneke et al., 1997b). The overall normalization has been treated di!erently in the literature. In the approach of Dokshitzer et al. (1996), it is suggested to parametrize the normalization of all 1/Q power corrections by a single process-independent number, to be extracted from the data once. Stein et al., 1996 originally suggested to "x the overall normalization parameter-free by the normalization of the renormalon ambiguity. This turned out to "t the data poorly and the authors subsequently also treated the overall normalization as a free parameter (Maul et al., 1997). In Beneke et al. (1997b) it is
A common overall normalization is omitted, because it plays no role in what follows. See Section 4.2.2 for de"nitions and the derivation of the result for F in terms of UV properties of twist-4 distributions. *
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suggested that the normalization should be adjusted in a process-dependent way and only the shape of the x-distribution taken as a prediction of the model. Because of di$culties in constructing the gluon contribution in the model, one may think of adjusting the normalization of quark and gluon contributions separately. The &renormalon model' of twist-4 corrections has drawn much of its inspiration from Fig. 15 "rst shown by Dokshitzer et al. (1996) (see also Dasgupta and Webber, 1996; Maul et al., 1997). The shape of the twist-4 correction to the structure function F calculated from the model reproduces the shape required to "t experimental data very well. Note that the renormalon model contains only the non-singlet contribution to F , which is expected to dominate except for small values of x. Encouraged by this observation, Stein et al. (1996) and Dasgupta and Webber (1996) considered the longitudinal structure function F , while Dasgupta and Webber (1996) and Maul et al. (1997) * considered the structure function F . The polarized structure function g has been analysed by Dasgupta and Webber (1996) and Meyer-Hermann et al. (1996) and Maul et al. (1997) Other polarized structure functions were examined by Lehmann-Dronke and SchaK fer (1998) and the transversity distribution h by Meyer-Hermann and SchaK fer (1997). Recently, Stein et al. (1998) added a model prediction for the singlet contribution to F , which modi"es Figs. 15 and 16 at small x, below those x for which comparison with data is possible. It is interesting to compare this prediction with other model parametrizations of twist-4 corrections at small x. The treatment of singlet contributions is more di$cult and ambiguous in the renormalon model than non-singlet contributions. The calculation relies on singlet quark contributions, which are then reinterpreted as gluon contributions according to the procedure suggested by Beneke et al. (1997b). In any case, the renormalon model cannot be applied at x so small that logarithms of x need to be resummed. One may naturally wonder whether there is an explanation for why the model seems to work in cases where it can be compared with measurements. Several hints are provided by the comparisons shown in Figs. 15}17. We recall that the model for twist-4 corrections is target-independent in the sense that all target-dependence enters trivially through the target dependence of the twist-2 distribution functions. In terms of moments M , Eq. (5.35) implies L MU/MU" "MU/MU" L L L L
(5.36)
exactly. Hence the model is useful only if the genuine twist-4 target dependence is small compared to the magnitude of the twist-4 correction itself. Fig. 16 shows that this is indeed the case for F of protons against deuterons, in particular in the region of large x. It is known that higher-twist corrections (as well as higher order perturbative corrections) are enhanced as xP1 (see for example Bodwin et al., 1989). This is in part an e!ect of kinematic restrictions near the exclusive region and the renormalon model reproduces such enhancements.
Note, however, that Meyer-Hermann and SchaK fer (1997) did not consider the correlation functions of physical currents and therefore the result is not applicable to a measurable deep inelastic scattering process. See Section 5.3.1 for a discussion of this point in the context of fragmentation. This is seen most easily in the dispersive approach discussed in Section 3.5.2, in which the radiated gluons acquire an invariant mass that modi"es the phase space boundaries.
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Fig. 15. Relative twist-4 contribution D (x) (called C(x) here) de"ned by Eq. (5.35) to the structure function F in the &renormalon model' compared with the data analysis of Virchaux and Milsztajn (1992). Plot taken from Dokshitzer et al. (1996).
Fig. 16. Relative twist-4 contribution D (x) (called C (x) here) de"ned by Eq. (5.35) to the proton (deuteron) structure N B function F in the &renormalon model' (dashed line) compared with proton ("lled circles) and deuteron (empty circles) data (Virchaux and Milsztajn, 1992). Plot taken from Maul et al. (1997). The solid curve shows the literal estimate of the renormalon ambiguity.
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Fig. 17. Twist-4 correction to xF as extracted from the (revised) CCFR data. The three plots show the e!ect of including leading order (LO), next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) QCD corrections in the twist-2 term. The data points are quoted from the analysis of Kataev et al. (1997). Overlaid is the shape obtained from the &renormalon model' for the 1/Q power correction.
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For the structure functions it is found that power corrections related to renormalons are of order
K L , Q(1!x)
(5.37)
at least those related to diagrams with a single gluon line (Beneke and Braun, 1995b). This provides some insight into the kinematic region in which the twist expansion breaks down. It also tells us that the increase of the twist-4 correction towards larger x seen in the model and the data in Figs. 15 and 16 may to a large extent be the correct parametrization of such a kinematic e!ect. Note that Eq. (5.37) can be understood as following from the fact that the hard scale in DIS is Q(1!x at large (but not too large) x. It is also possible that both the experimental parametrization of higher-twist corrections and the model provide e!ectively a parametrization of higher-order perturbative corrections to twist-2 coe$cient functions. As far as data are concerned, it should be kept in mind that it is obtained from subtracting from the measurement a twist-2 contribution obtained from a truncated perturbative expansion. As far as the renormalon model is concerned, it is best justi"ed by the &ultraviolet dominance hypothesis' (Beneke et al., 1997b) (see Section 5.1.3). Since UV contributions to twist-4 contributions can also be interpreted as contributions to twist-2 coe$cient functions, a &perturbative' interpretation of the model prediction suggests itself. Note that higher-order corrections in a (Q) vary more rapidly with Q than lower-order ones, and may not be easily distinguished from Q a 1/Q behaviour, if the Q-coverage of the data is not rather large. An interesting hint in this direction is provided by the analysis of CCFR data on F of Kataev et al. (1997), reproduced in Fig. 17. The "gure shows how the experimentally "tted twist-4 correction gradually disappears as NLO and NNLO perturbative corrections to the twist-2 coe$cient functions are included. At the same time, the renormalon model for the twist-4 corrections reproduces well the shape of data at leading order, and hence parametrizes successfully the e!ect of NLO and (approximate) NNLO corrections. This is an important piece of information, relevant to quantities for which an NNLO or even NLO analysis is not yet available. Note that whether the model is interpreted as a model for twist-4 corrections or higher order perturbative corrections is insigni"cant inasmuch as renormalons are precisely related to the fact that the two cannot be separated unambiguously. The model clearly cannot be expected to reproduce "ne structures of twist-4 corrections. Its appeal draws from the fact that it provides a simple way to incorporate some contributions beyond LO or NLO in perturbation theory, which may be the dominant source of discrepancy with data at accuracies presently achievable. 5.3. Hard QCD processes II In this section we summarize results on hard processes that do not admit an OPE. We do not follow the historical development and begin with fragmentation functions in e>e\ annihilation, The possibility to use renormalons for this purpose was "rst noted by Aglietti (1995). However, the result of this paper was not con"rmed by Beneke and Braun (1995b) and Dokshitzer et al. (1996). Compared to Kataev et al. (1997) we have rescaled the renormalon model prediction (solid curve) by a factor 1.5. As mentioned above we treat the overall normalization as an adjustable parameter.
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which provide a continuation of Section 5.2.4. We then turn to hadronic event shape observables in e>e\ annihilation and deep-inelastic scattering. These are the simplest observables with 1/Q power corrections and renormalon-inspired phenomenology has progressed furthest in this area. Soft gluons play an important role for 1/Q power corrections. The issue of soft gluon resummation near the boundary of partonic phase space and power corrections is taken up in Section 5.3.4, where the Drell}Yan process is studied from this perspective. Finally, in Section 5.3.5 we summarize work related to renormalons on other hard processes not covered so far. 5.3.1. Fragmentation in e>e\ annihilation Inclusive single particle production in e>e\ annihilation, e>e\PcH, ZPH(p)#X, is the time-like analogue of DIS. The double di!erential cross section can be expressed as 3 dp& 3 dp& dp& (e>e\PHX)" (1#cos h) 2 (x,Q)# sin h * (x,Q) 8 dx 4 dx dx dcos h 3 dp& # cos h (x,Q) . 4 dx
(5.38)
We de"ned the scaling variable x"2p ) q/q, where p is the momentum of H, and q the intermediate gauge boson momentum; Q"q denotes the centre-of-mass energy squared and h the angle between the hadron and the beam axis. In the following, we will not be concerned with the asymmetric contribution and with quark mass e!ects. Neglecting quark masses, (1/p ) dp& /dx 2 * (where p is the Born total annihilation cross section) is independent of electroweak couplings and the longitudinal cross section is suppressed by a . We drop the superscript &H' in the following and Q imply a sum over all hadron species H. At leading power in K/Q, the formalism that describes the fragmentation structure functions dp&/dx is analogous to that for DIS. The structure functions are convolutions of perturbative . coe$cient functions and process-independent parton fragmentation functions de"ned for example in the MS scheme. The formalism treats logarithmic scaling violations in Q. In addition, there exist power-like scaling violations (&power corrections') due to multi-parton correlations (Balitsky and Braun, 1991). However, contrary to DIS, the moments of these multi-parton correlations are not related to matrix elements of local operators and the OPE cannot be applied to fragmentation. This provides the motivation for the renormalon analysis. In the standard leading order analysis of diagrams with a single chain of vacuum polarizations (formally, the &large-b ' approximation) there are two contributions to the fragmentation process, shown in Fig. 18. We refer to the left diagram as &primary quark fragmentation' and to the right diagram as &secondary quark fragmentation', because in the "rst case the fragmenting quark is connected to the primary hard interaction vertex, while in the second case the fragmenting quark arises from gluon splitting gPqq . The gluon contributions are pure counterterms, except at order a , and therefore are of no relevance to power corrections in the present approximation. The Q secondary quark contribution is not inclusive over the cut quark bubble, because it is one of those quarks that fragments into the registered hadron H. As a consequence, when one uses the dispersive method described in Section 3.5.2 to compute the diagrams, the calculation is not the
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Fig. 18. Primary (left) and secondary (right) quark fragmentation diagrams (in cut diagram representation) in the large-b approximation or the approximation of single gluon emission. Note that the "gure to the right appears to have two chains of fermion loops, but should nonetheless be interpreted as a single chain diagram.
same as a one-loop calculation with "nite gluon mass. (They do coincide for the primary quark contribution.) Renormalons in fragmentation were considered in Dasgupta and Webber (1997) and Beneke et al. (1997b) for longitudinal and transverse components separately. In the "rst paper a simpli"ed prescription was adopted in which all contributions were calculated with a "nite gluon mass. In the second paper the diagrams of Fig. 18 were evaluated exactly. While the "nite gluon mass prescription is certainly unsatisfactory, because it does not account for gluon splitting, it is not clear whether the exact evaluation is more realistic, because it accounts only for gPqq , but not for gPgg, which is more important. The problem is connected with the fact that one computes fermion loops, but usually argues that they trace contributions that should naturally be written in terms of the full QCD b-function coe$cient b . This argument is di$cult to justify for a non inclusive process such as secondary quark fragmentation, because restoring the full b does not allow us to extrapolate from gPqq to gPgg. The conclusion is that the renormalon model for power corrections is more ambiguous, as far as the x-dependence is concerned for non-inclusive processes. These ambiguities are discussed in detail in Beneke et al. (1997b). The result for the x-dependence of K/Q power corrections to the longitudinal fragmentation cross section dp /dx from Beneke et al. (1997b) is shown in Fig. 19. The function H(x) is de"ned as * * in Eqs. (5.34) and (5.35) except that the scale K in Eq. (5.35) is omitted, so that H is dimensionless, G * and F is replaced by dp /dx. The vertical scale in the "gure is arbitrary and the overall . * normalization should be adjusted to data on power corrections, once the LEP1 analysis becomes available. We note that the secondary quark contribution (which we will shortly interpret as
For deep-inelastic scattering this has to be taken into account, too, for singlet, as opposed to non-singlet, quantities. See Stein et al. (1998) for a calculation of singlet contributions to DIS.
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Fig. 19. Shape of K/Q power corrections to the longitudinal fragmentation cross section as a function of x. Primary quark fragmentation (dashed line), secondary quark fragmentation (dotted line) and their sum (solid line).
a gluon contribution) exceeds the primary quark contribution at x(0.1, while the latter dominates in the region where the registered hadron takes away a sizeable fraction of the available energy. This is as expected. We also observe that the higher-twist corrections become large for small and large energy fraction x. The twist expansion breaks down in these regions. For large x the situation is similar to DIS, but the behaviour at small x has no analogue in DIS and will be discussed more below. Because the primary quark contribution is less ambiguous than the secondary quark contribution, we consider the model more reliable in the large x region. However, for the longitudinal cross section it turns out that the small-x region is not very di!erent in the massive gluon model. The actual calculation requires the expansion of the distribution function that enters the dispersive representation (3.83) at small values of the dispersion variable m. For the secondary quark contribution to longitudinal fragmentation, one "nds
4 1 dpO Q C a * " $ Q 2 !6x#2x#6 ln x#m ln mAO Q (x)#O(m) . (5.39) ¹ (m,x), * * x 2p p dx The coe$cient AO Q (x) is the function that determines the shape of the 1/Q power correction and * enters Eq. (5.35). One then notes that
4 3 !6x#2x#6 ln x "2 [CE *P ](x) , (5.40) * E O x N D where CE (x)"4(1!x)/x is the gluon coe$cient function at order a and P the gluon-to-quark Q E O * splitting function. The asterisk denotes the convolution product. This suggests (Beneke et al., 1997b) that one can reinterpret the secondary quark contribution as a gluon contribution } to be folded with the gluon fragmentation function } by &deconvoluting' the gluon-to-quark splitting function. The power correction to the gluon contribution, AEO(x), is then de"ned through . [AEO*P ](x)"AO Q (x) . (5.41) . E O . 2
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The result can be compared with the result obtained from the "nite gluon mass calculation (Dasgupta and Webber, 1997). Analysing the various ambiguities in restoring the gluon contributions, Beneke et al. (1997b) suggested the following parametrization of twist-4 corrections:
1 GeV dz 2 1!z dpU * (x,Q)" c d(1!z)# D (x/z,k)#c D (x/z,k) , E * z E z O * Q z O dx V dpU 1 GeV dz 2 1 *>2(x,Q)" c ! #1# d (1!z) D (x/z,k) O dx z O *>2 Q [1!z] 2 V > 1!z # c #d D (x/z,k) , E *>2 z E
(5.42)
(5.43)
where D denotes the leading-twist fragmentation function for parton i to decay into any hadron, G &L#T' the sum of longitudinal and transverse fragmentation cross sections and the plus distribution is de"ned as usual. The power corrections are added to the leading-twist cross sections as dpU dpU dp . (x,Q)" . (x,Q)# . (x,Q) . dx dx dx
(5.44)
The constants c and d are to be "tted to data and depend on the order of perturbation theory and I factorization scale k adopted for the leading-twist prediction. The parametrization can be used only for x'K/Q, owing to strong singularities at small x. It is worth noting that the renormalon model predicts no 1/Q power corrections for the fragmentation functions at "nite x. This is at variance with fragmentation models implemented in Monte Carlo simulations, which lead to 1/Q power corrections (see e.g. Webber, 1994b), but consistent with Balitsky and Braun (1991). Owing to energy conservation, the parton fragmentation functions disappear from the second moments
dp& 1 dx x . , (5.45) p , . dx 2 & which can therefore be calculated in perturbation theory up to power corrections. (With this de"nition p #p coincides with the total cross section e>e\P hadrons.) The power expansion of 2 * the fragmentation cross section has strong soft-gluon singularities and the expansion parameter relevant at small x is K/(Qx). This can be related to the fact that in perturbation theory the hard scale relevant to gluon fragmentation is not Q, but the energy Qx of the fragmenting gluon. Dasgupta and Webber (1997) and Beneke et al. (1997b) noted that these strong singularities lead to a linear K/Q correction to the second moment. This can be seen from
K
/
K L K 1 & dx x Q 2 Qx
(5.46)
A K/Q correction to p was already reported in Webber (1994a). However, the calculation there, which takes into * account a gluon mass only in the phase space, is not complete.
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for any n, which also tells us that the correct 1/Q power correction is obtained only after resumming the power expansion at de"nite x to all orders. The strong singularities at small x occur only in the secondary quark (gluon) contribution. The result for the distribution function that enters Eq. (3.83) is
a 5n p (5.47) ¹ (m), *" Q 1! (m#2 , * p p 32 and, according to Section 3.5.2, the (m-term in the small-m expansion indicates a K/Q power correction. The total cross section in e>e\ annihilation into hadrons is given by the sum of the transverse and longitudinal cross section. In p #p all power corrections of order 1/Q cancel, * 2 compare Section 5.2.1. The sizeable linear power correction to the longitudinal (and transverse) cross section also leads to large perturbative corrections, comparable to those in other event shape observables. The perturbative corrections to p in the large-b approximation can be found in Beneke et al. (1997b). * Manohar and Wise (1995) noted that hadronic event shape observables can have any power correction if one chooses an arbitrarily IR sensitive but IR "nite weight on the phase space. The moments of fragmentation functions provide a simple example of a set of quantities that can have fractional power corrections (Beneke et al., 1997b). The leading power behaviour of
1 1 dp * 2 (5.48) dx xA 2 p dx is corrected by terms of order (K/Q)A, where c can be arbitrarily small and positive. This should be compared with the moments of DIS structure functions, which can be described by the OPE, and which receive only 1/Q power corrections for any moment as long as the moment integral exists. Nason and Webber (1997) also considered heavy quark fragmentation in e>e\ annihilation. Although secondary heavy quark fragmentation exists, it does not contribute to power corrections in K/Q at leading order, because the gluon that splits into the heavy quark pair must have an invariant mass larger than 4Me\ annihilation Hadronic event shape variables in e>e\ collisions can be used to measure the strong coupling, in particular as they are more sensitive to a than the total cross section. Event shape variables are Q If one evaluates the longitudinal cross section with a "nite gluon mass, the coe$cient of (m is 2n/3. We emphasize again that the "nite gluon mass calculation cannot be related to renormalons for quantities like p . *
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computed theoretically in terms of quark and gluon momenta and measured in terms of hadron momenta. Apart from a correction for detector e!ects, the comparison of theory and data requires a correction for hadronization e!ects. It is believed that hadronization corrections are power suppressed in K/Q (where Q is the centre-of-mass energy) and it is known experimentally for quite some time that these corrections are substantial (see, for instance, Barreiro, 1986 for an early review). Until recently, the traditional method to take them into account has been hadronization models, implemented in Monte Carlo programs that also simulate a parton shower. A hadronization correction that scales with energy as K/Q provides a good description of the data. In this section we review recent developments that relate hadronization corrections to power corrections indicated by renormalons in the perturbative prediction for the event shape variable. This connection was suggested by Manohar and Wise (1995) for a toy model and by Webber (1994a) for some QCD observables, although within a simpli"ed prescription that was re"ned later. These papers provided the "rst theoretical indications that hadronization corrections should scale (at least) as K/Q. Subsequent, more detailed analyses (Dokshitzer and Webber, 1995; Akhoury and Zakharov, 1995; Nason and Seymour, 1995) con"rmed this conclusion. Korchemsky and Sterman (1995a) also found K/Q power corrections, potentially enhanced by inverse powers of the jet resolution parameter, to the 2-jet distribution in e>e\ annihilation. Below we consider the following set of event shape variables: the observable &thrust' is de"ned as "p ) n" G G , (5.49) n "p " G G where the sum is over all hadrons (partons) in the event. The thrust axis n is the direction at which 2 the maximum is attained. An event is divided into two hemispheres H by a plane orthogonal to the thrust axis. The heavier (lighter) of the two hemisphere invariant masses is called the heavy (light) jet mass M (M ). The jet broadening variables are de"ned through & * "p ;n " 2 . (5.50) B " GZ&I G I 2 "p " G G In terms of these the total jet broadening is de"ned by B "B #B and the wide jet broadening 2 by B "max(B ,B ). Furthermore, from the eigenvalues of the tensor 5 (p?p@)/"p " G G G G (5.51) "p" G the C-parameter C"3(j j #j j #j j ) is de"ned. All these event shape observables are IR safe, i.e. insensitive to the emission of soft or collinear partons at the logarithmic level. As a consequence they have perturbative expansions without IR divergences. It is relatively easy to understand that event shape observables are linearly sensitive to small parton momenta and are hence expected to receive long-distance contributions of order K/Q. For illustration we consider the average value of 1!¹ in somewhat more detail. At leading order, this quantity has no virtual correction, and we require only the matrix element for cHPqq g. We have seen in several instances before, that in the context of leading-order renormalon calculations, the gluon acquires an invariant mass squared, which we denote by mQ. To make the connection with hadronization, it is natural to think of this invariant mass as of that of a virtual gluon at the end of ¹"max
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a parton cascade, before hadronization into a light hadron cluster with mass of order K sets in. For a con"guration where all momentum is taken by the qq pair and the virtual gluon is produced at rest, we have 1!¹"(m&K/Q, as compared to 1!¹"0 for the analogous con"guration with a zero-energy massless gluon. In a more physical language, the production of a light hadron at rest changes the value of 1!¹ by an amount linear in the hadron mass over Q. For the purpose of illustration we follow Webber (1994a) and compute the average 11!¹2 with a "nite gluon mass (mQ, emphasizing however (Nason and Seymour, 1995; Beneke and Braun, 1995b) that this is not equivalent to the computation of renormalon divergence, as the de"nition of thrust is not inclusive over gluon splitting gPqq (see also Sections 3.5.2 and 5.3.1 for a discussion of this point). The average of 1!¹ is given by
11!¹2" PS[p ]"M "(1!¹)[p ] . G OOE G
(5.52)
Introducing the energy fractions x "2p ) q/q, and reserving x for the gluon energy fraction, we G G have
x#x 2(x #x ) 1 ! #m "M ""8C N g $ A Q (1!x )(1!x ) OOE (1!x )(1!x ) (1!x ) 1 2m 2 ! # P8C N g . $ A Q (1!x )(1!x ) (1!x ) (1!x )(1!x )
(5.53)
For the leading correction of order (m, one may in fact set m"0 in the matrix element and x "x "1 in the non-singular terms, as done in the second line of the above expression. In terms of the energy fractions thrust is given by 2 max(x ,x ) , (5.54) ¹" 2!x #(x!4m where we anticipated that x is small in the region of interest. Note that the leading correction comes from x of order (m and hence m cannot be dropped in this expression. The thrust variable can also be de"ned with "p "PQ in the denominator of Eq. (5.49). Then ¹"max(x ,x ) instead G G of Eq. (5.54). The two de"nitions agree to all orders in perturbation theory, but di!er nonperturbatively by hadron mass e!ects. The phase space is
PS[p ]" dx dx h(x #x !(1!m))h G
1!x !m !x . 1!x
(5.55)
We then "nd (Beneke and Braun, 1995b) C a 11!¹2" $ Q (0.788!7.32(m#2). p
(5.56)
If we use the alternative de"nition of thrust mentioned above, the coe$cient 7.32 is replaced by 4. This value has been adopted in phenomenological studies initiated by Webber (1994a), Dokshitzer and Webber (1995) and Akhoury and Zakharov (1995). The di!erence constitutes an ambiguity due
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to the simpli"ed gluon mass prescription. One may wonder how (m enters the answer, because the phase space boundaries do not contain a square root of m. If we change one of the integration variables to x , we "nd that x '2(m and the linear power correction can be seen to arise from the fact that the integral over gluon energy fraction is dx and restricted as indicated. The pattern of gluon radiation leads to energy integrals dx /x . IR "niteness implies that the phase space weight, here 1!¹, is constructed so as to eliminate the logarithmic divergence as x P0. The generic situation with event shapes is a linear suppression of soft gluons. An important conclusion is that in the approximation considered so far the K/Q power correction arises neither from the emission of collinear but energetic partons nor from soft quarks, but only from soft gluons. This is consistent with the analysis of fragmentation in Section 5.3.1, where the leading 1/Q power correction to the longitudinal cross section was seen to originate only from soft gluon fragmentation. As a consequence we obtain the qualitative prediction 11!¹2 / 22"const;a (Q) [exp: 0.54$0.16] . (5.57) Q 11!¹2 / In the numerator the 2-jet region ¹+1 is excluded. Hence a hard gluon has to be emitted, which causes an additional suppression in a (Q). The number in brackets quoted from DELPHI collabQ oration (1997) shows some suppression, although not as large as expected. A slightly smaller number is obtained in Wicke (1998b). However, the constant that multiplies a has not been Q estimated theoretically, and details of the experimental "t procedure, for which the reader should consult DELPHI collaboration (1997), constitute an important source of uncertainty. Because in 1(1!¹)2 the soft gluon region is suppressed by two powers of x , one also expects the 1/Q power correction to this quantity to be suppressed by one power of a (Q). In particular, one obtains only Q a 1/Q power correction from the one gluon emission process discussed above. In both cases, however, this does not imply that the hadronization correction relative to the perturbative correction is small, because the perturbative coe$cients at order a are also reduced in Q and 1(1!¹)2 relative to 11!¹2. A recent analysis of experimental data at 11!¹2 / 22 various centre-of-mass energies (Wicke, 1998b) reports that the power correction to the second moment 1(1!¹)2 is consistent with a 1/Q behaviour. For the third moment a 1/Q behaviour is found, which is surprising, because for all 1(1!¹)L2 with n52 one expects a 1/Q behaviour. No matter how strong the suppression of the soft gluons, there should be a 1/Q power correction from hard collinear partons. Dokshitzer and Webber (1995) and Akhoury and Zakharov (1995) (DWAZ) (see also Korchemsky and Sterman, 1995b) suggested that the leading power correction to average event shape observables may be described by a single (&universal') parameter multiplied by an observabledependent, but calculable, coe$cient. For an event shape S, de"ned such that its average is of order a , we can write Q k K (k) a (k)#2# 1 #O(1/Q) , (5.58) 1S2"A a (k)# B !A b ln 1 1 Q Q 1 Q Q
see also the introductory discussion in Section 5.1.3. Dokshitzer and Webber (1995) parametrize the coe$cient of the power correction in the form
4C c k K K (k)" $ 1k a (k )!a (k)! !b ln # !2b a (k) , 1 ' ' Q k 2p Q p '
(5.59)
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where k is an IR subtraction scale (typically chosen to be 2 GeV), a (k ) is the non-perturbative ' ' parameter to be "tted and K"(67/18!p/6)C !5N /9. The remaining terms approximately D subtract the IR contributions contained in the perturbative coe$cients A and B up to second order. The universality assumption can be tested by "tting the value of a (k ) or, equivalently, K (k) to ' 1 di!erent event shape variables. Extensive analyses of the energy dependence of event shape variables and power corrections to them have been carried out by DELPHI collaboration (1997) and members of the (former) JADE collaboration (Movilla FernaH ndez, 1998a). In Fig. 20 we compare the energy dependence of 11!¹2 and 1M /Q2 with the prediction based on second order perturbation theory with and & without a 1/Q power correction. It is clearly seen that (a) the second-order perturbative result with scale k"Q is far too small and (b) the di!erence with the data points is "tted well by a 1/Q power correction. In addition to the two quantities reproduced here, the energy dependence of three jet fractions, the di!erence jet mass and the integrated energy-energy correlation can be found in DELPHI collaboration (1997). The jet broadening variables are analysed in Movilla FernaH ndez (1998a). In Table 9 we reproduce the "tted values of a for some of these variables. For the central values of a shown in the Table the coe$cients c in Eq. (5.59) are taken to be c "1, c & "1, 1 \2 + Q c 2 5"1 (Dokshitzer and Webber, 1995; Webber, 1995). The theoretical status of these coe$cients is somewhat controversial, as we discuss below. Nevertheless, the measurements indicate that the parameter for 1/Q power corrections is not too di!erent for the set of event shapes analysed so far. The jet broadening observables are special, because one expects an enhanced (ln Q)/Q power correction, which has not been taken into account in the experimental "ts. In absolute terms the power correction added to thrust and the heavy jet mass is about 1 GeV/Q. This is a sizeable correction of order 20% even at the scale M , because the perturbative 8 contribution is of order a (M )/p. The "t for a is sensitive to the choice of renormalization scale Q 8 k and in general to the treatment of higher order perturbative corrections. There is nothing wrong with this, because the very spirit of the renormalon approach is that perturbative corrections and non-perturbative hadronization corrections are to some extent inseparable. Hence we "nd it plausible that the 1/Q power correction accounts in part for large higher order perturbative corrections, which are large precisely because they receive large contributions from IR regions of parton momenta. It was noted in Beneke and Braun (1996) that choosing a small scale, k"0.13Q, reduces the second-order perturbative contribution and power correction signi"cantly for 11!¹2. In Fig. 20 (dashed curve) we have taken a very low scale, k"0.07Q, to illustrate the fact that the running of the coupling at this low scale can fake a 1/Q correction rather precisely (a straight line in the "gure). Campbell et al. (1998) performed an analysis of 11!¹2 in the e!ective-charge scheme. This scheme selects the scale k"0.08Q. Campbell et al. (1998) "t a , a third-order perturbative coe$cient and a 1/Q power correction simultaneously and Q "nd a reduced power correction of order (0.3$0.1) GeV/Q consistent with Beneke and Braun (1996).
In their second publication, Movilla FernaH ndez et al. (1998b) performed "ts to the jet broadening measures, taking into account the logarithmic enhancement. We refer the reader to this work, but do not quote their numbers in the table, since they adopt a normalization of the power correction di!erent from Eq. (5.59), following Dokshitzer et al. (1998b).
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Fig. 20. Energy dependence of 11!¹2 (upper) and the heavy mass 1M /Q2 (lower) plotted as function of 1/Q. Data & compilation from Movilla FernaH ndez (1998a), see references there. Dotted line: second order perturbation theory with scale k"Q. Solid line: second order perturbation theory with power correction added according to Eq. (5.59) and with k"Q, k "2 GeV. For a (2 GeV) the "t values 0.543 for thrust and 0.457 for the heavy jet mass from Movilla FernaH ndez ' (1998a) are taken. The dashed line shows second order perturbation theory at the very low scale 0.07Q with no power correction added. For both observables a (M ) has been "xed to 0.12. I thank O. Biebel for providing me with the data Q 8 points.
Table 9 Fits of a (M ) and the power correction parameter a (2 GeV) de"ned in Eq. (5.59) taken from DELPHI collaboration Q 8 (1997) and Movilla FernaH ndez (1998a). See there for details of the error breakdown. DELPHI does not include the LEP2 data points S
a (2 GeV)
a (M ) Q 8
11!¹2 [DELPHI] 11!¹2 [JADE] 1M /s2 [DELPHI] & 1M /s2 [JADE] & 1B 2 [JADE] 2 1B 2 [JADE] 5
0.534$0.012 0.543> \ 0.435$0.015 0.457> \ 0.342> \ 0.264> \
0.118$0.002 0.120> \ 0.114$0.002 0.112> \ 0.116> \ 0.111> \
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The DWAZ model relies on the assumption of universality of power corrections, i.e. the assumption that all non-perturbative e!ects can be parametrized by one number. Di!erent motivations for this assumption have been given in Dokshitzer and Webber (1995), in Akhoury and Zakharov (1995), and in Korchemsky and Sterman (1995b). The nature of this assumption has not been completely elucidated so far. In the formulation of the model of Dokshitzer and Webber (1995) the 1/Q power correction to 11!¹2 and M /Q are predicted to be equal, but the & power correction to the light jet mass M/Q is predicted to be suppressed by a factor of a (Q). * Q Akhoury and Zakharov (1995) argued that, in the two-jet limit, a universal hadronization correction is associated with each quark jet and hence the 1/Q power correction to 11!¹2 should be twice as large as that to M /Q, while the 1/Q power correction to M/Q should be as large as & * that to M /Q. The data reported above appears to favour near-equality for 11!¹2 and M /Q. & & On the other hand, the very small value of the 1/Q term for the di!erence mass M"M !M & * observed in DELPHI collaboration (1997) seems to favour the picture of Akhoury and Zakharov (1995). Nason and Seymour (1995) considered the e!ect of gluon splitting gPqq on power corrections to various event shape observables and argued that neither of the two answers is correct and that universality in the sense of the DWAZ model is unlikely to hold. They observe that thrust and the heavy jet mass are related by 1!¹"M /Q, if, in the two-jet limit, a soft gluon splits into two & collinear quarks, both of which go into the same hemisphere; however, the relation is 1!¹"2M /Q if the quarks are emitted from the gluon back-to-back. As a consequence 1!¹ & and M /Q provide di!erent weights on the four-parton phase space and the coe$cients of their & linearly IR sensitive contributions are not related in a simple way. Beneke and Braun (1995b) arrived at a similar conclusion, noting that event shapes resolve large angle soft gluon emission at the level of 1/Q power corrections. If collinearity of the emission process is not required, the association of the power correction to a particular jet is di$cult to maintain. The situation can be clari"ed either by "nding an explicit operator parametrization of the 1/Q IR sensitive contribution valid to all orders in perturbation theory, or by explicit next-to-leading order calculations that take into account the emission of two gluons. The "rst approach was taken by Korchemsky et al. (1997), extending earlier work on jet distributions (Korchemsky and Sterman, 1995a) to averaged event shapes. Let us de"ne the operator
dy "y "y( H (yI) , (5.60) P(y( )" lim G G (2p) y with H the energy momentum tensor and y( a unit vector, as the measure of momentum (energy) of IJ soft partons (hadrons) deposited at asymptotic distances (for instance, in the calorimeter of the detector) in the direction of y( . Close to the two-jet limit, the soft partons are emitted from a pair of almost back-to-back quarks. For event shape weights that have (at least) a linear suppression of Recently, Dokshitzer et al. (1998b) introduced a distinction of &single-jet' and &whole-event' properties, which revises the original formulation of Dokshitzer and Webber (1995) towards the formulation of Akhoury and Zakharov (1995) as far as thrust and jet masses are concerned. This distinction has to be carefully taken note of when one compares for example the "ts of a (k ) in Movilla FernaH ndez (1998a) with those in Movilla FernaH ndez et al. (1998b). '
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soft particles the standard eikonal approximation can be used for the fast quark propagators and the quark propagation can be described by a product of Wilson line operators = with v and TT v light-like vectors pointing in the direction of the outgoing fast quarks. Squaring the matrix elements, the energy #ow of soft radiation from the qq system is described by the distribution E(y( )"10"=R P(y( )= "02 . TT TT In terms of these quantities, Korchemsky et al. (1997) "nd
1 dX(y( ) f (X(y( )) E(y( ) , 1S2 " / Q 2n 1
(5.61)
(5.62)
where the integral extends over the full solid angle. The integral has a transparent interpretation as an observable-dependent (and calculable) weight of the non-perturbative energy #ow distribution E(y( ). There are corrections to this result from multi-jet con"gurations. These corrections are suppressed by factors of a (Q). Note that Eq. (5.62) embodies universality in terms of a universal Q distribution function E(y( ). But since every event shape takes a di!erent integral of E(y( ), their 1/Q corrections are not related through the same non-perturbative parameter. The DWAZ model can be recovered, when E(y( ) is approximated by a constant. Operators similar to Eq. (5.60) were also introduced by Sveshnikov and Tkachov (1996) and Cherzor and Sveshnikov (1997). They stress that event shape variables in general are most naturally de"ned in terms of calorimetric energymomentum #ow (rather than the energy-momentum of particles) and note that such a de"nition would lend itself more easily to an analysis of power corrections. The second approach was followed by Dokshitzer et al. (1998a,b) who presented a detailed analysis of IR-sensitive contributions to the matrix elements for the emission of two partons. For event shape observables with a linear suppression of soft partons, the matrix elements can be evaluated in the soft approximation. Dokshitzer et al. (1998b) "nd that for 11!¹2, the jet masses, and the C-parameter the coe$cient of the 1/Q power correction that is obtained for one gluon emission is rescaled by the same factor 1.8. This implies that these observables take the same section of the distribution function (5.61) to leading and next-to-leading order. This conclusion follows from the fact that Dokshitzer et al. (1998b) assume that the nearly back-to-back quark jets acquire an invariant mass that is large compared to K (but small compared to Q) as a consequence of perturbative soft gluon radiation. In this case a soft gluon with energy of order K, which is of interest for power corrections, cannot determine which hemisphere becomes heavy and which becomes light. It is important that the correction factor 1.8 has no parametric suppression, because the coupling constant in diagrams with soft gluon emission with momenta of order K should be considered of order 1. In renormalon terminology this is related to the fact that the overall normalization of renormalon divergence receives contributions from arbitrarily complicated diagrams. As a consequence one can expect further unsuppressed rescalings, not necessarily equal for the event shapes mentioned above, in still higher orders. Dokshitzer et al. (1998a,b) argue that there are no corrections to the rescaling factor 1.8 from the emission of three and more partons. This is due to the fact that they parametrize the non-perturbative parameter for 1/Q power corrections as an integral of an e!ective coupling a (Dokshitzer et al., 1996). In this language more complicated diagrams would necessitate the introduction of integrals of aL and hence, new non-perturbative parameters. Since from general considerations these parameters cannot be expected to be small,
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these parameters presumably violate the simple universality hypothesis in terms of a single non-perturbative parameter. One can also consider power corrections to event shape distributions, rather than averaged event shapes (Korchemsky and Sterman, 1995b; Dokshitzer and Webber, 1997). Recall that at leading order in a the thrust distribution is Q dp/d¹"d(1!¹) . (5.63) It is not di$cult to see that in the approximation of one-gluon emission discussed earlier, the 1/Q power correction to the Nth moment of thrust is given by 1¹,2 "!N11!¹2 ,N(a K/Q) (a '0) , / / 2 2 which implies dp a K "d(1!¹)# 2 d (1!¹)#2 . d¹ Q
(5.64)
(5.65)
It is suggestive but not rigorous to interpret the correction as the "rst term in the expansion of
a K d 1! ¹! 2 Q
,
(5.66)
so that the main e!ect results in a non-perturbative shift of the thrust value. Qualitatively, such an e!ect is expected on purely kinematic grounds from hadron mass e!ects. In writing Eq. (5.66) we have to assume that the power correction of order NK/Q exponentiates exactly in moment space (Korchemsky and Sterman, 1995b; Dokshitzer and Webber, 1997). Whether exponentiation occurs in this sense has not yet been established. In a more general framework one would introduce a non-perturbative distribution function that resums the power corrections of order (NK/Q)I and write the thrust distribution as a convolution of its perturbative distribution with this distribution function. This is analogous to the introduction of shape functions in the heavy quark e!ective theory to describe the endpoint regions of certain energy spectra (Neubert, 1994b; Bigi et al., 1994a). The kth moment of this distribution function is related to the coe$cient of (NK/Q)I, which need not, however, be related to a . Such a distribution function would not be universal, i.e. it 2 would be di!erent for di!erent event shapes. Dokshitzer and Webber (1997) assume a distribution function of the form (5.66) and arrive at dp/d¹"F
(¹!d¹) , (5.67) where F (¹) denotes the perturbative thrust distribution and d¹ a non-perturbative shift of order K/Q. They "nd that the data on thrust distributions at various energies are well described by the ansatz (5.67) down to rather small values of 1!¹ (see also Wicke, 1998a). The C-parameter distribution has also been successfully "tted with this parametrization (Catani and Webber, 1998). A non-perturbative distribution function in analogy with heavy quark decays as described above has been introduced by Korchemsky (1998), to which we refer for more details on the factorization of perturbative contributions and the evolution equations for the moments of the distribution function. Just like average event shapes, the distribution function can also be expressed in terms of the universal distribution E(y( ). But again a complicated weight is taken, which forbids a straightforward relation of di!erent event shape variables. Korchemsky (1998) uses a simple three-parameter
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ansatz for the distribution function and obtains excellent agreement between the predicted and measured thrust distributions at all centre-of-mass energies between 14 and 162 GeV. 5.3.3. Event shape observables in deep inelastic scattering Event shape variables can also be measured in DIS. Compared to e>e\ annihilation, DIS o!ers the advantage that an entire range of Q can be covered in a single experiment. Event shape variables in DIS are usually de"ned in the Breit frame, where the gauge boson momentum that induces the hard scattering process is purely space-like: q"(0,0,0,Q). In leading order the target remnant moves into the direction opposite to q and the struck parton moves into the direction of the virtual gauge boson. This direction de"nes the ¤t hemisphere', which is in many ways similar to one hemisphere in e>e\ collisions. DIS event shape variables are then de"ned in close analogy to those for e>e\ annihilation, but with the sum over hadrons (partons) restricted to the current hemisphere. As for event shape variables in e>e\ annihilation, K/Q power corrections are expected for their DIS analogues. Dasgupta and Webber (1998) computed the coe$cient using the "nite gluon mass prescription for the one gluon emission diagrams. The predicted event shape average is then represented in the form (5.58) and (5.59). The H1 collaboration (1997) compared the prediction to their data over a range of momentum transfers Q from 7 to 100 GeV. Their "t to the energy dependence using the parametrization (5.58) is shown in Fig. 21 and the corresponding values of a (2 GeV) are reproduced in Table 10. It is remarkable that a (2 GeV), the parameter for the 1/Q power correction, comes out nearly identical for the four event shapes shown in Fig. 21, and, moreover, that its value is the same within errors as for event shapes in e>e\ annihilation. This supports the idea that hadronization of the current jet in DIS is similar to hadronization in one hemisphere in e>e\ annihilation. From a theoretical point of view the universality between DIS and e>e\ annihilation is not obvious, because the factorization of the remnant and the current jet cannot be expected beyond leading power in K/Q, since soft gluons can connect the two. It is important to note that the data teach theorists an interesting fact, but that the numerical agreement for a (2 GeV) cannot be considered as signi"cant to the accuracy at which it appears. For tests of universality it would be more useful to "t all event shapes with a common value for a (M ). The fact that the "tted a (M ) is di!erent for the observables in Table 10 introduces Q 8 Q 8 a systematic uncertainty in a (2 GeV). In addition, the parameter that enters the prediction is c a (2 GeV). The value for a (2 GeV) follows once c is computed in a particular prescription. The 1 1 ambiguities in theoretical calculations of c are large and the fact that the gluon mass prescription 1 gives consistent results may also be an interesting coincidence. Dasgupta et al. (1998) have extended their calculation for event shapes in DIS to fragmentation processes in DIS. Data on the energy fraction dependence of power corrections to fragmentation functions would be highly interesting, as the same e!ects as discussed in Section 5.3.1 for fragmentation in e>e\ collisions are expected to occur in DIS. At the same time, an entire range in Q can be scanned in ep collisions. So far the theoretical calculation has been done only for quark-initiated DIS. At energies of the HERA collider one expects a large contribution from gluon-initiated DIS. In the leading-order renormalon model the gluon contribution can be reconstructed from quark-singlet contributions by the deconvolution method Beneke et al. (1997b) discussed in Section 5.3.1.
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Fig. 21. Energy dependence of 11!¹ 2, 11!¹ 2/2, the current jet broadening 1B 2 and the current jet hemisphere A X A invariant mass 1o 2 in DIS compared to NLO perturbation theory with and without 1/Q power correction. Figure taken A from H1 collaboration (1997).
Table 10 Fits of a (M ) and the power correction parameter a (2 GeV) (de"ned in Eq. (5.59)) to DIS event shape variables taken Q 8 from H1 collabora (1997). See there for de"nitions of the quantities listed. The error is almost entirely theoretical S
a (2 GeV)
a (M ) Q 8
11!¹ 2 A 11!¹ 2/2 X 1o 2 A 1B 2 A
0.50> \ 0.51> \ 0.52> \ 0.41> \
0.123> \ 0.115> \ 0.130> \ 0.119> \
5.3.4. Drell}Yan production and soft gluon resummation We have considered power corrections to hadronic "nal states in e>e\ annihilation and to DIS. Renormalon divergence also appears in the hard scattering coe$cients in hadron}hadron collisions. The simplest hadron}hadron hard scattering process is Drell}Yan production of a lepton
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pair or a massive vector boson, A#BP+cH,=,Z,(Q)#X, where X is any hadronic "nal state. At leading power dp/dQ"p =(q,Q), where p is the Born cross section, q"Q/s, and dx dx G H f (x ,Q) f (x ,Q) u (z,a (Q)) , (5.68) =(q,Q)" H H GH Q x x G G G H G H with z"Q/(x x s) and s is the centre-of-mass energy squared of A and B. In the following we are concerned with renormalon divergence and long-distance contributions to the hard scattering factor u (z,a (Q)). It is convenient to work in moment space, in which GH Q =(N,Q), dq q,\ =(q,Q)"f (N,Q) f (N,Q) u (N,a (Q)) , (5.69) Q OO O O where the right-hand side is expressed in terms of moments of the parton distributions (hard scattering factor) with respect to x (z). G When Q is large, one can consider large moments 1;N;Q/K. Conventional, "xed-order perturbation theory fails for high moments, because one encounters corrections aL lnKN with m up Q to 2n. The physical origin of these corrections is that there exist three scales Q, Q/(N and Q/N and the logarithms are ratios of these scales. These scales appear because, for large N, the moment integral is dominated by Q&s, which leaves little phase space for the hadronic system X. In a perturbative calculation, the energy available for real emission is constrained to be of order Q/N and the IR cancellation between virtual and real correction becomes numerically ine!ective. The logarithmically enhanced contributions can be resummed systematically to all orders in perturbation theory (Sterman, 1987; Catani and Trentadue, 1989). The result has the exponentiated form
(5.70) u (N,a (Q))"H(a (Q)) exp[E(N,a (Q))]#R(N,a (Q)) , Q Q Q Q OO where R(N,a (Q)) vanishes as NPR, H(a (Q)) is independent of N, and the exponent is given by Q Q E(N,a (Q)) Q z,\!1 /\Xdk R A(a (k ))#B(a ((1!zQ))#C(a ((1!z)Q)) . " dz 2 (5.71) Q R Q Q k 1!z / \X R The function A is related to soft-collinear radiation and also referred to as &cusp' or &eikonal' anomalous dimension. The function B relates to the DIS process, which enters when the parton densities are factorized. The function C, not needed for the resummation of next-to-leading logarithms, relates to the Drell}Yan process (see Sterman (1987) for details). The arguments of the coupling constants re#ect the physical scale relevant to the respective subprocess. Renormalon divergence is also related to soft gluons and one may ask what the precise relation to soft gluon resummation is. This question has guided the work on renormalons in Drell}Yan production. Note that the integrals in Eq. (5.71) are formal, because they include integration over the Landau pole of the coupling. It was already noted in Collins et al. (1989) that this implies
In the remainder of this section we restrict attention to the qq annihilation subprocess.
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sensitivity to the large-order behaviour in perturbation theory. Contopanagos and Sterman (1994) performed the "rst quantitative analysis and found that the ambiguity due to the Landau poles in Eq. (5.71) in conventional leading or next-to-leading-order resummations scales as K/Q. Leading order resummations of logarithms of N need only keep the "rst-order term in a of A(a )" Q Q a a #2 . At this order B and C can be set to zero. One then "nds for the Borel transform Q (de"ned by Eq. (2.5) and using u"!b t as usual) of the exponent 4(N!1) B[E ](N,u) S " a . (5.72) ** 1!2u The pole at u"1/2 leads to an ambiguity of order K/Q in de"ning the exponent at leadinglogarithmic accuracy, which was noted by Contopanagos and Sterman (1994). The question arises of whether this ambiguity indicates a power correction of order K/Q to the hard scattering factor of the Drell}Yan cross section or whether the ambiguity appears as the consequence of a particular implementation of soft gluon resummation that was not designed to be accurate beyond leading power. This question has been studied by Beneke and Braun (1995b) at the level of one gluon virtual and real corrections with vacuum polarization insertions and accounting for gluon splitting into a qq pair. Even in this approximation the functions A, B and C that enter the exponent become in"nite series. The large-order terms in these series account for highly subleading logarithms in N and are not needed for the resummation of such logarithms to a given accuracy. On the other hand, the Borel transform of the exponent becomes B[E](N,u) S "
1 4(N!1) B[A](1/2)! B[C](1/2) , 4 1!2u
(5.73)
and the residue of the pole at u"1/2 involves the series expansion of A and C to all orders. Beneke and Braun (1995b) found that, when all orders are taken into account, the expression in square brackets is zero, and the pole is cancelled. After this cancellation the leading power correction to Drell}Yan production turns out to be of order NK/Q, at least in the approximation mentioned above. Note that the function B, related to the DIS process, does not appear in Eq. (5.73). This is due to the argument of the coupling, which is larger, (1!zQ, in this case. In general, the terms introduced by performing collinear factorization in the DIS scheme are found not to be relevant to the discussion of potential K/Q corrections. This is expected, because higher-twist corrections scale only as K/Q in DIS. The physical origin of the cancellation becomes more transparent in terms of the sensitivity of the one-gluon emission amplitude to an IR cut-o!. To this end we choose a cut-o! k and require the energy and transverse momentum of the emitted gluon to be larger than k. We are interested in terms of order k in the cut-o!. To this accuracy the one-gluon emission contribution in moment space can be written as
/\X dk 1 C a \I / R . (5.74) dz z,\ = (N,k)"2 $ Q k p ((1!z)!4k /Q I R R The expansion at small k of this integral starts with logarithms of k. They would be cancelled by adding the virtual correction and collinear subtractions, both of which can be seen not to be able to
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introduce a linear dependence on k. Expanding the square root in k /Q, one "nds the following R expression for the term of order k/Q in the expansion at small k: C a k (!1)I C(1/2) k = (N,k) U 4 $ Q (N!1) "0 ) . (5.75) k! C(3/2!k) p Q Q I Hence there is in fact no linear sensitivity to an IR cut-o!. One needs all terms in the expansion of the square root to obtain this cancellation. This means that to linear power accuracy the collinear approximation k ;k &Q(1!z)/2, where k is the transverse momentum and k the energy of the R R emitted gluon, is not valid. It is essential to consider also large angle, soft gluon emission with k &k . This conclusion (Beneke and Braun, 1995b) is general and extends beyond the Drell}Yan R process. For the resummation of leading (next-to-leading, etc.) logarithms of N an expansion in k /k is R justi"ed. The leading logarithms are obtained by neglecting k under the square root of Eq. (5.74). R This leads to the "rst term only in the sum of Eq. (5.75) and a non-vanishing coe$cient of k/Q in agreement with the pole at u"1/2 in Eq. (5.72) obtained in the same approximation. The fact that the exact phase space for soft gluon emission is required to determine the coe$cient of power corrections correctly relates to the fact that all terms in the expansion of the functions A and C in the exponent have to be kept for this purpose. In particular the function C, not related to the eikonal anomalous dimension, is needed and this rules out the possibility discussed in Akhoury and Zakharov (1995) that the universal parameter for 1/Q power corrections is given by the integral over the eikonal anomalous dimension A(a (k )). Another implication is that the Q R angular ordering prescription, according to which the emission angles of subsequent emissions in a parton cascade decrease, and which generates the correct matrix elements to next-to-logarithmic accuracy in N (see for example Catani et al., 1991), cannot be applied to power corrections. The intuitive argument that partons emitted at large angles can resolve only the total colour charge of the previous branching process does not hold true beyond leading power. This argument also resolves a paradox raised by Korchemsky and Sterman (1995b), who noted that 1/Q power corrections at large N and to 1!¹ close to ¹"1 should be related, because the corresponding resummation formulae for logarithmically enhanced terms in perturbation theory are related. At present such a relation is known only to next-to-leading logarithmic accuracy (Catani et al., 1993). The fact that all orders in the exponent are needed for power corrections explains that it is consistent to expect K/Q power corrections to thrust but not to the Drell}Yan process. Is it possible to organize the resummation of leading, next-to-leading, etc., logarithms in N without introducing undesired, because spurious, power corrections of order K/Q? (Catani and Trentadue, 1989) noted that one may substitute
e\A# !z z,\!1P!H 1! N
(5.76)
in Eq. (5.73) to next-to-leading logarithmic accuracy. Then, for N;Q/K, which one must require for a short-distance treatment, the integration in Eq. (5.73) does not reach the Landau pole and Recall that the expansion parameter for power corrections is NK/Q. For N&Q/K the Drell}Yan process ceases to be a short-distance process, and factorization breaks down.
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there are no power corrections to the exponent, unless the series expansions for A, B and C are themselves divergent. Beneke and Braun (1995b) addressed the above question in the fermion bubble approximation, which provides a useful toy model, because the functions A, B and C are in"nite series expansions in a . Ignoring complications from collinear subtractions, the partonic Drell}Yan cross section Q factorizes into p( (N,Q)"H(Q,k) S(Q/N,k) up to corrections that vanish as NPR, where "7 H depends only on the &hard' scale Q and S on the &soft' scale Q/N. Following Korchemsky and Marchesini (1993), the soft part is expressed as the Wilson line expectation value
S(Q/N,k,a )" Q
Q dy e W/\X 10"¹M ;R (y)¹ ; (0)"02 , dz z,\ "7 "7 2 2p \
(5.77)
where
k
ds pI A (p s#x) P exp !ig ds pI A (p s#x) , (5.78) I I Q \ \ and p denote the momenta of the annihilating quark and anti-quark. The &soft part' S satis"es a renormalization group equation in k, which can be used to sum logarithms in N, because S depends only on the single dimensionless ratio Q/(Nk). The solution to the RGE equation ; (x)"P exp ig Q "7
R R kN #b(a ) ln S(Q/N,k,a (k))"C (a ) ln #C (a ) Q Ra Q Q "7 Q Rk Q Q
(5.79)
reads
kN / dk R C (a (k )) ln R #C (a (k )) , (5.80) Q R "7 Q R Q k / , R where S(a (Q/N)) denotes the initial condition for the evolution and in the end we have set k"Q. Q From the analysis in the fermion loop approximation, one can draw the following, more general, conclusions. The anomalous dimensions C (a ) and C (a ) have convergent series expansions when de"ned Q "7 Q in the MS scheme. Since the integrations in the exponent of Eq. (5.80) exclude the Landau pole for all moments N in the short-distance regime, it follows that the resummation, embodied by the exponent, can be carried out without ever encountering the divergent series and power corrections implied by them. The conclusion is then that the renormalon problem is a problem separate from soft gluon resummation. Renormalons and power corrections enter in the hard part H and the initial condition S. Because S depends only on Q/N, the parameter for power corrections to S is NK/Q. One "nds that all power corrections of order (NK/Q)I to the Drell}Yan cross section are correctly reproduced in the soft part. In the approximation considered in Beneke and Braun (1995b), terms with k"1 do not exist. Note that if the exponentiated cross section is written in the &standard form' (5.70) and (5.71), the initial condition S(a (Q/N)) is absorbed into the exponent at Q the expense of a rede"nition of C (C ). With this rede"nition the functions in the exponent are "7 divergent series. As always, there is the question of whether the absence of renormalon divergence that would correspond to a K/Q power correction is speci"c to the (essentially abelian) approximation of p( "H(a (Q)) ) S(a (Q/N)) exp "7 Q Q
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Beneke and Braun (1995b) and persists to more complicated diagrams. The answer to this question is still open. Akhoury and Zakharov (1996) and Akhoury et al. (1998,1997) put the cancellation of 1/Q corrections to Drell}Yan production in the more general context of Kinoshita}Lee}Nauenberg (KLN) cancellations. Knowing that any potential 1/Q correction would come from soft particles, but not collinear particles, they consider KLN transition amplitudes, which include a sum over soft initial and "nal particles degenerate with the annihilating qq pair. The KLN transition amplitudes have no 1/k (where k stands for the energies of the soft particles) contributions (collinear factorization is implicitly assumed). As a consequence, the amplitude squared, integrated unweighted over all phase space, is proportional to dk k , which by power counting implies at most 1/Q power corrections. To make connection with a physical process, one has to demonstrate that the sum over degenerate initial states can actually be dispensed of. The authors above use the Low theorem to show this for Drell}Yan production in an abelian theory. For QCD this still remains an open problem. Korchemsky (1996) argued that non-abelian diagrams (involving the three-gluon vertex) at two-loop order would give a non-vanishing contribution to a certain Wilson line operator introduced in Korchemsky and Sterman (1995a) to parametrize 1/Q corrections to Drell}Yan production. It would be very interesting to carry out the two-loop calculation to see whether a non-zero linear infrared contribution is actually present in these diagrams. Qiu and Sterman (1991) extended collinear factorization for Drell}Yan production to 1/Q corrections and showed that the same twist-4 multi-parton correlations enter as in DIS. The factorization is carried out at tree-level and hence may not be conclusive on the issue of a 1/Q power correction, which would require a demonstration that soft gluon interactions cancel to all orders in perturbation theory to the level of 1/Q accuracy. This is, at present, the missing element in a proof that there are no 1/Q long-distance sensitive regions in the Drell}Yan process to all orders in perturbation theory. Korchemsky and Sterman (1995a) have also considered power corrections to the transverse momentum (impact parameter) distributions in Drell}Yan production. In impact parameter space, they "nd that ambiguities in de"ning the perturbative contribution to the exponent require power-suppressed contributions of the form (bK)(a ln Q#b)
(5.81)
with b the impact parameter. The leading correction is quadratic in K and consistent with the parametrization of long-distance contributions suggested by Collins and Soper (1981). 5.3.5. Other hard reactions Renormalon divergence and the corresponding power corrections have been investigated for several other hard QCD processes: Hard-exclusive processes. Mikhailov (1998) and Gosdzinsky and Kivel (1998) considered the Brodsky}Lepage kernel that determines the evolution of hadron distribution amplitudes in the large-b approximation. In the MS de"nition, the series expansion of the kernel is convergent as expected for anomalous dimensions. The form factor for the process cH#cPp was analysed in detail by Gosdzinsky and Kivel (1998). One "nds two sources of renormalon divergence and power
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corrections. The "rst is power corrections in the hard coe$cient function, which are present independently of the form of the hadron wave function. These correspond to higher-twist corrections in the hard scattering formalism. Additional power corrections are generated after integrating with the hadron wave function over the parton momentum fractions and these depend on the details of the wave function. These power corrections arise from the region of small parton momentum fraction and can be associated with power corrections due to the &soft' or &Feynman' mechanism for exclusive scattering. For the form factor of the above process, both power corrections are of order 1/Q or smaller. Belitsky and SchaK fer (1998) considered deeply virtual Compton scattering cH#APc#B. For this process and the cHcp form factor there exist only two IR renormalon poles at u"1,2 in the hard coe$cient functions. This is analogous to the GLS sum rule (5.29) and indeed the same diagrams are considered here and there, except for di!erent kinematics. Small-x DIS. Renormalons in the context of small-x structure functions were discussed by Levin (1995) and Anderson et al. (1996). To be precise, renormalons are understood there as a certain prescription to implement the running coupling in the BFKL equation. There appears to be a 1/Q correction to the kernel, but in Anderson et al. (1996) it is argued that this correction is suppressed after convolution with the hadron wave function such that the correction to the structure function is only of order 1/Q. The next-to-leading order BFKL kernel has now been calculated (Fadin and Lipatov, 1998; Ciafaloni and Camici, 1998). Kovchegov and Mueller (1998) separate a &conformally invariant' part from a &running coupling' part and investigate the series expansion of the solution to the BFKL equation when the exact one-loop running coupling is kept in the running coupling part. Ignoring overall factors, the result is a series expansion of the form (aya)L C(n/2) , (5.82) Q L where a"42f(3)b/p and y is the (large) rapidity that characterizes a scattering process in the BFKL limit. If we take the Borel transform with respect to a, the above series leads to a typical Q renormalon pole. The unusual feature is that the location of the renormalon pole depends on the kinematic variable y, and not only in overall prefactor. When aya&1 the series diverges from the Q outset and no perturbative approximation is possible. This leads to the interesting constraint y(1/(aa) for rapidities to which the BFKL treatment can be applied. The same constraint has Q been found independently by a di!erent method (Mueller, 1997). The inclusive cHcH cross section into hadrons was analysed by Hautmann (1998). In this case one "nds a 1/Q power correction. 5.4. Heavy quarks In this section we consider hard processes for which the large scale is given by the mass of a heavy quark. We "rst deal with the notion of the (pole) mass of a heavy quark itself and its relation to the heavy quark potential. We then discuss renormalons in heavy quark e!ective theory, their implications for exclusive and inclusive semi-leptonic B decays, and close with brief remarks on renormalons in non-relativistic QCD.
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5.4.1. The pole mass The pole mass of a quark is de"ned, to any given order in perturbation theory, as the location of the pole in the quark propagator. It is IR "nite, gauge independent and independent of renormalization conventions (Tarrach, 1981; Kronfeld, 1998). Quarks are con"ned in QCD and quark masses are not directly measurable. The binding energy of quarks in hadrons is of order K and it is natural to expect that the notion of a quark pole mass cannot be made more precise. Nevertheless, for heavy quarks with mass m
cI(p. #k. #m)c dBk I a (k e\ ) m !m (k)"(!i)C gkC $ Q Q k((p!k)!m) (2p)B +1
. (5.83) NK For p"m the integral scales as dk/k for small k. This implies that the series expansion obtained from Eq. (5.83) leads to an IR renormalon singularity at t"!1/(2b ) (u"1/2) with a correspond ing ambiguity of order K. The integral (5.83) can be done exactly or the leading divergent behaviour can be extracted from the expansion of the integrand at small k as in Section 2.2. The asymptotic behaviour of the series expansion in a "a (k) is Q Q C e m !m (k)" $ k (!2b )Ln!aL> . (5.84) Q p +1 L The linear IR sensitivity of the pole mass has a transparent interpretation in terms of the static quark potential, discussed in the present context in Bigi and Uraltsev (1994), Bigi et al. (1994b) and Beneke (1998). An in"nitely heavy quark interacts with gluons through the colour Coulomb potential
The MS mass is related to the bare mass by subtraction of pure ultraviolet poles in dimensional regularization and contains no IR sensitivity at all. This statement will be made more precise in the following section.
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just as the pole mass of a stable quark. The large width C L Q +1 +1 4p +1 +1 L C a (5.85) "m (m ) $ Q [d (!b )L#d ]aL> L L Q +1 +1 4p L in the large-b approximation (Beneke and Braun, 1995a; Neubert, 1995b; Ball et al., 1995a; Philippides and Sirlin, 1995) and beyond it (Beneke, 1995). The Borel transform of the mass shift, B[dm/m](u)" d uL/n!, in the large-b limit is not just given by the Borel transform of L L Eq. (5.83), but has to take into account the correct UV subtractions in the MS scheme, see Section 3.4. The complete result is (Beneke and Braun, 1994; Ball et al., 1995a)
B[dm/m](u)"
m \S C(u)C(1!2u) GI (u) eS 6(1!u) # , k C(3!u) u
(5.86)
where the expansion coe$cients of GI (u) are given by g /n! if the expansion coe$cients of G (u) in L u are given by g and where L C(4#2u) 1 . (5.87) G (u)"! (3#2u) C(1!u)C(2#u)C(3#u) 3 The resulting series coe$cients are shown in Table 11 for the scale choice k"m , which will be +1 assumed in what follows. For comparison we also show the contribution to r from the subtraction L term (3#GI (u))/u and the separate contributions from the "rst IR and UV renormalon poles to d . The subtraction contribution is a convergent series and is practically negligible already at n"2. L Furthermore, the series in the large-b approximation is very rapidly dominated by the "rst IR
The &3' is added to GI to cancel the pole in u.
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Table 11 Perturbative corrections to dm: the &large-b limit' from Beneke and Braun (1995a) in comparison with the contribution from the subtraction function and a breakdown of d into contributions from the "rst renormalon poles (location L indicated in parantheses). Renormalization scale k"m n
d L
0 1 2 3 4 5 6 7
4 18.7446 70.4906 439.435 3495.70 35358.7 423257 5939874
d [sub.] L !2.5 1.458 1.251 0.083 !0.233 !0.083 0.009 0.012
IR(1/2)
IR(3/2)
IR(2)
UV(!1)
9.2039 18.408 73.631 441.78 3534.3 35343 424116 5937622
!6.091 !4.061 !5.414 !10.83 !28.88 !96.26 !385.0 !1796
7.008 3.504 3.504 5.256 10.51 26.28 78.83 275.9
!0.7555 0.7555 !1.511 4.533 !18.13 90.66 !544.0 3807.7
renormalon. The coe$cient d reproduces the exact one-loop correction. One may also compare d (!b )"12.43 [N "4] with the exact two-loop result (Gray et al., 1990): r "8.81. More D over, the C-term in the exact result, which is not reproduced in d (!b ), is rather small and the $ non-abelian term C C is large. These evidences together suggest that the relation between the pole $ mass and the MS mass is dominated by the leading IR renormalon already in low orders and may even be well approximated in the large-b limit. The numbers in Table 11 imply that this relation begins to diverge at order a for charm quarks and at order a or a for bottom quarks. Q Q Q One can make use of the fact that the leading IR renormalon singularity at u"1/2 is related to the linear UV divergence of the self-energy R of a static quark (see also Section 5.4.3) to determine the singularity exactly up to an overall constant (Beneke, 1995). The derivation is analogous to that in Section 3.2.3: the linear UV divergence leads to a non-Borel summable UV renormalon singularity at u"1/2 in the Borel transform of R , if the UV divergences are regulated dimensionally. Following Parisi (1978), the imaginary part of the Borel integral I[R ] is proportional to the insertion of the dimension-3 operator hM h (with h a static quark "eld) and can be written as (5.88) Im I[R ](a ,p,k)"E(a ,k) RM (a ,p,k) , Q Q Q FF where RM is the static self-energy with a zero-momentum insertion of hM h. The coe$cient function FF satis"es a renormalization group equation, which is simpli"ed by the fact that hM h has vanishing anomalous dimension. One then shows that
Im I[dm]"!E(a ,k)"const;k exp Q
?
dx Q
1 "const;K . 2b(x)
(5.89)
It is more conventional to quote r for the di!erence m !m (m ), in which case r "11.41, in better agreement +1 with the large-b approximation with respect to which the two scale choices are equivalent.
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The a -dependence of the imaginary part of the Borel integral determines the large-order behaviour Q of the perturbative expansion of dm according to Eqs. (3.46) and (3.48). The present case is particularly simple, because the large-order behaviour is completely determined in terms of the b-function coe$cients. Since b is now known (van Ritbergen et al., 1997), the result of Beneke (1995) can be extended to 1/n corrections to the leading asymptotic behaviour. The result is
s s r "const;(!2b )LC(n#1#b) 1# # L n#b (n#b) (n#b!1)
s # #2 , (n#b) (n#b!1) (n#b!2)
(5.90)
where b"!b /(2b ) and 1 b b s " ! ! # , (5.91) 2b 2b 2b 1 b b bb b b b b ! # ! ! # # . (5.92) s " ! 2b 8b 2b 4b 2b 8b 4b Numerically, one has b"+0.395,0.370,0.329, for N "+3,4,5, and the coe$cients in the square D bracket in (5.90) that correct the leading asymptotic behaviour are very small:
s "+!0.065,!0.039,0.008, , (5.93) s "+!0.057,!0.064,!0.072, , (5.94) s "+0.054#0.111b ,0.046#0.162b ,0.034#0.246b , . (5.95) The known b-function coe$cients are all negative and between 0 and !1. It is natural to assume that b is of order 1 to obtain an estimate of the 1/n term. The smallness of the pre-asymptotic corrections leads again to the conclusion that the series is close to its asymptotic behaviour already in low orders. Note that the present considerations do not assume the large-b approximation. Corrections to Eq. (5.90) are of order 1/n and 1/2L from the next IR and UV renormalon pole. Encouraged by this observation, we extrapolate Eq. (5.90) to n"1,2. For N "4 we obtain D 1.00#0.14b r , "3.14 (5.96) 0.70!0.51b r r 0.98#0.01b "4.47 . (5.97) r 1.00#0.14b The large dependence on b in the "rst relation indicates that n"1 is too small for the asymptotic formula to apply. However, already at n"2 the asymptotic formula (5.90) may become useful. Since only r is known exactly at present, we cannot yet make use of this result.
s is given numerically below.
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5.4.2. The heavy quark potential In this section we discuss renormalon divergence in the perturbative expansion of the heavy quark potential, that is the non-abelian Coulomb potential. It turns out that there is a close relation between the potential and the pole mass, as far as their leading IR renormalon divergence is concerned. The static potential in coordinate space, <(r), is de"ned through a Wilson loop = (r,¹) of spatial ! extension r and temporal extension ¹ with ¹PR. In this limit = (r,¹)&exp(!i¹<(r)). The ! potential in momentum space,
where v"(1,0) and v ) q"0. To "nd the leading contribution from k&K ;q and /!" k#q&K ;q, which is left over after the IR divergence is cancelled as described above, we /!" expand the integrand in k (the contribution from small k#q is identical). The integrals in each term of the expansion depend only on the vector v. Hence, in a regularization scheme that preserves Lorentz invariance all odd terms vanish because v ) q"0. The long-distance contribution is again of relative order K /q. A similar argument holds for all other one-loop diagrams. /!"
Fig. 22. One-loop corrections to the heavy quark potential.
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The argument generalizes to an arbitrary diagram. Because v ) q"0 and because there is no other kinematic invariant linear in q, it follows from Lorentz invariance that the leading power correction to the potential in momentum space cannot be K /"q", but has to be quadratic: /!"
K 4pC a (q) $ Q 1#2#const; /!"#2 .
(5.99)
The corresponding leading IR renormalon is located at t"!1/b (u"1). Let us emphasize that we are not concerned with the long-distance behaviour of the potential at q&K , but with the /!" leading power corrections of the form (K /q)I, which correct the perturbative Coulomb potential /!" when q is still large compared to K . Renormalons cannot tell us anything about the potential at /!" con"ning distances. When one considers the coordinate space potential, given by the Fourier transform of
B[<(r)](u)"(ke\!)S
dq q r!4pC $ e (2p) (q)>S
C C(1/2#u)C(1/2!u) "! $ e\S! (kr)S . pC(1#2u) r
(5.100)
There is now a pole at u"1/2, which implies C a (1/r) <(r)"! $ Q (1#2# const;K r#2) /!" r
(5.101)
for the coordinate space potential. The long-distance contributions to the coordinate space potential are parametrically larger than for the momentum space potential and its series expansion diverges much faster. In absolute terms the long-distance contribution amounts to an rindependent constant of order K. The leading power correction to <(r) originates only from small q in the Fourier integral and one can set e q r to 1 to obtain it. Because
(5.102)
where
1 dm(k )"! D 2
dq
The rapid divergence has been noticed in a di!erent context by Jezabek et al. (1998).
(5.103)
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The notation is suggestive, because the subtraction can be interpreted as a mass renormalization. De"ne the potential-subtracted (PS) mass (Beneke, 1998) m (k )"m !dm(k ) . (5.104) .1 D D Comparing the Borel transform of the pole mass (5.86) with Eq. (5.100), we note that the singularity at u"1/2 is cancelled in the di!erence (Beneke, 1998; Hoang et al., 1998). Hence the PS mass de"nition is less IR sensitive in this approximation than the pole mass. As a consequence its relation to the MS mass de"nition is better behaved (less divergent) than the corresponding relation for the pole mass. It can be shown that the cancellation extends beyond the large-b approximation used here for illustration (Beneke, 1998). The relation of the PS mass to M,m (m ) at the two-loop order is given by +1 +1 4a (M) k m (k )"M 1# Q 1! D .1 D 3p M
#
a (M) k k Q K ! D K #4pb ln D !2 p 3M M
#2 ,
(5.105)
where K "13.44!1.04n is the two-loop coe$cient in the relation of m to M (Gray et al., D 1990) and K "10.33!1.11n the one-loop correction to the Coulomb potential in momentum D space. The PS mass has a linear dependence on the IR subtraction scale. Mass de"nitions of this kind have been advocated by Bigi et al. (1994b), see also the review by Bigi et al. (1997). These authors favour a subtraction based on integrals of spectral densities of heavy-light quark current two-point functions. Czarnecki et al. (1998) have computed the subtraction term for this de"nition to two-loop order. The precise form of the subtraction di!ers from the above at order a and higher, Q because the de"nitions of the subtracted masses are di!erent. We can use the PS mass and subtracted potential instead of the pole mass and the Coulomb potential to perform Coulomb resummations for threshold problems. The bene"t of using an unconventional mass de"nition is that large perturbative corrections related to strong renormalon divergence associated with the coordinate space potential are obviated. Physically, the crucial point is that, contrary to intuition, heavy quark cross sections near threshold are in fact less long-distance sensitive than the pole mass and the coordinate space potential. The cancellation is made explicit by using a less long-distance sensitive mass de"nition. 5.4.3. Heavy quark ewective theory and exclusive B decays We now turn to heavy quark e!ective theory (HQET) in the context of which renormalons in heavy quark decays and the pole mass have been discussed "rst (Beneke and Braun, 1994; Bigi et al., 1994b). HQET is based on the idea that heavy hadron decays involve the large scale m
At order a the factor 1!k /M in Eq. (5.105) is replaced by 1!4k /(3M). Q D D
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non-perturbative matrix elements that capture the physics on the scale K. New spin and #avour symmetries emerge below the scale m, which relate the matrix elements for di!erent decays. This is what makes HQET useful (see the review by Neubert (1994a) for references to the original literature). In a purely perturbative context, HQET can be viewed as the expansion of Green functions with heavy quark legs around the mass shell. We begin with the expansion of the heavy quark propagator in this perturbative context (Beneke and Braun, 1994), since it provides a nice example of how factorization introduces new renormalon poles and how they are cancelled over di!erent orders in the expansion in the sum of all terms. We de"ne p"mv#k, with m the parameter of the heavy mass expansion, k the small residual momentum k;m, and v"1. We then consider the inverse heavy quark propagator in full QCD, S\, projected as 1#v/ 1#v/ 1#v/ S\(vk,m )" S\(p,m) . . / 2 2 2
(5.106)
The Borel transform of the inverse propagator can be calculated exactly in the approximation of one-gluon exchange with vacuum polarization insertions. The expansion of the result can be cast into the general form B[S\](k,m;u)"md(u)!B[m ](m/k;u)#B[C](m/k;u)夹(vkd(u)!B[R ](vk/k;u)) . #O((vk)/m,k/m) .
(5.107)
The asterisk denotes the convolution product of the Borel transforms. The second term on the right-hand side is the Borel transform of the series that relates the pole mass to m. It is given by (see Eq. (5.86))
C m \S C(u)C(1!2u) #subtractions . (5.108) e\S!6(1!u) B[m ](m/k;u)"m d(u)# $ 4p k C(3!u) The subtraction function may be di!erent from Eq. (5.87), if m is not the MS mass. The second line is the convolution of a coe$cient function that depends only on the scale m and the inverse (static) propagator of HQET that depends only on vk. The Borel transform of the latter is given by
C(!1#2u)C(1!u) 2vk \S C e\S!(!6) #subtractions . B[R ](vk;u)" $vk ! C(2#u) k 4p
(5.109)
The subtraction function has no poles in u and can be omitted for the present discussion. The Borel transform of the unexpanded inverse propagator S\ has IR renormalon poles only at positive . integer u. Compared to this, every term in the expansion around the mass shell has new renormalon
Another perturbative example of this kind is given by Luke et al. (1995a), who consider a toy e!ective Lagrangian with four-fermion interactions and higher-dimension derivative operators obtained from integrating out a heavy particle. Here we take the Borel transform of the series including the term of order a. This gives rise to d(u). Q
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poles at half-integer u, and in particular at u"1/2. Close to u"1, the Borel transform of S\ is . 1 k B[S\](k,m;u)J , (5.110) . v ) k#k/(2m) 1!u implying an ambiguity of order K/v ) k. The residue of the pole at u"1 becomes divergent on-shell (k"0), which causes the singularity structure of the Borel transform to change, when one expands in the residual momentum k. Because there is no singularity at u"1/2 in the unexpanded inverse propagator, the singularity has to cancel in the sum of all terms in the expansion. Inspection of Eqs. (5.108) and (5.109) shows that the singularity at u"1/2 in the pole mass cancels with a singularity at the same position in the self-energy of a static quark. It is of conceptual importance that the pole at u"1/2 in the static self-energy is an ultraviolet pole, which comes from the fact that the self-energy of an in"nitely heavy quark is linearly divergent. Similar cancellations take place for other poles (e.g. at u"3/2) over di!erent orders in the heavy quark expansion. This is just a particularly simple example of how IR poles in coe$cient functions (depending only on m/k and not on k) cancel with UV poles in Green functions (depending only on k and not on m) with operator insertions at zero momentum in HQET. The general nature of such cancellations has already been emphasized, see also the more complicated example in Section 4.2.2. What happens if one chooses the pole mass as the renormalized quark mass parameter? Then the "rst and second terms on the right-hand side of Eq. (5.107) cancel each other and one is left with an apparently uncancelled pole at u"1/2 on the right-hand side. This simply tells us that one has to be careful not to absorb long-distance sensitivity into input parameters, if one wants to have a manifest cancellation of IR renormalons. Up to now we considered the limit K;k;m, in which HQET amounts to the expansion of Green functions around the mass shell. For a physical heavy-light meson system, the residual momentum k is of order K and the long-distance parts of the factorized expressions for heavy hadron matrix elements in QCD are non-perturbative matrix elements in HQET. Then we have the usual situation that IR renormalon ambiguities in de"ning the coe$cient functions must correspond to UV ambiguities in de"ning matrix elements in HQET. Since several processes may involve the same matrix elements, this leads to consistency relations on the IR renormalon behaviour in the coe$cient functions of di!erent processes. In the remainder of this section we brie#y consider several implications and applications, the latter mainly in the large-b approximation, of renormalons in HQET. The binding energy of a heavy meson (Bigi and Uraltsev, 1994; Beneke and Braun, 1994; Bigi et al., 1994b). In HQET the mass of a meson can be expanded as j #3j #O(1/m) . (5.111) m "m #KM ! @ @ 2m @ To be speci"c, we have taken the pseudoscalar B meson. The parameters KM and j are the same for all members of a spin-#avour multiplet of HQET. KM is interpreted as the binding energy of the meson in the limit mPR. j denotes the contribution to the binding energy from the heavy quark kinetic energy and j the contribution from the spin interaction between the heavy and the light quark. (We follow the conventions of Neubert (1994a).) The fact that the pole mass expressed in
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terms of, say, the MS mass (or another mass related to the bare mass by pure ultraviolet subtractions) has a divergent series expansion with an ambiguity of order K, leads to the conclusion that the binding energy KM , which is also of order K, is not a physical concept. Since KM "m !m #2, it depends on how the pole mass is de"ned beyond its perturbative @ expansion and di!erent de"nitions can di!er by an amount of order K, that is by as much as the expected magnitude of KM itself. Physically, this is not unexpected: because quarks are con"ned, the meson cannot be separated into a free heavy and a light quark relative to which the binding energy could be measured. In decay rates KM appears at subleading order in HQET, while the quark mass does not appear at leading order. We may ask whether this would allow us to determine KM , by-passing the argument above. This is in fact not possible, because a term of order KM /m appears always in conjunction with a coe$cient function at order (K/m) with a renormalon ambiguity of order K/m. One can rewrite Eq. (5.111) as m "[m !dm(k)]#[KM #dm(k)]#2 @
(5.112)
with a residual mass dm(k) that subtracts the (leading) divergent behaviour of the series expansion for the pole mass. In order to obtain a decent heavy quark limit, the residual mass term should stay "nite in the in"nite mass limit. At the same time, it must be perturbative to subtract the perturbative expansion of m . This leads to a linear subtraction proportional to a factorization scale k
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Of course, we can avoid KM altogether by eliminating it from the relations of physical observables, but in practice this is often not an option that is easy to implement. (It is for the same reason that one usually works with renormalized rather than bare parameters, although all divergences would drop out in the relation of physical quantities.) The kinetic energy. The matrix element of the chromomagnetic operator, j in (5.111), is related to the mass di!erence of the vector and pseudoscalar meson in the heavy quark limit and therefore physical and unambiguous. For the matrix element j of the kinetic energy operator hM (iD )h , the T , T situation is not obvious. Curiously enough, there is no IR renormalon at u"1 in the Borel transform of the pole mass (5.108) and therefore it follows from Eq. (5.111) that there is no ambiguity in j at this order in the #avour expansion. Beneke et al. (1994) speculated that the kinetic energy may be protected by Lorentz invariance from quadratic divergences, just as Lorentz invariance protects this operator from logarithmic divergences to all orders in perturbation theory (Luke and Manohar, 1992). The problem was discussed further in Martinelli et al. (1996) and it seems to have been settled "nally by Neubert (1997), who showed that even for a Lorentz-invariant cut-o! a quadratic divergence exists at the two-loop order. The kinetic energy operator mixes with the operator hM h and its matrix element is not physical in the same sense as KM is not physical. T T Exclusive semi-leptonic heavy hadron decays. As already mentioned, the predictive power of HQET derives from the fact that the heavy quark symmetries relate di!erent decays and reduce the number of independent form factors. According to our general understanding of IR renormalons in coe$cient functions, they are related to the de"nition of non-perturbative matrix elements at subleading order in the 1/m expansion. Since there is a limited number of such matrix elements in semi-leptonic decays, the IR renormalon behaviour of the coe$cient functions satis"es certain consistency relations, which simply express the fact that if the matrix elements are eliminated to a given order in 1/m and physical quantities are related directly, there should be no ambiguities left to that order in 1/m. The decay K PK ll is a particularly simple example to illustrate this point, @ A because the form factors, at subleading order in 1/m, are proportional to the same Isgur-Wise function that appears at leading order. Hence, we can write (setting m PR and keeping m large @ A but "nite) 1K (v )"c cIb"KM (v)2"u (v )[F (w)cI#F (w) vI#F (w)v k]u(v) A @
(5.113)
with w"v ) v and F (w)"m(w)(C (m ,w)#D (m ,w)(KM /m )#O(1/m)) . G G @ A G @ A A A
(5.114)
The large-order behaviour of the series for the pole mass determines the renormalon ambiguity of KM . Because the unexpanded form factors F are observables, the large-order behaviour of the series G expansion of C is determined by the requirement that its renormalon ambiguity matches with D KM . G G Neubert and Sachrajda (1995), and Luke et al. (1995a) checked, in the large-b limit, that this is indeed the case. The situation is more complicated for semileptonic BPD decays and has been considered in detail by Neubert and Sachrajda (1995). Numerical results in the large-b approximation. The Borel transforms of some coe$cient functions in HQET are known exactly in the large-b approximation and we give a brief overview of these results.
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(i) The HQET Lagrangian reads C (a ) 1 L "hM iv ) Dh # hM (iD)h # Q hM p g GIJh #O(1/m). T T IJ Q T T T 2m T 4m
(5.115)
It involves only one non-trivial coe$cient function C (a ) to this order in 1/m. Grozin and
Q Neubert (1997) computed the coe$cient function in the large-b approximation. So far, this is the only exact result in the large-b limit that involves diagrams with a three-gluon vertex. As expected, a rapid divergence of the series is found and an IR renormalon pole at u"1/2, which can be related to higher-dimension interaction terms in the e!ective Lagrangian. An interesting point is that the renormalization group (RG) improved coe$cient function requires both the anomalous dimension and matching relation to be computed. However, because anomalous dimensions are convergent series in the MS scheme, while the expansion of matching coe$cients is divergent, the contribution from the anomalous dimension is almost insigni"cant in higher-orders. In su$ciently large orders the &leading logarithms' are in fact smaller than the factorially growing constant terms. This allows Grozin and Neubert (1997) to conclude that the RG improved coe$cient is already known accurately to next-to-next-to-leading order despite the fact that the three-loop anomalous dimension is not known. (ii) Neubert (1995a,b) considered the matching of bPc currents at zero recoil, c C
b"g hM A C h@#O(1/m ) , (5.116) 4 4 T 4 T A @ for the vector and the axial-vector current. The decay rate for BPDHll is proportional to "< "F(v ) v ), where A@ (5.117) F(1)"g (1#d ) . K This leads to a precise determination of "< " from the measured rate near zero-recoil, provided the A@ perturbative correction to g and the leading power correction d are under control. The K large-b approximation provided the "rst estimate of the second (and higher) order corrections to the matching coe$cient and their magnitude was found to be moderate (with a normalized at the Q &natural scale' (m m ). Meanwhile the two-loop correction is known exactly (Czarnecki and @ A Melnikov, 1997a). It turns out that the large-b approximation to the two-loop coe$cient is very accurate for the axial vector case g , but not accurate at all for g . One reason seems to be that the 4 N a-term is anomalously small for g . The main point, however, is that the two-loop correction D Q 4 to the matching of the currents is not large. But if there are no large corrections associated with the running coupling, no improvement due to a large-b resummation should be expected. Further information on the form factor F(1) at zero recoil can be obtained from sum rules based on the spectral functions of heavy quark currents (Bigi et al., 1995). In addition to the purely virtual form factors, real gluon emission has to be allowed for. The relevant calculation in the large-b limit can be found in Uraltsev (1997).
The exact one-loop and two-loop coe$cients are both somewhat less than a factor 1/2 smaller for g than for g . 4 However, the N -term is a factor of 15 smaller. D
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5.4.4. Inclusive B decays Inclusive semi-leptonic decays BPX ll, where X is an inclusive "nal state without or with S A charm, can be treated in an expansion in K/m (Chay et al., 1990; Bigi et al., 1992). Contrary to @ exclusive decays the leading non-perturbative corrections are suppressed by two powers of m and @ involve the parameters j introduced in Eq. (5.111). The leading term in the expansion is the hypothetical free quark decay. From a phenomenological point of view, the main result of the heavy quark expansion is to a$rm that non-perturbative corrections are in fact small, less than 5%. Therefore, the main uncertainty in the prediction of the decay rate, which for decays into charm can be used to measure the CKM element "< ", comes from unknown perturbative A@ corrections to the free quark decay beyond the one-loop order. Traditionally the free quark decay is expressed in terms of the quark pole mass. This seems indeed to be the natural choice for the decay of a free particle. The series expansion of the free quark decay is
a (m ) G"< "m C(BPX el)" $ S@ @ 1!C g Q @ [1#D] , $ p S 192p
(5.118)
where D" r a (m )L" a (m )L[d (!b )L#d ] (5.119) L Q @ Q @ L L L L parametrizes perturbative corrections beyond one loop. The heavy quark expansion clari"es that it is not a good idea to use the pole quark mass parameter. When the pole mass is used, the series coe$cients diverge rapidly due to an IR renormalon at t"!1/(2b ). It produces an ambiguity of order K/m in summing the series, which does not correspond to any non-perturbative parameter in @ the heavy quark expansion. The resolution is that the rapid divergence appears only because the pole mass has been used. If we express the pole mass as a series times the MS mass and eliminate it from Eq. (5.118), then the leading divergent behaviour of the r cancels with the series that relates L the pole mass to the MS mass (Bigi et al., 1994b; Beneke et al., 1994). The leftover divergent behaviour corresponds to power corrections of order K/m, consistent with the heavy quark expansion. In the large-b limit, it is found (Beneke et al., 1994; Ball et al., 1995b) that the divergent behaviour leftover corresponds in fact to a smaller power correction of order K/mln(K/m) consistent with the observation that in this approximation all matrix elements at order 1/m are unambiguous, see Section 5.4.3. Numerically, the e!ect of the cancellation is dramatic beyond two-loop order and we have illustrated it in the large-b limit (as explained in Section 5.1.1) in Table 12. Up to two-loop order we may note, however, that g (see Eq. (5.118)) in the MS scheme is in fact larger than in the on-shell scheme and that dM is not a small correction. The example of inclusive B decays illustrates the fact that pole masses are not useful bookkeeping parameters, say for the Particle Data Book. Either their value, extracted from some process
For simplicity of notation we consider the decay into the massless u quark in the general discussion. The explicit demonstration of this cancellation has now been extended, by purely algebraic methods, to two-loop order (Sinkovics et al., 1998).
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Table 12 Higher-order coe$cients to bPu decay in the large-b approximation together with partial sums for !b,Da (m )"0.14. 2nd and 3rd columns: Decay rate expressed in terms of the b pole mass. 4th and 5th columns: Q @ Decay rate expressed in terms of the b MS mass. The last line gives the principal value Borel integral computed according to Eq. (3.83) together with an estimate of the uncertainty due to renormalon poles. In the MS scheme this uncertainty is very small, because the leading term in the expansion of the one-loop correction with a massive gluon expanded in the gluon mass j is of order j/mln(j/m). Table from Ball et al. (1995b) @ @ n
d L
1#D
dM L
1#DM
0 1 2 3 4 5 6 7 8 R
1 5.3381702 34.409913 256.48081 2269.4131 23679.005 289417.40 4081180.2 65496131.0 *
1 1.747 2.422 3.126 3.997 5.271 7.450 11.75 21.42 2.314$0.615
1 4.3163 8.0992 26.680 82.262 421.33 1656.1 12135 52862 *
1 1.604 1.763 1.836 1.868 1.890 1.903 1.916 1.924 1.925$0.012
(e.g. B decays, if we knew the CKM matrix elements) would depend sensitively on the loop order of the theoretical input calculation or one would assign to it a large error due to higher-order corrections. Another process predicted in terms of the pole mass would also seem to be poorly predicted, because of large higher-order corrections. However, because the theoretical errors are correlated with those in the pole mass input parameter, the actual uncertainties are much smaller. It is preferable to use book-keeping parameters that do not introduce such correlations. The optimal choices are book-keeping parameters that themselves are less long-distance sensitive than any process in which one would use them. The MS mass, which is basically a bare mass minus UV subtractions, is such a parameter, although only in a perturbative setting. We may also eliminate the quark mass in favour of the physical B meson mass. In this case we get
a (m ) KM G"< "m 1!C g Q @ [1#D]!5 #O(1/m) . C(BPX el)" $ S@ $ S p m 192p
(5.120)
Now the large perturbative corrections in D are cancelled by the fact that KM has to be speci"ed as one-loop, two-loop, etc., and di!ers with loop order, such as to cancel the large corrections to D. Apart from the fact that, beyond two loops, the magnitude of the so-de"ned KM is far larger than that of K, the delicate cancellation of all K/m e!ects that has to be arranged in this way seems a high @ price to pay, in comparison to using a quark mass de"nition without large long-distance sensitivity, together with a better-behaved series expansion. Ball et al. (1995b) performed a detailed numerical analysis of higher order corrections to inclusive semi-leptonic decays into charm in the large-b limit in the MS scheme and the on-shell scheme for the bottom and charm quark mass. They calculated the distribution function ¹(m) that enters Eq. (3.83) analytically for bPu transitions and numerically for bPc transitions. The renormalon
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problem is less severe for bPc than for bPu, because the leading IR renormalon cancels in the di!erence m !m and, in the rate for bPc, the quark masses appear numerically in this @ A combination to a certain degree. Ball et al. (1995b) found that, after higher-order corrections are taken into account, one obtains values for "< ", from the calculation in the MS scheme and the @A on-shell scheme, which are consistent with each other, contrary to what is found in one-loop calculations (Ball and Nierste, 1994). The corrections in the MS scheme are not small at one- and two-loop order, which re#ects the fact that the MS mass at the scale of the mass, is relatively small and too far away from the natural range for a &physical' quark mass given by m $K. Instead of a full resummation of (some) higher-order corrections we can also optimize the choice of scale and quark mass de"nition to avoid large corrections to some extent (Shifman and Uraltsev, 1995; Uraltsev, 1995). Since the analysis of Ball et al. (1995b) there has been some progress in the calculation of the exact 2-loop correction to inclusive bPc transitions and it is interesting to compare the large-b limit with these results. Up to order a the series of radiative corrections in the on-shell scheme is Q (Ball et al., 1995b)
a (m ) a (m ) a (m ) G"< "m C(BPX el)" $ @A @ f (0.3) 1!1.67 Q @ !14.2 Q @ !173 Q @ #2 , A p p p 192p (5.121) where f (m /m ) is a tree level phase space factor, and the numerical values in square brackets A @ assume m /m "0.3. (f (0.3)"0.52.) The exact two-loop correction is still unknown. However, the A @ di!erential decay rate dC/dq, where q is the invariant mass of the lepton pair, has been computed analytically at three special kinematic points q"(m !m ) (Czarnecki and Melnikov, 1997a), @ A q"0 (Czarnecki and Melnikov, 1997b) and the intermediate point q"m (Czarnecki and A Melnikov, 1998). The authors then interpolate the three points by a second-order polynomial in q and obtain !(12.8$0.4)
a (m ) Q @ p
(5.122)
for the second-order correction, to be compared with !14.2 above. The large-b limit has worked well in the on-shell scheme. Note that in this case the two-loop correction is large. This should be contrasted with the situation for exclusive semi-leptonic decays, see the end of Section 5.4.3. The large-b approximation has also been applied to the top decay tP=#b with the = assumed to be on-shell in the approximation that m /m "0 (Beneke and Braun 1995a) and 5 R for "nite m /m (Mehen, 1998). Not unexpectedly, the convergence of the series is again improved 5 R if one does not use the top pole mass in the decay rate, except for the hypothetical limit m Pm . 5 R
The second-order correction was "rst obtained by Luke et al. (1995b). The di!erence to the value 15.1 quoted there comes from the fact that we use N "4 rather than N "3(#1.14) and the remaining di!erence (!0.2) is probably due D D to numerical errors.
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5.4.5. Non-relativistic QCD Quarkonium systems, like heavy-light mesons, can be treated with e!ective "eld theory methods (Caswell and Lepage 1986; Bodwin et al., 1995). The e!ective theory is non-relativistic QCD (NRQCD). The expansion is done in v, where v is the typical velocity of a heavy quark in an onium. This is somewhat di!erent from the expansion we encountered before in HQET or DIS, which are expansions in K/m and K/Q, respectively. The renormalon structure in the matching of QCD currents on non-relativistic currents and in the velocity expansion of some quarkonium decays has been considered by Braaten and Chen (1998) and Bodwin and Chen (1998). Since v need not be connected with the QCD scale K } for example one could have the hierarchy m<mv
1 10"sRDt"g2 , 10"QM cIc Q"g2"dI C(a )10"sRt"g2# Q 2m
(5.123)
where t and s are non-relativistic two-spinor "elds. A leading IR renormalon pole in the Borel transform of C(a ) at u"1/2 is found, which corresponds to an ambiguity of relative order K/m. Q There is a UV renormalon pole at u"1/2 in the matrix element of the higher-dimension operator in square brackets. However, to obtain a complete cancellation of the singularity at u"1/2, one also has to take into account the fact that the "rst matrix element in square brackets has a renormalon ambiguity proportional to itself. This somewhat unfamiliar situation arises, because, due to insertions of higher-dimension operators in the NRQCD Lagrangian, the matrix element is expressed as a series in v, and there exist power-UV divergences from these insertions. In HQET, on the contrary, it is conventional to parametrize the contributions from insertions of higher-dimension operators in the Lagrangian as separate &non-local' operators. The second &peculiarity' is that the Borel transforms of the coe$cient functions also have an infrared renormalon pole at negative u"!1/2. Recall that if a one-loop integral has the small-k behaviour dk/k>L, an IR renormalon pole at u"n is obtained. The pole at u"!1/2 is therefore due to the fact that the integrals that contribute to the matching coe$cient are linearly IR divergent. This divergence would be regulated by a small relative momentum of the heavy quark and anti-quark and then give rise to the Coulomb divergence 1/v. To compute the coe$cient function, the relative momentum is set exactly to zero. In dimensional regularization the power-like IR divergence is set to zero at every order in perturbation theory, but it leads to a Borel-summable IR renormalon at u"!1/2. (Recall that linear ultraviolet divergence gives rise to an unconventional non-Borel summable singularity at positive u"1/2.)
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6. Connections with lattice 5eld theory One may be surprised to "nd renormalons discussed in connection with lattice gauge theory, as we emphasized that renormalons are &artefacts' of performing a short-distance expansion. If the exact, non-perturbative result could be computed, one would never concern oneself with renormalons. The connection arises from the fact that it is di$cult to simulate quantities on the lattice that involve two very di!erent scales Q
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residual mass term dm is generated by quantum corrections in the lattice regularization of HQET. The residual mass counterterm can be de"ned non-perturbatively as
tr S (x,t#a) 1 F dm "!lim ln , (6.2) tr S (x,t) a F R where S (x,t) is the static quark propagator in the Landau gauge at the point (x,t) and the trace is F over colour. In perturbation theory dm &a /a. The binding energy E of the ground state meson in Q a given channel is computed from the large-time behaviour of the two-point correlation function " Z exp(!Et), 10"J(x,t)JR(0,0)"02 R
(6.3)
x
where J is the heavy-light current in HQET with the appropriate quantum numbers. The binding energy is linearly divergent, but the linear divergence is the same to all orders in perturbation theory as that of dm . Hence, Martinelli and Sachrajda (1995) de"ne KM ,E!dm ,
(6.4)
which is "nite as aP0 and of order K. The lattice calculation of Crisafulli et al. (1995) and GimeH nez et al. (1997) gives KM "(180>) MeV. One can then de"ne a &subtracted pole mass' m "M ! \ 1 KM #O(K/m), which replaces the naive perturbative expression m "M !KM . The subtracted pole mass is still a long-distance quantity, and useful only if it can be related to another mass de"nition such as the MS mass M "m (m ). But then M can be computed @ @ +1 +1 directly from a lattice measurement of E. To see this, let M "m (1# c a (M)L>), then to @ L L Q a given order N in perturbation theory, the relation is
, 1 , M " 1# c a (M )L> m !dm (a)# r a (a)L> L Q L Q @ 1 @ a L L , 1 , " 1# c a (M )L> M !E(a)# r a (a)L> , (6.5) L Q @ L Q a L L where dm and E are evaluated non-perturbatively for a given a, and r a (a)L> is the L L Q perturbative evaluation of the linear divergence of dm or E. (They coincide.) The renormalon divergence cancels asymptotically between the two series in Eq. (6.5) and the linear divergence also cancels up to order a,>. However, because the series is truncated, one cannot take a too small. Q Note that the subtraction is done perturbatively and it is not necessary to de"ne KM or m to obtain 1 M as illustrated by the second line. But because the (leading) renormalons cancel, a nonperturbative subtraction is not necessary. In terms of Borel transforms the cancellation near the leading singularity at u"1/2 looks, schematically, 1 1!2u
m \S 1 1 \S ! , k ma ka
(6.6)
The relation of dm to dm can be found in Martinelli and Sachrajda (1995), but the distinction is not relevant to the present discussion.
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if the Borel transform is taken with respect to a (k). In practice, the cancellation may be numerically Q delicate if m and a\ are very di!erent. Using the procedure explained here, GimeH nez et al. (1997) quote M "(4.15$0.05$0.20) GeV, where the second error has been assigned as a consequence @ of the unknown second-order coe$cient r . In physical units the inverse lattice spacings in these simulations are between 2 and 4 GeV. There are corrections of order K/M from higher dimension operators in HQET, see Eq. (5.111). These are smaller than the error due to the unknown perturbative subtraction terms. The important conclusion is that the MS mass can be reliably determined from the B meson mass and a lattice measurement of E(a), provided the r are known to L su$ciently high order in lattice perturbation theory. An extended subtraction procedure for the kinetic energy (Martinelli and Sachrajda, 1995) has also been studied numerically (Crisafulli et al., 1995), but the accuracy of the subtraction is not yet su$cient to reach physically interesting values. 6.2. The gluon condensate Power divergences are even more severe in the calculation of the gluon condensate, because the operator a GIJG is quartically divergent. On the lattice the gluon condensate is computed from Q IJ the expectation value of the plaquette operator ; . Classically, we have . p a 1 1 QGG " . (6.7) 1! tr ; ? . 36 p 3 a Quantum #uctuations introduce corrections to the unit operator, and the above relation is modi"ed to
c p a 1 #O(a) , (6.8) 1P2, 1! tr ; " L # C (b)a Q GG . bL 36 %% p 3 L where b"6/(4pa(1/a)) denotes the lattice coupling at lattice spacing a and a(1/a) the bare lattice Q Q coupling. Note that there is no term of order a, because there is no gauge-invariant operator of dimension 2. For aK;1, the "rst series is far larger than the gluon condensate, which one would like to determine and therefore has to be subtracted to high accuracy. Not only has it to be subtracted, it has to be de"ned in the "rst place. The series has an IR renormalon, and the coe$cients c are expected to diverge as L 3b L (6.9) c J ! C(n!2b /b ) , L 4p
as follows from adapting Eq. (3.51) with d"4 to the present convention for the expansion parameter. The ambiguity or magnitude of the minimal term of the series is of order (aK) as the gluon condensate term in Eq. (6.8) itself. Again we emphasize that in principle one need not subtract the power divergence and one can consider a\ as a hard factorization scale. Using the Langevin method (Parisi and Wu Yongshi, 1981), Di Renzo et al. (1995) calculated the "rst eight coe$cients c in pure SU(3) gauge theory to good accuracy: L c "+1.998(2), 1.218(4), 2.940(16), 9.284(64), 34.0(3), 135(1), 567(21), 2505(103), . (6.10) L
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According to Eq. (6.9) the ratio of subsequent coe$cients is expected to be 0.21n for large n. The coe$cients (6.10) of the series expressed in the lattice coupling grow much more rapidly than this. The behaviour of Eq. (6.9) is expected for series expressed in terms of an expansion parameter whose b-function is convergent, see Section 3.4 on scheme-dependence of large-order estimates. We expect this to be true in the MS scheme. We do not know the large-order behaviour of the b-function in the lattice scheme and we will assume that the relation between the lattice and the MS coupling does not diverge factorially. In this case (6.9) should hold in both schemes asymptotically. However, the lattice coupling is related to the MS coupling by large "nite renormalizations unrelated to renormalons. This causes series expansions in the lattice coupling to be badly behaved generally and to be irregular, basically because the scale parameter is unnaturally small in the lattice scheme: K "K /28.8. As a consequence it may be expected that the asymptotic behav +1 iour (6.9) is obscured in low/intermediate orders of perturbation theory in the lattice scheme. Di Renzo et al. (1995) suggest to assume that Eq. (6.9) holds in a well-behaved continuum scheme R and then use a three-loop relation b "b!r !r /b (6.11) 0 to express Eq. (6.9), assumed to hold for b , in terms of b. They "nd that the set of coe$cients (6.10) 0 is well described if the continuum scheme is chosen such that r "3.1 and r "2.0 (values quoted from Burgio et al. (1998). In the MS scheme, with b normalized at p/a, we would have r "1.85 +1 and r "1.67 (LuK scher and Weisz, 1995). The preferred values of the "t can be understood as a change of scale: in terms of b (0.706/a) one obtains r "3.1 and r "2.1 in Eq. (6.11). +1 Since IR renormalon divergence arises from large-size #uctuations, the asymptotic behaviour (6.9) does actually not appear on any "nite lattice. According to the estimate (2.24) the asymptotic behaviour is a!ected by "nite volume e!ects at a critical order n "4ln N#c, where N is the number of lattice points in each direction and c is a constant in the limit of large N. For the values N"8, 12 that pertain to the calculation of Di Renzo et al. (1995) the precise value of c is important to establish whether the IR renormalon contribution to the coe$cients c is already a!ected by the L "nite volume. An analysis of the situation in the O(N) p-model (Di Renzo et al., 1997) suggests that c is large enough to leave the 8-loop coe$cients una!ected. The conclusion of Di Renzo et al. (1995) is therefore that the factorial growth (6.9), with an ambiguity of order (aK) corresponding to the gluon condensate, is con"rmed by the pattern of the lattice coe$cients c . L Can the gluon condensate be obtained by subtracting the series to 8-loop order? Ji (1995b) suggested various procedures to extrapolate the 8-loop truncated series to a sum. Subtracting this sum from Monte Carlo data for the plaquette expectation value, he obtained the value 1(a /p)GG2+0.2 GeV, which is at least a factor 10 larger than the &phenomenological value' Q quoted in Eq. (5.13). Burgio et al. (1998) went further and examined the remainder as a function of b (and hence a). The result is shown in Fig. 23. The left plot shows Monte Carlo data of the plaquette expectation values from which the one-loop, two-loop etc., perturbative terms in Eq. (6.8) are consecutively subtracted. According to (6.8) one expects the remainder to scale as (aK), if all terms in the perturbative series up to the minimal term are subtracted. In this case the series of curves in the left plot should approach the line marked K/Q (a,1/Q) in the plot. Contrary to the
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Fig. 23. (a) The subtracted plaquette expectation value as a function of loop order compared to the scaling of a 1/Q and 1/Q term. (b) Comparison of the all-order subtracted plaquette MC data with the scaling of a 1/Q and 1/Q term. Figure taken from Burgio et al. (1998).
expectation, the remainder approaches a clear a"1/Q behaviour. The right plot checks that this is not due to the fact that not all terms up to the minimal have been subtracted. What is shown is a subtraction based on a Borel-type resummation of the higher-order terms in the series, assuming that it follows the asymptotic behaviour (6.9). The resultant remainder has again a clear a behaviour, despite the fact that such a term is not present in (6.8). The observation of K/Q terms in the subtracted plaquette expectation value has led to speculations that there might be sources of power corrections of UV origin that give rise to 1/Q power corrections (Grunberg, 1997; Akhoury and Zakharov, 1997a). Because they are of UV origin, they would not be in contradiction with the OPE according to which only a a"1/Q term can appear in Eq. (6.8). These ideas can be motivated by considering the integral
1 / dk a (k), (6.12) Q where a (k) is supposed to be a physical de"nition of the coupling. The integral receives a contribution of order K/Q from k&K and of order a (Q) from k&Q. But if the e!ective coupling has a term of order K/k in its own short-distance expansion, then this gives rise to a power correction of order K/Q from large k&Q. The problem with the argument is that the In fact, tentative evidence for an unexpected a behaviour in the plaquette expectation value and a certain Creutz ratio derived from it was already reported by Lepage and Mackenzie (1991) several years earlier. These authors had only second-order perturbation theory available.
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de"nition of an e!ective coupling is to a large extent arbitrary and it is not clear how the argument could be applied to the lattice calculation above, where we assumed explicitly that the coupling de"nition does not contain power corrections. Furthermore, if one uses a coupling with larger power corrections than the observable under investigation, then one obtains additional power corrections not parametrized by matrix elements that appear in the short-distance expansion of that observable, but related only to the short-distance expansion of the coupling itself. These power corrections are, however, &standard'. One can always choose a coupling without power corrections by de"nition. Then the question is whether with such a de"nition of the coupling there exist power corrections that are not parametrized by matrix elements of operators in the OPE. An analysis of the 1/N expansion in the p-model (Beneke et al., 1998) "nds a negative answer in that case. Before a de"nite conclusion can be drawn on the signi"cance of lattice data above, one may consider the possibility that the observed a scaling is a pure lattice artefact and does not indicate any unconventional power correction beyond the OPE. One point of concern is that Eq. (6.8), which is assumed by Di Renzo et al. (1995) and Burgio et al. (1998), does not make the dependence of the plaquette expectation value on a completely explicit. One can view lattice gauge theory at small values of aK as an e!ective theory, i.e. an expansion around the continuum limit. The plaquette operator has the expansion (6.13) P,1!tr ; "C (ln a) ) 1#C (ln a)a(a /p)GG#O(a) , . %% Q in which there is no term of order a. This does not yet imply that the matrix element of the plaquette operator does not contain an a term. The lattice Lagrangian in pure gauge theory can be expanded as L (a)"L #a C (ln a)OG #O(a) , (6.14) G G with dimension-6 operators OG . Hence the vacuum expectation value of the plaquette has the small-a expansion
(6.15) 1P2"C (ln a)112#a C (ln a)C (ln a) dx1¹(1,OG (x))2#O(a) , G G where the vacuum expectation values are now taken in the a-independent vacuum of the continuum theory, contrary to the vacuum average in Eq. (6.8), which refers to the lattice vacuum. The a correction in the form of a time-ordered product can be interpreted as a correction due to the fact that the vacua in the lattice and the continuum theory are di!erent at order a. Such terms are not in contradiction with the operator product expansion of the plaquette operator. However, the connected part of the time-ordered product in Eq. (6.15) is zero, and it remains unclear whether a higher-dimension operator in the e!ective lattice action is responsible for the remainder of order a, which Burgio et al. (1998) "nd after their subtraction procedure. In the continuum theory the dimension-6 operators in the Lagrangian are suppressed by the ultraviolet cut-o! K of QCD. Hence, they are arbitrarily small in the operator product expansion 34
I thank S. Sharpe for this remark.
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in K/Q of a physical process with K;Q;K . It is only because in the lattice simulation one has 34 identi"ed a\"K "Q that they become relevant. This conclusion is general and applies to the 34 calculation of any power divergent quantity in lattice gauge theory. Note that the dimension-6 operators on the right-hand side of Eq. (6.14) can be eliminated by working with a (non-perturbatively) improved action. Thus a lattice simulation with an improved pure gauge theory action should "nd a reduced a term, if it is due to higher-dimension operators in the e!ective lattice action.
7. Conclusion In this review we have described in detail the physics of renormalons from a predominantly phenomenological point of view. This has been a very active area of research over the past six years and the understanding of large-order behaviour and power corrections to particular processes in QCD has expanded enormously. In general, the renormalon phenomenon deals with the interface of perturbative and non-perturbative e!ects in observables that involve a large momentum scale compared to K. Such observables cannot be treated easily even in lattice gauge theory. If we were forced to distill a single most important and general conclusion from the work reviewed here, it would be this: Since the conception of QCD the emphasis of perturbative QCD has been on constructing IR "nite observables or to isolate the collinearly divergent contributions, for example in parton densities. This leads to perturbative expansions with "nite coe$cients. The study of IR renormalons and the power corrections associated with them calls on us to extend the notion of IR xniteness to the notion of IR insensitivity. For quantities that are perturbatively less sensitive to small loop momenta are not only expected to have smaller non-perturbative corrections, but also smaller higher order corrections in their perturbative expansions, and are therefore better predictable in a purely perturbative context. At the present times of precise experimental QCD studies, this is an issue of direct phenomenological relevance. The concept of IR insensitivity should be applied "rst of all to the fundamental parameters of the QCD Lagrangian, the coupling constant and the quark masses. In this respect we have concluded that the pole mass de"nition should be abandoned even for heavy quarks, because it is more sensitive to long distances than many processes involving heavy quarks. On the other hand, the MS de"nition of the strong coupling, which has become the accepted standard for perturbative calculations, has very good properties from this point of view. The MS scheme seems indeed to be a fortunate choice. In addition to "xed-sign IR renormalon divergence, which is related to physical and scheme-independent power corrections, there exist also UV renormalons related to irrelevant operators in the in"nite UV cut-o! limit. The corresponding divergent behaviour is universal, sign-alternating, and does not lead to physical power-suppressed e!ects. The minimal term of the series due to UV renormalons is scheme-dependent and it seems that in the MS scheme the UV renormalon behaviour is generally suppressed and therefore of little relevance to accessible perturbative expansions in low or intermediate orders. Once infrared-insensitive input parameters are "xed, the infrared properties of any particular observable are manifest in its perturbative expansion. Perhaps one of the most interesting outcomes of IR renormalons is the prediction, based only on basic properties of QCD, that most
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observables that probe hadronic "nal states } such as &event shape' observables in e>ePhadrons } have large K/Q power corrections and large higher order perturbative corrections. The study of these power corrections has been pursued with vigour, theoretically and experimentally. Even though the theoretical interpretation of the results may turn out to be very di$cult, the experimental studies are extremely important, not only to guide further theoretical developments. Since QCD has matured beyond the stage of qualitative &tests', the prediction of QCD (background?) processes with high precision has become crucial. Meeting this challenge requires the understanding of power corrections and higher order perturbative corrections. A review that leaves no open questions may be a cause of satisfaction for its author, but it would also re#ect sad prospects for its subject. Because of this, we would like to conclude with 11 problems, the solution of which we consider important (the numbers in paranthesis refer to those sections relevant to the problem): Formal and diagrammatic problems: 1. Is the expansion of the b-function in the MS scheme convergent? (3.4). 2. Prove diagrammatically to all orders in 1/N that the large-order behaviour in QCD is D determined by b after a partial resummation of the #avour expansion. What is the explicit structure of singularities at next-to-leading order in the #avour expansion of QCD? (3.2.2). 3. Can one classify the IR renormalon singularities of on-shell Green functions and min-kows-kian observables with the same generality as UV renormalon singularities? What are the universal elements in this classi"cation? Determine the strength of IR renormalon singularities in on-shell Green functions. (3.3). 4. Are there singularities in the Borel plane other than renormalon and instanton singularities? If not, why not? (2.4). Phenomenological questions: 1. Are there 1/Q corrections to Drell}Yan production beginning from two-loop order? (5.3.4). 2. Which operators parametrize the K/Q power correction to the longitudinal cross section in e>e\ annihilation? (5.3.1). 3. Can one construct &better' event shape variables, that is observables with reduced or no K/Q power correction, which are sensitive to a at the same time? (5.3.2). Q 4. Demonstrate that one can combine perturbative series at leading power and a lattice calculation of the "rst power correction with an accuracy better than the "rst power correction. (4.2.1,6) Beyond renormalons: 1. What is the large-order behaviour of the series of power corrections? There are compelling arguments (Shifman, 1994) that this series also diverges factorially. But what is the precise behaviour in QCD? 2. Are there power corrections to time-like (minkowskian) processes related to the fact that parton-hadron duality is only approximate? Can one quantify &violations of parton-hadron duality'?
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3. If large-size (o&1/K) instantons play an important role in the QCD vacuum, how do they a!ect properties of short-distance expansions (Chibisov et al., 1997)? We hope that the answers to these questions will some day necessitate another review.
Acknowledgements I would like to thank "rst of all P. Ball, N. Kivel, L. Magnea, V.A. Smirnov, and in particular V.M. Braun and V.I. Zakharov for their collaboration on topics discussed in this report. In addition I have bene"tted from discussions with L. Lellouch, G. Marchesini and M. Neubert, and many other colleagues. I thank V.M. Braun, G. Buchalla, G. Marchesini, M. Neubert and C.T. Sachrajda for their comments on the manuscript.
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A. Leike/Physics Reports 317 (1999) 143}250
THE PHENOMENOLOGY OF EXTRA NEUTRAL GAUGE BOSONS
A. LEIKE Ludwigs-Maximilians-Universita( t, Sektion Physik, Theresienstr. 37, D-80333 Mu( nchen, Germany
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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The phenomenology of extra neutral gauge bosons A. Leike Ludwigs-Maximilians-Universita( t, Sektion Physik, Theresienstr. 37, D-80333 Mu( nchen, Germany Received November 1998; editor: R. Petronzio Contents 0. Introduction 1. Preliminary considerations 1.1. Parameters describing extra neutral gauge bosons 1.2. Extra neutral gauge bosons in E models 1.3. Extra neutral gauge bosons and contact interactions 1.4. Four fermion interactions and form factors 1.5. Model dependence of Z constraints 1.6. Extracting Z limits from data 2. Z search at e>e\, e\e\ and k>k\ colliders 2.1. Z search in e>e\Pf fM 2.2. Z search in e>e\Pe>e\ and e\e\Pe\e\ 2.3. Z search in e>e\P=>=\ 2.4. Z search in other reactions
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3. Z search at pp and pp colliders 3.1. Born cross section of pp(pp )PZ Pf fM 3.2. Higher-order processes and background 3.3. Observables 3.4. Z constraints 4. Z search in other experiments 4.1. ep collisions 4.2. Atomic parity violation 4.3. Neutrino scattering 4.4. Cosmology 5. Summary and conclusions Acknowledgements Note added Appendix A. Notation Appendix B. Available FORTRAN programs for Z "ts References
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Abstract The phenomenological constraints on extra neutral gauge bosons at present and at future colliders are reviewed. Special attention is paid to the in#uence of radiative corrections, systematic errors, and kinematic cuts on the Z constraints. Simple estimates of the Z constraints from di!erent reactions are derived. They make the physical origin of these constraints transparent. The results existing in the literature are summarized and compared with the estimates. The consequence of model assumptions on the Z constraints is
E-mail address: [email protected] (A. Leike)
0370-1573/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 3 3 - 1
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discussed. The paper starts with an overview of Z parameters and the possible links between them by model assumptions. It continues with a discussion of Z limits and Z measurements in di!erent reactions at e>e\ and k>k\ colliders. It follows an overview of the corresponding limits at proton colliders. Possible Z constraints from other reactions as ep collisions, atomic parity violation, neutrino scattering and cosmology are brie#y mentioned. 1999 Elsevier Science B.V. All rights reserved. PACS: 14.70.Hp
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0. Introduction A neutral gauge boson is a spin-one particle without charge. It transmits forces in gauge theories. One well-known neutral gauge boson is the photon. In 1923, A. H. Compton found the direct experimental con"rmation that the photon is an elementary particle. The photon is connected with the ;(1) gauge symmetry of electrodynamics. Noether's theorem states that this symmetry must correspond to a conserved quantity, the electric charge. The ;(1) gauge symmetry is exact. Therefore, the mass of the photon is zero. It can mediate interactions to in"nite distances. Matter interacts not only through electromagnetic forces. In particular, weak decays as the b-decay violate quantum numbers respected by electrodynamics. These processes indicate that there must be additional fundamental interactions. In 1961, S.L. Glashow, S. Weinberg and A. Salam proposed the uni"ed description of electromagnetic and weak phenomena in a gauge theory based on the gauge group S;(2) ;;(1) . At low energies, we can only observe the ;(1) * 7 symmetry. Therefore, the S;(2) ;;(1) gauge symmetry must be broken at some energy scale * 7 E . Processes with energies E;E feel only the ;(1) symmetry of electrodynamics. At energies E'E , the interaction has the full S;(2) ;;(1) gauge symmetry. In the language of * 7 particles, this means that the weak interaction must be mediated by gauge bosons, which have masses of the order of E . We know experimentally that E "O(100) GeV. The theory of weak interactions explains b decay by the virtual exchange of a heavy positively or negatively charged gauge boson, the =!. These charged gauge bosons are associated to the S;(2) * gauge group. The S;(2) gauge symmetry has also one neutral (diagonal) generator. The particle * associated with this neutral generator must be a neutral gauge boson with a mass of the order of E . Because the electromagnetic symmetry ;(1) is di!erent from the ;(1) symmetry, the mass 7 eigenstates c and Z of the neutral gauge bosons are a linear combination of the two neutral gauge bosons associated to the S;(2) and ;(1) gauge groups. Their properties are predicted in the * 7 electroweak theory. The predictions for the Z boson are con"rmed by the experiments at LEP and SLC with an incredible precision. In addition to electroweak interactions, there exist at least two other fundamental interactions, the strong interaction and gravity. Many physicists believe that all fundamental interactions must have one common root. They do not like the complicated gauge group of the SM. They suppose that strong and electroweak interactions can be described by one simple gauge group G at very high energies E'E . Such %32 theories are called grand uni"ed theories (GUTs) [1]. For energies E;E the gauge group %32 G must be broken [2] to retain the SM gauge symmetry S;(3) ;S;(2) ;;(1) . One can imagine A * 7 this symmetry breaking similar to the breaking of the S;(2) ;;(1) symmetry to ;(1) in * 7 the SM. As was shown by H. Georgi and S.L. Glashow in 1974, the smallest simple gauge group G, which can contain the SM, is G"S;(5). The number n of neutral gauge bosons of a GUT is given by n"rank[G]. We have rank[S;(5)]"4. Therefore, there is no room for additional neutral gauge bosons in the S;(5) GUT. GUTs make predictions which can be tested in experiments. In particular, they predict that the proton must decay. This decay is mediated by the exchange of gauge bosons with a mass O(E ). %32 It is the analogue of the b decay described in the electroweak theory. To be consistent with present experiments on proton decay, we get the condition E '10 GeV. This energy is much larger %32
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than E . It is important that it is smaller than the Planck mass, M "( c/G +1.2;10 GeV. . , At energies above the Planck mass, gravity is expected to become as strong as the other interactions. At energies well below M , as it happens in GUTs, the e!ects of gravity can be . neglected. E is also predicted as the energy where the three running gauge coupling constants of %32 the SM gauge group become equal. The value of E obtained experimentally by this matching %32 condition predicts a proton lifetime, which contradicted the measurement already several years ago. Only the recent precision measurements at LEP and SLC could prove that the three running gauge couplings do not meet in one point if they run as predicted in the S;(5) GUT. Therefore, one must add something else if one wants to describe all SM interactions by one simple gauge group. One popular direction of research is supersymmetry. We are interested here in another solution of the problem, the consideration of larger uni"cation groups. All GUTs with gauge groups larger than S;(5) predict at least one extra neutral gauge boson (Z ). It was shown by H. Fritzsch and P. Minkowski in 1975 that the next interesting gauge group larger than S;(5) is SO(10). The SO(10) theory predicts one extra neutral gauge boson because rank[SO(10)]"5. It is a non-trivial fact that all SM fermions of one generation "t in only one multiplet of SO(10). To complete the multiplet, one new fermion with the quantum numbers of the right-handed neutrino must be added. The SO(10) GUT is not in contradiction with present experiments. GUTs with gauge groups larger than SO(10) predict more than one extra neutral gauge bosons and many new fermions. These new (exotic) fermions must be heavy to make the theories consistent with present experiments. The mass of the Z is not constrained by theory. A priori, it can be anywhere between E and E . As was shown in Refs. [3,4], it has naturally a mass of about one TeV in some supersymmet%32 ric GUTs. Then, a Z can be observed at the next generation of colliders. An observation of a Z would provide information on the GUT group and on its symmetry breaking. It is of special interest for experimental physicists because a Z would serve as a calibration point for future detectors. The study of the Z phenomenology is therefore an important part of the scienti"c program of every present and future collider. As the SM Z boson, the Z is expected to be a very short-lived particle. It can only be observed through its decay products or through indirect interference e!ects. It can be detected either in very high-energy processes or in high precision experiments at lower energies. In the "rst kind of processes, the energy of the colliding particles must be high enough to produce a Z . The decay products of the Z must be then detected above the SM background. Such a background is always present because the SM Z boson or the photon are produced by the same processes, which create a Z . In precision experiments, the experimental errors and the errors due to the theoretical predictions of the observables must be smaller than the expected deviations due to a Z . There are several previous reviews on Z physics. Refs. [5,6] give an overview of Z physics in the E GUT. In Ref. [7] the search for a Z in high precision experiments is reviewed. A recent short review is given in [8]. In the recent years, extensive studies on the sensitivity of future e>e\ [9,193,269], pp(pp ) [10] and e!p [11,243,245] colliders to a Z have been completed. Among the new developments are E the inclusion of radiative corrections, realistic experimental cuts and systematic errors in Z studies and the development of codes allowing direct Z "ts to experimental data; E the discussion of a Z not only in the context of a model but also without model assumptions;
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E the experimental successes during the last years of precision measurements, heavy #avour tagging and highly polarized beams allow the investigation of new observables. In this paper, we review the main results of these new developments. Special attention is paid to the mechanisms leading to Z limits in the di!erent reactions. The resulting approximate formulae make the dependence of Z constraints and Z measurements on experimental conditions and model parameters transparent. The approximate formulae are compared with the existing results in the literature. We divide the paper into "ve sections and several appendices. In the "rst section, we introduce the parameters describing extra neutral gauge bosons. The E GUT is considered as an example of a theory containing extra neutral gauge bosons. Far below the Z resonance, the Z can be described as a special case of four fermion contact interactions. A general formalism of the inclusion of Z e!ects in Standard Model cross sections are form factors. We emphasize that it is necessary to distinguish between model-dependent and model-independent Z analyses. We conclude Section 1 with some general remarks on the data analysis. In Section 2, we list Z constraints at e>e\, e\e\ and k>k\ colliders. The constraints from various reactions are considered in individual sections. Every section is organized in the same pattern. First, the relevant observables are discussed in the Born approximation. It is followed by a discussion of radiative corrections. Finally, di!erent Z constraints are considered. If necessary, di!erent cases of the centre-of-mass energy are distinguished. In Section 3, the Z constraints obtained at hadron colliders are considered. Z constraints from other experiments; electron}proton collisions, atomic parity violation, neutrino scattering and cosmology are mentioned in Section 4. Section 5 contains our summary and conclusions. Two appendices complete this paper. The main notation is collected in Appendix A. Appendix B contains an overview of the available FORTRAN codes suitable for Z "ts in di!erent experiments.
1. Preliminary considerations 1.1. Parameters describing extra neutral gauge bosons Information on extra neutral gauge bosons can be obtained through experimental constraints on or measurements of its parameters. These Z parameters are introduced in this section. After "xing our assumptions and notation, we consider e!ects of gauge boson mixing. We then deal with the Z couplings to SM fermions and with Z decays to SM particles. 1.1.1. Assumptions and limitations Suppose that a Z exists in nature. Then, its parameters depend on many unknown details of the theory. We assume that at most one extra neutral gauge boson is light enough to give the "rst signal in future experiments and that no other signals from a GUT are found at that time. In the case where there are additional signals of new physics, they would give interesting extra information on the parameters of the new theory.
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We assume that the underlying e!ective gauge group at low energies is S;(3) ;S;(2) ;;(1) ;; (1) . (1.1) A * 7 We further assume that the couplings of the Z to fermions are universal for all generations. In models where this is not the case, one has to be careful about the suppression of #avour changing neutral currents. See Ref. [12] for recent constraints on such models and for further references. A GUT containing a Z predicts many new particles as, for example, additional (exotic) fermions, additional charged gauge bosons or additional Higgs bosons. In the simplest case of a SO(10) GUT, the only additional fermion is one right-handed neutrino. The number of exotic fermions rises drastically for larger gauge groups. The couplings of these fermions to gauge bosons are "xed in the GUT but their masses are not constrained by the theory. Under our assumption that the Z gives the only signal of the GUT, the decays of the Z to exotic fermions and Higgs bosons are kinematically suppressed [13]. We neglect the mixing of SM fermions with exotic fermions [14] and possible simultaneous mixing of gauge bosons and fermions [15]. In a SUSY GUT, many interactions involving supersymmetric particles are predicted. We do not consider these e!ects [16]. We do not discuss either the special case of leptophobic Z models [17,18], which were constructed to explain the discrepancy of R at LEP with the SM expectations. @ BESS models [19] also predict extra neutral gauge bosons. The corresponding Z limits derived for hadron [20,21] and lepton [20,22,23] colliders will not be discussed here. 1.1.2. Mixing There are no quantum numbers which forbid a mixing of neutral gauge bosons. However, the ZZ mixing arises naturally in many models. 1.1.2.1. Kinetic mixing. We "rst consider the case where all neutral gauge bosons are massless. Assume diagonal kinetic terms of the gauge "elds. The general, tree level parametrization of the neutral current Lagrangian for the gauge group (1.1) is then [24,25] (1.2) !L "fM c@+g¹D= #g >DB #g >DZ #g Q DB #g Q DZ , f , @ @ @ @ @ ,! where the summation over the fermions f is understood, ¹D is the third component of the SM isospin, >D is the hypercharge and Q D is the charge due to the new ; (1). The particles associated with the ;(1) and ; (1) gauge groups are denoted as B and Z , correspondingly. = is the same 7 @ "eld as de"ned in the SM. The non-zero contributions proportional to g and g can arise during the evolution from GUT energies to the weak scale. The two by two matrix g can be made triangular by a rotation of GH the two abelian gauge bosons around the angle h , c "cos h , s "sin h , ) ) ) ) ) B c !s B @ " ) ) @ . (1.3) Z s c Z @ ) ) @ The angle h describes the kinetic mixing. After mixing (1.3), Lagrangian (1.2) transforms to ) !L "fM c@+g¹D= #g >DB #g >DZ #g Q DZ , f (1.4) ,! @ @ @ @ with g "g c #g s , g "!g s #g c , g "g c #g s ,0 and g "!g s #g c . ) ) ) ) ) ) ) )
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If there is no additional symmetry requiring g "0, the term proportional to g is needed to ensure the renormalizability of the theory [25]. Kinetic mixing plays an important role in some leptophobic models [26] and can communicate SUSY-breaking to the visible sector [27]. Kinetic mixing between the gauge bosons = and B is forbidden in the SM by S;(2) gauge @ @ * invariance. 1.1.2.2. Mass mixing. Gauge symmetry must be broken at low energies to describe massive gauge bosons. In the SM, the entries of the mass matrix
B M !M M 1 5 , M " L " (B, = )M 1+ = 1+ + 2 !M M M 5 5 are related to the vacuum expectation value v of the Higgs "eld
U"
>
,
1 2"gv/(2, M "gv/2, M "g v/2 . 5
(1.5)
(1.6)
M is diagonalized by a rotation of the symmetry eigenstates around the Weinberg angle 1+ h , c "cos h , s "sin h leading to the well-known mass eigenstates of the photon and the 5 5 5 5 5 Z boson,
c
c
s B 5 5 . Z !s c = 5 5 The masses of the mass eigenstates are "
(1.7)
M"M #M , M"0 . (1.8) 8 5 A The mass of the photon is zero due to the exact ;(1) gauge symmetry at low-energies re#ecting charge conservation. This symmetry protects the photon from further mixing. The Weinberg angle is related to the entries of the mass matrix (1.5) tan 2h "2M M /(M !M) . 5 5 5 We get the following relations between the Weinberg angle and the mass values:
(1.9)
t "s /c "(M!M)/(M!M)"M/M and M M "s c (M!M) 5 5 5 A 8 5 5 5 5 8 A "s c M . (1.10) 5 5 8 In a theory with the gauge group (1.1), the mass matrix of the Z and Z receives non-diagonal entries dM, which are again related to the vacuum expectation values of the Higgs "elds,
Z M dM 1 8 , M " . L " (Z, Z )M 88Y Z 88Y + 2 dM M 8Y
(1.11)
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We assume that the vacuum expectation values of the Higgs "elds are real. The mass matrix (1.11) is diagonalized by a rotation of the "elds Z and Z around the mixing angle h , c "cos h , + + + s "sin h leading to the mass eigenstates Z and Z , + + Z c s Z " + + . (1.12) Z !s c Z + + The masses M and M of the mass eigenstates Z and Z are (1.13) M "[M#M $((M!M )#4(dM)] . 8Y 8 8Y 8 It follows that
M (M (M , and hence o "M /Mc 'M /Mc "o "1 . (1.14) 8
5 5 5 8 5 We have M "M and M "M for h "0. LEP 1 and SLC performed precision measurements 8 8Y + of the mass eigenstate Z . Similar to the SM, the mixing angle h is related to the entries of the mass matrix (1.11), + tan 2h "2dM/(M!M ) . (1.15) + 8 8Y We get the following relations between h and the mass values: + t "s /c "(M!M)/(M!M) (1.16) + + + 8 8 and dM"s c (M!M) . (1.17) + + For "xed M and M , Eq. (1.16) relates h and M independently of the Higgs sector. This 8 + constraint on h is called the mass constraint [5,6]. It predicts h &1/M for large M . For a "xed + + Higgs sector, dM is also "xed leading to the constraint on h given in Eq. (1.17). It is called the + Higgs constraint [5,6]. For large M , it is stronger than the mass constraint predicting the asymptotic behaviour h &1/M [5,6]. + 1.1.2.3. Alternative parametrization of the mixing. In left}right models [28] with the gauge group S;(2) ;S;(2) ;;(1) , (1.18) * 0 \* the mass eigenstates c, Z and Z are obtained by a one-step mixing through an orthogonal 3;3 matrix [29]
Z c c s c c #s s s c s !s c = 5 + 5 + *0 + *0 5 + *0 + *0 * , Z " !c s !s s c #c s !s s s !c c = 5+ 5 + *0 + *0 5 + *0 + *0 0 c !s c c c s > 5 5 *0 5 *0 \*
(1.19)
depending on the three mixing angles h , h and h with s "sin h , c "cos h . In Eq. 5 + *0 *0 *0 *0 *0 (1.19), we adapted the notation to our de"nitions. Note that the "elds = , = , > commute in * 0 \* the gauge group (1.18).
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In previous sections, we described the mixing of the neutral gauge bosons associated with the gauge group (1.1) by two mixing angles h and h . To apply this procedure to left}right models, 5 + one has to "rst break the gauge group (1.18) to S;(2) ;;(1) ;; (1). This breaking will gene* 7 rate non-diagonal terms in the associated gauge "elds through fermion loops [26] if (QDQ D)O0. These non-diagonal terms can be described by a non-zero g in D Eq. (1.4). In the case g "g , no non-diagonal terms exist (g "0) and the mass matrix can be 0 * diagonalized by the two angles h and h [28] in both mixing procedures. 5 + In this paper, we prefer the two-step mixing procedure. It has advantages in the model independent approach where one starts with the gauge group (1.1) and the corresponding symmetry eigenstates B and Z . 1.1.3. Couplings to SM fermions The couplings of the symmetry eigenstate Z to fermions are "xed in a GUT. Experiments are sensitive to the couplings of the mass eigenstates Z and Z . 1.1.3.1. Couplings of the symmetry eigenstates. Lagrangian (1.4) can be written in the form !L "eA J@#g Z J@ #g Z J@ . ,! @ A @ 8 @ 8Y It contains the currents
(1.20)
J@" fM c@v (0)f, J@ " fM c@[v !c a ] f, J@ " fM c@[v !c a ] f . (1.21) A D 8 D D 8Y D D D D D To "x the notation, we give here the coupling constants of the photon and the SM Z boson to the SM fermion f, v (0)"QD, a (0)"0 , g ,e"(4pa"gs , 5 D D (1.22) g "e/2s c "((2G M) , v "¹D!2QDs , a "¹D I 8 D 5 D 5 5 with ¹C "!, QC"!1. Di!erently from the SM, it is useful to de"ne the electric charge as QD,¹D#(>D in GUTs. The comparison of the two Lagrangians (1.4) and (1.20) then gives g "( e/c . 5 The couplings g , v and a depend on the particular Z model. In the sequential standard model D D (SSM), all couplings of the Z to SM fermions are equal to those of the SM Z boson, g "g , v "v , a "a . Although it is hard to obtain the SSM in GUTs, it is a popular D D D D benchmark model, which is limited in di!erent experiments. Starting from Lagrangian (1.4) and identifying g "g , one obtains v and a as a function of D D the charge Q D and the hypercharge >D [25], v "Q D*#Q D0#(g /g )(>D*#>D0)"Q D*#Q D0#(g /g )((!¹D#2QD) , D (1.23) a "Q D*!Q D0#(g /g )(>D*!>D0)"Q D*!Q D0#(g /g )((!¹D) . D We de"ne the couplings to left- and right-handed fermions as ¸ "(v #a ), D D D
R "(v !a ) . D D D
(1.24)
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A possible charge Q JA*"!Q J0 is not considered here because we count the right-handed neutrino as an exotic fermion, which is assumed to be heavy. Therefore, v !a does not enter our J J analysis. Hence, only seven of the eight couplings v and a are independent. D D For extra neutral gauge bosons arising in the gauge group (1.1), we have [Q D,¹D]"0 . Then, the two relations Q S*"Q B*,Q O*
(1.25)
and Q C*"Q J*,Q J*
(1.26)
must be ful"lled to preserve S;(2) gauge invariance. Relations similar to Eq. (1.26) are ful"lled for * the hypercharges >D in the SM. Therefore, relations (1.26) remain valid if the charges Q D are replaced by the corresponding Z couplings to fermions. It follows that v and a can be D D parametrized by the following xve independent couplings: ¸ "¸ ,¸ , ¸ "¸ ,¸ , R , R , R . S B O C J J S B C
(1.27)
1.1.3.2. Couplings of the mass eigenstates. From Eqs. (1.23) and (1.12), we deduce the couplings of the mass eigenstates Z and Z to fermions a (1)"c a #(g /g )s a ,a (1!y ) , D + D + D D D v (1)"c v #(g /g )s v ,a (1)[1!4"QD"s (1!x )] , D + D + D D 5 D a (2)"c a #(g /g )s a , v (2)"c v #(g /g )s v . (1.28) D + D + D D + D + D The functions x and y give the ZZ mixing in terms of form factors, D D v #t v g /g v g a v /a !v /a + D ! D +h D D D D D, x "(1!v /a )\ D D D D + g a 1!v /a a #t a g /g a D + D D D D D y "!s g a /g a #(1!c )+!h g a /g a . (1.29) D + D D + + D D They are convenient for a simultaneous description of ZZ mixing and electroweak corrections at the Z peak [30]. We neglect terms of higher order in the mixing angle in the last approximation in Eq. (1.29). For small mixings, x and y are small being proportional to h . D D +
1.1.4. Decay width We consider here only Z decays to SM particles. We refer, for example, to Ref. [6] for the decay widths to other particles present in a GUT and to [31] for Z decays to bosons. 1.1.4.1. Born approximation. The partial decay width of the Z , n"1, 2 to a fermion pair f fM is [32] L C(Z Pf fM ),CD"N kM (g/12p)+[v (n)#a (n)](1#2m/M)!6a (n)m/M, , L L D L L D D D L D D L (1.30) N is the colour coe$cient, i.e. N "1(3) for f"l(q). k is the phase space factor due to the massive D D "nal fermions, k"(1!4m/M. D L
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A Z originating from the gauge group ; (1) given in Eq. (1.1) or the Z arising after kinetic mixing (1.3) have no couplings to = bosons. However, in the case of a non-zero mass mixing, the Z component contained in the mass eigenstates Z and Z interacts with = bosons, "g c ,g "g s , where g "ec /s is the SM coupling between ='s and g 558 + 558 558 + 558 5 5 558 the SM Z boson. The structure of the interaction is essentially the same as in the SM. The partial decay width of the Z to a =>=\ pair is then [31] C(Z P=>=\),C5 "(g M /192p)s (M /M )(1!4M /M) 558 + 8 5 ;(1#20M /M#12M /M) . (1.31) 5 5 In GUTs, C(Z P=>=\) is kept in a reasonable range for M PR because the potentially dangerous factor M is compensated by small mixing angles s +1/M, due to the Higgs + constraint (1.17). The decays Z Pf fM <, <"Z, = are of higher order. However, they are logarithmically en hanced by soft and collinear radiation. In the limit M/M;1, the decay width becomes [33] 8 C(Z Pf fM Z),CDD8 "(M gg/192p)[R(2)R(1)#¸(2)¸(1)] D D D D [ln(M/M)#3 ln(M/M)#5!p/3#O(M/M)] . (1.32) 8 8 8 The decay width C(Z Pf fM =),CDD5 can be obtained from Eq. (1.32) by an appropriate replacement of the couplings and masses. The decays Z Pf fM c and Z P=>=\c are considered as radiative corrections to the decays Z Pf fM and Z P=>=\. All other decays of the Z to SM particles are expected to be small. Decays of the Z to non-standard particles depend on additional model parameters. We assume that they are kinematically suppressed. The total decay width C of the Z is de"ned as the sum over all partial decay widths. It is L L convenient to combine M and C in the complex mass, L L m"M!iM C . (1.33) L L L L 1.1.4.2. Higher-order ewects. The Z width can be neglected in experiments with typical energy transfers much lower than M as far as C ;M . C can be measured in pp(pp ) collisions for s'2(3)M or in e>e\ collisions for s5M. To reach the experimental accuracy, the inclusion of higher-order e!ects is necessary in these experiments. Radiative corrections to C are absolutely necessary for precision measurements at the Z peak in e>e\ or k>k\ collisions. Of course, tree level decays to non-standard particles have to be included if they are not suppressed. In this case, the corresponding parameters must be measured in independent experiments. We concentrate here on known higher-order e!ects of the decays to SM particles. Radiative corrections: Radiative corrections to CD can be deduced from the results known for SM Z decays [34], CD"CDR R , n"1, 2 , L L /#" /!" R "1#(a/p)(QD) , /#"
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R "0 for f"l , /!" R "1#a (M)/p#1.405a(M)/p!12.8a(M)/p!((QD)/4)aa (M)/p for f"q . /!" Q Q Q Q (1.34) For C , the top mass should also be taken into account in the radiative corrections [35]. See Ref. [36] for O(aa ) corrections in that case. Q The SM weak corrections to SM Z decays are calculated in Ref. [37] in terms of form factors. See also Section 2.1.2.4. The concept can be generalized to SM weak corrections of Z decays to SM fermions. The full one-loop corrections in the GUT depend on the details of the theory. They can only be calculated if all relevant new parameters are known from independent experiments. The main corrections to C5 are QED corrections from radiation o! the "nal state and the Coulomb singularity. See Section 2.3.2 or [38] for further details and references. Radiative corrections to CDD4 are expected to be a small correction to a small quantity. The known SM corrections to four fermion "nal states are summarized in Ref. [39]. Mass mixing: Mass mixing changes the partial decay widths of the Z because it induces changes in the couplings. The e!ect is of the order h and is therefore small due to the present experimental + limits on the mixing. It is interesting for precision measurements in e>e\ or k>k\ collisions at s+M. It must be taken into account together with weak corrections of the full GUT. Energy-dependent width ewects: At lowest order, a particle has no width. It is obvious that a width is needed to describe a resonance. The simplest approximation is to use the constant width in the propagator, which is calculated in the previous sections. The next step of precision is to take into account that the width is a function of the energy. In general, the inclusion of a "nite width violates gauge invariance because it partially takes into account e!ects, which are of higher order in perturbation theory. It is shown in [40] that the gauge violating terms can be enhanced by large kinematic factors&s/m in some processes with four C fermion "nal states, i.e. in e>e\Pe>=\l . The problem can be cured by the inclusion of C additional higher-order contributions, which restore the gauge invariance, see [40,41] for further details. The e!ect is under control in fermion pair production where one should take the s-dependent width [42]. It is C (s)+(s/M) C (M) if the vector boson can decay only into light particles. The s-dependence of C leads to a shift *E "(C /M )M of the Z peak position [42]. For s'M the width should be taken s-dependent too because it in#uences the radiative tail as explained in Section 2.1.2.1. 1.1.5. Summary of Z parameters Under the assumptions of Section 1.1.1, we are left with the following Z parameters, M ,C h + g v (2), a (2) D D
the mass and width of the mass eigenstate Z , the ZZ mixing angle , the coupling strength , the vector and axial vector couplings to fermions.
(1.35)
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One can choose [43] the following quantities to parametrize the "ve independent Z couplings (1.27) describing v (2) and a (2): D D s s g g "a(2) , e ,[¸ (2)!R (2)] C 4pa M !s J C 4pa M !s 8Y 8Y PC ,(¸ (2)#R (2))/(¸ (2)!R (2))"v (2)/a (2) , 4 J C J C C C PO ,¸ (2)/(¸ (2)!R (2))"(v (2)#a (2))/2a (2) , * O J C O O C PG ,R (2)/¸ (2)"(v (2)!a (2))/(v (2)#a (2)), i"u, d . (1.36) 0 G O G G G G In collisions of unpolarized protons, one is insensitive to the relative sign of the Z couplings. Then, an alternative set of parameters is convenient [44], cJ ,¸(2)/(¸(2)#R(2))"(v #a )/2(v #a ) , * J J C J J C C cO ,¸(2)/(¸(2)#R(2))"(v #a )/(2(v #a )) , * O J C O O C C ;I ,R(2)/¸(2)"(v !a )/(v #a ) , S O S S O O DI ,R(2)/¸(2)"(v !a )/(v #a ) . B O B B O O e can be added to Eq. (1.37) as the "fth parameter. Both parameter sets are related, see Ref. [45].
(1.37)
1.2. Extra neutral gauge bosons in E models In an E GUT [6,46], the "ve independent charges Q D are constrained [45] in addition to (1.26), (1.38) Q CA*"Q SA*"Q O*, Q J*"Q B*A . Therefore, the charges Q D of a general Z in an E GUT can be parametrized by two independent parameters. However, the three conditions (1.38) lead to only two relations between the Z couplings [45] as far as the ratio g /g in Eq. (1.23) is unknown, 2¸ #R #R "0, ¸ !R !¸ #R "0 . (1.39) O S C O B J C The experimental check of conditions (1.39) allows to determine whether a Z can belong to the E breaking scheme or not. There are many breakings possible in the E group. See [47] for an extensive discussion. Let us not consider the general E case but restrict ourselves to the gauge breaking scheme E PSO(10);;(1) PS;(5);;(1) ;;(1) , (1.40) R Q R where the linear combination Z (b)"s cos b#t sin b
(1.41)
is assumed to be light. The special case g"Z (!arctan(5/3 ) is often considered. b is the free parameter in the breaking scheme (1.40). For the s arising in the breaking of the SO(10)
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Table 1.1 The vector and axial vector couplings of the Z to SM fermions in the E and LR models f
E
a D
l
v D
3 @# @ ( ( @# @ ( ( ! @# @ ( ( @# @ ( (
e u d
¸R 3 @# @ ( ( 2 @ ( 0
!2( @
a D
v D
? ? !? ?
? !? ? ! #? ? ! !? ?
[28,32,48,49] to S;(5) [50] and the t arising in the breaking of the E to SO(10), we get !Q O*"Q J*"( , Q O*"Q J*"( , Q Q R R
g "g (s , 5
g "0 .
(1.42)
Eq. (1.23) now de"nes the couplings of the Z to SM fermions as given in Table 1.1. One has to be careful with the di!erent notations for b in the literature. For example, the model parameter h in Ref. [6] is related to our b as b"h#p/2. Consider the gauge breaking scheme [51,52] SO(10)PS;(3) ;S;(2) ;S;(2) ;;(1) A * 0 \* as a second example. Now the Z couples to the current
(1.43)
J@ "( (aJ@ !(1/2a)J@ ), a,((c /s )g /g!1 , (1.44) *0 0 \* 5 5 0 * where g and g are the couplings to the left- and right-handed gauge groups. J is the cur* 0 0 rent associated with the S;(2) group and B and ¸ are baryon and lepton numbers, 0 B!¸"2(Q!¹ !¹ ). a is constrained to lie in the range (2/34a4(c /s !1. Within 5 5 * 0 our conventions, we have
2 a#1 g 1 3a!2 , g "g s , " . Q D"Q D 5 g *0 Q 5 a a (15
(1.45)
The case a"(2/3 is identical to Z"s. Again, Eq. (1.23) gives all Z couplings to SM fermions shown in Table 1.1. The decay width of a Z to fermion pairs is small in E theories. Typical values for CD/M are D 8Y between 0.5% and 2% if only decays to SM fermions are kinematically allowed. If the decays to all exotic fermions and Higgs bosons are possible, C becomes roughly three times larger [31,51]. The entries of the mass matrix (1.11) are completely de"ned in a "xed model. For example, in a model with a gauge symmetry breaking by two Higgs doublets and one Higgs singlet [31],
> U " , U " , U " ,
\
1 2"gv /(2, i"1,2,3, v"v#v , G G
(1.46)
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one gets [31] 1 [Q ( )v ] . (1.47) M "4Ms G G 8Y 8 5 v G Q and Q are the ; (1) charges of the two Higgs doublets. The entry M of the mass matrix (1.11) is 8 known from the SM. Eq. (1.47) gives an example for the Higgs constraint (1.17). The values of Q ( ) and Q ( ) depend on the particular symmetry breaking [53]. The Higgs constraint can be combined with the formula for C5. It gives C5+CC in left-right models [54]. dM"2Ms 85
Q ( )v!Q ( )v , v
1.3. Extra neutral gauge bosons and contact interactions Far below the resonance, Z e!ects can be described by four fermion contact interactions. The interaction of a Z with the massless fermions f and F is then given by the amplitude g M(Z )" (¸ ¸ u c u u c@u #R R u c u u c@u D $ D * @ D * $ * $ * D $ D 0 @ D 0 $ 0 $ 0 M 8YD $ #R ¸ u c u u c@u #¸ R u c u u c@u ) . D $ D 0 @ D 0 $ * $ * D $ D * @ D * $ 0 $ 0 This can be compared with four fermion contact interactions [55]
(1.48)
gD$ gD$ ** u c u u c@u # 00 u c u u c@u M(contact)"4p D * @ D * $ * $ * (KD$) (KD$ ) D 0 @ D 0 $ 0 $ 0 ** 00 D $ gD$ gD$ . (1.49) # 0* u c u u c@u # *0 u c u u c@u (KD$ ) D 0 @ D 0 $ * $ * (KD$ ) D * @ D * $ 0 $ 0 *0 0* KD$, i, j"¸, R are the scales of the new physics and gD$"$1. GH GH Constraints on four fermion contact interactions can always be interpreted as constraints on extra neutral gauge bosons, g¸ ¸ /4pM ,¸,¸,/s"gKL /(KKL ), m, n"f, F . K L 8Y K L ** ** We use the de"nition g (s . ¸,,¸ D D (4p M 8Y
(1.50)
(1.51)
Relations similar to Eq. (1.50) can be written for the other helicity combinations. Note that Z interactions far below the Z peak can only constrain the ratios ¸,, R, and not the Z couplings D D and the Z mass separately. Interactions of extra neutral gauge bosons are not as general as four fermion contact interactions. Lagrangian (1.48) describes the interaction between fermions of #avours f and F by the four couplings ¸,, ¸,, R, and R,. The same process is described in contact interactions by the 12 D $ D $ parameters KKL, i, j"¸, R, mn"f f, fF, FF. GH
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If a future helicity conserving experiment shows small deviations from the SM predictions, one can always parametrize the deviation in terms of contact interactions. If the new interaction is due to a Z , the parameters of the contact interactions ful"ll the relations gLL "gLL "1, ** 00
gKL "gKL , (KKL )"(KKL )"((KKL )(KKL ), mn"+, fF, FF , *0 0* *0 0* ** 00
(1.52) (KD$)"((KDD)(K$$), i"¸, R . GG GG GG The normalized Z couplings can then be calculated according to Eq. (1.50). Under the assumption of the gauge group (1.1), the Z couplings must satisfy the additional relations (1.27). Of course, Z e!ects near the Z resonance cannot be described by contact interactions. 1.4. Four fermion interactions and form factors In many experiments, a Z can be detected through four fermion interactions. In addition to the Z exchange, we always have the SM contributions where the Z is replaced by the photon or the Z boson. The contributions due to the exchange of extra neutral gauge bosons can be absorbed into the couplings of the SM Z boson [56]. The following discussion is general for four fermion interactions. It can be directly applied to Bhabha and M+ller scattering, pp, pp or ep scattering. Consider the four fermion interaction e>e\Pf fM ( fOe) as an example. The amplitude for this process is g L v (e)c [v (n)!c a (n)]u(e) ) u ( f )c@[v (n)!c a (n)]v( f ) . (1.53) M" @ C C D D s!m L L In the SM, the summation runs over the photon and the Z boson (n"0,1). In a theory including a Z , the extra amplitude with the Z exchange, M "M(n"2), has to be added. It is important # that M has the same structure as the SM amplitude. Then one can formally include the # contribution of M in the couplings of the SM Z boson leaving the couplings of the photon # unchanged. Consider the amplitude with Z and Z exchange only g L v (e)c [v (n)!c a (n)]u(e)u ( f )c@[v (n)!c a (n)]v( f ) M" @ C C D D s!m L L ,[g/(s!m)]v (e)c u(e)u ( f )c@v( f )v (1)v (1)(1#e ) @ C D TT !v (e)c u(e)u ( f )c@c v( f )v (1)a (1)(1#e ) @ C D T? !v (e)c c u(e)u ( f )c@v( f )a (1)v (1)(1#e ) @ C D ?T #v (e)c c u(e)u ( f )c@c v( f )a (1)a (1)(1#e ) , @ C D ?? with x (2)y (2) g(s!m) C D , x,y"a,v, s " . e "s VW 8 8Y x (1)y (1) 8Y 8 g(s!m) C D
(1.54)
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The coe$cients e contain all information of the amplitude M . Various additional amplitudes VW # M arising, for example, from weak corrections or ZZ mixing, can be written in the form (1.54) if # the quantities e are speci"ed [57]. VW Following the tradition of electroweak corrections, we want to parametrize the contributions e by (complex) form factors o ,i ,i ,i , which are introduced by replacements of the couplings VW CD C D CD v (1)v (1)Pa (1)a (1)[1!4"QC"s i !4"QD"s i #16"QCQD"s i ] , C D C D 5 C 5 D 5 CD v (1)Pa (1)[1!4"QC"s i ], v (1)Pa (1)[1!4"QD"s i ] , C C 5 C D D 5 D a (1), a (1)Punchanged, g"Pgo . (1.55) C D CD Comparing with Eq. (1.54), the form factors can be expressed through e , VW o "1#e , CD ?? 1 i " [1#[e v (1)!e a (1)]/[v (1)!a (1)]] ?T D ?? D D D D o CD e v (1)v (1)#e a (1)a (1)!e a (1)v (1)!e v (1)a (1) 1 ?? C D ?T C D T? C D 1# TT C D . (1.56) i " CD o [v (1)!a (1)][v (1)!a (1)] C C D D CD In particular, the additional Z amplitude can be included in the Z couplings specifying e as given VW in Eq. (1.54)
o "1#s (a (2)a (2))/(a (1)a (1)) , CD 8Y 8 C D C D a (2)[v (2)!a (2)] 1 C D D 1#s , i " 8Y 8 a (1)[v (1)!a (1)] D o C D D CD a (2)[v (2)!a (2)][v (2)!a (2)] 1 C D D C C 1#s . (1.57) i " 8Y 8 a (1)[v (1)!a (1)][v (1)!a (1)] CD o C D D C C CD The result agrees with the formulae given in Ref. [58]. As mentioned there, this method of the inclusion of the Z contributions has the advantage that it can be easily implemented in computer codes designed for SM calculations. The form factors (1.56) and (1.57) work equally well for any four fermion process. They include the Z contribution without any approximation. Of course, they are s-dependent, in general, not small and even resonating near the Z peak. Consider the case where two additional amplitudes are added to M , 1+ M"M #M #M . (1.58) 1+ Suppose that both additional amplitudes are parametrized in the way described above, i.e. assume that the form factors oG , iG , iG , iG , i"1, 2 are known. Then, the combined form factors can be CD C D CD calculated by taking into account eR "e #e , VW VW VW io #io !1 i o #i o !1 D CD CD CD oR "o #o !1, iR" D CD , iR " CD CD . (1.59) CD CD CD D CD o #o !1 o #o !1 CD CD CD CD
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The summation rules (1.59) are exact. In many applications, the form factors are not very di!erent from one. In this case, the approximate summation rules (1.60) oR "o o , iR "ii, iR "i i CD CD CD D D D CD CD CD are often used. In their derivation, contributions proportional to e e are neglected. VW VYWY Case (1.58) arises, for example, in the simultaneous description of ZZ mixing and electroweak corrections. The functions eK describing the ZZ mixing are VW eK "s (c #(g /g )s x /x )(c #(g /g )s y /y )!1, x, y"a, v, 8Y 8 + C C + 8Y 8 + D D VW 8 8 + (1.61) s "(s!m)/(s!m) . 8 8 8 Again, the resulting form factors can be calculated [57] using Eq. (1.56), oK "s (c #(g /g )s a /a )(c #(g /g )s a /a ) , 8Y 8 + C C + 8Y 8 + D D CD 8 8 + iK"[1#(s g v !a )/(c g v !a )]/[1#(s g a )/(c g a )] , D + 8Y D D + 8 D D + 8Y D + 8 D iK "iKiK . (1.62) CD C D The form factors oK , iK and iK are related to the functions x and y introduced in Eq. (1.28), CD D CD D D oK "(1!y )(1!y ), iK"1!x . (1.63) CD C D D D 1.5. Model dependence of Z constraints Future experiments will either be consistent with the SM or show deviations from the SM predictions. In the "rst case, the data can be used to constrain extensions of the SM, for example, contact interactions or theories predicting extra neutral gauge bosons. In the case of a deviation, one can try to interpret it in terms of Z parameters. This procedure could either fail or favour some Z models compared to others. The analysis can be done with or without assumptions on the Z model. We call these procedures model-dependent and model-independent analyses. Of course, there are several steps from the model-independent Z analysis to the modeldependent Z analysis. As far as the model assumptions are consistent with the experimental data, a model-dependent analysis is justi"ed and welcome to learn more details about the underlying theory. For example, helicity conserving processes can be parametrized by four fermion contact interactions. Only little can be learned about the origin of the new interaction in this case. If there are deviations from the SM and conditions (1.52) and (1.27) are ful"lled, the new interaction is consistent with a Z coming from the gauge group (1.1). This theoretical assumption increases our knowledge about the origin of the new interaction. If the couplings ful"l relations (1.39), the new interaction is consistent with a Z coming from the E group. The subsequent assumption that the new interaction is due to a Z from an E breaking, increases the model dependence but allows to probe further details of the assumed model. The experimental veri"cation that !3¸ "¸ is also O J ful"lled supports the hypothesis that the interaction is due to a Z from a SO(10) breaking. Finally, the new interaction could be tested for compatibility with a de"nite Z , e.g. Z "s. This hypothesis contains most model assumptions but allows more detailed tests of the theory and the best "ts to the remaining free parameters.
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Table 1.2 Advantages (#) and disadvantages (!) of a model-dependent and model-independent Z analysis Model-dependent Z analysis
Model-independent Z analysis
! The constraints are a mixture of experimental results and theoretical assumptions. ! A separate data analysis is needed for every new Z model. # Single Z parameters can be constrained.
# The constraints result from data only. They are not biased by theoretical assumptions. # Z limits for a new Z model can be deduced without a new data analysis. ! Only combinations of Z parameters can be constrained. ! Z limits from di!erent experiments cannot always be compared.
# Z limits from di!erent experiments can always be compared.
Examples of model-dependent Z constraints are the lower bounds on M or the regions of 8Y h allowed for certain E models quoted in the Particle Data Book. + For a model-independent analysis, one has to pay a price. Usually, one cannot constrain single Z parameters but only certain combinations of them. Often, only a limited set of observables is useful for the model-independent analysis. Examples of model-independent Z constraints are the allowed regions of v+"v h and a+"a h from LEP 1 data [59}61], the constraints on v,+v /M D D + D D + J J 8Y and a,+a /M from LEP2 data [62] or the constraints on pJ "p(pM pPZ ) ) BrJ from Tevatron J J 8Y 2 data [63]. Model-independent Z constraints can always be converted into model-dependent Z constraints specifying the Z model. The constraints on model parameters obtained in such a two-step procedure are, in general, weaker than the constraints that would be obtained by a direct model-dependent "t to the same data. Model-dependent and model-independent Z analyses are complementary. Both analyses have advantages and disadvantages, which are summarized in Table 1.2. Throughout this paper, we explicitly mention whether a constraint is obtained in a model-dependent or in a model-independent analysis. 1.6. Extracting Z limits from data In any analysis, one selects observables O from the data and compares them with the predictions G O (SM#Z ) in a theory including a Z . This allows to exclude or to con"rm Z models at a given G con"dence level. The procedure is di!erent in experiments with indirect and with direct Z limits. 1.6.1. Indirect Z limits In experiments with indirect Z signals, for example in e>e\Pk>k\ below the Z resonance, the SM already predicts a large number of events of the given signature. A Z gives a signal in the observable O if it produces a deviation *8YO from the SM prediction O (SM), which is larger than G G G the experimental error *O . Here and in the following, *O stands for the error of an asymmetry, the G G error of a ratio of cross sections or for the relative error of a cross section. Because of the large
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number of events, these errors can be assumed to be Gaussian distributed. Neglecting correlations, we de"ne
O !O (SM#Z ) G G s" . (1.64) *O G G For s's #s , the considered model is excluded at a certain con"dence level depending on s .
In a real experiment, all possible observables would be measured and contribute to the "nal result. In a theoretical analysis, the additional information due to measurements of two observables O and O related by theory (i.e. O (SM#Z )"O (SM#Z )) could be taken into account by the inclusion of one of these observables but with a smaller e!ective error *OR. This error can be estimated as
O !O (SM#Z ) O !O (SM#Z ) G G " , s" *O *OR G G 1 1 " . (1.65) (*O ) (*OR) G G The generalization to more observables is straightforward. Experimental errors consist of statistical and systematic contributions. We assume in the estimates of the following sections that the combined error is the quadratic sum: *O"((* O)#(*O)"* O(1#r .
(1.66)
We de"ne the ratio r"*O/* O. Optimal observables [64,65] can be constructed to get the maximum sensitivity to Z parameters. To measure (small) Z parameters j(Z ) in indirect Z analyses, one can look for deviations in di!erential cross sections, ( )#d(*8Yp( ))"dp ( )#j(Z ) dp( ) . (1.67) 1+ 1+ Examples for j(Z ) are h or (s/M . The Z parameter is measured by an integration over the + phase space d with a weight function f ( ) [64], dp( )"dp
f (j(Z ))" f ( ) dp( ) .
(1.68)
The weight function can be chosen in such a way that the sensitivity to j(Z ) becomes maximum. For one parameter, one gets [64], f ( )"*8Yp( )/p ( ). The generalization to several parameters 1+ is given in Ref. [65]. 1.6.2. Direct Z limits Direct Z production is possible in e>e\ or k>k\ collisions for s+M or in hadron collisions for s'M. In direct production in e>e\ or k>k\ collisions, many Z events are expected at future colliders. It can be assumed that the events are Gaussian distributed allowing a s analysis (1.64). In contrast to indirect Z limits, the number of Z events at the Z peak is much larger than the SM background allowing precision measurements.
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In direct Z production at hadron colliders, the number of SM background events is expected to be very small or zero. A few Z events serve as a signal. Then, one cannot assume that these events are Gaussian distributed. Constraints on Z models at a given con"dence level are obtained for all models, which predict the same number of Z events. For example, in the case where the signal is Poisson distributed, the SM background is zero and zero events are observed, all theories predicting N "3 events are excluded at 95% con"dence [66]. For a non-zero background, 8Y N depends on it [66]. 8Y 2. Z search at e>e\, e\e\ and l>l\ colliders Lepton colliders have the advantage that di!erent observables can be detected above a small background. As has been shown by the LEP and SLC experiments, di!erent fermions as e,k,q,c,b can be tagged in the "nal state. The polarization of q s and likely of top quarks can be measured. Highly polarized electron beams are available. One can also hope for a reasonable positron polarization. At muon colliders, one expects some polarization of both beams [67]. e>e\ and k>k\ collisions yield several interesting reactions which can probe di!erent properties of extra Z bosons. Fermion pair production allows for a measurement of a large number of di!erent observables. It is assumed that the "nal fermions are not electrons or electron neutrinos. All couplings of the Z to charged SM fermions can be constrained separately. This is a unique property of this reaction. Bhabha and M~ller scattering have much larger event rates than fermion pair production. In addition, M+ller scattering could pro"t from two highly polarized electron beams. Of course, these reactions are sensitive to gauge boson couplings to electrons only. The sensitivity to these couplings competes with fermion pair production. W pair production is very sensitive to ZZ mixing. This sensitivity is enhanced for large energies because a non-zero ZZ mixing destroys the gauge cancellation between the di!erent amplitudes present in the SM. The sensitivity to other Z parameters cannot compete with fermion pair production. All other reactions in e>e\ or e\e\ collisions cannot add useful information on extra neutral gauge bosons. Although we will explicitly mention mostly e>e\ and e\e\ collisions, all results presented in this section are also applicable to k>k\ or k\k\ collisions. One important di!erence arises for measurements on and beyond the Z peak. As known from Z physics, a precise measurement of the mass and the width of the resonance relies crucially on the accurate monitoring of the beam energy. Here, a muon collider would have clear advantages compared to an electron collider [67] because the latter su!ers from a large beam energy spread. 2.1. Z search in e>e\Pf fM The sensitivity of fermion pair production to di!erent Z parameters depends crucially on the ratio of the centre-of-mass energy (s to the masses M and M of the gauge bosons Z and Z . We
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distinguish four di!erent cases, Case 1:
s+M . Case 2: MOs(M . Case 3: s+M . Case 4: s'M , (2.1) where s+M means (M !C )(s((M #C ). L L L L L Case 1 is very important in setting constraints on the ZZ mixing angle. The Z propagator, i.e. the sensitivity to M is usually suppressed by a factor C /M . Case 2 would give the "rst signals of a Z in e>e\ collisions. The SM parameters are already precisely known from measurements at the Z peak leading to accurate predictions for observables at higher energies. Information about a Z is obtained from the di!erences to these predictions. If there is agreement with the SM predictions, lower bounds on the Z mass can be set for a "xed model. The sensitivity to the ZZ mixing angle is suppressed by a factor C /M compared to measurements at the Z peak. Case 3 is certainly the best possibility to get precise and detailed information about a Z . The corresponding measurements have much in common with the LEP and SLC experiments at the Z peak. Case 4 is interesting because it can constrain a Z with couplings to SM fermions much weaker than predicted in the usual GUTs. Such a Z could escape detection in experiments below its resonance. This section is organized as follows. After giving the relevant observables in the Born approximation, we discuss the di!erent radiative corrections. A discussion of constraints on Z parameters in the four cases (2.1) follows in di!erent subsections. Model dependent and model independent constraints are distinguished. Every subsection on Z constraints is organized by the same pattern. For every constrained Z parameter, the physical origin of the constraint is explained "rst and a simple estimate is given. The estimate is then confronted with Z constraints obtained from present experiments and with constraints obtained in theoretical analyses for future colliders. 2.1.1. Born approximation 2.1.1.1. Amplitude. The amplitude for e>e\P(c, Z, Z ,2)Pf fM , fOe is g L v (e)c [v (n)!c a (n)]u(e) u ( f )c@[v (n)!c a (n)]v( f ) . (2.2) M" @ C C D D s!m L L The summation runs over the exchanged gauge bosons. In contrast to Eq. (1.48), we use here the parametrization in vector and axial vector couplings. This is useful because the photon has a pure vector coupling while the Z boson has almost a pure axial vector coupling to the e>e\ pair in the initial state. Only products of an even number of couplings of the Z to fermions appear in amplitude (2.2). Hence, the process e>e\Pf fM cannot distinguish between Z models, which di!er only by the signs of all Z couplings to fermions.
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Measurements at one energy point su$ciently far away from the Z and Z resonances can only restrict the normalized couplings a, and v, [68], D D
g s g s and v,,v (2) (2.3) a,,a (2) D D D D 4p m!s 4p m!s and not the Z couplings a (2),v (2) and the Z mass m separately. Therefore, a heavy Z with D D large couplings cannot be distinguished from a light Z with small couplings as far as M<s. Below the resonance, fermion pair production by a Z can be described by e!ective four fermion contact interactions, see Section 1.3. Note that de"nition (2.3) di!ers slightly from de"nition (1.51). We will use de"nition (2.3) in this section because it is natural in fermion pair production. For comparisons with other reactions de"nition (1.51) must be used. Note that the di!erence between the two de"nitions is O(s/M). This is a small quantity for Z masses at the detection limit M +3(s to 7(s. 8Y Several widths C below the M resonance, the width C does not in#uence the Z limits and can be neglected. Therefore, the indirect Z limits from fermion pair production remain valid for extra Z bosons which have for some reason a width much larger than predicted in usual GUTs. This is an important di!erence with Z limits from hadron colliders, which depend on C through branching ratios as BrI . The amplitude can be decomposed as M"M #g h M on the Z resonance in the limit of 1+ + + small ZZ mixing. According to Eq. (1.28), M is proportional to either a or v . Therefore, + D D measurements at the Z peak constrain the combinations a+"h g a and v+"h g v (2.4) D + D D + D and not a , v and h separately [18]. Similar to the o!-resonance case, a Z with large couplings D D + and small ZZ mixing cannot be distinguished from a Z with small couplings and large ZZ mixing as far as h ;1. + 2.1.1.2. Cross section. The total and the forward}backward cross sections are de"ned as
dp dp dp (2.5) dc , p " dc ! dc , c"cos h . $ dc dc dc \ \ h is the angle between the outcoming anti-fermion fM and the incoming positron. At the Born level, the cross sections pD , A"T, FB are , pa , pD "d p (s; m, n)"d N C (m, n) s (s) sH(s) (2.6) D s K L K L K L with d " and d "1. N is due to color, N "1(3) for f"l(q). 2 $ D D The summation runs over all interferences. s (s) is the propagator of the vector boson Z with the L L invariant energy squared s p " 2
s g . s (s)" L L 4pa s!m L
(2.7)
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C (m, n) and C (m, n) contain the vector and axial vector couplings of the gauge boson Z to the 2 $ L fermion f, v (n) and a (n), and the helicities of the initial (j ,j ) and "nal (h ,h ) fermions, D D > \ > \ C (m, n; j , j , h , h )"C (initial fermions);C ("nal fermions) , 2 2 2 C (initial fermions)"j [v (m)vH(n)#a (m)aH(n)]#j [v (m)aH(n)#a (m)vH(n)] , 2 C C C C C C C C C ("nal fermions)"h [v (m)vH(n)#a (m)aH(n)]#h [v (m)aH(n)#a (m)vH(n)] , 2 D D D D D D D D C (m,n; j ,j ,h ,h )"C (m,n; j ,j ,h ,h ) (2.8) $ 2 with j "1!j j , j "j !j , > \ > \ (2.9) h "(1!h h ), h "(h !h ) . > \ > \ The unpolarized case corresponds to j "j "h "h "0, i.e. j "h "1 and j "h "0. > \ > \ The couplings of the vector bosons to fermions are given in Section 1.1.3. The left}right cross sections are de"ned as p "p (j "!1, j "0)!p (j "#1, j "0) , *0 2 \ > 2 \ > p "p (j "!1, j "0)!p (j "#1, j "0) , (2.10) *0 $ $ \ > $ \ > where a summation over the polarizations of the "nal states is assumed. The cross section p "p (h "!1, h "0)!p (h "#1, h "0) 2 \ > 2 \ > is useful if the polarization of the "nal state can be measured. The di!erential cross section can be calculated from p and p as 2 $ dp "(1#c)p #cp . 2 $ dc
(2.11)
(2.12)
We would like to mention here that cross sections depending on transverse polarizations can be considered if the polarization of the "nal state is measurable or if transverse beam polarization is available, see [69] for a general discussion and [70] for applications to LEP 1. Simple transverse asymmetries are suppressed as m /(s, where m is the mass of the polarized particle, while double D D transverse asymmetries do not have such a suppression. We do not consider these potentially interesting observables here because they su!er from experimental di$culties. For b} and top quark production, the "nite fermion mass m must be taken into account in the D cross section (2.8), C ("nal fermions)"[v (m)vH(n)#a (m)aH(n)][1#h #(b/2)(!2#h )]b 2 D D D D !a (m)aH(n)(2#h )(1!b)b#[v (m)aH(n)#a (m)vH(n)]h b , D D D D D D (2.13) C ("nal fermions)"[v (m)aH(n)#a (m)vH(n)]h b#v (m)vH(n)h b!a (m)aH(n)h b . $ D D D D D D D D We have k"4m/s and b"(1!k. D
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2.1.1.3. ZZ mixing. The e!ect of ZZ mixing is already included in the couplings v (n), a (n), D D n"1, 2 by de"nition (1.28). One can rewrite Eq. (2.8) in terms of the unmixed couplings and form factors x ,y , which take into account the mixing, D D v (1)Pa (1)[1!4"QD"s (1!x )] , D D 5 D a (1)Pa , D D pa/2s c P(pa/2s c )(1!y )(1!y ) . (2.14) 5 5 5 5 C D This formalism is very useful at the Z peak where ZZ mixing and electroweak corrections must be described simultaneously [30]. Replacements (2.14) must also be applied to the couplings in the total width C appearing in the propagator. 2.1.1.4. Observables. Helicity conservation allows non-zero cross sections for only the four di!erent spin con"gurations j "!j "$1 and h "!h "$1. Furthermore, we see from Eq. > \ > \ (2.8) that the forward}backward cross section p is uniquely related to the total cross section p . $ 2 Therefore, only four of the cross sections and asymmetries, which one can construct from pD and 2 pD , are independent. An example of independent observables are the total cross section and simple $ asymmetries pD total cross section , 2 AD "pD /pD forward}backward asymmetry , $ $ 2 D A "pD /pD left}right asymmetry , *0 2 *0 AD "pD /pD polarization asymmetry of the "nal state . (2.15) 2 They can be constructed for every "nal fermion #avour f. Not all observables listed above are measurable. All four observables require the detection of the #avour f. Forward}backward asymmetries require discrimination between particles and antiparticles. Left}right asymmetries require beam polarization. AD requires a measurement of the polarization of one "nal particle. One can measure combined asymmetries for some "nal states. AD combined left}right forward}backward asymmetry , *0 $ AD combined polarization forward}backward asymmetry , $ AD combined left}right polarization asymmetry . *0 As mentioned above, the combined asymmetries must be related to simple asymmetries,
(2.16)
AD "AD , AD "AD , AD "AD . (2.17) *0 $ $ *0 *0 $ Assuming lepton universality, we "nd a further relation between observables with leptons in the "nal state AJ "AJ . *0
(2.18)
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Of course, relations (2.17) and (2.18) do not hold for m O0. They are modi"ed by radiative D corrections. However, as we will see in the next section, they are a good approximation for light "nal state fermions and radiative corrections with appropriate kinematic cuts. In a theoretical analysis, the additional information due to measurements of several related observables as AJ and AO could be taken into account by the inclusion of only one of these *0 observables but with a smaller e!ective error, see Eq. (1.65). Optimal observables [64,65] can be constructed to enhance the sensitivity to Z parameters. Two parameters, the axial vector and the vector coupling of the Z to charged leptons, a , v , are J J measured by three independent observables. Contradictory signals in all three observables could disprove a Z as the origin of these signals. A consistent measurement of non-zero a or v allows the J J measurement of couplings to the other fermions f by the four additional independent observables (2.15). Therefore, two additional relations between the four observables must be ful"lled if the interaction is due to a Z . At the Z peak, they are AD "AD AC and AD "AC for m "0 in *0 *0 D $ *0 the Born approximation. Again, the hypothesis of a new vector boson could be disproved. If the deviation turns out to be inconsistent with Z interactions, one can try to describe the new physics in the more general framework of four fermion contact interactions (1.49). To eliminate the systematic errors from the luminosity measurement and #avour detection and to reduce the sensitivity to radiative corrections, ratios of total cross sections are considered as well as observables depending on the sum of all light "ve quark #avours, R "pS>B>Q>A>@/pI , A "AS>B>Q>A>@, 2 2 *0 *0
R "p@ /pS>B>Q>A>@, R "pA /pS>B>Q>A>@ . @ 2 2 A 2 2 (2.19)
Similar observables are de"ned by the sum over all lepton #avours (considering only the s-channel contribution for "nal electrons). We denote them with the #avour index l. At the peak of a gauge boson Z , the partial decay widths CD"C(Z Pf fM ) are important L L L additional observables. As mentioned before, we do not consider asymmetries involving transverse polarizations. There is one potentially interesting observable, the P and ¹ odd transverse-normal spin correlation, which is proportional to the imaginary part of the product of the propagators of the exchanged gauge bosons. This observable gives bounds on M from measurements at the Z peak, which are not suppressed by the factor C /M . Unfortunately, this potential sensitivity is completely killed by the loss of statistics in the measurements of the two transverse correlations of the "nal q s [71]. Contributions proportional to the imaginary part of the Z propagator are suppressed as C /M o! the Z peak. 2.1.2. Radiative corrections All observables entering the Z search must be predicted with theoretical errors smaller than the expected experimental error. This demands the inclusion of radiative corrections. Fortunately, not all radiative corrections are of equal importance. This allows simplifying approximations. The O(a) corrections to the SM process are presented in Refs. [72}76], for a review see [77]. 2.1.2.1. QED corrections. Among the complete O(a) corrections to e>e\P(c, Z, Z ,2)Pf fM , the numerically largest QED corrections are a gauge invariant subset. Furthermore, initial state
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corrections, "nal state corrections, and the interference between them are separately gauge invariant. The QED corrections can be calculated in a model-independent way. They depend on the kinematics as the scattering angles and the energies of all "nal particles. We focus here on QED corrections to light fermion production. The "nal state corrections and the interference between initial and "nal state corrections to the new Z interferences can be obtained from the SM result. The initial state corrections to the Z interferences are calculated in [78,79] for massless and in [80] for massive "nal fermions. We discuss here the initial state radiation because it is of major importance for Z tagging. Initial state corrections can be calculated in the structure function approach [81],
p'10(s)"
dx \
dx D(x , s)D(x , s)p(x x s) .
(2.20)
\
V V The structure function approach assumes that the colliding electron and positron have energies degraded by radiated photons, which is described by the structure function D(x, s). The structure function is independent of the particular observable. To calculate the corrections to distributions, one has to boost the "nal particle pair to the laboratory system for every choice of x and x . This is easy in Monte Carlo algorithms but impossible in analytical calculations. Alternatively, initial state corrections can be calculated in the #ux function approach
p'10(s)"[1#S(e)]p (s)#
dv p (s(1!v))HC (v) . (2.21) C The #ux functions HC (v) depend on the particular observable. They describe the probability of the emission of a photon with a certain energy fraction. v is the energy of the emitted photon in units of the beam energy. For p ,p and dp/dc they can be found in Refs. [73,82,76]. To order O(a), we 2 $ have 1#(1!v) b , HC (v)"HM C (v)# C"b C 2 2 2v v
a 1#(1!v) 1!v 1!v b ¸ !1!ln , HC (v)"HM C (v)# C" (QC) C $ $ (1!v/2) p v (1!v/2) v b "(2a/p) (QC) (¸ !1), ¸ "ln (s/m ), QC"!1 . (2.22) C C C C The quantity [1#S(e)] in Eq. (2.21) describes the Born term plus corrections due to soft and virtual photons. To order O(a), we have S(e)"SM #b ln e"b (ln e#)#(a/p) (QC) (p/3!) . (2.23) C C The structure function and the #ux function approach give an equivalent description of QED corrections. See Refs. [38,83] for an extensive discussion and further references. Starting from convolution (2.21), the origin of the radiative tail and its magnitude can be estimated. We do this ignoring details of the radiator functions S(e) and HC (v). The s dependence of the Born cross section is
1 s s s 1 ! . p + s (s ) sH(s )" L s K mH!m s s !mH s !m L L K K
(2.24)
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For m"n, the "rst factor of the last expression becomes s iM s L "! . (2.25) mH!m 2 C M L L L L This imaginary quantity must be met by another imaginary multiplier to give contributions to the cross section. Because we average over transverse polarizations, it can arise only from the v integration (2.21) over the remaining factors of Eq. (2.24). Keeping only the relevant term after partial fraction decomposition, one gets
i M s M i M mH/s!1#* 1 L L dv L ln L p'10(s)!p(s)+ + . (2.26) 2 2 2 C M s mH/s!1 1!v!mH/s 2 C L L L L L The real part of the argument of the logarithm is negative for s'M and *'1!M/s. These are L L the necessary conditions for the development of the radiative tail; the centre-of-mass energy must be larger then the mass of the resonance, and the radiation of photons must be allowed, which are su$ciently hard to ensure a `radiative returna to the resonance. We now estimate the magnitude of the radiative tail by restoring the missing factors rad. tail+p'10(s)!p(s)+p(s; n, n)b (p/2)M /C . (2.27) 2 2 2 C L L Only the contribution of the exchange of the vector boson n appears in Eq. (2.27). Therefore, the other interferences are not enhanced by the radiative tail. Putting s " and M /C "M /C , one L L 5 gets pJ (s; 0, 1)"0 and pJ (s; 1, 1)/pJ (s; 0, 0)" in the limit s<M. This gives 8 2 2 2 rad. tail+7pJ (s; 1, 1)+ pJ , (2.28) 2 2 which is in reasonable agreement with the exact calculation and with Fig. 2.1. For b quark production, where the Z boson exchange p@ (s; 1, 1) dominates over the photon exchange p@ (s; 0, 0), 2 2 the e!ect of the radiative tail is much more pronounced. The radiative tail enhances SM cross sections, while for M '(s the Z signal is not enhanced. 8Y Therefore, the radiative tail must be removed for a Z search below the Z resonance. This can be done by removing events with hard photons by demanding *(1!M/s. The dependence of the cross section on * is shown in Fig. 2.1 for two di!erent energies. The upper curve corresponds to an energy above the Z peak but below the Z peak, the lower curves to an energy above the Z and Z peaks. One recognizes the step-like behaviour for photon energies where the radiative tail(s) are `switched ona. We see that the radiatively corrected cross section is numerically similar to the Born prediction only for a certain cut, which rejects all hard photons from the radiative returns to resonances. This is the reason why Z analyses at the Born level give limits, which are numerically similar to those obtained with radiative corrections and appropriate kinematic cuts. Of course, radiative corrections must be included in "ts to real data. The radiative tail is due to the emission of photons with energies E in the (narrow) interval A *\"1!M/s!M C /s(E /E (1!M/s#M C /s"*> . (2.29) L L L A L L L Therefore, experiments with energies above resonances have sharp peaks in the photon energy spectrum. This is illustrated in Fig. 2.2. There, the Z and Z resonances are represented by peaks
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Fig. 2.1. The total cross section pI as function of the cut on the photon energy * in units of the beam energy for 2 M "M "800 GeV. The upper (lower) set of curves corresponds to (s"200(1000) GeV. This is Fig. 1 from Ref. [57]. 8Y E Fig. 2.2. dpI /d* as function of the photon energy * in units of the beam energy. For the solid line, we choose 2 M "M "800 GeV and (s"1000 GeV. The dashed line is the SM. The dotted line is the function pIb /[*(1!*)], 8Y E 2 C compare estimate (2.85).
at E /E "* "1!M/s with * "0.992 and * "0.36. All "nal states f fM contribute to the A L L peaks. Their heights and widths depend on the width of the related gauge boson as indicated in Eqs. (2.26) and (2.29). 2.1.2.2. Weak corrections. The precision of present and future e>e\ colliders is high enough to be sensitive to weak corrections. They can be implemented by form factors [58,84,85] applying the following formal replacements of the coupling constants in the Born cross section: v (0)Pv (0)F (q) , D D v (1)v (1)Pa (1)a (1)[1!4"QC"s i !4"QD"s i #16"QCQD"s i ] , C D C D 5 C 5 D 5 CD v (1)Pa (1)[1!4"QC"s i ], v (1)Pa (1)[1!4"QD"s i ] , C C 5 C D D 5 D (2.30) a (1), a (1)Punchanged, g"4pa/2s c P(2G Mo . I 8 CD C D 5 5 The complex functions o , i , i and i contain all information about the SM weak corrections, CD C D CD while the complex function F (q) takes into account the e!ects of the vacuum polarization of the photon. Although the largest contribution to F (q) comes from QED, we feel that it should be mentioned in this section. If the new functions are set to one, we recover the Born formulae. i is CD di!erent from i i due to box contributions, which enforce the additional replacement rule for the C D combination v (1)v (1). The conjugated couplings in the Born cross section have to be replaced with C D
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the corresponding complex conjugated relations (2.30). The functions i , i and i can be C D CD absorbed in e!ective Weinberg angles. ZZ mixing e!ects and weak corrections have to be treated simultaneously for predictions at the Z peak. According to Eqs. (2.14) and (1.60), it can be done [30] by the following replacements in Eq. (2.30): i Pi+"i (1!x ) , D D D D i Pi+ "i (1!x )(1!x ) , CD CD CD C D o Po+ "o o (1!y )(1!y ) , CD CD CD C D MPM . 8
(2.31)
In Eq. (2.31), we have neglected terms, which are proportional to the SM weak corrections times the mixing angle h . The multiplier o "M/M takes into account that replacement (2.30) +
8 of g is valid only for the mass of the symmetry eigenstate M . Alternatively, one could calculate 8 the form factors (2.31) by the sum rule (1.59) where i, i and o are the weak form factors i , i D CD CD D CD and o and i, i and o are the mixing form factors iK, iK and oK given in Eq. (1.62). CD D CD CD D CD CD Replacements (2.31) must also be made to the Z width entering the propagator of the mass eigenstate. Non-standard one-loop corrections cannot be calculated without knowledge about the underlying theory. If the Z is the "rst signal of physics beyond the SM, these corrections will probably not play an important role. They are expected to be a small correction to a small deviation from the SM prediction. See Refs. [25,86] for the renormalization of S;(2) ;;(1) ;; (1) gauge theories. * 7 2.1.2.3. QCD corrections. QCD corrections have to be taken into account in the case of hadronic "nal states. They do not feel the gauge boson exchanged before. Therefore, the known SM results for massless [34] and massive [35,87,88] "nal state fermions can be used. The O(a ) QCD Q corrections can be obtained from the O(a) "nal state QED corrections by the replacement aPa . Q They depend on the mass of the "nal state fermion and on the maximum allowed energy E "* (s/2 of the radiated gluon. For m "0 and * "1, the lowest-order QCD corrections D E E E vanish for forward}backward asymmetries and reduce to the well-known factor 1#a /p for total Q cross sections. As known from SM calculations, radiative corrections due to spin asymmetries of the "nal state need special care. Due to spin-#ip contributions induced by gluon radiation, the result of a calculation with zero fermion masses m "0 does not coincide with the result calculated for D m O0 in the limit m P0 [89]. The problem also arises in "nal state QED corrections, however, it D D is numerically less important because a is numerically smaller than a . Q 2.1.2.4. Corrections to C . As is known from the experiments at LEP and SLC, the measurements on the Z resonance are sensitive to radiative corrections to the Z width. In contrast to the corrections to C described in Section 1.1.4.2, we give here the formulae including the weak
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corrections to C , N G (2M o8kR R (M)+[v (1)#a (1)](1#2m/M)!6a (1)m/M, . CD" D I D /#" /!" D D D D D 12p (2.32) The functions R and R describing the QED and QCD corrections [34] are given in Eq. (1.34). /#" /!" The function o8 absorbs *r arising during the replacement of the weak coupling constant by the D muon decay constant G (2M G (2M 4pa 8" I 5. " I g" s c 1!*r (1!*r)c 5 5 5
(2.33)
As in e>e\Pf fM , the remaining weak corrections can be taken into account [37] by a replacement of the vector couplings v (1)Pa (1)[1!4"QD"s i ] . D D 5 D
(2.34)
The functions i are often absorbed in s de"ning e!ective Weinberg angles. In comparison to the D 5 weak corrections to e>e\Pf fM , we have no box contributions in the Z decay. Therefore, a special replacement rule (2.30) for the product v (1)v (1) is absent. C D In the case of a non-zero ZZ mixing, we have to apply additional replacements similar to rule (2.31), oDPoDo (1!y ), i Pi "i (1!x ) . 8 8 D D D D D
(2.35)
2.1.3. Z constraints at s+M The Z signal in measurements at the Z peak is a deviation of the couplings of the mass eigenstate Z to fermions from the SM prediction. This prediction depends on the SM parameters, which are also de"ned by measurements at the Z peak. If one wants to constrain Z parameters from the same data, the cleanest analysis would be a simultaneous "t of SM and Z parameters. We call this procedure direct Z analysis. Alternatively, the data distributed around the Z peak can be "tted "rst in a model-independent procedure. The results of such a "t are M , partial and total decay widths CD and cross sections and asymmetries at the peak. This output is the input of a second "t, which determines the SM and Z parameters. We call this procedure indirect Z analysis. The "rst direct analysis was done in [30]. The indirect analyses are shown to agree with this direct analysis. As mentioned in the introduction, we will not consider the mixing of the new fermions with SM fermions. See Ref. [90] for such an analysis based on LEP 1 data. In the following subsections, we discuss di!erent Z constraints. We always start with a derivation of a simple estimate. This estimate shows the scaling of the constraint with di!erent parameters of the experiment as integrated luminosity, centre-of-mass energy and systematic errors. We then discuss present constraints and possible constraints from future experiments and prove the quality of our estimates.
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2.1.3.1. Model-independent constraints on v+ and a+. Estimate. The quantities v+ and a+ de"ned D D D D in Eq. (2.4) can be constrained at the Z peak independently of the Z model. Consider the partial decay width and di!erent asymmetries at the Z peak v v+#a a+ g D D "CD#*8YCD , [v(1)#a(1)]N +CD 1#2 D D CD"M D D 12p D g (v#a) D D AD "A A +AD#A*A #A*A "AD#*8YAD , $ D C C D $ $ $ C D AD "A +A#*A "A #*8YA , *0 C C C *0 *0 "A +A#*A "AD #*8YAD (2.36) AD "A D D D *0 $ v a+#a v+ av a+#va v+ 2a (1)v (1) D D !4 D D D D D D . D D , and *A +2 D D (2.37) with A , D D a (1)#v (1) g (v#a) g (v#a) D D D D D D The index zero denotes the observables without mixing. If the deviation *8YO in the observable O is larger than the experimental error *O , G G G *8YCD'*CD , *8YAD '*AD , *8YAD '*AD , *8YAD '*AD , (2.38) $ $ *0 *0 one can see a signal. Relations (2.38) and (1.60) predict that the di!erent observables are blind to Z models predicting v+ and a+ between two parallel lines. Neglecting systematic errors, the Z D D constraints (2.38) scale like 1/(¸. For f"e,l and s ", Eqs. (2.36) and (2.38) transform to 5 "a+"'(g /4)*CC /CC , "a+#v+"'(g /2)*CJ /CJ , C J J "v+"'(g /6)*AC /A , "v+"'(g /4)*AC . (2.39) C $ C C *0 The constraint from AC linear in h exists only for s O leading to AO0. Eqs. (2.39) allow an C $ + 5 independent constraint or measurement of a+#v+"¸+ and ¸+ and therefore an experimental J J J C check [59,60] of relations (1.27). Present constraints: Model independent Z constraints based on the combined data of LEP and SLC [91] are discussed in Refs. [23,59}61]. Before confronting CD, AD and AD with the data, $ *0 radiative corrections have to be included. This implies a substitution of g with G in expression I (2.30) for CD. The substitution induces a dependence on o "M/M. This spoils the model
8 independent limits on v+, a+ making them dependent on the additional Z parameter M . D D The problem is solved in Refs. [59,60] by considering special combinations of observables where the leading dependence on o and o drops out. As a result, these special combinations are also CD
much less sensitive to the top and the Higgs mass. With the recent experimental data [91] on M ,G , s , a(M) and M , one can follow another 5 I 5 8 R procedure and calculate o according to Eq. (2.33). One gets o "1$0.003, compare [92].
The main sources of the uncertainty of o are the experimental error of the M measurement and
5 the theoretical error of *r arising due to the unknown Higgs mass and to a less extent due to the experimental error of the top mass. In the two-p errors, the experimental error in the M measurement dominates because the theoretical error in *r is not doubled. We assumed that 5 the symmetry is broken by one Higgs doublet only. In general, one would obtain di!erent results for extended Higgs sectors. In this sense, the error of o still contains a model dependence, which
is expected to be small. The expression of CD is now independent of M . The price one has to pay is that the uncertainty of o must be added to the experimental error of CD.
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Fig. 2.3. Areas of (a+, v+), for which the extended gauge theory's predictions are indistinguishable from the SM (95% J J C¸). Models between the dashed (dotted) lines cannot be detected with C , (AJ and AJ together). The regions J $ *0 surrounded by the solid lines cannot be resolved by all three observables combined, see text. The numbers at the straight line indicate the value of h in units of 10\ for the s model. The dots for h (0 are not labelled. The rectangle is + + calculated from Fig. 2.18. Fig. 2.4. Areas of (g a /(g a ), h ) and (g v /(g v ), h ), for which the extended gauge theory's predictions are indistinguish J J + J J + able from the SM (95% C¸). This "gure is an update of Fig. 4 in Ref. [61].
Fig. 2.3 illustrates the constraints on a+ and v+ obtained from the data [91], CJ " J J (83.91$0.11) MeV, AJ "0.0174$0.0010, A "0.1401$0.0067. In the SM, we have A " $ O O AO "A . The plotted regions correspond to a s"s #5.99. In contrast to the demonstration *0
in Ref. [61], we take into account the deviations of the measurements from the theoretical prediction in Figs. 2.3 and 2.4. The constraints from C and from AJ and A combined are shown J $ O separately for o "1. The quantitative agreement with estimates (2.39) is very good. The excluded
regions are slightly rotated relative to the axes because s O. The deviation of the experimental 5 value from the theoretical prediction leads to a parallel shift of the exclusion region of the corresponding observable. All three observables combined cannot exclude the region inside the ellipse. The uncertainty of o yields a shift of the ellipse, which results in the larger solid region.
This shift is possible only in one direction because we always have M (M according to 8 Eq. (1.14). Future improved measurements of M and M and a determination of M would reduce 5 R & the shift. The present data are consistent with assumption (1.25), i.e. with ¸+"¸+. One could then C J interpret constraint (2.39) from the invisible width as a constraint to ¸+. Unfortunately, this gives C no improvement to the combined region shown in Fig. 2.3. The model-independent constraint shown in Fig. 2.3 can be interpreted as a constraint on the ZZ mixing angle h for any "xed model with known couplings a and v . Varying the mixing angle, one + J J
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moves on a straight line in Fig. 2.3. This line intersects the model independent exclusion limit for certain values of h . The values of h at the intersection points de"ne the excluded regions of h . + + + Graphically, one obtains !0.006(h (0.0025 for the s model. The model independent con+ straints obtained from a one-parameter "t are expected to be stronger. Model-independent limits on v+, a+ can be obtained in a similar procedure, see Ref. [61]. O O A global "t to LEP data would allow to constrain the "ve independent couplings ¸+, R+, ¸+, J C O R+ and R+ simultaneously. These couplings are de"ned in analogy to v+, a+, S B D D ¸+"h g ¸ , D + D
R+"h g R . D + D
(2.40)
Future constraints: The present constraints on a+ and v+ obtained from LEP data can be C C improved by future measurements of the reaction e>e\P=>=\. The rectangle in Fig. 2.3 is calculated from Fig. 1 of Ref. [93]. See Section 2.3 for details. The constraints on a+ and v+ from the Z peak cannot be improved by measurements at O O MOs(M. Future measurements at the Z peak (if it exists) allow a separate measurement of the couplings a ,v and of the mixing angle h . D D + 2.1.3.2. Model independent constraint on h . Fig. 2.3 is independent of h as far as h is small. This + + + allows the derivation of a model independent constraint on h . In the simplest approximation, + where only the linear terms in h are kept in Eq. (1.28), we obtain the estimate + "h "((*c/c)g c/g c , where c"a , v and c "a , v . + J J J J
(2.41)
*c is the bound on a+ or v+ taken from Fig. 2.3. In particular, estimate (2.41) gives "h "(0.003 for J J + g a /(g a )"0.62 as it is the case in many GUTs. J J The exact numerical result for h as a function of g a /(g a ) is shown in Fig. 2.4. The + J J approximate bound (2.41) is recovered at large g c /(g c). In contrast to Eq. (2.41), the exact calculation gives a constraint on h also for a Z with zero coupling, i.e. for g c /(g c)"0. It is + "h "(0.035 for c"a . This can easily be understood from Eq. (1.28) where the deviations of the + J couplings a (1) or v (1) from a or v with increasing h eventually become larger than the D D D D + experimental error even for the case g "0. If one allows a large ZZ mixing, there is one particular Z with all couplings proportional to those of the SM Z boson and a "ne-tuned overall coupling strength, g c "g c (1!c )/s , c"a, v, f"l, c, b , D D + +
(2.42)
which will not produce a deviation of c (1) from c . The beginning of this region of insensitivity can D D be recognized in Fig. 2.4. Models with "ne tuning (2.42) in all couplings can only be detected by e!ects of the Z propagator. 2.1.3.3. Model-dependent constraint on h and M . Estimate: h can be much better constrained in + + particular Z models because they link the Z couplings to leptons and to quarks by model parameters. Assuming v +v and a +a , we obtain from Eq. (2.39) D D D D "h "(h +(g /2g )*CD/CD . + +
(2.43)
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Constraint (2.43) scales with the integrated luminosity ¸ as
h & +
1#r . ¸
(2.44)
r is the ratio of the systematic and statistical error as de"ned in Eq. (1.66). Consider now the sensitivity to M . For simplicity, we assume h "0 identifying Z "Z and + Z "Z. Again, we assume that the Z and Z couple with the strengths g and g to SM fermions setting v +v , a +a . Only the ZZ interference is important near the Z resonance, Then, we can D D D D approximate the relative shift of cross sections, *8YO/O "*p /p by a ratio of propagators, 1+ 2 2 *8YO g "Res sH " g s!M 8 8Y " 8. + (2.45) O g "s " gM !s 1+ 8 8Y The deviation *8YO has to be compared with the experimental error *O. Choosing (s"M #C /2, we conclude that a Z with a mass 8 8 g *O g O C 1 C 8 8 ,M 1# , *o" (2.46) M (M "M 1# 8 8Y 8Y 8 g O *o M g *O M 8 8 would be detected by the observable O. The factor C /M re#ects the suppression of the ZZ 8 8 interference relative to the resonating contribution. One exception is the transverse-normal spin asymmetry mentioned in Section 2.1.1.4. It is proportional to the imaginary part of the product of the propagators. Hence, it has no suppression factor C /M . Unfortunately, this potential sensitivity is compensated by the loss of statistics in the measurement of this asymmetry [71]. The result is a net sensitivity to M weaker than Eq. (2.46). Present constraints: Measurements at the Z resonance give the best present limits on ZZ mixing [66]. The analysis of LEP data requires the inclusion of weak corrections. This induces a dependence of the limits on the Higgs and the top mass. Sometimes, h and M are "tted as independent + parameters, which corresponds to a Higgs sector consisting of an arbitrary number of Higgs doublets and singlets. One should note that in this case the weak corrections calculated within the minimal SM are only approximate. For a speci"ed Higgs sector, h and M are related by the + Higgs constraint (1.17). This relation transforms the tight limits on h to limits on M , which are + much better than those obtained for unconstrained Higgs sectors, compare also Fig. 7 in Ref. [94]. The present two-dimensional constraint of the parameter space h , M from L3 data is shown in + Fig. 2.5. The "gure is based on LEP data published in 1997 [95]. A s "t to the observables pJ , AJ , AO , R and A@ is performed. The bound on h is dominated by the data from the 2 $ @ $ + Z peak, while the bound on M is dominated by the data above the Z peak. Constraints on h for a Z in E models obtained in the same analysis [103] are shown in + Fig. 2.6. We see that the data give tight constraints on h for all models considered. + Several indirect [23,59,60,90,94,96}99] and direct [30,62,100,101,103] Z analyses have been performed recently. We can only comment on some of them and compare the results with the naive estimates derived in the previous section. Ref. [94], DELAGUILA92. This analysis is based on LEP data published in [104] and on lq, l e, eH data [105], atomic parity violation data [106] and on data on M [105,107], see Table 1 I 5
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Fig. 2.5. The 95% C¸ allowed regions in the h !M plane for the s model. The input of this "gure is + 8Y M "(91.1863$0.0019) GeV, M "(175$6) GeV, M "150 GeV and a "0.118$0.003. This "gure is a preliminary 8 R & Q result taken from reference [102]. Fig. 2.6. The 95% C¸ bounds on the ZZ mixing angle, h , as a function of the E parameter b"h . The input of this + "gure is M "(91.1863$0.0019) GeV, M "(175$6) GeV, M "150 GeV, a "0.118$0.003 and M '550 GeV. 8 R & Q 8Y This is Fig. 1 from Ref. [103].
of Ref. [94]. The limit on h , see Fig. 7 of Ref. [94], is dominantly set by the LEP data. Special + attention is paid to the constraints on the breaking parameters of the E theory by the data. Furthermore, the correlation between M and M is considered in detail. We show in Table 2.1 the 8Y R bounds taken from Fig. 7 for M "700 GeV, M "130 GeV and M "100 GeV. We select 8Y R & C "(2485$9) MeV from the data [104] for estimate (2.43) of h . We multiply the 90% C¸ + numbers given in [94] by 1.195 to estimate the 95% C¸ bounds. Ref. [30], LEIKE92. This analysis is based on the data from Ref. [108], which includes the LEP data from 1989 and 1990. It is the "rst direct Z analysis. It is shown there that the results of earlier indirect analyses [60,109,110] agree with the results of the direct analysis. The values of h given in + Table 2.1 are taken from Fig. 3 of [30] for M "700 GeV, M "150 GeV and M "300 GeV. We 8Y R & use C "(2487$10) MeV based on the data [108] for our estimate. 8 Ref. [96], ALTARELLI93. This analysis is based on LEP data [111] and on CDF and UA2 data. The limits on h are dominated by the LEP data. In addition, the data are interpreted in +
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Table 2.1 The 95% C¸ ranges of the ZZ mixing angle h for di!erent Z models obtained in the analyses listed in the text. The + estimate h is calculated using Eq. (2.43) with g /g "(s +0.62 + 5 Analysis
s
t
g
¸R
h +
[94] [30] [96] [101] [3] [102] [60] 2.1.3.1
!0.007, 0.005 !0.006, 0.008 !0.0035, 0.0035 !0.0070, 0.0078 !0.0029, 0.0011 !0.0036, 0.0017 !0.022, 0.012 !0.006, 0.003
!0.007, 0.006 !0.009, 0.006 !0.0051, 0.0043 !0.0075, 0.0095 !0.0022, 0.0026 !0.0039, 0.0029 !0.008, 0.014 !0.011, 0.006
!0.006, 0.008 !0.011, 0.009 !0.013, 0.0090 !0.029, 0.029 !0.0055, 0.0021 !0.0049, 0.0055 !0.024, 0.040 !0.018, 0.014
!0.008, 0.003 !0.004, 0.008 !0.0029, 0.0033 !0.0057, 0.0077 !0.0013, 0.0021 !0.0053, 0.0033 !0.006, 0.011 !0.008, 0.005
$0.006 $0.007 $0.005 $0.005 $0.002 $0.002
terms of the e parameters de"ned in [112] and in models with speci"ed Higgs sectors. Note that the angle h parametrizing the E models introduced there is connected with b introduced in Eq. (1.41) by the relation b"h !arctan(5/3. We take h from Table 5 for M "150 GeV and + R M "100 GeV choosing h "0, 50,!30 degrees for Z "g, s, t. These values h are only approx& imate for Z "s(t), where the exact values should be 52.23(!37.83). Furthermore, we exploited the insensitivity of the reaction e>e\Pf fM relative to the simultaneous change of the signs of all Z couplings to fermions, i.e. to a shift h Ph #p. We multiplied the numbers from Table 5 in Ref. [96] by 1.96 to estimate 95% C¸ numbers from the values given at one-p. We select C "(2489$7) MeV from the data [111] for our estimate. 8 Ref. [101], ABREU95. This direct Z analysis of the DELPHI collaboration is based on LEP data from 1990 to 1992. The limits on h are obtained for M "150 GeV and M "300 GeV. The + R & e!ect of a is small as far as it is chosen between 0.118 and 0.128. Q Ref. [3], CVETIC[ 97. This analysis is based on LEP and SLD data [113,114]. The CDF constraint M "(175$6) GeV is included. We take the 95% C¸ bounds on h from Table 2. Our R + estimate is based on C from the data [113,114]. 8 Ref. [102], S.RIEMANN97. This analysis is based on L3 data [115] and LEP data [116] published in 1996 and 1997. It includes the recent measurements beyond the Z peak. The measurements on the Z peak de"ne the limits on h , while the measurements beyond the + Z peak de"ne M . The values for h shown in Table 2.1 are obtained for 8Y + M "(175$6) GeV, a (M )"0.118$0.003 and M '550 GeV. R Q 8 8 We now compare the results of the analyses listed above with the results on h obtained from + a model-independent analyses based on leptonic observables only and speci"ed to speci"c models in a second step. Ref. [60], LAYSSAC92. This analysis is based on LEP data published in [117]. We show in Table 2.1 the bounds on h taken from Fig. 3 in Ref. [60]. We multiply the bounds given there at + one-p with 1.96 to estimate the 95% C¸ bounds. This paper, Section 2.1.3.1. This analysis is based on combined LEP and SLC data from 1990 to 1995 published in [91]. We show in Table 2.1 the bounds on h obtained from Fig. 2.3 by the same + procedure as explained for the s model.
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Table 2.2 The lower bounds on Z masses M in GeV excluded with 95% C¸ by the analysis explained in the text. The estimate 8Y M from Eq. (2.46) is added with g /g "(5/3s +0.62 8Y 5 Analysis
s
t
g
¸R
M 8Y
[30]
148
122
118
}
138
The limits from the model independent analyses are weaker than those obtained in the model-dependent analyses. The main reason of this di!erence is that the model-independent analyses are based on data from leptons in the "nal state only. In the model-dependent analyses, all couplings of the Z to fermions are linked by model parameters. This allows the inclusion of leptonic and hadronic observables in the analysis. We list only the results of one analysis on M because the limits from precision measurements at the Z peak alone are rather poor in the case of an unconstrained Higgs sector. Ref. [30], LEIKE92. See description of limits on h above for details of this analysis. We show in + Table 2.2 the limits on M quoted in Section 4.2 of Ref. [30]. Estimate (2.46) is based on 8Y C "(2487$10) MeV selected from the data [108] used in [30]. 8 We see that the predictions of formulae (2.43) and(2.46) are in reasonable agreement with the numbers obtained in the exact analyses of real data. Of course, they cannot reproduce details of the di!erent models. Future constraints: The limits on h from the data [91] could be improved only by future + measurements of the reaction e>e\P=>=\. For details, we refer to Section 2.3. The present limits on M for models with unconstrained Higgs sectors are much weaker than those from LEP 2 or from the Tevatron. Future e>e\ and pp(pp ) experiments will further improve the limits on M . With constrained Higgs sectors, the indirect limits on M from the measurements at the Z peak compete with the limits from the other experiments [66]. 2.1.4. Z limits at MOs(M Below the Z resonance and o! the Z peak, di!erent cross sections and asymmetries are predicted by the SM. The SM parameters are known very precisely from the measurements at the Z peak. If future measurements di!er from these predictions, one can try to interpret the di!erences as e!ects due to an extra Z boson. The Z signal arises through interferences of the Z with the photon or Z boson. The deviations due to these interferences can be detected if they are larger than the experimental error. Compared to measurements at the Z peak, the sensitivity to h is suppressed by a factor C /M + due to statistics. Therefore, the dependence on h can be neglected putting h "0 and identifying + + Z "Z and Z "Z. Early Z analyses can be found in Refs. [32,46,118,119]. We consider here results for Z constraints obtained by di!erent recent analyses.
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2.1.4.1. Model independent constraints on v, and a,. Estimate: As shown in Section 2.1.1, the J J amplitude of o!-resonance fermion pair production depends only on the normalized couplings v, and a, and not on a ,v and M separately. D D D D 8Y Consider the constraints arising from the independent observables pJ , AJ and AJ introduced 2 $ *0 in Section 2.1.1. The measurement of each of these observables excludes a certain domain of the Z couplings v, and a,. This domain can be calculated analytically in the Born approximation taking J J into account the contributions of the cZ and ZZ interferences, neglecting the Z Z contribution. For simplicity, we set s ". The three considered observables detect a signal if the following 5 conditions are ful"lled [68]: pJ : 2 AJ : $ AJ : *0
v, a, s (s) as pJ *pJ J # J 8 51, H " 2 2, 2 H H 4 2 pJ (QED) pJ 2 2 2 2
v, a, (3!AJ s (s)) as /2(pJ /pJ (QED))*AJ J $ 8 51, H " 2 2 $ , ! J $ AJ ! s (s) H H AJ (!(3/16)s (s)) $ 8 $ $ $ 8 v, J H T
a, as /2(pJ /pJ (QED))*A J 51, H " 2 2 *0. *0 H 1#(1/4)s (s) *0 8
(2.47)
s (s) is de"ned in Eq. (2.7) and pJ (QED)"4pa/3s. pJ detects a Z with couplings outside an ellipse 8 2 2 above the Z peak and outside a set of hyperbolas below the Z peak. The forward}backward asymmetry is sensitive to a Z with couplings outside a set of hyperbolas above the Z peak and to a Z with couplings outside an ellipse below the Z peak. The left}right asymmetry detects a Z with couplings outside a di!erent set of hyperbolas for all energies. As explained in Section 2.1.2.1, the quantitative prediction (2.47) is only changed a little by radiative corrections if appropriate kinematic cuts are applied. The axes of the ellipse and the hyperbolas H ,H and H do not 2 $ *0 depend on the Z model. They are proportional to the root of the experimental errors *pJ , *AJ 2 $ and *A . For further details, we refer to [68]. *0 Alternatively, the diwerential cross sections of left- and right-handed beams can be considered. The Cramer}Rao bound [120] for the reaction e>e\Pk>k\ is derived in the second reference of [121] in the limit M <s<M and s ". In our notation, we have 8Y 8 5 s +(¸/s)p[10(v,)#22(v,)(a,)#7(a,)] . C C C C
(2.48)
Bound (2.48) gives the theoretical limit of a constraint of the Z parameters by the observables considered. For comparison, we give bound (2.47) deduced from pJ in the same limit as (2.48) 2 s+(¸/s)p[(v,)#(a,)] . C C
(2.49)
As it should be, it is worse than Eq. (2.48). To conclude, we remark that expressions similar to Eq. (2.47) were obtained some time ago in Ref. [122] to constrain the interactions of the SM Z boson. Present constraints: Present constraints on v, and a, are available from measurements at J J TRISTAN and at LEP 2.
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Fig. 2.7. Combined regions allowed by pI and AI for s"s #4 in the (A ,< ) plane for TRISTAN, LEP1, LEP1.5 2 $
J J and LEP2 colliders. Two bounds are shown for LEP2 corresponding to ¸"150 pb\ (one year of running) and ¸"500 pb\ (three years of running). Radiative corrections are included. This is Fig. 4 of Ref. [125]. Fig. 2.8. The 95% C¸ allowed regions in the a , v plane from ¸3 data now (large region) and at the end of LEP2 (small J J region). The input of this "gure is M "(91.1863$0.0019) GeV, M "(175$6) GeV, M "150 GeV, a "0.118$0.003 8 R & Q and M "500 GeV. This is Fig. 2 of Ref. [103]. 8Y
Ref. [123], OSLAND97. This analysis is based on measurements of pI and AI at TRISTAN 2 $ [124] (¸"300 pb\, (s"58 GeV) and LEP1.5 (¸"5 pb\, (s"130!140 GeV). The constraints on a, and v, are shown in Fig. 2.7. The constraints from future measurements at LEP2 J J ((¸"150(500) pb\, (s"190 GeV) are predicted too. The systematic error of both observables is assumed to be 1% at all colliders (see Table 1 in Ref. [123]. The conventions are A "a,,< "v,. J J J J Ref. [103], S.RIEMANN97. This analysis is based on LEP data published in 1997 [95]. A s "t to the observables pJ , AJ and AO is performed to give the allowed regions in the a !v plane. 2 $ J J The resulting constraints are shown in Fig. 2.8. All radiative corrections and systematic errors are included in this analysis. Future constraints: Predictions for future constraints on a, and v, of a 500 GeV collider including J J radiative corrections are shown in Fig. 2.9. The analysis includes statistical and systematic errors. The resulting combined errors of the observables entering Fig. 2.9 are *pJ /pJ "1%, *AJ "1% 2 2 $ and *AJ "1.2%. We refer to [126] for further details. As predicted by the Born analysis (2.47), *0 the regions indistinguishable from the SM are approximately ellipses for pJ and areas between 2 hyperbolas for AJ and AJ . The remaining two parts of the hyperbolas from AJ are outside the $ *0 $ "gure. A gives only a marginal improvement to the Z exclusion limits. *0 A quantitative comparison with Eq. (2.47) shows that the error of the Born prediction for the exclusion regions is below 10%. Of course, this number depends on the kinematic cuts as explained in Section 2.1.2.1. Eq. (2.47) can be used to predict the changes in Fig. 2.9 for di!erent errors of the observables.
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Fig. 2.9. Areas of (a,, v,) values, for which the extended gauge theory's predictions are indistinguishable from the SM J J (95% C¸) for (s"500 GeV and ¸"20 fb\. Models inside the ellipse cannot be detected with pJ measurements. 2 Models inside the hatched areas with falling (rising) lines cannot be resolved with AJ (AJ ). I thank S. Riemann for $ *0 providing this "gure. Fig. 2.10. The normalized leptonic vector and axial vector couplings v, and a, for M "3(s in typical GUTs. For the J J 8Y s model, M is varied in units of (s. 8Y
The model-independent exclusion region predicted for "nal LEP2 data ((s"190 GeV, ¸"0.5 fb\) is shown in Fig. 3a of Ref. [126]. It looks very similar to Fig. 2.9 and agrees with Fig. 2.7. For illustration purposes, the domains of the normalized couplings a,,v, are shown in Fig. 2.10 J J for E models. For a "xed Z model, i.e. known couplings v and a , the limits on a, and J J J v, transform to Z mass limits. This is illustrated in Fig. 2.10 for Z"s. Superimposing Figs. 2.9 J and 2.10, we predict M +4(s. For a "xed M , Fig. 2.9 can be drawn for a and v as done 8Y J J Q in Fig. 2.8. The couplings of the Z to quarks a, and v, can only be constrained if the Z couplings to leptons O O are non-zero. This is di!erent from the constraints on a+ and v+ at the Z peak, which were O O possible also in the case v+"a+"0. J J In models, where the couplings of the Z to leptons are considerably smaller than those to quarks, one can have a signal in the quarkonic observables without a signal in leptonic observables. See Ref. [127] for a discussion of such a possibility. Assuming relations (1.27), the measurements at future colliders would constrain the "ve couplings ¸ (2), R (2), ¸ (2), R (2), R (2) . J C O S B
(2.50)
In the case of an agreement of these measurements with the SM predictions, the allowed regions of couplings (2.50) contain zero.
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The measurement of non-zero couplings in the case of a Z signal is investigated in the literature and will be discussed in Section 2.1.4.4. Finally, we mention that a,, v, and a+, v+ are uniquely related by the Higgs constraint (1.17) in D D D D models where the Higgs sector is speci"ed. An appropriate scaling and a superposition of Figs. 2.3 and 2.9 would then allow a direct comparison of the model-independent limits from e>e\Pf fM for s+M and for MOs(M. 2.1.4.2. Model-dependent constraints on M . Estimate: To obtain an estimate for M , we "x the Z 8Y 8Y couplings v +v , a +a . Consider the cZ interference in pD. We derive D D D D 2 s *8YO g "Res sH " g 1 A 8Y + . (2.51) + , which gives M "(s 1# 8Y O g "s " g M !s *o A 1+ 8Y Models with M (M would give a signal in the observable O. Comparing the two expressions 8Y 8Y for measurements on and o! the Z peak given by Eqs. (2.46) and (2.51), we see that Eq. (2.46) has an additional suppression factor C /M . 8 8 We usually have 1/*o,g/gO/*O<1. This allows the further approximation in Eq. (2.51),
(2.52) M +(s/*o . 8Y Approximation (2.52) is in agreement with estimates (2.47), which contains more details of the model. Taking into account the scaling of the error *o&(s/¸, one obtains the well-known scaling law of the Z mass limit [68,128,129], M +(s¸) . It is valid in the absence of systematic errors. 8Y A scaling including the systematic errors is obtained substituting ¸ by ¸/(1#r), M &[s¸/(1#r)] . (2.53) 8Y The in#uence of (not too large) systematic errors on Z exclusion limits is therefore rather moderate. Radiative corrections give cross sections depending on kinematic cuts. We already noticed the moderate dependence of cross sections on the photon energy cut * in the case where the radiative return is forbidden, see Fig. 2.1. The in#uence of * on M is further reduced by the fourth root in 8Y Eq. (2.53). Present constraints: The best present mass limits on extra neutral gauge bosons predicted in E models come from proton collisions. As an example of limits on M from e>e\ collisions, we 8Y quote the numbers in Table 2.3 read o! from Ref. [102] S. RIEMANN97. More details of the analysis are given in Section 2.1.4.1. The estimate of M is calculated using the statistical error of 8Y pJ . Formula (2.51) must be used because of the small numbers of events. 2 Future constraints: Many analyses investigate the Z mass limits reachable at future colliders. The minimal input of these analyses are assumptions about the centre-of-mass energy, the integrated luminosity and a list of observables used in the "t. Optionally, systematic errors are included. Radiative corrections have to be included into "ts to real data. However, they introduce only small changes in theoretical investigations, where the `dataa can be generated in the Born approximation and then be "tted by Born formulae [68,130]. See Section 2.1.2 for the reasons why this works well. We now comment on the recent theoretical analyses, the results of which are collected in Table 2.4.
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Table 2.3 The lower bound on Z masses M excluded with 95% C¸ by the analysis explained in the text 8Y Analysis
s
t
g
¸R
SSM
M 8Y
[102]
300
220
230
310
520
240
Table 2.4 The lower bound on Z masses M in TeV excluded by the di!erent analyses described in the text. The estimate (2.51) for 8Y M is calculated with g /g "0.62 8Y Analysis
(s/TeV
¸ (fb)
s
t
g
¸R
SSM
M (E ) 8Y
[126] [126] [126] [126] [131] [132] [132]
0.19 0.19 0.5 0.5 0.5 0.5 0.5
0.5 0.5 20 20 50 50 50
0.99 1.10 2.8 3.1 3.2 5.2 4.0
0.56 0.64 1.6 1.8 1.8 2.5 2.3
0.62 0.69 1.7 1.9 2.3 2.9 2.5
1.10 1.30 3.2 3.8 3.7 4.2 3.7
1.50 1.70 4.0 4.7 4.0 6.9 6.9
0.95 1.10 2.6 3.1 2.6 3.9 3.9
#syst. stat. #syst. stat. e>e\ k>k\
Ref. [126], LEIKE97. Theoretical analysis for LEP2 and a future linear collider. The observables pJ , AJ , AJ , AO , R , A , R "p@ /p , A@ , A@ (2.54) 2 $ *0 *0 @ 2 2 $ *0 are included in the "t. Eighty percent polarization of the electron beam is assumed. The e$ciency of #avour tagging is included in the systematic errors. The full SM radiative corrections are included. The numbers giving limits at 95% C¸ are taken from Table 3 of Ref. [126]. The mass limits without and with inclusion of systematic errors are shown. We take the error of the most accurately measured observable for our estimate of M , i.e. *A in all scenarios. The one-p errors of *A 8Y *0 *0 with and without systematic errors are 0.8% and 0.6% (0.7% and 0.5%) for LEP2 (LC500). Ref. [131], RIZZO96. Theoretical analysis for new gauge boson searches at di!erent future colliders. The observables pD, AD , AD , AD , AO and AO (2.55) 2 $ *0 *0 $ $ for di!erent "nal state fermions f"l, c, b, t are included in the "t. Ninety percent polarization of the electron beam is assumed. Initial state radiation, "nite identi"cation e$ciencies and systematic errors associated with luminosity and beam polarization uncertainties are taken into account. We selected for Table 2.4 the numbers given in Table 3 of Ref. [131], which are given at 95% con"dence. Ref. [132], GODFREY96. Analysis of new gSauge boson searches at di!erent future colliders. Compared to the older analysis [133], more observables and the option of a k>k\ collider are included. The 18 observables pI , AD , AD , f"k, q, c, b; AD , f"k, c, b; AO , R , A D $ *0 *0 $ *0
(2.56)
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are included in the "t for e>e\ colliders. Ninety percent electron polarization is assumed. Only the 10 observables, which do not demand beam polarization, are included for k>k\ colliders. Detection e$ciencies are taken into account. Radiative corrections and systematic errors are not included. The numbers quoted in Table 2.4 are taken from Fig. 1 of Ref. [133] multiplied with 1.15 to transform the 99% C¸ limits given there to 95% C¸ limits. Estimate (2.52) in Table 2.4 is obtained with the statistical error of A as input. *0 The estimate for M in Table 2.4 gives a good prediction of the exact exclusion limits. Of course, 8Y it cannot describe the di!erences between the E models. Scaling (2.53) predicts that systematic errors of the same magnitude as the statistical errors, i.e. r"1, should change M PM /(2, 8Y 8Y which is a reduction of 16% only. This is in agreement with Table 2.4. Observables, which require #avour tagging, have systematic errors, which usually dominate the statistical errors. However, these observables give only a moderate contribution to M keeping the dependence of M on 8Y 8Y their systematic errors small. For details, we refer to [126,131]. The price one has to pay for model-independent limits on v, and a, can be estimated comparing J J the exclusion limit for M "2.8 TeV quoted in Table 2.4 with the value M "4(s"2 TeV Q Q obtained in the model-independent analysis explained in Section 2.1.4.1. Both analyses are based on the same assumptions on the data. The di!erence occurs because the model-independent analysis is based on a two parameter "t to pJ , AJ and AJ , while the numbers quoted in Table 2.4 2 $ *0 are based on a one-parameter "t to many more observables. The di!erence between the values for M from both analyses is not too large. This re#ects the importance of pJ , AJ and AJ in the 8Y 2 $ *0 M constraint. 8Y 2.1.4.3. Constraints on g . GUTs are the main motivation for the search for extra neutral gauge bosons. In GUTs, all gauge interactions are uni"ed at high energies. In general, it is not expected that the renormalization group equations change the gauge couplings drastically during the evolution from GUT energies down to low energies. Therefore, one expects g +g at low energies. On the other hand, it is useful from the experimental stand point of view to "nd all possible constraints on new particles, which can be derived from the data. This is the reason why we consider limits on a Z with small couplings to all SM fermions. The best present constraints on a Z with small couplings in the case of a non-zero ZZ mixing come from measurements at the Z peak, see Fig. 2.4. Here, we assume that there is no ZZ mixing. Estimate: Eq. (2.51) can be inverted to give an estimate of the sensitivity to the coupling strength g M !s *O 8Y . (2.57) g "g s O
Models with couplings g 'g would give a signal in the observable O. Formula (2.57) is not true near the Z resonance. If one assumes that measurements with an accuracy of 1% are performed at energies (s, (2s, (4s,2, a Z with g 'g /7 is excluded below its resonance with 95% con"dence. Alternatively, the bounds on a g and v g follow from the model-independent limits on a,, v,, J J J J compare de"nition (2.3), a g "a,(4p("m!s"/s . J J
(2.58)
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Fig. 2.11. Present (thin lines) and future (thick lines) upper bounds (95% C¸) on a g as function of the Z mass. Every J bound is shown for C /M "0.01 (solid lines) and C /M "0.1 (dashed lines). See text for the input.
Formula (2.58) is also valid for energies above the Z resonance. As expected, the sensitivity to a weakly coupled Z increases for the centre-of-mass energies approaching the Z mass. As it should be, estimate (2.58) agrees with Eq. (2.57) after substituting a, according to Eq. (2.47). D Present and future constraints: The present bounds on a, and v, from TRISTAN can be read o! J J from Fig. 2.7, those from 1996 L3 data are given in Refs. [62,102], TRISTAN: a,"0.025 v,"0.03, J J LEP 1.5, (s+130 GeV: a,"0.095 v,"0.16, J J (2.59) LEP 2, (s"170 GeV: a,"0.10 v,"0.11. J J The bounds on a, and v, expected from a next linear collider can be taken from Fig. 2.9, J J N¸C, (s"500 GeV, ¸"20 fb\: a,"0.0093, v,"0.011. (2.60) J J The constraints on a g resulting from inputs (2.59) and(2.60) are shown in Fig. 2.11. The limits J on g for models with other couplings are expected to be similar because the constraints on a, and J v, are comparable. The inclusion of PEP and PETRA data would improve the bound on g for J small M . 8Y Although, C is related to g in GUTs, we consider it as a free parameter in Fig. 2.11. This "gure 8Y illustrates that the constraint is not very sensitive to C /M . The same is true for the mass exclusion limits obtained in the previous section. This insensitivity to C /M is an important di!erence to Z limits from hadron collisions. For models, where a g is known, mass limits M can be derived from Fig. 2.11. For example, J 8Y Z "t has pure axial couplings to electrons, a "1/(6, (a g +0.094). It follows that J J
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M "180 GeV from Fig. 2.11. The value M "220 GeV quoted in Table 2.3 is based on the 8Y 8Y same data set. It is better because hadronic observables also enter this number, while only leptonic observables enter the limit from Fig. 2.11. As we will see later, the limit on g for M 's can be further improved for M (s by searching 8Y 8Y for photons from the radiative return to the Z resonance. 2.1.4.4. Errors of model measurements. In the case of deviations from the SM predictions, one has to prove experimentally that these deviations are due to an extra neutral gauge boson. If this is the case, the three observables pJ , AJ and AJ depend on the two couplings a and v only. Therefore, 2 $ *0 J J there must be a relation between these observables. Z theories occupy only a two-dimensional subspace of the three-dimensional space spanned by the values of these three observables. See Ref. [134] for a detailed discussion of this point. If the new interaction is due to a Z , its couplings to all fermions should be measured. These measurements can de done with or without model assumptions as far as these are consistent with the data. In contrast to the Z exclusion limits, the systematic errors of future experiments have a signi"cant in#uence on measurements of Z model parameters. Estimate: Suppose that there exists a Z with M "f M (M . We "rst estimate the 8Y K 8Y 8Y experimental bounds on the Z mass M\(M (M>, which can be set by the observable O. 8Y 8Y 8Y Considerations similar to those of the exclusion limits in Section 2.1.4.2 give
M!"M 8Y 8Y
1G*o(1!*o/f ) M K 8Y . + 1Gf $*o (1Gf K K
(2.61)
The last approximation is valid under the conditions *o;1!f (1 and *o;f , which are K K ful"lled in a reasonable model measurement. For small f , we can further approximate, K M>!M\ *M 1 1 M 8Y 8Y+ 8Y , f . 8Y+ (2.62) M>#M\ 2 K 2 M M 8Y 8Y 8Y 8Y Similar considerations can be used for a measurement of the coupling strength g assuming that the Z mass is known
*g /g +[ f !*o(1!f )]+ f . (2.63) K K K Again the last sequence of the approximations relies on *o;f . In practice, estimates (2.62) and K (2.63) work satisfactorally for f (. K Estimates (2.62) and (2.63) give a general relation between Z exclusion limits and relative errors of Z model measurements: They relate the amount of (¸s) one has to pay for the detection of a Z of a certain model to the amount of (¸s) which is necessary for a model measurement with the C accuracy e, of the same model by the same observables at the same con"dence level [135], (¸s) +(1/4e)(¸s) . (2.64) C The in#uence of systematic errors on model measurements is predicted by estimates (2.62), (2.63) and (2.53), *M /M , *g /g &[(1#r)/s¸] . 8Y 8Y
(2.65)
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Relations (2.65) are scaling laws similar to Eq. (2.53). Compared to exclusion limits, the in#uence of the systematic error is now more pronounced. Present measurements: There are no experimental indications for extra neutral gauge bosons. However, in the PEP [136], PETRA [137] and TRISTAN [138] experiments the couplings (and the mass) of the SM Z boson were constrained by measurements below its resonance. These experimental results allow the test of our estimates (2.62) and (2.63). In fact, these estimates give a correct prediction of the experimental error of the Z coupling measurement *a /a of all these I I experiments within a factor of two. Let us demonstrate the estimates with the results of the AMY collaboration [138]. In the "rst step, we calculate M +350 GeV using estimate (2.51). We took the most accurate observable 8 AI "!0.303$0.028 for *o adding the statistical and systematic errors in quadrature. Estimates $ (2.62) and (2.63) predict *M /M , *g/g+3.4%. This can be compared with the result of the AMY 8 8 analysis, *g/g"*(a#v/(a#v+*a /a "0.024/0.476+5%. J J J J J Future measurements: Fig. 2.12 shows typical results of a "t to a, and v, in the case of a Z signal. J J It includes the full SM corrections and a cut on the photon energy. As in the case of exclusion limits shown in Fig. 2.9, di!erent observables shrink di!erent regions in the parameter space. A twofold sign ambiguity remains as long as fermion pair production is the only process which detects the Z . We would be left with a fourfold sign ambiguity without AJ because only AJ or related *0 *0 observables are sensitive to the sign of v,a,. This underlines the essential role of beam polarization J J in Z model measurements. A superposition of Figs. 2.10 and 2.12 allows us to estimate errors of model measurements. Assume a measurement of the overall coupling strength c,+((a,)#(v,) and M "1.5 TeV. 8Y J J J One "nds from Fig. 2.12 the errors *c,/c,"0.27, 0.23 and 0.11 for Z "¸R, s and SSM. To J J compare this result with estimate (2.62), one should take M obtained from leptonic observables 8Y only, which are 2.0, 2.6 and 2.7 TeV according to Table 2.4 in [126]. The errors predicted by estimate (2.62) are then 0.28, 0.17 and 0.15. The agreement of the numbers is reasonable. In Ref. [126], the couplings of the Z to b quarks are constrained "xing the leptonic couplings to the values predicted in certain E models. Assuming that the couplings to quarks are predicted by the same model, one can constrain the allowed region for a , v by observables with b quarks in @ @ the "nal state. The resulting regions are clearly o! the point (a , v )"(0, 0) as shown in Fig. 2.13. @ @ Compared to Fig. 2.12, the constraints on v, and a, have a larger error, which is due to the larger O O systematic errors of b-quark observables. Alternatively, one can assume Z couplings to leptons, which are inside the combined region of Fig. 2.9 for a 500 GeV collider. Such models do not give a signal in the leptonic observables. Taking di!erent sets of leptonic couplings satisfying these conditions, one can estimate the limits on Z couplings to quarks. Such an analysis is performed in Ref. [102] for LEP2 data and in Refs. [127] for future colliders. Assuming relation (1.25), all couplings of the Z to SM fermions are described by the "ve parameters (1.36). One can then try to "t these parameters by all available observables simultaneously. This is done in Ref. [43] for an e>e\ collider with (s"0.5 TeV, ¸"20 fb\ and M "1 TeV for di!erent models using the 18 di!erent observables 8Y pD , f"l, c, b, t . (2.66) pJ , R , A , RD" 2, AD , AD , AD *0 *0 $ 2 *0 pJ $ 2
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Fig. 2.12. Resolution power of LC500 (95% C¸) for di!erent models and M "1.5 TeV based on a combination of all 8Y leptonic observables. This is Fig. 4b from Ref. [126]. Fig. 2.13. Resolution power of LC500 in the (a,, v,) plane (95% C¸) based on a combination of all b-quark observables. @ @ Di!erent Z models are considered, M "1.5 TeV. This is Fig. 7 from Ref. [126]. 8Y Table 2.5 Values of the couplings (1.36) for typical models and statistical errors as determined from probes at the NLC ((s"500 GeV, ¸"20 fb\, M "1 TeV). 100% heavy #avour tagging e$ciency and 100% longitudinal polarization 8Y of the electron beam is assumed. The numbers are taken from table 4 of Ref. [45] Analysis
s
t
g
¸R
PJ 4 PO "P@ * J PS 0 PB "P@ 0 0 e
2.00$0.08 !0.50$0.04 !1.00$0.15 3.00$0.24 0.071$0.005
0.00$0.04 0.50$0.10 !1.00$0.11 !1.00$0.21 0.121$0.017
!3.0$0.5 2.0$0.3 !1.00$0.15 0.50$0.09 0.012$0.003
!0.148$0.018 !0.143$0.037 !6.0$1.4 8.0$1.9 0.255$0.0016
In Refs. [43,8], one can "nd three-dimensional "gures of possible constraints. We select the result of such a "t from Table 2.5 of the newer analysis [45] and present it in Table 2.5. No radiative corrections and no systematic errors are included in the analysis [45]. The complete Table 4 in Ref. [45] underlines again the crucial role of beam polarization in model measurements. It is shown there that the errors of model measurements without beam polarization are several times larger. Having made measurements of the Z couplings, one can go one step further and check whether the signal is compatible with a Z originating from an E GUT, i.e. check whether relations (1.39) are ful"lled. If this is the case, one can try to de"ne the underlying parameters g /g and b of the breaking of the E group from the data [45]. Alternatively, one can try to constrain [45,126,130] the parameters a or b parametrizing the E models as given in Eqs. (1.40) and (1.44). Inyuence of systematic errors: Let us compare scaling (2.65) with the results of an exact calculation shown in Fig. 2.14. We "rst note that the di!erent scenarios change mainly the size of
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Fig. 2.14. In#uence of luminosity, Z mass, and systematic error on contours of Z bbM couplings. A Z in the s model is assumed; (s"500 GeV. This is Fig. 3 of Ref. [139]. Table 2.6 Z coupling combinations PJ , P@ and P@ and their one-p errors derived from all observables with and without systematic 4 * 0 errors for (s"500 GeV and M "1 TeV. This is Table 5 of Ref. [126] 8Y
PJ , no syst. err. 4 PJ , syst. err. included 4 P@ , no syst. err. * P@ , syst. err. included * P@ , no syst. err. 0 P@ , syst. err. included 0
s
t
g
¸R
2.00$0.11 2.00$0.15 !0.500$0.018 !0.500$0.070 3.00> \ 3.00> \
0.00$0.064 0.00$0.13 0.500$0.035 0.500$0.130 !1.00$0.29 !1.00> \
!3.00> \ !3.00> \ 2.00> \ 2.00> \ 0.50$0.11 0.50> \
!0.148> \ !0.148> \ !0.143$0.033 !0.143$0.066 8.0> \ 8.0> \
the constrained region but not its shape. The change of the size can be characterized by one number. Let us normalize the size of the region without systematic errors to unity. Then the sizes of the regions of the "ve scenarios in Fig. 2.14 are (legend from top to down) 1.9, 5.6, 2.5, 1 and 2.9. Combining the curves with *"0 and *"1.5% for ¸"50 fb\ and using estimate (2.65), we calculate */* "r"2.3, and therefore * "0.65%. This is exactly the value, which would follow from the expected number of b quark events. We now estimate (2.65) the sizes of all remaining curves in Fig. 2.14 as 1.8, 4.2, 2.5, 1, 2.8. The agreement except for the second curve is surprisingly good. The disagreement of the second curve arises because the accuracy e of the model measurement is well above , which means that Eq. (2.65) is a bad approximation. In a similar way, the di!erent scenarios of the other "gures in Ref. [139] are reproduced by relation (2.65). Estimate (2.65) can be confronted with the measurements of model parameters quoted in Table 2.6. The in#uence of systematic errors is predicted by Eq. (2.65). For r"1, as expected for leptonic
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observables in Ref. [126], *PJ are predicted to relax by (2, which reproduces the tendency in 4 Table 2.6. P@ given in Table 2.6 is only measured by observables with b quarks in the "nal state. It is * therefore dominated by the systematic errors of b quark observables. In Ref. [126], these systematic errors are roughly four times as large as the statistical errors. According to estimate (2.65), r"4 should enlarge *P@ and *P@ by a factor (17. Indeed this happens for some models in Table 2.6. * 0 To conclude, the systematic errors have a large in#uence on errors of model measurements. Combining measurements at several energies: The estimates obtained above for measurements at one energy point can be generalized to measurements, which are distributed over several energy points s with integrated luminosities ¸ . Observing that only the combination s¸ enters all G G estimates, one can rescale all measurements to one energy s with the luminosity ¸ " s ¸ /s G G G and assume one measurement at s with ¸ . A di!erence occurs, if one wants to use several energy points for a simultaneous measurement of the Z mass and the Z couplings by a "t to the line shape as proposed in Refs. [140,141]. Such a measurement was demonstrated historically for the SM Z boson at the PEP and PETRA colliders. It demands measurements of high accuracy at several well separated energy points. The luminosities ¸ should be chosen in such a way that the statistical signi"cance G *8YO/*O+(s ¸ G G
(2.67)
is about equal for the di!erent energy points s . G Although the error of the simultaneous "t of M , v , a scales with the product s¸ like the error of 8Y J J a "t to v,, a,, the prefactor is very di!erent. One needs much more luminosity to measure the J J couplings and the Z mass separately compared to a measurement of v, and a, only. Here is one D D example of the amount of s¸ needed for di!erent experiments ((s(M "1.6 TeV, 95% C¸): Q Detection: measurement of g/M or v,, a, with 15% error: 8Y J J measurement of g and M separately with 15% error: 8Y
s¸+0.7 TeV/fb s¸+8 TeV/fb s¸+260 TeV/fb
The "rst two numbers are obtained from Table 2.4 using scalings (2.53) and (2.62) with r"0, the third number is taken from Ref. [140]. 2.1.5. Z measurements at s+M A precision measurement on top of the Z resonance in e>e\ or k>k\ collisions would be the best experiment to study the properties of an extra neutral gauge boson. See for example [6,142}144] for related studies. Such an experiment would have much in common with the measurements at the Z -peak at LEP 1 and SLC, however, there are also important di!erences. In the case of a non-zero ZZ mixing, in addition to the decay Z Pf fM other decay modes such as Z P=>=\ or Z PZ H are allowed. Because there are no experimental hints for a non-zero ZZ mixing, we discuss the decay to =>=\ in a di!erent section. If Z -decays to exotic fermions are kinematically allowed, the number of observables at the Z -peak is even larger than that at the Z -peak yielding important additional information on the breaking scheme of the underlying GUT.
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A second di!erence to the Z -peak is the e!ect of beamstrahlung. The resulting energy spread of the beams is expected to be between 0.6% and 2.5% for the discussed 500 GeV e>e\ colliders and even larger for higher energies [145]. Some precision measurements would demand an energy spread as low as 0.2% [146]. A precision scan of the Z peak is among these measurements. Here, a k>k\ collider would have clear advantages over an e>e\ collider. In k>k\ collisions, the energy spread of the beams is naturally between 0.04% and 0.08% [67]. It is expected that it can be further reduced to 0.01% [67]. The absolute calibration of the centre-of-mass energy is expected to be of the same accuracy. As in the previous cases, Z measurements at the Z peak can also be made with or without model assumptions. 2.1.5.1. Model-independent Z measurements. A model-independent measurement on the Z reson ance can be done generalizing the model-independent approach to the Z resonance [147,148]. It has the advantage that no unknown radiative corrections are needed. We follow here a notation close to Ref. [149], where also a more extensive discussion and further references may be found. The ansatz for the four helicity amplitudes is MDG(s)"es([RD/s#RDG/(s!m )#RDG/(s!m )#FDG(s)], i"0, 1,2, 3 . (2.68) 8 A 8Y FDG(s) is an analytic function without poles. In general, expression (2.68) is not unique. However, it makes sense, if the resonances are well separated. This is the case because we have m ;m due to the present experimental constraints. The QED parameter RD can be measured at such a low energy A that all other contributions are unimportant. Therefore, a model-independent "t to the Z reson ance can be performed xxing the QED term. See Ref. [148] for a "rst "t and [150] for recent experimental results. There are no hints for a Z at the Z resonance. Therefore, the "ts to the Z resonance are independent of a possible term of the Z pole. By the same scheme, a model independent "t of the Z resonance can be performed with amplitude (2.68) xxing the QED and the Z terms. Such a procedure is as unique as are the present model-independent "ts to the Z resonance. The complex constants RD and RDG can be derived within a given theory to a certain order of A 8 perturbation series. At the Born level, we have RD"QCQD , A RD"4¸ ¸ g/e, RD"4¸ R g/e, RD"4R R g/e, RD"4R ¸ g/e , 8 C D 8 C D 8 C D 8 C D RD"4¸ ¸ g/e, RD"4¸ R g/e, RD"4R R g/e, RD"4R ¸ g/e . (2.69) 8Y C D 8Y C D 8Y C D 8Y C D The complex mass m "M M !iMM CM is slightly di!erent from the on-shell mass m because it G G G G G contains a constant width de"ning the complex pole in the ansatz (2.68). If the vector boson Z can L only decay into light fermion pairs, we have [42] MM "M !C/2M , CM "C !C/2M . (2.70) L L L L L L L L From amplitude (2.68), total cross sections can be calculated by the standard procedure,
pD"(1/2s) "MDG(s)" dC""MDG(s)"(1/(4p))p/s . G
(2.71)
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The four Born cross sections pD, pD, pD and pD introduced in Eqs. (2.10) and (2.11) are obtained 2 $ *0 by linear combinations of pD G pD"#pD#pD#pD#pD, pD "#pD!pD#pD!pD , 2 $ pD "!pD!pD#pD#pD, pD "!pD#pD#pD!pD . (2.72) *0 Combining Eqs. (2.71) and (2.68), one arrives at a model-independent formula for cross sections
M )jD sr D#(s!M M )j D 4pa rAD srD #(s!M # # , A"T, FB, ¸R, pol . (2.73) pD(s)+ s "s!m " "s!m " 3 In formula (2.73), the functions FDG(s) and terms without poles are neglected for simplicity. At the Born level, the SM coe$cients are [148] 1 rAD" N [$1]"RD" , A 4 D G 1 CM [$1]"RDG"#2 ImCD , rD "N 8 D 4 M M G jD "N +2ReCD !2(CM /M M )ImCD , , D 1 CD " (RD)H [$1]RDG . (2.74) 4 A 8 G The signs [$1] are the same as the cross sections pD in Eq. (2.72). The parameters of the Z peak are
1 CM r D"N [$1]"RDG"#2 Im(C D!C D) , D 4 8Y M M G j D"N +2ReC D!2(CM /MM )Im C D#2(1!M M /M M )ReC D#2(CM /M M )Im C D, , D 1 C D" (RD)H [$1]RDG , 8Y 4 A G 1 (2.75) C D" [$1]RDG(RDG)H . 8 8Y 4 G M /M is a small parameter because the Z and Z peaks are well separated. We therefore neglected terms of the order M/M, M C /M or smaller. This keeps formulae (2.75) relatively simple. Before formula (2.73) can be used for "ts to data, QED (and QCD) corrections must be taken into account [148]. Initial state corrections can be calculated by the convolution (2.21). Final state radiation and the interference between initial and "nal state radiation can be included by a di!erent convolution [148]. However, these corrections do not change the pole structure. Therefore, they could be absorbed into e!ective coe$cients (2.74) and (2.75). 2.1.5.2. Model-dependent Z measurements. Z measurements at the Z peak are precision measurements. They require radiative corrections. Unfortunately, these corrections depend on all
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the parameters of the whole theory. If these are poorly known, theoretical uncertainties arise. This is similar to LEP 1 and SLC where theoretical errors of the radiative corrections at the Z peak arise through the unknown Higgs mass and through the experimental errors of the top- and the = mass. At a Z peak, the situation is much more uncertain. Today it is not known whether a Z exists at all. It is even more speculative to predict the underlying gauge group. If it is known, one would still need some idea about the breaking scheme, the particle content and particle masses to calculate radiative corrections. We have shown in the previous sections that some information can be obtained by future experiments below the Z peak. Because of the di$culties mentioned above, we have to constrain ourselves to general conclusions and estimates in the Born approximation. The cross section at the resonance peak can be expressed through branching ratios p (e>e\Pf fM )+pD(M; Z , Z )"(12p/M)(CC /C )(CD/C )"(12p/M )BrC BrD . (2.76) 2 2 For a Z originating in usual GUTs, one expects millions of muon pairs and tens of millions of hadron pairs per year from Z decays at the proposed electron or muon colliders. This is similar to the situation at LEP 1 and SLC. The systematic errors at future colliders are expected to be at the same level as at LEP 1 of SLC. Therefore, it can be expected that the Z couplings to fermions could be measured with a similar precision as the Z couplings. The measurements of Z couplings constrain the ZZ mixing angle. The constraints from the Z peak are expected to be stronger than those from the Z peak because in a GUT the couplings of the Z to fermions are in general larger than the couplings of the Z to fermions. A sensitivity to h +10\ is derived in Ref. [143]. + The measurements of Z couplings constrain parameters of the GUT. In a naive estimate, one would expect an accuracy of a measurement of cos b of an E GUT comparable to the accuracy of the sin h measurement at the Z peak. 5 The measurement of M and C is limited by the beam energy spread and by the error of the energy calibration *(s; *M , *C '*(s. 2.1.5.3. Limit on g . The best limit one can put on a weakly interacting Z is obtained for M "(s. The cross section at the Z peak (2.76) is independent of the ratio g /g . The sensitivity to g /g is limited at an e>e\ collider because it has a "nite beam energy spread *(s/(s. Only cross sections of e>e\ pairs with (s"M $C are enhanced. The observed number of Z events can be approximated for *(s'C , C (s g C (s 12p N8Y +¸ p(M; Z , Z )+ ¸ BrC BrD . DD M *(s 2 g M *(s M
(2.77)
In the last step of the approximation, we assumed C /M +gC /gM . A Z produces a signal of n standard deviations if N N8Y "n (N1+"n (¸pD(M) . N 2 DD N DD
(2.78)
(2.79)
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An estimate for g can now be obtained from Eq. (2.77), g +g (n N
*(s M 1 pD(M)M 2 . C 12pBrC BrD ¸ (s
(2.80)
Assuming *(s/(s"1%, BrC "BrI "3.36%, ¸"80 fb\ and M "1 TeV , we get g +g /140 with 95% con"dence (n "2) for the reaction e>e\Pk>k\. N
(2.81)
2.1.6. Z limits at s'M Above the Z resonance, the sensitivity to ZZ mixing is much lower than on the resonance. We therefore neglect the mixing identifying the Z and Z with the Z and Z . Suppose that a Z was missed below its resonance because it has very weak couplings. The question we want to discuss here is, for which coupling strengths is it possible to detect such a Z above its resonance. If the couplings to all fermions are very small, the Z eventually escapes detection. A vector boson, which couples to quarks only, can still have quite large couplings and be consistent with the present data. See Ref. [151] for a discussion of this point and for further Ref. [152] for bounds on such a Z from di!erent experiments. It turns out that the error of the photon energy measurement *E /E is an important input of the A A bounds because they arise from events with a fermion pair and one hard photon in the "nal state. At a k>k\ collider, this parameter limits the constraints. However, as mentioned in the previous section, the proposed e>e\ colliders su!er from a large beam energy spread *(s/(s. Then, the error of the photon energy *E in the estimates has to be replaced by *E #*(s. A A 2.1.6.1. Model-independent limits on g . Starting from relation (2.51), an upper bound g on the coupling strength g of the Z to SM fermions can be derived, (2.82) g +g ((*O/O)(s!M )/s . 8Y It follows that a one percent cross section measurement of the reaction e>e\Pf fM can exclude models with g 'g /7 at 95% con"dence for all M ((s. 8Y The sensitivity to a Z is considerably larger than Eq. (2.82) if one considers the photon energy spectrum of the reaction e>e\Pf fM c. As discussed in Section 2.1.2.1 and shown in Fig. 2.2, the spectrum of the photons radiated from the initial state has sharp peaks for energies which set the f fM subsystem back to the resonance. The energy E "* ) E of the photons responsible for the A radiative tail is distributed in the narrow range *>(*(*\ with *!"1!M /s$M C /s. 8Y 8Y 8Y The number of events with these photons can be estimated from the magnitude of the radiative tail (2.27) N8Y+¸p (s; Z , Z )b (p/2)M /C +(g/g)¸p (s; Z, Z)b (p/2)M /C . A 2 C 8Y 8Y 2 C 8 8 In the last step of approximation (2.83), we used estimate (2.78) and p (s; Z , Z )/p (s; Z, Z)+g/g . 2 2
(2.83)
(2.84)
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Of course, these approximations can be improved if more details of the model are known. The SM background also contributes photons to the interval (*\, *>). The number of these events can be estimated by convolution (2.21) N1+"¸[p'10(s)!p(s)]"¸ A 2 2
>
\
dv p(s(1!v))HC (v)+¸p(s)b (*>!*\)/*H(1!*H) . 2 2 C (2.85)
*H is some value between *\ and *>. We assumed that p(s)&1/s for energies between 1/s and 2 1/[s(1!*H)]. Estimate (2.85) can be compared with the SM result in Fig. 2.2. It gives a satisfactory prediction away from resonances. The ratio of the Z signal and the SM background for photon energies between *\ and *> can now be estimated as
N8Y p M p(s; Z, Z) M A + 8 2 1! 8Y . (2.86) N1+ 4 C p(s) s A 8 2 Again, approximation (2.78) is used. Note that ratio (2.86) is independent of g /g . Numerically, we get N8Y/N1++40 for A A pI (1 TeV) and M "800 GeV . (2.87) 2 E This estimate is in a good agreement with Fig. 2.2. To detect the Z signal, two additional conditions must be ful"lled. The luminosity must be high enough to produce a reasonable number of events and the error of the photon energy *E /E must A A be small enough to detect the signal above the background. Let us "rst assume an arbitrarily good photon energy resolution. Assume that the events are Poisson distributed. Then, zero observed events exclude all theories with 95% con"dence, which predict N8Y53. This can be interpreted as a limit g on g , A N8Y p(s) 2 C A 2 8 g "g . (2.88) N1+ p(s; Z, Z) pb M DD 2 C 8 N1+ is the number of fermion pairs expected in the Born approximation as de"ned in Eq. (2.79). DD Estimate (2.88) gives the best bound on g , which could be reached with a given luminosity in the reaction e>e\Pf fM c. Numerically, we get g +g /45 under assumptions (2.87) and ¸"80 fb\ and N8Y"3 . (2.89) A Unfortunately, the energy resolution *E /E +0.1/(E /GeV of real detectors is "nite [153]. As A A A a result, all photons with energies *"(1!M /s)(1$*E /E ) are observed in the experiment. For 8Y A A the expected luminosities at future colliders, there are photons N1+ from the background even in A the narrowest bin of the photon energy. Numerically, we get N1++30 from estimate (2.85) under A assumptions (2.87), (2.89) and *E /E "1%, which is in good agreement with Fig. 2.2. With such A A an event number, we can assume Gaussian statistics in our estimates. One expects a n !p signal N for theories predicting
N8Y"n (N1+ . A N A
(2.90)
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The resulting expression for g can easily be derived from Eqs. (2.90), (2.85) and (2.83) g "g c [(8/pb )C/M] +g ) 0.266c , P C 8 8 P c "[(*E /E )1/N1+] [n [p(s)/p(s; Z, Z)](s/M ] . (2.91) P A A DD N 2 2 8Y Numerically, we get g +g /24 with 95% con"dence (n "2) under the same conditions as before N and a photon energy resolution of 1% and (s/M +1. 8Y As we see, the consideration of fermion pair events with one additional hard photon gives a considerable improvement of g obtained from o!-resonance fermion pair production. Present upper bounds on g from fermion pair production without additional photons are displayed in Fig. 2.11. Fermion pair events accompanied with hard photons are investigated at LEP [154]. This allows to reconstruct cross sections and asymmetries of e>e\Pk>k\ for energies lower than (s. 2.1.6.2. Measurements of M . If a Z signal with M (s is found, the Z mass can be measured, 8Y 8Y M "(s(1!E /E ), *M /M "(*E /2E )(s/M !1) . (2.92) 8Y A 8Y 8Y A A 8Y We exploited formula (2.29) in the derivation of this estimate. E is the photon energy of the hard A photons from the radiative return. Using the knowledge of M , one can tune the energy to the 8Y resonance and perform precision measurements there. 2.2. Z search in e>e\Pe>e\ and e\e\Pe\e\ Bhabha and M+ller scattering can probe the Z couplings to electrons only. While Bhabha events serve as additional observables in e>e\ collisions, M+ller scattering requires the e\e\ option of a linear collider. Although the luminosity of e\e\ collisions is expected to be smaller than that of e>e\ collisions because of the anti-pinch e!ect, M+ller scattering has the advantage of two polarized beams and of a cleaner environment. Early Z analyses can be found in Refs. [155}157] for Bhabha scattering and in Ref. [157] for M+ller scattering. 2.2.1. Born approximation 2.2.1.1. Amplitude. In Bhabha (M+ller) scattering, electrons and positrons (only electrons) appear in the "nal state. The neutral gauge bosons are exchanged in the s and t (t and u) channels. The resulting angular distributions are very singular for small scattering angles. M+ller scattering has a symmetrical angular distribution. The angular distribution of Bhabha scattering is peaked in the forward direction. Considerations similar to fermion pair production in Section 2.1.1 show that these reactions can constrain only the model-independent parameters a, and v,. As o!-resonance fermion pair C C production, they are rather insensitive to ZZ mixing. We therefore neglect the mixing angle putting h "0. +
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2.2.1.2. Cross section. The Born cross section of Bhabha scattering including the Z exchange is, following the notation of Ref. [158]: dp pa " ( f #(j !j ) f #j j f ) > \ > \ dc 2s
(2.93)
with (1#c) ) [G (s, t)#G (s, t)] f "(1#c) ) G (s, s)#2c ) G (s, s)!2 1!c #2
(1#c)!4 (1#c)#4 ) G (t, t)#2 ) G (t, t) , (1!c) (1!c)
(1#c) (1#c) H(s, t)#4 H(t, t) , f "(1#c)H(s, s)!4 (1!c) (1!c) f "!f #[16/(1!c)][G (t, t)!G (t, t)]
(2.94)
and , G (s, t)"Re s (s)sH(t)[v (m)v (n)H#a (m)a (n)H], K L C C C C K L , G (s, t)"Re s (s)sH(t)[v (m)a (n)H#a (m)v (n)H] , K L C C C C K L , H(s, t)"Re s (s)sH(t)[v (m)v (n)H#a (m)a (n)H][v (m)a (n)H#a (m)v (n)H] K L C C C C C C C C K L with t"!(s/2)(1!c) .
(2.95)
The summation runs over the exchanged gauge bosons. See Section 2.1.1 for further de"nitions. The Born cross section of M~ller scattering including the Z exchange is [121], gg /(4pa) dp 16pa , L K " s (k !c)(kH!c) dc L K L K ;+4j (R RH#¸ ¸H)k kH#4j (R RH!¸ ¸H)k kH K L K L K L K L K L K L #j R ¸ RH¸H[k kH#(1#k kH#2k #2kH)c#c], K K L L K L K L K L
(2.96)
with k "1#2m/s and j "1#j j . (2.97) L L > \ Again, the summation runs over the exchanged gauge bosons. R and ¸ denote the left- and K K right-handed couplings to electrons, ¸ "¸ (m), R "R (m). Further de"nitions can be found in K C K C Section 2.1.1. Alternatively to formulae (2.95) and (2.96), the Z contributions can be included by form factors as explained in Section 1.4.
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2.2.1.3. Observables. Consider "rst Bhabha scattering. Only contributions proportional to f can be measured with unpolarized beams. With polarized electrons, one can measure the left}right asymmetry A (c)"(dp !dp )/(dp #dp ) . (2.98) *0 * 0 * 0 It is sensitive to contributions proportional to f . Two polarized beams allow a measurement of the asymmetry [155] A (c)"(dp !dp )/(dp #dp ) . (2.99) * ** 00 ** 00 It is sensitive to contributions proportional to f . This is di!erent from fermion pair production and = pair production where two polarized beams give no new information compared to electron polarization only. LEP 2 has naturally transverse beam polarization. Then, transverse asymmetries [155] can be considered,
dp( (>p dp A((c)"(dp(!dp(>p )/(dp(#dp(>p ) with " d . (2.100) 2 dc d dc (\p In contrast to Bhabha scattering, it is sure that in M~ller scattering both electron beams can be highly polarized. With two polarized beams, one can measure several angular distributions [121] (1/p)dp/dc, (1/p**)dp**/dc, (1/p00)dp00/dc, (1/p*0)dp*0/dc ,
(2.101)
which are all linear combinations of the three contributions (2.96) proportional to j , j and j . In "xed target M+ller scattering, the left}right asymmetry 1!y G Q p !p 0" I (1!4s ) A (e\e\Pe\e\)" * 5 *0 1#y#(1!y) p #p (2pa * 0 with Q"y(2m#2m E ) , y"!(p !p)/(p #p) (2.102) C C is of special interest. It can be measured with very high precision in future experiments [159]. The last sequence in Eq. (2.102) is valid [160] in the limit (s;M. p(p ) are the energy}momenta of 8 one initial ("nal) electron. 2.2.2. Radiative corrections The generalization of the SM radiative corrections to s-channel Z exchange is discussed in Section 2.1.2. No essential new problems arise due to the Z exchange in the t or u channel. We therefore limit ourselves to give the main references to the SM processes. QED corrections to SM Bhabha scattering can be found, for example, in [161,162]. QED corrections are universal allowing a generalization of the SM result to the whole process including additional Z contributions. See Refs. [158,162}164] for results of weak corrections. Weak corrections together with ZZ mixing could be taken into account as in the case of fermion pair production by replacements (2.30) and (2.31) of the couplings. In Ref. [165] one "nds the needed
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formulae. However, such a replacement is not necessary in Bhabha and M+ller scattering because these reactions are as insensitive to ZZ mixing as o!-resonance fermion pair production. QED corrections to M+ller scattering are calculated in Ref. [166], while the electroweak corrections to A are calculated in Ref. [160]. QED initial state corrections can be taken into *0 account [167] by the structure function approach [81]. QCD corrections to both processes enter as virtual corrections at one loop as described in Refs. [160,164]. 2.2.3. Z Constraints 2.2.3.1. Model-independent constraints on v, and a,. Estimate: In contrast to fermion pair producC C tion, the total cross section and simple asymmetries are not very sensitive to a Z . The best sensitivity is achieved by "ts to angular distributions of polarized cross sections. For M+ller scattering, the Cramer}Rao minimum variance bound [120] is given in the second reference of [121] in the limit M;s;M , s "1/4. We have in our notation, 8 8Y 5 s +256p(¸/s)[(R,)#(¸,)]"32p(¸/s)[(v,)#6(v,a,)#(a,)] . (2.103) C C C C C C This bound corresponds to the sensitivity of an experiment with an in"nite number of angular bins and no systematic errors. It can be compared with constraints (2.47) and (2.48) obtained for di!erent observables in fermion pair production. In particular, estimate (2.103) predicts the widths of the bands in the R , ¸ plane allowed by leftC C and right-handed electron scattering alone "R,", "¸,"([s s/256p¸] . (2.104) C C Future constraints: Model-independent Z limits from M+ller scattering are studied in [121] at the Born level. Fig. 2.15 shows the regions of Z couplings to electrons, which could be excluded. ¸ and R in Fig. 2.15 are related to our conventions as ¸ "2R g /e, R "2¸ g /e. R and ¸ are C C restricted independently in experiments with both beams right-handed or both beams left-handed polarized. Estimate (2.104) is in good agreement with Fig. 2.15. For one left- and one right-handed beam, one is sensitive to Z models where the combination R ¸ exceeds a certain value. This property can immediately be read o! from the cross section (2.96). The allowed region for unpolarized beams is also shown in Fig. 2.15. The distributions of two left-handed or two right-handed scattered electrons contain almost all information on a Z . In contrast to fermion pair production, polarized beams give important improvements already to the Z exclusion limits. The exclusion limits from M+ller scattering are compared with those from Bhabha scattering and fermion pair production in Fig. 2.16. We have v "2v g /e, a "2a g /e in our conventions. 8Y C 8Y C Under the assumptions made in [167], M+ller scattering gives the best Z constraints to the model-independent Z exclusion limits. After the inclusion of observables with electrons and q's in the "nal state, the exclusion limits of M+ller scattering and fermion pair production become comparable. The exclusion limits of Bhabha scattering would improve with polarized positron beams. The in#uence of systematic errors due to the polarization error, the angular resolution and the luminosity error on Z exclusion limits are studied in Ref. [167]. Z limits can also be obtained from e\k\ scattering [169]. Assuming generation universality, this reaction constrains the same parameter combination as M+ller- or Bhabha scattering. The
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Fig. 2.15. Areas of the leptonic Z couplings ¸ and R , which are excluded with 95% con"dence by M+ller scattering for di!erent beam polarizations. (s"500 GeV, ¸"10 fb\ and M "2 TeV were assumed. The results are obtained by 8Y collecting events with "c"(0.985 in 10 equal bins. This is Fig. 1 from the "rst reference of [121]. Fig. 2.16. Exclusion limits (95% C¸) from fermion pair production, Bhabha and M+ller scattering for the couplings of the Z to leptons including systematic errors. The numerical input is (s"500 GeV, ¸"50 fb\(¸"25 fb\) for e>e\(e\e\) scattering. The electron polarization is P "90%, *P /P "1%, *¸/¸"0.5%, "cos h"(0.985, *h"10 mrad. C C C Ten equal bins in cos h are chosen. This is Fig. 5 from Ref. [167].
exchange of neutral gauge bosons in e\k\ scattering is possible only in the t channel. However, it seems to be much more di$cult to create a highly polarized muon beam of high luminosity then an electron beam with the same properties. 2.2.3.2. Model-dependent constraints on M . Estimate: An estimate of Z limits from Bhabha and 8 M+ller scattering can be obtained by considerations similar to those which lead to estimate (2.51). The observable O is now the relative number of events in a certain angular bin. Comparing the shift *8YO due to a Z with the SM prediction O , one gets 1+ t *8YO g "Re+s s ," g " + . g "s " g t!M O 8Y 1+
(2.105)
It follows that
M (M " s ) 8Y 8Y
1!c g O 1!c 1 " s) 1# 1# 2 g *O 2 *o
(2.106)
produces a signal in the observable O. A similar estimate of M can be derived from constraint 8Y (2.103).
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Fig. 2.17. Contours of resolvability at 95% con"dence of the Z couplings around several possible true values marked with a &#'. The assumptions are (s"0.5 TeV, ¸"40(20) fb\ for e>e\(e\e\) collisions. Ten bins in the scattering angle between 103 and 1703 are used. The Z mass is 2 TeV. This is Fig. 5 from the third reference of [121].
Comparing estimates (2.106) and (2.51), we conclude that the Z mass limits from Bhabha and M+ller scattering could be competitive to e>e\Pf fM with leptons in the "nal state only. For completely speci"ed models where the annihilation into quarks contributes to M , the mass 8Y exclusion limit from fermion pair production is better. Future constraints: Future constraints on M can be obtained from Figs. 2.15 and 2.16 using 8Y scaling (2.52). The limits from M+ller scattering are better than those from e>e\Pk>k\ and e>e\Pe>e\. However, e>e\Pf fM gives better limits on M if observables with quarks in the "nal 8Y state are included. A measurement of A (2.102) in a "xed target experiment at SLAC is expected to have the *0 precision *A "1.4;10\, while the SM prediction is A "1.8;10\ [160]. A Z from the *0 *0 E group (1.41) would multiply A by the factor [168] *0 1#7(M/M )(cos b#(sin bcos b) . 8 8Y
(2.107)
Therefore, the experiment is sensitive to M (870 GeV. Q 2.2.3.3. Errors of model measurements. The error of a Z model measurement in Bhabha and M+ller scattering is given by estimates (2.62) and (2.63). Of course, these reactions can only constrain observables involving the Z couplings to electrons. Fig. 2.17 shows the result of a corresponding analysis. Fermion pair production and M+ller scattering are complementary in a model measurement. Fermion pair production removes a sign ambiguity present in the measurements of M+ller scattering.
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2.3. Z search in e>e\P=>=\ The symmetry eigenstate Z does not couple to the = pair due to the S;(2) gauge symmetry. * The process e>e\P=>=\ is sensitive to a Z only in the case of a non-zero ZZ mixing. The individual interferences of = pair production rise proportional to s in the limit of large centre-ofmass energies (s. In the SM, the sum of all interferences scales like ln s/s in the limit of large s due to a delicate gauge cancellation. In the case of a non-zero ZZ mixing, the couplings of the Z di!er from the SM predictions for the Z. Then, the gauge cancellation present in the SM is destroyed. The result is a huge magni"cation of new physics e!ects at large energies. Unitarity is restored at energies s<M independently of details of the large gauge group. Similar to the reaction e>e\Pf fM , it is useful to distinguish di!erent cases, Case 1:
s(M , Case 2: s+M , Case 3: s'M , (2.108) where s+M means (M !C )(s((M #C ). Case 1 gives stringent exclusion limits on ZZ mixing. Case 2 allows the best exclusion limit or the most accurate measurement of the ZZ mixing angle if a Z exists. Case 3 cannot add new information compared to cases 1 and 2. We assume here no mixing of the SM = bosons with extra charged gauge bosons. Furthermore, we neglect a possible mixing between SM fermions and exotic fermions. These e!ects are considered, for example, in Ref. [170]. A measurement of the = polarization would be very useful to separate ZZ and lepton mixing e!ects [125] and for a simultaneous constraint of many anomalous couplings [171,172]. An early analysis of Z e!ects in = pair production can be found in [173]. 2.3.1. Born approximation 2.3.1.1. Amplitude. Following Ref. [93], we write the amplitude as M"M #M , where the s and R Q t channel contributions including a Z are (2.109) MH\"((j !1)/4t s );TH\(s, c) , R \ J5 g [v (1)!j a (1)] g g [v (2)!j a (2)] g e g \ C # 558 C \ C MH\" ! 55A # 558 C ;GH\(s, c) . Q s!M s!M s
j ("!j )"$1 is the electron (positron) helicity as de"ned in Section 2.1.1, (s is the total \ > centre-of-mass energy of the e>e\ pair, c"cos h where h is the angle between the => and the positron and the invariant t is de"ned in Eq. (2.120). The extra neutral gauge boson changes only J the s channel amplitude. The functions TH\(s, c) and GH\(s, c) are not important in the following discussion. They are the same as in the SM and can be found, for example, in Ref. [171]. The amplitude of = pair production is linear in the Z couplings to the electron. This makes the = pair production sensitive to the absolute sign of these couplings. Their measurement can remove
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the sign ambiguity present in fermion pair production where the Z couplings to fermions always appear in pairs. The coupling constants of the interactions between three gauge bosons are "e(c /s )c , g "e(c /s )s . (2.110) "e, g 5 5 + 558 5 5 + 55A 558 The contribution of the extra neutral gauge boson can be absorbed in two s-dependent anomalous couplings [174] g
gH "e(1#d ) and gH "e(cot h #d ) . 5 8 55A A 558 The s channel amplitude including a Z is then
(2.111)
g [v !j a ] gH e gH \ C ;GH\(s, h) , MH\" ! 55A # 558 C (2.112) Q s!M s where d and d contain contributions due to the Z exchange and due to the ZZ mixing in the A 8 Z exchange [93], v g a (1) v (1) c v g a (2) v (2) c s( # 5s C C ! C s( , d " 5c C C ! C A s + e s + e a a v v C C C 5 C 5 c c a (1)s( c g a (2)s( # 5s C d "! 5# 5c C (2.113) 8 s s + a s( s + g a s( 5 5 C 8 5 C 8 and
s( "s/(s!m), n"Z, 1, 2 . (2.114) L L Because = pair production is studied su$ciently far away from the Z peak, we can neglect the Z and Z widths putting m "M and m "M . We know from present measurements [92] that 8 8 M !M (150 MeV. This allows the approximation 8 2M (M !M ) s( +1 . +1! 8 8 (2.115) s!M s( 8 8 The expressions for d and d can then be written as A 8 s( c a+ s( c v a+ v+ (2.116) d " 5 C C ! C 1! s( , d " 5 C 1! , 8 8 s g a A s e a v s( s( C C 8 5 C 8 5 where v+ and a+ are de"ned in Eq. (2.4). The terms proportional to s( dominate in the case s+M C C but can be neglected in the case s;M. Relation (2.116) shows that measurements of = pair production below the Z peak constrain the same combinations a+, v+ as do measurements of C C fermion pair production on the Z resonance. Experimental constraints on the anomalous couplings gH and gH bound the parameters 55A 558 d and d in a model-independent way. Constraints on d and d can be interpreted as a constraint A 8 A 8 to the combinations of Z parameters given in Eq. (2.113). Far below the Z resonance, the Z mass, the ZZ mixing angle and the Z couplings to fermions cannot be constrained separately.
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2.3.1.2. Cross section. The Born cross section with N exchanged gauge bosons is, following the notation of Ref. [175]:
, e (j dpH\H> Re s (s)sH(s)C (initial fermions) G " 2 !! dc pss s s K L K L e(g 2c ) (g 2c ) , 5 sH(s)[vH(n)#aH(n)] GD (j #j )# 5 GDD (j #j ) # Re L C C !! !! 8s 32 L (2.117)
with /4pa)s/(s!m) . (2.118) j"(s!s !s )!4s s and s (s)"(g g L L L 558L The de"nition of the propagator s (s) is slightly di!erent from Eq. (2.7) to absorb the coupling L constants from one triple gauge boson vertex and from one gauge boson}fermion vertex. The invariant masses squared s and s of the two ='s are equal to M for on-shell = production. The 5 de"nitions of C (initial fermions) and of the helicity combinations j , j are the same as 2 introduced in Section 2.1.1. The kinematic G-functions are [175,176] G (s; s , s ; c)" [jC #12s s C ] , !! GD (s; s , s ; c)"[(s!s !s )C !4s s (s(s #s )!C )/t ] , J !! GDD (s; s , s ; c)"[C #4s s C /t] J !!
(2.119)
with (2.120) C "2s(s #s )#C , C "(j/4)(1!c), t "(s!s !s !c(j) . J The notation used in Eqs. (2.117)}(2.120) allows a generalization to o!-shell = pair production and the inclusion of other 4-fermion background diagrams [175]. In the case where the helicities q (q ) of the =>(=\) can be measured, the cross sections > \ dpH>\H\>/dc should be considered. A corresponding analysis for on-shell = production can be found O O in Refs. [125,172]. 2.3.1.3. Observables. Our starting point for the construction of observables are the di!erential cross sections dpH>\H\>/dc. They allow for the measurement of total cross sections and asymmetries O O [177,178],
> dpH\H> , dc dc \ dpH\H> dpH\H> dc ! , pH\H>AH\H>" dc 2 $ dc dc \ p A "pH\\!pH\,p\!p> 2 2 2 2 2 *0 pH\H>" 2
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p A " 2 *0 $
dp\ dp> dp\ dp> ! ! dc # , dc dc dc dc dc dc dc \ \
pH\H>AH\H>(z)" 2 !#
X
dc \X
dpH\H> dpH\H> \X dpH\H> ! dc ! . dc dc dc dc X \
(2.121)
We omit the indices numbering the polarizations of the ='s to simplify the notation. The observables de"ned above can be understood as summed over the = polarizations or as written for "xed = polarizations. As in the reaction e>e\Pf fM , the helicities j "!(#)1 stand for a left ! (right)-handed electron or positron. Missing polarization indices of the electrons or positrons mean the average over initial polarizations. A real detector cannot measure from c"!1 to 1. The correction for this e!ect can be trivially taken into account in observables (2.121). Furthermore, an integration over only a part of the range of c is sometimes recommended to obtain maximum sensitivity to new physics as pointed out in the second reference of [143]. The unpolarized cross section is dominated by the scattering of left-handed electrons, p +p\>. The cross section of right-handed electrons excludes the neutrino exchange giving 2 2 a cross section, which is symmetric in the scattering angle. This induces a relation [177] between two observables, A "A . Cross sections with two left-handed or two right-handed beams are *0 $ $ zero. In the LEP 2 storage ring, the electrons and positrons have naturally transverse polarization. The asimutal asymmetry A is then an interesting alternative observable [172] 2
p dp d(p A ) 2 2 "2 d
cos (2 ) . 5dc d
5 dc 5
(2.122)
The = is an unstable particle, which can only be identi"ed through its decay products. A hadronic = decay allows a measurement of the ='s energy}momentum but not an identi"cation of its charge. A leptonic = decay allows a charge identi"cation but not a measurement of the energy}momentum because a part of it is carried away by the neutrino. If one = decays leptonically and one = decays hadronically, the most complete information about both ='s can be extracted. Only a part of the produced ='s can be reconstructed in the detector leading to an e!ective reduction of the luminosity. If the = polarization can be measured, a more detailed analysis is possible. For details, we refer to [39,179]. For the de"nition of optimal observables [64] in e>e\P=>=\, see Ref. [65]. 2.3.2. Background and radiative corrections The expected accuracy of the measurement of the total =>=\ cross section at future e>e\ colliders is about 1%. It has to be met by the theoretical prediction. Therefore, radiative corrections have to be considered. A short overview can be found in [180], for details and extensive original references see [39]. All present Z analyses of = pair production are done in the Born approximation. An analysis including all radiative corrections relevant to LEP 2 could in principle be done with any of the
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codes described in Ref. [39] if the code allows a setting of the anomalous couplings gH 558 and gH . 55A 2.3.2.1. Background. = bosons are unstable particles, which can be detected only through their decay products e>e\P(=>=\)Pf f f f . (2.123) We have also non-resonant (background) processes to the same order of perturbation theory e>e\Pf f f f , (2.124) which go directly to the same "nal state. Their contribution has to be added coherently to process (2.123) with o!-shell ='s to get a gauge invariant result. Di!erent FORTRAN codes calculating the complete process (2.124) are compared in Ref. [39]. 2.3.2.2. QED corrections. The QED corrections to the amplitude with Z exchange can be deduced from the SM results. We therefore give here only a short description of the related SM corrections. Initial state radiation: The QED corrections to on-shell = pair production are calculated in Ref. [181]. One would like to separate initial state corrections from "nal state corrections and the interference between them in the calculation for o!-shell = pair production because it is much more involved. Unfortunately, this cannot be done in a gauge invariant way. The reason is a charge #ow from the initial state to the "nal state in = pair production. This problem can be treated by the current splitting technique [182], in which the chargeless neutrino exchanged in the t channel is divided into two charge #ows of opposite sign. Now the charge #ows of the initial and the "nal state are separated, and the gauge invariance of initial state QED corrections is ensured as it is in the case of Z pair production. Initial state QED corrections to o!-shell = pair production are calculated in [182]. They reach several % near the == threshold. The corrections can be split into universal contributions, which are described by the #ux function (2.21) or structure function (2.20) approaches with the same functions HC (v) or D(x,s) derived for fermion pair production, and into non-universal contributions depending on the particular process. The non-universal contributions to o!-shell = and Z production are numerically suppressed by a factor s s /s [182]. The initial state QED corrections to o!-shell = pair production calculated in the #ux function (2.21) or structure function (2.20) approach are therefore a good approximation within the expected accuracy of the data. The generalization of these SM results to cross sections including Z exchange is straight forward. The radiative corrections to background (2.124) are a small correction to a small contribution. QED corrections to the background are usually taken into account by convolutions (2.20) or (2.21). Coulomb singularity: The Coulomb singularity [183] arises from long-range electromagnetic interactions between the produced massive charged particles. We get the correction p! "p (1#ap/2j) (2.125) 2 for = pair production, which diverges near threshold where the velocity j of the ='s approaches zero. It indicates that perturbation theory is not applicable in this region. Fortunately, the non-zero
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width C and a slight o!-shell production of the ='s regularize [184] the Coulomb singularity. 5 Nevertheless, the numerical e!ect can exceed 6% for the total cross section near threshold [184]. Coulomb singularities also arise in QED and QCD corrections to pair production of massive fermions [87]. There, the singularity can be avoided by a calculation in the limit of massless fermions or by a cut on the invariant mass of the massive fermion pairs. Such a cut is desirable for quark pairs in any case to avoid non-perturbative bound state regions. 2.3.2.3. Weak corrections. The SM one-loop correction to on-shell =>=\ production is calculated in Ref. [185]. The calculation of the complete SM one-loop correction to process (2.124) is very complex [186] and not done. If it is known in the future, the weak corrections in the presence of ZZ mixing can be treated as described in Section 2.1.2. 2.3.2.4. QCD corrections. The QCD corrections give sizeable contributions to distributions and cross sections. They enter the width of the =, where they can reach several %, see [187]. The QCD corrections can be naively implemented multiplying the cross section by the factor 1#a /p for Q every gauge boson decaying into a quark pair. Such a procedure is only a rough guess for background diagrams and for cross sections with kinematic cuts. The QCD corrections are calculated to O(a ) for on-shell = pair production including the "nal Q = polarizations and kinematic cuts in Ref. [188]. The calculation for the complete process e>e\Pkl udM can be found in Ref. [189]. The results of O(a ) corrections to e>e\Pq q q q , the I Q CC10, CC11 and CC20 processes, can be found in Ref. [190]. See also Ref. [191] for further references. No new problems arise in QCD corrections to processes with Z exchange. 2.3.3. Z constraints at s(M To order h , a Z would modify the cross section due to changes in the Z e>e\ couplings and + due to the Z exchange contribution. Modi"cations due to the mass shift *M"M !M in the 8 Z propagator are small, see discussions above Eq. (2.115), and of the order h . Far below + the Z resonance, the contribution of the Z exchange can also be neglected because it has the additional suppression factor s/M. Therefore, the Z signal considered here arises due to the modi"cation of the Z e>e\ coupling. All Z e!ects can be absorbed in the anomalous couplings d and d de"ned in Eq. (2.113). A 8 A search for a Z in =>=\ production is therefore a special case of a search for anomalous couplings [171,174,192,193]. Far below the Z peak, one has to take special care to separate Z models from other theories predicting anomalous couplings. One hint for a Z would be a non-zero d , which is usually absent in other theories due to the ;(1) gauge invariance. In a general data A analysis, one should try to constrain many anomalous couplings simultaneously and show that all combinations perpendicular to d are zero. If at least one of these perpendicular combinations is 8 not zero, this will indicate that there is other new physics in addition to a Z . See Ref. [194] for a corresponding analysis. An additional check of the Z hypothesis would be a comparison with deviations in e>e\Pf fM below the Z peak. The "nal proof of the hypothesis would be a measurement at the Z peak. We assume in the following that all deviations from the SM are due to a Z and ignore the possible confusion with other physics.
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2.3.3.1. Model-independent constraints on gH , gH or v+, a+. Estimate: Consider the cross A55 855 C C section e\e>P=>=\. Only the s-channel contributes to the scattering of right-handed elec0 trons. Expanding the total cross section p> in the limit of large s, we get 2 p>+(apc /12s)[1!(2s/M )(d !d s /c )] . (2.126) 2 5 8 A 85 5 The "rst term is the leading SM contribution, the second and third terms are the leading contributions in the parameters d and d . The two leading powers in s have cancelled in the SM A 8 term, while only the leading power in s has cancelled in the contributions proportional to d and d . A 8 It follows that the observable p> will see a signal if 2 *p>/p>(*8Yp>/p>+(2s/M)"d !d s /c " , (2.127) 2 2 2 2 8 A 85 5 where *p>/p> is the relative experimental error and *8Yp> is the deviation due to a Z . 2 2 2 Similar considerations can be used for the scattering of left-handed electrons. Unfortunately, the region of forward scattering c+1 gives a much reduced sensitivity to a Z . In this region most of the SM events are produced (making the cross section proportional to ln s/s), while the leading contributions in d and d are not logarithmically enhanced. One can avoid this pollution e!ect by A 8 "ts to angular distributions. Alternatively, the forward region can be excluded from the integration. The sensitivity to a Z depends on details of this procedure as shown in Fig. 2 of the second Ref. [143]. We estimate the sensitivity of the observable p\ to Z e!ects doing an expansion of 2 dp\(c)/dc around the point c"0, 2 dp\(c"0) 1 1 s 1/2!s 5 . & #2c ! 5 d #d (2.128) 5 8 s c dc s 4 M A 8 5 5 It follows that the observable p\ will see a signal, if 2 s *p\ *8Yp\ s 1/2!s 5 2( 2+ 5 . d #d (2.129) A 8 p\ M (1/4#2c ) p\ s c 5 2 2 8 5 5 As expected from an inspection of amplitude (2.112), the scattering of left- and right-handed electrons constrains di!erent combinations of d and d , A 8 *8Yp!/p!&"d !d (g /e)(v Ga )" . (2.130) 2 2 A 8 C C The scattering of unpolarized electrons gives constraints similar to those from e\ scattering. Any * single observable selected from p , p> or p\ is blind to an in"nite band in the d , d plane. 2 2 2 A 8 Combining the results from di!erent cross sections, one is insensitive only to a closed region in this plane. For later use, it is instructive to give a rough estimate of the size of this region using (2.129)
*p M 1/4#2c *p M 5 2 8+6.2 2 8 . (2.131) "d ", "d "(d , d + A 8 A 8 p s p s s 2 2 5 Assuming that the experimental error consisting of statistical and systematic errors scales like the statistical error, *p/p+1/(N+(s/¸, we get a scaling [143] of these constraints with the centre-of-mass energy and the integrated luminosity d , d , "v+", "a+"&((1#r)/s¸ . A 8 C C
(2.132)
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It is the same as derived for anomalous couplings [195]. As before, r is the ratio of the systematic and statistical errors. At the proposed colliders, the statistical errors dominate the error of the observable p>, while the error of p\ is usually dominated by systematic errors depending on the 2 2 cut on c. Let us make a remark regarding the comparison with e>e\P+M . A Z signal will arise there, if the Z interferences give deviations larger than the experimental error. Consider the deviation due to the cZ interference, *8Yp +(1/s) ) s (s)s (s)+(1/s) ) 1 ) s/(s!M)+1/M. 2 A It follows *8Yp /p +s/M . (2.133) 2 2 The constraint has the same dependence on s as constraints (2.127) and (2.129). However, the important di!erence is that the constraint from fermion pair production is normalized to M, while the constraint from = pair production is normalized to M. The ratio s/M is small far below the 8 Z peak, while s/M is large at future colliders independently of M. This di!erence is responsible 8 for the enhanced sensitivity of = pair production to the ZZ mixing angle. Present constraints: Present constraints on anomalous couplings from LEP data are given in Ref. [196]. Unfortunately, the constraints given there do not allow a derivation of an excluded region of d and d . A 8 Future constraints: Future constraints on d and d are given in Ref. [93]. This analysis is A 8 done at the born level and based on (s"0.5 TeV and ¸"50 fb\ assuming 90% polarization of the electrons, 30% detection e$ciency of the = bosons and 2% systematic errors. The di!erential cross section is considered in 10 equal bins in c for "c"(0.98. The resulting constraints are shown in Fig. 2.18. As expected from the estimates, the cross sections p> and p\ or p alone are 2 2 2 insensitive to bands in the d , d plane. The quantitative agreement with estimates (2.127) and A 8 (2.129) is good. The model-independent limit on d and d can be easily converted into limits on the ZZ mixing A 8 angle for any "xed Z model. For a "xed h , every model is represented by a dot in the d , d plane. + A 8 We show the region of the E and LR models for h "0.002 in Fig. 2.18. The ratio of d and d is + A 8 determined by the couplings of the Z to fermions only, independent of the ZZ mixing angle and the Z mass e 1 v a \ d 8" . (2.134) 1! C C g v s a v d C 8 C C A If one varies the mixing angle h for a "xed model, one moves on a straight line in the d , d plane. + A 8 The corresponding line is shown in Fig. 2.18 for Z "s. Those values of h , for which one hits the + model-independent bound, de"ne the constraint on h for that speci"c model. The limits on + h obtained directly from a one-parameter "t for a previously "xed model are expected to be + stronger. In the case of a deviation of d and d from zero, relation (2.134) between the Z couplings to A 8 electrons, can be tested in fermion pair production. Such a cross check [194] would help to verify that the deviation is due to a Z . In the case of a disagreement, the deviation cannot be due to a Z alone.
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Fig. 2.18. Upper bounds (95% C¸) on non-standard couplings (d , d ) from p>, p\ and p in e>e\P=>=\. See the A 8 2 2 2 text for the numerical input. The bands containing (d , d )"(0,0) cannot be excluded by the observables. The thin lines A 8 (the ellipse and the line from the s to the LR model) are the regions of the E and LR models for h "0.002. The straight + thin line is the region for Z "s for di!erent values for h varied in steps of 0.001. This an update of Fig. 1 of Ref. [93]. +
The model-independent constraints on d and d can be converted into model-independent A 8 constraints on v+ and a+ using relations (2.116). The mass dependence introduced by the C C propagator s( can be neglected far below the Z resonance. Applying this procedure to the limits presented in Fig. 2.18, one gets the constraints on v+, a+ shown in Fig. 2.3. C C In contrast to measurements of fermion pair production, = pair production cannot constrain Z couplings to fermions fOe. Therefore, the sensitivity of = pair production to ZZ mixing is reduced for models where the Z has small couplings to electrons. 2.3.3.2. Constraints on h . Estimate: Combining de"nition (2.116) of d and d and estimate (2.131), + A 8 we derive an estimate for h . Assuming v +v and a +a , we get from d + C C C C 8
s( \ *p M g . "h "(h +3.4 ) 2 ) 8 1! + + s g s( p 8 2
(2.135)
We have s( /s( +0 far below the Z resonance. The scaling with s, ¸ and r is the same as for d 8 A and d . 8 Estimate (2.135) can be compared with estimate (2.43) derived for fermion pair production at the Z peak. We see that the sensitivity of = pair production to h becomes eventually better for large + energies. Future constraints: Limits on h at future colliders are presented in Ref. [93]. It is an update of + the older analyses [125,143]. Fig. 2.19 shows the future limit on h as a function of M for Z "t. Remembering that estimate + (2.135) ignores details of the Z model and is based on a crude approximation of the excluded
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Fig. 2.19. Allowed domains (95% C¸) of h , M for the t model. The region below the solid curve can be excluded with + e>e\P=>=\ at (s"0.5 TeV and ¸"50 pb\. See the text for further inputs of the analysis. The current limit on M (h ) and the expected exclusion limit on M from e>e\Pf fM are indicated by the thick solid (dashed) lines. The thin + dotted (dashed) lines correspond to the mass constraint (1.16) with *M"0.2 GeV (the Higgs constraint (1.17)). This is an update of Fig. 2 from Ref. [93].
region, it gives a reasonable prediction of the constraint on h in the limit M PR. The present + limit on M and the expected improvement from fermion pair production at the same collider are R also indicated. The limit M from e>e\Pf fM is obtained from Table 2.4. We took the entry of 8Y analysis [126] for 20 fb\ with systematic errors but scaled from ¸"20}50 fb\ by relation (2.53). Also shown are the relations between h and M from the mass and the (model dependent) Higgs + constraint. If s approaches M , the in#uence of s( /s( on d and d , in Eq. (2.135) becomes dominant leading 8 A 8 to an additional enhancement to be discussed in the next section. The limit on h can be compared with the present constraint !0.0022(h (0.0026 for the + + t model taken from Table 2.1. We see from Fig. 2.18 that the reaction e>e\P=>=\ at a 500 GeV collider can add only little to the limits on the t from e>e\Pf fM . However, estimate (2.135) predicts a considerable improvement of the sensitivity for higher energies. The analysis of Fig. 2.19 can be repeated for di!erent Z models. The constraint on h (M PR) + for di!erent E models is plotted as function of cos b in Fig. 4 of Ref. [125]. 2.3.3.3. Constraints on M . The interference of the Z exchange with the SM contributions is sensitive to the Z mass. Consider the change in the observable O due to the Z Z interference ignoring details of the Z model (a +a , v +v ), C C C C s!M g s *8YO g g + h + 558 . (2.136) s!M g + s!M g g O 558 1+
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We took into account that h ;1 and that s<M at future colliders in the last step of the + approximation. It follows that M (M "(s(1#h (g /g )O/*O) (2.137) 8Y + would give a signal in the observable O. Compared to estimate (2.51) derived for fermion pair production, the sensitivity to M from = pair production is suppressed by the ZZ mixing angle. Remembering the discussion in Section 2.1.3.2, h is constrained to be small by measurements at + the Z peak independently of the model. Therefore, the Z mass bound from = pair production cannot compete with that from fermion pair production. The resulting indirect bounds on M , which one obtains combining the constraint on h with 8Y + the Higgs constraint depend on the extended Higgs sector. They are worse than the constraints from fermion pair production (compare Fig. 2.19). Figures similar to Fig. 2.19 are shown for other E models and (s"0.5 and 1 TeV in Ref. [93]. 2.3.3.4. Model measurements. Assume that there are non-zero anomalous couplings d and d . A 8 Then, these couplings can be measured in future experiments. The errors of such measurements can be estimated taking into account that the deviations of cross sections are linear in d and d . One A 8 gets *d , *d +d , d . A 8 A 8 The estimate for the error of a h measurement is + *h +h . + +
(2.138)
(2.139)
2.3.4. Z measurements at s+M The production of = pairs near the Z peak is essentially di!erent from the production far below the resonance. At s+M we de"nitely know that there exists a Z . We also know its mass, its width and its couplings to fermions. This information is provided best by fermion pair production on the resonance due to the large statistics of this reaction. Unfortunately, a description of == production has the same problem as the description of f fM production on the Z resonance. The needed radiative corrections within the GUT depend on many unknown parameters. 2.3.4.1. Constraints on h . The production of =>=\ pairs on the Z resonance is suppressed + because it is proportional to h . For (s+M $C /2, the Z Z interference, being proportional + to h , is most sensitive to ZZ mixing because it is enhanced by the Z propagator. + An estimate of the sensitivity to h is given by Eq. (2.135) as in the case s(M with the + important di!erence that the ratio s( /s( , where the width of the Z must now be taken into 8 account, gives the dominant contribution [93], "h "(h +3.4(*p /p )(M/s)g /g "Res /s " . (2.140) + + 2 2 8 8 Compared to the o!-resonance case, we have the additional enhancement factor "Res /s "+2C /M with 2C /M +(1/20}1/50) depending on the particular Z model and on the 8
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Fig. 2.20. Upper limits (95% C¸) for U"h as function of the E model parameter cos b at (s"M $C /2 for 90% + left (right)-handed polarized electrons solid (dotted) solid line. Positrons are unpolarized. The input is M "1 TeV, ¸"50 fb\ and a systematic error of 2%. I thank A.A. Pankov for providing this "gure.
number of the exotic fermion generations to which the Z can decay. The gain in the sensitivity due to all factors in Eq. (2.140) is so large that it overcompensates the loss in the sensitivity due to the poor statistics. This is the reason why = pair production near the Z resonance is much more sensitive to h than fermion pair production. See Ref. [143] for a further discussion of this e!ect. + The increase in the sensitivity for (s approaching M can be seen in Fig. 2.19. Repeating the procedure for di!erent E models, one arrives at Fig. 2.20. The constraint on h given there agrees + with estimate (2.140) derived for E models with 2C /M "1/20. Fig. 2.20 again demonstrates the essential role of beam polarization for exclusion limits in e>e\P=>=\. The sensitivity to h becomes stronger for higher energies according to scaling (2.132). + 2.3.4.2. Measurements of h . In the previous section, we assumed that there is a Z but that the ZZ + mixing angle is so small that only an upper bound can be set in the experiment. We now assume h is large enough to give a signal. Then, the error of a h measurement is given by Eq. (2.139) + + where now h must be taken from estimate (2.140). + 2.3.5. Z Constraints at s'M Consider the constraint on h (2.135), which transforms to + *p Mg *p M g s "3.4 2 8 . (2.141) "h "(3.4 2 ) 8 + s g M p Mg p 2 2 in the limit of high energies, s<M. We see that there is no further enhancement of the sensitivity with rising s. All contributions to the cross sections (2.126) and (2.128) are proportional to 1/s. Unitarity is restored independently of the details of the large gauge group. This can be understood treating the Z and Z as massless particles in the limit s<M. Then, one can consider the
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Fig. 2.21. p(e>e\Pk>k\q>q\) for (s"500 GeV as function of the lower cut s\ on the invariant energies s and s of the "nal muon and tau pairs. The solid (dotted) curves show total cross sections with (without) initial state QED corrections.
unmixed states Z and Z instead of Z and Z and remember that the Z does not couple to ='s. The resulting cross section for e>e\P=>=\ obviously behaves like the SM for large s. Estimate (2.141) of the sensitivity to h is always much worse than case 2 where additional + enhancement factors were present. Possible Z signals from the radiative return to the Z resonance cannot compete with fermion pair production due to lower statistics. 2.4. Z search in other reactions The only SM processes in e>e\ and e\e\ collisions with two particles in the "nal state not yet considered are e>e\PZZ and e>e\PZH. They are similar to = pair production. However, they do not have the enhancement factors of new physics. Therefore, these reactions are much less sensitive to a Z than = pair production. Higher-order processes cannot compete in setting Z limits due to statistics [197]. Some higher-order processes have cross sections comparable to those of two particle "nal states. Resonating gauge bosons or collinear radiation of light particles can be responsible for this enhancement . However, these enhancement mechanisms do not enhance the Z contributions. Therefore, they `pollutea a potential Z signal and should be removed by appropriate kinematic cuts. With these cuts, the resulting cross sections are too small to compete with two particle "nal states. The e!ect is demonstrated [198] in Fig. 2.21. A cut s\ on the invariant masses s and s of the k>k\ and q>q\ pairs suppresses "rst the photon exchange (s\ are small). For s\ 'M, the 8 Z exchange is suppressed too. The resulting cross section is approximately a factor a smaller than
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cross sections of particle pair production. As expected, the e!ect is not altered by radiative corrections. The reactions e>e\PZ Z and e>e\Pe>e\Z are considered in Refs. [200,201]. The second process is di$cult to observe above the background [201]. The "rst process needs very high energies. It was proposed [200] to use this process to resolve ambiguities in the experimental determination of the E breaking parameter h. However, this ambiguity can already be resolved by measurements below the Z peak [130]. The individual interferences of == scattering (e>e\Pe>e\=>=\) scale like s while the total cross section scales in the SM like 1/s for very high energies. As in e>e\P=>=\, a non-zero ZZ mixing angle would destroy this gauge cancellation. Therefore, it could be interesting to investigate the potential of == scattering for a Z search at TeV colliders. 3. Z search at pp and pp colliders The Z signal at hadron colliders comes from direct production. This is a principle di!erence compared to e>e\ and ep collisions. Therefore, the mass of a detectable Z must be smaller than the centre-of-mass energy of the colliding protons. In practice, the Z must be at least two times lighter because it is produced in collisions of partons. The Z is detected through its decay products, which must be separated from the SM background. Unfortunately, the background in a hadronic environment makes it di$cult or sometimes impossible to measure potential interesting observables. The decay of a Z to a fermion pair would probably give the "rst Z signal in hadron collisions. The invariant mass of the "nal state fermion pair is centred around the Z mass. This allows for a good separation of the signal from the background and for a measurement of the Z mass. The signature was exploited in the past to measure the properties of the SM Z boson at the UA1 and UA2 experiments [202]. Di!erent fermions in the "nal state can be tagged providing various cross sections and asymmetries as observables. See Refs. [49,203}205] for some old analyses. Rare Z decays, Z Pf fM < (with <"=, Z and f , f are higher-order processes. However, they are enhanced by large logarithms due to collinear and soft radiation. They give interesting complementary information in a Z model measurement. The Z decay to = pairs is possible only in the case of ZZ mixing. Unfortunately, this decay mode su!ers from the SM background of the associated production of a = and two jets [206]. Associated production of a Z together with another gauge boson, ppPZ <, <"Z, =, c is of higher-order compared to the production of a single Z . However, for s<M these processes are 8 logarithmically enhanced similar to soft photon radiation. They add independent information about Z models. There are no other processes known in hadron collisions, which are useful to add further information on a Z . 3.1. Born cross section of pp(pp )PZ PffM The considered reaction is much less sensitive to ZZ mixing than e>e\ experiments at the Z peak. Therefore, any ZZ mixing e!ects can be neglected putting h "0 and identifying the + Z with the Z and the Z with the Z .
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The Born cross section of the production of a Z , which decays to a fermion pair is p (pp(pp )P(c, Z, Z )XPf fM X),pD " dx dx p (sx x ; qq Pf fM )GO (x , x , M )h(x x s!MR) 8Y O M Q WK?V dQ dy p (sx x ; qq Pf fM )GO (x , x , M )h(x x s!MR) , " 8Y s R K?V + \W O
(3.1)
where MR is the sum of the masses of the "nal particles, x "((Q/s)e!W and y is the rapidity. The functions GO (x , x , M ), A"T, FB depend on the structure functions of the quarks, 8Y GO (x , x , M )"q(x , M )q (x , M )#q (x , M )q(x , M ) , 2 8Y 8Y 8Y 8Y 8Y (3.2) GO (x , x , M )"q(x , M )q (x , M )!q (x , M )q(x , M ) . $ 8Y 8Y 8Y 8Y 8Y The expressions for p (sx x ; qq Pf fM ), A"T, FB can be easily derived from Eqs. (2.6)}(2.9). These formulae also contain the dependence on the helicity of the initial and "nal fermions. If the polarization of the "nal fermion is measurable, one can observe pD "pDt!pDs . (3.3) 2 2 If polarized proton beams are available [207], the `left}righta cross section is of interest [208], (3.4) pD "pD!pD , 0 * *0 GO (x , x , M )"[qt(x , M )!qs(x , M )]q (x , M )#[q t(x , M )!q s(x , M )]q(x , M ). *0 8Y 8Y 8Y 8Y 8Y 8Y 8Y p are the total production cross sections with one left-or right-handed initial quark. As in the * 0 case of e>e\ collisions, the di!erence of p for a left- and right-handed quark in the initial state $ can be considered [208], pD "pD !pD , (3.5) *0 $ * $ 0 $ GO (x , x , M )"[qt(x , M )!qs(x , M )]q (x , M )![q t(x , M )!q s(x , M )]q(x , M ). *0 $ 8Y 8Y 8Y 8Y 8Y 8Y 8Y More complicated cross sections can be measured if both proton beams are polarized. 3.2. Higher-order processes and background 3.2.1. Rare Z decays The decay modes Z Pf fM < (with <"=, Z and f , f ordinary fermions) [209] are enhanced by logarithms due to collinear and soft radiation, (cf. Eq. (1.32)). The decays Z PZl>l\Pl>l\l>l\ and Z P=ll can be separated from the background [210,211]. The "rst process shows only J a weak dependence on the Z couplings serving as a consistency check of the experiment, while the second `gold-plateda decay mode yields useful and complementary information about the Z couplings [211}213]. The rare decays where f and f are quarks can also be measured, although with a larger systematic error [211].
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3.2.2. Associated Z production The production of a Z in association with another gauge boson, ppPZ <, <"Z, =, c is logarithmically enhanced for high energies similar to soft photon radiation from the initial state. The total cross section of the partonic subprocess is [44]
1#m 1!m #j gg >ln > !2j , (3.6) p (qq PZ <)" [¸¸ #RR ] O O O O 2 1!m 1!m !j 4ps > > with m "M /s#M /s and j"j(1, M /s, M /s), where j is the kinematic function (2.118). The > 8Y 4 8Y 4 cross section (3.6) is known from electron positron collisions [214]. Associated Z production gives complementary information and is free of SM backgrounds [44,215]. It can compete with other processes in the determination of the parameters of a Z model [211]. 3.2.3. Radiative corrections In a "rst approximation, the tree level Z contributions can be added to the SM cross section. This approach neglects radiative corrections to the new physics treating them as a small correction to a small e!ect. This approximation is probably true for a "rst Z discovery but has to be checked in a Z model measurement. In the following we brie#y mention the main Standard Model corrections. 3.2.3.1. QCD corrections. QCD corrections are numerically most important. They increase the lowest order cross section of vector boson production at the Tevatron by 20}30%. These corrections are often called K factors. Corrections to the unpolarized Drell}Yan process are known to order O(a) for the invariant Q mass distribution [216]. For rapidity- and x-distributions, they are calculated to order O(a ) [217] Q and partly to order O(a) [218]. Soft gluon contributions can be treated to all orders by Q exponentiation [219]. QCD corrections to polarized hadron scattering are known to order O(a ) Q [220]. 3.2.3.2. QED and weak corrections. QED corrections are model independent. As in e>e\ collisions, initial state corrections, "nal state corrections and the interference between them are separately gauge invariant for neutral current processes. Numerically, the dominant corrections come from "nal state radiation [221]. The corrections from initial state radiation and the interference between initial and "nal state radiation are small after factorizing the collinear singularities into the parton distribution functions [221]. As in e>e\ collisions, "nal state corrections do not feel the neutral gauge boson exchanged before. Pure weak corrections are expected to already be very small for the Standard Model Z production [221]. The corrections to Z production are expected to be even smaller. 3.2.4. Background The fermions coming from Z decay have an invariant mass, which is peaked around M . 8Y Fermion pairs with the same invariant mass could also be produced by gluon, photon or Z exchange.
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The experience of existing hadron colliders shows that b-quarks [222] and dijets [223] can be detected [224]. However, the sensitivity to new gauge bosons from quark pairs is reduced compared to muon and electron pairs due to the QCD background. The background to q pairs is considered in [225] and found to be manageable for a Z originating in an E GUT. It is shown there that the signal can be distinguished from the background of =>=\ production and from the background coming from misidenti"ed jets, which could be accidentally recognized as q decay products. Furthermore, the background from top pairs decaying to q pairs can be managed. Even the background from Drell}Yan production of q's can be removed although this is harder [225] than in the case of muon pairs. 3.3. Observables Similar to the case of e>e\ collisions, the total cross sections pD and di!erent asymmetries serve 2 as observables, pD, AD "pD /pD, AD "pD /pD, 2 $ $ 2 2
AD "pD /pD, *0 *0 2
AD "pD /pD . *0 $ *0 $ 2
(3.7)
Note that A in our notation is A in Ref. [208], while we reserve A for "nal state *0 $ $ polarization asymmetries following the notation of Ref. [225]. Not all observables (3.7) can be measured in a real experiment. In addition to the constraints mentioned in Section 2.1.1.4, complications arise because the signal has to be detected above the background of the hadronic environment. As in e>e\ collisions, AD is independent of the couplings to initial fermions. In hadron collisions, all dependence on the quark structure functions also drops out [225]. For lepton pairs in the "nal state, rapidity ratios can be de"ned [211], W (dpJ /dy)dy ( !W)(dpJ /dy) dy \W 2 $ r " \W . W (\W #W )(dpJ /dy) dy, A$ W"(\W !W )(dpJ /dy) dy 2 $ \W W \W W
(3.8)
The observable r is useful in distinguishing between di!erent Z models, while A being `a W $ W re"nement of a re"nementa is less sensitive to di!erent Z models [211]. Rare Z decays Z Pf f < allow for the de"nition of the ratios, r ,Br(Z Pl>l\Z)/Br(Z Pl>l\), r ,Br(Z PllZ)/Br(Z Pl>l\) , JJ8 JJ8 r ,Br(Z Pl!l=)/Br(Z Pl>l\) . JJ5
(3.9)
The index l refers to a summation over e and k and the index l in r to the summation over l ,l JJ8 C I and l . The ratios r and r , where the fermions f and f are quarks can be de"ned O 8 5 analogously. r depends only on the Z couplings to leptons. JJ5 The cross sections of associated Z production enter the ratios p(ppPZ <)BrJ , <"Z, =, c . R " 8Y4 p(ppPZ )BrJ
(3.10)
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3.4. Z constraints 3.4.1. Model-independent constraints on pD 2 3.4.1.1. Estimate. The signal of extra neutral gauge bosons in hadron collisions is a number N of 8Y excessive fermion pairs with an invariant mass around M . In the case of absence of a signal, the 8Y observable pD is constrained independently of the Z model. 2 We give here an approximation of pD to make the dependence of Z exclusion limits and Z model 2 measurements on the centre-of-mass energy, the integrated luminosity and model parameters transparent [226]. Such an estimate is useful to extrapolate from Z limits known for one collider and Z model to other colliders and Z models. Consider pI neglecting the background. Eq. (3.1) can be approximated treating the resonating Z 2 propagator of pI (Q; qq Pf fM ) in the narrow width approximation, 2 pM Q Pd(Q!M) . MC "Q!M#iMC"
(3.11)
We obtain
4p C (s 8Y BrI BrO f O , M pI " 8Y 2 3s M M 8Y 8Y O
with f O (r , M )" X 8Y
(3.12)
dx 1 GO x, , M x 2 xr 8Y PX X
(s and r " . X M 8Y
(3.13)
The function f O(r , M ) has only a very weak dependence on M in the region we are interested X 8Y 8Y in. We therefore can drop the second argument approximating f O(r , M )+f O(r ). X 8Y X The inspection of Z limits at the proposed colliders shows [133] that the functions f O(r ) are X needed only in a narrow interval of r , i.e. 3(r (5 for pp collisions and 2(r (3.5 for pp X X X collisions. Under these conditions, the functions for di!erent quarks q"u, d di!er mainly by a constant factor. Hence, we can make the following replacement in Eq. (3.12):
1 (s (s , M +f S BrS # BrB (3.14) BrO f O 8Y C M M SB 8Y 8Y O with C "f S(r )/f B(r )+2 (+25) at pp (pp ) colliders. We see that pI has a reduced sensitivity to SB X X 2 Z ddM couplings. The integral de"ning the function f S(r ) could be approximated by the function r?(r !1)@, which X X X takes into account the parametrization of the structure functions. However, we prefer an approximation by an exponential function because it can be inverted analytically. We get f S(r )+Ce\ PX, C"600 (300), A"32 (20) for pp(pp ) collisions. X
(3.15)
Approximation (3.15) and the exact calculation (3.13) of f S(r ) are shown in Fig. 3.1. We use the X structure functions [227]. The dependence of our results on this choice is negligible. Note that the
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Fig. 3.1. The function f S((s/M , 25 TeV) and the approximation (3.15). The curves of f S((s/M , Q) for Q"1 TeV 8Y 8Y could not be distinguished from Q"25 TeV. This is Fig. 1 from Ref. [226].
"t works satisfactorally up to r "10. It cannot describe SM Z production at the Tevatron where X we have r +20. 8 Collecting all approximations, pI can be written as 2 N pI , 8Y+(1/s)c C exp+!AM /(s, , 2 8Y 8Y ¸
(3.16)
with c "(4p/3)(C /M )BrI [BrS #(1/C )BrB ]. 8Y 8Y 8Y SB All details of the Z model are collected in the constant c . For convenience, we list the value of 8Y c for some Z models: 8Y Model:
s
t
g
¸R
SSM
1000 ) c (pp): 8Y 1000 ) c (pp ): 8Y
1.17
0.572
0.712
1.35
2.27
0.40
0.437
0.556
0.77
1.41
(3.17)
The approximate exponential dependence of pI (and of N ) on M can be recognized, for 2 8Y 8Y instance, in Figs. 1}5 of Ref. [49]. It even holds in associated Z production, ppPZ =, ppPZ Z, as can be seen from Fig. 3 of Ref. [44]. The reason is that associated production happens for very constrained values of Q+(M #M ) only; for smaller Q, the process is forbidden by kin8Y 4 ematics, for larger Q we have an exponential suppression due to structure functions. Eq. (3.16) is the starting point for several useful estimates regarding Z constraints in hadron collisions.
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Fig. 3.2. The limit on pJ as a function of the dilepton mass, as well as the expectation for Z "SSM. This is Fig. 3 from 2 Ref. [63].
Table 3.1 The lower bounds (95% C¸) on the Z mass from the analysis [63] compared to the estimate (3.18). The "nite detection e$ciencies e ,e and the K-factor as given in Section 3.4.1.2 are taken into account in the estimates C I Model
s
t
g
¸R
SSM
M /GeV from [63] 8Y M /GeV from (3.18) 8Y
595 552
590 560
620 582
630 611
690 665
3.4.1.2. Present constraints. The best present constraints on pD come from the Tevatron experi2 ments. They give a constraint on pD,pD(ppPZ ) ) BrD as a function of the invariant mass of the 2 2 "nal fermion pair f fM in the "nal state. The constraint from CDF [63] for electrons and muons in the "nal state (¸"0.110 pb\ and (s"1.8 TeV) is shown in Fig. 3.2. The corresponding constraints from D0 can be found in [228]. The CDF data constrain pJ (0.04 pb for large invariant masses. The deviation of the experimental 2 curves from this number for smaller Z masses are due to the SM background. The total detection e$ciencies [63] for electron and muon pairs are e +47% and 20%. The K factor from QCD C corrections is K+1.3. (see Table 3.1). The experimental results can be confronted with estimate (3.16). The exponential dependence of pJ on M predicted by the estimate can clearly be seen in the "gure. Taking into account 2 8Y the detection e$ciencies and the K factor, estimate (3.16) predicts (pD) +0.031 pb. This is in 2 reasonable agreement with the exact result.
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The SM background for dijets is much larger due to QCD e!ects. In Ref. [229], events with dijets did not allow constraints on extra neutral gauge bosons from the E GUT. 3.4.2. Model-dependent constraint on M 8Y 3.4.2.1. Estimate. Inverting approximation (3.16), we get a constraint on M , 8Y ¸ ) fb 1 0.386# ln 1000c pp 8Y 32 N s/TeV ¸c C (s 8Y 8Y +(s (3.18) ln M 'M + 8Y 8Y A s N ¸ ) fb 1 8Y 1000c pp . 0.583# ln 8Y 20 N s/TeV 8Y For M (M more than N additional events are expected. 8Y 8Y 8Y A formula similar to Eq. (3.18) was quoted in Ref. [230] some time ago. However, there the dependence of M on Z couplings and collider parameters is given only numerically. 8Y Relation (3.18) describes the scaling of M with the centre-of-mass energy and the integrated 8Y luminosity. M depends on ¸ only logarithmically. Therefore, the dependence of M on detector 8Y 8Y e$ciencies or event losses due to background suppression is only marginal. The logarithmic dependence of M on ¸ can be recognized in Fig. 3 of Ref. [131] or in Fig. 2.33 8Y in Ref. [63]. The reduction of M due to a decrease of the event rate by a factor of two is predicted 8Y by relation (3.18) to be 9% (7%) for the proposed pp (pp ) colliders. These numbers, which do not discriminate between Z models, s, and ¸, are in agreement with the last line of Table 1 in Ref. [131]. The model-dependent constant c enters (3.18) only under the logarithm leading to the weak 8Y model dependence of Z exclusion limits in pp and pp collisions. The physical origin of this e!ect is hidden in the properties of the structure functions de"ning (3.13) the function f (r ). Therefore, X relation (3.18) obtained for pI is qualitatively true for other observables too. 2 Radiative corrections lead to deviations of N from the Born prediction. The e!ect on M is 8Y 8Y moderate because N enters this limit only under the logarithm. 8Y Scaling (3.18) is the complement of relation (2.53) derived for e>e\Pf fM . The strong model dependence of Z exclusion limits from e>e\ collisions is due to their direct dependence on the square of the coupling constants of the Z to fermions, see Eq. (2.47). In the last step, Eq. (3.18) is written in a form, which makes the logarithm nearly zero for GUTs at the proposed colliders. We see that the numerical in#uence of the logarithm is suppressed by a small prefactor. The constant terms 0.386 and 0.583 give an estimate for the average sensitivity of pp(pp ) collisions to a Z in units of (s. For practical purposes, it is useful to rewrite Eq. (3.18) as
M (s, ¸) (s s ¸ 8Y + 1#m ln M (s , ¸ ) (s s¸ 8Y
¸ ) c C \ , with m" ln 8Y s )N 8Y
(3.19)
where now all model dependence is hidden in the constant m. Normalizing at one collider, Eq. (3.19) predicts the limits at another collider. All Z exclusion limits published in Fig. 1 of Ref. [133] can be reproduced by estimate (3.19) with an accuracy of 10% for m"0.13 (0.10) for pp (pp ) collisions.
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Table 3.2 The lower bound on the Z mass M in TeV excluded by the di!erent analyses described in the text. Estimate (3.18) is 8Y added for Z "SSM. This is Table 1 from Ref. [226] Analysis
(s TeV
¸ ) fb
s
t
g
¸R
SSM
Estimate (3.18)
[133] [133] [131] [131]
2(pp ) 14(pp) 60(pp) 200(pp)
10 100 100 1000
1.04 4.38 13.3 43.6
1.05 4.19 12.0 39.2
1.07 4.29 12.3 40.1
1.10 4.53 13.5 43.2
1.15 4.80 14.4 44.9
1.06 4.47 13.7 49.3
3.4.2.2. Present constraints. No Z signal is found in present experiments. This negative search result can be interpreted as exclusion limits M in di!erent models. 8Y The limits quoted in Ref. [63] are in a good agreement with prediction (3.18). 3.4.2.3. Future constraints. The minimal input of the di!erent analyses are the integrated luminosity ¸ and the centre-of-mass energy (s of the colliding particles (pp or pp ), a list of observables entering the "t and the number of Z events N demanded for a signal. If applied, kinematic cuts 8Y and radiative corrections must be speci"ed. It follows a list of di!erent analyses. Ref. [133], GODFREY95. This is a theoretical analysis for di!erent future colliders based on pC and pI with N "10. No Z decays to exotic fermions are assumed. one-loop QCD corrections 2 2 8Y are included in the Z production. The Z decay is calculated including two-loop QCD, one-loop QED corrections and top-quark decays. We selected two scenarios to present them in Table 3.2. The numbers are taken from Table 2 of Ref. [8] and from Fig. 1 of [133]. Ref. [131], RIZZO96. This is a theoretical analysis for di!erent future colliders based on pC 2 and pI with N "10. No Z decays to exotic fermions are assumed. No radiative corrections are 2 8Y included. We selected two scenarios for our Table 3.2. The numbers are taken from Table 1 of [131]. The results of the di!erent analyses are compared with estimate (3.18). We see that the prediction (3.18) agrees with the exact results within 10% in a wide range of ¸ and s. 3.4.3. Model-independent constraint on g /g Fig. 3.2 gives constraints on g /g . According to estimate (3.16), pI scales as C /M &g. The 2 8Y 8Y missing factor depends on the Z model. It is given in Eq. (2.78) for Z "SSM if no decays to exotic fermions are allowed. The resulting constraint g (g can be obtained graphically from Fig. 3.2 by an appropriate shift of the signal cross section. We get g +g /4.5 for Z "SSM and M (400 GeV. 8Y 3.4.4. Errors of model measurements 3.4.4.1. Estimate. Model parameters can be measured if some of the observables O introduced in Section 3.3 give a signal. A reasonable model measurement requires enough events to assume that they have a normal distribution. The one-p statistical error can then be estimated using Eq. (3.16) *AJ +1/(N +((s/¸)(1/c C) exp+AM /2(s, . 8Y 8Y $ 8Y
(3.20)
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Fig. 3.3. The estimate of *O (95% C¸) as a function of M /M as given in Eq. (3.22). M is the Z exclusion limit 8Y 8Y 8Y obtained from the same observable O alone. Shown are the predictions for the scenarios listed in Table 3.2 for Z "g. For (s"14 TeV, the dependence is shown for Z "t,g,SSM (from top to down). The thick solid line is obtained from relation (2.63).
Relation (3.20) relies on approximation (3.15), which becomes inaccurate for too large (s/M . The 8Y error of other observables also scales as Eq. (3.20) with s and ¸, however, the prefactors di!er. Compared to the exclusion limit M , the error of a model measurement is much more model 8Y dependent because the in#uence of the constant c is no longer logarithmically suppressed. The 8Y dependence on the integrated luminosity is the same as in Eq. (2.65). Therefore, the dependence of model measurements on systematic errors in hadron collisions is as pronounced as in e>e\ collisions. Combining Eqs. (3.18) and (3.20), we can predict *O for a given M (M if we know 8Y 8Y M from the observable O alone for the same collider, 8Y (3.21) *O+(N )\DK ) ((s/¸) ) 1/c C) \DK, f "M /M . 8Y K 8Y 8Y 8Y It is the complement to estimates (2.62) and (2.63) derived for e>e\ collisions, which relate exclusion limits and measurements of the same con"dence level. Relation (3.21) relates exclusion limits from N expected Z events to one-p errors of model measurements. To relate exclusion 8Y limits of 95% con"dence (N "3) to measurements of 95% con"dence, Eq. (3.21) modi"es to 8Y *O+2 ) 3\DK ) ((s/¸) ) 1/c C) \DK . (3.22) 8Y Both estimates (3.22) and (2.63) are shown in Fig. 3.3. Estimate (3.22) depends on collider parameters, while estimate (2.63) is universal for e>e\ collisions. Note that both estimates do not work for too small f . K 3.4.4.2. Future measurements. Estimate (3.20) can be confronted with results of the theoretical analysis [44], which assumes (s"14 TeV ¸"100 fb\ and M "1 TeV. The measurement of 8Y
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Table 3.3 Expected errors of measurements of *AC from Ref. [44] and from estimate (3.20) $ Analysis
s
t
g
¸R
SSM
*AC from [44] $ *O from (3.20)
0.007 0.008
0.016 0.012
0.014 0.011
0.006 0.008
* 0.006
Table 3.4 Values of parameters (1.37) and its statistical error-bars for typical models determined from probes at the LHC ((s"14 TeV, ¸"100 fb\). M "1 TeV. This is Table 3 of Ref. [8] 8Y
cJ * cO * ;I DI
s
t
g
¸R
0.9$0.016 0.1 1$0.16 9$0.057
0.5$0.02 0.5 1$0.14 1$0.22
0.2$0.012 0.8 1$0.08 0.25$0.16
0.36$0.007 0.04 37$6.6 65$11
O"AC is investigated there. Having in mind the crude approximations, which lead to estimate $ (3.20), the agreement is reasonable (see Table 3.3). As mentioned in Section 3.3, the polarization asymmetry of q's in the "nal state depends on the couplings of the Z to the q only, AO "v a /(v #a ) . (3.23) O O O O Therefore, it allows a model-independent measurement of this combination of coupling constants. The structure functions and branching ratios of the Z in#uence only the event rate and therefore the error of AO . For E GUTs, it is estimated [225] as *AO +1.5/(N . 8Y A measurement of the observables r ,A ,A ,r ,R ,R and R (3.24) W $ $ W JJ5 8Y8 8Y5 8YA would give model-independent information on the Z parameters cJ , cO , ;I and DI de"ned in Eq. * * (1.37). For M "1 TeV, the expected accuracy of such a measurement at LHC is 5% for cJ and 8Y * between 20% and 30% for cO ,;I and DI [211]. Estimate (3.20) predicts errors of about 1%. The * estimate is considerably smaller because it relies on AI . The measurements of cJ , cO , ;I and DI are $ * * based on all observables (3.24). Unfortunately, the measurements of the observables involving associated Z production or rare Z decays su!er from smaller statistics. See Table II of Ref. [211] for details. This explains the di!erence between the results in Table 3.4 and the estimate. The measurements of cJ ,cO ,;I and DI can be used to get information on the symmetry breaking * * sector [45]. Similar to a model measurement in e>e\ collisions described in Section 2.1.4.4, a veri"cation of relations (1.39) allows to check, whether the Z comes from the breaking of the E or SO(10) groups. In any case, the breaking parameters can be determined. Under the assumptions of Ref. [45], the statistical errors of such a measurement are around 10%. The
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breaking parameters cJ ,cO ,;I de"ne the Z couplings to SM fermions with a 16-fold sign ambiguity. * * This ambiguity can be removed by measurements at e>e\ colliders. Hadron colliders alone can reduce the sign ambiguity by collisions of polarized beams or by a measurement of observables, which are sensitive to the polarization in the "nal state as AO . The measurement of g /g in left}right symmetric models is considered in Ref. [52]. For 0 * M "1 TeV, this ratio could be measured at the LHC with a statistical error of about 1%. 8Y If a Z signal is found in hadron collisions, M and C can be de"ned by a "t to the invariant 8Y 8Y mass distribution of the "nal fermion pairs from the Z decay. M can be measured with an 8Y accuracy of C detecting only a few Z events. This is proven by the early measurements of the SM 8Y =- and Z-masses by the UA1 and UA2 experiments at CERN [202]. For larger event numbers, the systematic errors become important. See Ref. [231] for details.
4. Z search in other experiments There are other experiments not yet mentioned, which can give bounds on extra neutral gauge bosons. For completeness, we brie#y comment on some of them in the next sections. 4.1. ep collisions Neutral current electron}proton scattering occurs through photon, Z or Z exchange in the t channel. The Z is detected by indirect e!ects similar to e>e\ collisions. The additional contributions due to Z exchange lead to deviations of observables from their SM predictions. Compared to e>e\ collisions, ep collisions su!er from the hadronic background, in which these deviations must be detected. ep collisions are as insensitive to ZZ mixing as o!-resonance fermion pair production. Therefore, we put the ZZ mixing angle h to zero in this section and identify + Z and Z with Z and Z . Some early Z analyses can be found in Refs. [204,232}234]. 4.1.1. Born cross section The amplitude of ep scattering depends only on the ratio of the Z couplings and the Z mass. The six couplings a , v with f"e, u, d are always involved simultaneously because the Z must couple D D to the initial state. The cross section of the reaction e\ pPe\ X including extra neutral gauge bosons is * 0 * 0 s dp(e\ ) * 0 "2pa p(m, n) (4.1) Q dx dy K L with p(m,n)"s (Q)sH(Q)[[1#(1!y)]F* 0(x, Q)#[1!(1!y)]xF* 0(x, Q)] , K L F* 0(x, Q)"x[C (e)$C (e)] [C (q)(q(x, Q)#q (x, Q))] , 4 4 O
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xF* 0(x, Q)"$x[C (e)$C (e)] [C (q)(q(x, Q)!q (x, Q))] , 4 O C ( f )"v (m)vH(n)#a (m)aH(n), C ( f )"v (m)aH(n)#a (m)vH(n) . (4.2) 4 D D D D D D D D The propagator s (Q) is given in Eq. (2.7) and the couplings v (n) and a (n) are de"ned in Eqs. L D D (1.23) and (1.28). q(x, Q) and q (x, Q) are structure functions of the proton. The cross section for e> scattering is given by Eq. (4.1) with the replacements F* 0PF0 *, F* 0P!F0 * in Eq. (4.2). * 0 The kinematic variables Bjorken-x and y are de"ned as x,Q/(2P ) q), y,(P ) q)/(P ) p )"Q/xs with Q"!q"!(p !p ), C C J and s,(p #P) . (4.3) C p (p ,P) are the energy}momenta of the incoming electron (scattered electron, proton). s is the C J centre-of-mass energy squared and Q is the momentum transfer squared. We treat all initial and "nal particles as massless. Among the di!erent observables, the total cross sections are rather insensitive to a Z because the contributions from photon exchange are very large. Charge (A\>, A\>), polarization (A>>, A\ \) ** 00 *0 *0 and mixed (A\>, A\>) asymmetries can be de"ned [233,234] *0 0* AKL "(dp(eK)!dp(eL ))/(dp(eK)#dp(eL )), X, >"¸, R; m, n"#,! . (4.4) 67 6 7 6 7 Most of the systematic errors drop out in these asymmetries. 4.1.2. Radiative corrections Presently, there are no hints for a Z at HERA. We expect that possible new Z contributions to the cross section are very small. It is therefore su$cient to take into account only the QED corrections in the leading log approximation (LLA) [235] to these contributions. For the other contributions, one has to take into account the SM corrections. See Refs. [236,237] for an overview of SM corrections to ep collisions and for further references. QED corrections can be taken into account in a model independent way in the LLA. They consist of initial and "nal state radiation [238] and the Compton peak [239]. The full O(a) QED and weak corrections can be found in Refs. [37,240]. See also Section 2.1.2.2 for further references to weak corrections. They can be taken into account [235] by form factors [85] as described in Section 2.1.2.2. As discussed in Ref. [235], the electroweak corrections are of the same size as the Z e!ects. Therefore, they must be taken into account in a Z analysis at ep colliders. The M - and M -dependence of weak corrections in presence of Z production is R & discussed in Ref. [241]. The QCD corrections to Z production are the same as in the SM. See Ref. [242] for a short review and for further references. 4.1.3. Z constraints A model-independent Z analysis in ep collisions always involves the six Z couplings a , v , f"e, u, d. If condition (1.26) is assumed, the model-independent analysis can constrain D D the "ve combinations (1.36). ep collisions are sensitive to the relative sign of the Z couplings.
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Table 4.1 The 95% C¸ predictions for M from HERA with (s"314 GeV and the integrated luminosities quoted in the table. 8Y The "rst two rows are taken from Table 3 of Ref. [246] M /GeV 8Y
s
t
g
¸R
¸"0.5 fb\ ¸"1.0 fb\ Ratio
390 470 1.21
210 260 1.24
240 290 1.21
420 500 1.19
A model-independent analysis for HERA is carried out in Ref. [243]. Unfortunately, HERA cannot compete with hadron colliders [244,248] in setting Z bounds for usual GUTs. Model-dependent Z limits at electron proton colliders can be obtained by considerations similar to those explained for e>e\ collisions. We get the following scaling of M with ¸ and s, 8Y
s¸ 1 & . M +(Q 1# 8Y 1#r *o
(4.5)
The scaling is the same as in e>e\ collisions. Therefore, the dependence on systematic errors is the same. The di!erence is that the error *o depends on the kinematic variable Q. Reasonable statistics are obtained for Q well below s. As a result, ep collisions are sensitive to extra neural gauge bosons with masses comparable to the centre-of-mass energy (s. The model dependence of these bounds is large as in e>e\ collisions. See Ref. [245] for a "rst analysis of future exclusion limits including radiative corrections. A recent analyses can be found in Ref. [246]. A comparison of the numbers in Table 4.1 with those of Table 3.2 demonstrates that HERA cannot compete in setting mass limits to a Z predicted in typical extended gauge theories. Scaling (4.5) describes nicely the changes with the luminosity. Recently, the anomalous high Q events observed at HERA [247] received much attention. As pointed out in Ref. [248], these deviations from the SM cannot be explained by a Z coming from typical extended gauge theories, which is compatible with the present LEP and Tevatron data. Two models, the excited weak boson model [249] and the BESS model [19] are not ruled out by the present data. 4.2. Atomic parity violation The measurement of parity-non-conserving transitions in atoms has reached a precision [250,251], which allows constraints to extra neutral gauge bosons competitive [109,110,252,253] to those from collider experiments. Parity violating transitions occur due to the exchange of vector bosons with axial couplings. The experimental results are usually given in terms of the weak charge, Q "!2C (2Z#N)!2C (Z#2N), C "2a (1)v (1), q"u, d . 5 S B O C O
(4.6)
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Table 4.2 Values of the parameters c and c computed for s "0.2334 for di!erent Z models. The numbers are taken from Table 5 IX of Ref. [109]
c c
s
t
g
¸R
!138.0 65.6
37.2 0.0
!114.0 !16.4
!46.9 74.7
It arises due to the coherent interaction of the electron with all Z(N) protons (neutrons) in the nucleus of the considered atom. To calculate the SM prediction of Q , radiative corrections [254] must be applied, 5 C "!o (!s (0) ), C "!o (!#s (0) ) . (4.7) S .4 5 B .4 5 For Cs, we have Q "!376C !422C . The resulting SM prediction [255] for 5 S B M "175 GeV and m "100 GeV is Q1+"!73.04. R & 5 This can be compared with the present experimental value [250], Q "!72.11(27) (89) . (4.8) 5 The agreement with the SM prediction is used to constrain possible new physics. The exchange of extra neutral gauge bosons would give additional contributions to parity violating transitions in atoms. For Cs, the predicted change in the weak charge is [109] g *Q "(M/M!1)(Q1+#73.8)!h (Q#2a (2)Q1+)#2(M/M)(g/g)a (2)Q 5 C 5 C 5 5 8 5 +g (4.9) with Q"!376v (2)!422v (2). The "rst contribution is numerically negligible [109] due to an 5 S B accidental cancellation between the two contributions and due to the present experimental constraint on M/M!1"o !1(0.003 already mentioned in Section 2.1.3.1. Constraint 8
(4.9) can therefore be parametrized as [109] M (4.10) *Q +c h #c . 5 + M The coe$cients c and c are given for some models in Table 4.2. The present constraints on the ZZ mixing angle h from measurements at the Z peak are + stronger then those, which would result from atomic parity violation. As a result, measurements of atomic parity violation constrain mainly M /M . Note however, that the weak charge Q can receive compensating contributions from more than 5 one new physics source, which would relax the Z limits. Such cancellation e!ects are explicitly demonstrated in Ref. [256]. 4.3. Neutrino scattering High-energy neutrino scattering experiments provided interesting limits on Z parameters in the past. Today, they cannot compete with the limits from collider experiments. See Refs. [6,7] for
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Table 4.3 Present 95% C¸ limits on M in GeV in di!erent Z models from neutrino scattering experiments 8Y
Ref. [258] Table XI of Ref. [259] Fig. 2 Ref. [261]
s
t
g
¸R
262 215 500
135 54 155
100 87 190
253 220
a review and references to older experiments. Recent results of neutrino scattering experiments can be found in Refs. [257,258], a recent review is given in Ref. [259]. Neutrino-electron and neutrino-nucleon scattering experiments measure the couplings of the neutrino to the Z boson [259], v "!0.035$0.012$0.012, a "!0.503$0.006$0.016 . (4.11) J J These measurements are complementary to experiments at the Z peak because they measure the couplings at much lower centre-of-mass energies. The weak- and QED corrections to neutrino scattering are given in the "rst reference of [238] and in Ref. [260]. As e>e\ collisions at the Z peak, the agreement of these measurements with the SM prediction constrains physics beyond the SM. The constraints on extra neutral gauge bosons obtained by the CHARM II Collaboration [258] are given in Table 4.3. Comparisons of the excluded ranges in the h !M plane with L3 measurements can be found in Fig. 46 of Ref. [259]. + 8Y The CHARM, CCFR and CDHS collaborations quote model-independent results on neutrinonucleon scattering [259]. These results can be converted into constraints on di!erent extra neutral gauge bosons using the formalism of Ref. [7]. The resulting numbers for an unconstrained Higgs sector are given in Table 4.3. Recently, low-energy neutrino scattering experiments (l ePl e) are proposed [261]. The propoC C sal foresees to place a strong neutrino source in the centre of a neutrino detector. Such a neutrino source with an activity of 1.67$0.03 MCi based on Cr was already used to calibrate the GALLEX neutrino experiment. A l source based on Pm is proposed to have an activity of C 5!15 MCi [261]. All these sources emit neutrinos with energies well below 1 MeV. New detectors are proposed to measure the small recoil energy of the scattered electrons with high precision [261]. The experiment measures the neutrino couplings. This information can be used to set limits on extra neutral gauge bosons. The possible constraints on the ZZ mixing angle cannot compete with the present LEP measurements, while the possible bounds on M are interesting. They are shown 8Y in Figs. 1 and 2 of Ref. [261]. We produce mass limits from Fig. 2 of that reference neglecting systematic errors and present them in Table 4.3. 4.4. Cosmology The number of light neutrinos interacting with the SM Z boson is known from experiments at the Z -resonance to be N "2.989$0.012 [91]. J
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GUTs containing extra neutral gauge bosons also predict the existence of additional (right handed) neutrinos. The number of these neutrinos, which do not interact with the SM Z boson, is not constrained by the experiments at LEP and SLAC. The big bang nucleosynthesis of neutrons and the related abundance of He in the universe is sensitive to any particles with a mass lighter or about 1 MeV [262], the mass di!erence between the proton and the neutron. Assuming a primeval He abundance > "0.242$0.003, one gets [262] . a 95% CL interval N "3.0!3.7 assuming the D#He lower bound to the baryon density, and J N "3.0!3.2 assuming (D/H) "(2.5$0.75);10\ [263]. J . This measurement of N can be interpreted as a constraint on theories predicting light neutrinos J and light Z s [264]. The resulting bounds on M are stronger than those from collider experiments 8Y but they contain more assumptions on the model. Constraints on extra neutral gauge bosons from the supernova SN 1987A are considered in Refs. [265}268]. Models of stellar collapse predict an energy release of 4;10 W s. The measured neutrino events suggest an energy release exceeding 2;10 W s within 10 s. Therefore, at most 2;10 W s could be emitted by other particles. In particular, additional neutrinos lighter than about 50 MeV, the core temperature, would carry away a part of the energy. They can be produced through nucleon}nucleon bremsstrahlung (NNPNNl l ) in presence of a light Z . 0 0 The agreement between the models of stellar collapse and the neutrino observation puts constraints on extra neutral gauge bosons if there are additional light neutrinos present in the model. These constraints are considerably stronger than present collider limits except for models where the Z coupling to right-handed neutrinos vanishes [265]. Of course, these limits are based on more model assumptions than the collider constraints. A non-zero ZZ mixing would only strengthen the limits [266]. The in#uence of radiative corrections on the limits is small [266].
5. Summary and conclusions In this review, we have investigated the phenomenology of extra neutral gauge bosons. We have considered in detail the Z constraints, which can be obtained at e>e\, e\e\ and k>k\ colliders. At these machines, fermion pair production, Bhabha scattering, M+ller scattering and = pair production can contribute to a Z analysis. The constraints from lepton colliders are compared with those from pp and pp colliders. In the case of the absence of a Z signal, lepton and hadron colliders give complementary Z constraints. Lepton colliders give the best constraints on the ZZ mixing angle and on weakly interacting Z s. The exclusion limits from lepton colliders are almost insensitive to C /M but they are rather model dependent. Hadron colliders give the best present constraints on the Z mass for Z s predicted in popular GUTs. These constraints are rather insensitive to the Z model. They become worse for an enlarged Z width, which can arise if decays to exotic fermions are kinematically allowed. The complementary role of lepton and hadron colliders is demonstrated in Table 5.1. In the case of a Z signal, lepton and hadron colliders are complementary in a Z model measurement. The proposed hadron colliders with unpolarized beams measure the couplings of the Z to SM fermions with smaller errors than the proposed lepton colliders but with a 16-fold sign
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Table 5.1 Present and future limits on M (95% C¸) for di!erent E models and the SSM in GeV. The last line is obtained from the 8Y present Tevatron bounds using relation (3.19)
Tevatron 1997 [63] LEP 1997 [102] end of LEP (190 GeV, 0.5 fb\) [126] Tevatron after run II (2 TeV, 2 fb\)
s
t
g
¸R
SSM
595 300 990 940
590 220 560 930
620 230 620 970
630 310 1100 970
690 520 1500 1040
ambiguity. Lepton colliders with polarized electron beams measure the same couplings with a 2-fold sign ambiguity only. The Z limits from electron}proton colliders cannot compete with those from lepton and hadron colliders. We emphasized the importance of model-independent and model-dependent Z analyses. Both analyses are complementary. Model-independent constraints are useful to restrict any present and future Z model. For this universality, one has to pay the price that not always all observables are useful for a model-independent analysis and that model-independent constraints from di!erent reactions are not always comparable. On the other hand, additional model assumptions bias the limits. However, as far as they are consistent with the data, they help to tighten the exclusion limit or to reduce the error of the measurement of the remaining model parameters. We reviewed the status of the radiative corrections, discussed the importance of the di!erent radiative corrections in detail, and described how they can be included in theories including a Z . QED corrections can be calculated in a model independent way. QCD corrections to Z processes are the same as in the SM. Weak corrections to the new Z contributions cannot be calculated independently of the model. We assume that the Z e!ects arise "rst at the tree level and not in loops. Then, the higher-order corrections including new GUT particles are a small correction to a small e!ect and can be neglected. However, they cannot be neglected in precision measurements at the Z peak. Computer programs with these corrections are required for Z analyses. We listed o$cially released FORTRAN programs relevant for a Z search and indicated where these programs have already been used in an analysis of experimental data. In contrast to Z model measurements, Z exclusion limits are rather robust against details of systematic errors. For di!erent reactions, we discussed kinematic cuts, which enhance the sensitivity to extra neutral gauge bosons. We now comment on the Z limits from di!erent processes in more detail. They are collected in Table 5.2, which summarizes the main results of the di!erent sections of this review in a telegraphic style. The "rst column refers to the considered reaction. If necessary, di!erent cases of the centre-ofmass energy are distinguished. In the next three columns, the status of the Z search is indicated. The simplest analysis could be done at the Born level. The next step would be the investigation of radiative corrections needed to meet the accuracy of future data. We put a # there if the radiative
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Table 5.2 Summary table of the Z limits Reaction, c.m. energy
Born analysis exists
Main RCs known
Program with RCs exists
Typical 95% CL constraint and scaling with c.m. energy s and integrated luminosity ¸
#
#
#
"a+"(0.0005, "v+"(0.001 C C "a+", "v+"(0.02, q"c,b O O "h "(0.003 + "a,", "v,"(0.01 J J M 'M "(3 to 8)(s 8Y g (g /7 for M ((2s *g *M 1 , + (M /M ) g M 2 8Y
e>e\Pf fM s+M
M(s(M
#
#
#
s+M
#
!
!
s'M
#
!
!
e!e\Pe!e\
#
#
!
s(M
#
#
#
s+M pp, pp PZ X
#
!
!
Z PllM , bbM
#
#
#
e>e\P=>=\
g (g /140 *C , *M +*E *a (2)/a (2), *v (2)/v (2) D D D D
&¸\ &¸\ &¸\ &(s¸)\ &(s¸) &(s¸)\ &(s¸)\
&¸\
&¸\
*E s A g (g /24 for M +(s& 8Y E M A 8Y see e>e\Pf fM , M(s(M "a+", "v+"(0.00025 TeV/(s C C "h "(0.001 TeV/(s + "h "(0.0001 TeV/(s +
&(s¸)\
pI (3/¸ 2
&¸\
&(s¸)\ &¸\
1 ¸ ) fb ) 1000c 8Y (pp) M 'M +(s 0.4# ln 8Y 32 3s/TeV 1 ¸ ) fb ) 1000c 8Y (pp ) M 'M +(s 0.6# ln 8Y 20 3s/TeV pe!, p e!Pe!X
#
#
#
M '(0.7 to 1.6)(s
&(s¸)\
corrections are known with an accuracy comparable or better than the expected experimental errors. This column is more subjective because neither the future experimental errors nor the magnitude of the radiative corrections are precisely known in advance. The existence of o$cially released computer programs containing all radiative corrections needed for a direct "t to data is indicated in the fourth column.
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The last column of the table contains typical bounds on di!erent model parameters or combinations of them and a scaling of these bounds with the integrated luminosity and the centre-of-mass energy. Of course, the input of this column is only representative depending on the assumptions and limitations not given in the table but described in the corresponding sections of this review. The scaling with the luminosity assumes that the systematic error decreases proportional to the statistical error. It follows a short comment on every row: Z e!ects in fermion pair production at the Z peak arise mainly through deviations in the couplings of the mass eigenstate Z to fermions compared to the SM prediction for the Z boson. On-resonance Z production gives the best present limits on the ZZ mixing angle h . The number in the table is a typical experimental bound for GUTs. It is + almost independent of the Z mass. Without assumptions on the Z model, only v+ and a+, which D D are the product of the ZZ mixing angle and the Z couplings, can be constrained. The present constraints on v+ and a+ with f"u, d could only be improved by fermion pair production at the D D Z resonance. The strongest improvements on the present limits on v+ and a+ will come from C C = pair production at e>e\ or k>k\ colliders at TeV energies. The Z e!ects in o!-resonance fermion pair production arise mainly through interferences of the Z amplitude with the SM amplitudes. O!-resonance fermion pair production is sensitive to Z masses considerably larger than the centre-of-mass energy. The limits have a strong dependence on the Z couplings to fermions. They are insensitive to ZZ mixing. The exclusion limits in the table are given for typical GUTs and for colliders with ¸"80 fb\s/TeV. If no information on the Z model is available, one can only constrain the parameters v, and a,, which are proportional to D D ratios of the Z couplings and the Z mass. If the Z couplings to all SM fermions are very small, the Z could escape detection even for energies not far below its mass. However, it is hard to obtain such Z s in a GUT. The errors of Z model measurement and of Z exclusion limits scale di!erently with the integrated luminosity and centre-of-mass energy. Experiments on top of the Z resonance would certainly allow the most accurate measurements of the Z mass, of the Z width and of the Z couplings to SM fermions. In such measurements, muon colliders are clearly favored against electron positron colliders because they have a much smaller beam energy spread. The accuracy of the measurements of the Z couplings to fermions is expected to be comparable to the precision presently achieved at the Z resonance. As mentioned before, a weakly coupled Z can be missed in experiments below its resonance. If its couplings are not too small, such a Z can be observed in experiments above its resonance. In those experiments, the Z signal arises through the hard photons, which come from the radiative return to the Z resonance. These photons appear by the same mechanism, which is responsible for the hard photons from the radiative return to the Z resonance at LEP 2 energies. The Z limit from experiments above the Z resonance is sensitive to the photon energy resolution. Bhabha and M+ller scattering set bounds on the model independent parameters v, and a,, C C which are comparable to o!-resonance fermion pair production. Of course, Bhabha and M+ller scattering can only constrain the Z couplings to electrons. Model assumptions link Z couplings to leptons and quarks. In model dependent analyses, fermion pair production pro"ts from its additional observables with quarks in the "nal state. Therefore, the model dependent Z limits from fermion pair production are better than those from Bhabha and M+ller scattering. = pair production is very sensitive to changes of the Z couplings to fermions. Such changes destroy the gauge cancellation present in the SM. The result are large factors, which amplify the Z
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e!ects. = pair production can give the best model independent constraints on the parameters v+, a+. For models where the Z couplings to electrons are not zero, the resulting bounds on h are C C + better than those from fermion pair production. The best limits can be obtained in measurements near the Z peak where the Z c and Z Z interferences dominate. At energies above the Z resonance, unitarity is restored and Z e!ects are no longer enhanced by large factors. Proton colliders can see a Z if it is directly produced by a quark-anti-quark pair. Therefore, these colliders can detect only Z s with masses considerably smaller than the centre-of-mass energy of the colliding protons. The Z mass exclusion limits are rather insensitive to details of the model as far as the signal can be separated from the background. This becomes harder for Z s with small couplings or for Z s with small branching ratios to SM fermions. The Z exclusion limits scale non-symmetrically with the centre-of-mass energy and with the integrated luminosity. To improve the limits, an increase of the centre-of-mass energy is favored against an increase of the luminosity. For a "rst Z discovery, Z decays to muon pairs are the favored process. In the case of a Z signal, there are many other useful observables, which can help to measure the model parameters. The errors of Z model measurements at hadron colliders scale with the integrated luminosity as in e>e\ collisions but have an enhanced sensitivity to the centre-of-mass energy. Some other experiments can provide Z limits. We could only brie#y comment on ep collisions, atomic parity violation, neutrino scattering and cosmology. Many di!erent bounds on extra neutral gauge bosons can be obtained from various experiments. In the foreseeable future, we shall learn from the new experiments whether the Z boson has one or several massive partners as predicted by most uni"ed theories. Let us hope for surprises.
Acknowledgements It is a pleasure to thank Tord Riemann for many years of fruitful collaboration. With him, I started to work on extra neutral gauge bosons. He encouraged me to write this review. I had innumerable discussions with him. Many of his ideas entered this review. I am happy to thank Sabine Riemann for several years of pleasant collaboration, during which I learned a lot of details on experiments. I am grateful to Wolfgang Hollik, who always had time to discuss with me questions on extra neutral gauge bosons and on radiative corrections. Further, I would like to thank Francesco del Aguila and Claudio Verzegnassi for many discussions and continuous encouragement. I am grateful to G. Altarelli, K. Ellis, S. Godfrey, P. Langacker, P. Minkovski, T.G. Rizzo and P. Zerwas, for discussions of parts of this paper, valuable hints and warm hospitality and F. Berends, M. Cvetic\ , W.T. Giele, N. Lockeyer, K. Maeshima, A.A. Pankov, M. Peskin, M. Zra"ek for interesting discussions. I thank S. Riemann for providing the Figs. 2.5 and 2.9 and A.A. Pankov for providing Fig. 2.20. I bene"tted greatly from H. Fritzsch and R. RuK ckl due to their continuous support, many discussions and due to the stimulating working conditions at their institute. I thank S. Godfrey for the careful reading of the manuscript and for his many useful comments. Finally, I would like to thank my wife Ines for her patience while this paper was written and for hints concerning the manuscript. This work was partially supported by the German Federal Ministry of Research and Technology under contract No. 05 GMU93P, the Deutsche Forschungsgemeinschaft, and the EC contracts CHRX-CT-92-0004 and CHRX-CT940579.
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Note added In the time between submission and acceptance of this paper, interesting new results regarding the phenomenology of extra neutral gauge bosons appeared. The data of Super-Kamiokande [275] shows a clear evidence for a non-zero neutrino mass. In the context of GUTs, this is a strong support of uni"cation groups, which predict extra neutral gauge bosons. More accurate data of precision measurements appeared [276]. They lead to an improvement of some Z limits reported in this paper. Recent studies of extra neutral gauge bosons in di!erent models are published in [277]. The scattering of polarized protons would allow to measure more details of a Z model also at hadron colliders [278]. E!ects of very light extra neutral gauge bosons are discussed in [279]. The discrimination between Z e!ects and R-parity violating processes at the NLC is considered in [280].
Appendix A. Notation For the convenience of the reader, we collect the main notation in the following Tables A.1}A.3. The notation appearing in Section 1 has mainly to do with the de"nition of the Z parameters. It follows the notation of kinematic parameters and observables introduced in Section 2. We conclude with the notation relevant in the remaining chapters.
Table A.1 Main conventions and notation Symbol
Meaning
Z s, t, g, ¸R, SSM h + Z ,Z CD, C5, CDD4 L L L M ,C 8Y 8Y M ,C L L m L BrD L e, g , g
Vector particle associated with the extra ; (1) group in Eq. (1.1). Particular Z models, see Sections 1.2 and 1.1.3. Mixing angle between the symmetry eigenstates Z and Z . Mass eigenstates resulting from the mixing of the Z and Z . Partial decay widths of the Z as de"ned in Section 1.1.4. L Mass and width of the Z . Masses and total widths of the Z , n"0, 1, 2, Z "c. L Complex mass, m"M!iC M . L L L L Branching ratio of the Z decay to +M , BrD"CD/C . L L L L Coupling strengths of the c, Z and Z to fermions, see Eq. (1.20), e+0.31, g "g"e/(2c s )+0.37. 55 Vector and axial vector couplings of the Z(Z ) to the fermion f. Left- and right-handed couplings of the Z(Z ) to the fermion f. Vector and axial vector couplings of the Z to the fermion f. L Left- and right-handed couplings of the Z to the fermion f. L Sign-dependent coupling combinations, see Eq. (1.36). Sign-independent coupling combinations, see Eq. (1.37).
v , a (v , a ) D D D D ¸ , R (¸ , R ) D D D D v (n), a (n) D D ¸ (n), R (n) D D e , PC , PO , PS , PB 4 * 0 0 e , cJ , cO , ;I , DI * *
Section 1
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Table A.2 Continuation of conventions and notation Symbol
Meaning
s(s ) ¸ j ,j ,j c"cos h
Centre-of-mass energy squared of the considered (sub)process. Integrated luminosity Helicity combinations (2.9), (2.97) of the initial particles. h is the angle between the initial electron and the "nal fermion f (or "nal =\). Propagator as de"ned in Eq. (2.7). Propagator in e>e\Pe>e\ as de"ned in Eq. (2.114). Normalized couplings of the Z , as de"ned in Eq. (2.3). Mixing dependent couplings (2.4) of the Z . Shifts (2.111) in the couplings g and g due to a Z . 55A 558 Some observable (total or di!erential cross section or asymmetry), its experimental error and its SM prediction. Ratio of the systematic and statistical errors, see Eq. (1.66) g -depending error, of an observable, see Eq. (2.46). Shift of the observable O from its SM prediction due to a Z . Total cross section of fermion pair production in di!erent processes, see Eqs. (2.6) and (3.1). Forward}backward, left}right and polarization asymmetries for fermion pair production in di!erent processes, see Eqs. (2.15), (3.7) and (4.4). Combined asymmetries, see Eqs. (2.16) and (3.7). Ratios and asymmetries as de"ned in Eq. (2.19). Largest M , which can be detected. Ratio of M and M , see Eq. (2.62). 8Y Smallest mixing angle, which can be detected. Smallest coupling strength, which can be detected. Smallest shifts d and d , which can be detected, see Eq. (2.131). A 8 Flux function (2.21) describing QED initial state corrections. Cut on the photon energy in units of the beam energy. Boundaries of the considered range of the photon energy. Photon energy resolution of the experiment.
s (s) L s( (s) L v,, a, D D v+, a+ D D d ,d A 8 O, *O, O 1+ r *o *8YO pD 2 AD , A , AD $ *0 AD , AD , AD *0 $ $ *0 R , A , R , R *0 @ A M 8Y f K h + g d , d A 8 HC (v) * *>, *\ *E /E A A
Section 2
Appendix B. Available FORTRAN programs for Z 5ts Several computer programs are available allowing either theoretical investigations or direct "ts to data. All o$cially released programs for Z analyses known to the author are collected in Table B.1. In the "rst column, the name of the program is listed. The second column contains references related to the program. If there exists a program description, its reference is printed in bold. References to original papers describing the underlying physics of the program are printed in roman. Examples of references, where the program was used in theoretical or experimental Z studies, are printed in italics. In the last column the location of the program is given. It follows a short description of every program listed in the table.
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Table A.3 Continuation of conventions and notation Symbol
Meaning
F (q) x ,y D D i ,i ,o D CD CD
Form factor taking into account the vacuum polarization of the photon, see Eq. (2.30). Form factors of the ZZ mixing, see Eq. (1.29). Form factors taking into account the weak corrections, see Section 2.1.2.2. Form factors (1.62) taking into account the ZZ mixing. Form factors (2.31) taking into account the weak corrections, and the ZZ mixing.
iK, iK , oK D CD CD i+, i+ , o+ D CD CD
Section 2
Sections 3 and 4 r ,A W $ W r ,r ,r JJ8 JJ8 JJ5 R , <"Z,=, c 8Y4 q (x, Q) D c 8Y N 8Y f O(r , Q) X f S(r ) X C SB x, y, Q AKL 67 Q 5
Rapidity ratios as de"ned in Eq. (3.8). Ratios from rare Z decays, see Eq. (3.9). Ratios from associated Z production, see Eq. (3.10). Structure function de"ning the distribution of partons q in the p(p ). D Model-dependent constant de"ned in Eq. (3.16). Number of expected Z events, compare Eq. (3.16). Integrated product of two structure functions, see Eq. (3.13). Approximation (3.15) of f O(r , Q) for large Q and q"u. X Is de"ned below Eq. (3.14), C "f S(r )/f B(r ). SB 8 X Kinematic variables (4.3) in ep scattering. Asymmetries in Eq. (4.4) ep scattering. Weak charge (4.6) measured in atomic parity violation.
Table B.1 O$cially released FORTRAN programs designed for a Z search Program
Main references
Location
ZCAMEL ZEFIT ZFITTER GENTLE/4fan PYTHIA v.6.1
[78,79], [130] [30], [62,100,101,126] [58], [76,85,270], [271] [199], [175,182,272,198], [273] [274]
ftp://gluon.hep.physik.uni-muenchen.de/zcamel.f http://www.ifh.de/ riemanns/uu/ze"t5}0.uu distributed together with ZEFIT http://www.ifh.de/ biebel/gentle2.f http://www.thep.lu.se/tf2/sta!/torbjorn
See the Pythia web-page fort the main references. Pythia is used by many collaborations at e>e\, ep and hadron colliders.
ZCAMEL: The program describes the pair production of massless fermions in e>e\ collisions including the full O(a) QED corrections and the exponentiation of soft photons radiated from the initial state. It is a fast stand-alone program designed for theoretical studies. Use ZEFIT for "ts to data. ZEFIT: The program must be used together with ZFITTER. It describes the fermion pair production in e>e\ collisions. ZEFIT contains all additional Z contributions needed for "ts to
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data. Special attention is paid to the simultaneous treatment of electroweak corrections and ZZ mixing needed for direct "ts to data distributed around the Z peak. It is designed for "ts to data above the Z peak too. ZFITTER: describes fermion pair production in e>e\ collisions. It contains all known SM corrections needed for "ts to data. It is designed for SM studies. It is required by ZEFIT and distributed together with this code. The description of ZFITTER is needed to work with ZEFIT. This is the reason why we list this code in Table B.1. GENTLE/4fan: The stand-alone program describes the production of four fermions in e>e\ collisions. It contains all known SM corrections needed for "ts to data. It is originally designed for SM studies. It allows a study of Z e!ects in = pair production using the branch of anomalous couplings, see Appendix C of the program description. An on-line description can found at http://www.ifh.de/theory/publist.html. PYTHIA: is a general-purpose event generator for a multitude of processes in e>e\, ep and pp physics. The emphasis is on the detailed modeling of hadronic "nal states, i.e. QCD parton showers, string fragmentation and secondary decays. The electroweak description is normally restricted to improved Born-level formulae. It contains physics beyond the SM as supersymmetry, extra neutral gauge bosons or leptoquarks. Pythia was used to obtain the present Z limits at hadron colliders.
References [1] P. Langacker, Phys. Rep. 72 (1981) 185. [2] R. Slansky, Phys. Rep. 79 (1981) 1. [3] M. Cvetic\ , P. Langacker, Phys. Rev. D 54 (1996) 3570; Int. J. Mod. Phys. A 11 (1996) 1246; hep-ph/9707451, Univ. of Pennsylvania preprint UPR-0761-T, in: G.L. Kane (Ed.), Perspectives in Supersymmetry, World Scienti"c, Singapore, to appear. J.D. Lykken, hep-ph/9610218. [4] M. Cvetic\ , P. Langacker, Phys. Rev. D 54 (1996) 3570; M. Cvetic\ , P. Langacker, Mod. Phys. Lett. A11 (1996) 1247; J.R. Espinosa, Nucl. Phys. Proc. Suppl. 62 (1998) 187, hep-ph/9707541. [5] F. Zwirner, Int. J. Mod. Phys. A 3 (1989) 49. [6] J.L. Hewett, T.G. Rizzo, Phys. Rep. 183 (1989) 193. [7] P. Langacker, M. Luo, A.K. Mann, Rev. Mod. Phys. 64 (1992) 87. [8] M. Cvetic\ , S. Godfrey, in: T. Barklow, S. Dawson, H. Haber, J. Siegrist (Eds.), Electroweak Symmetry Breaking and Beyond the SM, World Scienti"c, Singapore, 1995. [9] T.G. Rizzo et al., SLAC-PUB-7365, hep-ph/9612440, Proc. 1996 DPF/DPB Summer Study on New Directions for High Energy Physics-Snowmass96, to appear. S. Godfrey et al., SLAC-PUB-7440, hep-ph/9704291, Proc. 1996 DPF/DPB Summer Study on New Directions for High Energy Physics-Snowmass96, to appear. E. Accomando et al., DESY 97-100, Phys. Rep., subm. and references therein. [10] D. Amidei, R. Brock (Eds.), Future electro weak physics at the Fermilab Tevatron, Report of the TEV}200 Study Group, FERMILAB-Pub-96/082, p. 182 and references therein. [11] R. RuK ckl, in: G. Jarlskog, D. Rein (Eds.), Proc. Large Hadron Collider Workshop, Aachen, Oct. 1990, CERN 90}133, vol. I, 1990, p. 229. [12] A.A. Andrianov, P. Osland, A.A. Pankov, N.V. Romanenko, J. Sirkka, hep-ph/9804389. [13] J.L. Rosner, Phys. Lett. B 387 (1996) 113; H. Georgy, S.L. Glashow, Phys. Lett. B 387 (1996) 341. [14] P. Langacker, D. London, Phys. Rev. D 38 (1988) 886; E. Nardi, Phys. Rev. D 48 (1993) 1240; D. London, P. Langacker, in: P. Langacker (Ed.), Precision Tests of the Standard Electroweak Model, World Scienti"c, Singapore, 1995, p. 883. p. 951. [15] U. Cotti, A. Zepeda, Phys. Rev. D 55 (1997) 2998.
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R. Balian et al./Physics Reports 317 (1999) 251}358
VARIATIONAL EXTENSIONS OF BCS THEORY
R. BALIAN , H. FLOCARD, M. VED NED RONI CEA/Saclay, Service de Physique TheH orique, 91191 Gif-sur-Yvette Cedex, France Groupe de Physique TheH orique, Institut de Physique NucleH aire, 91406 Orsay Cedex, France
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Variational extensions of BCS theory R. Balian , H. Flocard *, M. VeH neH roni CEA/Saclay, Service de Physique The& orique, 91191 Gif-sur-Yvette Cedex, France Groupe de Physique The& orique, Institut de Physique Nucle& aire, 91406 Orsay Cedex, France Received November 1998, editor: E. BreH zin Contents 1. Introduction: statistical mechanics of "nite systems 1.1. The two types of equilibrium data 1.2. Inference in statistical mechanics 1.3. Two di$culties of mean-"eld approximations 1.4. Broken symmetries in in"nite and "nite systems 1.5. Outline 2. The variational principle 2.1. A general method for the construction of variational principles 2.2. The action-like functional and the stationarity conditions 2.3. General properties of variational approximations 2.4. Cumulants of conserved observables 2.5. Connection with the standard variational principle 2.6. Variational principle for DK (u) alone 2.7. Comments 3. Extended "nite-temperature HFB approximation for characteristic functions 3.1. Generalities and notations 3.2. The variational ansaK tze and the reduced action functional 3.3. The coupled equations 3.4. Formal results 4. Expansion of the extended HFB approximation: #uctuations and correlations 4.1. The HFB approximation recovered
254 254 255 256 257 259 260 261 262 265 269 270 271 272 273 274 275 278 279 281 282
4.2. Fluctuations and correlations of conserved observables 4.3. Fluctuations and correlations of non-conserved observables 4.4. Diagrammatic interpretation 4.5. Comments 5. Projected extension of the thermal HFB approximation 5.1. Generalities and notations 5.2. The variational ansaK tze and the associated functional 5.3. The coupled equations 5.4. Unbroken PK -invariance 6. Projection on even or odd particle number 6.1. Characteristic function in the N-parity projected HFB approximation 6.2. The partition function in the N-parity projected HFB approximation 6.3. The N-parity projected BCS model: generalities 6.4. The N-parity projected BCS model: limiting cases 6.5. The N-parity projected BCS model: a numerical illustration 6.6. Discussion 7. Summary and perspectives Acknowledgements Appendix A. Geometric features of the HFB theory A.1. The reduced Liouville space A.2. Expansion of the HFB energy, entropy and grand potential
* Corresponding author. e-mail: #[email protected]. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 3 4 - 3
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R. Balian et al. / Physics Reports 317 (1999) 251}358 A.3. The HFB grand potential and the RPA equation A.4. Riemannian structure of the HFB theory A.5. Lie}Poisson structure of the time-dependent HFB theory
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Appendix B. Liouville formulation of the projected "nite-temperature HFB equations References
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Abstract A variational principle is devised which optimizes the characteristic function at thermodynamical equilibrium. The Bloch equation is used as a constraint to de"ne the equilibrium state, and the trial quantities are an unnormalized density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity yield for the latter a Bloch-like equation with an imaginary time running backwards. General conditions for the trial spaces are given that warrant the preservation of thermodynamic relations. The connection with the standard minimum principle for thermodynamic potentials is discussed. We apply our variational principle to the derivation of equations which are tailored for (i) the consistent evaluation of #uctuations and correlations and (ii) the restoration through projection of broken symmetries. When the trial spaces are chosen to be of the independent-quasi-particle type, we obtain an extension of the Hartree}Fock}Bogoliubov theory which optimizes the characteristic function. The expansion of the latter in powers of its sources yields for the #uctuations and correlations compact formulae in which the RPA kernel emerges variationally. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative di!erences between even and odd systems, depending on the temperature ¹, the level density and the strength of the pairing force, are investigated analytically and numerically. When the single-particle level spacing is small compared to the BCS gap D at zero temperature, pairing correlations are e!ective, for both even and odd projected cases, at all temperatures below the BCS critical temperature ¹ . There exists a crossover temperature ¹ such that odd}even e!ects disappear for " ¹ such that ¹ (¹(¹ . Below ¹ , the free-energy di!erence between odd and even systems decreases " " quasi-linearly with ¹. The low temperature entropy for odd systems has the Sackur}Tetrode form. When the level spacing is comparable with D, pairing in odd systems is predicted to take place only between two critical temperatures, thus exhibiting a reentrance e!ect. 1999 Elsevier Science B.V. All rights reserved. PACS: 05.30.Fk; 21.60.-n; 74.20.Fg; 74.80.Bj; 74.25.Bt Keywords: Variational principles; Fluctuations; Broken symmetry restoration; Projection; Even}odd e!ects; Heavy nuclei; Superconducting grains
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1. Introduction: statistical mechanics of 5nite systems 1.1. The two types of equilibrium data Let us consider a "nite physical system in thermodynamic equilibrium. We want to evaluate variationally the thermodynamic functions of this system along with the expectation values, #uctuations and correlations of some set of observables QK . The equilibrium state of A the system is de"ned from our knowledge of conserved quantities such as the number of particles, the energy, the total angular momentum, the parity. However, because our system of interest is "nite, these equilibrium data should be handled with care. Indeed, depending on the circumstances, their values can be given in two di!erent ways, either with certainty or statistically, and this allows us to classify the conserved quantities into two types. Take for instance a "nite, closed system thermalized with a heat bath. Among the variables that determine its state, the particle number N is given with certainty; we shall call N a conserved quantity of type I. On the other hand, the energy is known only statistically as only its expectation value is "xed; let us call it a conserved quantity of type II. The Gibbs argument, which identi"es a heat bath with a large set of copies of the system, entails that this system is in a canonical Boltzmann}Gibbs equilibrium characterized by a Lagrangian multiplier b de"ning the inverse temperature. Apart from N, our information is speci"ed by b or equivalently by the expectation value of the Hamiltonian. Take now an open system which communicates with heat and particle baths. In this case, both the energy and the particle number are conserved quantities of type II, and the Gibbs argument leads to the grand canonical distribution. For other equilibrium data one has likewise to recognize, according to the circumstances, whether they are of type I or type II. Note that, aside from the truly conserved quantities, the type II equilibrium data may also include nearly conserved quantities (or order parameters) which are here assumed to have all been identi"ed beforehand. For macroscopic systems, this distinction between types I and II is immaterial because the relative #uctuations that occur for observables of type II are negligible in the thermodynamic limit. Canonical and grand canonical distributions then become equivalent. When dealing with xnite systems, however, it is important to specify carefully which statistical ensemble is relevant to the physical situation under consideration. This problem is encountered in various systems of current interest. In nuclear physics, depending on the experimental conditions and/or the accuracy of the description, we may have to describe nuclei having either a well-de"ned or a #uctuating particle number, angular momentum or energy. Similar situations may occur for atoms or molecules. Another example is that of metallic or atomic clusters which can be prepared with a given particle number [1]. Let us also mention mesoscopic rings for which a proper description of the conductivity properties requires that there be no #uctuations (i.e., a canonical distribution) in the number of electrons [2]. Likewise, the description of mesoscopic superconducting islands [3] or metallic grains [4}6] demands a distribution with a given parity of the number of electrons. The recent discovery of Bose}Einstein condensates in dilute alkali atomic gases is currently arousing great interest [7]; in this case also, the theoretical formulation requires us to use states with well-de"ned numbers of particles.
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1.2. Inference in statistical mechanics A standard formal method to assign a density operator DK to a system characterized by data of both types proceeds as follows [8]. The type I data, referring to some conserved observables, determine a Hilbert space, or a subspace H (for instance the subspace with a given number of particles) in which the density matrix DK operates. On the other hand, type II data are the expectation values 1XK 2 of the remaining conserved observables which act in H . As a conseI quence, they impose on DK the constraints 1XK 2"Tr XK DK /Tr DK , (1.1) I I where the trace Tr is meant to be taken on the Hilbert subspace H . As we shall see, it is convenient to leave DK unnormalized or, equivalently, not to include the unit operator among the set XK . Within the subspace H the state DK is determined, up to a normalization factor, by I maximizing the von Neumann entropy Tr DK ln DK #ln Tr DK , (1.2) S,! Tr DK under the constraints (1.1). Associating a Lagrangian multiplier j with each datum 1XK 2, one I I recovers the generalized Boltzmann}Gibbs distribution
(1.3) DK Jexp ! j XK , I I I where j and 1XK 2 are related to each other through I I R 1XK 2"! ln Tr exp ! j XK . (1.4) I I I Rj I I In what follows, we take the form (1.3) of the density operator for granted, irrespective of the mechanism that led to it. Equilibrium distributions of the form (1.3) can be the outcome of some transport equation, the detailed nature of which is beyond the scope of this work. The form (1.3) can also be justi"ed by various theoretical arguments. One of them is the above-mentioned maximum entropy criterion which has its roots in information theory [9]. Alternatively, the Gibbs argument provides a direct statistical derivation of the Boltzmann}Gibbs distribution. An extension to non-commuting observables XK , relying as in the classical case on Laplace's principle of I indi!erence, has been worked out in Ref. [10]. Formally, the assignment of the density operator (1.3) to the system solves our problem since the thermodynamic functions, expectations values, #uctuations, correlations of the observables and all their cumulants are conveniently obtained from the characteristic function
u(m),ln Tr AK (m)DK , (1.5) where the operator AK (m), depending on the sources m associated with the observables QK , is de"ned as A A
AK (m),exp ! m QK . A A A
(1.6)
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Indeed, by expanding Eq. (1.5) in powers of the sources, 1 u(m)"u(0)! m 1QK 2# m m C !2 (1.7) A A A B AB 2 A AB one gets at "rst order the expectation values 1QK 2 of the observables QK . At second order appear A A their correlations C ,1QK QK #QK QK 2!1QK 21QK 2 (1.8) AB A B B A A B and #uctuations *Q,C , and in higher orders appear the cumulants of higher rank. Moreover, A AA since we chose not to enforce the normalization of DK , the generalized thermodynamic potential is given by the zeroth-order term:
u(0)"ln Tr DK "ln Tr exp ! j XK "S! j 1XK 2 . I I I I I I
(1.9)
1.3. Two dizculties of mean-xeld approximations Unfortunately, in most physical cases, the calculation above cannot be worked out explicitly. Indeed, the statistical operator (1.3) is tractable only if all the operators XK are of the single-particle I type. For systems of interacting particles, this is no longer the case since, in general, the data 1XK 2 I include the average energy 1HK 2. Recourse to some approximation scheme then becomes unavoidable. Many approaches start with the replacement of the density operator (1.3) by one with the form DK Je\+K ,
(1.10)
where M K is a single-particle operator M aRa , or more generally a single quasi-particle operator GH GH G H which includes also pairs aRaR and a a . (In this article, we shall mainly deal with systems of G H G H fermions; for the boson case, terms linear in the operators aR and a should be added to M K .) G G However, the form (1.10) entails two dizculties, both of which will be adressed in this article. (i) The "rst di$culty concerns the optimum selection of the independent-quasi-particle approximate state (1.10). One often uses a variational criterion, and this is an approach that we shall follow also. [Conventional many-body perturbation theory is not suited to the problems we have in mind since no small parameter is available.] The choice of this criterion, however, raises a crucial question. A standard option, which leads to the Hartree}Fock (or Hartree}Fock}Bogoliubov) solution, consists in choosing the minimization of the free energy or, at zero temperature, of the energy. This procedure is clearly adapted to the evaluation of the thermodynamic quantities that are given by u(0), the value of Eq. (1.5) when the sources m vanish (see Eq. (1.9)). However, the resulting A approximate state is not necessarily suited to an optimization of the characteristic function u(m) for non-zero values of the sources m . For restricted trial spaces such as (1.10) (or (1.11)), a variational A method designed to evaluate u(m), rather than u(0), is expected to yield an optimal independentparticle density operator which, owing to its dependence on the intensity of the sources, is better adapted to our purpose than the (m-independent) Hartree}Fock solution. (ii) The second di$culty is more familiar. Let us for instance consider a system for which the particle number is of type I and the energy of type II. It should be described by a canonical equilibrium distribution DK Jexp(!bHK ) built up in the N-particle Hilbert space H . Since our
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system is "nite, the results are expected to di!er from those of a grand canonical equilibrium. However, perturbative and variational approaches are often worked out more conveniently in the entire Fock space H. In particular, even an independent-particle state such as Eq. (1.10) where M K " M aRa resides in this Fock space H. Thus the simple ansatz (1.10) requires leaving the GH GH G H space H to which the exact canonical state belongs; as a consequence, the standard variational treatment introduces spurious components into the trial state. Unphysical thermodynamic yuctuations of the particle-number thus arise from the form of the trial state (1.10), which would be better suited to a grand canonical than to the canonical equilibrium being approximated. These #uctuations should be eliminated, and it is natural to improve the approximation by introducing a projection onto states with the right number of particles. This is achieved by replacing Eq. (1.10) by the more elaborate ansatz (1.11) DK JPK e\+K PK , , , where PK is the projection onto the subspace H with the exact particle number N. The , characteristic function (1.5) can now be evaluated by taking the trace Tr in the full Fock space H. Likewise, when the exact state has a non-#uctuating angular momentum or spatial parity, any trial state of the form (1.10) introduces thermodynamic #uctuations for these quantities, and adequate projections PK are required so as to suppress these #uctuations. However, if the operator M K commutes with PK , it is not necessary in Eq. (1.11) to insert the projection PK on both sides of the exponential. 1.4. Broken symmetries in inxnite and xnite systems Projections turn out to be very useful in another circumstance, namely when a symmetry is broken by the operator M K . Since in this case M K does not commute with PK , the projection should appear on both sides as in Eq. (1.11). Although projection matters mostly for "nite systems, let us "rst recall, for the sake of comparison, the situation in the in"nite systems. Consider a set of interacting particles which, in the thermodynamic limit, presents a spontaneously broken symmetry. One can think of a Bose liquid or a superconducting electron gas where the NK -invariance is broken. In such a system an essential property of the exact statistical operator DK is the long-range order displayed by the associated n-point functions. For an interacting Bose #uid at low temperature, if we denote by tR(r) the creation operator at the point r, expectation values over DK such as 1tR(r )t(r )2, 1tR(r )tR(r )t(r )t(r )2 factorize in terms of a single function u(r) as u(r )Hu(r ), u(r )Hu(r )Hu(r )u(r ) when the various points all move apart from one another. On the other hand, since for any size of the system its exact density operator DK commutes with NK , as does the Hamiltonian HK , the quantity 1t(r)2 vanishes. In order to give a meaning to the factors u(r), a standard procedure consists of adding to HK small sources which do not commute with NK : for instance terms in a and aR for bosons, or in aRaR and a a for fermions. The resulting equilibrium G G G H G H density operator DK displays therefore an explicit broken invariance. Despite the fact that such sources do not exist in nature, the remarkable properties of DK make it an almost inevitable substitute for DK . Indeed, in the limit when one "rst lets the size of the system grow to in"nity and then lets the sources tend to zero, DK is equivalent to DK as regards all the observables that commute with NK , such as tR(r )t(r ) or tR(r )tR(r )t(r )t(r ); however, the expectation value 1t(r)2 over DK does not any longer vanish and is interpreted as an order parameter. Furthermore, DK satis"es
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the clustering property for all operators, even those which do not commute with NK ; in particular, we now have 1tR(r )t(r )2 P1tR(r )2 1t(r )2 when "r !r "PR so that 1tR(r)2 can be identi "ed with u(r). The clustering property of DK as well as the equivalence of DK and DK for the physical observables were recognized for the pairing of electrons as early as the inception of the BCS theory [11]. Thus, for su$ciently large systems, the same physical results are obtained not only for di!erent canonical ensembles, but also with or without the inclusion of sources that break invariances. The situation is di!erent for a "nite system governed by the same interactions as in the in"nite case above, under similar conditions of density and temperature. Here again the expectation value 1t(r)2 is zero in the exact equilibrium state, but now no analogue to DK can be devised. Indeed, we can no longer de"ne an order parameter by means of factorization of n-point functions since the relative coordinates cannot grow to in"nity. Nor can we introduce sources which would produce non-vanishing expectation values 1t(r)2 without changing the physical n-point functions, as this requires the sources to tend to zero only after the thermodynamic limit has been taken. Even the translational invariance of a nucleus, an atom or a cluster cannot be broken in the exact ground state or in thermal equilibrium states if the Hamiltonian includes no external forces and commutes with the total momentum. Returning to the NK -invariance, for a nucleus with pairing or for a superconducting mesoscopic island or metallic grain, it is no longer without physical consequences to break the exp(i NK ) gauge invariance. This not only introduces spurious o!-diagonal elements in NK for the density matrix DK , but also a!ects observable matrix elements in the diagonal blocks in NK . Thus, the breaking of invariances does not have the same status for "nite and in"nite systems. In the latter case, it stands as a conceptual ingredient of the theoretical description which correctly implements the clustering property of the n-point functions. For "nite systems it is also essential, but now as a basic tool for building approximation schemes which economically account for some main features of the correlations. Both in perturbative and variational treatments, the starting point is most often an approximate density operator of the form (1.10) which does not commute with conserved observables such as NK . This yields tractable and sometimes remarkably accurate approximations, for which the clustering of the correlation functions arises naturally from Wick's theorem. Remember the success of the BCS theory in describing pairing correlations between nucleons in "nite nuclei. (One may wonder why nuclear physicists, who had acknowledged the existence of pairing e!ects long before 1957, did not anticipate the BCS theory; it may be that, too obsessed by the conservation of the particle number, they were inhibited in developing models which would break the NK -invariance.) Nonetheless, more realistic descriptions of "nite systems require the restoration of the broken invariances. A state of the form (1.10) involves #uctuations in the particle number that must be eliminated, whether they arise from a grand canonical description, or from unphysical o!-diagonal elements of NK connecting Hilbert spaces with di!erent particle numbers. A natural procedure to account for pairing correlations and at the same time enforce the constraint on the particle number consists in using a projection. The use of Eq. (1.11) arises naturally in this context. Likewise, breaking translational or rotational invariance introduces spurious #uctuations of the linear or angular momentum, as well as spurious o!-diagonal elements in the trial density operator (1.10). To remedy these defects we need again a trial density operator of the form (1.11) involving the appropriate projection PK on both sides.
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The discussion above pertains to the breaking of a conserved quantity of type I, that is, one which is known with certainty , such as NK in a canonical ensemble. For a quantity of type II, known only on average, such as NK in a grand canonical ensemble, the invariance breaking again introduces o!-diagonal elements of NK , which are spurious for a "nite system. One can suppress them, while letting NK free to #uctuate, by means of a di!erent type of projection, exhibited below in Eq. (5.11). The problem of the restoration of broken symmetries has already a long history. Within the nuclear context, in the zero-temperature limit, a general review of the litterature (up to 1980) may be found in [12]. An early reference on parity projection at "nite temperature is [13]. More recent works include Refs. [14}17], where additional references can be found. For a Monte-Carlo approach to the temperature dependence of pair correlations in nuclei, see Refs. [18,19]. In condensed matter physics, recent experimental studies about isolated mesoscopic metallic islands [3,20] or small metallic grains [4}6] have motivated several theoretical works about modi"ed BCS theories with well de"ned number parity [21}28].
1.5. Outline The two problems (i) and (ii) that we have stated in Section 1.3 will be worked out within a general variational setting described in Section 2. We shall write an action-like functional, Eq. (2.5), which yields as its stationary value the characteristic function (1.5). This corresponds to the static limit of a variational principle proposed elsewhere to evaluate multi-time correlation functions [29]. The approach uses a method [30,31], recalled in Section 2.1, which allows the construction of variational principles (and variational approximations) tailored to the optimal evaluation of the quantity of interest (here the characteristic function). This systematic and very general method encompasses several known variational principles, including the Lippmann}Schwinger formalism [32] (the quantity of interest being the S-matrix). An important ingredient is the incorporation in the trial functional of the Lagrangian multipliers associated with the primary equations, in our case the Bloch equation determining the canonical density operator (1.3). As a result, the number of variational degrees of freedom is doubled. Here, the variational quantities entering our functional (2.5) consists of two matrices: one akin to a density operator and the other to the observable AK (m) de"ned in Eq. (1.6). Within this framework, one can state some general conditions on the trial spaces that ensure the preservation of the thermodynamic identities by the variational approximations. Starting in Section 3, we apply the variational principle of Section 2 to systems of fermions for which pairing e!ects are signi"cant. Section 3 (as well as Section 4) deals with grand canonical equilibrium. The two variational operators introduced in Section 2 are taken as exponentials (3.4) of single-quasi-particle operators and the resulting expression of the approximate action-like functional is given in Section 3.2. The equations which express the stationarity and determine the optimal characteristic function, are derived in Section 3.3. They couple, with mixed boundary conditions, the variational parameters characterizing the two trial matrices and they generalize the Hartree}Fock}Bogoliubov (HFB) approximation [33]. In Section 4, the coupled equations obtained in Section 3 are expanded in powers of the sources m . The HFB approximation is recovered at the zeroth order, while the "rst order gives back the A
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usual HFB formula for the average values of observables (Section 4.1). Fluctuations and correlations of observables are provided by the second order. In the ensuing formulas for conserved and non-conserved observables, the RPA (random-phase approximation) kernel plays a central ro( le. In Section 4.4, through a diagrammatic analysis, we show the correspondence between our results and the summation of the RPA (bubble) diagrams. Section 5.1 addresses the projection problem stated in (ii) in Section 1.3. The functional and the coupled equations that result from variational ansaK tze of the type (1.11) are derived in Sections 5.2 and 5.3, respectively. Special attention is devoted to the case where the variational approximation does not break the PK -invariance (Section 5.4). Section 6 specializes the formalism of Section 5 to the projection on even or odd particle number. The more detailed study of this relatively simple case is motivated by its relevance to the description of heavy nuclei and to the interpretation of recent experiments on isolated superconducting islands [3] or small metallic grains [5]. Equations are obtained in Section 6.3 which should replace the BCS ones. Ensuing results are compared to those of Ref. [22]. In Section 6.4 we analyze some limiting situations with a special emphasis on the low temperature limit. Numerical applications are presented in Section 6.5; they illustrate physical situations encountered in nuclei, metallic islands or aluminium grains. In the last section we summarize our main results and present some suggestions for other applications and for extensions. Appendix A introduces our notation for the Liouville representation of the reduced fermion quasi-particle space. In particular, it gives a proof of the factorization of the RPA kernel (used in Section 4) into the stability matrix and the matrix whose elements are the constants associated with the Lie}Poisson structure of the HFB equations. In Appendix B, using the Liouville-space notation, we reexpress the equations derived in Section 5 in a fashion which exhibits their formal similarity with those obtained in Section 3. The length of this paper is partly a consequence of our wish to allow several reading options. Section 2 presents and discusses the variational principle for characteristic functions at thermodynamic equilibrium which underlies all the other sections. It can be read for itself since its scope is more general than the applications to fermion systems explored in the following Sections. The contents of Sections 5 and 6, which deal with projections, are largely independent of that of Section 4 which focusses on correlations. The reader primarily interested by the variational projection method can therefore bypass Section 4, except for the brief Section 4.1. If he is mostly concerned by the projection on even or odd particle-number, and ready to admit a few formulas demonstrated in Section 5, he can directly jump to Section 6. If his interest is focused upon the improvements over BCS introduced by parity-number projection, he may even begin with Section 6.3. Alternatively, the reader mainly interested by the variational evaluation of correlations and #uctuations in the grand canonical formalism can skip Sections 5 and 6. 2. The variational principle Our purpose in this section is to write a variational expression adapted to the evaluation of the characteristic function u(m) de"ned in Eq. (1.5), exp u(m),Tr AK (m)DK .
(2.1)
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The operator AK (m), de"ned in Eq. (1.6), involves the observables of interest QK and depends on the A associated sources m . The density operator DK describes thermodynamic equilibrium. It has matrix A elements in the Hilbert space H de"ned by the conserved quantities of type I. Therefore, the trace Tr in Eq. (2.1) is taken on this Hilbert space H . According to the Gibbs argument, the operator DK is an exponential of the conserved quantities XK of type II that are given on average (Eq. (1.3)). I From now on we assume that the Hamiltonian HK is one of these conserved quantities, with b as its conjugate variable. We thus write DK in the form DK "e\@)K ,
bKK , j XK . I I I
(2.2)
We have not introduced the unit operator among the set +XK ,. Then u(0) does not vanish but its I exponential is identi"ed (Eq. (1.9)) with the partition function Tr exp(!bKK ), for instance the canonical partition function for KK ,HK , or the grand canonical partition function for KK ,HK !kNK . This will allow us to evaluate variationally not only the expectation values and the cumulants of the set +QK ,, but also the thermodynamic quantities. A According to the circumstances, the space H can be the full Fock space, as in Sections 3 and 4, or a more restricted one such as the N-particle space for canonical equilibrium. Sections 5 and 6 resort in principle to the latter case; nonetheless, the introduction, as in Eq. (1.11), of a projection onto H makes it possible to use again the trace Tr in the full Fock space. 2.1. A general method for the construction of variational principles We shall rely on a methodical procedure for devising variational principles, and hence variational approximations, aimed at the evaluation of some quantity of interest. We brie#y recall the method; for a more detailed exposition see, for instance, Refs. [30,31]. Given a function f +xG, of a set of variables +xG,, we wish to estimate the value f +xG ,, the quantity of interest, at a point +xG , which is in principle completely determined by the set of equations g +xG,"0. In practice, we are concerned by the (common) situation where either the equations H g +xG,"0 are untractable and cannot a!ord the exact solution +xG , and/or where f +xG, cannot be H calculated at the point +xG ,. It is natural to associate with each equation g +xG,"0 a variable - H akin to a Lagrangian H multiplier, although we do not look here for a stationary value of f under constraints since our constraints are in su$cient number to determine f +xG ,. Then, a variational procedure suited to an approximate evaluation of f +xG , rests upon the introduction of the functional I+xG, - H,,f +xG,! - H g +xG, , H H
(2.3)
which involves both the variables xG and the additional quantities - H. The stationary value I of the functional I+xG, - H, is obviously equal to the exact value f +xG , for unrestricted variations of the point +xG, - H,. In fact, in this unrestricted case, letting I+xG, - H, be stationary with respect to the set +- H, ensures that g "0 and hence that xG"xG , so that I+xG , - H,"f +xG ,; it is therefore unnecessH ary to write the stationarity conditions with respect to the set +xG,.
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With respect to all the variables +xG,- H,, the stationary point +xG , - H, is a saddle point (Fig. 1). When we restrict the trial spaces of these variables, the stationarity of I provides a point +xG, - H, which can be close to the exact saddle point +xG , - H, if these spaces have been judiciously chosen. Then, since the surface I+xG, - H, is yat near its saddle point, the di!erence between f +xG , and the stationary value of I in the restricted space will be much smaller than the error on +xG , - H,. Several known variational principles enter the framework above [30,31]. Let us, for instance, recall how the Lippmann}Schwinger formulation can be retrieved. The quantity f +xG , of interest is here the transition amplitude 1W"U(t )2 from the state "U2 at the initial time t to the state "W2 at the "nal time t . The variables +xG, are the components of the ket "U(t)2 for t (t4t (the function f +xG , depends only on the part t"t of them). The constraints g +xG,"0 that determine "U(t)2 are H the components of the SchroK dinger equation (R/Rt#iHK )"U(t)2"0. Denoting as 1W(t)" the associated Lagrangian multipliers - H, we "nd the scattering amplitude as the stationary value of
I+U(t) , W(t),"1W"U(t )2!
R
R
dt1W(t)
R #iHK U(t)2 , Rt
where "U(t)2 is subject to the initial boundary condition "U(t )2""U2. The additional conditions RI/RxG"0 provide the SchroK dinger equation for 1W(t)" which must be solved backward in time with the "nal condition 1W(t )""1W". In the original work of Lipmann}Schwinger [32], the evolution of "U(t)2 is expressed in the interaction representation, and the times t and t are taken to !R and #R, respectively. When the elimination of one set of variables by means of the exact conditions RI/RxG"0 is feasible, a new variational principle may emerge, under suitable conditions, such that the quantity of interest f +xG , is obtained as a maximum or a minimum, and not merely as a stationary value. This is visualized in the schematic one-dimensional case of Fig. 1 by the dashed line. We shall encounter such situations in Sections 2.5 and 2.6. 2.2. The action-like functional and the stationarity conditions In our problem, the quantity of interest f +xG , is the characteristic function (2.1), which depends on the matrix elements of the operator DK "exp(!bKK ). We wish to characterize DK by means of some set of simple constraints that will be identi"ed with the equations g +xG ,"0 of the general H method. To this aim we remark that a set of linear equations is provided by the Bloch equation. More precisely, we choose to construct the canonical form DK "exp(!bKK ) as the solution of the symmetrized equation 1 dDK # [KK DK #DK KK ]"0 , du 2
(2.4)
where DK (u) is a u-dependent operator (04u4b). This equation is subject to the initial condition DK (0)"1) where 1) denotes the unit operator in the subspace H of the Fock space. The variables +xG, to be determined are thus the u-dependent matrix elements of DK (u) for 0(u4b, while the constraints g +xG,"0 are identi"ed with the matrix elements of Eq. (2.4). H
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Fig. 1. A geometric picture of the variational method. The x-axis stands for the variables +xG,, the y-axis for the variables +- H,. The level lines of the functional (2.3) show the existence of the saddle point +xG , - H,, the altitude of which is the quantity f +xG , of interest. The level lines corresponding to this altitude include the straight line +xG"xG ,, which visualizes the stationarity of I+xG,- H, with respect to the unrestricted set +- H,. On the other hand, the dashed line visualizes the stationarity of the functional I+xG, - H, with respect to the unrestricted set +xG,, and it meets the line +xG"xG , at the saddle point +xG , - H,. When +xG, - H, is restricted to some trial set, only part of the surface I+xG, - H, is explored. Stationarity of I with respect to both +xG, and +- H, in this subset provides a variational approximation in which the error on the altitude of the saddle point is of second order with respect to the deviation of +xG, - H, from the exact point +xG , - H,. Elimination of half the variables by means of RI/RxG"0 leads to a new functional represented by the dashed line, which may provide f +xG , as a maximum or a minimum under some convexity conditions.
The exact solution of Eq. (2.4) is exp(!uKK ), the matrix elements of which, for any u, constitute the set +xG ,. Here the quantity of interest f +xG ,"ln Tr AK (m)DK depends only on a subset of the variables +xG ,, namely on the matrix elements of DK (b). The Lagrangian multipliers - H associated with the constraints (2.4) are now the matrix elements of a u-dependent operator AK (u), and the functional (2.3) takes the form I+DK (u), AK (u),,Tr AK (m)DK (b) @ dDK (u) 1 ! du Tr AK (u) # [KK DK (u)#DK (u)KK ] . (2.5) du 2 Thus, the characteristic function (2.1) is the stationary value of the action-like functional (2.5) under arbitrary variations of the trial operators AK (u) and DK (u), the latter being subject to
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the boundary condition DK (0)"1) .
(2.6)
The speci"cs of our problem enter the functional (2.5) through the temperature 1/b , the operator KK and through the operator AK (m) (a functional of the observables and their sources). A variational approximation for exp u(m) will be generated by restricting the trial class for AK (u) and DK (u) and by looking for the associated stationary value of Eq. (2.5). The error will be of second order in the di!erence between the optimum trial operators and the exact ones. The doubling of the number of variational degrees of freedom, which include not only the trial state DK (u) but also the Lagrangian multiplier operator AK (u), is only bene"cial when approximations are made; indeed, as already indicated in Section 2.1 for the general procedure, for unrestricted variations of AK (u) the stationarity conditions on AK (u) alone yield the Bloch equation (2.4) and are su$cient to ensure that I+DK , AK , provides the exact value for exp u(m). When DK (u) and AK (u) are restricted to vary within some subset, the stationarity conditions of the functional I+DK , AK , with respect to AK (u) are read directly from Eq. (2.5): Tr dAK
dDK 1 # [KK DK #DK KK ] "0 . du 2
(2.7)
As regards the variation with respect to DK (u), an integration by parts allows us to rewrite the functional (2.5) in the form I+DK (u), AK (u),"Tr AK (0)#Tr DK (b)(AK (m)!AK (b)) @ dAK (u) 1 # duTr DK (u) ! [AK (u)KK #KK AK (u)] . 2 du From Eq. (2.8), for 04u(b, one obtains the stationarity conditions:
dAK 1 ! [AK KK #KK AK ] "0 , Tr dDK du 2
(2.8)
(2.9)
and for u"b: Tr dDK (b)(AK (m)!AK (b))"0 . (2.10) In Eqs. (2.7) and (2.9) we have omitted the u-dependence of DK , of AK , and of the allowed variations dDK and dAK . By construction of the functional (2.5), Eq. (2.7) gives back the symmetrized Bloch equation (2.4) for unrestricted variations of AK . For unrestricted variations of DK , Eqs. (2.9) and (2.10) yield (in the Hilbert space H ) the Bloch-like equation for AK (u) dAK 1 ! [AK KK #KK AK ]"0 (2.11) du 2 in the interval 04u(b, and the boundary condition AK (b)"AK (m)
(2.12)
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at the "nal value u"b. The variable u is thus evolving backward from b towards 0 in Eq. (2.11), the solution of which is (2.13) AK (u)"exp[(u!b)KK ]AK (m)exp[(u!b)KK ] . The exact equation (2.11) merely duplicates the Bloch equation (2.4). However, in any physical application, the trial spaces for DK (u) and AK (u) have to be restricted to make the calculations tractable. Then, in general, Eqs. (2.7) and (2.9) are coupled and must be solved simultaneously. Moreover, we saw that the boundary conditions (2.6) on DK (0) and (2.10) on AK (b) imply that Eq. (2.7) should be solved starting from u"0 up to u"b while Eq. (2.9) should be solved in the opposite direction. 2.3. General properties of variational approximations The stationary value I of the functional (2.5) depends on an ensemble of parameters g. Among them are obviously the source parameters m associated with the observables QK [the dependence is A A through the boundary condition (2.10)] and the temperature 1/b. The parameters g also include those, denoted as u, which can occur in the operator KK . In particular an external xeld u, coupled to an observable JK of the system, gives rise to a term !uJK in the Hamiltonian HK . For instance, u may represent a magnetic "eld coupled to the component JK of the magnetic moment operator along the "eld. Moreover, bu may be one of the Lagrangian multipliers j of Eq. (2.2). For instance, for a I grand canonical ensemble where KK "HK !kNK , the chemical potential k is regarded as one of the parameters u; likewise, for a system in uniform rotation, u may be an angular velocity associated with a component JK of the total angular momentum. The symbol g will therefore stand in this subsection for the set +m, b, u,. By construction, our quantity of interest expu(m) is equal to the stationary value I of the functional (2.5) (`sa stands for stationary); we shall use any one of the three notations ePK"I (g),I +AK (m), b, KK ,,I+DK , AK , g, . (2.14) E E The second notation speci"es some of the g parameters on which this stationary solution depends (parameters of the u-type are implicit within KK ). The last notation emphasizes that the optimized result depends on the parameters g both directly and through the optimal solution +DK , AK , E E determined within our variational class by the conditions dI+DK , AK , g,/dDK "0 and dI+DK , AK , g,/dAK "0. Let us now state with these notations a general property of variational principles that we shall often use: Owing to the stationarity conditions, the full xrst-order derivative of I with respect to any g reduces to its partial derivative evaluated at the stationary point:
d R d I (g), I+DK , AK , g," I+DK , AK , g, . E E D dg Rg dg K DK E AK AK E
(2.15)
The contributions to the derivative occuring through DK and AK cancel due to the stationarity. The E E relation (2.15) holds not only for the exact solution but also within any trial class for AK and DK . Another general property holds whatever the variational space for DK (u) and AK (u). The derivatives dDK /du and dAK /du belong to the set of allowed variations dDK and dAK , respectively. Writing Eqs. (2.7) and (2.9) for these variations and subtracting one equation from the other, we obtain
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along the stationary path the conservation law: d Tr KK [AK (u)DK (u)#DK (u)AK (u)]"0 . E E E E du
(2.16)
2.3.1. Characteristic function We shall always use for the operator AK a trial space such that it contains the unit operator and that variations dAK proportional to AK (dAK JAK ) are allowed. We then "nd from Eq. (2.7) that the integrand of the functional (2.5) vanishes, so that the stationary value ePK"I (g)"Tr AK (m)DK (b) (2.17) E does not involve the integral term. The exact relation (2.1) is thus preserved by the variational approximation. Moreover, since we do not normalize DK , its trial space will allow variations dDK proportional to DK (dDK JDK ). Another helpful property then emerges (which of course holds for the exact solution). Indeed, substituting dAK JAK in Eq. (2.7) and dDK JDK in Eq. (2.9) and combining these equations, we obtain d Tr AK (u)DK (u)"0 . E du E
(2.18)
Using dDK JDK in Eq. (2.10), we "nd ePK"Tr AK (m)DK (b)"Tr AK (b)DK (b) , (2.19) E E E even when AK does not belong to the variational space for AK . The relations (2.18) and (2.19) imply that the approximation (2.17) for exp u is equal to Tr AK (u)DK (u) for any value u, and in particular E E that ePK"Tr AK (0) . E
(2.20)
2.3.2. Partition function and expectation value of observables From Eq. (2.17) it follows (when dAK JAK ) that the partition function Tr DK (b), where the K subscript m"0 is shorthand for g,+m"0, b, u,, is given by the stationary value of the functional (2.5) in the absence of sources (AK "1) ): eP,I +1) , b, KK ,"Tr DK (b) . (2.21) K Let us now consider arbitrary values of the source parameters m . Using Eq. (2.15) where g stands A for one of these parameters, we "nd d dAK (m) du(m) , I +AK (m), b, KK ,"Tr DK (b) . E dm dm dm A A A According to Eqs. (1.7) and (2.21), in the limit where the m's vanish, Eq. (2.22) yields ePK
du Tr QK DK (b) Tr QK DK (b) ! ,1QK 2" A K " A K . A dm I +1) , b, KK , Tr DK (b) A K K
(2.22)
(2.23)
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This result tells us that the calculation of the expectation value of any observable requires only the solution of Eqs. (2.7) and (2.9) for AK "1) . Eq. (2.23) shows therefore that the density operator DK (b) suited to the calculation of the partition function is also suited to the variational evaluation of K the expectation values of the observables QK . A The knowledge of the m-dependence of DK (b) is only needed for the calculation of the higherE order cumulants. The use of Eq. (2.22) will allow us to gain one order when expanding u in powers of the sources. 2.3.3. The energy To "nd the derivative of I with respect to the inverse temperature b, we rewrite the functional (2.5) as
du Tr AK DK (u)d(u!b)!AK (u)
dDI (u) 1 # [KK DK (u)#DK (u)KK ] h(b!u) . du 2
(2.24) We then apply Eq. (2.15) with g replaced by b. Noting that dDK (u)/du" belongs to the variational S@ class allowed for dDK (b) and using Eq. (2.10), we obtain I"
ePK
d 1 du(m) , I "! Tr KK [AK (b)DK (b)#DK (b)AK (b)] , E E E E db 2 db
(2.25)
in which AK (b) can be replaced by AK when the variational space for AK contains AK . When AK "1) , Eq. E (2.25) becomes d Tr KK DK (b) du(0) K "1KK 2,E , "! ln Tr DK (b)" (2.26) ! K db Tr DK (b) db K where we have used Eq. (2.23) and introduced the notation E for the equilibrium energy in the canonical case (KK ,HK ), or for the expectation value 1HK !kNK 2 in the grand-canonical case. Thus, in the framework of our variational method (for any trial space allowing dAK JAK ), the derivative with respect to b of the approximate thermodynamic potential (AK (m)"1) ) provides the same result as the expectation value of KK obtained from Eq. (2.23) with QK "KK (i.e. AK (m)"exp !mKK ). 2.3.4. Thermodynamic potential and entropy We can also de"ne an approximate thermodynamic potential F ,!b\u(0) , % and an approximate entropy
(2.27)
du(0) S,b(E!F )"u(0)!b % db Tr KK DK (b) K , "ln Tr DK (b)#b K Tr DK (b) K which, owing to Eq. (2.26), satisfy the relation RF S"! % . R¹
(2.28)
(2.29)
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Our approximation will thus be consistent with the fundamental thermodynamic relations between the partition function, the temperature, the energy and the entropy. However, the approximate entropy (2.28) is not necessarily linked by the von Neumann relation (1.2) to the variational quantity DK (b) which optimizes F . K % Note that the entropy does not directly enter the functional (2.5), in contrast with the usual maximum principle for the thermodynamic potential, which requires the preliminary calculation of the entropy associated with the trial density operator. In fact, within our formalism, the status of the entropy S is that of a subordinate quantity: it can be calculated, using Eq. (2.28), from the b-dependence of the stationary value of the functional (2.5) with AK "1) . This is advantageous since we can thus bypass the explicit calculation of Tr DK ln DK , which is unfeasible when the trial state DK is, for instance, a projected independent particle state. 2.3.5. External xelds If we use the relation (2.15) where g stands for the external "eld u in a term !uJK entering the operator KK , we have
d 1 @ du du Tr JK [DK (u)AK (u)#AK (u)DK (u)] . (2.30) , I" E E E E 2 du du When AK "1) , i.e. when m "0, we shall see in Section 2.5 that under some rather general conditions A (satis"ed for instance in Sections 4.1, 5.4.2, 6.2 and 6.3), the products DK (u)AK (u)"AK (u)DK (u) do E E E E not depend on the variable u. Then Eq. (2.30) reduces to eP
1 d Tr JK DK (b) K "1JK 2 . ln Tr DK (b)" (2.31) K b du Tr DK (b) K In this case, as for Eq. (2.26), our variational approximation is consistent with the thermodynamic identities expressing the expectation values of observables JK as derivatives of the thermodynamic potential. 2.3.6. Evolution versus u of the operators AK DK and DK AK In Sections 3 and 4, AK (u) and DK (u) will be chosen within the trial class of exponentials of quadratic forms of creation and annihilation operators. Apart from the properties used above, namely that dAK and dDK belong to the classes of AK and DK respectively, we note that in this case the quantities dAK AK \, AK \ dAK , dDK DK \ or DK \ dDK span the set of quadratic forms. Let us more generally consider the case when the trial classes for DK and AK are such that DK \dDK and dAK AK \ belong to a common set of operators. Letting then XK "DK \ dDK "dAK AK \ in Eqs. (2.7) and (2.9) and subtracting one equation from the other, we "nd for any XK within the considered set
(2.32)
(2.33)
Tr XK
1 dAK DK # [AK DK , KK ] "0 . 2 du
Likewise, if we let >K "AK \ dAK "dDK DK \, we "nd Tr >K
dDK AK 1 ! [DK AK , KK ] "0 . du 2
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We shall encounter below these equations (Eqs. (3.52) and (5.41)), with arbitrary quadratic forms of creation and annihilation operators for XK and >K . For unrestricted variations of DK and AK , the operators XK and >K are arbitrary and Eqs. (2.32) and (2.33) imply that AK DK and DK AK both satisfy exact equations of the Liouville-von Neumann type (with a Hamiltonian $iKK /2): 1 dAK DK " [KK , AK DK ], 2 du
dDK AK 1 "! [KK , DK AK ] . du 2
(2.34)
Note however that the boundary conditions (2.6) and(2.12) do not involve DK and AK evaluated at the same argument but DK (0) and AK (b) at the initial and "nal limits of the evolution interval. A large arbitrariness is involved in the solution of the equations (2.34) with these mixed boundary conditions, and they are not su$cient to entail the form exp (!uKK ) for DK (u) and the form (2.13) for AK (u). However the simple structure of Eqs. (2.34) makes them useful. 2.4. Cumulants of conserved observables The variational principle (2.5) provides in general rather complicated expressions for the correlations, and more generally for the higher order cumulants of the observables QK . In this A section we consider a single observable QK which is conserved, that is, which commutes with the operator KK . (The extension to several observables commuting with KK , but not necessarily among themselves, is readily performed by replacing mQK by m QK .) In the exact case, one has the obvious A A A relation (2.35) ePK"Tr e\K/K e\@)K "Tr e\@)K >K/K @ , which tells that the exponential of the characteristic function (1.5) associated with the observable QK is equal to the partition function corresponding to the shifted operator KK (m),KK #mQK /b .
(2.36)
We shall now show that the relation (2.35) remains valid in variational approximations such that the trial sets for the operators AK and DK are invariant under left or right multiplication by exp (!jQK ), where j is any c-number and QK the considered conserved observable. This invariance property will be satis"ed in Sections 3 and 4, where AK and DK are taken as exponentials of quadratic forms, provided QK is a conserved single quasi-particle observable. We shall encounter the relation (2.35) in a di!erent context in Sections 5 and 6. Let us indeed consider a trial set +AK (u), DK (u), satisfying the boundary conditions (2.6) and (2.12) with AK (m)"exp (!mQK ). We note that the two operators AK (u), DK (u) de"ned by AK (u)"eSK/K @AK (u)eSK/K @,
DK (u)"e\SK/K @DK (u) e\SK/K @ ,
(2.37)
belong to our trial set and obey the boundary conditions (2.6) and (2.12) with AK "1) . Moreover the value of the functional (2.5) calculated with AK (u), DK (u), AK "1) and KK is equal to the value calculated with AK (u), DK (u), AK (m)"exp (!mQK ) and KK . By sweeping over the trial sets +AK (u), DK (u), for AK (m)"exp (!mQK ) and +AK (u), DK (u), for AK "1) , we obtain exp u+e\K/K , b, KK ,"I +e\K/K , b, KK ,"I +1) , b, KK #mQK /b, ,
(2.38)
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which, within our variational approach, is the counterpart of the relation (2.35). As a consequence, for trial spaces satisfying the invariance above, all the cumulants at temperature 1/b of a conserved observable can be obtained from a calculation of the partition function for the shifted operator KK de"ned in Eq. (2.36). In particular, the mean value 1QK 2 and the variance *QK are given by
1 R u+1) , b, KK #jQK , "! , b Rj K H R 1 R u+e\K/K , b, KK , u+1) , b, KK #jQK , *QK , " , (2.39) Rm b Rj K H which are consistent with exact thermodynamical relations. An example of special interest concerns the particle-number operator NK in the grand-canonical equilibrium (KK ,HK !kNK ). To obtain the cumulants of NK (i.e. AK "exp (!mNK )) it su$ces to consider the partition function in which k is replaced by k!j with j,m/b. Then Eqs. (2.39) become R 1QK 2,! u+e\K/K , b, KK , Rm
1 R u+1) , b, HK !kNK ,, 1NK 2" b Rk
1 R *NK " 1NK 2 . b Rk
(2.40)
These equations, valid in any variational space that satis"es the aforementioned invariance, coincide with the usual thermodynamic relations for the expectation value and the #uctuations of the particle-number operator. Similar relations arise for any conserved observable such as the momentum or the angular momentum. 2.5. Connection with the standard variational principle While for DK (u) the exact solution exp (!uKK ) has a trivial, exponential dependence on u, the exact solution (2.13) for AK (u) depends on u in a more complicated fashion since ln AK (u) is a linear function of u only if the operators AK and KK commute, as was the case in Section 2.4. For restricted variational spaces, due to the coupling between the di!erential equations (2.7) and (2.9), we expect in general both the approximate solutions DK (u) and AK (u) to display a non-trivial u-dependence. In fact, this feature endows the method with a versatility which partly compensates for the unavoidable restrictions of the trial spaces. Nevertheless, for m "0 (AK "1) ), that is, when we only want a variational evaluation of the A thermodynamic potential u(0), a simpli"cation occurs whenever the trial spaces satisfy the two properties: (i) the trial sets for the operators DK (u) and AK (u) are identical; (ii) if DK belongs to this set, so does cDK ? where c and a are any positive constants. These properties are obviously satis"ed for independent quasi-particle density operators (Sections 3 and 4). They shall also hold in Sections 5.4, 6.2}6.5, where we shall consider projections of these density operators onto subspaces of the Fock space, because the power a of the operator DK is well-de"ned in a basis where it is diagonal, even when DK has vanishing eigenvalues. Thus, within our variational space, we can make the ansatz DK (u)"e\S)K ,
AK (u)"e\@\S)K ,
(2.41)
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where the trial operator KK does not depend on u. The trial choice (2.41) for AK (u) is suggested by the simple form taken by Eq. (2.13) when AK "1) . As a consequence, we have AK (u)DK (u)"DK (u)AK (u)"DK (b)"e\@)K ,ZDK , (2.42) where we de"ne Z as the normalization Tr e\@)K . The variational expression (2.5) then reduces to I+KK ,"Z(1!ln Z)!Z Tr DK (ln DK #bKK ) . (2.43) By maximizing Eq. (2.43) with respect to Z, we obtain the restricted variational quantity ln I+DK ,"ln Z"!Tr DK ln DK !b Tr DK KK , where we recognize the von Neumann entropy
(2.44)
S"!Tr DK ln DK (2.45) associated with the normalized DK . The standard variational principle for thermodynamic potentials is thereby recovered since we now "nd that u(0) is the maximum with respect to DK (under the constraint Tr DK "1) of the right-hand side S!b1KK 2 of Eq. (2.44). This, however, takes place only for thermodynamic potentials or expectation values (2.23), and for particular trial spaces which satisfy the conditions (i) and (ii). We shall encounter in Sections 5.2 and 5.3 a special case of interest in which our variational principle cannot be reduced to the standard one, even for AK "1) , because the condition (ii) is violated: indeed, since in that case a symmetry is broken, the trial density operator is of the form (1.11) and involves on both sides a projection which does not commute with the operator M K . Here the entropy (2.28) resulting from our variational approximation can be identi"ed with Eq. (2.45). It also coincides with Eq. (1.2) where DK is replaced by DK (b), and it is the Legendre transform of the approximation for ln Z. One notes that the variational principle (2.5), which involves a saddle-point, has been transformed into the maximum principle (2.43). We already noted in Section 2.1 and Fig. 1 that the elimination of part of the variables xG and - H in the functional (2.3) by means of the conditions RI/RxG"0 may, under suitable conditions, lead to a principle of maximum. Here, the constraints on DK (u) and AK (u) imposed by the ansatz (2.41) also reduce the number of independent variables, though in a di!erent fashion. This restriction on the variational space turns out to be su$cient to ensure that u(0) is obtained as a maximum of I+KK ,. The analysis of Section 2.4 showed that for an operator QK commuting with KK , the characteristic function u(m) is equal to the thermodynamic potential associated with the shifted operator KK "KK #mQK /b when the trial spaces are invariant under multiplication by exp (!jQK ). In this case the cumulants of any order, and in particular the correlations, can be deduced from the standard variational principle (2.44) with KK replaced by KK . 2.6. Variational principle for DK (u) alone The ansatz (2.41), which restricts the parametrization of the two u-dependent variational quantities DK (u) and AK (u) to a single operator KK , is not suited to the evaluation of the characteristic
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function u(m) for m O0 (and therefore of cumulants of non-conserved quantities). In the followA ing Sections we shall therefore exploit the freedom allowed by the full set of the variables DK (u) and AK (u). Nevertheless the form of the exact solution (2.13) for AK (u) suggests another possibility: instead of letting AK (u) and DK (u) vary independently, one can constrain them according to AK (u)"DK (b!u)AK DK (b!u) ,
(2.46)
where AK stands for AK (m). Inserting this trial form into the functional (2.5) leads to the variational expression
@ dDK (u) 1 # [KK DK (u)#DK (u)KK ] , (2.47) du Tr DK (b!u)AK DK (b!u) du 2 in terms of a single variational quantity DK (u) obeying the boundary condition (2.6). Had we not symmetrized the Bloch equation, we would have found the alternative functional Tr AK DK (b)!
@ dDK (u) #KK DK (u) , (2.48) du Tr AK DK (b!u) du whose stationary value with respect to DK (u) yields the wanted quantity Tr AK DK . A special case of this variational principle has already been proposed and applied in connection with the Monte Carlo method [34]; it is obtained from Eq. (2.48) in the case of canonical equilibrium KK "HK by replacing AK by the dyadic "R21R ", where R is a point in the 3N-dimensional space. The stationary value of Eq. (2.48) is then the density matrix itself in the R-representation. If we further specialize DK (u) to depend exponentially on u, as does the exact solution, in the form Tr AK DK (b)!
DK (u)"e\MK S @ ,
(2.49)
both functionals (2.47) and (2.48) become
@ (2.50) du eMK S @ KK e\MK S @ . We thus recover the variational expression (3.20) of Ref. [31]. In Eq. (2.50) the trial quantity is the operator MK . For unrestricted variations of MK , the stationarity condition is MK "bKK and the stationary value is the characteristic function Tr AK e\@)K . It has been shown in [31] that, under rather general conditions, the functional (2.50) is maximum for MK "bKK . Tr AK e\MK 1) #MK !
2.7. Comments All subsequent sections are based on the variational functional (2.5) and on the resulting equations (2.7) and (2.9) which express its stationarity in restricted trial spaces. This approach presents several advantages. Within the same variational scheme, the optimization of the characteristic function u(m) provides approximate thermodynamic quantities, average values, correlations or #uctuations and (at least in
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principle) higher cumulants. We have seen in Sections 2.3 and 2.4 how the results are consistent with general requirements, in particular with thermodynamic relations. We have also seen how the stationarity of Eq. (2.5) entails through Eq. (2.22) cancellations in the expansion of the characteristic function in powers of the sources. This will greatly simplify the forthcoming calculations. We have relied on the Bloch equation to characterize the density operator of the system and to devise the action-like functional (2.5). The simplicity of the Bloch equation is re#ected by this functional, which will allow, for instance, to use for the operator DK (u) trial forms including projections (Sections 5 and 6). Such trial forms are di$cult to implement in the usual variational principle for thermodynamic potentials, since the associated von Neumann entropy cannot be written explicitly. In the present formalism, the entropy can be evaluated a posteriori through Eqs. (2.27) and (2.29). It is only under special conditions that the elimination of AK (u) from the action (2.5) results in the standard variational principle of minimum thermodynamic potential (Section 2.5). The variational principle (2.5) involves a stationarity rather than a maximum or minimum property which, admittedly, would yield a better control. For instance, the existence and uniqueness of the solution for the coupled equations (2.7) and (2.9) with boundary conditions (2.6) and (2.10) are not warranted for an arbitrary choice of the trial spaces. On the other hand, we feel that the internal consistency allowed by the inclusion of sources is an important feature, even if the variational approximation only involves a saddle point. This is exampli"ed by Section 4.5. From now on, we apply the variational expression (2.5) to systems of fermions. However, the formalism of Section 2 is general and, owing to its #exibility, can be adapted to a wider class of problems.
3. Extended 5nite-temperature HFB approximation for characteristic functions We now consider systems of interacting fermions for which pairing e!ects are important. In Section 3.1, after some notations and de"nitions, we recall some properties of operators of the independent-quasi-particle type (Eq. (3.4)) which play a central ro( le throughout this article. In Section 3.2 we take the form (3.4) as the trial class for both the operators DK (u) and AK (u) that enter the functional (2.5). Inserting this choice in (2.5) we derive the explicit form of the reduced action (Eq. (3.32)). In Section 3.3 we write the self-consistent coupled equations (3.36)}(3.37), (3.42)}(3.43) which express the stationarity of the reduced functional (3.32). These equations are suited to the optimization of the characteristic function (2.1); they account for pairing correlations and constitute a generalization of the Hartree}Fock}Bogolyubov equations. In this section (as in Section 4), we deal with grand canonical equilibrium, letting KK ,HK !kNK and taking traces in the full Fock space. The formalism will also apply if KK includes additional constraints of the single-quasi-particle type. For instance, for a nucleus rotating around the z-axis with angular velocity u, the constraint is !uJK where JK is the third component of the angular momentum. For a "nite system, one may also want to control its position in space by including in the set XK the center-of-mass operator, or, if it is deformed, its orientation by including the I quadrupole-moment tensor.
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3.1. Generalities and notations For the sake of conciseness, it is convenient to introduce 2n-dimensional column and line vectors de"ned as
a $
a L , aR"(aR 2aR a 2a ) , L L aR $
a"
(3.1)
aR L where aR and a are the creation and annihilation operators associated with a "xed single-particle G G basis (i"1, n). If we denote by a the jth component +j"1, 2n, of the column vector a, the H fermionic anticommutation rules are given by either one of the relations [a , aR ] "d , [a , a ] "p , H HY > HHY H H > HHY where p is the 2n;2n matrix
p"
0
1
1
0
(3.2)
.
(3.3)
As in the HFB theory, let us introduce operators with the form ¹K "exp (!l!aRLa) , (3.4) where l is a c-number and L is a 2n;2n matrix. As a consequence of the anticommutation relation (3.2), we can restrict ourselves to matrices L which satisfy the relation pLp"!L2 ,
(3.5)
where L2 denotes the transposed matrix of L. Instead of l and L one can alternatively use the c-number q and the 2n;2n matrix T de"ned by q"e\J,
T"e\L ,
(3.6)
to specify the operator (3.4). Another equivalent parametrization involves the normalization f, f,Tr ¹K "e\J[det(1#e\L)] "q exp+tr ln(1#T), , L and the generalized reduced density matrix R (or matrix of the contractions),
1 Tr a ¹K aR H HY" L R , HHY e #1 Tr ¹K
(3.7)
T " . (3.8) T#1 HHY HHY In Eq. (3.7), the symbol tr denotes a trace in the space of 2n;2n matrices while Tr is the trace in L the Fock space. The property (3.5) implies the relations: pT\p"T2 ,
pRp"1!R2 .
(3.9)
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As a consequence, the 2n;2n matrix R can be written in the form
R"
o i
i \ , 1!o2
(3.10)
> where the n;n matrices o, i and i correspond to the normal and abnormal contractions: \ > o "Tr a ¹K aR/Tr ¹K , GH G H i "Tr a ¹K a /Tr ¹K , (3.11) \GH G H i "Tr aR¹K aR/Tr ¹K . >GH G H The matrices i and i are antisymmetric. When the operator ¹K is hermitian, one has \ > o"oR, i "iR and the matrix R"RR reduces to the usual HFB form. > \ Operators of the type (3.4) allow the use of a generalized Wick theorem, even when they are not hermitian [35]. They also form a non-abelian multiplicative group, the algebra of which is characterized by the relations (3.20)}(3.24). These simple algebraic properties (more details can be found in [35] and the Appendix of [36]) will be used extensively below.
3.2. The variational ansaK tze and the reduced action functional Let us come back to our problem. We assume that the observables QK are operators of the A single-quasi-particle type (this restriction is not essential and can be removed at the cost of some formal complications (see [29])): QK "q #aRQ a , A A A so that the operator AK (m) (see Eq. (1.6)) belongs to the class (3.4). We thus write it as AK (m),¹K ,exp (!l !aRL a) ,
(3.12)
(3.13)
with l , m q , L , m Q . (3.14) A A A A A A Now we choose the trial spaces for the operators DK (u) and AK (u) entering the functional (2.5). This choice determines our variational approximation. We take as trial sets: DK (u)"¹K B (u) ,
AK (u)"¹K ? (u) ,
(3.15)
with (3.16) ¹K B (u),exp(!l B (u)!aRL B (u)a) , ¹K ? (u),exp(!l ? (u)!aRL ? (u)a) . (3.17) The 2n;2n matrices L B (u) and L ? (u) (constrained to satisfy the relation (3.5)) and the c-numbers l B (u) and l ? (u) constitute a set of independent variational parameters. Using Eqs. (3.6)}(3.8), we de"ne the coe$cients q ? , q B , the normalization factors f ? , f B , the matrices T ? , T B and the reduced density matrices R ? , R B associated with the operators ¹K ? and ¹K B .
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The boundary condition (2.6) on DK (0) implies the relations: l B (0)"0 ,
L B (0)"0 ,
(3.18)
or equivalently q B (0)"1 ,
T B (0)"1 ;
f B (0)"2L ,
R B (0)" . (3.19) The products DK (u)AK (u) and A K (u)DK (u) occur in the functionals (2.5) and(2.8). Taking advantage of the group properties [35] of the operators (3.4), we write AK DK "¹K ?B ,
DK AK "¹K B? ,
(3.20)
with ¹K ?B "exp (!l ?B !aRL ?B a) , (3.21) ¹K B? "exp (!l B? !aRL B? a) , (3.22) where the u-dependence of the c-numbers l ?B and l B? , and of the matrices L ?B and L B? has not been indicated; using the notation (3.6), one has the relations: e\J ?B "q ?B "e\J B? "q B? "e\J ? \J B "q B q ? ,
(3.23)
e\L ?B "T ?B "T ? T B ,
(3.24)
e\L B? "T B? "T B T ? .
According to Eqs. (3.7) and (3.23)}(3.24), the normalization of DK AK and AK DK , which we denote in shorthand by Z, is given at each `timea u by Z,Tr AK DK "f ?B "Tr DK AK "f B?
"e\J ? >J B exp +tr ln(1#e\L ? e\L B ), L "q ? q B exp +tr ln(1#T ? T B ), . (3.25) L As we already know from Section 2.3, the normalization Z does not depend on u at the stationary point and it is a variational approximation for the quantity exp u that we are looking for (see below Eqs. (3.48)}(3.50)). It thus plays the ro( le of a generalized partition function which reduces to the grand canonical partition function when the sources m are set to zero. A We can now write the functional (2.5) (which provides the characteristic function (2.1)) in terms of the variational parameters q B , T B , q ? , T ? . The term Tr AK dDK /du is obtained by di!erentiating q B and T B in Eq. (3.25) with respect to u, while holding q ? and T ? "xed, which yields Tr AK
d lnq B 1 dT B
dDK "Z # tr T B \R B?
. L du 2 du du
(3.26)
The last two terms in the integrand of the functional (2.5) are supplied by a straightforward application of the generalized Wick theorem. They involve the operator KK ,HK !kNK where the Hamiltonian HK of the system is assumed to have the form L 1 L < aRaRa a , HK " B aRa # GHIJ G H J I GH G H 4 GHIJ GH
(3.27)
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and where L (3.28) NK " aRa . G G G The matrix elements B of the one-body part of HK and the antisymmetrized matrix elements < of GH GHIJ the two-body interaction satisfy the relations B "BH , < "!< "!< "
Tr KK DK AK "Z E+R B? , ,
(3.30)
with 1 (3.31) E+R," (B !kd ) o # < (2o o !i i ) . GHIJ IG JH >GH \IJ GH GH HG 4 GHIJ GH When o"oR and i "iR , the expression (3.31) is formally identical to the HFB energy. However, > \ the quantities E+R ?B , and E+R B? , are not necessarily real since R ?B and R B? do not have to be hermitian. Altogether, for the trial choices (3.16) and (3.17), the functional (2.5) reads I+AK , DK ,"Z !
@
du Z
dlnq B 1 dT B 1 ; # tr T B \R B?
# [E+R B? ,#E+R ?B ,] , du 2 L du 2
(3.32)
where use has been made of the relation T B \R B? "R ?B T B \ .
(3.33)
According to Eq. (3.25), the boundary term Z is given by Z ,Tr AK (m)DK (b),Tr ¹K DK (b)"q q B exp +tr ln(1#T T B (b)), , (3.34) L where the c-number q and the matrix T , which specify the characteristic function we are looking for, are expressed in terms of the sources m by Eqs. (3.12)}(3.14) and (3.6). Similarly the A form (2.8) of I reads
I+AK , DK ,"Z !Z(b)#Z(0)#
;
@
du Z
dlnq ? 1 dT ? 1 # tr T ? \R ?B
! [E+R ?B ,#E+R B? ,] . du 2 L du 2
(3.35)
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In the functionals (3.32) and (3.35), the u-dependence has been omitted in the variational parameters q B , T B , and q ? , T ? , as well as in the normalization Z given by Eq. (3.25) and in the contractions R B? , R ?B associated with T B? , T ?B , respectively; the latter quantities are related to T ? and T B through Eq. (3.24). 3.3. The coupled equations The next step is to write the equations expressing the stationarity of the functional (3.32) or (3.35). We derive them directly from these expressions, but could also have used the general equations (2.7) and (2.9)}(2.10). Requiring the stationarity of (3.32) with respect to q ? (u) (in the functional (3.32), q ? (u) only appears as a proportionality factor in Z(u)) gives us the equation of evolution for l B (u)"!ln q B (u): dT B 1 dl B 1 " tr T B \R B?
# [E+R B? ,#E+R ?B ,] , 2 L du 2 du
(3.36)
which we shall solve explicitly below (Eq. (3.46)). We now require the stationarity of the functional (3.32) with respect to T ? (u), which appears through R ?B (u), R B? (u) and Z(u). Eq. (3.36) implies that we can disregard in this calculation the variation of Z(u). If we note that the variations of R ?B (u) and R B? (u) resulting from those of T ? (u) satisfy dR B? "T B dR ?B T B \ as a consequence of Eq. (3.33), we readily "nd the di!erential equation dT B
1 "! [H+R B? ,T B #T B H+R ?B ,] , du 2
(3.37)
where H+R, has the same formal de"nition, (3.38) dE+R,"tr H+R, dR , L as the HFB Hamiltonian, although H+R, is not necessarily hermitian. For the Hamiltonian (3.27), the 2n;2n matrix H takes the form
H+R,"
h
D
\ , D !h2 >
(3.39)
with h "(B !kd )# < o , GH GH GH GIHJ JI IJ 1 D " < i , \GH 2 GHIJ \IJ IJ 1 D " i < . >GH 2 >IJ IJGH IJ
(3.40)
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As a consequence of Eq. (3.29), the matrices D et D are antisymmetric. This entails for H the \ > same relation as Eq. (3.5): pHp"!H2 ,
(3.41)
which is consistent with pdRp"!dR2 and Eq. (3.38). Likewise, the stationarity conditions of the functional (3.35) with respect to q B (u) and T B (u) provide, for 04u(b, the di!erential equations dT ? 1 dl ? 1 " tr T ? \R ?B
! [E+R ?B ,#E+R B? ,] , 2 L du 2 du
(3.42)
for l ? (u)"!ln q ? (u) and dT ? 1 " [H+R ?B ,T ? #T ? H+R B? ,] , du 2
(3.43)
for T ? (u). Finally, by rendering the functional (3.35) stationary with respect to q B (b) and T B (b) (see Eqs. (3.13) and (3.6)), one obtains the boundary conditions for AK (u): q ? (b)"q ,
T ? (b)"T .
(3.44)
The main equations to be solved are the di!erential equations (3.37) and (3.43) for T B (u) and T ? (u) which are coupled through the matrices H+R ?B , and H+R B? ,. Moreover, one has to deal with the mixed boundary conditions T B (0)"1 , T ? (b)"T .
(3.45)
All our variational quantities depend upon the sources m (see Eqs. (3.12)}(3.14) and(3.6)) through A the boundary conditions (3.44). One notes that the evolution equation (3.43) of T ? can be deduced from Eq. (3.37) for T B by an overall change in sign of the right hand side and by the exchange a d, as could have been inferred from the comparison between the forms (3.32) and(3.35) of the functional I. (The same is true for the equations (3.36) and (3.42) governing l B and l ? if one takes account of Eqs. (3.37) and (3.43).) Once Eqs. (3.37) and (3.43) are solved, the c-numbers l B (u) and l ? (u) are obtained by integration of Eqs. (3.36) and (3.42) whose r.h.s. depend only on T B and T ? . Indeed, using Eqs. (3.37) and (3.43) with the boundary conditions l B (0)"0 and l ? (b)"l , we "nd
l B (u)"
S
du +!tr [R B? H+R B? ,#R ?B H+R ?B ,]#[E+R B? ,#E+R ?B ,], , L
l ? (u)"l #l B (b)!l B (u) .
(3.46) (3.47)
3.4. Formal results From the equations for T B , T ? , l B , l ? , together with the de"nition (3.25) and the relation (3.38), one can readily verify the conservation laws (2.18) and (2.16) which now
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read, respectively: dZ "0, du
d (E+R ?B ,#E+R B? ,)"0 . du
(3.48)
One can also verify that the integrands of the functionals (3.32) or (3.35) vanish. As was shown in Section 2, these properties directly result from the feature that variations dAK JAK and dDK JDK are allowed in the trial classes (3.16)}(3.17). We are looking eventually for the characteristic function u+AK (m), b, KK , which is given by the logarithm of the stationary value of the functional (3.32). This value is provided by the boundary term Z de"ned in Eq. (3.34). In this formula, the matrix T B (b) and the c-number q B (b)"exp [!l B (b)] are furnished by the solution of the coupled Eqs. (3.37), (3.43) and by (3.46), respectively. Since, according to Eq. (3.48), the normalization Z (de"ned in (3.25)) does not depend on u we have the relation u(m),u+AK (m), b, KK ,"ln Z "ln Z(u) ,
04u4b .
(3.49)
In particular, Z can be evaluated at u"0 (see Eq. (2.20)); from Eqs. (3.25) and (3.19) we "nd ln Z"!l ? (0)#tr ln(1#T ? (0)) , (3.50) L where l ? (0)"l #l B (b) is explicitly given by Eqs. (3.46)}(3.47) once the coupled equations (3.37) and (3.43) which yield T ? (0) have been solved. Let us add a few remarks about the properties of these last equations. In the derivations of this section (as well as in Section 2), the hermiticity of the operators DK (u) and AK (u), as de"ned in Eqs. (3.15)}(3.17), was nowhere assumed. If, for some value of u, the operators DK and AK are hermitian, i.e. if T B "T B R, T ? "T ? R, q B "q B H and q ? "q ? H, it follows that R ?B "R B? R and hence that H+R ?B ,"H+R B? ,R. When the operator AK (m) is hermitian, the inspection of Eqs. (3.37), (3.43) and (3.45)}(3.47) shows that these equations preserve the hermiticity of T B and T ? and the reality of q B and q ? . In this case, the stationary value Z given by (3.50) is also real. If AK (m) is not hermitian, as would happen for the characteristic functions with imaginary m's, the boundary condition (3.44) prohibits AK (u) and DK (u) from being hermitian. Combining Eqs. (3.37) and (3.43), one obtains the evolution equations for the products T ?B "T ? T B and T B? "T B T ? , and hence for the HFB-like contraction matrices R ?B and R B? associated (through Eq. (3.8)) with the operators ¹K ?B and ¹K B? . The calculation relies on the property dR"(1!R)dT(1!R) ,
(3.51)
and it leads to dR ?B 1 dR B?
1 " [H+R ?B ,, R ?B ] , "! [H+R B? ,, R B? ] . du 2 du 2
(3.52)
These equations, the single-quasi-particle reductions of Eqs. (2.34), involve commutators. They are reminiscent of the time-dependent Hartree}Fock}Bogoliubov equations (see Section A.5 of Appendix A), $i u replacing the time t. However, here H+R ?B , and H+R B? , are in general not hermitian and u is real. A consequence of the simple form (3.52) is that the eigenvalues of R ?B and
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R B? remain constant with u, in contrast to those of R ? and R B . It also follows from Eqs. (3.52) and(3.38) that each of the quantities E+R ?B , and E+R B? , is conserved, a "ner result than Eq. (3.48). In spite of the form of Eqs. (3.52), one should not infer from them that the evolutions of the matrices R ?B and R B? are decoupled. Actually the boundary conditions (3.19) and(3.44) on R B (0) and R ? (b) do not entail any explicit boundary conditions for R ?B and R B? . Moreover, Eqs. (3.52) are not equivalent to Eqs. (3.37) and (3.43) since R ? and R B are not determined uniquely by the knowledge of R ?B and R B? .
4. Expansion of the extended HFB approximation: 6uctuations and correlations In this section, starting from the characteristic function (3.49) as expressed by either one of the formulae (3.34) or (3.50), we derive the thermodynamic quantities, the expectation values of the observables, and their #uctuations and correlations. In principle this demands the determination of u(m) and its expansion up to second order in the sources. However, we shall see that the full solution of the coupled equations (3.37), (3.43), (3.46) and (3.47) for "nite values of the sources m is not A required. Indeed all the quantities of interest can be obtained through a prior expansion of the coupled equations in powers of the sources m , followed by a solution of the resulting equations A order by order. Moreover, owing to the variational nature of the approach, we shall need to expand the coupled equations to "rst order only. This procedure turns out to overcome the di$culties associated with the mixed boundary conditions. As expected, the usual HFB equations are recovered in the case AK "1) where the sources m vanish (Section 4.1.1). In agreement with Eq. (2.23) these equations are not only relevant for the A thermodynamic potential and for the ensuing quantities, but also for the calculation of the average values of single-quasi-particle operators (Section 4.1.2). In Sections 4.2 and 4.3 we work out the next order in the sources; this yields an explicit variational expression for the correlations C between two single-quasi-particle observables AB QK and QK (and for the #uctuations when c"d). Two di!erent situations are discussed. The "rst one A B corresponds to the case where the observables QK or QK commute with the operator KK . One then A B recovers the formula (4.27) which involves the HFB stability matrix F . When QK and QK do not A B commute with KK , one is led to the more general formula (4.63) which involves both the stability matrix F and the RPA kernel K . In Sections 4.2.3 and 4.3.3 we discuss the e!ect of the vanishing of some eigenvalues of F and of K that may in particular occur when an invariance is broken. We shall also evaluate the Kubo correlations, Eq. (4.37), and give a diagrammatic interpretation (Section 4.4) of the results obtained here variationally. Although the trial operator DK describes independent quasi-particles, its dependence on the sources yields expressions for the cumulants of order two and beyond which di!er from what might have been naively inferred from the standard mean-"eld HFB approximation. Indeed, the latter is only suited to the evaluation of expectation values; in particular the approximations that it provides for correlations and #uctuations violate the usual thermodynamic relations. This defect is cured in the present fully variational approach, as illustrated in Section 4.2.1 by the example of the #uctuations of the particle number in the grand canonical ensemble. In the present section, the increasing orders of the expansion with respect to the sources m will be A indicated by lower indices (0, 1,2) assigned to the relevant quantities.
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4.1. The HFB approximation recovered 4.1.1. Thermodynamic quantities To zeroth order in the sources (i.e., AK (m)"1) ) our problem amounts to evaluating the partition function Z"Tr DK (b). We expect that the coupled equations of Section 3.3 will yield the standard HFB approximation for Z, since the trial spaces for DK (u) and AK (u) satisfy the conditions required in the discussion of Section 2.5. Let us verify this point. If we associate with the contraction matrix R the single-particle Hamiltonian H de"ned by 1 1!R , R , H , (4.1) bH ,ln e@ #1 R the usual optimum HFB matrix R is given, at temperature 1/b, by the self-consistent equation H "H+R , , (4.2) where H+R, was de"ned in Eqs. (3.39)}(3.40). Because the relation (4.2) entails the commutation of H+R , with R , Eqs. (3.52) are satis"ed by the u-independent solution R ?B (u)"R B? (u)"R . (4.3) In the equations (3.37) and (3.43) for T B (u) and T ? (u), the e!ective Hamiltonians do not depend on u since H+R ?B ,"H+R B? ,"H . Moreover, the boundary conditions (3.45) reduce to T B (0)"1, T ? (b)"T "1. We thus "nd H (4.4) T B (u)"e\S , T ? (u)"e\@\SH . Eqs. (4.3)}(4.4) are the single-quasi-particle reductions of the general relations (2.42) and (2.41). One also notes that R is equal to R B (b), the contraction matrix associated with the state DK (b). Using the solution (4.4) of the coupled equations, we can now calculate l ? (0) from Eqs. (3.46)}(3.47), and hence ln Z from Eq. (3.50). The integrations over u, which can be performed explicitly, yield ln Z from which, according to Eqs. (2.27) and (2.28), we de"ne the thermodynamic grand potential F and the entropy S+R , % ln Z"S+R ,!bE+R ,,!bF +R , . (4.5) % The entropy is given by (4.6) S+R,"!tr [R ln R#(1!R)ln(1!R)] . L It is thus the entropy of a quasi-particle gas characterized by the contraction matrix R. Owing to the symmetry relation (3.9), the two terms under the trace in Eq. (4.6) are equal. The HFB energy E+R ,"1HK !kNK 2 is given by the expression (3.31). As expected from the discussion of Section 2.5, Eq. (4.5) coincides with the outcome of the variational principle (2.44), which reads here (4.7) ln Z"ln Tr DK (b)"MaxR[S+R,!bE+R,]"!b Min F +R, . % The stationarity condition of Eq. (4.7) is obtained from Eq. (3.38) and from the "rst-order variation 1!R 1 dR dS+R," tr ln R 2 L
(4.8)
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of the reduced entropy (4.6); it thus gives back the usual self-consistency condition (4.2). Therefore, at zeroth order in the sources, we have recovered the standard self-consistent HFB approximation for the thermodynamic potential. 4.1.2. Expectation values of observables The variational approximation for the expectation values 1QK 2 in grand canonical equilibrium is A in principle given by the "rst-order terms in the sources m . However, the knowledge of the A zeroth-order solution (4.1),(4.2) is here su$cient since, according to Eq. (2.23) and to the de"nitions (3.12) and (3.8), we have 1 Tr QK DK (b) A "q # tr Q R . (4.9) 1QK 2" A 2 L A A Tr DK (b) For the expectation value of the operator KK , we can apply Eq. (2.26) which was obtained by derivation with respect to the parameter b. Using Eqs. (3.30) at zeroth order and (4.3), we "nd R 1KK 2"E+R ,"! ln Z , Rb
(4.10)
where, as in Eq. (4.9), the expectation value is taken over DK (b). For the calculation of 1NK 2, we can use either (4.9) or the discussion in Sections 2.3 and 2.4. From Eq. (2.31) (or (2.40)) we get 1 R n 1 ln Z , 1NK 2"tr o" # tr NR " L L b Rk 2 2
(4.11)
where, as in Eq. (3.12), the operator NK can be represented by the c-number n/2 and the 2n;2n matrix
N"
1
0
0 !1
.
(4.12)
The formulas (4.9)}(4.11) are those given by the standard approximation, which consists in evaluating expectation values in the HFB state DK (b). Our variational procedure thus does not yield anything new when the characteristic function is expanded to "rst order in the sources. This is not the case at the next order. 4.2. Fluctuations and correlations of conserved observables 4.2.1. The particle number operator As a "rst example of a conserved observable (i.e., which commutes with KK ), we consider the particle number NK and we evaluate its variance *NK . A naive method would consist in taking the di!erence between the expectation value 1NK 2 and the square 1NK 2, both evaluated by means of the Wick's theorem applied to the state DK (b). This would yield 1 (4.13) 1NK 2!1NK 2" tr N(1!R )NR , 2 L where R is the contraction matrix (4.1)}(4.2) corresponding to DK (b).
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However, the state DK (b) is variationally "tted to the evaluation of the thermodynamic potential F , not to the evaluation of *NK . In agreement with our general philosophy, we should rather rely % on a variational principle suited to the evaluation of the characteristic function u. The variational approximation for the #uctuation *NK is then supplied by the second derivative of u with respect to the source associated with the observable NK . The result is expected to di!er from Eq. (4.13), since the optimal values of the trial operators DK (u) and AK (u) are thus dependent on the sources m . A Actually, under conditions on the trial classes that are obviously satis"ed here, we have already determined in Section 2.4 the result of this evaluation: 1 RF 1 R1NK 2 %, "! *NK " b Rk b Rk
(4.14)
where F and 1NK 2 are the outcomes of the variational calculation at zeroth and "rst orders in the % sources, given by Eqs. (4.5) and (4.11), respectively. The identities (4.14), satis"ed by our approximation for *NK , agree with exact thermodynamic relations which are violated by the naive result (4.13). Inserting Eq. (4.11) into (4.14), we have 1 RR *NK " tr N . L 2b Rk
(4.15)
The derivative RR /Rk is given by Eq. (4.2) which de"nes R self-consistently; the parameter k enters this equation through the contribution !kN to H+R ,. The expression (4.15) reduces to (4.13) only for a system of non-interacting particles, with [N, R ]"0, and is otherwise non trivial due to self-consistency. We shall write it explicitly below (Eq. (4.27)) in a more general context. The result (4.15) arising from of our variational treatment is more satisfactory than the naive expression (4.13). We noted that it meets the thermodynamic identities (4.14). Moreover, for a system whose pairing correlations do not disappear at zero temperature, Eq. (4.13) has a spurious "nite limit as b goes to R, whereas the variational estimate (4.15) for the particle-number dispersion vanishes in this limit as 1/b, with a coe$cient equal to the density of single-particle states at the Fermi level. It is therefore consistent with the expectation that, for "nite systems with or without pairing, results from the grand-canonical and canonical ensembles must coincide at zero temperature. 4.2.2. Characteristic function for conserved single-quasi-particle observables The ideas above are readily extended to correlations between any pair of single-quasi-particle observables QK and QK commuting with the operator KK . Indeed, it follows from the discussion of A B Section 2.4 that the evaluation of the characteristic function, and therefore of the cumulants, can be reduced in this case to the calculation of a thermodynamic potential when the trial spaces for AK and DK are invariant with respect to the transformation (2.37). Within the spaces (3.16) and (3.17) considered in this section, this invariance is satis"ed for observables QK of the type (3.12). A The identity (2.38) then shows us that the variational evaluation of the characteristic function u(m) amounts to the variational evaluation of the partition function associated with the shifted Hamiltonian KK "KK # m QK /b. The calculation is thus the same as in Section 4.1.1, except for A A A the change of KK into KK , and Eq. (4.7) is replaced by
u(m)"MaxR S+R,!bE+R,! m (q #tr Q R) . A A L A A
(4.16)
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The optimum matrix R is characterized by the extremum condition (4.16), which yields the self-consistent HFB equations 1 H +R ,,H+R ,# m Q . (4.17) A A b A The characteristic function (4.16) depends on the m's both directly and through the matrix R . From its "rst derivatives we recover the expectation values (4.9). The second derivatives, taken at m"0, yield the correlations between the QK 's. In order to write their explicit expression, setting A R "R #dR, we shall expand S+R ,, E+R , and F +R , around R up to second order. To this % aim, we introduce a condensed notation. Up to now, we have regarded R , H as well as other quantities denoted by a script HHY HHY capital (Q, L, T, dR, H , etc.) as 2n;2n matrices. We "nd it convenient to consider them alternatively as vectors in the Liouville space, with the pair (jj ) playing the ro( le of a single index (see Appendix A, Section A.1). Thus, the "rst-order variations (4.8) of S+R, and Eq. (3.38) of E+R, appear now as scalar products that we shall denote by the colon sign : which stands for tr . L Likewise the second-order variations will generate matrices in the Liouville space, with two pairs of indices (j j ). With this notation we can write, for R "R #dR, the expansions of the HFB entropy S+R,, energy E+R, and grand potential F +R,, de"ned by Eqs. (4.6), (3.31) and (4.5), respectively, as (see % Appendix, Section A.2) 1 , R " H +R , e@ Y Y #1
S+R ,KS+R ,#bH : dR#dR : S : dR , where bH stands for ln(1!R )/R as in Eq. (4.1); E+R ,KE+R ,#H+R , : dR#dR : E : dR , where H+R , is de"ned by Eq. (3.38); F +R ,KF +R ,#dR : F : dR , % % where we made use of the stationary condition (4.2) and where 1 F ,E ! S b
(4.18)
(4.19)
(4.20)
(4.21)
is the stability matrix around the HFB solution. In Eqs. (4.18)}(4.20), the second derivatives S , E and F , as well as the "rst derivatives bH and H+R ,, are functions of the HFB density R . In the j-basis of Section 3.1, S , E and F are 4-index tensors; we regard them here as (symmetric) matrices in the Liouville space (jj ). The explicit expression of S is given in Eq. (A.10). HHYIIY Contraction of such a matrix with a vector proceeds as 1 dR . (4.22) (S : dR) , S HHYIIY IYI HHY 2 IIY In Appendix A (Section A.4), we endow !S with a geometric interpretation. It appears naturally as a Riemannian metric tensor which de"nes a distance between two neighbouring states characterized by the contraction matrices R and R #dR.
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The correlation C between the two observables QK and QK (de"ned as in Eqs. (1.7)}(1.8)) is AB A B obtained from the expansion of the expression (4.16) around R : u(m)Kln Z! m 1QK 2!Min R[b dR : F : dR# m Q : dR] . A A B A A A A The extremum condition,
(4.23)
b F : dR"! m Q , A A A determines dR as function of the m's. From Eq. (4.23), one has
(4.24)
Ru R C " "!Q : . dR AB Rm Rm A Rm A B K B K Inserting into Eq. (4.25) the formal solution of Eq. (4.24),
R 1 dR "! F\ : Q , B Rm b K B one obtains "nally 1 C " Q : F\ : Q . AB b A B
(4.25)
(4.26)
(4.27)
We shall discuss the problem of the vanishing eigenvalues of F in Section 4.2.3. The formula (4.27) encompasses the #uctuations *QK , obtained for c"d. A Remark. The above results have been obtained indirectly by using the identity (2.38). We could also have found them directly by solving the coupled equations of Section 3.3. This is feasible, but not straightforward. Indeed, having de"ned R and H+R , by Eq. (4.17), we can "rst check that the solution of Eqs. (3.52) is given by R ?B (u)"e\SL @ R eSL @, R B? (u)"eSL @ R e\SL @ .
(4.28)
The proof uses the identity (4.32) below where U is replaced by exp uL . We can then solve the coupled equations (3.37) and (3.43), together with the required boundary conditions T B (0)"1 and T ? (b)"exp(!L ), which results in T B (u)"eSL @ e\SHY+RY, eSL @ , T ? (u)"e\SL @ e\@\SHY+RY, e\SL @ .
(4.29)
We "nally recover the characteristic function (4.16) by means of (3.46)}(3.47) and (3.49)}(3.50). If L commutes with R , and therefore with H +R ,, the expressions (4.28) and (4.29) simplify. Otherwise, when some QK -invariances are broken, the u-dependence in the solution (4.29) of the A evolution equations (3.37) and (3.43) is no longer as simple as it is for m"0 or for the exact solution
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[given by DK (u)"exp(!uKK ) and by Eq. (2.13) for AK (u)]. In particular ln T B (u) is not proportional here to u, but contains contributions in u, u,2 Although, in the case of commuting observables, we were able to indirectly solve the coupled equations (3.37) and (3.43), it would not have been easy to guess a priori the complicated u-dependence of the solutions (4.28)}(4.29). 4.2.3. Broken invariances Since equilibrium is described by the minimum of the grand potential F +R,, given by Eq. (4.20), % the eigenvalues of the matrix F de"ned in Eq. (4.21) are non-negative. We have, however, to consider the case of vanishing eigenvalues for which the inversion of F in Eqs. (4.24), (4.26) raises a problem. The relation (4.49) below shows that such vanishing eigenvalues of F give rise to the usual spurious modes of the RPA. A characteristic situation where this happens is that of broken invariance. For instance, in the mean-"eld description of pairing correlations, the particle-number NK symmetry is broken. For convenience, in this Subsection, we shall use this operator as the representative of all broken symmetries, although the whole discussion applies to any conserved single-quasi-particle observable. By de"nition of a conserved observable, NK satis"es the commutation relation [KK , NK ]"0. When the NK -invariance is not broken, NK commutes with the approximate density operator DK (b) and we have both [R , N]"0 and [H , N]"0, where the matrix N has been de"ned in Eq. (4.12). However, when the NK -invariance is broken, N does not commute with R and H despite the commutation of NK with KK in the Fock space. In the HFB theory, this non-commutation manifests itself by the occurence of non-zero o!-diagonal elements in Eqs. (3.10) and (3.39). Let us now start our analysis of the meaning of Eq. (4.27) by recalling how the NK -invariance, whether or not it is broken, is re#ected on our variational objects. We can associate with NK a continuous set of unitary transformations ;K ,exp ieNK which leave KK invariant. They are represented in the 2n-dimensional j-space of Section 3.1 by unitary matrices U,e CN .
(4.30)
From the de"nition E+R,"Tr DK KK /Tr DK and the commutation [KK , NK ]"0, we "nd Tr ;K DK ;K \KK E+R," "E+UR U\, , Tr ;K DK ;K \
(4.31)
for any R satisfying the symmetry condition (3.9). We also have S+R,"S+UR U\, since S+R,, de"ned by Eq. (4.6), depends only on the eigenvalues of R. Thus, if F +R, reaches its minimum for % some R satisfying (4.2) and if the NK -invariance is broken, the transformation UR U\ generates a set of distinct solutions which give the same value F +UR U\,"F +R ,. This implies that the % % second derivative F of F +R, has a vanishing eigenvalue. To "nd the associated eigenvector, we % note that the "rst-order variation of Eq. (4.31) with respect to R, together with the de"nition (3.38) of H+R,, yields UH+R,U\"H+URU\, .
(4.32)
Hence, letting R"R , taking an in"nitesimal transformation U and using dH"E : dR (a consequence of the de"nition (4.19) of E ), we "nd [N, H ]"E : [N, R ] . (4.33)
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The same argument holds when E+R, is replaced by F +R,, but then the "rst derivatives of % F vanish at R"R and Eq. (4.33) is replaced by % F : [N, R ]"0,
[N, R ] : F "0 .
(4.34)
These equations express that the commutator [N, R ], which does not vanish when the NK invariance is broken by R , is a right or left eigenvector of the matrix F for the eigenvalue 0. We are now in position to discuss the expression (4.27) for the correlation C of two conserved AB observables QK and QK when some invariance (here the NK -invariance) is broken. Depending on the A B nature of the observables, we can distinguish three cases. E (1) If both QK and QK are such that A B Q : [N, R ]"1[QK , NK ]2"0 ,
(4.35)
in particular if QK and QK commute with NK (as well as with KK ), their expectation values (4.9) are A B well de"ned, although R is not. Indeed, all the density matrices UR U\ for which F +R, is % minimum, will provide the same value for 1QK 2. The condition (4.35) ensures that the equation A (4.24) for RdR/Rm " has a "nite solution, that we can denote !b\F\ : Q . Due to Eq. (4.34) B K B this solution is not unique; it is only de"ned within the addition of ie [N, R ], where e is arbitrary. The condition (4.35), however, implies that this addition is irrelevant. Hence the correlation (4.27) is xnite and well-dexned, in spite of the vanishing eigenvalue (4.34) of F . This conclusion holds in particular for the #uctuation *N K obtained when QK "QK "NK . A B E (2) If QK , but not QK , satisxes Eq. (4.35), the expectation value 1QK 2 depends on the speci"c B A A solution R of Eq. (4.2). This feature is characteristic of a situation with broken NK -invariance. Moreover, the additive arbitrary contribution ie[N, R ] to RdR/Rm " also contributes to Eq. B K (4.27), so that the correlation between QK and QK is xnite, but ill-dexned. A B E (3) Finally, if both QK and QK violate Eq. (4.35), Eq. (4.24) has no "nite solution. The #uctuation of A B QK , in particular, is inxnite, consistently with the fact that 1QK 2 is itself ill-de"ned. A A These conclusions hold without any changes for the breaking of invariances other than NK . 4.3. Fluctuations and correlations of non-conserved observables In this section we consider the general case of single-quasi-particle observables QK which do not A commute with the operator KK . 4.3.1. Kubo correlations When the observables QK do not commute with KK , the expression (4.16) is still the variational A approximation for ln Tr exp[!bKK ! m QK ]. By expansion in powers of the m 's, this expression A A A A generates cumulants of the Kubo type. In particular, while its zeroth and "rst-order contributions coincide with Eqs. (4.7) and (4.9), respectively, the second-order contribution, Tr e\@)K @ du eS)K QK e\S)K QK A B!1QK 21QK 2 , C) "C) , A B AB BA b Tr e\@)K
(4.36)
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yields the Kubo correlations between the QK 's, which are thus given variationally by Eq. (4.27), A that is 1 C) " Q : F\ : Q . B AB b A
(4.37)
4.3.2. Standard correlations: derivation In order to obtain our variational approximation for the true correlations C " AB 1QK QK #QK QK 2!1QK 21QK 2 between QK and QK , we have to return to the coupled equations of B A A B A B A B Section 3.3, facing their u-dependence and expanding them in powers of the sources m . Section 4.1 A gave the zeroth-order solution, which happened to furnish both the thermodynamical potential and the expectation values. Fluctuations and correlations are provided by the next order that we now work out. The "nal result is expressed by Eq. (4.63). We start from the fundamental relation (2.22) that involves the stationary operator DK (b) and the operator AK (m) de"ned in Eq. (1.6); more explicitly it reads: Tr DK (b) dv e\T BKB/K BQK e\\T BKB/K B Ru A "! . (4.38) Tr DK (b) e\ BKB/K B Rm A As discussed in Section 4.1.2, Eq. (4.38) taken at m"0 yields 1QK 2 as the expectation value (4.9) of A QK in the state DK (b) already determined in Section 4.1.1. The correlations C and the #uctuations A AB *QK "C are obtained by expanding Eq. (4.38) up to "rst order in the m's which appear both A AA explicitly and through the expansion DK (b),DK (b)#DK (b)#2,DK (b)# m DK (b)#2 B B B of the stationary state DK (b). For shorthand we denote with an index d the coe$cient
(4.39)
RDK (b) (4.40) DK (b), B Rm B K of m in the "rst-order contribution to DK (b). Note that, thanks to the variational nature of the A method, only this "rst-order contribution is needed to evaluate the wanted second derivatives of the characteristic function. By using the formalism of Sections 3.1 and 3.2, we can rewrite Eq. (4.38) as
Ru "!q ! tr R ?B (b) dv e\TL Q eTL
A L A Rm A "!q ! tr R B? (b) dv eTL Q e\TL . (4.41) A L A The correlation C is the derivative of this expression with respect to m , evaluated at m"0. AB B Contributions arise here from the dependences in m of both R ?B (b) (or R B? (b)) and L , B m Q . As in Eq. (4.39), we expand the matrices R G (m) and T G (m) around m"0, introducing the A A A same notations R G ,RR G /Rm " and T G ,RT G /Rm " as in Eq. (4.40) (the symbol [i] stands A A K A A K
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for [a], [d], [ad] or [da]). We obtain C "!tr Q R ?B (b)#tr [Q , Q ]R AB L A B L B A "!tr Q R B? (b)!tr [Q , Q ]R L A B L B A "!tr Q +R ?B (b)#R B? (b), . (4.42) L A B B We now work out the coupled equations of Section 3.3 so as to "nd an explicit expression for Eq. (4.42). First we note, from the boundary conditions (3.44), that we have T ? (b)"1, T ? (b)"!Q , B B and hence, using Eqs. (3.8), (3.24) and (4.3):
(4.43)
R ?B (b)"R B (b)!(1!R ) Q R , B B B (4.44) R B? (b)"R B (b)!R Q (1!R ) . B B B The insertion of Eq. (4.44) into the expression (4.42) of C yields AB (4.45) C "tr [Q R Q (1!R )#Q (1!R )Q R ]!tr Q R B (b) . A B L A B AB L A B The "rst term in the right-hand side is what would emerge from a naive calculation using Wick's theorem with respect to the state DK (b). The departure from this result, given by the last term, arises from the variational nature of the calculation which modi"es DK (b) and hence the associated contraction matrix R B (b) when sources are present. Rather than evaluating R B (b) we shall transform Eq. (4.42) so as to bring the `timea u down B to 0 and thus "nd a more explicit expression for the correlation C . To this aim we expand AB the equations of motion (3.52) for R ?B and R B? around the lowest-order contribution (4.3). This leads to dR B?
1 dR ?B 1 B "! K : R B? , B " K : R ?B , 2 B du 2 B du
(4.46)
where the kernel K is de"ned, for R"R #dR, by K : dR,d[H+R,, R] RH+R, "[H+R ,, dR]# dR , R . (4.47) IIY RR IIY RR IIY The (2n;2n);(2n;2n) tensor K is the usual RPA tensor associated with the HFB Hamiltonian H+R, and customarily introduced to describe small deviations around the equilibrium state R . As in Eq. (4.22), the colon symbol in K : dR stands for half a trace (with a twist in the indices) involving the last pair of indices of K and the pair of indices of dR; equivalently, K : dR can be considered as a scalar product in the 2n;2n vector space of dR. It is shown in the Appendix (Section A.3) that K is related simply to the matrix F which, according to Eq. (4.20), characterizes the second-order variations of the thermodynamic potential F +R, around R . More precisely if % we de"ne, for any matrix M, the tensor C as the commutator with the HFB density matrix R , C : M,[R , M] , (4.48)
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the matrix K is (in the Liouville space) the product K "!C F ,!C : F , (4.49) again de"ned as in Eq. (4.22). The occurrence of the simple relation (4.49) and its meaning are explained in Sections A.3 and A.5 of Appendix A. The boundary condition T B (0)"1 implies R ?B (0)"R B? (0)"R ? (0),Y . B B B A formal solution of Eqs. (4.46), together with the boundary condition (4.50), gives
(4.50)
R B? (u)"e\SK : Y , R ?B (u)"eSK : Y, B B where Y is still unknown. The correlation (4.42) can now be written as
(4.51)
C "!Q : cosh b K : Y . AB A With the aim to "nd Y, using Eq. (4.44) and again Eq. (4.51), we can obtain
(4.52)
(4.53) R ?B (b)!R B? (b)"[R , Q ]"2 sinh bK : Y . B B B If K had no vanishing eigenvalue, we could eliminate Y between Eqs. (4.52) and (4.53) and write (4.54) C "!Q : coth b K : [R , Q ] . B AB A Unfortunately, due both to the vanishing eigenvalues of C associated with all one-quasi-particle operators that commute with aRH a and to the possible vanishing eigenvalues of F associated with broken invariances (see Section 4.2.3), the operator K "!C F has no inverse. In order to determine the unknown quantity Y,R ? (0) entering Eq. (4.52), we return to the B equation (3.43) which governs the evolution of T ? (u). Eq. (4.43) gives the boundary condition on B T ? (b) while, in agreement with Eq. (3.51), T ? (0) is related to Y through B B T ? (0)"(1!R )\Y(1!R )\ . (4.55) B We now relate T ? (b) to T ? (0) by expanding Eq. (3.43) to "rst order in the sources around the B B zeroth-order solution (4.1)}(4.4). This yields the di!erential equation 2
dT ?
B !H T ? #T ? H "(E : R ?B )e\@\SH#e\@\SH(E : R B? ) , B B B B du
(4.56)
where the u-dependence has been omitted in T ? , R ?B and R B? . We used the expansion B B B H+R #dR,KH #E : dR, entailed by the de"nition (4.19) of the matrix E . The quantities R ?B and R B? are given by Eq. (4.51) in terms of Y and as functions of u. In order to transform the B B left-hand side into a derivative, we multiply Eq. (4.56) by exp (b!u)H on the left and on the right. As a result, terms of the form eTHMe\TH appear on the r.h.s. of this equation. Using the fact (see Appendix, Section A.3) that the matrix S C in the Liouville space describes the commutation with bH , we rewrite these terms as (4.57) eT HMe\T H"eT @ SC : M . Relying now on Eq. (4.51) we note that, after this transformation, the two terms on the r.h.s. of Eq. (4.56) depend on u only through exponentials of S C and K . In addition, these terms can be
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explicitly integrated over u by means of the identity d S C K S C K K K [(e! @\S e! @\S @ : E : e! S "G2 @ !1)C\e!S #C\(e!S !e!@ )] , du
(4.58)
which is checked by using Eqs. (4.21) and (4.49). Although C has no inverse, the r.h.s. is nevertheless well de"ned because C\ is always multiplied either on the left side by S C or on the right side by C F when the exponentials are expanded. This amounts to extending to operators the natural convention (e?V!e@V)x\"a!b for x"0. Integrating the transformed di!erential equation for T ? (u), with the boundary condition T ? (b)"!Q given in Eq. (4.43), we "nd B B B H H e@\S T ? (u) e@\S #Q B B @\S S C K S C K )C\eS !(1!e\ @\S "[(1!e @ @ )C\e\S #2C\(sinh bK !sinh uK )] : Y . (4.59) For u"0, Eqs. (4.55) and (4.59) provide the required equation for Y. We can simplify the result by rewriting Eq. (4.55) as e@H T ? (0) e@H"4 cosh[bH ] Y cosh[bH ] B (4.60) "!2 (sinh S C )C\ : Y , an equality that is derived by using the expressions (A.11) and (A.19) of S and C in the basis where H is diagonal. Again the r.h.s. of (4.60) is well de"ned, despite the vanishing eigenvalues of C . We eventually "nd Q "2C\ sinh(bK ) : Y . (4.61) B Assuming that F\ exists, Eq. (4.61) can be inverted in the Liouville space; using the relation (4.49), this gives 1 K Y"! F\ : Q . B 2 sinh b K
(4.62)
4.3.3. Standard correlations: xnal expression It remains only to insert Eq. (4.62) into Eq. (4.52), which yields (4.63) C "Q : (K coth b K )F\ : Q . B AB A We recall that F is the stability matrix of the HFB theory, de"ned by Eq. (4.20), and that K is the RPA matrix, de"ned by Eq. (4.47). These two matrices are related to each other, and to the HFB reduced density operator R , through Eqs. (4.48) and (4.49). The "nal result (4.63) is our variational approximation for the correlations C or for the AB yuctuations *QK . The symmetry C "C is a simple consequence of the relation K "!C F , A AB BA whereas this symmetry was not obvious from our starting equations (4.42), (4.45) or (4.54). The formula (4.63) gives a meaning to the formal expression (4.54) which, due to the vanishing eigenvalues of the commutator C in K "!C F (Eq. (4.49)), was not de"ned. As for the vanishing eigenvalues of F , our discussion of Section 4.2.3 still holds for Eq. (4.63) as well as
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for Eq. (4.27), and all its conclusions remain valid when QK and QK are arbitrary single-quasiA B particle operators. Finally, when one of the observables, say QK , is a conserved quantity (i.e., [QK , KK ]"0), we "nd A A from Eq. (4.34) that Q : K ,!Q : C F "0 , (4.64) A A even if the QK -invariance is broken (if it is not, we have Q : C "0). Then the left-action of the A A operator K coth b K on Q amounts to a multiplication by the c-number 2/b, so that from Eq. A (4.63) we recover the result (4.27) of Section 4.2.2 for the correlations when one of the two observables QK or QK is conserved. A B 4.4. Diagrammatic interpretation The HFB e!ective Hamiltonian H and the RPA kernel K which occur in the results above also appear, as well known, in perturbative expansions. The former is associated with instantaneous self-energy insertions, the latter with the iteration of a pair of quasi-particle or quasi-hole lines. It is thus instructive to interpret in terms of diagrams the expressions obtained here variationally. A partial result was already obtained in [37] in the special case when (i) there is no broken invariance and (ii) the observables QK and QK are conserved. Moreover that derivation disregarded A B the di$culty caused by the vanishing eigenvalues of K . We sketch below a more complete proof. Let us "rst consider the perturbative expansion of the generalized grand potential ln Z ,ln Tr e\@)K Y ,
(4.65)
where 1 KK ,HK !kNK # m QK A A b A includes source terms. By splitting KK into
(4.66)
KK "KK #
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Fig. 2. A diagram contributing to ln Z in the HFB approximation. Each line represents the propagator associated with KK which includes the sources. Such diagrams sum up into the grand potential associated with an independent quasi-particle system governed by the self-consistent HFB Hamiltonian.
the matrix element Q . The same partial summation which, for ln Z, eliminated all the diagrams AH HY of the type of Fig. 2, now leaves for 1QK 2 the single diagram where Q is closed onto itself by the A A self-consistent propagator which accounts for all insertions. For equal times, this propagator reduces to R . Thus, the expression (4.9) for 1QK 2 is again represented by the sum of the A contributions of the diagrams with instantaneous self-energy insertions only. A new feature occurs if we expand ln Z up to second order in the sources m . We then obtain the A Kubo correlation (4.36) as
R ln Z . C) " AB Rm Rm K A B
(4.68)
Two dots Q and Q are now inserted in each diagram, but they may or may not lie on the same A B closed loop. In the "rst case, if we restrict ourselves to the diagrams retained above for ln Z , the summation of the self-energy insertions gives rise only to a non-interacting contribution, evaluated with the self-consistent Hamiltonian H . In the second case, this summation leads to the diagrams of the type of Fig. 3, which exhibit the irreducible skeleton attached to the dots Q and Q . A B From the topology of the primitive diagrams of Fig. 2, we thus obtain for C) the set of bubble (or AB open ring, or chain) diagrams (Fig. 3). These diagrams start from Q and end up at Q . They involve A B any number n"0, 1, 2,2 of vertices
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Fig. 3. Bubble diagram contributing to the correlation between the observables QK and QK . A pair of lines represents the A B two-quasi-particle propagator C expressed by Eqs. (4.69), (4.70). A vertex represents the two-body interaction, written as a tensor E . For Kubo correlations C) the `timea u associated with QK is integrated over the interval [0, b]; it is "xed at AB A 0 for standard correlations C . AB
bubble diagrams. In the latter case, the interaction
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provides 1 C , (uO0) C(u)"! b\C S #u
(4.71)
C(0)"!bS\ .
(4.72)
The expressions (4.70) and (4.72) for the limiting value C(0) are consistent with the value (A.26) of the metric tensor !S\ for R "R . I H Bubble diagrams iterate a vertex associated with the two-body interaction < by means of the propagator C(u). Regarded as a matrix in the 2n;2n Liouville space, the interaction < can be identi"ed with the matrix E (see Eq. (4.19)) of the second derivatives of E+R,, as seen from the expression (3.31). Finally, the initial and "nal vertices (Fig. 3) associated with the observables QK and QK (the correlation of which we want to calculate) give rise to factors Q and Q , regarded A B A B here as vectors of the Liouville space. Altogether, the contribution of the n-th order bubble diagram to the Kubo correlation C) , AB evaluated with DK (b) as the unperturbed density matrix, is obtained by iterating n times !E with the propagator C. However, we have to associate a `timea u, running from 0 to b, with the "nal observable QK of the chain of bubbles. This amounts to keeping only the value u"0 in all the A propagators C. We thus obtain, after summation over n, the contribution of the bubble diagrams to C) : AB C) " Q : b\[!C(0)E ]LC(0) : Q AB A B L 1 :Q . (4.73) "b\Q : B A C(0)\#E It then su$ces to take into account Eq. (4.72), together with the de"nition E !b\S of F , to identify C) with our variational expression (4.37). AB For the standard correlations C , we have now to set to zero the `timesa u associated with AB QK and QK . This is accounted for by a summation over u"2nib\ m where m3Z. We thus have A B C" Q : b\ [!C(u)E ]LC(u) : Q AB A B L S 1 C(u) : Q . "b\Q : B A 1#C(u)E S
(4.74)
The ordering of QK and QK is relevant only in the evaluation of the zeroth-order term, where the A B summand behaves as !C /u for uP$R as seen from Eq. (4.71); the average over both orderings as in Eq. (1.8) is achieved by taking a principal part in the summation over u for uP$R. By using the expression (4.71) of C(u) we "nd for uO0: 1 1 1 C(u)"! C "! C , b\C S #u!C E u#K 1#C(u)E
(4.75)
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where we utilized the relations (4.21) and (4.49) which introduces K . The summation over u"2nib\ m is then performed by means of 1 1 1 1 1 " coth bz! , (4.76) b u#z 2 2 bz S where is meant as a principal part for uP$R, the value u"0 being excluded. The S expression (4.76) also holds for z"0 where it is continuous. We still have to add the contribution u"0, already written in Eq. (4.73). We thus obtain (4.77) C"Q : ([! coth bK #(bK )\]C #[bF ]\) : Q . B AB A The combination of the "rst two terms is de"ned even if K has vanishing eigenvalues, since these give no contribution to the bracket. The last term may diverge for vanishing eigenvalues of F , in agreement with the discussion of Section 4.2.3, but it otherwise cancels the second term. Finally, using again Eq. (4.49) so as to exhibit the "nite factor (coth bK )K , we "nd for the contribution of the bubble diagrams, C"Q : ( coth bK )K F\ : Q , AB A B the same expression as our variational result (4.63).
(4.78)
4.5. Comments The formula (4.63), obtained by inserting the trial choice (3.15)}(3.17) in the variational principle (2.5), covers the general case where the single-quasi-particle observables QK and QK do not commute with the A B operator KK . Within the same variational framework, we have also found the simpler expression (4.37) or (4.27), either for the Kubo correlations or in the case where one of the two observables is conserved. We may regard all these formulae as quantal extensions of the Ornstein-Zernike approximation. Indeed, the latter can be derived [38] from a classical version of the variational principle (2.44) where KK is replaced by the operator KK which includes the sources (see Eq. (2.36)). The stability matrix F of the HFB theory (Eq. (4.21)) as well as the associated RPA kernel K (Eq. (4.47)) have emerged naturally in a variational frame (which, however, is not of the Rayleigh-Ritz type) without any other assumption than the independent quasi-particle trial choice (3.15)}(3.17) for the operators DK (u) and AK (u). In the present formulation, the matrices F and K have automatically come out in a speci"c way, entailed by the variational approach; this should be contrasted with the various fashions in which the so-called RPA approximation can be implemented either in static or in dynamic many-body problems. In particular, both matrices F and K appear in the general expression (4.63) while only F enters Eqs. (4.27) and (4.37). The results of Section 4.1 show that, in our variational scheme, the evaluation of the thermodynamic quantities and of the expectation values 1QK 2 requires only the use of the standard HFB A theory, without any RPA-type corrections, in contrast to the evaluation of the correlations and #uctuations. Far from being an inconsistency, this disparity is needed to ful"l the thermodynamic relations, as illustrated by Section 4.2.1. These would be violated, either by a naive use of HFB to evaluate the #uctuations as in Eq. (4.13), or in apparently more sophisticated approaches where thermodynamic quantities (such as the energy) or expectation values 1QK 2 would include RPAA type corrections.
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We have given in Section 4.4 the diagrammatic interpretation of our variational results. Thermodynamic quantities and expectation values 1QK 2 are represented by the sum of all the A diagrams (of the type of Fig. 2) in which any self-energy insertion is instantaneous; this sum reduces to the term without interactions when the unperturbed Hamiltonian is chosen to be the HFB self-consistent one aRH a. On the other hand, with the same choice of the unperturbed Hamil tonian, our variational approximation for the correlations is represented by the sum of the bubble diagrams (of the type of Fig. 3). These diagrams, which also appear in the RPA, involve the interaction, not only through H but also directly. Let us stress again that they do not contribute to the variational expressions for ln Z and 1QK 2. This di!erence in the diagrammatic A expansions is in fact logical, as shown in Section 4.4. Indeed, it is a consistent scheme to retain, for the perturbative expansion of the characteristic function ln Z , the set of diagrams of the type of Fig. 2 and then to expand the outcome in powers of the sources. At the zeroth and "rst order in these sources, this yields the HFB result for ln Z and 1QK 2. At the second order, in this perturbative A treatment as well as in our variational one, the bubble diagrams of Fig. 3 thus emerge from the HFB diagrams of Fig. 2 contributing to ln Z . Note again that the systematic insertion of bubble diagrams (a procedure sometimes called `RPA approximationa or `standard RPAa) in the calculation of all the physical quantities (such as the energy) would not be consistent in the framework of generating functions, and would not preserve thermodynamic consistency, as discussed for instance in Ref. [39]. The diagrammatic interpretation of the variational approximation raises also the question of its status with respect to the Pauli principle. Indeed, the subset of diagrams like the one in Fig. 2 for ln Z (or ln Z) or the one in Fig. 3 for the correlations violate the Pauli principle in the intermediate states, since exchanging the end points of two lines may lead to diagrams which have a di!erent topology and which are not included in the initially retained subseries. Nevertheless, the Pauli principle is properly taken into account at each step of our variational approach. Thus, deciding whether an approximation violates the Pauli principle or does not is an ambiguous question, as the answer depends on the path which leads to this approximation. Another variational principle, based on the maximization of the entropy rather than on the Bloch equation (2.4) for the characterization of the state, has been previously used [37] to evaluate the correlations C . While the formula obtained for the Kubo correlations was the same as Eq. AB (4.37), the result for ordinary correlations di!ered from Eq. (4.63). Its perturbative interpretation also involved the set of bubble diagrams. However, starting from second order, only the u"0 contribution to each diagram was included (see Section 4.1 of [37]). In contrast, our result (4.63) includes the summation over all values u"2nib\m. This appears as another advantage of the present variational principle based on Bloch's equation.
5. Projected extension of the thermal HFB approximation 5.1. Generalities and notations In Sections 3 and 4 we have dealt with equilibriums in a grand-canonical ensemble. However, as discussed in the Introduction, many problems involve data of type I that prohibit #uctuations for the associated conserved operators. For instance, in the case of the canonical ensemble, the state is
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required to have a well-de"ned particle number N . Trial states of the type (3.4) that were used in Sections 3 and 4, namely exponentials ¹K of single-quasi-particle operators, violate this requirement. We want, nonetheless, to keep the convenience associated with the operators ¹K and the Fock space. To resolve this dilemma, in conjunction with the variational functional (2.5) we shall use density operators DK involving projections, such as in Eq. (1.11). For instance, a natural choice for a trial state with a well-dexned particle number is (5.1) DK "PK ¹K B PK , , , where PK is the projection , 1 L PK " (5.2) dh e\ F, e F,K , , 2n onto the subspace with the particle number N . As in the preceding sections, the variational parameters associated with the choice (5.1) are contained in the operator ¹K B which belongs to the class (3.4). We shall consider successively the two types of situations that were analyzed in the Introduction. (i) The "rst type (Sections 5.2 and 5.3) corresponds to the case where the choice (5.1) involves an operator ¹K B which breaks a symmetry of the Hamiltonian. An example is that of a system in canonical equilibrium when pairing sources are added. Then, the operator ¹K B does not commute with PK , although the exact state DK "e\@)K does. In this case, projections are necessary on both , sides of ¹K B . (ii) In the second type (Section 5.4), no symmetry is broken by the operator ¹K B . This situation is illustrated by the thermal Hartree-Fock approximation. Then, the operator ¹K B does not violate the N-invariance and DK can be de"ned with only one projection on either side of ¹K B ; this projection is needed because ¹K B acts in the unrestricted Fock space. Whatever the case, projected trial states are not easy to handle in the form (5.1). In particular, they do not satisfy directly Wick's theorem which has been an important tool in Sections 3 and 4. In order to retain the simple properties of the exponential operators ¹K , we wish to express each projection as a sum of such exponentials. This is feasible when the projection is associated with some underlying group of operators ¹K E having single-quasi-particle operators as generators. In this case the projection can be expressed as a weighted sum of the operators ¹K E over the elements g of this group. For instance, the projection PK in Eq. (5.2) is built from the group elements , ¹K E ,e F,K (g,h) which are indexed by the parameter 04h(2n. As a consequence, our projected trial states will appear as sums of exponential operators of the ¹K type, each of which is easy to handle. Likewise, if we only wish to eliminate the odd (or even) particle-number components, we should introduce a projection on even (g"1) or odd (g"!1) numbers, which can be expressed as the sum
PK "(1#ge L,K ) , g"$1 E over the two elements of the group e F,K with h"0 and h"n. The projection on a given value M of the z-component of the angular momentum,
1 L dh e\ F+ e F(K X , PK " + 2n
(5.3)
(5.4)
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is similar to Eq. (5.2). To project in addition on a given value of the total angular momentum JK , we should introduce the rotation operators ¹K E "RK (X) in the Fock space; here the group index g denotes the three Euler angles X, and RK (X) has again the form of an exponential of single-particle operators. Integrating over the Euler angles with the rotation matrices D( (X) yields the operator +) 2 J#1 dX D(H (X)RK (X)" "cJM21cJK" , (5.5) PK ( " +) +) 8n A where "cJM2 denotes a basis in the Fock space labelled by J, M and another set of quantum numbers c. When K"M, Eq. (5.5) de"nes the projection on given values of J and M. The projections PK over the spaces with a given spatial parity can be expressed as ! (5.6) PK "(1$e L,K \) , ! where the single-particle operator
1 NK , dr [tR(r)!tR(!r)][t (r)!t (!r)] , (5.7) \ 4 N N N N N counts the number of particles lying in the single-particle states which are odd in the exchange r, pP!r, p (the factor in Eq. (5.7) accounts both for the normalization of the basic antisymmet ric states [tR(r)!tR(!r)]"02/(2 and for the double-counting in the integration over r). N N In order to encompass formally all these situations, we shall denote by
(5.8) PK " ¹K E , E the decomposition of our relevant projection as a linear combination of the group operators ¹K E . The symbol 2 denotes a weighted sum or integral over the elements of the group. In particular, E gH. The operators for Eq. (5.2), it stands for (2n)\Ldhe\ F,2 while for (5.3), it stands for H ¹K E have the form (3.4) of an exponential of a single-quasi-particle operator, which we shall denote as !l E !aRL E a. The coe$cients l E and L E depend on the group index g; for instance, in Eq. (5.2), we have l F "!ihn/2 and L F "!ihN (04h(2n), and in Eq. (5.3), l H "!ijnn/2 and L H "!ijnN ( j"0, 1), where N is the 2n;2n matrix de"ned by (4.12). As in Section 3.1, we denote the 2n;2n matrix exp(!L E ) as T E and the scalar exp(!l E ) as q E . We shall have to deal with products of two to four ¹K -operators, such as ¹K E ¹K B ¹K EY or ¹K E ¹K B ¹K EY ¹K ? ; from the algebra properties (3.23)}(3.24), these products have still the same ¹K -form and they are characterized by matrices such as T EBEY ,T E T B T EY ,
(5.9)
and scalars q EBEY "q E q B q EY . There is a contraction matrix R EBEY associated with the matrix T EBEY through Eq. (3.8); R EBEY , as T EBEY , depends on the ordering of the original operators. The equations that we shall derive in the next Subsections can also handle cases which are more general than pure projections. For instance, in the description of rotational bands of axial nuclei, it is often a good approximation to assume that the quantum number K associated with the
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component of the angular momentum on an internal axis of inertia has a well-de"ned value K . Then rotational states with given J and M values can be sought with the trial form (5.10) DK "PK ( ¹K B PK ( , )+ +) where the operators PK ( de"ned by Eq. (5.5) are not projections when MOK . Thermal +) properties of triaxial nuclei, which involve sums over the K quantum number, can also be handled by our method. Another example is the case where a symmetry (such as NK ) is broken. Suppose that we still want to work in the grand canonical ensemble but also want to suppress the o!-diagonal contributions between subspaces with di!erent values of N. We are led to consider operators of the form
L 1 L PK DK PK " d
d eG(\(Y,e\G(,K DK eG(Y,K , , , 4n , , 1 L " d e\G(,K DK eG(,K , 2n which are also amenable to our treatment.
(5.11)
5.2. The variational ansaK tze and the associated functional Now we must choose the trial classes for the operators DK (u) and AK (u) to be inserted into the functional (2.5) which, at its stationary point, supplies the characteristic function of interest. For the operator DK (u) we already made the choice
¹K E ¹K B (u)¹K EY , ¹K EBEY (u) , EEY EEY where ¹K B (u) is de"ned by Eq. (3.16). For the operator AK (u) we keep the same choice, DK (u)"PK ¹K B (u)PK "
AK (u)"¹K ? (u) ,
(5.12)
(5.13)
as in Eq. (3.17). Again, as in Section 3.2, the quantities l B (u), L B (u), l ? (u), L ? (u) form a set of independent variational parameters. The exact density operator DK satis"es PK DK "DK PK "DK , and when de"ned in the full Fock space the operator KK commutes with PK . Therefore, it makes no di!erence if we project DK (u) or AK (u) in the functional (2.5); the choice DK (u)"¹K B (u), AK (u)"PK ¹K ? (u)PK is equivalent to (5.12)}(5.13). The results of Section 3.2 are readily adapted through a mere replacement of ¹K B by ¹K EBEY and a summation over the group indices g and g . Thus the normalization Z[ , given by
Z[ "TrAK (u)DK (u)"
ZEEY(u) EEY
(5.14)
with (5.15) ln ZEEY(u)"![l ? (u)#l B (u)#l E #l EY ]#tr ln[1#T ? (u)T E T B (u)T EY ] , L replaces Z as de"ned by Eq. (3.25). We recall that, according to Eq. (2.18), Z[ does not depend on u when AK (u) and DK (u) are stationary solutions.
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The action-like functional (3.32), is now replaced by
@
I"Z[ !
du
dl B (u) 1 dT B 1 # tr T B \R BEY?E
# [E+R BEY?E , ZEEY ! L du 2 du 2 EEY
#E+R ?EBEY ,] .
(5.16)
The boundary term Z[ "TrAK DK (b) is obtained from Eqs. (5.14)}(5.15) by letting u"b in l B (u) and T B (u) and replacing l ? (u) and L ? (u) by the given quantities l and L (see Eq. (3.14)). We have used the analogue of the property (3.33) to transform T E \R EBEY? T E into R BEY?E . Moreover, from the commutation [KK , ¹K E ]"0 and in agreement with Eq. (4.31), we "nd that for any R the energy (3.31) satis"es E+T E RT E \,"E+R, ,
(5.17)
a property that we have also used to write E+R EBEY? , instead of E+R BEY?E , (similarly, one has E+R ?EBEY ,"E+R EY?EB ,). The expression (5.16) exhibits the fact that we might as well have projected AK (u) instead of DK (u). Indeed, taking into account the property (5.17), this expression does not change under the replacement of gdg by d and of a by g ag. Actually, we can readily check that if one projects AK instead of DK , Eq. (3.32) is directly transposed into Eq. (5.16) through the replacement of a by g ag. Likewise, by an obvious extension of Eq. (3.35) where d is replaced by gdg , we "nd for the functional I the alternative form:
I"Z[ !Z[ (b)#Z[ (0)#
@
du
dl ? (u) 1 dT ?
# tr T ? \R ?EBEY
ZEEY ! du 2 L du EEY
1 ! [E+R BEY?E ,#E+R ?EBEY ,] . 2
(5.18)
The expressions (5.16) and (5.18) generalize (3.32) and (3.35) through the averaging over g and g with the weight ZEEY/Z[ . They lead us to introduce the matrices
RM B ? ,Z[ \
ZEEYR BEY?E , EEY
RM ? B ,Z[ \
ZEEYR ?EBEY , EEY
(5.19)
and the pseudo-energies
EM B?,Z[ \
ZEEYE+R BEY?E , , EEY
EM ?B,Z[ \
ZEEYE+R ?EBEY , , EEY
(5.20)
(remember that E+R EBEY? ,"E+R BEY?E , and that E+R ?EBEY ,"E+R EY?EB ,). With these notations we can rewrite Eqs. (5.16) and (5.18) in a more condensed fashion, formally closer to the functionals
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(3.32) and (3.35):
I"Z[ !
@
dl B 1 dT B 1 du Z[ ! # tr T B \RM B ?
# [EM B?#EM ?B] , L du 2 du 2
I"Z[ !Z[ (b)#Z[ (0)#
@
dl ? 1 dT ? 1 du Z[ ! # tr T ? \RM ? B
! [EM B?#EM ?B] , du 2 L du 2 (5.21)
where RM B ? , RM ? B , EM B? and EM ?B replace R B? , R ?B , E+R B? , and E+R ?B , while Z[ replaces Z. An important change between the present functional and the one of Section 3.2 lies in that EM B? di!ers in general from E+RM B ? , (as well as from E+RM B ? ,) when the operator KK contains a two-body interaction. It implies that the variational outcome Z[ yielded by the functional (5.21) will di!er from that which would be obtained by projecting the optimum state found in Section 3; this holds even for the case (AK "1) ) of thermodynamic potentials. 5.3. The coupled equations As in Section 3.3 we "nd the equations governing the evolution of l B (u) and T B (u) by requiring that the functional (5.16) be stationary with respect to l ? (u) and T ? (u). The resulting di!erential equation for l B (u), dT B 1 dl B 1 " tr T B \RM B ?
# [EM B?#EM ?B] , 2 L du 2 du
(5.22)
is similar to Eq. (3.36), with RM B ? , EM B? and EM ?B replacing R B? , E+R B? , and E+R ?B ,. The di!erential equation for T B (u) di!ers from Eq. (3.37) by an explicit summation over the group indexes. It reads:
dT B 1 # [H BEY?E T B #T B H EY?EB ] ZEEYT E (1!R BEY?E ) du 2 EEY ;(1!R EY?EB )T EY "0 ,
(5.23)
with kEYE , H G ,H+R G ,# B 1!R G
(5.24)
where i stands for dg ag or g agd and where the c-number kEYE is de"ned by B dT B
1 #E+R BEY?E ,!EM B?#E+R EY?EB ,!EM ?B . kEYE, tr T B \(R BEY?E !RM B ? ) B du 2 L (5.25)
To derive the equation (5.23) we made use of Eqs. (3.51), (5.22) and of the fact that H+R, satis"es for any R the relation H+T E RT E \,"T E H+R,T E \ ,
(5.26)
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a consequence of the de"nition (3.38) and of Eq. (5.17). Note that Eq. (5.22) does not imply the vanishing of the integrand of Eq. (5.16) before summation over g and g . Likewise the di!erential equation for l ? (u) is obtained by varying the functional (5.18) with respect to l B (u): dl ? 1 dT ? 1 " tr T ? \RM ? B
! [EM ?B#EM B?] , du 2 L du 2
(5.27)
while the equation for T ? (u) is provided by varying Eq. (5.18) with respect to T B (u) for 04u(b:
ZEEYT EY (1!R ?EBEY ) EEY
dT ? 1 ! [H ?EBEY T ? #T ? H EBEY? ] (1!R EBEY? )T E "0 , du 2 (5.28)
where H ?EBEY and H EBEY? are de"ned according to Eq. (5.24) with i standing for agdg or gdg a and with the c-number kEYE replaced by kEEY given by B ? dT ?
1 #E+R ?EBEY ,!EM ?B#E+R EBEY? ,!EM B? . kEEY, !tr T ? \(R ?EBEY !RM ? B ) L ? du 2 (5.29)
Note that l ? #l B disappears from Eqs. (5.23) and (5.28) if we divide them by Z[ . These equations thus couple only the matrices T B and T ? . Once they are solved, l B and l ? are obtained by integration of Eqs. (5.22) and (5.27). In Appendix B, using the Liouville formulation and the notation (5.19)}(5.20), Eqs. (5.23) and (5.28) are rewritten in the forms (B.10)}(B.11) which look similar to Eqs. (3.37) and (3.43). Finally, in agreement with Eq. (2.19), the stationarity of Eq. (5.18) with respect to l B (b) and T B (b) provides the equations Z[ (b)"Z[ ,
R BEY?E (b)"R BEY E (b) ,
(5.30)
which are satis"ed by the same boundary conditions as in Section 3: l ? (b)"l ,
T ? (b)"T .
(5.31)
The boundary conditions at u"0 are l B (0)"0, T B (0)"1 as in Section 3. Again, all our variational quantities (l ? (u), l B (u), T ? (u), T B (u)) depend upon the sources m through the A boundary conditions (5.31). Despite their somewhat complex form, the coupled equations (5.23) and (5.28) entail some simple consequences. First, using Eqs. (5.22) and (5.27), one notes that ZEEY varies with u according to d ln ZEEY"kEYE!kEEY B ? du 1 d tr ln(1!R BEY?E )!c (u) , "! 2 du L
(5.32)
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where the function c (u) is given by the average
1 d c (u)"! ZEEY tr ln(1!R BEY?E ) . L du 2Z[ EEY Summation over g and g of Eq. (5.32) leads to dZ[ "0 , du
(5.33)
(5.34)
which was expected from Eq. (2.18). Thus the ultimate quantity of interest, the characteristic function u provided by the stationary value of ln I, is equal to ln Z[ (u) for any u. One can also check from Eqs. (5.23), (5.28) and (5.32) that d (EM B?#EM ?B)"0 , du
(5.35)
which is a special case of the general relation (2.16). The relations (5.34) and (5.35) generalize Eq. (3.48). When the operator AK is hermitian, the coupled equations (5.23) and (5.28) admit solutions with hermitian matrices T B (u) and T ? (u), and the resulting c-numbers l B (u) and l ? (u) are real. This is seen by grouping in all the equations the terms corresponding to ¹K E , ¹K EY respectively with the associated terms corresponding to ¹K EY \, ¹K E \, as is shown at the end of Appendix B. 5.4. Unbroken PK -invariance The approach above is general and covers situations in which the trial operators ¹K B (u) and ¹K ? (u) do not commute with the projection PK . It applies for instance to "nite systems of fermions in canonical equilibrium in which pairing correlations are operative. In this case the operator KK denotes the Hamiltonian HK alone without the term !kNK , the projection PK is given by (5.2) and the independent quasi-particle trial state ¹K B (u) as well as the trial operator ¹K ? (u) do not commute with NK , nor with the operations ¹K F "e F,K of the associated group. We specialize below to the simpler case when PK commutes with ¹K B (u). 5.4.1. Notation and general formalism As discussed in Section 5.1 and in the Introduction, there exist situations where projection is required while the PK -invariance is not broken by the variational approximation. In these cases ¹K B (u) and ¹K ? (u) commute with all the group elements ¹K E and therefore with PK . Only one projection is then needed on either side of ¹K B (u) and, instead of Eq. (5.12), the trial state can be written as
DK (u)"PK ¹K B (u)PK "¹K B (u)PK "PK ¹K B (u)" ¹K E ¹K B (u) . E A single summation over the group index g is su$cient in Eq. (5.14):
Z[ "TrAK (u)DK (u)" ZE(u) , E
(5.36)
(5.37)
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while Eq. (5.15) becomes (5.38) ln ZE(u)"![l ? (u)#l B (u)#l E ]#tr ln [1#T ? (u)T E T B (u)] . L Likewise, the double averaging in Eq. (5.19) or (5.20) is replaced by a single one over g with the weight ZE/Z[ ; for instance, we have
RM ?B,Z[ \ ZER ?BE . (5.39) RM B?,Z[ \ ZER B?E , E E The expressions of the functional I and of the coupled equations are readily derived from those obtained in Sections 5.2 and 5.3 by noting that the matrix T E commutes with T B (u) and T ? (u) and that the factor T EY can everywhere be replaced by T "1. Since g disappears and the ordering of g with respect to a and d is irrelevant, there remain only two (instead of four) di!erent contraction matrices: R B?E "R BE? "R EB? and R ?BE "R ?EB "R E?B . The functional (5.16) then simpli"es to
I"Z[ !
@
dl B (u) 1 dT B 1 # tr T B \R B?E
# [E+R B?E ,#E+R ?BE ,] . du ZE ! du 2 L du 2 E (5.40)
As before, the stationary value of ln I (our variational approximation for the characteristic function) is equal to ln Z[ which does not depend on u. Accordingly, the coupled equations of Section 5.3 take a simpler form. Note, however, that because in general kE (Eq. (5.25)) di!ers from kE (Eq. (5.29)), the matrix H EB? is di!erent from B ? H B?E and H ?EB di!erent from H ?EB (Eq. (5.24)). By combining the forms taken by Eqs. (5.23), (5.28) and (5.32) in this commutative case and by using Eq. (3.51), we "nd that the averaged contraction matrices (5.39) satisfy 1 d RM ?B" du 2Z[
ZE[H+R ?BE , , R ?BE ] ,
E 1 d RM B?"! du 2Z[
ZE[H+R B?E , , R B?E ] .
(5.41)
E This was expected from the general equations (2.32) and(2.33), since their validity conditions are satis"ed when the operators ¹K B (u) and ¹K ? (u) commute with PK . While Eqs. (3.52), which are appropriate to the unprojected variational treatment, had the same form as the time-dependent HFB equation (with an imaginary time), Eqs. (5.41) deal with the larger set of matrices R B?E and R ?BE and involve a summation over the group index g. The apparent decoupling of Eqs. (5.41) is again "ctitious because the boundary conditions cannot be expressed explicitly in terms of RM ?B and RM B?. 5.4.2. Partition function, entropy In the rest of this section, still assuming that the ¹K E -invariance is not broken, we focus on the case AK "1) for which our variational principle provides an approximation for the exact
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partition function (5.42) Z "Tr PK e\@)K . . We have shown in Section 2.4 that by shifting appropriately the operator KK , the variational principle (2.5) used with AK "1) also yields the characteristic function and therefore the cumulants of the one-body operators which commute both with KK and with the projection PK . As seen in Section 2.5, the boundary condition AK (b)"AK "1) allows us to restrict the trial classes (5.12) and (5.13) to AK (u)"e\@\S)K ; (5.43) DK (u)"PK e\S)K , the subscript 0 refers to the zero values of the sources m . The trial operator KK is a (u-independent) A single-quasi-particle operator commuting with PK , (5.44) KK "h #aRH a , which depends on the parameters h , a real number, and H , a 2n;2n hermitian matrix obeying (3.41) and commuting with T E . The choice (5.43) entails DK (u)AK (u)"PK e\@)K "AK (u)DK (u)"DK (b) , and the u-integration becomes trivial in the functional (2.5), which reduces to
I"Tr PK e\@)K (1#bKK !bKK )" ZE [1#bh #btr H RE !bE+RE ,] , L E
(5.45)
(5.46)
with ln ZE "!bh !l E #tr ln [1#e\@HT E ] , (5.47) L where the trial quantities h and H must be determined variationally. In the generalized contraction matrices R T E
T E
" , (5.48) RE , H e@ #T E 1#R (T E !1) the matrices H , T E and R all commute; for the unity element of the group (¹K "1) ; l "0, T "1), the matrix R given by Eq. (5.48) reduces to the contraction matrix R of the usual "nite-temperature HFB formalism (Eq. (4.1)). As in Eq. (2.43), we recognize in the simple expression (5.46) the standard variational principle associated with the maximization of the entropy compatible with the constraint on 1KK 2. If one introduces the quantities
1 1 RM , ZE RE , EM , ZE E+RE , , Z[ , ZE , Z [ Z [ E E E the functional (5.46)}(5.47) takes the more compact form I"Z[ [1#bh #btr H RM !bEM ] . L
(5.49)
(5.50)
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Within the selected variational space, the best approximation for the exact partition function Z (Eq. (5.42)) is supplied by the maximum of I since the r.h.s. of Eq. (5.46) is always smaller than . Z . The maximization with respect to the number h yields . (5.51) h "!tr H RM #EM . L This equation ensures that, according to the discussion of Section 2.5, the maximum of I reduces to Z[ . Requiring now the functional I to be maximum with respect to the matrix H provides
ZE RE (H !HE)(1!RE )"0 , E
(5.52)
with kE , HE,H+RE ,# 1!RE where
(5.53)
kE,!tr H (RE !RM )#E+RE ,!EM . (5.54) L Eq. (5.52), together with the de"nitions (5.53) and (5.54), determines the trial matrix H to which RE is related through Eq. (5.48). The j-basis where H and RE are diagonal (with eigenvalues H and RE , respectively) is thus de"ned by the equations H H
ZE RE H+RE , (1!RE )"0, jOk H HI I E while the diagonal eigenvalues of the matrix H are given by
(5.55)
ZE +RE H+RE , (1!RE )#RE (E+RE ,!EM ), , H " (M\) H HH H H I IH E H where M is the symmetric matrix
(5.56)
M , ZE +d RE (1!RE )#RE (RE !RM ), . (5.57) HI HI H H H I I E Therefore the projection generates a more complicated relation (Eqs. (5.52) and (5.48)) between H and the RE 's than the relation between H and R (Eqs. (4.1) and (4.2)) given by the grand canonical HFB formalism. Eqs. (5.52)}(5.54) can also be obtained directly from the evolution Eqs. (5.23) and (5.28) when one takes into account the simpli"cations induced by the choice (5.43)}(5.44) and by the commutation properties resulting from Eqs. (5.41). Indeed, kE is then the u-independent common value of the quantities kE"kE"kE introduced in Eqs. (5.25) and (5.29), while the matrices H G de"ned in B ? (5.24) also become u-independent and all reduce to H ?EB "H ?EB "H B?E "H EB? "HE. Several consistency properties can be derived from the formulation above. When the functional (5.50) is stationary with respect to the number h , the approximate entropy S[ associated with the approximate normalization Z[ and projected HFB energy EM by the thermodynamic relation S[ "ln Z[ #bEM ,
(5.58)
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(a specialisation of Eq. (2.28) to the restricted variational spaces de"ned by Eqs. (5.43) and (5.44)) is also related to the approximate density operator DK (b)"PK exp(!bKK ) by the von Neumann relations (1.2) and (2.45). Using Eqs. (5.47), (5.49) and (5.51), one obtains the more explicit form S[ "!tr [RM ln R #(1!RM ) ln(1!R )]#ln L
e\J E [det+1#R (T E !1),] . E
(5.59)
As expected, S[ reduces to the HFB entropy S+R , (Eq. (4.6)) when the group of operators T E is restricted to its unity element. Furthermore, in accordance with Eq. (2.26), the stationarity of Eq. (5.46) with respect to the matrix H ensures that the equilibrium energy, !R ln Z[ /Rb, is equal to EM . According to the discussion of Section 2.3, this guarantees that, in agreement with thermodynamics, the projected entropy S[ also satis"es the relation (2.29) where F ,!¹ ln Z[ . % Remark. Through the ansatz (5.43) we have restricted the trial class by choosing an exponential u-dependence for the operators DK (u) and AK (u) and by relating their exponents. Therefore, it is not obvious that the maximum of I as given by Eq. (5.46) is equal to the stationary value of the functional (5.40) which is de"ned within the more general class (5.12)}(5.13) of operators whose u-dependences are not limited a priori. In the light of Section 2.5, the commutation of PK with ¹K ? and ¹K B is su$cient to ensure this equality. Indeed, if we associate a projection PK to AK (u) as well as to DK (u), the conditions (i) and (ii) of Section 2.5 are both satis"ed. Nevertheless let us give a more explicit proof along the line already followed in Section 4.1 for the HFB case. We have to check that the ansatz (5.43) with h and H given respectively by Eqs. (5.51) and (5.52) satis"es the set of coupled equations (5.22) and (5.23), or (5.27) and (5.28) which express the unrestricted stationarity. We "rst recall that, for the choice (5.43), the matrix R B?E (u) is u-independent and given by Eq. (5.48). As l B (u)"uh and l ? (u)"(b!u)h , the functional (5.40) is identi"ed with Eq. (5.46) and both Eqs. (5.22) and (5.27) are identical with Eq. (5.51). Eq. (5.23) then becomes
ZE RE (H !HE!eSHHEe\SH) (1!RE )"0 . E
(5.60)
Moreover, when H is solution of Eq. (5.52) (which implies Eq. (5.55) in the basis where H is diagonal) one has the commutation relation
H ,
E
ZE RE HE(1!RE ) "0 ,
(5.61)
so that Eq. (5.60) is satis"ed for all values of u. This last property can also be demonstrated by means of Eqs. (5.41). Likewise, within the replacement of u by b!u, the stationarity condition (5.28) also has the form (5.60) and this is a consequence of Eq. (5.52). Therefore, in the case AK "1) and unbroken PK -invariance, the result of the optimization of the functional (5.16) is identical to the approximation Z[ of the exact parition function Z (Eq. (5.42)) obtained by the maximization of . the functional (5.46).
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6. Projection on even or odd particle number In this section we apply the formalism of Section 5 to a speci"c problem with the purpose to illustrate the usefulness of the method. In su$ciently small systems of fermions, such as atomic nuclei or metallic grains, for which the Hamiltonian favors pairing correlations, there exists a qualitative diwerence between the properties of systems with even and with odd numbers of fermions, whereas systems with N, N$2, N$4 2 particles display strong similarities. At zero temperature the HFB approximation, which is known to provide a good description of pairing correlations in nuclei, involves a sum of components with the same particle-number parity. On the other hand, at non-zero temperature, any number, odd and even, of quasi-particle con"gurations are allowed by the standard HFB solution. As a consequence the HFB thermal equilibrium state has both even and odd particle-number components. A realistic description of "nite systems with "xed particle number N would require a full projection as de"ned by Eqs. (5.1)}(5.2). (See Chapter 11 of Ref. [12] for the litterature, starting with Refs. [40}42], about the particle-number projection at zero temperature.) Nevertheless, the similar properties of systems with the same particle-number parity suggest that a "rst signi"cant step towards this goal consists in projecting the independent quasi-particle trial state onto the space with even or odd particle (or equivalently quasi-particle) number by means of the projection PK de"ned in Eq. (5.3). E Using the formulation of Section 5 we want therefore to approximate variationally the characteristic function u (m) of a system with a given parity of the particle number (or N-parity). In E particular, for AK "1) , we shall thus obtain an approximation for the number-parity projected grand-partition function (6.1) Z , e@I, Tr e\@&K , E , , where the prime indicates that the sum runs only over even (g"1) or odd (g"!1) values of N. We still enforce the particle-number average by means of the chemical potential k entering the operator KK "HK !kNK . 6.1. Characteristic function in the N-parity projected HFB approximation In the specialization of the equations of Section 5 to the present case, the sum in the general E formula (5.8) reduces to a sum of two terms: PK "(¹K #g¹K ),(1) #ge L,K ) . E In the 2n;2n representation, we have simply
(6.2)
l "0, T "1; l "!inn, T "e LN"!1 . (6.3) The commutation of T and T with any matrix T of the algebra introduced in Section 3.1 is obvious. This re#ects the fact that all operators of the single-quasi-particle type (3.12), or of the trial forms ¹K B or ¹K ? given by Eq. (3.4), commute with ¹K and ¹K and with the projection PK because E they involve only an even number of creation or annihilation operators. Thus, although a BCS type
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of state (3.16) breaks the particle-number invariance eGF,K , it does not break the N-parity invariance eGL,K , and we can work with the variational spaces DK (u)"PK ¹K B (u), AK (u)"¹K ? (u) , (6.4) E where only one projection PK is needed. E Therefore, in the present section we shall not need the general formalism of projection developed in Sections 5.2 and 5.3; it shall be su$cient to rely on Section 5.4 which is suited to unbroken invariances. The reader who has chosen to skip Section 5 is requested to accept Eqs. (5.37)}(5.40). With the replacement of by a sum and the simpli"cations associated with Eq. (6.3), they furnish E the expression of the action functional corresponding to the N-parity projection. Let us "rst transpose the results obtained for characteristic functions in Section 5.4.1. The matrices R ?BE (u) and R B?E (u) which occur in the functional (5.40) can here be written explicitly: R G "R G ,
R G
, R G " 2R G !1
(6.5)
with i"ad or da and with R ?B or R B? de"ned as in Section 3.2. The weight factors ZE, given by Eq. (5.38), take now the compact form (6.6) Z"e\J ? \J B +det (1!R ?B ),\ , L Z"rZ, r"+det [N(1!2R ?B )], , (6.7) L and hence the normalization Z[ (Eq. (5.37)) reads E (6.8) Z[ "Z(1#gr) . E According to the general formalism of Section 2, the evaluation of the approximate characteristic function u(m)"ln Z[ "ln Tr AK (m)DK (b) requires the solution of the coupled equations which deterE mine DK (u) and AK (u). To this aim we will use Eqs. (5.23)}(5.25), (5.28)}(5.29) and(5.32)}(5.33) with the following simpli"cations: (i) suppression of the integration on the group index g which is everywhere set to 0 (¹K EY "¹K "1) ), (ii) replacement of by the sum with j"0 or 1, (iii) E H commutation of T H with T ? and T B whenever it is required. Thanks to these simpli"cations we can express the evolution equations in a more explicit form than in Section 5.3 since the group elements T E are now trivial. For instance, Eq. (5.23) for T B (u) writes: dT B
1 1 1 dT B
!gr "! [H M B?T B #T B H M ?B] , B B 2R B? !1 du 2R ?B !1 2 du
(6.9)
where we have introduced the generalized HFB Hamiltonians H M ?B and H M B? de"ned by B B 1 1 H M ?B,H ?B !gr H ?B
, B 2R ?B !1 2R ?B !1 1 1 H B?
, H M B?,H B? !gr B 2R B? !1 2R B? !1
(6.10)
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in terms of the matrices given in Eq. (5.24). Taking into account that the quantities kEYE de"ned in B Eq. (5.25) now satisfy the relation k"!grk B B
(6.11)
with
1 R B? (1!R B? ) dT B
k" 2tr T B \ B L 2(1#gr) 2R B? !1 du #E
R B?
R ?B
!E+R B? ,#E !E+R ?B , , 2R B? !1 2R ?B !1
(6.12)
one can also express the matrices H M ?B and H M B? as B B
1 R G
1 2grk B , H M G"H+R G ,!gr H # B 2R G !1 2R G !1 2R G !1 2R G !1
(6.13)
where i stands as above for ad or da. Likewise, the evolution equation (5.28) for T ? (u) can be put in a form similar to Eq. (6.9). In Eqs. (6.9)}(6.10) and (6.13), the subscript d in the matrices H M G accounts for the fact that the c-numbers k and k and therefore the matrices H M G and B ? B B H M G are in general di!erent. ? For g"g "0, the specialization of Eqs. (5.32) and (5.33) to the N-parity projection yields a compact form for the evolution equation of ln Z,
1 dR B?
d ln Z gr "! tr , du 1#gr L 2R B? !1 du
(6.14)
in terms of R B? (or of R ?B ). This equation could also have been obtained from the general property dZ[ /du"0, using Eq. (6.7) and the derivative of r drawn from Eq. (6.8). To complete the E set of equations which may be useful for the calculation of the characteristic function, we can rewrite Eqs. (5.41) in terms of the matrices R B? and R ?B . For the evolution of R B? we get 1 R B? (R B? !1) d ln Z dR B?
1 dR B?
!gr #2 "![H M B?, R B? ] , B 2R B? !1 du 2R B? !1 2R B? !1 du du (6.15) where d ln Z/du is given by Eq. (6.14). A similar equation, with da replaced by ad and a plus sign before the commutator in the right-hand side, holds for R ?B . These equations for R B? and R ?B are coupled through the boundary conditions, as were Eqs. (3.52) which they generalize. Remark. Due to the square roots occuring in Eqs. (6.6)}(6.7), we must carefully "x the determinations which control the signs of Z and Z. In Eq. (6.6) we can rely on the fact that, for small values of the sources, the contraction matrix R ?B is nearly real, lying between 0 and 1, which makes the determination unambiguous. In Eq. (6.7) we have introduced the factor N to account for the term !l of Eq. (5.38), but both the matrices N and 1!2R ?B have positive and negative eigenvalues;
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moreover, they do not commute when pairing takes place. In order to overcome this di$culty and to de"ne unambiguously the sign of r, we proceed by continuity, starting from the unperturbed state without pairing. In this case, when N and R ?B are simultaneously diagonalized, the two diagonal blocks of N(1!2R ?B ) are identical, due to Eq. (3.9), and provide twice the same factors in Eq. (6.7). We then choose the determination of r as r" (1!2R ?B ) , (6.16) H H where R ?B are the diagonal elements of R ?B lying in the block such that the eigenvalues of N in H Eq. (4.12) equal #1. Hence, r is positive if the number of eigenvalues of the single-particle unperturbed Hamiltonian that are smaller than the chemical potential k is even, while r is negative if this number is odd. In the low temperature limit, for AK "1) and arbitrary k, one obtains for the grand potential F "!b\ ln Z[ the expected result, namely the minimum of (E!k1NK 2) for % E either even N (g"#1) or odd N (g"!1). When the interactions are switched on and pairing takes place, we note that within the determinant of Eq. (6.7) the particle number matrix N can be replaced by the matrix N (associated with the quasi-particle number) which commutes with R ?B . We can thus de"ne r by continuity, starting from Eq. (6.16) and keeping the same form (6.16) where the R ?B are now the eigenvalues of R ?B associated with the subspace where N equals #1. H 6.2. The partition function in the N-parity projected HFB approximation If one is only interested in the evaluation of the N-parity projected grand partition function Z (obtained for AK "1) ), or more generally of the characteristic function for conserved singleE particle observables QK (obtained for AK "exp(! m QK ) with [QK , KK ]"0), it is not necessary to A A A A A solve the coupled equations of Section 6.1, which are suited to the evaluation of the most general characteristic function. Simpli"cations occur as in Section 5.4.2 by relating to each other the trial forms of DK and AK . We therefore introduce a trial operator KK of the form (5.44) which depends on the c-number h and the 2n;2n matrix H ; for the restricted variational spaces, we take DK (u)"PK e\S)K , E
AK (u)"e\@\S)K .
(6.17)
The matrices T B (u) and T ? (u) are therefore equal to exp(!uH ) and exp(!(b!u)H ), respectively, while the u-independent contraction matrices R ?B and R B? are expressed in terms of H by 1 "R . R ?B (u)"R B? (u)" 1#e@H
(6.18)
We shall also need the specialization of the matrices RE , de"ned in Eq. (5.48), to the simple projection group associated with PK : E R 1 " R"R , R" . 2R !1 1!e@H
(6.19)
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As in Section 5.4.2, the c-number h is given by Eq. (5.51) where RM and EM , de"ned by Eqs. (5.49), E E take now the explicit forms 1 RM " (R #gr R) , (6.20) E 1#gr 1 (E+R ,#gr E+R,) . (6.21) EM " E 1#gr In these equations the quantity r "Z/Z is given by b b r " det N tanh H (6.22) " tanh H , 2 2 H H where N stands for the quasi-particle-number operator and H denote the eigenvalues of the H matrix H . The sign of r is obtained by continuity according to Eq. (6.16). The matrix H is the solution of the self-consistent Eq. (5.52) which, after a suitable reorganiza tion, can be written
b M . H 1!gr coth H "H 2
(6.23)
Using Eqs. (6.19) and (5.53) to de"ne H from R ,R and H from R, the matrix H M is given by b b H M "H!gr coth H H coth H . (6.24) 2 2
In terms of the matrices R and R, the operator H M can be expressed as b b b H M "H+R ,!gr coth H H+R, coth H !2 gr kcoth H , 2 2 2
(6.25)
in which the quantity k (see Eq. (5.54) and note that k"!gr k) is 1 1 H #E+R,!E+R , . k" tr (6.26) 1#gr 2 L sinh bH In this projected variational scheme, Eq. (6.23) replaces the standard HFB self-consistent equation H "H+R ,. The logarithm of the exact N-parity projected grand-partition function Z (Eq. (6.1)) is approximated by ln Z[ , the stationary value of the functional (5.46), as E E H (6.27) ln Z[ "tr ln[1#e\@ ]#tr [bH RM ]!bEM #ln[(1#gr )] . L E E E L Equivalently, this quantity can be written as
(6.28) ln Z[ "ln Z!gr b k#ln[(1#gr )] , E where ln Z is the logarithm of the HFB partition function, S+R ,!bE+R ,, which is associated with the independent-quasi-particle contraction matrix R (see Eq. (4.5)).
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The entropy S of the exact projected state is approximated by the specialization to the present E case of the quantity S[ given in Eq. (5.59): E S[ "!tr [RM ln R #(1!RM )ln(1!R )]#ln[(1#gr )] . (6.29) E L E E After some rearrangement, S[ takes the form E bH gr #ln[(1#gr )] , tr (6.30) S[ "S+R ,! E 2(1#gr ) L sinh bH where S+R , is the usual HFB entropy given by Eq. (4.6). As stressed in Section 5.4.2, the Eqs. (6.23)}(6.26) satis"ed by H ensure that the approximate self-consistent projected entropy S[ obeys E the standard thermodynamic relations (2.28) and (2.29) with F "!b\ ln Z[ . % E Since our trial operators DK (u) and AK (u) satisfy the validity conditions of Eq. (2.23), the density matrix which optimizes the partition function can be used to obtain the variational approximation for the expectation value of any single-quasi-particle operator QK of the type (3.12). For instance, the transposition of Eq. (2.23) gives (6.31) 1QK 2"q#tr QRM , L E which generalizes the HFB formula (4.9). Altogether, to obtain the variational approximation Z[ for the N-parity projected grandE partition function Z de"ned by Eq. (6.1), one "rst determines the matrix H (jointly with the E contraction matrix RM ) from the self-consistent Eq. (6.23) which contains (through H M ) the cE numbers r and k, given by Eqs. (6.22) and (6.26), respectively. One then evaluates Z[ by means of E Eq. (6.28). Approximations for the thermodynamic quantities such as the entropy (6.29)}(6.30), or for the expectation values (6.31) of single-quasi-particle observables, in particular of NK , involve the averaged contraction matrix RM de"ned in Eq. (6.20); this matrix is positive since, from Eqs. (6.18), E (6.19) and (6.22), its diagonal eigenvalues are given by 1!gr coth @H H . (6.32) RM "R EH H 1#gr According to the general discussion of Section 2.4, our variational approximation for the #uctuation of NK is given by the thermodynamic relation 1 RRM 1 R1NK 2 E. " tr N *NK " 2b L Rk b Rk
(6.33)
The derivative RRM /Rk is deduced from RH /Rk, itself evaluated from Eq. (6.23). E For large systems, r is exponentially small while k is proportional to the volume, as shown by Eqs. (6.22) and (6.26); these behaviours imply that the projection does not modify the HFB results. However, for su$ciently small systems and at su$ciently low temperature, the factors entering r (Eq. (6.22)) may be close to $1, except near the Fermi surface; the terms involving r may thus be signi"cant compared to the dominant terms, although "r " is always less than 1. The results of this section could have been obtained directly from the formalism developed in Section 6.1 by using the speci"c u-dependence of T B and the commutation of H with H M . The latter property, already established in the Remark of Section 5.4.2 for a more general case, can also
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be proven from Eqs. (6.14) and (6.15) once the u-independence (6.18) of the contraction matrix R B? is acknowledged. 6.3. The N-parity projected BCS model: generalities We now use the formalism of the previous section by specializing to the BCS-type pairing. In this case, the n creation operators aR can be arranged (possibly by means of a unitary transformation) in G a set of couples +(aR, aR ), 14p4n/2, such that the Hamiltonian (3.27) takes the form N N (6.34) HK " e (aRa #aR a )! G aRaR a a . NO N N O O N N N N N NO N This may encompass models for electrons in a superconductor (p then denotes the momentum and a spin pointing upwards, while p denotes the opposite momentum and spin), or for nucleons in a deformed nuclear shell (the couple momentum-spin is then replaced by the component of the total angular momentum along the principal axis of inertia). For the sake of simplicity we have not introduced a magnetic "eld, which would lift the degeneracy between the states p and p . We have also assumed that the diagonal matrix elements G are zero (a non-zero value of G entails a small NN NN state-dependent shift of the single-particle energy e , which leads to more complicated formulas but N does not modify the conclusions). Moreover, we consider the usual case where the quantities G are real and symmetric in the exchange pq. NO Remark. The reader who wishes to begin with this Section 6.3 without having followed the derivations of Sections 5, 6.1 and 6.2 can do so if he is ready to adopt the de"nitions and accept the expressions and equations that we now recapitulate. Our variational space was introduced in Section 6.2, Eq. (6.17), where the projection PK was given in Eq. (6.2) and where the trial operator E KK (see Eq. (5.44)) involved a c-number h (which will play no ro( le in the following Sections) and a 2n;2n hermitian matrix H . The contraction matrices R and R were de"ned in Eqs. (6.18)}(6.19) from the matrix H which is itself determined self-consistently by Eqs. (6.23) and (6.25). The matrix RM and the scalar quantities r , EM and k were given in Eqs. (6.20), (6.21), (6.22) E E and (6.26), respectively. The approximate projected partition function Z[ , entropy S[ and expectaE E tion values 1QK 2 were given in Eqs. (6.28), (6.30) and (6.31). 6.3.1. The variational space Within the BCS scheme the operator KK can be put into the familiar diagonal quasi-particle form
KK " h ! e # e (bRb #bR b ) , N N N N N N N N by means of the canonical transformation
(6.35)
b "u a #v aR , a "u b !v bR , N N N N N N N N N N (6.36) b "u a !v aR , a "u b #v bR . N N N N N N N N N N (Note that two di!erent sign conventions are currently used in the literature for this transformation.) The parameters u and v can be chosen real and non-negative, with u#v"1. The N N N N variational quantities are thus e , v and h . N N
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All the 2n;2n matrices that enter our formalism can then be decomposed into a number n/2 of 2;2 symmetric matrices labelled by p and corresponding to the operators a and aR , plus another N N set of matrices labelled by p which di!ers from the previous one through a change in sign of v . In N particular the p-components of the matrix H (Eq. (5.44)) are H "e ; , N N N
(6.37)
where ; is the real symmetric orthogonal matrix N
u!v N ;, N N 2u v N N
2u v N N . v!u N N
(6.38)
Accordingly, the p-components of the contraction matrix R (Eq. (6.18)) read
aRa 1 R " Tr ¹K B (b) N N N Tr ¹K B (b) aRaR N N
aa N N "(1!; t ) , N N a aR N N
(6.39)
where we have introduced for shorthand the notation t ,tanh b e . N N
(6.40)
Likewise, the matrix R (Eq. (6.19)) has the p-components R "(1!; t\) . N N N
(6.41)
The energy E+R ,, associated with the operator KK "HK !kNK and evaluated for Eq. (6.39), is now given by E+R ," (e !k)[1!(u!v)t ]! G u v u v t t . N N N N NO N N O O N O N NO
(6.42)
The corresponding matrix H +R , has the p-components N
e !k H +R ," N N D N
D N k!e
,
(6.43)
N
with D, G u v t . N NO O O O O
(6.44)
By substituting t\ to t in Eqs. (6.42) and (6.43) we obtain the energy E+R, and the matrix O O H+R,. In particular, D is replaced by N D, G u v t\ . N NO O O O O
(6.45)
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The c-numbers r (Eq. (6.22)) and k (Eq. (6.26)) are respectively equal to r " t , N N 1 k" (t\!t )+e !(e !k)(u!v)!u v (D#D), . N N N N N N N N N N 1#gr N
(6.46) (6.47)
6.3.2. The variational equations The trial parameters e and v must be determined from the self-consistent matrix equation N N (6.23), (6.25). By means of the canonical transformation (6.36), let us write this equation in the basis where H is diagonal. From the diagonal element of Eq. (6.23) we thus obtain e (1!gr t\)#2gr kt\ N N N " (u!v)(e !k)(1!gr t\)#2u v (D!gr Dt\) , (6.48) N N N N N N N NN and from the o!-diagonal element: 0"2u v (e !k)(1#gr t\)!(u!v)(D#gr Dt\) . (6.49) N N N N N N N NN Together with the preceding de"nitions of D, D, t , r and k, Eqs. (6.48) and (6.49) are the N N N self-consistent equations which determine the variational parameters e , u and v . N N N Let us further work out these equations. By introducing the weighted averaged pairing gap D#gr Dt\ NN , D , N (6.50) EN 1#gr t\ N and making use of the relation (u!v)#(2u v )"1, we can simplify Eq. (6.49) into N N N N 2u v u!v 1 N N" N N" , (6.51) D e !k e EN N N where e has the familiar BCS form N e ,((e !k)#D . (6.52) N N EN Except for the de"nition of D , Eq. (6.51) is the same as the usual BCS equation for the coe$cients EN u , v of the canonical transformation (6.36). As regards Eq. (6.48), it appears as an extension of the N N BCS relation e "(u!v)(e !k)#2u v D ; actually, we can rewrite Eq. (6.48) as N N N N N N N 2gr 2gr k e "(u!v) (e !k)#2u v D ! (D!D) ! , (6.53) N N N N N N EN t!rt\ N N t !gr t\ N N N N and hence, by using Eq. (6.51) to eliminate u and v , as N N 2gr k D (D!D) 2gr EN N N . e "e ! ! (6.54) N N t !gr t\ t!rt\ e N N N N N The equations above reduce to the usual BCS ones if g is replaced by zero. Then both D and N D become identical with the BCS gap D , while D becomes irrelevant. However here we have EN !1 N N
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g"1 for even systems, g"!1 for odd systems, so that several quantities D, D and D now occur. N N EN The quantities e are the energies which appear in the diagonal form of the density operator. Eqs. N (6.52) and (6.54) show that these e have no longer the BCS form. We can eliminate u and v from N N N the de"nitions (6.44), (6.45) and (6.50) of D, D and D . These quantities thus satisfy the equations N N EN D (t\!t ) 1 EO O O , (6.55) D!D" G NO N N 2 e O O 1 D t 1#gr t\ t\ N O . EO O D " G (6.56) EN 2 NO e 1#gr t\ N O O Eq. (6.56) exhibits the projection correction to the so-called BCS gap equation, while Eq. (6.55) obviously has no BCS counterpart. Finally, by using (6.53) and the identity u v (Dt !D t\)"0, we can write the expression (6.47) for k in a more explicit form: N N N NN N N t\!1 t\!t 1#gr t\ D (D!D) 2gr N N N EN N N . k 1# "! N (6.57) 1!gr t\ 2(1#gr ) 1!gr t\ e 1#gr N N N N N Altogether, we have to solve the self-consistent Eqs. (6.54) and(6.56) for the two sets e and D . N EN The quantity e is indirectly involved through t ,tanh(be /2). The two c-numbers r and k are N N N expressed by Eqs. (6.46) and(6.57), respectively. The di!erence D!D, which enters (6.54) and N N (6.57), can be eliminated by means of Eq. (6.55). The coe$cients u and v are given explicitly by N N Eq. (6.51).
6.3.3. Comments In Eq. (6.56) the last fraction, which takes care of the projection, can also be written as 1#(t\!1) f (gr t\) with f (x)"x/(1#x). The positive term (t\!1) di!ers signi"cantly from O N O zero only when be (1. For an even-number projection (g"1), the function f is positive and O varies between 0 and 1/2 while for an odd-number projection (!1(gr (0) f is negative and varies from !R to zero. The following conclusions can therefore be drawn: E in the sums of Eqs. (6.56)}(6.57), the modi"cations due to projection a!ect predominantly the terms corresponding to the single-particle states near the Fermi surface. E the even-number projected gap is larger than the BCS one while the odd-number gap is smaller. E larger di!erences with BCS are expected for the projection on an odd number of particles. These features will be con"rmed below, analytically (Section 6.4) and numerically (Section 6.5). In particular the fact that the odd-N projection produces more important e!ects than the even-N projection, especially at low temperatures, can be understood as follows. Consider the ordinary grand canonical equilibrium. The associated grand partition function Z"Z #Z is the sum of > \ the projected functions Z and Z de"ned by Eq. (6.1). In grand canonical equilibrium, the > \ probability of "nding an odd number of particles is Z /(Z #Z ), and we obtained for it \ > \ the approximation Z[ /(Z[ #Z[ ) where Z[ is given by Eq. (6.28). In the BCS grand canonical \ > \ E equilibrium, the ground state is built by putting either zero or two particles in each level (p, p ). Hence, in the zero-temperature limit, Z[ /Z[ tends to 0. More generally, at low temperature, Z[ is \ > \ smaller than Z[ since ln Z then behaves as !b(E!kN). The probability for odd N is then smaller > than that for even N, and it even vanishes as ¹P0.
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A technical consequence is the following. When building up an approximation for an odd-N system, one must carefully enforce the elimination of the even-N components. This was achieved in Section 6.3.2 since all our calculations for g"!1 are completely decoupled from those for g"#1. In contrast, Ref. [43] is based on the approximate evaluation of Z (by the BCS approximation) and of the staggered partition function denoted as Z , which is Z[ !Z[ in our > \ language. Two gap functions, D and D , are thus introduced, satisfying equations similar to Eqs. (6.44) and (6.45), respectively. The self-consistent equations which yield the values of D and D are then derived by optimizing independently Z"Z[ #Z[ and Z "Z[ !Z[ . Since at low > \ > \ temperature Z and Z are both dominated by Z[ , it is not surprising that D and D are close to > each other and become equal at ¹"0. The evaluation of Z[ as a di!erence between the \ independently optimized values of Z and Z is however not reliable. In our approach, we optimize directly the partition function Z[ for odd-N systems. This leads to the coupled Eqs. (6.50), \ (6.55) and (6.56) for D, D and D , whose results signi"cantly di!er from those of Ref. [43]. They N N \N underline the importance, within a variational approach, to optimize the quantity of interest itself. When the system becomes su$ciently large, both r and r k become small. Hence the above set of coupled equations reduces to the standard BCS equations. The projection on the particlenumber parity therefore produces e!ects only for su$ciently small systems. 6.3.4. Thermodynamic quantities Once Eqs. (6.54) and (6.56) are solved, the thermodynamics is governed by the Eqs. (6.30), (6.21) and (6.28) which yield S[ , EM and Z[ , respectively. To compute the approximation S[ given by Eq. E E E E (6.30) to the exact N-parity projected entropy S , we take into account the twofold degeneracy of E the eigenvalues (e!@CN#1)\ of the contraction matrix R . We "nd that S[ di!ers from the E independent quasi-particle entropy S , !1 b b (6.58) S ,S+R ," 2 ln 2 cosh e !be tanh e , N N !1 2 2 N N according to
gr be (t\!t )#ln[(1#gr )] . S[ "S ! (6.59) E !1 1#gr N N N N It results immediately from Eqs. (6.31), (6.39) and (6.51) that the expectation value of the particle number is given in terms of the chemical potential by
e !k t #gr t\ N N . (6.60) 1NK 2" 1! N 1#gr e N N This expression is also the derivative with respect to k of F "!ln Z[ /b"EM !S[ /b, which is our % E E E approximation to the logarithm of the exact N-parity projected grand-potential !ln Z /b. The E expectation value 1HK !kNK 2 is obtained from Eqs. (6.21) and (6.42) as
t #gr t\ N 1HK !kNK 2,EM " (e !k) 1!(u!v) N N N E N 1#gr N t t #gr t\t\ N O , ! G u v u v N O NO N N O O 1#gr NO
(6.61)
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while, using Eqs. (6.50), (6.51), (6.60) and (6.61), the energy 1HK 2 is given by
e (e !k)#D t #gr t\ N EN N . (6.62) 1HK 2" e ! N N N e 1#gr N N Here again, the deviation from the BCS energy is exhibited by the two terms that contain r . In accordance with the general properties of our variational scheme, the thermodynamic relations (2.28) and (2.29) are satis"ed by the approximate entropy S[ . The expectation values of the E single-particle observables are given by Eq. (6.31). The speci"c heat, as expected, is equal both to !bdS[ /db and to the derivative of 1HK 2 with respect to the temperature. In Eq. (6.62), the E temperature appears directly in t "tanh(be /2), and indirectly through e , D , r and also through N N N EN k since 1NK 2 should be kept "xed in the derivative. Finally, the general arguments at the end of Section 6.2 show that the characteristic function for any single-quasi-particle operator commuting with KK , such as NK , is variationally obtained by adding a source term to the operator KK . We "nd therefore that the #uctuation satis"es as it should 1 R1NK 2 , *N" b Rk
(6.63)
where in Eq. (6.60) the derivative should be taken both explicitly and through e , D and r . N EN A somewhat similar extension of the BCS theory, also involving a projection on the spaces with a given parity of the particle number, has been worked out in Ref. [22]. The results agree with ours for the partition function Z[ . The comparison between our variational Eqs. (6.48)}(6.49) and those E of Ref. [22] appears more di$cult. In particular it is not clear to us why the entropy found in Ref. [22] does not satisfy, in contrast to ours, the thermodynamic relations (2.27) and (2.28). On the other hand, the general discussion of Section 2 as well as Eq. (6.63) show that #uctuations and correlations calculated with our method must de"nitely be di!erent; for instance we "nd here that the #uctuations of the particle number tend to zero with the temperature, as expected from general grounds. 6.4. The N-parity projected BCS model: limiting cases At zero temperature, the BCS (and HFB) solutions for the ground and excited states are exact eigenstates of the number-parity operator, and their eigenvalues $1 depend on the number of quasi-particles added to the even BCS vacuum. We thus expect the ¹"0 limit of the even-N projected solution (g"#1) of Eq. (6.56) to coincide with the BCS ground state. We shall also see below that this same equation produces the lowest one-quasi-particle BCS state when g"!1. This odd-N BCS solution with lowest energy assigns a speci"c role to the twofold-degenerate individual level whose energy e is closest to the Fermi energy k. In this section we use the index p"0 to label any quantity pertaining to this level. For odd-N systems, we shall later on denote by the index p"1 the level with second smallest quasi-particle energy. The ¹"0 odd-N ground state is obtained by creating one p"0 quasi-particle on the BCS vacuum, that is by e!ecting in all sums the substitution (u , v )P(v ,!u ) for only one of the two quasi-particle states 0 or 0 associated with e . Under this transformation the product u v changes sign and the contributions of these
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two quasi-particle states to the gap cancel exactly. Thus for pO0, the odd-N gap equation is D G \O , (6.64) NO e O O O$ both in the odd-N BCS and projected cases. The only formal di!erence with the usual BCS equation, which at ¹"0 describes the even-N systems, is the suppression of the q"0 term in the sum. Nevertheless, through self-consistency, depending on the level density at the Fermi surface, this di!erence may lead to a signi"cant decrease of the pairing gaps that we shall investigate below both analytically and numerically. Since creating a quasi-particle with index p"0 in the BCS vacuum amounts to creating exactly one particle on the level p"0 or p "0 , the approximation above is usually referred in nuclear physics as the blocking approximation. Note that Eq. (6.64) should be solved with the constraint v"N!1, on the average occupancy of the levels N$ N pO0. To describe odd-N systems at "nite temperature, one could think of the following naive generalization of Eq. (6.64); we shall refer to it as BCS1. One of the two states at the Fermi surface (p"0 or p "0 ) is again assumed to be occupied exactly by one particle while the remaining N!1 particles are a!ected by the pairing correlations. The occupations of the pO0 levels are then calculated according to the standard "nite-temperature BCS procedure. The temperature dependence enters the equation for the pO0 gaps only through the usual thermal factor. These prescriptions lead to the equation: 1 D " \N 2
D !1 Ot . G (6.65) NO e N O O O$ In this BCS1 approximation, the unpaired particle which occupies the level at the Fermi surface forbids the formation of the pair (0, 0 ). In contrast, at "nite ¹, our odd-N projected equation (6.56) does not compel any speci"c level to be blocked by the unpaired particle. As the temperature grows, the statistical #uctuations make it possible that other levels than p"0 be concerned. We shall see in Section 6.5.1 (Fig. 4) that this can lead to sizeable di!erences with Eq. (6.65). However, we show below how, at ¹"0, Eq. (6.64) is recovered from Eq. (6.56) when g"!1. D
1 " !1N 2
6.4.1. The extreme low temperature regime In this section, we consider the situation where the single-particle level spacing at the Fermi surface is at the same time larger than the temperature and smaller than the ¹"0 pairing gap. Nuclei and small metallic grains with pairing provide physical illustrations of such a situation. We defer to the next Section the analysis of the case when the single-particle level density is much larger than b and such that it is legitimate to replace discrete sums on level indices by integrals weighted by the level density. In Section 6.5.4, we shall consider still another situation in which the single-particle level spacing and the BCS pairing gap are almost equal. Let us "rst recall the low-temperature limits of some quantities and equations of interest. For large values of b, the usual BCS gap equation becomes D
D 1 !1 O (1!2e\@CO) . " G NO e !1 N 2 O O
(6.66)
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In Eq. (6.66) the terms proportional to exp(!be ) lead to a decrease of the gaps versus the O temperature ¹"1/b. In the same limit, the BCS entropy (6.58) is approximated by "2 be e\@CN , !1 N N while the particle-number average (6.60) is given by S
(6.67)
e !k (1!2e\@CN) . (6.68) " 1! N !1 e N N In the N-parity projected extension of BCS, the quantity r (Eq. (6.46)) plays a central ro( le. In the low-temperature limit considered here, its value tends towards 1 and one has 1NK 2
ln r "!4 e\@CN , (6.69) N up to terms equal to, or smaller than exp(!3bD ). (However, for some odd-N systems, it can E happen that r vanishes with the temperature because one term of the product exactly vanishes; see Section 6.5.4.) Even particle number. Using Eq. (6.69), the expansion of Eq. (6.56) for g"#1 yields
1 D (6.70) D " G >O 1!2e\@CO!4 e\@CO>CP . NO e >N 2 O O P P$N O At ¹"0 Eqs. (6.66) and (6.70) are identical. In Eq. (6.70), the terms which control the lowtemperature corrections to the gaps are also negative. However, they are now of the order of, or smaller than, exp(!2bD ) while they were of the order of exp(!bD ) for BCS (near ¹"0, > !1 D KD ). This re#ects the fact that any excitation with "xed N-parity requires the creation of > !1 at least two quasi-particles and therefore an energy larger than 2D . Therefore the decrease of the > pairing gaps with increasing temperature will be slower than for the BCS case. The low temperature behaviour of the entropy (6.59) is given by (6.71) S[ "2 be e\@CN#2 b(e #e ) e\@CN>CO . N O > N NON$O N Like for all other thermodynamic quantities, the ¹-dependence is governed by terms smaller than exp(!2bD ). For instance, for the average particle number, the limit of large b is > e !k 1NK 2" 1! N 1!2e\@CN!4 e\@CN>CO . (6.72) e N N O O$N The analysis of Eqs. (6.54), (6.55) and (6.57) shows that, at ¹"0, the energy e of a state in the N vicinity of the Fermi surface is smaller than its BCS counterpart. For instance, assuming that all the matrix elements G (pOq) have the same value GI , one "nds NO D D D GI D > (1!d ) 3 >N! > #d > . (6.73) e "e ! N N e N N 6 e e e N
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In fact, in the numerical analysis of Section 6.5, we shall "nd that e and e di!er by only few N N percent. Odd particle number. For N odd (g"!1) and ¹ small, we must treat separately the gaps D associated with the quasi-particles pO0 and the gap D associated with the lowest \N \ quasi-particle energy. Let us "rst consider Eq. (6.56) for pO0. At low temperature both the numerator and the denominator in the right-hand-side fraction vanish as exp(!be ). After dividing out by this common factor, we obtain the well-conditioned equation:
D D D \O# G \!G \O e\@CO\C . G (6.74) NO e N e NO e O O O O$ As expected, at ¹"0, Eq. (6.74) is identical with Eq. (6.64). As bPR, Eq. (6.56) automatically singles out the level with the lowest quasi-particle energy and generates the odd particlenumber BCS ground state. The blocking approximation thus emerges naturally in the zerotemperature limit. The low-temperature correction to the gap D is positive and depends on terms \N which are proportional to the exponential of the di!erence e !e . Therefore, starting from the O value provided by Eq. (6.64), one expects the gap to "rst increase with temperature as exp(!b(e !e )). The case p"0 is slightly more complicated. One "nds that the temperature corrections involve no longer e but the second smallest quasi-particle energy e , the corresponding gap D and the \ degeneracy g : 1 D " \N 2
D 1 D 1 D D 1 \# \ e\@CO\C . G \O! G \O!G (6.75) D "! G O O \ e 2 e g e e 2 O O O O$ The ¹"0 limit of this equation provides an expression of D in terms of the D (qO0). \ \O However, D does not enter the equation (6.64) which, at ¹"0, determines all the other gaps, and \ it is thus relevant only at non-zero temperature. Despite their formally di!erent expressions, we shall "nd in the numerical application performed in Section 6.5 that the values of D and D are \ \N equal within a few percent. The low-temperature entropy (6.59) behaves as (6.76) S[ "ln 2# b (e !e ) e\@CN\C . \ N N N$ When bPR, the limit of S[ is equal to ln 2. This value re#ects the twofold degeneracy of the \ ground state. For the average number of particles, the low temperature limit of Eq. (6.60) yields
e !k 1! N (1!2e\@CN\C) . (6.77) 1NK 2"1# e N N N$ In contrast to the corresponding BCS formula (6.68), the temperature correction depends on the di!erences e !e . N
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Using Eqs. (6.54), (6.55) and (6.57) one "nds that, at ¹"0, e is larger than e . If one assumes N N again that all the matrix elements G (pOq) are equal to the same value GI , one obtains NO D GI D \ . (6.78) e "e # (1!d ) \N#d N e N e N N 2 N To derive this formula, we have taken g "2 as in the numerical examples presented in Section 6.5. In these applications, we shall also "nd that the magnitude of e !e is small (a few percent). N N
6.4.2. The condensed matter low temperature regime In the physics of mesoscopic superconductors, one often considers a situation in which the twofold degenerate unperturbed energies e are su$ciently dense near the Fermi surface to allow N the replacement of a sum over p by an integral over e,e !k with a weight w equal to the level N $ density. One also assumes that the pairing matrix elements are constant and equal to GI over an interval K around the Fermi energy k, and that they vanish outside. The width of this interval, taken of the order of the Debye energy, is assumed to be much larger than the pairing gap. It must be "nite, however, to ensure the convergence of the integrals. The ordinary BCS gap equation at zero temperature
K
w de $ , (6.79) \K (e#D then de"nes a p-independent quantity D that we shall take in this section as a reference energy. In this "eld of physics, the `low-temperaturea regime actually refers to an intermediate range of temperatures such that the conditions 2 " GI
1 1 ; ;b;w $ K D
(6.80)
are valid. Under these conditions Eqs. (6.66) and (6.67) give, after elimination of the cut-o! K by means of Eq. (6.79), analytical expressions for the low temperature dependence of the BCS pairing gap: D
!1
(b)"D 1!
2n e\@D , bD
(6.81)
and of the BCS entropy: (6.82) "2 w D (2nbD e\@D . !1 $ We shall check below (Section 6.5.2) that, in the limit w D<1, the pairing gap D obtained with $ EN the N-parity projection does not depend much on p and that it is almost identical with D (b). This !1 will be illustrated by the numerical results plotted in Fig. 5. Moreover, we shall see that the di!erence between e and e (e)"((e !k)#D K(e#D is small. Under the conditions N N EN (6.80), the approximation S
2nD be 3 D e\@CN\DKe\@C D 1# K d(e) 1# # d (e) , b 8D 8bD 2b
(6.83)
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holds. Hence the expression (6.69) becomes ln r "!4 w $
2n D 3 e\@D 1# . b 8bD
(6.84)
Even particle number. The gap equation (6.70) provides an analytical expression for the lowtemperature dependence of the even-N gap D (b): > w D (b)"D 1!4n $ e\@D . (6.85) > b
The extrapolation to zero temperature of D is equal to the BCS gap D. Under the same conditions, > one can rewrite the expression (6.71) as (6.86) S[ "8n (w D) e\@D . > $ The BCS entropy (6.82) is larger than S[ . Indeed, the disorder decreases when odd-particle > components are removed. Odd particle number. Here we cannot use Eqs. (6.74), (6.76) as a starting point since they are only valid when w 4b (while the validity of Eqs. (6.70), (6.71) rests on the condition 1/D;b). We must $ directly implement the conditions (6.80) and Eqs. (6.83)}(6.84) in the general equations (6.56) and (6.59). For the (p-independent) odd-N gap D (b), Eq. (6.56) can be reformulated as \ K w de 1 1 2 $ " ! # . (6.87) D (b) 2bD (b) GI K (e#D (b) \ \ \ \ The limit of this equation, in the regime (6.80), yields
1 1 D (b)"D 1! # . (6.88) \ 2w D 4bw D $ $ At zero temperature the odd-N gap D is therefore smaller than the BCS gap by the quantity \ 1/2w . This property has been shown to be related with the blocking of the p"0 level located at $ the Fermi surface, as described by the odd-N gap Eqs. (6.64) or (6.74). In the temperature range de"ned by the conditions (6.80), the gap grows linearly with temperature. In contrast, in the very low temperature regime investigated in Section 6.4.1 (b5w ), the gap grows exponentially as $ exp(!b(e !e ))Kexp(!b/2wD). $ In the regime de"ned by the inequalities (6.80) the odd-N projected thermodynamic quantities have a larger temperature variation than the corresponding BCS ones. This results from the fact that the excitation energies are then of the order of the inverse of wD and therefore much smaller $ than w\ and therefore D. $ For the entropy (6.59) we obtain
1 1 2n wD $ . S[ "ln 2# # ln \ b 2 2
(6.89)
It is the expression of the Sackur-Tetrode entropy (see Chap. 7, p. 323 of Ref. [8]) for one classical particle of mass D, with an internal twofold degeneracy, enclosed in a one-dimensional box
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of length 2n w . This result is natural since the elementary excitations concern only a $ single unpaired particle, with energy e KD#e/2D; if we identify e"e !k with the momentum N N of a particle in a box, the identi"cation of the level densities leads to the size 2n w for this $ box. While elementary excitations of even-N systems require energies of at least 2D, the e!ective spectrum for odd-N systems is nearly continuous, with the form m/2wD where m is an $ integer. This occurence of gapless elementary excitations is consistent with recent experiments on the conductance of odd-N ultrasmall aluminium grains [6]. The analogy with the Sakur-Tetrode entropy shows also that in the variational projected solution, contrary to the BCS1 one, the unpaired particle does not remain on the single-particle level of energy e "k. The projected solution is a coherent superposition of con"gurations in which this particle explores all singleparticle levels such that b"e !k"41. The fact that we recover a formula of classical thermoN dynamics is another indication that, strictly speaking, the regime associated with the conditions (6.80) is not a low temperature regime. Actually, Eq. (6.80) implies that the temperature is su$ciently large so that wD/b<1, which ensures the positivity of the entropy S[ as given by $ \ Eq. (6.89). Ordering of entropies. The inequalities S[ 'S 'S[ implied by Eqs. (6.82), (6.86) and (6.89) \ !1 > are easily understood. When the system is constrained to have an even particle number, we have seen that its lowest excited states are obtained from the ground state by the creation of two quasi-particles, which requires at least an energy 2D. For an odd system, the excitation of the (gapless) unpaired last particle is much easier, consistently with S[ 'S . \ !1 At non-zero temperature the BCS state is a weighted sum over even and odd components. The concavity of the entropy implies that S is larger than the smallest of the quantities S[ and S[ , !1 > \ that is S[ . The inequality S (S[ is made possible by the very small weight of the odd > !1 \ components entering the BCS state at low temperatures. The smallness of this weight also explains why, as already noted in Section 6.3.3, the odd-N projection a!ects the BCS results more than the even one. Indeed, we have met several qualitative di!erences between the properties of odd-systems and their BCS description, while BCS explains fairly well the even systems. This point will also be exhibited by the numerical applications of Section 6.5, especially on Figs. 4 and 9 which show the gaps at the Fermi surface as a function of temperature. The same considerations on the entropy also hold in the extreme low temperature regime of Section 6.4.1. Indeed, Eqs. (6.67), (6.71) and (6.76) again imply S[ 'S 'S[ . \ !1 > 6.4.3. Higher temperatures High temperatures relevant for superconductivity are of the order of the critical temperature, that is ¹ "0.57D. General conclusions on the behaviour of the equations can only be obtained when the density of single-particle states at the Fermi surface is signi"cantly larger than one, which entails w D<1. In this limit, the quantity t tends towards 0 for any value of p and q. Then, $ P$N O P Eq. (6.56) is equivalent to:
D 1 (6.90) D " G EOt 1#g(1!t) t . NO e O O P EN 2 O O P$N O This di!ers from the BCS equation by the factor within the braces, which deviates from 1 by a vanishingly small correction. The projected gap values converge (from above when g"1 or from
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below when g"!1) towards the BCS values as one reaches the critical temperature. In particular, the odd solution converges towards the standard BCS solution and not towards BCS1. In Section 6.5.4, we present a numerical case which does not pertain to the limit w D<1 and $ yields qualitatively di!erent results. 6.4.4. Large systems In large systems at non-zero temperatures, the product r decreases as the inverse of an exponential of the volume of the system. Then, an iterative solution of Eqs. (6.54),(6.56), (6.62) and (6.57) is attainable by starting from the BCS solution. Since in several places r appears multiplied by coth(be /2), the e!ect of the projection is surmised to be especially sizeable near the Fermi N surface. In this large system limit, the approximate projected entropy (6.59), for both values of g, di!ers from the BCS entropy by a constant: S[ "S !ln 2 . (6.91) E !1 This result shows that, for a given non-zero temperature, the even-N and odd-N grand-canonical states become equivalent when the size of the system increases. With respect to the BCS grand canonical state, the N-parity projection reduces the number of relevant microscopic states by a factor 2, thence the term !ln 2 in Eq. (6.91). The same feature occurs for a non-interacting system. Eq. (6.91) also shows that for large systems, pairing interactions a!ect the entropy similarly in BCS and N-parity projected grand-canonical states. The quantity k increases linearly with the volume, so that e !e is expected to be dominated N N by the second term in the r.h.s. of Eq. (6.54). Then, at the lowest non-trivial order, the quasi-particle energy can be expressed as D (t\!t ) D (t\!t ) gr t\ OY . EO O O EOY OY (6.92) e Ke # N G OOY N N e e 2 OY O OOY The self-consistency and the constraint on the particle number moreover shift k and D away from EN their BCS values. 6.5. The N-parity projected BCS model: a numerical illustration In this section we estimate the importance of the corrections to BCS introduced by the number-parity projection. To this aim, we have performed three schematic calculations that exemplify situations encountered in the physics of (i) superdeformed nuclear states around mass A"190, (ii) superconducting mesoscopic metallic islands and (iii) ultrasmall metallic grains. 6.5.1. Heavy nuclei We "rst consider an application to a "eld of nuclear structure physics which, recently, has been the subject of an intense experimental exploration. Superdeformed nuclear states are generally populated via fusion reactions between two heavy ions. Once the colliding nuclei have fused and a few neutrons have evaporated, the compound nucleus decays with a small probability (a few percent) into superdeformed metastable con"gurations via the emission of statistical c-rays. In
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heavy nuclei, the angular momentum dependence of the moment of inertia observed in superdeformed bands gives strong evidence for the presence of pairing correlations. Most of the presently available data concern the properties of superdeformed ground states or of the lowest excitations. However, it is expected that the next generation of c-arrays, in Europe and in the US, will soon allow investigations of the statistical behaviour of the spectrum and of the properties of heated nuclei. In view of the general scope of the present work, rather than performing a realistic nuclear calculation (see for instance Ref. [44]) that would require the solution of the full extended HFB equations written in Section 6.2, we prefer to investigate a simple model in which the nuclear context outlined above is used as a guidance for selecting the relevant parameter values. Thus we consider twofold degenerate single-particle levels of energies e , regularly spaced with a density N w "2 MeV\. Our active pairing space includes 21 levels and extends 5 MeV above and 5 MeV $ below the Fermi surface, so that the cut-o! K equals 10 MeV. The matrix elements of the interaction are taken to have the form G "GI (1!d ) with GI "0.23 MeV. For this value of GI , NO NO the BCS pairing gap D at zero temperature is 1 MeV, consistent with the data for the even heavy nuclei. In fact, it is partly because D is close to 1 MeV in medium and heavy nuclei that the MeV turns out to be a natural energy unit in nuclear dynamics despite the fact that binding energies are several orders of magnitude larger. For these values of the parameters w , K and GI (or equivalently D), we have solved the N-parity $ projected coupled equations (6.54) } (6.57) with g"$1 as function of the temperature. This provides us with the projected pairing gap D de"ned by Eq. (6.50). Although our interaction is EN very simple, it leads to state-dependent gaps for the BCS as well as for the projected cases. The numerics, however, shows that this state dependence is weak: we "nd that at all temperatures the variation of the gap D versus p never exceeds 9%. We thus content ourselves with plots of the EN ¹-dependent value of the gap associated with the quasi-particle having the lowest energy in either the even or odd system. Hereafter, we denote this quantity D where D "D for N-even projection E E > and D "D for N-odd projection; we denote moreover by D (¹) the ¹-dependent BCS gap E \ !1 associated with the lowest-energy quasi-particle (D (¹),D ). !1 E As a reference energy in all the "gures of the present section, both for the temperature in abscissa and for the ¹-dependent projected gaps D in ordinates, we take the zero-temperature BCS gap E associated with the lowest energy quasi-particle which we denote D (D,D (¹"0)). !1 We "rst consider (upper part of Fig. 4) the even-number case with 1NK 2"20. (Here, 1NK 2 should not to be confused with the numbers of neutrons and protons which, for a heavy nucleus, are of the order of 120 and 80, respectively. 1NK 2 corresponds to the particles within the active pairing space; it does not take into account the nucleons of the fully occupied levels whose energies are lower than k!K/2.) We have plotted the temperature dependences of the BCS gap D and of the projected !1 gap D . As expected from the discussion of Eq. (6.56) in Section 6.3.3, the even-N projected gap is > always larger than the BCS one. The two curves meet at ¹"0 and also at the critical temperature ¹ "0.57D, consistently with the analysis of Section 6.4.3. The lower part of Fig. 4 presents results for the odd-number case with 1NK 2"21. It compares the temperature-dependent gaps obtained with our odd-N projected method, with BCS and with BCS1. As explained in Section 6.4, in the BCS1 option the level p"0 (with e "k"0) is always occupied by exactly one nucleon and therefore not available for pairing. We have therefore chosen to show in all three cases the gap D (g"0, $1) associated with the lowest-energy quasi-particle E
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Fig. 4. Comparison of BCS and number-parity projected gaps at the Fermi surface versus temperature for w D"2, $ w K"20 and 1NK 2"20 or 21. These parameter values are illustrative of the physics of heavy nuclei. The upper $ (respectively lower) part of the "gure corresponds to an even (respectively odd) average number of particles. With respect to BCS, the pairing gap is enhanced by even-N projection and (more strongly) reduced by odd-N projection, except near ¹ . For the BCS1 curve, see Sections 6.4 and 6.5.1. The odd-N projected gap increases with ¹ up to 0.4D.
whose index is distinct from 0. The odd-N projected curve interpolates the BCS and BCS1 ones. In agreement with the discussion in the introduction of Sections 6.4 and 6.4.1, this curve meets the BCS1 curve at ¹"0 where they both yield the correct result. As discussed in Section 6.4.3, the projected curve terminates at the same critical temperature as BCS. The failure of BCS at low temperature is associated with the small weight of the odd-N components in the BCS state (see Sections 6.3.3 and 6.4.2). The BCS1 approximation, by enforcing the odd parity at zero temperature, approaches at low temperature the projected state better than does BCS. On the other hand, when the temperature increases, the even and odd components tend to have equal weights while the di!erences between even and odd systems fade out. In that limit the BCS1 approximation becomes inadequate since it prevents the formation of a pair in the level e . This inhibits a correct description of pairing correlations which, at high temperature, take place coherently on all levels in the energy range of the gap for the odd as well as for the even systems.
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When the temperature grows, the odd-N projected gap begins to increase as expected from (6.74)}(6.88), then reaches a maximum, decreases and converges asymptotically towards the BCS curve near the critical temperature. At low temperatures, there is a signi"cant di!erence between the usual BCS gap D and D . In nuclear physics, this di!erence should be taken into account when !1 \ mass di!erences between odd and even neighbour nuclei are used to extract pairing gap values. In the numerical example of Fig. 4, the condition w D<1 is not satis"ed. It is therefore not $ surprising that the results do not comply with Eqs. (6.85) and (6.88) derived under the conditions (6.80). When gaps and temperatures are measured in D units, the BCS curve is universal up to small di!erences resulting from the details of the spectrum. For instance, the critical temperature always comes out equal (with an accuracy better than 1%) to the value 0.57D which is derived for a continuous spectrum with constant level density. In contrast, as anticipated in Section 6.4, the N-parity projected curves show more diversity. In our model, their characteristics are mostly determined by the product w D and to a lesser degree by the product w K and the ratio K/D. The $ $ quantity w D, `the number of Cooper pairsa, is the number of twofold degenerate single-particle $ levels in an energy interval equal to the gap, while w K is the number of levels in the active pairing $ space. In the nuclear physics case of Fig. 4 we have taken w D"2 and K/D"10, the number w K $ $ of levels being 21. The data corresponding to condensed matter physics are expected to be quite di!erent. 6.5.2. Mesoscopic metallic islands For superconducting materials, K is of the order of the Debye energy while D is of the order of the critical temperature, so that K/D is larger than above. On the other hand, depending on the size of the metallic islands or grains, such as those considered in recent experiments [3}5,45], the product w D can vary between 0.5 and several thousands. $ Let us "rst consider the case w D<1, which corresponds to most of the experimental studies $ performed on superconducting metallic islands. In this limit the e!ects of the N-parity projection on the ¹-dependent pairing gaps D are weak. In order to be able to visualize them, we have ! considered in Fig. 5 not too large a value of w D, taking w D"10 with K/D"20 and hence $ $ w GI "0.34. Thus, the active space comprises 201 twofold-degenerate levels. The average particle $ numbers of the even and odd systems are 200 and 201, respectively. The BCS curve and the even-N projected curve D are almost indistinguishable. In agreement with the discussion of the projected > gap equation in Section 6.3.3, the odd-N projected gap di!ers from the BCS value more than the even-N. At low temperature, it increases and is represented with good accuracy by the linear ¹-dependence of Eq. (6.88). In Fig. 6 we have plotted the entropy of the odd system calculated in four di!erent ways: with or without (i.e. GI "0) pairing interaction and with or without projection on number parity. The odd-N projected entropy S[ tends to ln 2 when ¹P0; above the critical temperature it joins the \ non interacting odd-N projected entropy. In contrast, the BCS entropy vanishes at small temperature; above the critical temperature, it joins the grand-canonical entropy of the non-interacting system. For ¹ larger than 0.35D the odd-N projection lowers the BCS entropy, according to S !S[ " ln 2. In contrast, at small ¹, one has S[ <S and the projected entropy !1 \ \ !1 S[ approaches the Sackur-Tetrode regime (Eq. (6.89)). Although we are working with a discrete \ spectrum, for ¹/D50.1, the grand-canonical unprojected entropy of the non-interacting system is
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Fig. 5. Comparison of BCS and N-parity projected gaps at the Fermi surface versus temperature for w D"10, $ w K"200 and 1NK 2"200 (upper part) or 201 (lower part). Such parameter values are illustrative of superconducting $ mesoscopic islands. The e!ects are qualitatively the same as in Fig. 4 but weaker. Even}odd e!ects disappear above ¹ "0.3D. "
almost undistinguishable from the expression 2n S" w ¹ , 3 $
(6.93)
obtained by assuming a continuous spectrum with constant level density. The only e!ect of the odd-N projection is again a lowering of the entropy by ln 2. In Fig. 7 we show the specixc heat C "d1HK 2/d¹"¹dS/d¹ for the odd system. With or without 4 projection, above the critical temperature ¹ "0.57D, the speci"c heat of the interacting system follows the linear form (6.93). Both the critical temperature and the magnitude of the discontinuity of C are the same for the BCS and odd-N projected curves. The low temperature behaviour, 4 however, is a!ected by the projection. The enlarged scale inset in Fig. 7 shows that the BCS speci"c heat drops to zero at low temperature, while the odd-N projected curve approaches the value expected for a single classical particle moving in a one-dimensional box (see Section 6.4.2). As mentioned in Section 6.3.3, the odd-N speci"c heat C takes this value only over the intermediate 4
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Fig. 6. Entropy of an odd-number system for w D"10, w K"200 and 1NK 2"201. The solid curve corresponds to the $ $ variational solution with projection on an odd number of particles. Around ¹"0.1D it has a Sakur}Tetrode form and it tends to ln 2 for ¹P0. The dashed curve (BCS) is obtained with the standard "nite-temperature BCS formalism. The dashed-dotted and dotted curves are obtained by dropping the interactions. The latter unprojected non-interacting case coincides with the entropy of the Fermi gas.
range of temperatures where the quasi-particle spectrum e is well approximated by a quadratic N function of e !k. In Section 6.5.4 we shall investigate in more details its zero-temperature limit. In N the absence of pairing interaction C does not depend on whether odd-N projection is or is not 4 e!ected, and it is again given by the linear function (6.93). 6.5.3. The free-energy diwerence We now consider the di!erence dF between the free energies of the odd and even systems, a quantity which is experimentally accessible [45]. In our formalism this di!erence can be reached by means of the approximate free energies obtained by the Legendre transform, F (1NK 2)"F (k)#k1NK 2 , (6.94) E %E of the N-parity projected BCS grand potentials F "!b\ln Z[ given by Eqs. (2.27) and (6.28). %E E The free energies F do not pertain to canonical ensembles with non-#uctuating particle number, E but to N-parity projected grand canonical ensembles. Nevertheless, we expect that dF does not di!er signi"cantly from F (1NK 2"N#1)!F (1NK 2"N). \ > At low temperatures, the experimental projected free-energy di!erence dF decreases linearly (see Fig. 5 of Ref. [45]) and tends to zero for a temperature lower than the critical temperature. Let us see how this behaviour can be understood within our formalism. First, using the Legendre transform (6.94), we numerically calculate dF from the odd-even grand potential di!erence for given values of ¹ and k, 1 dF ,F (k)!F (k)"! (ln Z[ !ln Z[ ) , \ > % %\ %> b
(6.95)
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Fig. 7. Speci"c heat of an odd-number system for w D"10, w K"200 and 1NK 2"201. Symbols and abbreviations are $ $ the same as in Fig. 6. The solid curve reaches the classical value for one degree of freedom near ¹"0.1D while the BCS value tends to zero, as exhibited by the inset. The projected and unprojected curves with no interaction are identical and have the linear behaviour of the Fermi gas.
with Z[ given by Eq. (6.28). Fig. 8 shows the variation with temperature of dF, for the same E parameters w D"10 and K/D"20 as in Section 6.5.2 (we have taken as our unit scale D the value $ for the odd system 1NK 2"201, which di!ers from the even system value by less than 0.1%). In agreement with experiment, one observes that the projected and BCS curves are identical for temperatures above 0.4D while, at lower temperatures, dF decreases almost linearly from a zerotemperature value which, by extrapolation, can be estimated to lie close to D. In the BCS case, for a "nite system in grand-canonical equilibrium, the free energy F (1NK 2) is !1 a function of the expectation value 1NK 2 which may vary continuously from 1NK 2"N"200 to 1NK 2"N#1"201. Moreover, RF /R1NK 2"k(1NK 2) is the BCS chemical potential as func!1 tion of 1NK 2 at the considered temperature. Accordingly, we can obtain the di!erence dF "F (N#1)!F (N) by integration of k(1NK 2) between N and N#1. In the !1 !1 !1 present model with N#1 equidistant twofold degenerate single-particle levels, for symmetry reasons, k(N#1) is equal to the energy of the level at mid-spectrum (101th level) which we take equal to zero. The Fermi level k(1NK 2) lies halfway between two consecutive levels for even 1NK 2, and coincides with a level for odd 1NK 2 provided the occupation number is close to 1 (resp. 0) at the bottom (resp. top) of the active space. Finally, when w ¹<1, or w D<1, k(1NK 2) is expected to $ $ vary smoothly, without oscillations. Hence we "nd 1 (1NK 2!N!1) , (6.96) k(1NK 2)K 2w $ from which we derive the BCS di!erence of free energies: ,> 1 k(1NK 2) d1NK 2K! . (6.97) dF " !1 4w , $
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Fig. 8. Di!erence between the free energies of neighbouring odd (1NK 2"201) and even (1NK 2"200) systems as function of temperature for the same parameter values w D"10 and w K"200 as for Figs. 5}7. The solid and dashed curves $ $ correspond to self-consistent calculations with and without projection on N-parity, respectively. Above the crossover temperature ¹ "0.3D, the even}odd e!ect disappears, as for D in Fig. 5. Below ¹ the projected di!erence dF decreases " E " nearly linearly. It lies close to the dashed-dotted curve which displays the low-temperature perturbative approximation (6.99).
At this level of approximation one "nds that dF is temperature independent, a result supported !1 by the numerical evaluation, as illustrated by Fig. 8. It is also possible to obtain an analytic expression of dF for temperatures in the range (6.80). We "rst calculate dF perturbatively around the BCS solution by means of Eqs. (6.28) and (6.84). If we % neglect the contribution r (k #k ), we obtain \ > 1 1#r ¹ ¹ dF K1! ln 8n(w D) %K ln . (6.98) $ bD 1!r 2D D D In order to relate dF to dF by means of Eq. (6.94), we have in principle to control the variation of % k associated with the addition of one particle and the change of N-parity. However, we note that the derivative with respect to k of the approximate expression (6.98) is practically zero. (Indeed, RD/Rk" "0 for symmetry reasons.) This means that the relation between 1NK 2 and k is I the same for the even-N and odd-N projected solutions. Hence this relation is also the same as for BCS (Eq. (6.96)). Since k vanishes for 1NK 2"N#1, the di!erence dF coincides with % F (N#1)!F (N#1), as shown by Eq. (6.94). The extra contribution F (N#1)!F (N) to dF \ > > > is then evaluated by integration of RF /R1NK 2"k(1NK 2) as in Eq. (6.97). We "nd: > 1 ¹ 1 KD! ! ln(8nD w ¹) . (6.99) dF"dF ! $ % 4w 4w 2 $ $ This expression is represented on Fig. 8 by the curve labelled `proj. pert.a. It is very close to the full solution of the projected equations up to ¹"0.3D.
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Thus, our results agree with the observation [45] that, when the spacing between single-particle levels is small compared to the gap (w D<1), N-parity ewects disappear above some temperature $ ¹ which lies below the critical temperature. Actually, Fig. 8 shows that for w D"10 a sharp " $ crossover takes place around ¹ "0.3D (that is, around 0.5 ¹ ) between the N-parity dependent " and independent regimes. Below this value, projection strongly modi"es dF, while parity e!ects disappear above. Figs. 5}7 exhibit the identity of the various BCS, even-N and odd-N projected results above ¹"0.4D. More generally, for arbitrary values of w D much larger than 1, the di!erence dF is given at $ low temperature by the approximation (6.99), at higher temperature by the value !1/(4w ) $ associated with the same trivial parity e!ect than for a non-interacting system or for the BCS approximation. The crossover between the two regimes takes place around the solution of the equation 1 ¹ "K . D 8n ¹ " ln w D $ D
(6.100)
This crossover temperature is also visible on Fig. 5, which shows that D (¹) is the same for BCS and E for the two projected cases in the range ¹ (¹(¹ , where ¹ K0.3D for w D"10. " " $ When the size of the system decreases, w D also decreases. The crossover temperature ¹ thus $ " rises; from Eq. (6.100) one "nds that it reaches the critical temperature ¹ "0.57D for w DK1.5. $ Thus, if the distances between levels become comparable to the gap, one can expect that odd-even e!ects will take place over the entire range of temperatures where pairing occurs. Physical systems which satisfy this condition seem presently available since one can now prepare and study ultrasmall metallic grains [4,5]. 6.5.4. Ultrasmall metallic grains As an example of modelisation of ultrasmall metallic grains, we have chosen the parameter values w DK1.1 and w K"100. These are obtained for w GI "0.23, and for an active space $ $ $ having 101 twofold degenerate levels. For the odd (resp. even) system we take 1NK 2"101 (resp. 100). In Fig. 9 we present the gap D as function of the temperature for odd and even systems. The E critical temperature associated with the even-N projection comes out somewhat larger than the BCS value. Even more striking is the behaviour of the odd-N projected curve. Then the gap is zero at ¹"0 but takes a non-zero value above some xrst transition temperature up to a second one, lower than the BCS critical temperature. This peculiar behaviour can be explained as follows. At ¹"0, the unpaired particle occupies one of the two single-particle states e "k with a probability exactly equal to one. This blocks the formation of pairs utilizing the two degenerate states with energy e and creates an e!ective single-particle spectrum with a distance between the levels at the Fermi surface which is twice as large as D. Consequently, pairing is inhibited. When the temperature grows, the bachelor particle statistically populates other single-particle states, as indicated in the discussion above on the entropy (6.89). This allows pairing correlations on the strategic level at the Fermi surface (e "k), provided the temperature is not too low. One thus observes a reentrance ewect, with two transition temperatures: one at which the pairing switches on and, at a higher temperature, the usual one at which it switches o!.
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Fig. 9. Comparison of BCS and N-parity projected gaps at the Fermi surface versus temperature for w D+1.1, $ w K"100, 1NK 2"100 (upper part) or 101 (lower part of the "gure). Such parameters are illustrative of extremely small $ superconducting grains. The critical temperature is raised by even-N projection, lowered by odd-N projection. In the latter case, pairing disappears below a new, lower critical temperature 0.15D, thus displaying a reentrance phenomenon.
The entropy of the even system 1NK 2"100 is shown in Fig. 10. Projection strongly reduces the entropy below the critical point. This projection e!ect is seen to be more important for the interacting system. At small ¹, the four entropies vanish exponentially. As expected the decrease rate of the projected entropy is the fastest. For ¹'0.3D, the grand-canonical entropy of the non-interacting system agrees with the linear form of Eq. (6.93). It bends down below ¹"0.3D under the e!ect of the spectrum discreteness; in the domain ¹'0.3D, projection again induces only a lowering by ln 2. Fig. 11 displays the entropy of the odd system 1NK 2"101. When the pairing interactions are present, the BCS formalism, which only constrains the average number of particles, yields the result 0 instead of the right zero-temperature limit ln 2. This correct limit is obtained by the odd-N projection. When there is no interaction, for ¹'0.3D, the grand-canonical value (dotted curve) is again described by Eq. (6.93). However the entropy tends to ln 4 at small ¹. This comes from the fact that, at ¹"0, the Fermi energy exactly coincides with the energy of the last occupied level
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Fig. 10. Entropy of an even-number system for w D+1.1, w K"100, 1NK 2"100. The symbols and notations are the $ $ same as in Fig. 6. Projection strongly reduces the entropy.
Fig. 11. Entropy of an odd-number system for 1NK 2"101 and the same parameters w D+1.1, w K"100 as in Figs. $ $ 9 and 10. The symbols and notations are the same as in Fig. 6. The projection lowers the entropy above ¹ , but raises it at lower temperatures with the limit ln 2 at ¹P0.
whose average occupation is equal to 1/2. When this value is inserted in the standard formula for the grand-canonical entropy of a non-interacting fermion system (while taking into account the twofold level degeneracy), one obtains ln 4 instead of the expected value ln 2. Projection on odd-N parity decreases the entropy by ln 2 and restores the correct limit. Note that this is not a trivial
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result since the exact particle number is not restored by the projection; only its parity is. In Fig. 11 the reentrance phenomenon manifests itself by the fact that the entropy of the interacting system is larger than that of the non-interacting system over the temperature interval 0.14(¹/D(0.34; otherwise the e!ect of the interactions is rather small. The original features associated with the N-parity projected mean-"eld approximation when w D is close to one are even more visible on the diwerence of the specixc heats of neighbouring odd $ and even systems. In Fig. 12 one sees that the projected curve shows discontinuities for the even-system critical temperature, and also for the two odd-system critical temperatures. The comparison with the same curve with w D"10 shown in the inset of Fig. 12, which corresponds to $ the second derivative of the curves on Fig. 8, displays the striking qualitative change arising from the variation of w D over one order of magnitude. $ 6.6. Discussion We shall resume in the general conclusion the various results obtained above. We focus here on the change in the pairing properties when the size of the systems decreases. The dimensionless parameter characterizing this size appears to be w D, which takes values 10 in Fig. 5 (metallic $ islands), 2 in Fig. 4 (heavy nuclei) and 1.1 in Fig. 9 (aluminum grains). From this standpoint the pairing properties of heavy nuclei (Section 6.5.1) are intermediate between those of metallic
Fig. 12. Di!erence dC between the speci"c heats of neighbour odd and even systems 1NK 2"101 and 1NK 2"100 4 (multiplied by the ratio of the zero-temperature BCS gap D over ¹) as function of temperature for the same parameters w D+1.1, w K"100 as in Figs. 9}11. The solid and dashed curves correspond to self-consistent calculations with and $ $ without projection on particle-number parity, respectively. The critical temperature for the even system and the two critical temperatures for the odd system show up as discontinuities. The curve di!ers qualitatively from that of the inset, which for 1NK 2"200 and 1NK 2"201 corresponds to the parameters w D"10, w K"200 of Figs. 3}8. In this inset, the $ $ dashed, dashed-dotted and dotted curves are indistinghishable from the temperature axis.
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mesoscopic islands (Section 6.5.2) and those of ultrasmall aluminium grains (Section 6.5.4). We recall that D denotes the zero-temperature BCS gap associated with the lowest-energy quasiparticle, that is, D,D (¹"0). !1 Our analysis shows that even-N systems are sensitive to the parameter w D at temperatures not $ too far from the BCS critical temperature, and that their projected gap D (¹) is slightly stronger > than the BCS gap D (¹). The discrepancy with BCS is stronger for odd-N systems; as measured !1 by D (¹)/D, it signi"cantly increases when the size (i.e., w D) decreases. The di!erence is especially \ $ important when the temperature is much lower than D, since the p"0 level is completely blocked at ¹"0. Accordingly, for su$ciently low temperatures, the ratio D /D increases with ¹ as \ a consequence of the thermal excitation which depopulates this p"0 level (Figs. 4 and 5). The zero-temperature value of D /D decreases with w D: thus, for "xed D, odd nuclei or odd supercon\ $ ducting metallic grains display a weaker and weaker pairing gap D as their size decreases. When \ w D becomes su$ciently small, the pairing fades out, "rst at ¹"0. Then, for still smaller odd $ systems, the temperature domain over which pairing correlations are e!ective shrinks (Fig. 9) and it eventually disappears at a "nite temperature for odd-N systems. In contrast, for even systems whatever their size, D /D remains equal to 1 at zero temperature. > Of course, as suggested long ago by Anderson [46], superconductivity disappears at any temperature for a "xed coupling constant when the size of the system } directly re#ected on the level spacing } is su$ciently small (the BCS gap itself then vanishes at ¹"0 and hence at all temperatures due to the discreteness of the spectrum). As already pointed out by several authors [24}28] this disappearance (as characterized by the projected gap D ) takes place for larger sizes E when N is odd than when N is even. In both the BCS and N-parity projected results, the sharp transition from the superconducting to the normal phase arises because of the invariance-breaking mean-"eld nature of the approximation. In reality, the "nite size of the system prevents any sharp transition, and more elaborate treatments are required to determine to which extent the discontinuities that were found in the present section are smoothed out. Various methods have been devised to study this broadening. Several e!ects were already taken into account in Ref. [47], including #uctuations of the order parameter in the Ginzburg-Landau theory and quasi-particle contributions associated with discrete spectra. A revival of such ideas can be found in [26], where path integral techniques are used to account for the #uctuations, and in [16,17], where the static path approximation is combined with the projection on the particle-number, or on the N-parity, with applications to nuclear physics. It does not seem easy to merge these techniques with our variational approach so as to render the treatment of Section 6 more realistic around the transition lines. Moreover, some of them do not apply at low temperatures. Already proposed in [47], the projection on a well-de"ned particle number seems better suited to our purpose. One can surmise that it should account, at least partly, for the #uctuations and "nite-size e!ects which smooth out the transition. This conjecture is supported by the work of Ref. [14], where the rounding-o! of the BCS transition produced by the projection on the particle number was quantitatively exhibited on an exactly soluble model. The approximation is based on the minimization of the free energy after projection. However, the use at non-zero temperature of this standard variational principle requires the evaluation of the entropy of the projected independent quasi-particle state. The di$culty of such a calculation led the authors of Ref. [14] to treat only a degenerate BCS Hamiltonian of the form (6.34) with e "0 N
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for all p (with a constant G ). Another indication may be given in the zero-temperature nuclei by NO the smoothness of the superconducting to normal phase transition which is experimentally observed when the angular momentum increases and theoretically explained through the projection method (see for instance Refs. [12,44]). Returning to our problem and assuming that variational approximations in the canonical ensemble can describe the most important e!ects of the transition broadening, we can imagine to apply the method of Section 5 with the projection on the particle number (Eq. (5.2) rather than on the N-parity as we did in Section 6. This approach seems promising since our variational principle (5.16) avoids the explicit calculation of the trial entropy. It introduces instead the two u-dependent trial quantities DK (u) and AK (u) whose evolution must be determined over the interval [0, b]. The fact that the variation of I is performed after projection should, as suggested by the results of Ref. [14], entail the products u v to be non-zero at any temperature, thus producing N N a smooth transition. The projection on a well-de"ned particle-number should have a negligible incidence on the results obtained for mesoscopic metallic islands where 1NK 2 is large and hence has small relative #uctuations. In nuclei, on the other hand, the smoothing of the superconducting to normal phase transition due to the "nite number of neutrons and protons should be signi"cant; nevertheless the increase of D with ¹ occuring in odd-N systems (Fig. 4) may be real. For very small metallic \ grains, for which the smoothing e!ect is expected to be the largest, it is an open question whether or not the reentrance phenomenon, as it is for instance revealed in Fig. 12 by a maximum in the curve for the di!erence of speci"c heats, will survive.
7. Summary and perspectives A general variational method for evaluating the properties of "nite systems at thermodynamic equilibrium has been proposed and applied to "nite fermion systems with pairing correlations. Extensions of the standard mean-"eld theories were thus obtained for two types of problems: (1) the variational evaluation not only of thermodynamic quantities, but also of expectation values, #uctuations or correlations between observables and more generally of characteristic functions; (2) the restoration of some constraints on the physical space of states or/and of some broken symmetries violated by the variational AnsaK tze. To reach this double goal, we relied on an action-like functional (2.5) which is designed to yield as its stationary value the relevant characteristic function. Two dual trial quantities enter this functional. The operator DK (u) is akin to a statistical operator evolving from 1) to the actual density operator DK "e\@)K
(7.1)
as u goes from 0 to b, while the operator AK (u) is akin to the observable AK de"ned in Eq. (1.6). Such a doubling of the number of variational degrees of freedom is distinctive of a general class of variational principles built to optimize some quantity of interest de"ned by given constraints. Here the constraint is the (symmetrized) Bloch equation (2.4) satis"ed by DK (u), while AK (u) appears as a Lagrangian multiplier associated with this constraint. For unrestricted trial spaces, the stationar-
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ity of the functional (2.5) expresses the Bloch equation (2.4) for DK (u); in addition, it yields the equation (2.11) for AK (u) which is the Bloch analogue of a (backward) Heisenberg equation. These two equations are decoupled and duplicate each other. However, when the trial spaces are restricted (as is unavoidable for any application), the two equations generally become coupled, with mixed boundary conditions involving both ends of the integration range [0, b]. The fundamental thermodynamic relations are preserved by the variational approximations under rather general conditions on the choice of the trial spaces (Section 2.3); these conditions are satis"ed by the AnsaK tze that have been used throughout the present article. For conserved observables, the characteristic functions, and therefore all the cumulants, are obtained from the approximate partition function via a simple shift of the operator KK (Section 2.4). Moreover, in the limit when the sources m entering the characteristic function vanish, and provided the trial spaces satisfy two A additional conditions stated in Section 2.5, our variational principle reduces to the standard maximization principle for thermodynamic potentials. Beginning with Section 3, we used the functional (2.5) to study systems of fermions in which pairing correlations are e!ective. For the variational operators DK (u) and AK (u) we choose the trial forms (7.2) ¹K B (u),exp(!l B (u)!aRL B (u)a) , ¹K ? (u),exp(!l ? (u)!aRL ? (u)a) , (7.3) where a de"ned in Eq. (3.1) denotes the 2n operators of creation aR or annihilation a ; the trial H H quantities are the scalars l B and l ? and the 2n;2n matrices L B and L ? . The generalized Wick theorem then allows to write explicitly the resulting functional as Eq. (3.32). The stationarity conditions are expressed by the set of coupled equations (3.37), (3.43), (3.46) and (3.47) which constitute an extension of the Hartree}Fock}Bogoliubov approximation. The equation (3.37) proceeds forwards from 0 to b with respect to the variable u while Eq. (3.43) evolves backwards from b to 0, since the boundary conditions are "xed at u"0 for DK and at u"b for AK . In Section 4 thermodynamic quantities, #uctuations and correlations have been evaluated via the expansion of the coupled equations in powers of the sources m . The thermodynamic quantities A are given by the zeroth order term in the expansion while expectation values 1QK 2 of single-quasiA particle observables (7.4) QK "q #aRQ a A A A are given by the "rst order. In both cases, one recovers (Section 4.1) the standard HFB results as expected from the general discussion of Section 2. The #uctuations and correlations C ,1QK QK #QK Q 2!1QK 21QK 2 are given by the second B A A B AB A B order. In the case where at least one of the two observables QK and QK commute with the operator A B KK , we obtain (Eq. (4.27)) 1 C " Q : F\ : Q . AB b A B
(7.5)
The 2n;2n matrices Q and Q de"ned in Eq. (7.4) are regarded in Eq. (7.5) as vectors in the A B Liouville space, and the colon symbol stands as a scalar product in this space, or equivalently as
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half the trace (tr ) in the original space of the 2n;2n matrices (Section 4.2.2 and Eq. (A.1)). In the L Liouville space, the stability matrix F is de"ned in Eq. (4.20) as the second derivative with respect to the reduced density R of the HFB grand potential F around its minimum, the HFB solution % R . If both observables QK and QK do not commute with KK , Eq. (7.5) expresses their Kubo A B correlation (Section 4.3.1). It then appears as a generalization to the non-commutative case of the Ornstein}Zernike formula, here derived from a general variational principle. The ordinary correlation of two observables QK and QK that do not commute with KK is covered A B by the more general formula (Eq. (4.63)) C "Q : (K coth b K )F\ : Q . B AB A
(7.6)
The matrix K is the usual RPA kernel written in the Liouville space (Eq. (4.47) or (A.15)). The HFB stability matrix F is related to the dynamical RPA matrix K through Eq. (4.49), or (A.22)}(A.23). Notwithstanding the independent-quasi-particle nature of the operators DK (u) and AK (u), we have obtained non-trivial approximate expressions for the correlations. In particular, formulas (7.5) or (7.6) incorporate long-range e!ects. This was made possible because, through the optimization of the functional (2.5), the variational quantities DK (u) and AK (u) acquire a dependence on the sources m . A Despite the coupling inherent to the stationarity equations (3.37) and (3.43), we have found the explicit expressions (7.5) and (7.6) for the correlations. In this context, we have found a simple u-dependence for the solutions of these equations to "rst order in the sources. Although the "nal formulas (7.5) or (7.6) are compact, their derivation was complicated by the existence of zero eigenvalues in the kernel F , due to some broken invariance, and in the kernel K (Section 4.3.2). Indeed, for the formulas (7.5) and (7.6) to be meaningful, one must specify how to handle these zero eigenvalues. We showed in Sections 4.2.3 and 4.3.3 that the precise meaning to be given to these formulas depends on the commutation, or non-commutation, of the observables QK and QK with the A B conserved single-particle operator associated with the broken invariance. Analyzed in perturbation theory, the formulas (7.5) and (7.6) were shown to correspond to a summation of bubble diagrams (Section 4.4). Let us stress that the RPA kernel comes out from our variational approach without any additional assumption beyond the choice (7.2)}(7.3) of the trial AnsaK tze. In this way, the RPA acquires a genuine variational status (albeit not in the Rayleigh}Ritz sense). These various aspects of the RPA are discussed in Section 4.5. Sections 5 and 6 are devoted to the second type of problems posed in the Introduction, namely to those situations which require projections. Though the operator KK , and hence the exact density operator (7.1), commute with any conserved quantity, the corresponding invariance can be broken by the variational approximation if the commutation is not preserved by the trial operator DK (u). In this case, a projection PK over an eigenspace of the conserved quantity is required on both sides of DK to restore the symmetry. On the other hand, even when the invariance is not broken, a projection is needed if the trial operators act in a space which is wider than the physical Hilbert space H . (An example is the extension of the thermal Hartree}Fock approximation to a system with a well de"ned particle number.) Sections 5 and 6 cope with both situations. In the "rst case where the invariance is broken, we replace the trial expression (7.2) for the density operator by DK (u)"PK ¹K B (u)PK ,
(7.7)
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where ¹K B (u) still has the independent quasi-particle form (7.2); for AK (u) we keep the ansatz (7.3). In order to handle the product (7.7) we take advantage of the fact that the projection PK can be written as a sum of group elements ¹K E which have an independent quasi-particle form, so that Eq. (7.7) is also a sum of terms of the ¹K -type. The resulting functional (Eq. (5.16)) and coupled stationarity equations (Eqs. (5.22), (5.23), (5.27) and (5.28)) are written in Sections 5.2 and 5.3. Although these coupled equations are noticeably more complicated than the unprojected ones, the number of variational parameters, and therefore the dimension of the numerical problem, do not increase. The case of unbroken PK -invariance, where the trial operators ¹K B (u) and ¹K ? (u) commute with the group elements ¹K E , is discussed in Section 5.4. Since PK now commutes with ¹K B (u), the ansatz (7.7) reduces to DK (u)"PK ¹K B (u)"¹K B (u)PK ,
(7.8)
which entails several simpli"cations in the action-like functional (Eq. (5.40)) and in the stationarity equations. We have examined in some detail (Section 5.4.2) the limiting case AK (m)"1) which admits the explicit solution (5.43) for DK (u) and AK (u). This leads to further simpli"cations in the functional (Eq. (5.50)) and in the self-consistent equations (5.51)}(5.54). Several thermodynamic consistency properties were veri"ed. As an example the projected entropy, energy and grand potential satisfy the usual thermodynamic identities (Eqs. (2.26)}(2.29)). In Section 6 the theoretical framework elaborated in Section 5.4 was applied to study the e!ects of the particle-number parity in a "nite superconducting system at thermodynamic equilibrium. In the trial density operator (7.8), PK given by Eq. (5.3) is the projection on either even or odd particle numbers. This problem is of interest for recent or future experiments on mesoscopic superconducting islands, small metallic grains or heavy nuclei. The explicit form of the coupled equations (see Eq. (6.9)) was written in Section 6.1. The parity-projected grand partition function (or more generally the characteristic function for conserved single-particle operators QK ) was evaluated in A Section 6.2, where Eq. (6.23), together with the de"nitions (6.25)}(6.26), was shown to replace the standard HFB self-consistent equation (4.2). Section 6.3 deals with a system (electrons in a superconductor or nucleons in a deformed nucleus) governed by a BCS model Hamiltonian, where the non-interacting part involves twofold degenerate levels p with energies e , and the interaction scatters pairs from one level to another. The BCS N gap is now replaced by the quantities D de"ned in Eq. (6.50), and the usual BCS equation is EN replaced by the number-parity projected self-consistent equation (6.56), 1#gr t\t\ D t 1 N O , EO O (7.9) D " G NO((e !k)#D 1#gr t\ EN 2 N O O EO in which G is a pairing matrix element of the BCS Hamiltonian (6.34). The notation t stands for NO N t ,tanhbe , (7.10) N N where the quasi-particle energy e , given by Eq. (6.54), di!ers from the standard BCS expression N ((e !k)#D . The number r is de"ned (Eq. (6.46)) as N EN r , t . N N
(7.11)
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The formulae (7.9)}(7.11) depend on the parity of the particle number N through the value of g (g"#1 for even N or g"!1 for odd N). The e!ects of the projection appear indirectly through the occurence of e in Eq. (7.10), and directly through the last fraction of Eq. (7.9). N As discussed in Section 6.3.3, this fraction is larger than one for even-N systems, smaller for odd-N systems. As a consequence, the even-N projected gap is larger than the BCS gap, while the odd-N projected gap is smaller, in agreement with the idea that pairing is inhibited in small odd-N systems. The analysis of this fraction reveals, moreover, that the odd-N projection di!ers from BCS more than the even-N. In Section 6.4 we considered some limiting situations. At zero temperature the BCS and even-N projected gaps are equal. However, while the BCS quantities deviate from their zero-temperature limit by terms of the form exp(!bD), the deviations are of the form exp(!2bD) for the even-N projection. The di!erences are more striking between BCS and the odd-N projection. At zero temperature, the BCS gap is larger than the odd-N projected gap. When the temperature grows, the latter starts to increase before reaching a maximum. The odd-N entropy tends to ln 2 when ¹ tends to zero, re#ecting the twofold degeneracy of the ground state; in Section 6.5.4 we commented upon the fact that this result is not completely trivial. In Section 6.4.2, we had shown that, at temperatures larger than the single-particle level spacing and smaller than the pairing gap, the odd-N entropy coincides with the Sackur-Tetrode entropy of a single-particle with a mass D moving in a one-dimensional box. This is consistent with gapless elementary excitations in odd-N systems. When the single-particle level density is su$ciently large, the critical temperatures are equal for the BCS and the odd or even projected solutions. Moreover, at non-zero temperature, for a large system or when the level spacing becomes small with respect to the pairing gap at the Fermi surface, Eq. (7.9) reduces to the usual BCS equation. However, the value of the projected (odd or even) entropy is lower than the BCS value by ln 2. Note that the entropy is not involved in the action functional (2.5), in contrast with the standard variational principle for thermodynamic potentials. Here, the entropy comes out as a by-product of our variational approximation; it is obtained in the end through the di!erentiation (2.28). In order to push further the comparison between the BCS and the projected results we presented in Section 6.5 schematic calculations which exemplify three di!erent physical situations. A crucial parameter turns out to be the product w D where D is the zero-temperature BCS gap and w the $ $ single-particle level density at the Fermi surface. This parameter, which indicates how many single-particle levels lie within the energy range of the gap, is sometimes called the `number of Cooper pairsa. When w D is large, the projection e!ects are weak. Fig. 5 illustrates the case w D"10. The BCS $ $ and even-N projected gaps are almost undistinguishable. The relative di!erence between the BCS and odd-N gaps is 1/2w D at ¹"0. The di!erence between the free energies of the odd and even $ systems can be estimated perturbatively from the BCS solution. The outcome (Eq. (6.99)), equal to D!1/4w at ¹"0, decreases almost linearly with ¹ and displays a cross-over towards the $ asymptotic value !1/4w at a temperature ¹ smaller than the critical temperature (see Fig. 8). $ " Between ¹ and ¹ , the gap function is the same whatever the N-parity. These results are " consistent with experiments on mesoscopic superconducting islands which show that, when w D 1, the odd}even e!ects disappear at some temperature below ¹ [45]. $ The value w DK2 corresponds roughly to the case of heavy nuclei (Fig. 4). In the even system, $ the di!erence between the projected and BCS gaps becomes more appreciable than in Fig. 5. In the
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odd system, the projected gap at zero temperature is reduced by about 30% with respect to BCS; this result can also be obtained from a BCS calculation in which the pair formation is blocked for the level that coincides with the Fermi surface. As the temperature grows, so does the odd-N projected gap. The BCS, even and odd projected gaps merge again at higher temperature where they all display the same critical behaviour. Fig. 9, where w DK1.1, corresponds to a situation which could be encountered in ultrasmall $ Aluminium grains [4}6]. In the even-N projected state, the critical temperature is higher than for the BCS state. In the odd system a new phenomenon occurs: there is no gap at zero temperature. However, the phase diagram (¹, D) obtained by projecting on odd number exhibits a reentrance ewect: when the temperature increases, a gap appears at a "rst transition temperature, reaches a maximum and disappears at a second transition temperature smaller than the BCS one. This behaviour can be understood as follows: at ¹"0, at least one particle remains unpaired and it fully occupies one of the two single-particle states at the Fermi surface, forbidding the formation of a pair with components on this strategic level. When the temperature increases, the unpaired particle partially frees up this level, which then becomes available for the formation of Cooper pairs. According to Fig. 12, the existence of two transitions in odd ultrasmall metallic grains could be experimentally detected by speci"c heat measurements of odd and even systems. However, in more elaborate treatments going beyond the projection on the N-parity, the transitions will be smoothed out by additional "nite-size e!ects [47]; further studies are required to ensure that the reentrance phenomenon is not completely washed away by these e!ects. Besides its relevance for "nite superconducting systems, one can imagine using the variational method introduced in this paper for studying other physical problems. In particular the formulation of Section 5.4, which treats situations with unbroken PK -invariance, is applicable in various problems. Consider for instance a "nite fermion system at equilibrium without pairing correlations. In this case the exponents in the trial quantities (7.2) and (7.3) need only contain single-particle terms in aRa which commute with NK . The projection PK has the form (5.2). G H A variational method to evaluate the canonical partition function is then furnished by Section 5.4.2, which constitutes a projected extension (with a well-de"ned number of particles) of the thermal Hartree}Fock approximation. It su$ces in Eqs. (5.43)}(5.44) to take a trial operator KK of the single-particle type. This procedure also provides, along the lines of Section 2.4, a consistent variational estimate for the characteristic function of any conserved single-particle observable (commuting with both NK and HK ). For other single-particle observables (commuting with NK only) one should resort to the method of Section 5.4.1 where the trial operators ¹K B (u) and ¹K ? (u) have a more complicate u-dependence. Likewise, for "nite systems without odd-multipole deformations, the projection over a given space-parity can be performed by implementing the operator (5.6) in Sections 5.4.1 and 5.4.2. In case the symmetry is broken (Sections 5.2 and 5.3), the functional (5.16) takes a more intricate form, due to the occurence of two projections. One could then make use of a further approximation by using the scheme described in Ref. [48] where an integral such as (5.2) on the group elements is replaced by a truncated sum. This would generate approximate expressions for the canonical thermodynamic functions and correlations which should be easier to handle numerically than the general expressions (5.23)}(5.25) and (5.28)}(5.29). In the variational methods described above, one could imagine implementing trial quantities DK (u) and AK (u) with a more general form than Eqs. (7.2) and(7.3). For instance Ref. [49] has
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demonstrated, in the related context of time-dependent Hartree}Fock equations, the feasibility of extensions where T B (u) is multiplied by some polynomial in creation and annihilation operators. Using the methods developed here, one could obtain extensions of the HFB formalism, or of the projected HFB formalism, which would take into account a wider class of correlations. The variational setting of Section 2 could easily be adapted to problems of classical statistical mechanics. A "rst possibility would be to start from the classical version of the functional (2.5) in which traces, density operators and observables are replaced by integrals, distribution functions and random variables in phase space. The commutative nature of these quantities implies that a characteristic function would be obtained as a maximum. Alternatively, one could start from the quantal functional (2.5) and then apply a semi-classical limiting process, for instance through the use of the Wigner transform. The latter procedure would yield in addition the "rst quantum corrections to the classical limit. This could be of interest for heavy nuclei or for su$ciently large clusters of atoms and molecules. Minor modi"cations are needed to deal with systems of interacting bosons. Besides the change of the anticommutation rules (3.2) into commutation ones, linear terms in a and aR should be added to the observables (3.12) and to the exponents of the trial operators (7.2) and (7.3). This should result in a systematic variational approach of Bose}Einstein condensation in "nite interacting systems, a phenomenon which has become of direct experimental interest with the recent achievement of condensates in dilute atomic gases at ultra-cold temperatures. It is worthwhile to note that here again the projection on the right number of particles is an important requirement. Another physical problem where a projection is essential arises in particle physics, when mean-"eld-like approximations violate the colour invariance. One may imagine restoring it by means of a projection on colour singlets along the lines of Section 5. In conclusion we would like to stress the consistent character of the present method, besides its generality and its #exibility. We have veri"ed the agreement of our approximation scheme with thermodynamics. We have recovered naturally the RPA (the complete series of bubble diagrams). Finally, although we have not touched upon the subject in this already long article, we point out the complementarity of the present static approach with the variational treatment of dynamical problems of Ref. [29]. Indeed, the use of the Bloch equation as a constraint for the initial state combines coherently with the use of the (backward) Heisenberg equation as a dynamical constraint; a variational principle comes out which (within restricted trial spaces) optimizes both the initial state and the evolution of the system. This comprehensive static and dynamic variational method should provide a convenient tool for evaluating time-dependent correlations, and hence transport properties, either in the grand-canonical formalism as in Section 4 or with projections as in Sections 5 and 6.
Acknowledgements We are indebted to M.H. Devoret, D. Este`ve and N. Pavlo! for very instructive discussions, to S. Creagh and N. Whelan for reading parts of our manuscript, and to M.T. Commault and M. Trehin for their help in the preparation of the "gures. One of us (M.V.) gratefully acknowledges the support of the Alexander von Humbolt-Stiftung and the hospitality of the Physics Department of the Technische UniversitaK t MuK nchen.
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Appendix A. Geometric features of the HFB theory A.1. The reduced Liouville space The Liouville representation of quantum mechanics (see for instance [51], Section 3) relies on the idea that the algebra of observables, when regarded as a vector space, can be spanned by a basis of operators. Any observable is then written as a linear combination of the operators of the chosen basis, with coe$cients interpreted as (covariant) coordinates. Correlatively, for any state, the expectation values of the operators of the basis are regarded as the (contravariant) coordinates of this state. In the HFB context, it is su$cient to consider the subalgebra of quadratic forms of creation and annihilation operators a (equal to either a and aR). With the notation (3.12), keeping H G G aside the trivial c-number q, we are led to regard the coe$cients Q as the covariant coordinates of HHY the operator aRQ a . The factor and the constraint (3.5) imposed on Q are consistent with HHY H HHY HY the duplication aRa "!a aR#d "! p aRa p #d occuring in the operator basis. H HY HY H HHY IIY HYI I IY IYH HHY We shall thus consider Q both as a matrix in the 2n-dimensional j-space of creation and HHY annihilation operators (Section 3.1) and as a vector in the 2n;2n-dimensional reduced Liouville space of observables. Likewise, since in the HFB approximation we focus on density operators of the form (3.4) and since they are characterized by the contraction matrices R de"ned by Eqs. (3.8) and(3.10)}(3.11), HHY the contravariant coordinates of these density operators reduce to the set R , with the constraint HHY (3.9). Again, R is both a matrix in the j-space and a vector in the reduced Liouville space of states. HHY As in the full space of observables and states, the expectation value of an operator aRQa, in a state characterized by its contractions R , is given here by the scalar product of the vectors HHY R and Q in the Liouville space. We denote this scalar product as (A.1) Q : R"tr QR" Q R . HHY HYH L HHY In Eq. (A.1) the colon symbol : indicates therefore both a summation on a twisted pair of indices and a multiplication by the factor accounting for the duplication of coordinates both in R and Q. Among others, we shall use in this Appendix the following two matrices in Liouville space: "2d d , R "2p p . (A.2) HHYIIY HIY IHY HHYIIY HI IYHY They are invariant under the exchange (jj ) (kk ) (symmetry in Liouville space). One has also R"1. Moreover, the relation (3.9) implies that the variation dR of a contraction matrix satis"es 1
R : dR"pdR2p"!dR .
(A.3)
A.2. Expansion of the HFB energy, entropy and grand potential The HFB energy, de"ned by E+R,"Tr DK KK /Tr DK (including therefore the term !k1NK 2), is given by Wick's theorem as the function (3.31) of the Liouville vector R . Its partial derivatives HHY have been de"ned in Eqs. (3.38)}(3.40) as the 2n;2n matrix H+R, constrained by (3.41). In the Liouville space, in agreement with Eq. (3.38), H appears as the gradient of E, that is, a covariant HHY
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vector which, with the notation (A.1), reads dE+R,"H+R, : dR .
(A.4)
Using Eqs. (3.7) and (3.8), the HFB entropy is evaluated as S+R,,!Tr DK ln DK /Tr DK !ln Tr DK "!tr [R l n R#(1!R) ln (1!R)] , (A.5) L where the two terms under the trace are equal. Similarly to Eq. (A.4), its gradient, written by means of Eq. (3.9) so as to satisfy the same identity (3.41) as H, is given by dS+R,"L : dR ,
(A.6)
with L,ln [(1!R)/R]. The second derivatives of E+R, and S+R, de"ne the twice covariant tensors E and S entering the expansions E+R#dR,"E+R,#H+R, : dR#dR : E : dR , (A.7) S+R#dR,"S+R,#L : dR#dR : S : dR#2 . (A.8) Both matrices E and S are symmetric (for instance, E "E ). Moreover, they satisfy HHYIIY IIYHHY R : E : R"E , R : S : R"S , (A.9) so that E : dR obeys the same symmetry relation (A.3) as dR. The explicit form of the second derivative E of E+R, comes out from Eq. (3.31) and dH"E : dR. When R and dR are HHY HHY hermitian matrices in the j-space, the matrix S of the second variations of S+R, is always negative in Eq. (A.8). As shown in [51], it can be expressed as
S "!2 HHYIIY
dv
1 R#v(1!R)
1 R#v(1!R)
,
(A.10)
HIY IHY which, in a basis where R is diagonal (R "R d ), reduces to HHY H HHY L !L H I. (A.11) S "2d d HHYIIY HIY IHY R !R H I The ratio is equal to !1/R (1!R ) when R "R . H H I H The standard HFB variational principle (Section 4.1) provides the grand potential at grand canonical equilibrium as the minimum of 1 F +R,"E+R,! S+R, , % b
(A.12)
which is attained for R"R . From Eqs. (A.4) and (A.6), the stationarity condition yields the self-consistent equation 1 H+R ," L , b
(A.13)
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which de"nes the HFB solution R , in agreement with Eq. (4.1). Around this equilibrium, the expansion of the grand potential F +R, reads % (A.14) F +R #dR,"F +R ,#dR : F : dR#2 , % % where F ,E !S is a positive matrix, like !S . @ A.3. The HFB grand potential and the RPA equation The expansion of Eq. (3.52) around R ?B "R (or around R B? "R ) leads to Eq. (4.46) which involve the RPA kernel K de"ned in Eq. (4.47) as RH+R, dR , R . (A.15) K : dR,[H+R ,, dR]# IIY RR IIY RR IIY In Liouville space the matrix K is nonsymmetric; it depends on R and on the parameters of KK . We show here that K is directly related to the matrix F which enters Eq. (A.14), or Eq. (4.20). To this aim, we "rst introduce in the Liouville space a new matrix which generates the commutators with R. Its action on any vector M is de"ned by
C : M"!M : C"[R, M] .
(A.16)
Accordingly, its expression as a matrix is C "d R !d R #p (Rp) !p (pR) , (A.17) HHYIIY IHY HIY HIY IHY HYIY IH IH HYIY where the last two terms yield the same contribution as the "rst two when applied to a vector M satisfying Eq. (A.3) (i.e., R : M"!M). The matrix C is antisymmetric, and it moreover satis"es C : R"R : C"!C .
(A.18)
In a basis where R is diagonal, it reduces to C "2d d (R !R ) , (A.19) HHYIIY HIY IHY H I as it is obvious from Eq. (A.16). The commutators with the matrix L can then be represented in Liouville space by the product C : S which acts according to C : S : dR"S : C : dR"!dR : C : S"[L, dR] .
(A.20)
Indeed, in a basis where R is diagonal, this identity, analogous to Eq. (A.16), is a direct consequence of Eqs. (A.11) and (A.19). Note that the matrices C and S commute. The "rst term of Eq. (A.15) is thus proportional to Eq. (A.20), in which C and S have to be taken at the stationary point where the relation L ,ln (1!R )/R "bH+R , is satis"ed. To write the second term, we note that RH dR "E : dR , IIY RR IIY IIY
(A.21)
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a consequence of Eq. (A.7). Hence, using Eq. (A.16) and then adding Eq. (A.20), we "nd for any R and dR:
RH+R, 1 1 dR , R # [L, dR]"!C : E : dR# C : S : dR IIY RR b b IIY IIY "!C : F : dR .
(A.22)
The sought for expression of the kernel (A.15) is then simply K "!C : F , (A.23) where the matrices C and F are evaluated at the equilibrium point R"R . The RPA kernel K is thus directly related to the matrix F of the second derivatives of the grand potential. At the minimum (attained at R ) of the trial grand potential F +R,, the matrix F is non % negative. Therefore the eigenvalues of K , which are the same as those of !F : C : F in the space of Liouville vectors satisfying Eq. (A.3), are real. We thus recover the well-known consistency between the stability of the variational HFB state and the nature of the associated RPA dynamics in real time. Moreover, due to the antisymmetry of C , the non-vanishing eigenvalues of K come in opposite pairs. A.4. Riemannian structure of the HFB theory By relying on the very existence of the von Neumann entropy S(DK ), it has been shown that the set of density operators DK can be endowed with a natural Riemannian structure [51]. The basic idea is to introduce a metric such that the distance ds between two neigbouring states DK and DK #dDK is de"ned by ds"!dS .
(A.24)
With this metric the general projection method of Nakajima}Zwanzig appears as an orthogonal projection in the space of states. A reduction of the exact description through projection on a subset of simpler states, in our case the HFB states of the form (3.4), induces from Eq. (A.24) a natural metric on this subset. It is thus natural to de"ne !S, the positive symmetric matrix describing the second variations of the entropy (A.8), as a metric tensor in the reduced 2n;2n Liouville space of states. This de"nition allows us to introduce distances between neighbouring HFB states, and also to establish a correspondance between the covariant and contravariant components of the vectors of Section A.1. More precisely, this canonical mapping relates the variations of the observables aRLa to the variations of those states which are their exponentials; it reads: S : dR"dL . (A.25) We can also introduce the twice contravariant metric tensor !S\ in the subspace of single quasi-particle observables by inverting S in Liouville space (S\ : S"1). In a basis where R and L are diagonal, the matrix elements of S\ are R !R H I , S\ "2d d HHYIIY HIY IHY L !L H I
(A.26)
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where the ratio is !R (1!R ) when R "R . In an arbitrary basis, it can be written as H H I H L L eS e\S S\ "!2 du L . (A.27) HHYIIY e #1 eL#1 HIY IHY Finally, we can introduce as in Eq. (3.7) the reduced thermodynamic potential
1 1 ln f+L,"ln Tr exp ! aRLa " tr ln (1#e\L) . 2 2 L
(A.28)
The expression of the "rst-order term in the expansion in powers of dL, 1 ln f+L#dL,"ln f+L,!R : dL! dL : S\ : dL#2 , 2
(A.29)
exhibits the fact that the entropy (A.8) is the Legendre transform of the reduced grand potential (A.28), while the second-order term is seen to describe a distance between two neighbouring observables. This last feature is the counterpart in the reduced HFB description of a structure already recognized [51] in the full Liouville space. All these properties, on which we shall not dwell here, confer a geometric nature to the reduced HFB description and set it into the general projection method [51]. A.5. Lie}Poisson structure of the time-dependent HFB theory We shall now introduce another geometric type of structure, of a symplectic type. This will allow us to regard the reduced energy E+R, or the grand potential F +R, as a classical Hamiltonian for % the dynamics associated with the HFB theory. The evolution of the variables R is governed in the time-dependent mean-xeld HFB theory by HHY the equation i
dR "[H+R,, R] , dt
(A.30)
written in the 2n;2n matrix representation. In order to analyze this equation, we "rst note that the basic operators aaR of the Liouville representation are the generators of a Lie group with the algebra [a aR , a aR ]"d a aR !d a aR #p a a !p aR aR . (A.31) H HY I IY IHY H IY HIY I HY HYIY I H IH HY IY Following the arguments of Ref. [50], where a similar formulation was set up for the Hartree}Fock case, we note that the contractions R are through Eq. (3.8) in one-to-one correspondence with HHY the operators a aR . Regarding the R 's as a set of classical dynamical variables, we introduce H HY HHY among them a Poisson structure; indeed, the Lie-algebra relation (A.31) suggests to characterize the Lie}Poisson bracket between the dynamical variables by the Poisson tensor i +R , R ,"C , (A.32) HHY IIY HHYIIY where each C is the linear function of the R's inferred from the right-hand side of Eq. (A.31). As a matrix, C is readily identi"ed with the matrix introduced in Eqs. (A.16), (A.17) which described, in
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the Liouville space, the commutation with R. While the metric of Section A.4 was characterized by the symmetric Riemann tensor !S, the present Poisson structure is characterized by the antisymmetric Poisson tensor C. For more details on Poisson structures, see for instance Ref. [52]. In contrast to the standard Poisson brackets between canonically conjugate position and momentum variables, which are c-numbers, the Lie}Poisson brackets (A.32) depend on the dynamical variables R . In the Liouville representation, the bracket of two functions f and g of the HHY R's is evaluated by saturating their gradients with the tensor C according to Rg Rf . i + f, g," : C : RR RR
(A.33)
The compatibility of Eq. (A.33) with the de"nition (A.32) results from the property (A.18) of the tensor C and from the relation 1 RR HHY" (1!R) , (A.34) HHYIIY 2 RR IIY which itself is a consequence of Eq. (3.8). Using the rules (A.33) and (A.34), we identify the r.h.s. of Eq. (A.30) as a Lie}Poisson bracket since the time-dependent mean-"eld HFB equation reads dR 1 " [H+R,, R] i
dt 1 RE 1 RE 1 RE RR " : C" : C : (1!R)" :C: , i RR 2i RR i RR RR
(A.35)
that is dR "+E+R,, R, . dt
(A.36)
This equation has thus the form of a classical dynamical equation of motion; the HFB energy E+R, plays the ro( le of a classical Hamiltonian, while a Lie}Poisson structure is associated with the dynamical variables R by Eq. (A.32). Since the gradient (A.6) of S+R,, as a matrix in the j-space, commutes with R, the relation (A.16) implies that the Lie}Poisson bracket +S+R,, R, vanishes. Hence, we can alternatively regard the grand potential (A.12), instead of E+R,, as a classical Hamiltonian and write dR "+F +R,, R, . % dt
(A.37)
This remark is useful when the time-dependent mean-"eld equation (A.30) is used in the small amplitude limit to describe approximately collective excitations around the grand-canonical equilibrium state. By expanding R as R #dR, Eq. (A.30) then reduces to the RPA equation associated with the HFB approximation, ddR 1 " K : dR , dt i
(A.38)
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where the RPA kernel K is expressed by Eq. (A.15). We can now identify the r.h.s. of Eq. (A.38) with the "rst-order term +dF +R,, R , in the expansion of Eq. (A.37) in powers of dR around R . Using % the expansion (A.14) and the expression (A.32) of the Lie}Poisson bracket, we "nd that Eq. (A.38) is equivalent to i ddR/dt"!C : F : dR , (A.39) and thus recover the expression (A.23) for K . This new derivation of Eq. (A.23) provides us with a simple interpretation of the RPA equation (A.39) as an equation for small motions in ordinary classical dynamics. Indeed, small changes of canonically conjugate variables q , p around a minG G imum of a Hamiltonian H+q, p, are governed by !d.& +q, q, +q, p, .& .& dq d dq .N K! .O.O .O.N " , (A.40) dt dp d.& +p, q, +p, p, .& .& dp .N .N.O .N.N where one recognizes the same multiplicative structure as in the r.h.s. of Eq. (A.39), within the replacement of the ordinary Poisson bracket by Eq. (A.32) and of H+q, p, by F +R,. % For small deviations around an arbitrary TDHFB trajectory, the time-dependent RPA kernel K is de"ned by
RH+R, ddR "K : dR,[H+R,, dR]# dR , R . (A.41) IIY RR dt IIY IIY Since we no longer lie at the minimum of F +R,, this variation (A.41) involves not only the % expansion of F but also of R, which appears both directly and through the Poisson tensor C. We % thus get, in the product i
K : dR"H : dC!C : E : dR "!C : F : dR#(H!b\L) : dC ,
(A.42)
an additional term proportional to the deviation between bH+R, and L"ln[(1!R)/R]. Apart from this term, which arises from the dependence of the structure constants C on the variables R, the present classical dynamics di!ers from an ordinary Hamiltonian dynamics through the presence of vanishing eigenvalues in the tensor C. The Poisson structure generated by (A.32) therefore di!ers from the usual symplectic structure associated with canonically conjugate variables because some combinations of the dynamical variables R have a vanishing bracket with any variable. This property re#ects the existence of structural constants of the motion, which never vary whatever the ewective Hamiltonian in Eq. (A.36). In particular, the eigenvalues of R and S+R, always remain constant.
Appendix B. Liouville formulation of the projected 5nite-temperature HFB equations Using the Liouville-space formalism introduced in Appendix A, it is possible to recast the coupled equations (5.23) and(5.28) governing the evolution of T B (u) and T ? (u), de"ned in Eqs. (3.6) and(3.16)}(3.17), so as to make these equations formally similar to Eqs. (3.37) and (3.43).
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To this aim, we "rst introduce two operators T ? and T B whose action on a Liouville vector Q is given by T ? : Q,T ? QT ? \ ,
T B : Q,T B QT B \ .
(B.1)
They satisfy the relations R : T ? : R"T ? , T ? \"T ? 2,
R : T B : R"T B , T B \"T B 2 ,
(B.2) (B.3)
where the superscript T denotes the matrix transposition in the Liouville space. Next, we de"ne the matrix
MM B ? , HHYIIY
ZE EY+2([1!R EBEY? ]T E ) (T E \R EBEY? ) HIY IHY E EY #(R EBEY? !RM B ? ) (R BEY?E !RM B ? ) , , HHY IIY and the two Liouville vectors
GM B ? ,
E EY
(B.4)
ZE EY+(1!R EBEY? )H+R EBEY? ,R EBEY? #(E+R EBEY? ,!EM B?)(R EBEY? !RM B ? ), , (B.5)
GM ? B ,
ZE EY+(R ?EBEY H+R ?EBEY ,(1!R ?EBEY )#(E+R ?EBEY ,!EM ?B)(R ?EBEY !RM ? B ), .
E EY After multiplication to the right by T ? , Eq. (5.23) can be reformulated as MM B ? :
dT B
1 T B \ # (GM B ? #T ? \ : GM ? B )"0 . du 2
(B.6)
By introducing in the Liouville space the matrix MM ? B ,T ? : MM B ? : T B ,
(B.7)
namely
MM ? B , HHYIIY
ZE EY+2(R ?EBEY T EY \) (T EY [1!R ?EBEY ]) HIY IHY E EY #(R ?EBEY !RM ? B ) (R EY ?EB !RM ? B ) , , HHY IIY
(B.8)
and the vectors H M B ? "MM B ? \ : GM B ? ,
H M ? B "MM ? B \ : GM ? B ,
(B.9)
we can rewrite Eq. (B.6) as 1 dT B
"! [H M B ? T B #T B H M ? B ] , 2 du a form which resembles closely Eq. (3.37).
(B.10)
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In the same way Eq. (5.28) can be transformed into dT ? 1 " [H M ? B T ? #T ? H M B ? ] , 2 du
(B.11)
where the matrices H M ? B and H M B ? in the single-particle space are obtained from Eqs. (B.4)}(B.5) and(B.8)}(B.9) by the exchanges a d and g g . One major di!erence, however, between the coupled equations (B.10)}(B.11) and Eqs. (3.37) and (3.43) is that the former involve four di!erent quantities (H M B ? , H M ? B , H M ? B , H M B ? ) instead of two (H+R B? ,, H+R ?@ ,) for the latter. The matrices MM ? B and MM B ? required for the calculation of the quantities H M ? B and H M B ? appearing in Eq. (B.11) are related to MM ? B and MM B ? through MM ? B "MM ? B 2 ,
MM B ? "MM B ? 2 .
(B.12)
From the de"nition (B.4)}(B.5) and (B.7) of the matrices MM B ? and MM ? B and of the Liouville vectors GM B ? and GM ? B , one establishes the relations R : MM B ? : R"MM B ? , R : GM B ? "!GM B ? ,
R : MM ? B : R"MM ? B , R : GM ? B "!GM ? B .
(B.13) (B.14)
One deduces that, as the reduced HFB Hamiltonian H+R,, the vectors in Liouville space H M B ? and H M ? B verify the relation (3.41) which in the Liouville formulation becomes: R:H M B ? "!H M B ? ,
R:H M ? B "!H M ? B .
(B.15)
Through the exchanges a d and g g one extends the results (B.13)}(B.15) to the matrices MM ? B , MM B ? and vectors GM ? B , GM B ? , H M ? B , H M B ? . When the operator AK (de"ned in Eq. (1.6)) is hermitian, Eqs. (B.10), (B.11), (5.23) and(5.28), have hermitian solutions. Indeed, in such a case these equations preserve the hermiticity of T ? and T B and the reality of l ? and l B . To prove it, one "rst notices that the relations T E R"T E\ , T ? R"T ? , T B R"T B result in R EBEY? R"R ?EY\BE\ ,
(B.16)
ZE EYH"ZEY\E\ .
(B.17)
Then, since ( )H" \ \, one sees from Eq. (B.17) that Z is real. The use of this property along E EY EY E with the equality E+RR,"E+R,H in Eqs. (5.19) and (5.20) implies RM B ? R"RM ? B ,
RM B ? R"RM ? B ,
EM B?H"EM ?B .
(B.18) (B.19)
Finally, on formulae (B.4)}(B.5) and (B.8), one readily checks the relations MM B ? H "MM ? B
, MM B ? H "MM ? B
, HYHIYI HHYIIY HYHIYI HHYIIY GM B ? R"GM ? B , GM B ? R"GM ? B .
(B.20) (B.21)
They yield the equations H M B ? R"H M ? B ,
H M B ? R"H M ? B ,
which entail the hermiticity of dT B /du and dT ? /du.
(B.22)
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CONTENTS VOLUME 317 M. Beneke. Renormalons A. Leike. The phenomenology of extra neutral gauge bosons R. Balian, H. Flocard, M. VeH neH roni. Variational extensions of BCS theory
Nos. 1}2, p. 1 Nos. 3}4, p. 143 Nos. 5}6, p. 251
CONTENTS VOLUMES 311}316 I.V. Ostrovskii, O.A. Korotchcenko, T. Goto, H.G. Grimmeiss. Sonoluminescence and acoustically driven optical phenomena in solids and solid}gas interfaces R. Singh, B.M. Deb. Developments in excited-state density functional theory D. Prialnik, O. Regev (editors). Processes in astrophysical #uids. Conference held at Technion } Israel Institute of Technology, Haifa, January 1998, on the occasion of the 60th birthday of Giora Shaviv S.J. Sanders, A. Szanto de Toledo, C. Beck. Binary decay of light nuclear systems B. Wolle. Tokamak plasma diagnostics based on measured neutron signals F. Gel'mukhanov, H. Agren. Resonant X-ray Raman scattering J. Fineberg. M. Marder. Instability in dynamic fracture Y. Hatano. Interactions of vacuum ultraviolet photons with molecules. Formation and dissociation dynamics of molecular superexcited states J.J. Ladik. Polymers as solids: a quantum mechanical treatment D. Sornette. Earthquakes: from chemical alteration to mechanical rupture S. Schael. B physics at the Z-resonance D.H. Lyth, A. Riotto. Particle physics models of in#ation and the cosmological density perturbation R. Lai, A.J. Sievers. Nonlinear nanoscale localization of magnetic excitations in atomic lattices A.J. Majda, P.R. Kramer. Simpli"ed models for turbulent di!usion: theory, numerical modelling, and physical phenomena T. Piran. Gamma-ray bursts and the "reball model E.H. Lieb, J. Yngvason. Erratum. The physics and mathematics of the second law of thermodynamics (Physics Reports 310 (1999) 1}96) G. Zwicknagel, C. Toep!er, P.-G. Reinhard. Erratum. Stopping of heavy ions at strong coupling (Physics Reports 309 (1999) 117}208) F. Cooper, G.B. West (editors). Looking forward: frontiers in theoretical science. Symposium to honor the memory of Richard Slansky. Los Alamos NM, 20}21 May 1998 D. Bailin, A. Love. Orbifold compacti"cation of string theory W. Nakel, C.T. Whelan. Relativistic (e, 2e) processes D. Youm. Black holes and solitons in strong theory J. Main. Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra N. KonjevicH . Plasma broadening and shifting of non-hydrogenic spectral lines: present status and applications
311, No. 1, p. 1 311, No. 2, p. 47 311, Nos. 3}5, p. 95 311, No. 6, p. 487 312, Nos. 1}2, p. 1 312, Nos. 3}6, p. 87 313, Nos. 1}2, p. 1 313, No. 3, p. 109 313, No. 4, p. 171 313, No. 5, p. 237 313, No. 6, p. 293 314, Nos. 1}2, p. 1 314, No. 3, p. 147 314, Nos. 4}5, p. 237 314, No. 6, p. 575 314, No. 6, p. 669 314, No. 6, p. 671 315, Nos. 1}3, p. 1 315, Nos. 4}5, p. 285 315, No. 6, p. 409 316, Nos. 1}3, p. 1 Nos. 4}5, p. 233 316 No. 6, p. 339
