B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
SPIN PHYSICS AND POLARIZED STRUCTURE FUNCTIONS
Bodo LAMPE , Ewald REYA Sektion Physik der UniversitaK t MuK nchen, Theresienstr. 37, D-80333 MuK nchen, Germany Max-Planck-Institut fuK r Physik, FoK hringer Ring 6, D-80805 MuK nchen, Germany Institut fuK r Physik, UniversitaK t Dortmund, D-44221, Dortmund, Germany
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Physics Reports 332 (2000) 1}163
Spin physics and polarized structure functions Bodo Lampe , Ewald Reya * Sektion Physik der Universita( t Mu( nchen, Theresienstr. 37, D-80333 Mu( nchen, Germany Max-Planck-Institut f u( r Physik, Fo( hringer Ring 6, D-80805 Mu( nchen, Germany Institut f u( r Physik, Universita( t Dortmund, D-44221 Dortmund, Germany Received September 1999; editor: W. Weise Contents 1. Introduction 1.1. Polarization of a Dirac particle 2. The polarized structure functions 2.1. Basics of pure photon exchange 2.2. Quantitative formulae for pure photon exchange 2.3. A look at the forward Compton scattering amplitude 2.4. E!ects of weak currents 3. Polarized deep inelastic scattering (PDIS) experiments 3.1. Results from old SLAC experiments 3.2. The CERN experiments 3.3. The new generation of SLAC experiments 3.4. Future polarization experiments 4. The structure function g and polarized parton distributions 4.1. The quark parton model to leading order of QCD 4.2. Higher-order corrections to g 4.3. Operator product expansion for g 4.4. The behavior of g (x, Q) at small x 5. The "rst moment of g 5.1. The "rst moment and the gluon contribution
4 6 6 6 8 10 11 14 14 15 20 22 26 26 32 39 46 50 50
5.2. The "rst moment and the anomaly 5.3. Detailed derivation of the gluon contribution 5.4. The Bjorken sum rule 5.5. The Drell}Hearn}Gerasimov sum rule 6. Polarized parton densities and phenomenological applications 6.1. Deep inelastic polarized lepton}nucleon scattering 6.2. Heavy quark production in polarized DIS and in photoproduction 6.3. Heavy quark production in hadronic collisions 6.4. High p jets in high-energy 2 lepton}nucleon collisions 6.5. Semi-inclusive polarization asymmetries 6.6. Information from elastic neutrino}proton scattering 6.7. The OPE and QCD parton model for g and g > 6.8. Single-spin asymmetries and handedness 6.9. Structure functions in DIS from polarized hadrons and nuclei of arbitrary spin 6.10. Nuclear bound state e!ects 6.11. Direct photons and related processes in proton collisions using polarized beams
* Corresponding author. E-mail address:
[email protected] (E. Reya). 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 0 0 - 3
58 61 69 71 72 72 85 92 94 97 99 102 105 111 115 117
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163 6.12. Spin-dependent structure functions and parton densities of the polarized photon 7. Nonperturbative approaches to the proton spin 8. Transverse polarization 8.1. The structure function g
124 127 132 132
8.2. Transverse chiral-odd (&transversity') structure functions Acknowledgements Appendix. Two-loop splitting functions and anomalous dimensions References
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141 147 147 151
Abstract A review on the theoretical aspects and the experimental results of polarized deep inelastic scattering and of other hard scattering processes is presented. The longitudinally polarized structure functions are introduced and cross section fromulae are given for the case of photon as well as =! and Z exchange. Results from the SLAC and CERN polarization experiments are shown and compared with each other as well as their implications for the integrated g (x, Q) are reviewed. More recent experiments presently underway (like HERMES at DESY) and future projects (like RHIC at BNL, HERA-No and a polarized HERA collider at DESY) are discussed too. The QCD interpretation and the LO and NLO Q-evolution of g , i.e. of the longitudinally polarized parton densities, is discussed in great detail, in particular the role of the polarized gluon density, as well as the expectations for xP0. Particular emphasis is placed on the "rst moment of the polarized structure function in various factorization schemes, which is related to the axial anomaly, and on its relevance for understanding the origin of the proton spin. Sum rules (i.e. relations between moments of the structure functions) are derived and compared with recent experimental results. Various other phenomenological applications are discussed as well, in particular the parametrizations of polarized parton densities as obtained from recent data and their evolution in Q. Furthermore, jet, heavy quark and direct photon production are reviewed as a sensitive probe of the polarized gluon density, and the physics prospects of the future polarized experiments at RHIC (pl pl ) and a polarized HERA collider (el pl ) are studied. DIS semiinclusive asymmetries and elastic neutrino}proton scattering are reviewed, which will help to disentangle the various polarized #avor densities in the nucleon. The status of single- and double-spin asymmetries, and the observation of handedness in the "nal state, are discussed as well. Structure functions for higher spin hadrons and nuclei are de"ned and possible nuclear e!ects on high energy spin physics are reviewed. The theoretical concept of spin-dependent parton distributions and structure functions of the polarized photon is presented and possibilities for measuring them are brie#y discussed. Various nonperturbative approaches to understand the origin of the proton spin are reviewed, such as the isosinglet ; (1) Goldberger}Treiman relation, lattice calculations and the chiral soliton model of the nucleon. The physical interpretation and model calculations of the transverse structure function g are presented, as well as recent twist-3 measurements thereof, and the Burkhardt}Cottingham sum rule is revisited. Finally, the physics of chiral-odd &transversity' distributions is described and experimental possibilities for delineating them are reviewed, which will be important for a complete understanding of the leading twist-2 sector of the nucleon's parton structure. In the appendix the full two-loop anomalous dimensions and Altarelli}Parisi splitting functions governing the Q-evolution of the structure function g are given. 2000 Elsevier Science B.V. All rights reserved. PACS: 13.88.#e
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1. Introduction One of the most fundamental properties of elementary particles is their spin because it determines their symmetry behavior under space-time transformations. The spin degrees of freedom may be used in high-energy experiments to get information on the fundamental interactions which are more precise than those obtained with unpolarized beams. For example, the SLC experiment at SLAC is able to determine sin h with a higher precision by using polarized 5 e>e\ beams than current experiments at LEP with unpolarized beams (for a recent review see, e.g., [522]). Another aspect of polarization is the question how the spin of non-point-like objects like the nucleons is composed of the spins of its constituents, the quarks and gluons. This question can best be answered in high-energy experiments because the quarks and gluons behave as (almost) free particles at energy/momentum-scales Q
Fig. 1. The basic polarized deep inelastic scattering process.
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laboratories in the recent years. Whether or not there is agreement between the new CERN and SLAC data will be discussed in detail and commented in the course of this work. Furthermore, the Hermes experiment which takes place in the DESY-HERA tunnel [207,208,232] has, more recently, also provided us with precision measurements of proton and neutron spin structure functions [12,41]. In Section 2 we summarize all relevant expressions for polarized DIS cross sections and structure functions for neutral and charged electroweak currents. Previous and recent results of longitudinally polarized DIS experiments for g,(x, Q) are presented and compared with each other in Section 3. The LO and NLO QCD renormalization group evolution of g (x, Q) and of longitudinally polarized parton densities df (x, Q) are derived in Section 4, as well as their small-x behavior. Section 5 is devoted to the "rst moment (i.e. total helicities) of longitudinally polarized parton densities and of g in various factorization schemes which is related to the axial anomaly and its relevance for understanding the origin of the proton spin. Here, the Bjorken and Drell}Hearn}Gerasimov sum rules are derived and compared with recent experimental results as well. Section 6 includes most of the phenomenological aspects relevant for longitudinally polarized processes. We start in Section 6.1 with a brief historical review of &naive' parton model expectations for polarized parton densities; then we turn to recent developments for determining df (x, Q) in LO and NLO from recent data on gN L(x, Q) and the implications for their "rst moments (total helicities). Here we also discuss brie#y the present status of the orbital component ¸ "¸ #¸ in (1.1), such as the Q-evolution equations for ¸ (Q) X O E O E and how one might possibly relate them to measurable observables. In addition, hard processes initiated by doubly (singly) polarized hadron}hadron collisions such as the production of heavy quarks, of large-p photons and jets and of Drell}Yan dimuons will be also suitable 2 to measure the polarized parton distributions df (x, Q), f"q, q , g, in particular the gluon distribution dg(x, Q). Details will be discussed in the various subsections of Section 6. Furthermore, polarized ep and e>e\ collisions can also shed light on the so far unmeasured polarized parton densities of the photon which are theoretically formulated and discussed in Section 6.12. Various nonperturbative approaches to understand the origin of the proton spin are presented in Section 7, such as the isosinglet Goldberger}Treiman relation, lattice calculations and the chiral soliton model of the nucleon. Finally structure functions resulting from transverse polarizations are dealt with in Section 8. In Section 8.1 the theoretical concepts and model calculations of the transverse structure function g (x, Q) are presented as well as recent twist-3 measurements thereof, and the Burkhardt}Cottingham sum rule is revisited. The physics of the chiral-odd &transversity' distributions is described in Section 8.2 and experimental possibilities for delineating them are reviewed. It should be remembered that a complete understanding of the leading twist-2 parton structure of the nucleon requires, besides the unpolarized and longitudinally polarized parton densities f (x, Q) and df (x, Q), also the knowledge of the transversity densities d q(x, Q) which are experimentally entirely unknown 2 so far. The full two-loop polarized Altarelli}Parisi splitting functions dP(x) and their Mellin nGH moments (anomalous dimensions) dPL, governing the Q-evolution of g (x, Q) and df (x, Q), are GH summarized in the appendix.
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1.1. Polarization of a Dirac particle Let us start with a few basic facts about the polarization of a relativistic spin particle. A free Dirac particle of four-momentum p and mass m is described by a four-component spinor u(p, s) which satis"es the equation (p. !m)u(p, s)"0 ,
(1.2)
where p. "c pI. The polarization vector s is a pseudovector which ful"ls s,(s)!(s)"!1 I and sp"0. The projection operator onto a state with polarization s is known to be P(s)"(1#c s. ) . (1.3) The transformation properties of s are given in Table 1 where we consider two Dirac particles which move along the z-direction in the lab frame, one of it with transverse and the other one with longitudinal polarization. The transverse polarization vector is not changed when going from the rest frame to the lab frame, but the longitudinal is. The important point to notice is that at high energies E<m the product ms remains "nite and converges to p: * ms & p . * # This fact will be used repeatedly in later applications.
(1.4)
2. The polarized structure functions 2.1. Basics of pure photon exchange First we consider Fig. 1 with photon exchange only. We assume that both the incoming lepton and the incoming nucleon are polarized (polarization vectors sI and SI). We shall see below why this is important. The procedure for polarized particles is analogous to the case of unpolarized particles, i.e. the cross section is a product of a leptonic tensor ¸ which is known (cf. Fig. 2) and IJ a hadronic tensor =IJ which can be expanded into Lorentz covariants whose coe$cients de"ne the structure functions which are to be measured. In unpolarized e(k)N scattering one has the
Table 1 A transverse and a longitudinal polarized Dirac particle in their rest and laboratory frame
Rest frame p"(m, 0, 0, 0) Lab frame p"(E, 0, 0, (E!m)
Transverse polarization
Longitudinal polarization
s "(0, 1, 0, 0) 2 s "(0, 1, 0, 0) 2
s "(0, 0, 0, 1) * 1 s " ((E!m, 0, 0, E) * m
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
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Fig. 2. The de"nition of the lepton tensor.
well-known functions F and F whereas for polarized particles two additional functions g and g arise:
= " dxe OV1PS"J (x)J (0)"PS2 IJ I J
Sq M qM SNg (x, Q)# SN! PN g (x, Q) "= (sym.)#i e IJMN IJ Pq Pq
(2.1)
e is the totally antisymmetric tensor in four dimensions, e "1, e "!1, etc., and the IJMN polarization vector of the proton is normalized to S"!1. Note that there is no analogy F g because the contribution of g to the cross section vanishes in the limit of ultra relativistic on-shell quarks (SN&PN) since there are not enough four vectors available anymore to form an antisymmetric combination in Eq. (2.1). Furthermore Q"!q and x"Q/2Pq are the usual (Bjorken) variables of the DIS process. It is not entirely trivial to see that (2.1) is the most general form of the antisymmetric hadron tensor. One has to make use of the e-identity g?@eIJMN"g?Ie@JMN#g?JeI@MN#g?MeIJ@N#g?NeIJM@
(2.2)
to get rid of tensors like [(p e !p e )qM#p ) qe ]S?p@ . (2.3) I J?@M J I?@M IJ?@ The incoming lepton (Fig. 1) is assumed to be polarized too. Why is that necessary? To see that let us have a look at the lepton tensor ¸ "tr[(1#c s. ) ( k. #m )c (k. #m )c ] . (2.4) IJ J I J J Obviously, ¸ consists of a part independent of the lepton polarization s@ and a part linear in s@, IJ the former being symmetric in k and l, the latter antisymmetric: ¸ "¸ (sym.)#2im e q?s@ . (2.5) IJ IJ J IJ?@ The antisymmetry of the last term is due to the c in (1.3) and to the vector coupling of the photon to fermions. With the symmetric part alone in Eq. (2.5), g and g cannot be extracted from Eq. (2.1). One needs the antisymmetric part, i.e. the lepton polarization. From (2.5) it seems that all polarization e!ects are suppressed at high energy by a factor m . J However, in the case of longitudinal polarization one has m s@Pk@ [according to (1.4)] and thus J there is no suppression by factors of m . In the following we shall always presume the leptons to be J longitudinally polarized.
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Fig. 3. The basic form of the polarized experiments.
How to measure g and g ? The cross section p&¸ =IJ will be of the form IJ ¸ =IJ"¸ (sym.) = IJ(sym.) #¸ (antisym.) =IJ (antisym.) . IJ IJ IJ
(2.6)
One should try to get rid of the "rst term in (2.6) because it is the cross section for unpolarized scattering. One possibility is to consider di!erences of cross sections with nucleons of opposite polarization [161,356,375,63,512] as is depicted in Fig. 3. In both parts of the "gure one starts with a beam of high energetic leptons with left-handed helicity ("longitudinally polarized with spin vector antiparallel to the direction of motion). This beam is sent to two nucleon probes with opposite longitudinal polarization, i.e. with their spins along the direction of the lepton beam and opposite to it, and to two probes with opposite transverse polarization. In the di!erence of the cross sections [part (a) of Fig. 3] the unpolarized structure functions drop out and only g survives (with respect to the suppressed (2yxM/Q)g contribution, where y"Pq/Pk), i.e. g can in principle be uniquely determined from measuring this di!erence. Similarly, in the di!erence of cross sections obtained from part (b) of Fig. 3, the transverse polarization case (kS"k ) S"0), the sum (y/2)g #g appears. However, there is an overall suppression factor 2xM/(Q, where M is the nucleon mass so that g can be obtained only from rather low-energy experiments. The appearance of this factor has, of course, to do with the transverse polarization. We can take the fact that g appears only in cross sections with transverse polarized nucleons as a hint that it is di$cult to accomodate g in the parton model. There is no notion of transversality in the conventional parton model. We shall come back to this &transverse spin structure function' in Section 8. 2.2. Quantitative formulae for pure photon exchange To be more speci"c let us write down the most general cross section di!erence relevant for polarized deep inelastic "xed target lN scattering [375,63,549]:
d[p(a)!p(a#p)] 8a " cos a Q dx dy d
y yc y 1! ! g (x, Q)! cg (x, Q) 2 4 2
yc y ! sin a cos c 1!y! g (x, Q)#g (x, Q) 4 2
.
(2.7)
This formula comprehends all information from the antisymmetric part of the tensor Eq. (2.1) where a is the angle between the lepton beam momentum vector k and the nucleon-target polarization vector S, is the angle between the k}S plane and the k!k lepton scattering plane
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
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Fig. 4. The geometry of the polarized deep inelastic scattering process in the lab frame.
(cf. Fig. 4), c"2Mx/(Q and a scaling limit has not been taken. From Eq. (2.7) it is obvious that e!ects associated with g are suppressed (at least) by a factor 2M/(Q with respect to the leading terms. More convenient than di!erences of cross sections are asymmetries A(a)"(p(a)!p(a#p))/(p(a)#p(a#p))
(2.8)
as, for example, the longitudinal asymmetry A "(p (2.9) N !p = )/(p N #p =) * obtained for a"0, and the transverse asymmetry A obtained for a"p/2 and an asymmetric 2 integration over . A picks up the coe$cient of cos a in Eq. (2.7) whereas A picks up the coe$cient of sin a cos . * 2 Events with near p/2 or 3p/2 (where the nucleon spin is perpendicular to the scattering plane) can be obviously neglected in the determination of A ; they are not a good measure of g #yg /2. 2 The really interesting quantities are the virtual photon asymmetries A "(p !p )/(p #p )
(2.10)
and A "(2p )/(p #p ) (2.11) 2* where p and p are the virtual photoabsorbtion cross sections when the projection of the total angular momentum of the photon}nucleon system along the incident lepton direction is 1/2 and 3/2. Note that p "(p #p ) and that the term p arises from the interference between 2* 2 transverse and longitudinal amplitudes. The signi"cance of these quantities will be clari"ed in Section 2.3. A and A can be related, via the optical theorem, to the measured quantities A and A , or, * 2 equivalently, to the structure functions by means of the following relations: A "D(A #gA ) , * A "d(A !mA ) , 2
(2.12)
A "(g !cg )/F , A "c(g #g )/F .
(2.14)
(2.13)
with (2.15)
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The kinematic factors D, d and g, m are de"ned by y(2!y) D" , y#2(1!y)(1#R) g"2c(1!y)/(2!y) ,
d"D
2e , 1#e
m"g(1#e)/2e
(2.16)
e"(1!y)/(1!y#y/2)
(2.17)
with
being the degree of transverse polarization of the virtual photon. D and d can be regarded as depolarization factors of the virtual photon. Note that g and m are of order M/(Q. Finally, R is the ratio of cross sections for longitudinally and transversely polarized virtual photons on an unpolarized target, de"ned by 2xF (x, Q)"F (x, Q)(1#c)/(1#R(x, Q)) . Note that in the limit c,4Mx/Q;1,
(2.18)
R"F /2xF (2.19) * where F ,F !2xF . Furthermore, in the leading logarithmic order of QCD one arrives * asymptotically at the well-known Callan}Gross relation 2xF (x, Q)"F (x, Q). 2.3. A look at the forward Compton scattering amplitude Let us now brie#y discuss the implications of the optical theorem on polarized DIS. The hadron tensor, Eq. (2.1), is the absorptive part (imaginary part) of the forward Compton scattering amplitude. This amplitude in general has a decomposition into four independent amplitudes which one may choose as [¹(1#P1#)#¹(1!P1!)] , (2.20) (2.21) ¹(0#P0#) , [¹(1#P1#)!¹(1!P1!)] , (2.22) (2.23) ¹(0#P1!) . Their degrees of freedom correspond to the four structure functions F , F , g and g . More precisely, the combinations (2.20)}(2.23) correspond [356,186,357,440] via the optical theorem, to F , F , g !cg and c(g #g ), respectively. * There are rigorous theoretical limits on the virtual photon asymmetries A and A in Eqs. (2.10) and (2.11) namely "A "41,
(2.24)
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"A "4(R .
(2.25)
These inequalities follow from the hermiticity of the electromagnetic current J "J> I I
(2.26)
which implies aHa =IJ50 I J
(2.27)
for any complex vector a . Suitable choices of a lead to the above inequalities. I I 2.4. Ewects of weak currents Until now we have restricted ourselves to photon exchange only, i.e. to the processes eNPeX, kNPkX, at momenta Q;m , and have found two polarized structure functions 8 g and g . If we take into account =! and Z exchange, three parity violating polarized structure functions usually called g , g and g arise in addition. They will appear, for example, in the scattering of neutrinos on polarized nucleons lNPl X but become also important for the polarized extension of HERA at large values of Q. Explicitly, the hadron tensor has the form [218,62,432,571,472}474]
Sq q q M ie qMSNg #ie qM SN! PN g #qS !g # I J g = "= (S"0)# IJMN IJMN IJ IJ IJ Pq q Pq
Pq qS P ! q # q I Pq I #
1 2
Pq P ! q I q I
Pq P ! q g J q J
qS Pq S ! q # P ! q J q J J q J
qS S ! q I q I
g
.
(2.28)
In this expression g , g and g #g are the &longitudinal' structure functions which survive in the high energy limit, and g and g !g are the transverse ones. The appearance of symmetric tensors which are linear in SI is due to the axial vector component of the =! and Z couplings to fermions. Notice that terms proportional to qI or qJ can be dropped in the de"nition of = because they give no contribution in the limit m /EP0 when contracted with the appropriate IJ J lepton tensors ¸ . IJ If one considers longitudinally polarized nucleons (SI&PI), the structure functions g and g !g are clearly not of interest. The cross section di!erence p N !p = of part (a) of Fig. 3 will measure a linear combination [63,218,62] of g , g and g #g with coe$cients depending on the vector and axial vector couplings of the =/Z to the lepton. Let us "rst consider neutrino nucleon scattering. Here the lepton tensor is the same for the polarized and the unpolarized case: ¸ (long. pol.)"¸ (unpol.)"2(k k #k k !g kk #ie k?k @) IJ IJ I J J I IJ IJ?@
(2.29)
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because the l=>-interaction is E
(2.30)
i.e. one can formally get the polarization cross section (p !p = )J, by taking the well-known v lNPl\X cross section for unpolarized beams and replace the unpolarized structure functions F by the polarized ones (here g ,gJ, and F ,FJ, are the structure functions speci"c to G G G G G lN-scattering):
1 G s y xyM xyM d(p !p = )J, v " $ y 1! ! xgJ,! gJ, dx dy 2p (1#Q/m ) 2 2E E 5
xM xyM # yx 1# gJ,# 1!y! E 2E
xM 1# gJ,#gJ, E
, (2.31)
where s"2ME in the nucleon's rest frame. In contrast, if the nucleon is transversely polarized one "nds d(p O !p P )J,&¸ (long. pol.)=IJ(transv. pol.) , IJ
(2.32)
i.e.
d(p 1 y MG O !p P )J, $ " (xyM[2(1!y)E!xyM] !2xy gJ,#gJ, dx dy 2 16p (1#Q/m ) 5
xyM y # xygJ,# 1!y! gJ,! gJ, . 2E 2
(2.33)
To obtain this result one should make use of the relations kS"0 (transverse polarization), PS"0, kP"Pq/y, 2kq"q, P"k"0 and x"Q/2Pq. In Eq. (2.33) we recognize the term &(y/2)g #g which was mentioned already earlier for the case of pure c-exchange. Note that the transverse cross section is suppressed by M/Q with respect to the longitudinal cross section in (2.31). The corresponding cross section for antineutrinos, lNPl>X (=\ exchange) can be obtained by reversing the sign of g and g in Eqs. (2.31) and (2.33). Note that, to get a nonvanishing e!ect, one must have kS"0 but qSO0. In Sections 4 and 6 a physical (parton model) interpretation will be given for the structure functions g , g and g #g . It should be stressed that until today no satisfactory physical model exists for the &transverse' structure functions g and g !g . We now turn to neutral and charged current interactions initiated by charged leptons, since neutrino-induced reactions on polarized targets are not very realistic, because large nucleon targets are di$cult to polarize. For su$ciently large Q, charged (and neutral) current exchange occurs in l!-induced processes as well. As will become clear later, such an experiment would yield some very important information on the polarized parton densities and will therefore hopefully be carried out in the next century.
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One has separate cross sections for charged and neutral current exchange. For the charged current processes, lNPlN one can take over the results from above, Eqs. (2.31) and (2.33). For the neutral current the cross section consists of three terms, for c-, for Z-exchange and for c-Zinterference, dp&gAA¸AA =IJAA#gA8¸A8=IJA8#g88¸88=IJ88 , IJ IJ IJ where
(2.34)
¸AA "2(k k #k k !g kk !ije k?k @) , (2.35) IJ I J J I IJ IJ?@ ¸A8"(v !ja )¸AA , (2.36) IJ J J IJ ¸88"(v !ja )¸AA (2.37) IJ J J IJ v "!#2s and a "! are the Z}l\-couplings and j"$1 is the helicity of the incoming 5 J J lepton. Furthermore, we have de"ned gAA"1 ,
(2.38)
G m 1 gA8" $ 8 , 2(2pa (1#Q/m8 )
(2.39)
g88"
G m 1 $ 8 . 2(2pa (1#Q/m8 )
(2.40)
Note that G m /2(2pa"1/4s c +1.4 and 2ME"s. All three hadron tensors =AA, =A8 and $ 8 5 5 =88 have an expansion of the form of Eq. (2.28), but of course for =AA one has gAA"gAA"gAA"0 due to parity conservation. All in all, there are 12 free independent polarized structure functions, among them seven (gAA, gA8, g88, gA8, g88, gA8 , g88 ) with and "ve without a parton model > > interpretation. The neutral current cross section for longitudinally polarized leptons is given by [62]
a dpl, ,! (j, S"S )"4pMEy * dx dy Q
2 xyM gGCG 2xy FG # 1!y! (FG #gG ) y 2E GAA 88 A8 y xyM 2 xM xyM gG ! 1# 1!y! gG ! 2jx 1! FG !2jx 2!y! 2 E y E 2E
xM xM gG #2xy 1# gG # 4j E E
(2.41)
where, for negatively charged leptons, CAA"1, CA8"v !ja , C88"(v !ja ) (2.42) J J J J and for positively charged leptons one simply replaces a by !a . Notice that when the lepton #ips J J its helicity, j changes sign, and when the nucleon #ips its spin, all terms containing a polarized
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structure function g also change sign. Upon averaging over j and S one obtains the unpolarized cross section
1 dpl, a 2 xyM dpl, ,! (unp.)" ,! (j, S)"4pMEy gGCG 2xyFG # 1!y! FG , (2.43) 4 dx dy Q y 2E dx dy H 1 G where we have again used s"2ME appropriate for a "xed nucleon target in its rest frame. Alternatively, dp/dx dQ can be simply obtained from dp/dx dQ"(1/sx)dp/dx dy. In the case of nucleons with transverse polarization, i.e. with a spin orthogonal to the lepton direction (z-axis) at an angle a to the x-axis, one has
a 2 xyM y dpl, ,! (j, S"S )"2MEy gGCG 2xy FG # 1!y! FG !2jx 1! FG 2 y dx dy d
Q 2E 2 G #
(xyM[2(1!y)E!xyM] cos(a! ) E
x 1 2 xyM ; !2jxgG !4j gG # gG # 1!y! gG !2xgG y y y 2E
.
(2.44)
To experimentally unravel the whole set of independent structure functions one should make use of leptons of opposite charges and/or polarizations, in which cases the structure functions enter with di!erent weights. Furthermore, use can be made of the propagator structure 1/(1#Q/ m ) , so that one can separate the cc, cZ and ZZ components by measurements at di!erent 8 Q-values. We shall come back to these processes in Section 6.7 where the parton model interpretation and some phenomenological applications will be given.
3. Polarized deep inelastic scattering (PDIS) experiments 3.1. Results from old SLAC experiments After the famous Stern}Gerlach discovery it became possible to produce and use polarized atomic beams. However, experiments with polarized lepton beams are a fairly recent development because before 1972 it was not possible to produce a polarized electron beam. In 1972 physicists from the Yale university succeeded in polarizing electrons by photoionization from polarized alkali atoms. This was used afterwards for elastic scattering experiments between polarized electrons and polarized atoms. As time went by, beam energies increased from the eV level to the level where deep inelastic experiments can be performed. This is } in short } the prehistory of the SLAC experiments. The CERN experiments followed another route. They used polarized muons from high energetic pions. These muons are automatically polarized because of the weak V}A nature of the decay. But let us start with SLAC. High energetic electrons from a polarized alkali source were used in the SLAC experiment E80 which took place in 1976 [43,104]. The electrons in this experiment with
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energies between 6 and 13 GeV, scattered o! a polarized butanol target, were detected by the 8 GeV spectrometer built for all DIS experiments at SLAC. Average beam and target polarizations of experiment E80 were rather high (50% resp. 60%). In the butanol target only the hydrogen atoms contribute to the polarized scattering because carbon and oxygen are spin-0 nuclei. Therefore, the e!ective target polarization is reduced by a &dilution factor' f" ("ratio of number of hydrogen nucleons over total number of nucleons). This dilution e!ect together with other nuclear uncertainties is a problem of all present experiments. The HERMES experiment [207,208] at DESY is using a hydrogen (as well as deuterium or He) gas target [12,41] in which this problem is absent. The main de"ciency of the E80 experiment was that its polarized source was rather limited in beam current. Nevertheless, it was possible to select about two million scattered electron events and to determine the asymmetry A , Eq. (2.10), for the proton for several x values between 0.1 and 0.5, and rather low values of Q (about 2 GeV). The asymmetry turned out to be rather large, in rough agreement with expectations based on the quark-parton model [43,104]. The desire to reduce higher twist e!ects motivated a second SLAC experiment in 1983 (E130) [105]. This experiment was run at an electron beam energy of about 23 GeV. The beam polarization was increased to about 80% and a new MoK ller polarimeter was built which allowed for continuous beam polarization measurements during the experiment. The detector was improved as well so that the kinematic coverage extended in x from 0.2 to 0.65 and in Q from 3.5 to 10 GeV. The experiment concentrated on measurements of rather high x values and consequently collected only about one million events. In Fig. 5 the results for the proton asymmetry AN are shown for E80 and E130 together. Their average of [105] CN (1Q2K4 GeV)"0.17$0.05 (SLAC E80, E130) (3.1) is in good agreement with the value () of the static SU(6) quark model [357,186,440] and the Ellis}Ja!e &sum rule' [333,253,254] to be discussed in Section 5. Here we have de"ned the &"rst moment' of g by CN L(Q), gN L(x, Q) dx . (3.2) However, the result (3.1) is plagued by a large error whose main source comes from the extrapolation into the unmeasured x ranges, in particular as xP0. We shall see that a discrepancy with the CERN data exists which originates from data taken at small x (between 0.01 and 0.1).
3.2. The CERN experiments The CERN PDIS experiments were started as an addendum [490,77,78,549] to the unpolarized EMC deep inelastic muon-nucleon experiments. A polarized beam source of muons with energies E "100}200 GeV was available to hit a polarized ammonia (NH ) target. Due to the high muon I energy, x-values as low as 0.01 could be reached. The results from this experiment which was meant to supplement the SLAC data at small-x and to con"rm the Ellis}Ja!e &sum rule' [333,253,254] came as a major surprise. Actually, they
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Fig. 5. The spin asymmetry of the proton from the &old' SLAC data from 1976 and 1983 [365]. In the naive SU(6) model one has AN " and AL "0 [186].
con"rmed the SLAC measurements in the common large-x region but found too small asymmetries in the small-x region, in disagreement with the expectations of Gourdin, Ellis and Ja!e and the naive quark parton model. The high energetic muon beam is a low-luminosity beam because it is a secondary beam from the semileptonic decay of pions which are produced in proton collisions. Its main advantage is its high energy which not only allows to enter the small-x region but also guarantees higher Q-values in the region of intermediate-x and a corresponding suppression of possible higher twist e!ects. In the rest system of the pions the emitted muons are 100% left-handed, due to the V}A nature of the decay. In the laboratory frame the muons have a degree of polarization which depends on the ratio E /E . For the EMC experiment the beam was selected such that the polarization of the muon I p beam was about 80%. The muon beam polarization was determined from the muon event distribution via a Monte Carlo study of muon production } an indirect and not really satisfactory method. The ammonia target was quite large, with a length of about 2 m, and separated into two halves with opposite polarization. On the average only a fraction f, the dilution factor, of the target protons were polarized. The polarization of the target could be determined as a function of its length (by NMR coils placed along the target). The scattered muons were detected by a well-established muon tracking spectrometer. It was possible to reconstruct the muon scattering vertex to determine that half of the target in which the scattering took place. The target spins were reversed (once per week) 11 times in the experiment. Changes in the spectrometer acceptance from one spin reversal to another was the main systematic uncertainty in the experiment. In Fig. 6 the combined results from SLAC and CERN on the structure function gN are presented. The low values of the asymmetries at small-x translate into low values for the proton structure function and a low value for its "rst moment [490,77,78] CN (1Q2)"0.126$0.010$0.015 (EMC, SLAC) ,
(3.3)
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Fig. 6. Combined results of SLAC and EMC on gN [78]. The dashed curve is an EMC parametrization of the data.
where the "rst is the statistical and the second is the systematic error which includes, as always, uncertainties from Regge extrapolations to xP0; the average values of Q are 1Q2" 10.7 GeV (EMC) and 1Q2"4 GeV (SLAC). It should be emphasized that almost all of gN dx stems from the region x50.01 which gives [490,77,78], instead of (3.3),
gN (x,1Q2) dxK0.123 .
(3.4)
If, on the contrary, one does not combine the EMC and SLAC measurements, the EMC asymmetry measurements alone imply similar results, namely [490,77,78] CN (1Q2"10.7 GeV)"0.123$0.013$0.019 (EMC) .
(3.5)
The surprising EMC result triggered a second CERN experiment with polarization, the so-called SMC experiment [25,23,549,29]. Instead of ammonia a polarized butanol target has been used. In butanol the only polarized nucleons are the protons (K12%) and deuterons (K19%). The goal of this experiment was to infer information about the neutron from the di!erence gB !gN . Butanol allows for a much more rapid spin reversal than ammonia, this way reducing the large systematic uncertainty from the varying detector acceptance. With the alcohol targets, spin reversals took one hour and were implemented every 8 h. The main improvement of the SMC experiment was in the measurement of the beam polarization. It was obtained from the energy spectrum of positrons from muon decay. The positron energy spectrum is rather sensitive to the muon beam polarization and provides a direct measurement. In Figs. 7 and 8 the SMC results for AB and gB are shown which are plotted as a function of ln x to make the small-x results more transparent. For x less than 0.1 the results are compatible with zero although with rather limited statistics. The theoretically more interesting longitudinally polarized neutron structure function gL (x, Q) can be obtained via the relation gB (x, Q)"[gN (x, Q)#gL (x, Q)](1!w ) "
(3.6)
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Fig. 7. The virtual photon}deuteron cross section asymmetry as measured by SMC [23] and compared to the SLAC E143 data [3]. Only statistical errors are shown. The size of the systematic errors is indicated by the shaded area. Fig. 8. gB deduced from AB in Fig. 7 evolved to a common Q"5 GeV [23]. Only statistical errors are shown. The size of the systematic errors is indicated by the shaded area.
where w (K0.058) accounts for the D-state admixture in the deuteron wave function [25]. The " resulting [26] gL (x, Q"5 GeV) is shown in Fig. 9 where the &old' EMC [490,77,78] and SLAC [43,104,105] data have been used for gN , all reevaluated at Q"5 GeV under the assumption that the asymmetries AB N are independent of Q. (This latter assumption is theoretically questionable, at least in the small-x and larger-Q region where no data exist so far, as will be discussed in Section 6.) Also shown in Fig. 9 are the SLAC (E142) data [65], to be discussed next, which agree with the SMC results in the x region of overlap. The SMC results of Fig. 9 imply [25] CL (Q"5 GeV)"!0.08$0.04$0.04 (SMC) (3.7) which still deviates from the Ellis}Ja!e expectation !0.002$0.005. More recently, SMC has also measured gN (x, Q) [17,18,24] shown in Figs. 10 and 11, which results in CN (Q"10 GeV)"0.136$0.013$0.011 (SMC) . This result increases to [17,18,24]
(3.8)
CN (Q"5 GeV)"0.141$0.011 (SMC, EMC, SLAC) (3.9) if the &old' EMC and SLAC measurements are included which led to (3.3). Eq. (3.9) represents, for the time being, probably the best estimate of the full "rst moment (04x41) of gN (x, Q) at Q"5 to 10 GeV. All the above results are still signi"cantly below the Gourdin}Ellis}Ja!e [333,253,254] expectation (assuming a vanishing total polarization of strange sea quarks in the LO-QCD parton model expression for C in, e.g., Eq. (5.14) below) of about 0.18$0.01 which will be discussed in more detail in Sections 5 and 6.1.
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Fig. 9. gL as a function of x at Q"5 GeV. The full circles are the data from the SLAC E142 experiment. The dashed and solid curves show the extrapolations to low x using the E142 data and using the combined data, respectively [26]. Fig. 10. SMC/EMC results for gN at the average Q for each x bin [17,18,28]. Only statistical errors are shown. The size of the systematic errors is indicated by the shaded area.
Fig. 11. The solid circles show the SMC results for gN as a function of x, at Q "10 GeV. The open boxes show the integral from x to 1 (left-hand axis). Only statistical errors are shown. The solid square shows the result for the "rst moment (integral from 0 to 1), with statistical and systematic errors combined in quadrature [17,18].
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SMC [27,30] and HERMES [13] have also presented semi-inclusive p! measurements which give direct access to the polarized valence densities du (x, Q) and dd (x, Q). More details will be T T discussed in Section 6.5. Due to the use of (longitudinally) polarized high current electron beams, measurements of gN L(x, Q) with much higher statistics are obtained from the latest round of SLAC experiments (although at signi"cantly lower values of Q due to the lower beam energy) to which we turn now. 3.3. The new generation of SLAC experiments The SLAC E142 experiment [65] used a di!erent electron source than the old SLAC experiments [43,104,105] relying on developments on solid-state GaAs cathodes. They were able to produce high current electron beam pulses with energies 19.4, 22.7 and 25.5 GeV and through that a rather high statistics experiment. Altogether, 300 million events could be collected. In addition, reversal of beam spin direction could be implemented randomly. The target was polarized He. From the measurements on He the neutron structure function can be directly inferred because the polarization e!ects from the two protons inside the helium compensate each other, due to the Pauli principle. This is true to the extent that the helium nucleus is in its S-state. The probability for this is about 90%. The remaining 10% probability with which the two proton spins are parallel can be corrected for. There is another nuclear uncertainty at low Q (1Q2K2 GeV), namely the possible exchange of mesons (o's and p's) which is a bit more dangerous for He than for the deuteron because He has a larger binding energy per nucleon than d. Furthermore, its magnetic moment is a worse approximation to the free neutron than k is to k #k . B N L The He target is polarized by optical pumping. Circularly polarized near infrared laser light illuminates the target cell with He and rubidium vapor. The outer shell electrons in the rubidium become polarized and transfer their polarization to the He nucleus via spin exchange collisions. Once achieved, the polarization of the He is rather stable. The entire target chamber was placed in a constant magnetic "eld which holds the He spins in a "xed orientation. Target spin reversal was achieved several times per day and used to reduce the systematic error. The results for gL are shown in Fig. 12 and can be compared with the more recent SLAC E143 experiment [3] in Fig. 13. These experiments span the Q-range 1(Q(10 GeV, corresponding to 0.029(x(0.8 at their respective energies, where the E143 polarized electron beam energies refer to 9.7, 16.2 and 29.1 GeV. The E143 data for gL in Fig. 13 imply [3] CL (1Q2"2 GeV)"!0.037$0.008$0.011 (E143) (3.10) compared with !0.022$0.007$0.009 from the E142 data [65]. These results are consistent with the less accurate SMC measurement in (3.7). The E143 experiment [3,4] uses a deuterated ammonia (ND ) target, polarized by dynamic nuclear polarization in a 4.8 T magnetic "eld, in order to check the SMC experiment at lower Q. The comparison of their measured asymmetries AB (x, Q) in Fig. 7 demonstrates that these two experiments are consistent with each other although it should be kept in mind that the Q-value of SMC is about twice as large, for each speci"c x-bin, as of E143; the average Q of the latter experiment varies from 1.3 GeV (at low x) to 9 GeV (at high x). A comparison of the integral of gL , gL (x , Q) dx , for each x-bin as lower V integration limit is shown in Fig. 14 which eventually (as xP0) leads to the full &"rst moment' of
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Fig. 12. Result for gL at Q"5 GeV from SLAC (E142) [65]. The most recent SLAC (E154) results are also shown for comparison [7]. Fig. 13. xg for (a) the deuteron at Q"3 GeV and (b) the neutron at Q"2 GeV as measured by E142 and E143 [3]. Systematic errors are indicated by the shaded bands.
Fig. 14. Comparison between SLAC and CERN of the integral of gL at the average Q of the respective experiments.
gL as stated in Eqs. (3.7) and (3.10). The results shown in Fig. 14 are particularly interesting because they clearly demonstrate that the di!erence between SLAC and SMC/EMC comes mainly from the small-x region, x(0.03, where SLAC has no data points and thus has to fully rely on assumptions about the behavior of g as xP0. Furthermore the E143 experiment used also an ammonia target [2] in order to check, at lower Q, the SMC/EMC proton measurements for gN . Their results are again compatible with the ones
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Table 2 Experimental results for the g -integrated quantity C (Q) in Eq. (3.2) Experiment
C (Q/GeV)
Reference
EMC, SLAC SMC E143 SMC SMC SMC E142 E143 E154
CN (10.7)"0.126$0.010$0.015 CN (10)"0.136$0.013$0.011 CN (3)"0.127$0.004$0.010 CL (10)"!0.063$0.024$0.013 CL (5)"!0.08$0.04$0.04 CL (5)"!0.048$0.022 CL (2)"!0.022$0.007$0.009 CL (2)"!0.037$0.008$0.011 CL (5)"!0.041$0.004$0.006
[77,78] [17,18,24] [2] [20] [25] [24] [65] [3] [7]
of SMC in the x-region of overlap and give [2] CN (Q"3 GeV)"0.127$0.004$0.010 (E143) (3.11) to be compared with Eqs. (3.3)}(3.5) and (3.8). Since the latter SMC result/estimate (3.8) holds at Q"10 GeV, it is more appropriate to calculate the integral at Q"3 GeV, assuming gN /FN KAN to be independent of Q, which gives 0.122$0.011$0.011, instead of Eq. (3.8), and compares better with the E143 result (3.11). These various consistent measurements imply that by now, after 20 years of having performed polarized deep inelastic experiments, we have available a rather reliable and su$ciently precise result for gN (x, Q) dx which is con"dently more than two standard deviations below the Ellis}Ja!e}Gourdin sum rule expectation (no polarized strange sea [333,253,254]) of CN (Q"3 GeV) "0.160$0.006 [2]. We shall come back to this point in #( Section 5. The integrated quantities CN L(Q) which resulted from all polarization experiments discussed and performed thus far [574] are "nally summarized in Table 2. There are several new experiments at SLAC (see e.g. [366]), the experiments E154 [7] and E155 [66] with a He target [365] and an ammonia target [71], respectively, which run at 50 GeV beam energy and which are the follow-up experiments to E142 and E143. Clearly, with SLAC statistics, these measurements will provide us with a powerful test of the larger Q and small x (90.01) dependence of the spin structure functions gN L, as can be seen from the comparison with the "rst results of E154 with E142 in Fig. 12. 3.4. Future polarization experiments The HERMES experiment [207,208,89,12,41], being a "xed target experiment, takes place in the HERA tunnel with a longitudinally polarized electron beam of about 30 GeV incident on a polarized H, D or He gas jet target. Although this experiment is performed at a similarly low energy as present SLAC measurements, the novel technique of polarized gas-jet target technology is expected to allow for high-precision measurements since the target atoms are present as pure atomic species and hence almost no dilution of the asymmetry occurs in the scattering from unpolarized target
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material. The main advantages: HERMES will use a pure proton gas target in a thin wall storage cell, so that the dilution factor will be close to one and there is almost no background from windows e!ects. The thin targets will still provide very good statistics because the beam current is enormous. Even a measurement of the structure function g is conceivable. Finally, the spectrom eter allows for multiparticle identi"cation and thereby measurements of semiinclusive cross sections from which additional information on valence, sea and strange quark polarization can be obtained. In addition, polarized internal gas targets allow for a rapid reversal of the target spin [207,208] which will be crucial for understanding and minimizing systematic errors. This rapid spin reversal will be also crucial for measuring directly neutron asymmetries by using a He gas target where spins of the two protons are practically in opposite directions and thus the He target acts as an e!ective neutron target. Actual data taking has started in 1995/96 and the accessible kinematic region (0.024x40.8, 14Q410 GeV) will be similar to the one of present SLAC experiments. It is intended to measure gN L B and gN L B and in particular semi-inclusive asymmetries Ap from el pl (no )PepX which allows to extract separately the polarized valence quark densities N L dd (x, Q) and du (x, Q). More details will be discussed in Section 6. T T New experiments like COMPASS [106,486] (consisting partly of the former SMC) and HERMES [355] have been proposed for measuring semi-inclusive (D-mesons, etc.) reactions in deep inelastic kl pl (do ) scattering at 100}200 GeV ko -beam energies and 27.5 GeV eo -beam energies, respectively. These will be dedicated experiments for measuring, among other things, the polarized gluon distribution dg(x, Q) which plays a predominant role in understanding the nucleon spin structure and which so far is experimentally entirely unknown. It will be deduced from the production of heavy quarks (like charm) via the fusion process co H go Pcc responsible for open charm production which is one of the most promising and cleanest processes for extracting dg(x, Q) since it occurs already in the leading order (LO) of QCD with no light quark contributions present (see Section 6.2). It is expected to achieve a sensitivity for dg/g of about 15%. Note, however, that this cross section is not sensitive to the &anomalous' (""rst moment) part of the polarized gluon density, because the "rst moment contribution of polarized gluons to heavy quark production vanishes. These issues will be discussed in detail in Sections 5 and 6. So far we have concentrated on deep inelastic lo No reactions. Purely hadronic reactions are presently being studied at the Fermilab Spin Physics Facility which consists of a 200 GeV longitudinally polarized proton or antiproton beam incident on a "xed polarized proton target [14,584]. Apart from the non-uniquely polarized pentanol target, the beam energy is rather low for a purely hadronic reaction ((s"19.4 GeV). Nevertheless, "rst measurements resulted in a small, almost vanishing longitudinal spin asymmetry Ap for inclusive p-production at small pp ** 2 (between 1 and 4 GeV). This measurement is naively more consistent with a small polarized gluon component in the proton than with a large one, but has a very large error. An updated result is being prepared by E-704 as well as results on the totally inclusive polarized cross section and polarized hyperon, direct photon and J/t production [21,22]. All of these have limited statistics and thus will be of very limited use for perturbative QCD interpretations. Using the 200 GeV proton beam, E-704 have also measured transverse single-spin asymmetries in inclusive pion production, pt#pPp!#X and found appreciable asymmetries [16]. More details will be discussed in Section 6.8. On the other hand, there is the upcoming very promising experimental spin program of the Relativistiv Heavy Ion Collider (RHIC) Spin Collaboration (RSC) [148,585] at the Brookhaven
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National Laboratory, to which many of the E-704 physicists have switched. At RHIC both proton beams, with an average energy of 250 GeV each, will be polarized, using the &Siberian snake' concept [216,217,425,514], to an expected polarization of about 70%. Due to the high luminosity of the order of &10 cm\ s\ (corresponding to an integrated luminosity of about 800 pb\) the polarized RHIC pp collisions will play a decisive role in measuring the polarized gluon density. This can be achieved either via heavy quark production (gl gl PQQM , etc.) or via direct photon production (gl ql Pcq, etc.). The latter process is particularly promising, and a sensitivity of about 5% is expected for dg/g. The theory of these processes will be discussed in Section 6. Of course, polarization asymmetries for semi-inclusive pion production and jet production will be also measured, in particular for =! and Z production which give access to the various polarized quark densities du, du , dd and ddM separately (cf. Section 6). Furthermore, transversity distributions (see Section 8.2) will also be accessible. Some other theoretical aspects of RHIC are discussed in [136]. The build-up of the RSC experiment is depicted in Fig. 15. The protons are taken from a polarized H\ source and are succesively accelerated by a LINAC, a Booster and the Alternating Gradient Synchrotron (AGS) to an energy of 24.6 GeV. In the AGS a &partial snake' is built in to maintain the polarization. A successful partial snake test has been carried out in 1994 by the E880
Fig. 15. The built-up of the RHIC collider. Fig. 16. The layout of a Siberian snake.
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collaboration [120,364,329]. Within the RHIC tunnel two full Siberian snakes will be installed, more precisely four split Siberian snakes, two per ring, 1803 apart. This is expected to be complete in 2000, at which time the PHENIX [334] and STAR [348] detectors should also be completed. First data taking is scheduled for 2000 as well. For the nonexperts we include here a qualitative description of how a &Siberian snake' works. Siberian snakes [216,217] are localized spin rotators distributed around the ring to overcome the e!ects of depolarizing resonances. For spin manipulations on protons at high energy, one should use transverse magnetic "elds, because the impact of longitudinal "elds disappears at high energy. In a transverse B-"eld the spin of the high energy protons rotates i c times faster than motion N (c"E/m and i "1.79 is the anomalous magnetic moment of the proton). During acceleration, N a depolarizing resonance is crossed if this product is equal to an integer or equals the frequency with which spin-perturbing magnetic "elds are encountered. A Siberian snake turns the spin locally by an angle d (d"1803 for a full snake, dO0 for a partial snake) so that the would-be resonance condition is violated. For the two full Siberian snakes in the RHIC collider the number of 3603 spin rotations per turn is , i.e. the resonance condition can never be met. The evolution of the spin and orbit motion in the snake area are shown in Fig. 16. Coming back to the future of polarized experiments, we note that the possibility of polarized protons is also being considered at HERA. The concept for HERA is in principle very similar to RHIC, the role of the AGS being taken by DESYIII and PETRA which successively accelerate the protons to 40 GeV [123]. There have been further Workshops on the future (spin) physics at HERA [369,124] where the option of polarized high energy protons (besides polarized electrons) at HERA has been discussed. We refer to the proceedings of these Workshops for extensive discussions of this topic [123,124,369]. Of particular interest would be a fully polarized HERA (el pl ) collider. Such a el pl collider would be unique for studying &small-x' (x:10\) physics of g (x, Q) and df (x, Q), f"q, q , g which will be discussed in the next section, in particular Section 4.4; it will give access to the experimentally entirely unknown (partonic) structure functions of a resolved polarized photon to be discussed in Section 6.12; and last, but not least, it will enable measurements of the electroweak spin structure functions [124] in Section 2.4, as will be discussed in more detail in Section 6.7. Another experiment (&HERA-No ') utilizing an internal polarized "xed nucleon target in the 820 GeV HERA proton beam has also been examined, see also [423] and references therein. Conceivably, this would be the only place where to study high-energy nucleon}nucleon spin physics besides the dedicated RHIC spin program. An internal polarized nucleon target o!ering unique features such as polarization above 80% and no or small dilution, can be safely operated in a proton ring at high densities up to 10 atoms/cm [547]. As long as the polarized target is used in conjunction with an unpolarized proton beam, the physics scope of HERA-No would be focussed to &Phase I', i.e. measurements of single-spin asymmetries (to be discussed in Section 6.8). Once polarized protons should become available later, the same set-up would be readily available to measure a variety of double-spin asymmetries. These &Phase II' measurements would constitute an alternative "xed target approach to similar physics which will be accessible to the collider experiments STAR and PHENIX at the low end of the RHIC energy scale ((sK50 GeV). Furthermore, there are several polarized low energy (EC\ :5 GeV) facilities, such as AmPS NIKHEF, MIT-Bates, CEBAF-Newport News, ELSA-Bonn and MAMI-Mainz, some of which are already operating and will provide us with low-energy precision measurements of eo No reactions. The interested reader is referred to the respective review articles in [264].
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4. The structure function u and polarized parton distributions 4.1. The quark parton model to leading order of QCD The parton model is a very useful tool for the understanding of hadronic high-energy reactions. This is due to its simplicity and comprehensiveness as well as to its universality, i.e. applicability to any hadronic process. In the form of the QCD improved parton model it has had tremendous successes in the understanding of unpolarized scattering. One is therefore tempted to apply it to processes with polarized particles as well. In the following we will assume that the nucleon is longitudinally polarized. The notion of transversality is di$cult to adopt in the parton model, only at the price of losing many of its virtues. The main aim of this section is to represent the polarized structure functions in terms of &polarized' parton densities, in a similar fashion as the spin-averaged structure functions can be represented in terms of spin-averaged parton densities. For de"niteness, let us consider the process kpPkX at scales Q;m (only photon exchange needed), with a proton of positive longitudinal polariza5 8 tion ("right-handed helicity). Within this reference proton there are (massless) partons with positive and negative helicity which carry a fraction x of the proton momentum and to whom one can associate quark densities q (x, Q) and q (x, Q). The di!erence > \ dq(x, Q)"q (x, Q)!q (x, Q) (4.1) > \ measures how much the parton of #avor q &remembers' its parent proton polarization. Similarly, one may de"ne dq (x, Q)"q (x, Q)!q (x, Q) > \ for antiquarks. Note that the ordinary, spin-averaged parton densities are given by q(x, q)"q (x, Q)#q (x, Q) > \
(4.2) (4.3)
and q (x, Q)"q (x, Q)#q (x, Q) . (4.4) > \ In the quark parton model, to leading order (LO) in QCD, g can be written as a linear combination of dq and dq [266,54,45], 1 (4.5) g (x, Q)" e[dq(x, Q)#dq (x, Q)] , O 2 O where e are the electric charges of the (light) quark-#avors q"u, d, s. Notice that in the case of O spin-averaged structure functions F the negative helicity densities q , q enter with an opposite \ \ sign, e.g. F (x, Q)" e(q#q ). This has to do with the opposite charge conjugation property of O O c and c c . In the case of the polarized lN structure functions g , etc., the situation is reversed. I I There the dq (x, Q) enter with a negative sign, g &dq!dq , cf. Section 6; and for the unpolarized lN structure function F one has F &q!q . Furthermore, Eq. (4.5) can be decomposed into a #avor nonsinglet (NS) and singlet (S) component g (x, Q)"g (x, Q)#g (x, Q) , ,1 1
(4.6)
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where 1 (4.7) " (e!1e2)(dq#dq ) O 2 O with 1e2"(1/f ) e (e.g. for f"3 light u, d, s #avors 1e2") and O O 1 1 (4.8) g " 1e2 (dq#dq ), 1e2dR . 1 2 2 O In the last equation we have de"ned the singlet combination (sum of ) quarks and antiquarks by g
,1
dR(x, Q)" [dq(x, Q)#dq (x, Q)] , (4.9) O where the sum usually runs over the light quark-#avors q"u, d, s, since the heavy quark contributions (c, b,2) have preferrably to be calculated perturbatively from the intrinsic light quark (u, d, s) and gluon (g) partonic-constituents of the nucleon which will be discussed in Section 6.2. The QCD scale-violating Q-dependence of the above structure functions and parton distributions is inherently introduced dynamically due to gluon radiation (qPqg) and gluon-initiated (gPqq ) subprocesses depicted, to LO, in Fig. 17. To the leading logarithmic order (LO), these Q-corrections have been calculated in [54,36,37,520]. The next-to-leading order NLO (two-loop) results (Wilson coe$cients and in particular splitting functions) have recently been calculated [478,568,569] and will be discussed in detail in Section 4.2. One of the main ingredients from QCD (or, more generally, from any strongly interacting quantum "eld theory) is the appearance of gluon distributions in the nucleon in the form dg(x, Q) which is the longitudinally polarized gluon density, probed at a scale Q, and is de"ned as follows. Assume that in our reference proton of positive helicity the gluons have momenta of the form p "E(1, 0, 0, 1) [in the Breit-frame E P"((P#M, 0, 0, P), q"(0, 0, 0,!Q) in which E"xP]. The two possible polarization vectors of the gluon are eI"(1/(2)(0, 1,$i, 0) which correspond to positive and negative circular E polarization. To each state of polarization one can attribute a gluon density, g (x, Q) and >
Fig. 17. The parton subprocesses cHqPgq and cHgPqq .
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B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
g (x, Q). The ordinary, spin-averaged gluon density is given by g(x, Q)"g #g , whereas dg is \ > \ de"ned as dg(x, Q)"g (x, Q)!g (x, Q) . (4.10) > \ Since the individual parton distributions with de"nite helicity f (x, Q), f"q, q , g, in Eqs. (4.1)}(4.4) ! and (4.10) are by de"nition positive de"nite, their di!erence df has to satisfy the general positivity constraints "df (x, Q)"4f (x, Q) .
(4.11)
In LO the gluon distribution does not directly contribute to the structure function g (x, Q) in (4.5), but only indirectly via the Q-evolution equations. Furthermore, it is a pure #avor-singlet, as is dR(x, Q) in (4.9), because each massless quark #avor u, d, s is produced by gluons at the same rate. The LO Q-evolution (or renormalization group) equations are as follows. Only #avornonsinglet (valence) combination dq ["du!du , dd!ddM , (du#du )!(dd#ddM ), (du#du )# ,1 (dd#ddM )!2(ds#ds ), etc.], i.e. sea- and gluon-contributions cancel, evolve in the same way in LO: a (Q) d dq (x, Q)" Q dP dq , ,1 ,1 2p dt ,1
(4.12)
where t"ln Q/Q , with Q being the appropriately chosen reference scale at which dq is ,1 determined (mainly from experiment), and a (Q) 1 Q K (4.13) 4p b ln Q/K *with b "11! f and f is the number of active #avors. The convolution ( ) is given by dy x (P q)(x, Q)" P q(y, Q) (4.14) y y V which goes over into a simple ordinary product if one considers Mellin n-moments to be discussed later. The LO NS splitting function,
dP (x)"dP(x),P !P , (4.15) ,1 OO O> O> O\ O> where P ! > corresponds to transitions from a quark q with positive helicity to a quark q with > ! O O positive/negative helicity, is given by [54]
1#x dP(x)"P(x)"C OO OO $ 1!x
(4.16) > with C "4/3. The fact that dP turns out to be equal to the unpolarized splitting function $ OO P, i.e. OO P (x)"0 (4.17) O\ O>
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
29
Fig. 18. The leading-order splitting process qPqg.
in Eq. (4.15), is a consequence of helicity conservation, i.e. the fact that no transition between quarks of opposite helicity are allowed in massless perturbative QCD } at least to leading order. In suitable (chirality respecting) regularization schemes this statement, i.e. Eq. (4.17), can be generalized to higher orders as we shall see later. The LO diagram is shown in Fig. 18: The conservation of the quark helicity is a consequence of the vector-like coupling between quarks and gluons. Furthermore, since the gluon has spin #1 or !1, a "nite angular momentum between the quark and gluon is always produced in such a process. Finally, the convolution (4.14) with the ( ) > distribution [54] in (4.16) can be easily calculated using
dy x dy x g(y)" f f y y y y > V V
x V g(y)! g(x) !g(x) dy f (y) . y
(4.18)
In contrast to (4.12), the LO Q-evolution equations in the #avor-singlet section are coupled integro-di!erential equations,
2fdP dR a (Q) dP d dR(x, Q) OO OE " Q 2p dP dP dt dg(x, Q) dg EO EE
(4.19)
with dP(x) given by Eq. (4.16). The remaining longitudinally polarized splitting functions are OO de"ned, in analogy to (4.15), dP (x),P>
>
!P\
(4.20)
>
with A, B"q, g, which ful"l PG \ "P! > due to parity invariance of the strong interactions (QCD). The spin-averaged splitting functions are given by the sum P (x)"P> > #P\ > . Note that dR refers to the sum of all quark #avors and anti#avors and therefore the factor 2f in front of dP in (4.19). Besides dP in (4.16), the remaining polarized LO splitting function in (4.19) are OE OO given by [54] dP(x)"¹ [x!(1!x)]"¹ (2x!1) , OE 0 0 dP(x)"C EO $
1!(1!x) "C (2!x) , $ x
1 1 dP(x)"C (1#x) # EE x (1!x)
>
(4.21)
11 (1!x) f ! # ! d(1!x) , 6 x 9
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with ¹ " and C "N "3. The advantage of introducing the single set of di!erences of parton 0 A distributions (dR, dg) in (4.19), instead of using the individual positive and negative helicity densities q and g in (4.1) and (4.10), is that they evolve independently in Q of the set of the conventional ! ! unpolarized densities (R, g) where R, (q#q ). This is in contrast to the individual densities D q and g of de"nite helicity. The evolution equations (4.12) and (4.19) can be solved numerically ! ! by iteration directly in Bjorken-x space. In many cases it is, however, more convenient and physically more transparent to work in Mellin n-moment space where the LO as well as the NLO evolution equations can be solved analytically to a given (consistent) perturbative order in a . This Q is due to the fact that for moments, de"ned by
xL\f (x, Q) dx , (4.22) the convolution (4.14) appearing in (4.12) and (4.19) factorizes into simple ordinary products: f L(Q),
dy x f (y)g y y V (4.23) " dx xL\ dy dz d(x!zy) f (y)g(z)"f LgL . In moment space the LO nonsinglet and singlet evolution equations (4.12) and (4.19) are thus simply given by dx xL\f g,
dx xL\
a (Q) d dqL (Q)" Q dPLdqL (Q) , OO ,1 2p dt ,1
(4.24)
2fdPL dRL(Q) a (Q) dPL d dRL(Q) OO OE " Q , 2p dPL dPL dt dgL(Q) dgL(Q) EO EE where the dPL are simply the nth moment of Eqs. (4.16) and (4.21): GH 1 4 3 # !2S (n) , dPL" OO 3 2 n(n#1)
(4.25)
1 n!1 , dPL" OE 2 n(n#1) 4 n#2 dPL" , EO 3 n(n#1)
11 4 2f # !2S (n) ! . dPL"3 EE 6 n(n#1) 32
(4.26)
Here S (n), L 1/j"t(n#1)#c , t(n),C (n)/C(n) and c "0.577216. H # # The evolution equations (4.24) and (4.25) in n-moment space are usually referred to as LO renormalization group (RG) equations which were originally derived [181,155,420,338,
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
31
339,295,212,267,288] from the operator product (light-cone) expansion for unpolarized structure functions (for reviews see, for example, [45,492,509]). The moments dPL are called (or, more GH precisely, related to the) &anomalous dimensions' because they determine the logarithmic Qdependence of the moments of parton distributions and thus of g as we shall see later. The solution of the simple NS equation (4.24) is straightforward: OO dqL (Q ) dqL (Q)"¸\ @ B.L (4.27) ,1 ,1 with ¸(Q),a (Q)/a (Q ), b being de"ned after (4.13) and dqL (Q ) is the appropriate NSQ Q ,1 combination of polarized parton densities "xed (mainly) from experiment at a chosen input scale Q . The solution of the coupled singlet evolution equations (4.25) is formally similar to (4.27): dRL(Q) dRL(Q ) , "¸\ @ B.K L (4.28) dgL(Q) dgL(Q ) where dPK L denotes the 2;2 singlet matrix of splitting functions in Eq. (4.25). The treatment of this exponentiated matrix follows the standard diagonalization technique, see e.g. [338,339], where one projects onto the larger and smaller eigenvalues jL of dPK L with the help of the 2;2 ! projection matrices PK given by ! dPK L!jL 1 8 (4.29) PK ,$ ! jL !jL > \ with
1 jL " [dPL#dPL$((dPL!dPL)#8f dPLdPL ] . ! 2 OO EE OO EE OE EO
(4.30)
The projection matrices PK have the usual properties PK "PK , PK PK "PK PK "0 and ! ! ! > \ \ > PK #PK "1. Since dPK L"jL PK #jL PK , the matrix expression in (4.28) can be explicitly > \ > > \ \ calculated using f (dPK L)"f (jL )PK #f (jL )PK , > > \ \
(4.31)
i.e. ¸\ @ B.K L"¸\ @ HL> PK #¸\ @ HL\ PK . > \ Thus the solution of (4.28) is explicitly given by
(4.32)
dRL(Q)"[a dRL(Q )#b dgL(Q )]¸\ @ H L\#[(1!a )dRL(Q )!b dgL(Q )]¸\ @ H L> , L L L L (4.33)
The anomalous dimensions dcL, being usually de"ned as an expansion in a /4p, dcL"(a /4p)dcL#(a / Q Q Q 4p)dcL#2, in terms of the LO (one-loop) dcL and NLO (two-loop) dcL expressions, are related to the dPL via dPL"!dcL and dPL"!dcL, where the two-loop splitting functions dP will become important for the GH GH GH GH GH NLO evolutions to be discussed in Section 4.2.
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B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
a (1!a ) L dRL(Q ) ¸\ @ HL\ dgL(Q)" (1!a )dgL(Q )# L L b L a (1!a ) L dRL(Q ) ¸\ @ HL> # a dgL(Q )! L L b L
(4.34)
with a "(dPL!jL )/(jL !jL ) , (4.35) L OO > \ > b "2fdPL/(jL !jL ) . (4.36) L OE \ > Once the parton distributions are "xed at a speci"c input scale Q"Q , mainly by experiment and/or theoretical model constraints, their evolution to any Q'Q is uniquely predicted by the QCD-dynamics due to the uniquely calculable &anomalous dimensions' dPL in LO of a . Of GH Q course a LO calculation by itself is in general not su$cient since neither the K-parameter in a (Q) Q can be unambiguously de"ned nor can one prove the perturbative reliability of the results which requires at least a NLO analysis, to which we will turn in the next subsections. To obtain the x-dependence of structure functions and parton distributions, usually required for practical purposes, from the above n-dependent exact analytical solutions in Mellin-moment space, one has to perform a numerical integral in order to invert the Mellin transformation in (4.22) according to
1 (4.37) dz Im[e Px\A\X Pf LA>X P(Q)] , f (x, Q)" p where the contour of integration, and thus the value of c, has to lie to the right of all singularities of f L(Q) in the complex n-plane, i.e., c'0 since according to Eq. (4.26) the dominant pole of all dPL is GH located at n"0. For all practical purposes one may choose c+1, u"1353 and an upper limit of integration, for any Q, of about 5#10/ln x\, instead of R, which su$ces to guarantee accurate and stable numerical results [314,316]. Note that it is advantageous to use u'p/2 in (4.37) because then the factor x\X P dampens the integrand for increasing values of z which allows for a reduced upper limit in the numerical integration; this is in contrast to the standard mathematical choice u"p/2 which corresponds to a contour parallel to the imaginary axis. 4.2. Higher-order corrections to g The LO results discussed so far originated from calculating the logarithmic O(a ) contributions Q of the parton subprocesses cHqPgq and cHgPqq (Fig. 17) to the zeroth-order &bare' term cHqPq of g [45,54]: a (Q) dy x x 1 dq (y) tdP #df g (x, Q)" e dq (x)# Q OO y O y 2p y 2 O V O O a (Q) dy x x 1 # e Q dg (y) tdP #df , (4.38) OE y E y 2p y 2 O V O O
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
33
where dq , dg denote the unphysical and unrenormalized (i.e. scale independent) bare parton distributions and the function df and df are sometimes called &constant terms' or &coe$cient O E functions' because they are related to the ln Q-independent terms and to the Wilson coe$cients usually introduced within the framework of the operator product expansion. In the leading logarithmic order (LO) one assumes the dominance of the universal t"ln Q/Q terms which are independent of the regularization scheme adopted, in contrast to df . O E Eq. (4.38) as it stands is physically meaningless since Q is an entirely arbitrary scale which, among other things, serves as an e!ective cuto! for the emitted parton-k (or angular) phase space 2 integration in order to avoid collinear (mass) singularities when the emitted partons in Fig. 17 are along the intitial quark/gluon direction (k "0 or h"0). Therefore, only the variation with Q can be 2 uniquely predicted for the physical and renormalized (i.e. scale dependent) &dressed' quark distribution, a (Q) t(dq dP#dg dP) dq(x, Q),dq (x)# Q OO OE M 2p
(4.39)
and similarly for a suitably de"ned &dressed' polarized gluon density dg(x, Q). These rede"nitions result in the RG evolution equations (4.12) and (4.19) [54]. In NLO, i.e. if one goes beyond the leading logarithmic order, the &"nite' terms df in (4.38) O E have to be included, as well as the contributions of the two-loop splitting functions dP(x). These GH additional quantities have the unpleasant feature that they depend on the regularization scheme adopted. For calculational convenience one often chooses dimensional regularization and the 't Hooft-Veltman prescription for c [362,141,163]. In D"4!2e dimensions one obtains for the diagrams in Fig. 17 [478,503,565,566,128,466,259] 1 g (x, Q)" e+dq (x) 2 O O O x x Q 1 a (Q) dy dq (x) ln ! #c !ln 4p dP #dC # Q # OO O y y y k e 2p V 1 Q 1 a (Q) dy x x # e Q dg (y) ln ! #c !ln 4p dP #dC , (4.40) # OE y E y 2 O k e 2p y V O O where the dimensional regularization mass parameter k is usually chosen to equal Q, and
ln (1!z) 4 dC (z)" (1#z) O 1!z 3
1 9 p 1#z 3 ! ln z#2#z! # d(1!z) , ! 2 1!z 3 2 (1!z) > > (4.41)
1 1!z dC (z)" (2z!1) ln !1 #2(1!z) . E 2 z distribution is, as usual, de"ned by > f (x) f (x)!f (1) dx , dx (1!x) 1!x > and Eq. (4.18) is again useful for calculating the convolutions.
(4.42)
The ( )
(4.43)
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B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
The unphysical &bare' quark and gluon densities dq and dg in (4.40) have to be rede"ned in order to get rid of the singularities present (for e"0). The renormalized &dressed' quark distribution is de"ned by
Q 1 a (Q) ln ! #c !ln 4p (dq dP#dg dP) dq(x, Q),dq (x)# Q # OO OE k e 2p
(4.44)
and there is a similar equation for the rede"ned gluon density. Eq. (4.44) refers to the &modi"ed minimal subtraction' (MS) factorization scheme since also the combination c !ln 4p, being just # a mathematical artifact of the phase space in 4!2e dimensions, has been absorbed, together with !1/e, into the de"nition of dq(x, Q). Besides dimensional regularization one has of course the possibility to use masses of quarks and/or gluons to regulate the divergences. However, in those schemes NLO calculations become usually far more cumbersome. Furthermore, one obtains results for dC and dC which di!er from the ones in (4.41) and (4.42). This, however, is not O E a fundamental problem. It can be traced to a di!erence in the de"nition of parton densities and NLO splitting functions dP in the various schemes. We shall come back to this point at the end of GH Section 4.3 and in Section 5. It should be mentioned that the straightforward MS result for dC in (4.42), which gives rise to E a vanishing &"rst moment'
dz dC (z)"[!1#1]"0 , (4.45) E has been a matter of dispute during the past years [55,241,158,282,242,56,381,112,564,46,50,47,128, 283,284,466,259,531,103,165,510,511,370]. This vanishing is caused by the second term #2(1!z) in square brackets of dC (z) in (4.42) which derives from the soft nonperturbative collinear region E where k &m;K [462,565,566,460]; therefore it appears to be reasonable that this term should 2 O be absorbed, besides the dP piece in (4.40), into the de"nition of the light (nonperturbative) quark OE distribution dq(x, Q"Q ) in (4.44). This implies that, instead of dC (z) in (4.42), we have E 1!z 1 !1 (4.46) dCI (z)" (2z!1) ln E z 2 *C , E
which has a nonvanishing "rst moment *CI "!, in contrast to (4.45). It should be mentioned E that a "rst moment of ! for the Wilson coe$cient of the gluon can be obtained without any subtractions in the so-called o!-shell scheme, to be discussed at the end of Section 4.3 and in Section 5. This scheme therefore directly reproduces the ABJ anomaly contribution to the "rst moment of g . For more details see Section 5. However, from a technical point of view the o!-shell scheme is somewhat impractical, because two-loop splitting functions are very di$cult to calculate in this scheme, and only the MS results (4.41) and (4.42) comply with the NLO(MS) result [478,568,569] for the two-loop splitting functions dP(x) to be discussed next. GH Within the MS factorization scheme the NLO contributions to g (x, Q) are "nally given by 1 a (Q) g (x, Q)" e dq(x, Q)#dq (x, Q)# Q [dC (dq#dq )#2dC dg] . (4.47) O O E 2 2p O
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
35
Fig. 19. Diagrams relevant for two-loop splitting functions.
The NLO parton densities df (x, Q), f"q, q , g, evolve according to the NLO evolution equations where the (scheme dependent) two-loop splitting functions dP(x) enter and to which we now turn. GH In NLO the evolution equations (4.12) and (4.19) have to be generalized since, in contrast to the LO in a , the O(a) two-loop splitting functions dP allow for transitions between quarks and Q Q GH antiquarks and among the di!erent quark #avors as illustrated in Fig. 19. The situation is completely analogous to the unpolarized, i.e. spin-averaged case [45,212,267,288]. The NLO #avor non-singlet renormalization group equations read d dq (x, Q)"dP dq ,1! ,1! dt ,1!
(4.48)
where
a (Q) a (Q) dP(x)# Q dP (x) , dP " Q OO ,1! ,1! 2p 2p
(4.49)
and a (Q) 1 b ln ln Q/K Q K ! (4.50) 4p b ln Q/K b (ln Q/K) with b "102!f (e.g. b "9, b "64 for the f"3 light active #avors). There are two di!erent, independent NS evolution equations in NLO because of the additional transitions between di!erent, non-diagonal #avors (uPd, uPs , etc.) and qq -mixing (uPu , etc.) which start at 2-loop (a) order as illustrated in Fig. 19. Thus, opposite to the situation of unpolarized (spin-averaged) Q parton distributions [212,314], dq corresponds to the NS combinations du!du ,du and ,1> T dd!ddM ,dd , while dq corresponds to the combinations dq#dq appearing in the NS T ,1\ expressions (du#du )!(dd#ddM ) and (du#du )#(dd#ddM )!2(ds#ds ). The required 2-loop splitting functions dP (x) in (4.49) are the same [414,586] as the unpolarized ones, ,1! dP "P in the (chirality conserving) MS regularization scheme [212,267] } similarly to the ,1! ,1! LO splitting function in Eq. (4.16). The NLO #avor-singlet Q-evolution equations are similar to the LO ones in (4.19):
dR d dR(x, Q) "dPK dt dg(x, Q) dg
(4.51)
More explicitly, P (x) is given, in the notation of [212] by P "P "P$P according to Eqs. (4.8), (4.35) ,1! ,1! ! OO OO and (4.50)}(4.55) of [212], which are summarized in the appendix.
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with dR being de"ned in (4.9), and
a (Q) a (Q) dPK (x) , dPK (x)# Q dPK " Q 2p 2p
(4.52)
where the LO 2;2 matrix dPK of singlet splitting functions is as in Eq. (4.19) and the NLO 2-loop singlet matrix is given by
dP 2fdP OO OE . (4.53) dP dP EO EE The calculation of all these NLO singlet splitting functions dP(x), i, j"q, g, has been recently GH completed in the MS factorization scheme [478,568,569] and is summarized in the appendix. As in the LO case, the independent NS evolution equations (4.48) and the coupled singlet equations (4.51) can be solved numerically by iteration directly in Bjorken-x space in order to calculate g (x, Q) in (4.47) to NLO. It will, however, be more convenient and physically more transparent to work in Mellin n-moment space where the evolution equations can be solved analytically. In moment space the NLO evolution equations (4.48) and (4.51) are simply given by dPK (x)"
a a d dqL (Q)" Q dPL# Q dPL dqL (Q) , ,1! ,1! 2p ,1 2p dt ,1!
a a d dRL(Q) " Q dPK L# Q dPK L 2p 2p dt dgL(Q)
dRL(Q) dgL(Q)
(4.54) (4.55)
with the moments of the LO splitting functions given in (4.26) and the moments of the NLO(MS) #avor-NS splitting function dPL are again equal to the unpolarized ones. The moments of the ,1! NLO(MS) #avor-singlet splitting functions dPL appearing in dPK L, as de"ned in (4.52), have also GH been presented in [478]; however, in a form which is not adequate for analytic continuation in n as required for a (numerical) Mellin inversion to Bjorken-x space. The appropriate NLO(MS) anomalous dimensions can be found in [309] and are summarized in the appendix. The more complicated matrix equation (4.55) can be easily solved in a compact form by introducing [288] an evolution matrix (obvious n-dependencies are suppressed)
a (Q) ;K ¸\ @ B.K L , EK (Q)" 1# Q 2p
(4.56)
These are given by Eqs. (3.65)}(3.68) of [478] where dP"dP #dP with dP being given by Eq. (3.65) of OO ,1\ .1 OO .1 OO [478]. Note, however, that the dP have been de"ned relative to (a /4p) in [478], cf. footnote 1, and that the factor 2f in GH Q (4.53) has been absorbed into the de"nition of dP in [478]. OE In the notation of [267], the &anomalous dimensions' (see footnote 1) are given by dPL "!cL(g"$1)/8 with ,1! ,1 cL(g) given by Eq. (B.18) of [267] which are summarized in the appendix. ,1 In the notation of [478,309], dPL"!dcL/8 with dcL being given by Eqs. (A.2)}(A.6) of [309]. Moreover, GH GH GH 2fdPL"!dcL/8 since the factor 2f has been absorbed into the de"nition of dcL in [478,309]. OE OE OE
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
37
de"ned as the solution of the equation [cf. (4.51)] d EK "dPK LEK dt
(4.57)
with ¸(Q),a (Q)/a (Q ), the NLO a being given in (4.50), and where ;K accounts for the 2-loop Q Q Q contributions as an extension of the LO expression (4.28). It satis"es [288] (4.58) [;K , dPK L]"b ;K #RK with RK ,dPK L!(b /2b )dPK L, which yields 2 PK RK PK PK RK PK \ > > \ ;K "! (PK RK PK #PK RK PK )# # (4.59) > > \ \ b jL !jL !b jL !jL !b > \ \ > where the projectors PK and jL are given in (4.29) and (4.30). The singlet solutions are then given ! ! by
dRL(Q)
a (Q) a (Q ) " ¸\ @ B.K L# Q ;K ¸\ @ B.K L! Q ¸\ @ B.K L;K 2p 2p dgL(Q)
dRL(Q ) #O(a) , (4.60) Q dgL(Q ) where the remaining matrix expressions can be explicitly calculated using Eq. (4.31). For the #avor nonsinglet evolution equations (4.54), which do not involve any matrices, Eq. (4.58) simply reduces to ; "!(2/b )R and (4.60) obviously reduces to ,1 ,1 a (Q)!a (Q ) 2 b Q ! dqL (Q)" 1# Q dPL ! dPL ,1! ,1! 2b OO 2p b L ;¸\ @ B.OO dqL (Q )#O(a) . (4.61) ,1! Q These analytic solutions (4.60) and (4.61) in n-moment space can now be inverted to Bjorken-x space by numerically performing the integral in (4.37). The "nal predictions for g (x, Q) can then be calculated according to (4.47). Alternatively, one can use directly the moment solutions (4.60) and (4.61) and insert them into the nth moment of (4.47), ;
1 gL (Q)" e O 2 O where [415]
a a 1# Q dCL [dqL(Q)#dq L(Q)]# Q 2dCL dgL(Q) , E 2p O 2p
3 4 1 1 1 1 9 dCL " !S (n)#(S (n))# ! S (n)# # # ! O 3 2 n(n#1) n 2n n#1 2
(4.62)
(4.63)
is the nth moment of dC (z) in Eq. (4.41) with S (n) given after (4.26) and S (n), O L 1/j"(p/6)!t (n#1) where t (n)"d ln C(n)/dn. The nth moment of dC (z) in Eq. (4.42) H E is
n!1 1 2 1 (S (n)#1)! # dCL " ! E 2 n(n#1) n n(n#1)
(4.64)
38
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
which gives rise to the vanishing "rst (n"1) moment, dC"0. On the other hand, in the E factorization scheme of Eq. (4.46) we have 2 1 dCI L "dCL ! E E 2 n(n#1)
(4.65)
i.e. dCI "!. With these expressions at hand one can now obtain g (x, Q) from Eq. (4.62) by E performing a single numerical integration according to (4.37). It should be remembered that a theoretically consistent NLO analysis can be conveniently performed within the common MS factorization scheme where all dP are known [478,568,569], using dC of Eqs. (4.63) and (4.64). GH O E This is particularly relevant for the parton distributions which have to satisfy the fundamental positivity constraints (4.11) at any value of x and scale Q, as calculated by the unpolarized and polarized evolution equations, within the same factorization scheme. From the above solutions (4.60)}(4.62) it becomes apparent that, as usual, a NLO analysis requires two-loop O(a) splitting functions dP and one-loop O(a ) Wilson coe$cient, i.e. partonic Q GH Q cross sections, dC . Furthermore, in any realistic analysis beyond the LO, the coe$cient and G splitting functions are not uniquely de"ned to the extent that it is a mere matter of a theorist's convention of how much of the NLO corrections are attributed to dC and how much to dP [see, G GH for example, the discussion which led to Eq. (4.46)]. This is usually referred to as &renormalization/factorization scheme convention'. What is, however, important is that, to a given perturbative order in a , any physically directly measurable quantity must be independent of the Q convention chosen (&scheme independence') and that the convention dependent terms appear only beyond this order in a considered which are perturbatively (hopefully) small. The requirements Q of convention independence of our NLO analysis can be easily derived [288,302] from Eqs. (4.60)}(4.62): Choosing a di!erent factorization scheme in the NS sector (dC "dC ) ,1 O according to dCL PdCYL "dCL #DL , ,1 ,1 ,1 ,1 this change has to be compensated, to O(a), by an appropriate change of dP , Q ,1 b dPLPdPYL"dPL# DL . ,1 ,1 ,1 2 ,1
(4.66)
(4.67)
Similarly in the singlet sector, where we have to deal with 2;2 matrices, a change of the factorization scheme dCK LPdCK YL"dCK L#DK L
(4.68)
implies [288,302] dPK LPdPK YL"dPK L#(b /2)DK L![DK L, dPK L] (4.69) in order to guarantee convention (scheme) independence to order a. The upper row of dCK Q corresponds to our fermion (quark) dC ,dC and gluon dC ,dC Wilson coe$cients in an OO O OE E obvious notation. To keep the treatment as symmetric as possible we introduced in addition hypothetical Wilson coe$cients dC and dC in the lower row of dCK which do not directly EO EE
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
39
contribute to deep inelastic lepton-hadron scattering and indeed drop out from the "nal results relevant for g . In general, the transformation of splitting functions in (4.69) and of the gluon density is not "xed by the change of the physical "rst row coe$cient functions alone since the lower row of DK L remains undetermined. This is partly in contrast to the unpolarized situation [288,302,314] where the energy-momentum conservation constraint (for n"2) may be used together with the assumption of its analyticity in n. From these results it is clear that a consistent, i.e. factorization scheme independent NLO analysis of g (x, Q) in (4.62) requires the knowledge of all polarized two-loop splitting functions dPL [or dP(x)], besides the coe$cient functions dCL [or dC (x)]. Such an analysis can be GH GH O E O E conveniently performed in the common MS factorization scheme where all dPL are known GH [478,568,569] } a situation very similar to the unpolarized case. One could of course choose to work within a di!erent factorization scheme, in particular one which leads to (4.46) and (4.65), as for example in the chirally invariant (CI) or JET scheme [174,176,446,482], or any other speci"c scheme. In this case, however, one has for consistency reasons to calculate all polarized NLO quantities (dC , dPL, etc.), and not just their "rst (n"1) G GH moments, in these speci"c schemes as well as also NLO subprocesses of purely hadronic reactions to which the NLO (polarized/unpolarized) parton distributions are applied to. So far, complete NLO calculations have only been performed in the MS scheme. 4.3. Operator product expansion for g The phenomena discussed so far in the QCD improved parton model can also be understood in the framework of the operator product expansion (OPE). In contrast to the QCD parton model, which is universally valid, the OPE is designed exclusively to the understanding of deep inelasic lepton nucleon scattering. One starts with an expansion of the Fourier transform of the time ordered product of two currents, i.e., the virtual Compton amplitude
¹ "i dx e OV¹(J (x)J (0)) I J IJ
(4.70)
near the light-cone x&1/Q+0, i.e. in powers of 1/Q, and is led to a description of the moments of structure functions in terms of anomalous dimensions, Wilson coe$cients and matrix elements. The OPE has been used very successfully to derive results for unpolarized scattering [338,339,295,492,509], and results for polarized scattering exist as well [36,37,520,414,478]. The particular feature of polarized DIS is the appearance of antisymmetric (k l) terms in the expansion of ¹ . We shall start with a discussion of the structure function g for the case of photon IJ exchange. The analysis of g is complicated by the appearance of transverse e!ects and will be discussed in Section 8. We may then assume longitudinal polarization P "MS of the proton. N N One has ¹ (antisymm., em. current) IJ
2 L q 2q L\ RNI 2IL\ EL(Q/k, a ) , "ie qH G G Q I I IJHN Q G L 2
(4.71)
40
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
where RNI 2IL\ "iL\tM c c+NDI 2DIL\ ,j t G G with i"0,2, 8; n"1, 3, 5,2 and
(4.72)
(4.73) RNI 2IL\ "iL\e?@A+NG DI 2DIL\ ,GIL\ ? @A with n"3, 5, 7,2 are the relevant operators, j are the Gell}Mann matrices (j "unit matrix) G and + , denotes symmetrization on the indices pk 2k . In order to obtain operators of de"nite L\ twist and spin, the appropriate subtraction of trace-terms is always implied. R are the \ nonsinglet and R and R the singlet contributions. For each n"3, 5, 7,2 there are 8 NS operators and two singlet operators. For n"1 (the "rst moment case) there are 8 NS operators tM c c j t but only one singlet operator tM c c t, the axial vector singlet current. This operator is N \ N of particular interest because it carries the triangle anomaly. Through this anomaly the gluon enters the scene, via dCI in Eqs. (4.46) and (4.65) even though formally no gluon operator R exists E for n"1. The case of the "rst moment will be discussed in great detail in Section 5. We shall denote the matrix elements of the operators R by M (S "P /M): G G N N (4.74) 1PS"RNI 2IL\ "PS2"!MLS+NPI 2PIL\ , . G G
Furthermore, the imaginary part of 1PS"¹ "PS2 is related to the hadronic tensor = in Eq. (2.1) IJ IJ which determines the cross section and contains the structure functions [181,338,339,295,492]. Therefore, through the optical theorem, g can be related to the EL and the matrix elements ML: G G 1 (4.75) dx xL\g (x, Q)" MLEL(Q/k, a ) G G Q 2 G for n"1, 3, 5,2 Notice that the knowledge of the moments for n"1, 3, 5,2 together with the fact that g is even in x !x completely determines g by analytic continuation. The functions EL, the G &Wilson coe$cients', have a Q-dependence which is determined by the anomalous dimensions dcL de"ned in Section 4.2 namely
?Q /dcL(a ) Q da EL(Q/k, a )" EL(1, a (Q))¹ exp ! G Q H Q 2b(a ) Q Q GH ?Q I H where k is the renormalization point. ¹ indicates &time' ordering,
@
@
f (x)dx#
@
@
dy f (x) f (y)#2 . (4.77) ? ? ? V When one expands Eq. (4.76) in powers of a , it turns out that the two-loop b-function and Q anomalous dimensions enter the "rst-order correction. This will be shown explicitly in Eq. (4.80). Those two-loop quantities as well as the "rst-order Wilson coe$cients are in general scheme and convention dependent. By scheme dependence we mean a dependence on the regularization procedure as well as on the renormalization prescription. But this dependence cancels in the combination which enters Eq. (4.83) below, i.e. it cancels in the prediction for the physical quantities. This is complementary to what we said about di!erent de"nitions of quark densities in Section 4.2 and is exactly analogous to what happens in unpolarized scattering [288,509,302]. ¹exp
f (x)dx"1#
(4.76)
dx
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
41
Comparing (4.75) and (4.62) there is a one-to-one correspondence between the OPE and the parton model description. The matrix elements ML correspond to the moments of suitable G combinations of parton densities dq, dq and dg. For example,
1 ML " dx xL\[du#du #dd#ddM !2(ds#ds )] . 18
(4.78)
The anomalous dimensions of the OPE govern the Q-evolution of the parton densities and the Wilson coe$cients arise as the &constant terms' discussed in Eq. (4.40). This correspondence holds for general n. For n"1 there is the peculiar situation that only one singlet operator exists whereas in the parton model there are two degrees of freedom, the "rst moment of dR and the "rst moment of dg. Therefore, the translation between the two schemes is somewhat more subtle for n"1 than for general n and will be discussed in detail in Section 5. For each n"3, 5,2 the anomalous dimension matrix dcL decomposes into a 2;2 singlet and a 8;8 nonsinglet block. Let us "rst discuss the nonsinglet part in some detail. In the nonsinglet block the operators are multiplicatively renormalizable with e!ectively one anomalous dimension dcL , proportional to the moment of the evolution kernel dP introduced in the last section (see ,1 ,1 footnote 1). It can be expanded in powers of a Q
a a dcL " Q dcL# Q dcL#O(a) , ,1 Q ,1 4p ,1 4p
(4.79)
where dc L are given by the moments of the quantities dP and dP introduced in Eq. (4.49). ,1 OO ,1\ They are identical to the analogous quantities in unpolarized scattering. More precisely, one has dcL"!4dPL and dcL"!8dPL (see footnotes 1 and 4) because of the use of 4p instead of ,1 OO ,1 ,1\ 2p in the expansion parameter. dP plays no role for g but is only important for g and g #g ,1> in (2.28) (cf. Section 6.7). Since the OPE method provides only the values of the even or odd moments depending on the crossing parity of the particular structure functions, only speci"c NS splitting functions (dP or dP ) are allowed in the evolution kernel. This is in contrast to the ,1\ ,1> more general parton model method discussed before, which makes no restriction on the value of n. In other words, the parton model formulae, besides reproducing the OPE results, provide also the analytic continuation of the OPE results to those values of n which are arti"cially forbidden in the OPE. Inserting Eq. (4.79) into (4.76) one obtains
¹ exp !
"
?Q /dcL (a ) ,1 Q da Q 2b(a ) Q ?Q I
,1 @ a (Q) BAL a (Q)!a (k) dcL b dcL Q ,1 ! ,1 #O(a) , Q 1# Q Q a (k) 2b 4p 2b Q
(4.80)
where we have used
b(a ) a a Q "!b Q !b Q 4p 4p a Q
(4.81)
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B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
with b "11! f and b "102!f as in the previous section. This makes explicit that the two-loop anomalous dimensions and b-function enter the NLO one-loop analysis of cross sections, i.e. Wilson coe$cients. The Wilson coe$cient has the form a EL(1, a (Q))"1# Q dCL #O(a) Q Q 2p ,1
(4.82)
with dCL given by Eq. (4.63) in the MS-scheme. ,1 One can combine the above expansions to obtain
,1 @ a (Q) BAL a (Q) a (Q)!a (k) Q dx xL\g (x, Q)" Q 1# Q dCL # Q ,1 ,1 a (k) 2p 4p Q dcL b dcL 1 ; ,1 ! ,1 ML(k) (4.83) G 2b 2 2b G which should be compared with Eqs. (4.61) and (4.62). This equation shows how the O(a ) Q Wilson coe$cient dCL and the anomalous dimensions combine with the matrix element ,1 to form the nth moment of g . As discussed earlier, the scheme dependence of the quantities dCL and dcL must cancel in the combination in Eq. (4.83) because it gives a physical ,1 ,1 observable. Similar features arise in the singlet sector and in higher orders although the formalism becomes more complicated. In the singlet sector the anomalous dimensions form a nondiagonal 2;2 matrix and there are two coe$cients, one for the quark type operator R and the other one for the gluon operator R . To "rst order a the coe$cient for the quark type operator is the same as for the Q nonsinglet operators, and is of the form
EL (1, a (Q))"1#(a /2p)dCL #O(a) O Q Q O Q with dCL given again by Eq. (4.63). The coe$cient for the gluon operator is of the form O EL (1, a (Q))"(a /2p)dCL #O(a) E Q Q E Q
(4.84)
(4.85)
with dCL given in the MS scheme by Eq. (4.64). The singlet anomalous dimension matrix has an E expansion
a a dcL " Q dcL# Q dcL#O(a) 1 Q 1 4p 1 4p
(4.86)
where dcL , dcL and dcL are 2;2 matrices with indices ij"qq, qg, gq, gg. With these expansions 1 1 1 one can now calculate the singlet part of dx xL\g (x, Q), using the one-to-one correspondence between the parton model densities and the OPE matrix elements, with the same methods as described after Eq. (4.55). The result is exactly analogous to Eqs. (4.60) and (4.62). An alternative to present this somewhat cumbersome matrix result is the following. If one goes to a basis of the quark and gluon matrix elements, in which dcL is a diagonal matrix, one can calculate the exponential 1
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
43
matrix
¹ exp !
dE ?Q /dcL (a ) >> 1 Q da " Q dE 2b(a ) Q Q ? I \>
dE >\ dE \\
(4.87)
with
! @ a (Q) BAL a (Q)!a (k) dcL b dcL !! ! ! Q 1# Q (4.88) dE " Q !! a (k) 2b 4p 2b Q ! @ 8 @ a (k) a (Q) BAL dcL a (Q) a (Q) BAL Q Q Q !8 dE " ! Q , (4.89) !8 dcL!dcL!2b 4p a (k) 4p a (k) Q Q ! 8 where dcL"!4jL are the eigenvalues of dcL [cf. Eq. (4.30)], and dcL is assumed to be of the ! ! 1 1 form
dcL dcL >\ dcL" >> (4.90) 1 dcL dcL \> \\ in the new basis. The "rst-order Wilson coe$cients and the two-loop anomalous dimensions are scheme dependent. However, for consistent schemes the scheme dependence must be such that the predictions for the physical quantities, the moments of g , are scheme independent. This implies that the combinations of coe$cients and anomalous dimensions, which appear in the representation of g , are scheme independent as discussed at the end of the previous section. For example, from Eq. (4.83) we see that 2dCL #(cL/2b ) must be a scheme-independent combination. A similar ,1 ,1 scheme-independent combination exists in the singlet sector. In the following we want to compare some results for dCL and dcL in various schemes. Note that the "rst-order anomalous ,1 O E ,1 1 dimensions dcL [given in Eq. (4.26)] as well as the b-function coe$cients b and b are scheme independent. In all schemes singular collinear pole contributions arise, which are to be factorized into the quark distributions. The various schemes are: E The MS scheme in dimensional regularization using the reading point method [419] for the treatment of c (c is appearing due to the projector on the quark's helicity). In this scheme the factorized singular terms in the coe$cients are of the form dcL(!1/e!ln 4p#c ). # The reading point method is the more systematic generalization of the CFH [163] c prescription, i.e. a totally anticommuting c , in which the cHq-vertex is de"ned to be the starting point, from which the Dirac trace is read. From the calculational point of view this scheme is the most tractable one, because no extra mass parameters or counter terms have to be introduced. Indeed, it is this scheme, for which all coe$cients dCL are known and all anomalous dimensions G dcL have been recently calculated for the "rst time [478]. The coe$cients are given in GH Eqs. (4.63) and (4.64) and the two-loop anomalous dimensions are given in [478,309] as described in footnotes 3 and 5. It is interesting to note that for the "rst moment (n"1) dC "dC"!C "!2 , ,1 O $ dC"0 , E
(4.91) (4.92)
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B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
and dc "0 , ,1\
(4.93)
f dc"24C , OO $2
(4.94)
where dc ,c (g"!1) (see footnote 4). The relation (4.91) implies that the correction to ,1\ ,1 the Bjorken sum rule is saturated by dC , i.e. dc "0 in Eq. (4.83) due to (4.93). The relation ,1 ,1\ (4.94) implies that in this scheme the quark contribution to the "rst moment of g is not conserved (Q-dependent). It will be shown in Section 5 that this implies that in this scheme there is no gluon contribution to the "rst moment of g , i.e. dC"0 as made explicit in Eq. (4.92). The E same results for dCL and dcL are obtained [478] if one adopts the HVBM [362,141] method for G GH treating the c matrix in DO4 dimensions, provided an additional renormalization constant (counter term) [437] is introduced in order to guarantee the conservation of the NS axial vector operators R in (4.72). It is mandatory to keep these NS axial vector currents conserved [414] \ due to the absence of gluon-initiated triangle c -anomalies in the #avor nonsinglet sector which dictates the vanishing of dc . ,1\ It should be noted, however, when using &naively' the original HVBM prescription for the treatment of c one obtains [578,567] a result for dCL "dCL which implies for the "rst moment ,1 O dC"!C which is di!erent than the one in Eq. (4.91). This corresponds, however, to O $ a nonzero value for dc [on account of the scheme invariance of 2dC # ,1\ ,1 (dc /2b )"!3C , using Eqs. (4.91) and (4.93)] in contradiction to the conservation of the ,1\ $ NS axial vector current. E The regularization with (on-shell) massless quarks and o!-shell gluons. From the calculational point of view this scheme is di$cult, because the gluon o!-shellness is di$cult to handle in two loops, and consequently the dcL are not known for general n. However, this scheme is rather GH meaningful physically for polarized DIS, because it corresponds best to the notion of constituent quarks [56,50,47,511] as will be discussed in detail in Section 5. Furthermore, it avoids the fundamental di$culties with c present in dimensional regularization. The Wilson coe$cients in this scheme are given by [177,413]
9 3 3 2 1 # ! dCL "C ! ! # O $ 4 2n n#1 n (n#1) #
3 1 ! S (n)#(S (n))!3S (n) , 2 n(n#1)
1 2 1 2 dCL "2¹ ! ! # E 0 n n#1 n (n#1)
,
where, as previously, S (n), L 1/jI. In particular one has I H dC "dC"!2 , ,1 O dC"! E
(4.95) (4.96)
(4.97) (4.98)
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
45
and [50] dc "dc"0 . ,1\ OO
(4.99)
These results correspond to a nonvanishing gluon contribution to the "rst moment of g and to conserved, i.e. Q-independent, "rst moments of the polarized quark densities. A theoretically consistent NLO analysis of polarized structure functions and parton distributions cannot be performed for the time being, apart from their "rst (n"1) moments since it would require the knowledge of all dcL, or equivalently of dc (x), in this particular regularizGH GH ation/factorization scheme. Just as the MS scheme the o!-shell scheme naturally ful"lls dC"!2. This implies that the O coe$cient alone saturates the Bjorken sum rule and reproduces the standard correction factor 1!(a /p) for the "rst moment of the singlet contribution. Q E The regularization with massive quarks. Some of the collinear singularities in the coe$cient functions can be regularized by introducing a quark mass. The remaining singularities are again regularized by a nonzero o!shellness/mass of the gluon. For example, for the Wilson coe$cient of the quark "eld one obtains to one-loop order [557]
5 5 2 1 2 # ! dCL "C ! ! # O $ 2 2n n#1 n (n#1) #
7 1 # S (n)!3S (n)!(S (n)) . 2 n(n#1)
(4.100)
For the "rst moment this yields dC"!C , i.e. not directly the expected correction to the O $ Bjorken sum rule (cf. Section 5 for more explanations on the scheme dependence of the "rst moment). Although this scheme is not very tractable in a two-loop calculation, and for the light quarks is usually not considered very physical, its results are sometimes interesting for comparative reasons. For example, in [50] it was shown that the di!erence dc(massive quarks)} OO dc(massless quarks) is of such a form that it cancels the corresponding change in dC so that OO E a scheme independent result arises (for more details see Section 5). In the case of heavy quarks (m
3 2 2 2 3 # ! # S (n)!4S (n) dCL "C ! # O $ 2n n#1 n (n#1) 2
(4.101)
and dCL is as in Eq. (4.96). The "rst moments are again as in Eqs. (4.97) and (4.98), and the ones E for the quark anomalous dimensions are given by Eq. (4.99). Again the anomalous dimensions for arbitrary n are not known.
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4.4. The behavior of g (x, Q) at small x The behavior of g at small x is an important issue because the experimental determination of the "rst moment of g (x, Q), as de"ned in Eq. (3.2), depends on it rather strongly. Therefore it has an in#uence on tests of the fundamental Bjorken sum rule [118,119] and of the Gourdin}Ellis}Ja!e expectations for CN L, which will be discussed in detail in Section 5, and also on the question how large the contribution from the various parton species to the proton spin is, cf. Eq. (1.1). Due to the very di!erent polarized and unpolarized splitting functions (except for dPL"PL, dPL "PL ) one expects di!erent Q evolutions of g (x, Q) and F (x, Q), OO OO ,1! ,1! respectively, and thus di!erent small-x predictions for these two cases especially in the medium to small-x region (x(0.2) dominated by polarized #avor-singlet contributions dR and dg. According to our results in the two previous subsections, e.g. Eqs. (4.28) and (4.60), these quantum "eld theoretic renormalization group predictions depend on the input densities dR(x, Q ) and dg(x, Q ) to be "xed mainly by experiment. Unfortunately, the present polarization experiments [77,78,25,17,18,20,65,2}4] with their scarce statistics constrain these singlet input densities rather weakly, see, e.g. [52,53,293,313,309], in particular dg(x, Q ) remains almost entirely arbitrary, which is in constrast to unpolarized structure functions (see, e.g., [315,318,319]). One therefore has, for the time being, to rely on theoretical prejudices and guesses. There is a suggestion from Regge theory for the small-x behavior of g . Under the assumption that there is no spin-dependent di!ractive scattering, one may assume that the a (1260) trajectory dominates the Regge asymptotics [143,255] and obtains g (x, Q )& x\?? , V
!0.5:a (0):0 , ?
(4.102)
where a (0) is the intercept of the degenerate a (1260), f (1285) and f (1420) Regge trajectories. The ? scale Q where this asymptotic expectation is supposed to hold is entirely unrestricted by Regge arguments. A naive guess would be that g (x, Q ) should behave more or less like a constant as xP0 (g &x), to be compared with a "t [143,255] to the EMC data [77,78], which has been done \ . The main uncertainty lies of course in the for the region x(0.2 and yields gN (x, Q)&x > value of Q"Q where this Regge behavior is implemented. If, for example, we implement Eq. (4.102) at a scale Q K1 GeV then, according to the QCD evolution, gN (x, Q) as well as dq(x, Q), dq (x, Q) and dg(x, Q) derived from it will be steeper as xP0 than in Eq. (4.102) at Q'1 GeV, e.g. at QK10 GeV relevant for some recent experiments [77,78,25,17,18,20]. The only somewhat reliable conclusion we can draw from this is that g and thus dq , dq and dg will not T diverge at the same strength as xP0 as the unpolarized structure functions F and F /x since q &x\?M &x\ and q , g&x\?. &x\. The divergence of the unpolarized structure funcT tions for xP0 is driven by the 1/x singularity of the vacuum (Pomeron) exchange which is not present in dq (x, Q), dq (x, Q) and dg(x, Q). T Accepting a particular input behavior at Q"Q , the perturbative QCD prediction at Q'Q for the singlet sector follows from Eqs. (4.28)}(4.36) and (4.60). Even in the small-x limit one has to resort to the full solutions of the evolution equations since the &leading pole' or &asymptotic 1/x' approximation [248,113,192,87] to the polarized splitting functions is quantitatively (and partly even qualitatively) not appropriate [313,294], at least for x-values of experimental relevance,
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
47
x910\. For qualitative purposes it is nevertheless instructive to recall the leading pole (in n) or asymptotic 1/x approximation, presented here in LO for simplicity. In polarized scattering all dP GH in Eqs. (4.21) and (4.26) are less singular as xP0 than in unpolarized scattering: dPL&1/n GH [dP(x)&const.] as compared to PL&1/n!1 [P(x)&1/x] and PL&1/n [P(x)& GH EG EG OG OG const.]. This complicates the situation to some extent because polarized quark and gluon densities contribute at the same level. Thus the 2;2 matrix of splitting functions in Eq. (4.25), required for the LO solution (4.28), reduces to
1 C$ !2f ¹0 (4.103) dPK L& n 2C 4C V $ with C ", ¹ ", C "3. We "rst diagonalize this matrix and obtain for the eigenvalues $ 0 [cf. (4.30)] 1 jL " jM , ! n !
1 40 32 jM " $ ! 2 3 3
3f 1! 32
.
(4.104)
For f"3 active #avors we have jM "11.188<jM "2.145. This means that the second term in > \ Eq. (4.31) which involves the jL renormalization group exponent is subleading as xP0 and we \ obtain approximately
dRL(Q)
C !jM !2f ¹ dRL(Q ) 1 $ \ 0 e @ HM > LK , K (4.105) j M !j M dgL(Q) 2C 4C !jM dgL(Q ) > \ $ \ where m"m(Q),ln ¸\"ln[a (Q )/a (Q)]. Approximately the same result can be obtained, if Q Q one uses just the dominant dP contribution 4C and neglects the remaining entries in (4.103): EE In that case one has jM "4C "12 and jM "0, i.e. a su$ciently accurate approximation [192]. > \ Assuming the input densities to be #at in x as xP0 [cf. Eq. (4.102)], dR(x, Q )&const. and dg(x, Q )& const., i.e. dRL(Q )&1/n and dgL(Q )&1/n, Eq. (4.105) can be easily Mellin-inverted, cf. (4.37), on account of [299,467,215]
1 1 A> 1 A> 1 df (x, Q)" dn x\Ldf L(Q)& dn x\L e? L"I 2 a ln x 2pi 2pi n A\ A\
.
(4.106)
Using the asymptotic expression for the modi"ed Bessel function I (z)&eX/(2pz!O(1/z) for large z, we arrive at the &double leading log' (DLL) formula
1 2 jM m(Q) ln (4.107) dR(x, Q), dg(x, Q)& CR exp 2 E x b > V where all (partly unknown) constants are lumped into CR . This gives the dominant xP0 behavior E of the singlet component g of g "g #g which dominates g because the nonsinglet part 1 ,1 1 (4.27) leads, analogously to the above derivation using dPL&C /n, to OO $ 2 1 C m(Q) ln (4.108) dq (x, Q)& C exp 2 ,1 ,1 b $ x V
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B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
which is again subleading since jM
$ input (C '0) drives dR(x, Q) negative as xP0 and Q increases, due to the negative large matrix E element !2f ¹ in (4.105). Therefore gN is eventually driven negative as well for x:10\ and 0 Q'1 GeV [87,48]. Thus, estimates of small-x contributions to the "rst moment C (Q) of g (x, Q) using Regge extrapolations alone will be unreliable, underestimating their size, for a positive dg, and sometimes even giving them the wrong sign [48,87]. Eq. (4.107) may be compared with the asymptotic 1/x behavior of the unpolarized structure function F (x, Q): Assuming that the gluon with PL&2C /(n!1) dominates the evolution of EE the singlet quark combination R"q#q , the quantity which couples directly to F , one obtains the standard DLL result [299,467,215]
2 1 2C m(Q) ln (4.109) xg (x, Q) & exp 2 b x V which is in principle similar to the result in Eq. (4.107) but has an additional factor 1/x in front due to the dominant vacuum (Pomeron) exchange allowed for spin-averaged structure functions as discussed at the beginning of this subsection. This relatively simple exercise demonstrates explicitly that A (x, Q) in Eq. (2.14) will not be independent of Q in the small-x region, as commonly assumed [490,77,78,25,17,18,20,65,2,3] for extracting g (x, Q) [because jM K4C '2C according to Eqs. (4.107) and (4.109)], i.e. > 2 g (x, Q) 1 g (x, Q) K2x & x exp 2((2!1) 2C m(Q) ln . (4.110) A (x, Q)K b F (x, Q) x F (x, Q) V Note that A (x, Q)&const. as xP1 due to dPL"PL and dPL "PL which dominate in OO OO ,1! ,1! the large-x region. It should, however, be emphasized that the above asymptotic results in (4.107) and (4.108) are not even qualitatively su$cient for x as low as 10\ when compared with the exact LO/NLO results for A [313,294,309]. They might become relevant for x:10\, depending on the input densities [313,294]. In contrast to the unpolarized F (x, Q), measurements of g (x, Q) in the small-x region will be far more di$cult due to A (x, Q)K2xg /F P0 as xP0 according to (4.110). Here, a fully polarized HERA el pl collider [123,124,369] would be of great help to delineate the small-x behavior of g which is expected to be entirely di!erent in QCD [48,87,551,276,224,28] as compared to naive Regge extrapolations in (4.102) as discussed above after Eq. (4.108). In addition, the evolution of parton distributions f (x, Q) involve in general convolutions P f, see e.g. Eqs. (4.12), (4.14) and (4.19), which are sensitive to the shape of these distributions in the large-x region, even for (P f )(xP0). This can be easily envisaged by considering according to Eq. (4.102), for example,
? a xL f (x)"x\?(1!x)?"x\? n L which implies for the convolution, using P(x)"1/xN with p50,
a 1 x\? 1 (P f )(xP0)K ! #O(xN\?) . xN n n#p!a p!a L
(4.111)
(4.112)
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49
Thus, because of the remaining a-dependence, the behavior of P f in the small-x limit is closely correlated with the distributions in the large-x region! A more detailed quantitative analysis can be found in [294]. Therefore the mere knowledge of P(x) and f (x, Q) as xP0 is insu$cient to predict the small-x behavior of g (x, Q). It is thus apparent that reliable results can only be obtained by performing full LO and/or NLO analyses as described in the previous Sections 4.1 and 4.2 [313,309]. Anyway, the scale-violating Q-dependence of A (x, Q) is a general and speci"c feature of perturbative QCD as soon as gluon and sea densities become relevant. This is due to the very di!erent polarized and unpolarized splitting functions dP and P , respectively (except for xP1 as discussed above, after GH GH Eq. (4.110)). Finally, we turn to the more speculative nonperturbative approach to the small-x behavior of g . The result in Eq. (4.102) is obtained if there is no spin dependence in di!ractive scatter ing. The possibility of spin-dependent di!ractive scattering has been examined in [192,102] where it has been shown that a logarithmic rise of g at small x could in principle be induced g (x, Q)& ln (1/x) V
(4.113)
with the scale Q being entirely unspeci"ed. An explicit calculation [102], being based on the exchange of two nonperturbative gluons, has manifested this ln x behavior, i.e. g &2 ln (1/x)!1. The behavior in Eq. (4.113) is by no means compelling but it shows that there is a considerable amount of theoretical uncertainty, at least as far as the nonperturbative input distributions are concerned. A numerical analysis shows that this leads to an uncertainty in the determination of the "rst moment, de"ned in Eq. (3.2), typically about 10% [192]. This can be seen by taking the present experimental results with cut values x90.01, cf. Eq. (3.4), and "tting Eqs. (4.102) and (4.113) in the small-x region. Even extreme double-logarithmic contributions g &ln (1/x) are conceivable and, in fact, claimed to be present at x;1 [192,263,93,94], which are not included in the usual RG evolution equation. More precisely, in a simple ladder approach one has for each (except for the "rst) power of a terms of the form (a ln (1/x)). Summing up to arbitrary order in a one obtains Q Q Q g &(1/x)A (?Q . This is to be contrasted to the unpolarized structure function F for which one gets in this approach F &(1/x)A ?Q . The latter result is obtained by resumming single logarithms of 1/x where double logarithms are not present in the unpolarized case. Numerically, the exponent c (a Q turns out to be larger than 1 for the singlet contribution to g . Therefore in this approach the "rst moment of g does not exist. This probably signals a breakdown of the approximation used and suggests that other, yet unknown, terms in addition to the logarithms of x have to be summed as well. A recent more detailed and consistent analysis [126,125] has demonstrated, however, that such more singular terms are not present in the #avor nonsinglet contribution to g because potentially large lnLx contributions get cancelled by similar singular terms in the Wilson coe$cients. Such a conclusion cannot be reached for the singlet contribution to g since less singular terms of the NNLO (3-loop) singlet splitting functions have not yet been calculated [127].
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5. The 5rst moment of u1 5.1. The xrst moment and the gluon contribution Quantities of particular signi"cance are the "rst moments of polarized parton distributions df (x, Q),
*f (Q),
dx df (x, Q), f"q, q , g
(5.1)
since they enter the fundamental spin relation Eq. (1.1). According to Eqs. (4.1) and (4.2), *(q#q ) is the net number of right-handed quarks of #avor q inside a right-handed proton and thus *R(Q), cf. Eq. (4.9), is a measure of how much all quark #avors contribute to the spin of the proton. Similarly, *g(Q) in (1.1) represents the total gluonic contribution to the spin of the nucleon. Later we shall try to answer the important question of how much these "rst moments contribute numerically to the spin of the proton. The present experimental results on the "rst moment C (Q), dx g (x, Q), de"ned in Eq. (3.2) and reviewed in Section 3, have a better statistics than g (x, Q) itself, just because C (Q) is an average. It should, however, be kept in mind that the determination of dx g (x, Q) relies on theoretical assumptions concerning the extrapolations for xP0 and xP1 since the actual (x, Q) dependent data extend only over a limited range of x. Let us "rst study the scale (Q) dependence of *R(Q) and *g(Q) in LO. The n"1 moment of the singlet evolution equations in (4.25) is given by
, 2f*P a (Q) *P d *R(Q) OO OE " Q 2p *P, *P dt *g(Q) EO EE
*R(Q) *g(Q)
#O(a) Q
*R(Q) a (Q) 0, 0 " Q #O(a) (5.2) b Q 2p 2, *g(Q) 2 according to Eq. (4.26) , and b "(11N !2f ) is the coe$cient appearing in the renormalization A group equation for a , da /dt"!b a/4p#O(a), with N "3 and f"3 for the relevant &light' Q Q Q Q A u, d, s #avors. The Q-independence of *R is trivial due to *P"*P"0 which holds also for OO OE each NS combination in (4.24) due to d/dt *q (Q)"0#O(a). Thus the total polarization of ,1 Q each (anti)quark #avor is conserved, i.e. Q-independent in LO: *\ q (Q)"const. The vanishing of *P can be understood as follows: assume that a gluon of positive helicity #1 splits into a qq OE pair. Then the helicity of the quark will always be # and that of the antiquark !, and therefore *P"0. (Note that in this process angular momentum is produced.) *P "0 is a consequence of OE OO dP(x)"P(x) together with the so-called Adler sum rule [32] which says that the number of OO OO quarks of a certain #avor inside the proton is Q-independent. In chirality-preserving regularization schemes the Q-independence of *\ q holds true even beyond the leading order. Furthermore, Eqs. (5.2) imply for a *g Q d [a (Q)*g(Q)]"0#O(a) (5.3) Q dt Q
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51
and thus a (Q)*g(Q)K const., i.e. Q-independent in LO [431,304,305,55], but not in higher Q orders. This is a rather peculiar property of the n"1 moment of dg(x, Q) and derives formally from the appearance of b in *P. Therefore the product a *g behaves more like an object of EE Q order O(1), although strictly speaking it refers to a combination which enters only in NLO, and any contribution &a *g to C (Q) could be in principle potentially large, irrespective of the value of Q Q. From the theoretical point of view it is important to note that a (Q)*g(Q) becomes Q Q-dependent beyond the LO. However, for practical purposes the Q-dependence is too small to distinguish *R and a (Q)*g(Q) by examining this Q-dependence. Q In the LO-QCD parton model one has, using the S;(3) #avor decomposition Eq. (4.6), CN L(Q)"($A # A #A ) ,
(5.4)
where A "*u#*u !*d!*dM ,
(5.5)
A "*u#*u #*d#*dM !2(*s#*s ) ,
(5.6)
A ,*R" (*q#*q )"A #3(*s#*s ) , (5.7) O with A being related to ML in Eq. (4.75) in an obvious way. As discussed above, the two #avor G G nonsinglet combinations (A ) and the singlet A ,*R are independent of Q in LO QCD. As discussed already in Section 4.3 (in particular after Eq. (4.94)), the NS combinations have to remain Q-independent, i.e. conserved to any order in a due to the absence of the gluon-initiated Q c -anomaly in the NS sector. This is in contrast to the singlet component A which can and will become Q-dependent beyond the LO, depending of course on the speci"c factorization scheme chosen (e.g. due to the nonvanishing *c in (4.94) in the MS scheme). The fundamental conserved OO NS quantities A and A can be "xed by the Gamov}Teller part of the (#avor changing) octet hyperon b-decays (F, D values) [529,530,63]: A "F#D"g /g "1.2573$0.0028 , 4
(5.8)
A "3F!D"0.579$0.025
(5.9)
with the F, D values taken from [480,191]. It should be noted that the result for A relies only on the fundamental S;(2) isospin symmetry, i.e. is obtained just from the neutron b-decay. In order to obtain A one has, however, to extend the phenomenological analysis of the g /g ratios to the full 4 S;(3) baryon octet, the spin hyperons p, n, K, R ! and N \. The b-decay of some of these baryons or, more precisely, the transition of strange into nonstrange components, can be used to get information about A . Thus the main assumptions used are S;(3) symmetry and the D approximation of massless quarks } both are not very precise but are to some extent reasonable. Then the hyperon transition matrix elements of the octet of the axial vector currents are of the general form 1H PS"JG "H PS2"S (!if F#d D) , H I I I GHI GHI
(5.10)
52
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where the hyperons are denoted by H , i"1,2, 8 (H "R, etc.) and f and d are the totally G GHI GHI antisymmetric and symmetric S;(3) group constants, respectively. The expression (5.10) is completely "xed by the two constants F and D, with a recent experimental "t resulting in [480,191] F"0.459$0.008, D"0.798$0.008 ,
(5.11)
i.e. F/D"0.575$0.016, which gives A in Eq. (5.9). Moreover, Eqs. (5.8) and (5.9), together with (5.5) and (5.6), allow to express *(u#u ) and *(d#dM ) in terms of *(s#s ): *u#*u "(A #A )#*s#*s "0.92#*s#*s , (5.12) *d#*dM "(A !A )#*s#*s "!0.34#*s#*s . (5.13) The crude assumption *s"*s "0 essentially corresponds to the so-called Ellis}Ja!e sum rule to be discussed below. A "nite *s"*s O0 (as likely to be the case experimentally) obviously implies signi"cant changes of the total polarizations carried by u und d quarks in Eqs. (5.12) and (5.13). Apart from contributing di!erently to the nucleon's spin, such changes of *(u#u ) and *(d#dM ) from their &canonical' values (A $A ) in (5.12) and (5.13) may be of astrophysical relevance [249}252,336] as well as of substantial consequences for, e.g., laboratory searches of supersymmetric dark matter candidates, and for the #ux of neutrinos from supersymmetric dark matter annihilation in the sun (which will be reduced due to the reduced photino/neutralino trapping rate in the sun). The constraint equations (5.8) and (5.9) are the ones used in most analyzes performed so far and we shall refer to them as the S;(3) symmetric &standard' scenario. While the validity of (5.8) D is unquestioned since it depends merely on the fundamental S;(2) isospin rotation (u d) D between charged and neutral axial currents, the constraint Eq. (5.9) depends critically on the assumed S;(3) #avor symmetry between hyperon decay matrix elements of the #avor changing D charged weak axial currents and the neutral ones relevant for * f (Q). Although there are some arguments in favor of this latter full S;(3) symmetry [545], there are serious objections to it D [529,530,407,64,449], i.e. to the constraint Eq. (5.9). We shall come back to these S;(3) symmetry D breaking e!ects later. Inserting the constraints Eqs. (5.8) and (5.9) into Eq. (5.4) gives, using (5.7), 5 1 1 CN L"$ (F#D)# (3F!D)# (*s#*s ) 36 3 12
"
0.185$0.004
1 # (*s#*s ) , 3 !0.024$0.004
(5.14)
where the contribution from the strange sea remains unknown since the #avor changing (NS) hyperon b-decay data cannot constrain the singlet quantity A ,*R in Eq. (5.7). Assuming naively *s"*s "0, Eq. (5.14) gives CN K0.185 (5.15) #( which is the so-called Ellis}Ja!e &sum rule' originally derived in [333,253,254]. This &naive' theoretical expectation lies, however, signi"cantly above present measurements (cf. Table 2). This
B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
53
fact is usually referred to as the &spin crisis' or more appropriately &spin surprise' since a sizeable negative polarization of the strange sea is required [304,305,252,249] in Eq. (5.14) (*s"*s K !0.05 to !0.1) in order to reduce CN , i.e. to reduce signi"cantly the singlet contribution *R in #( (5.7) relative to A ; thus *R;, i.e. the quark #avors seem to contribute marginally to the spin of the proton, Eq. (1.1), which is surprising indeed! Alternatively, a negative polarization of the light sea quarks (*u K*dM (0, keeping *s"*s "0) could also account for a suppression of *R in Eq. (5.7) when S;(3) symmetry-breaking e!ects are taken into account [450,451,448], i.e. D when the constraint Eq. (5.9) does not hold anymore; or a negative a *g contribution to *R could Q equally account for the required reduction [55,241] of A in Eq. (5.7). The latter scenario of a large gluon contribution will be discussed in detail below. More detailed quantitative analyses and results will be discussed in Section 6. Here it su$ces to remark that one does not expect intuitively a large total polarization of strange sea quarks due to the fact that it is easier for a gluon to create a nonstrange &light' pair (uu , ddM ) than a heavier strange pair } a situation very similar to the unpolarized broken S;(3) sea [315,318,319] as observed by neutrino-nucleon scattering experiments [10,277,496]. It is perhaps also interesting to compare the above results with the expectations of a (nonperturbative) &constituent' quark model [357,186,440]. The constituent models generally ful"l *R"1, i.e. the entire nucleon spin is saturated by valence quark spins. Among the conventional &extreme' constituent models, the static S;(6) model [357,186,440] is the most favorable, because it is able to explain some of the static properties of nucleons. For example, the S;(6) model explains the measured ratio k /k to be about ! (experimental value "!0.685). L N However, it fails to predict g /g K1.26 correctly but gives g /g ". Since g /g is the triplet 4 4 4 ingredient of g (x, Q) dx one suspects that S;(6) will fail for g (x, Q) dx as well. In a static picture of the nucleon, in which the nucleon consists purely of valence quarks, its wave function can be found by counting all possible antisymmetric combinations of three quark states. If there is negligible ¸ in the system, one quark spin is always antiparallel to the other two. The X proton can then be described by a wave function which is a member of a 56-plet of S;(6), and the probabilities to "nd u , u , d and d in the proton turn out to be 5/3,1/3,1/3 and 2/3, > \ > \ respectively. For the neutron the role of u and d are to be interchanged. Thus *u"4/3, *d"!1/3 and *u "*dM "*s"*s "0, i.e. A13" , A13"A13"1
(5.16)
It is instructive to remind the reader that constituent quark models may be used to represent the nucleon magnetic moments k in terms of quark magnetic moments k "e/2m( where m( is the constituent quark mass, e.g. m( "336 MeV , O O O S [95,408,168,360]. This fact can be used to derive the successful relation k /k "!. Writing k "k (u !u )# N S > \ L N k (d !d )#k (s !s ) it is even possible to include strange quark contributions [95,408,360]. However, a real B > \ Q > \ understanding of CN within constituent models will never be possible. The reason for this is that in magnetic moments antiquarks count with opposite sign than in CN . Constituent quark models are able to account for antiquarks but informations from magnetic moment measurements cannot be used to get informations on CN because in magnetic moments combinations *q!*q "q !q !q #q enter, whereas CN is determined by combinations *q#*q " > \ > \ q !q #q !q . > \ > \
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so that CN 13" , CL 13"0 .
(5.17)
These numbers deviate only by about 30% in the NS sector from the experimental results, Eqs. (5.8) and (5.9), but the di!erence is huge in the singlet sector A ;A13"1 } again a rather surprising result. It should be kept in mind, however, that constituent quarks are strictly nonperturbative objects which cannot be reached by perturbative evolutions in contrast to partonic quarks which should rather be identi"ed with the current quark content of hadrons. It should, however, be kept in mind that in relativistic quark models such as the MIT-Bag Model, the Cloudy Bag Model or the Nambu}Jona}Lasinio model, the static SU(6) expectations in (5.16) are signi"cantly reduced, bringing them close to their physical, i.e. experimental values (for a recent review, see [101]). The NLO corrections to Eq. (5.4) in the MS scheme can be directly read o! Eq (4.62):
a (Q) CN L(Q)" 1! Q p
1 1 1 $ A # A # *R(Q) , 12 36 9
(5.18)
where we used Eqs. (4.91) and (4.92); the NS combinations A remain Q-independent due to the vanishing of *c in (4.93) in contrast to the singlet combination A (Q),*R(Q) since *cO0 ,1\ OO in (4.94). The total polarizations of the parton distributions themselves evolve according to Eqs. (4.54) and (4.55) with n"1 and the LO *P given in Eq. (5.2) and the NLO *P given by GH GH [478] (see footnotes 4 and 5) *P "0, *P +!0.3197 , ,1\ ,1> *P "!2f, *P"0 , OO OE 2f b 1 *P" *P" 59! , EE EO 3 4 3
(5.19)
where again b "102!f. Note that *g(Q) in Eq. (4.62) does not contribute to C in (5.18) in the MS factorization since here the "rst moment decouples due to *C ,dCL"0, cf. (4.92). It E E should be emphasized, however, that an actual analysis of present data which are available only in a limited x-range requires all moments in (4.62) where dgL(Q), or equivalently dg(x, Q), represent an important contribution to g (x, Q) as we shall see in Section 6. Furthermore such an analysis is, for the time being, possible only in the MS scheme where all two-loop dP are known GH [478,568,569], unless one allows for transformations (4.69) to other factorization schemes. Moreover, since the polarized and unpolarized parton densities originate from the same densities of de"nite positive and negative helicities, df"f !f and f"f #f according to Eqs. (4.1)}(4.4) > \ > \ and (4.10), it is also desirable to remain within the factorization scheme commonly used in all present NLO analyzes of unpolarized deep inelastic/hard processes } which is the MS scheme. This is of particular importance for a consistent implementation of the fundamental positivity constraints (4.11).
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For completeness it should be mentioned that the NLO(MS) result in Eq. (5.18) has been extended to even higher orders [438]:
1 1 CN L(Q)" $ A # A 12 36
a (Q) a (Q) 1! Q !3.5833 Q p p
a (Q) 1 a (Q) # *R(Q) 1! Q !1.0959 Q p 9 p
(5.20)
for f"3 light #avors. The evolution of *R(Q) involves now, in contrast to the NLO result (5.18), two- and three-loop anomalous dimensions of the singlet axial current [438]. The Q corrections to the NS coe$cient function for the A contributions have been calculated [439] even to O(a). Q This is relevant for the (nonsinglet) Bjorken sum rule to be discussed in Section 5.4. Although a complete NLO analysis for all moments is usually performed in the MS scheme, there is a factorization scheme for the xrst moment, which is closer to the constituent quark picture Eq. (5.16); it allows, at least in principle, for larger contributions of *R to the spin sum rule (1.1). This is the scheme corresponding to Eq. (4.46), or the o!-shell scheme resulting in Eqs. (4.97) and (4.98), where Eq. (4.62) yields
a (Q) CN L(Q)" 1! Q p a (Q) " 1! Q p
1 1 1 1 a (Q) $ A # A # *R ! f Q *g(Q) 12 36 9 9 2p
1 1 1 $ A # A # A (Q) #O(a) Q 12 36 9
(5.21)
for f"3 light quark #avors and where the singlet contribution is now given by a (Q) *g(Q) A (Q)"*R !f Q 2p
(5.22)
in contrast to the identi"cation (5.7) which appears also in (5.18). Here, *R is di!erent from the quantity *R(Q) appearing in (5.18) and is Q independent due to the vanishing of *c in (4.99). OO Whether this conserved *R is used to de"ne the actual polarized quark densities or alternatively the &renormalized' nonconserved quantity *R(Q),A (Q), is a matter of theoretical convention (choice of factorization scheme) or &intuitive' physical interpretation [46,47,510,511] as shall be discussed below. It should be remembered that only the products *C A and *C *g in Eq. (5.21) G G E are renormalization convention independent quantities to the O(a) considered, which can thereQ fore be meaningfully related to physical, i.e. experimental measurements. In this sense the second line in Eq. (5.21) is identical to the "rst one since the NNLO terms O(a) are disregarded. It is this Q latter expression which has been frequently used for "rst moment analyses in the past, for example in [56,52,53,293]. It should be again mentioned that in these (non-MS) factorization schemes only the n"1 moments of splitting functions (anomalous dimensions) have been calculated so far to which we will now turn. Originally, NLO contributions to the "rst moment C (Q) have been calculated by Kodaira [413] with the help of the operator product expansion near the light cone using an o!-shell
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B. Lampe, E. Reya / Physics Reports 332 (2000) 1}163
factorization scheme with all external lines kept o!-shell (p(0) and assuming massless quarks and gluons. They have been con"rmed and substantially reinterpreted in the framework of the QCD-based parton model in recent years [55,241,158,50]. The upshot of the result in the o!-shell scheme is that Eq. (5.4) goes over to
1 1 CN L(Q)" $ A # A 12 36
a (Q) 1 a (Q) 2f 1! Q # A 1! Q 1! p 9 p b
,
(5.23)
where A ,A (Q)[1!(a (Q)/p)(2f/b )] and A (Q) is given by (5.22) for f"3 light quark Q #avors. This result "xes the e!ect of the quarks and gluons and incorporates all scaling violations up to next-to-leading order. Remember that all the *\ q in *R are Q independent quantities due to Eq. (4.99). Several remarks are in order: the decomposition of A (Q) into a quark and a gluon contribu tion, Eq. (5.22), was not known to Kodaira [413] because he worked in the framework of the OPE where the basic objects, and in particular the singlet part A (Q), are matrix elements. The decomposition becomes only apparent in the QCD parton model where the gluon term is induced by the photon-gluon fusion process in Fig. 17. The coe$cient f"3 of *g in Eq. (5.22) is the number of light quarks because heavy quarks (with masses much larger than K ) can be shown to yield /!" a vanishing "rst moment contribution [577,433,311]. This is a consequence of the matrix element calculation which will be discussed in Section 5.3. The factors 1!(a (Q)/p) in Eq. (5.23) arise from the Wilson coe$cient of the quarks, both for the Q nonsinglet and singlet contribution. The fact that *R and *g appear in a factorized form in Eq. (5.22) has to do with the fact that for the singlet there is only one operator. Thus the expansion of the RG exponent is formally the same as for the nonsinglet case in (4.80) or (4.83) which contributes (a /4p)(*c/2b )"(a /p)(2f/b ) in the OPE scheme used by Kodaira [413], using Q OO Q *c"0. Since this two-loop term &2f/b in Eq. (5.23) has been explicitly factored out from the OO (fermionic) singlet matrix element, according to Kodaira's original calculation [413], the A in Eq. (5.23) [but not A (Q)] is Q-independent in this speci"c way of writing. The A in Eq. (5.23) should in fact depend on the renormalization (input) scale k. This dependence has been suppressed here but will be made explicit later [cf. Eq. (5.53)]. Note that the term &2f/b does not arise in the nonsinglet part in (5.23), but is inherently a singlet contribution and a two-loop e!ect of the triangle anomaly (Fig. 20). It is evident from the above results and discussions that the representation Eq. (5.22) of A (Q) in terms of *R and *g depends on the regularization scheme, despite the fact that the remaining Q-dependent corrections in Eq. (5.23) are scheme independent. The appearance of the gluon in (5.22), with the !a /2p coe$cient as calculated directly from the photon}gluon fusion diagram in Q Fig. 17 using an o!-shell regularization, could account for the reduction of the singlet contribution to C (Q) [55,241,158] required by experiment as discussed in particular after Eq. (5.15). For example, the recent experimental average result for CN in (3.9) at Q"5 to 10 GeV implies, using Eq. (5.21) together with (5.8) and (5.9) for A , a (Q) *g(Q)"0.27$0.13 A (Q),*R !3 Q 2p
(5.24)
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57
Fig. 20. The two-loop triangle diagram which contributes to the singlet *c. OO
at Q"5 to 10 GeV . Making the extreme and equally naive assumption that *s"*s "0 and disregarding any e!ects due to #avor S;(3) breaking, one obtains a large *R" D A #3(*s#*s )"A "0.579$0.025 according to Eqs. (5.7) and (5.9). If one wants to keep a large and conserved *RK0.6 and at the same time reduce the Ellis}Ja!e prediction (5.15) to the experimental measurement in Eq. (3.9), a value *g(Q"10 GeV)"3.4$1.5
(5.25)
is required by Eq. (5.24). This number is obtained with the NLO a (10 GeV)/p"0.061 Q for K "0.2 GeV. Similar results have been obtained recently by QCD-"ts performed directly +1 by the experimental E154 and SMC groups [9,224,31], for example, who arrived at *g(5 GeV)K1.7$1.1 although it is di$cult to estimate the total error on *g for the time being. Realistically, however, there is no reason for a vanishing total helicity of strange quarks and one expects a combination of both e!ects (*gO0 and *s"*s O0), probably with additional #avor S;(3) breaking e!ects, to account for the required singlet suppression in Eq. (5.24). As we shall see D in the next section, present data are far too scarce for distinguishing or con"rming the various scenarios. Before turning to a discussion of more detailed (x, Q) dependent analyses of present experiments, let us concentrate on a few theoretical aspects concerning the total helicities ("rst moments) of quarks and gluons for the remainder of this section. As already mentioned above, it is a matter of theoretical convention to interpret *R as the actual total helicity of the singlet quark densities or e!ectively the entire singlet expression (5.22) a (Q) *g(Q) *R(Q),*R !3 Q 2p
(5.26)
which, in contrast to *R , is Q-dependent and e!ectively has to be interpreted as the singlet contribution entering the MS result (5.18). The real question then arises is which *R enters, for example, Eq. (1.1). Intuitive arguments have been forwarded [46,47,510,50,434] in favor of the conserved Q-independent *R in Eq. (5.26): in this case the #avor singlet quark contribution can be kept sizeable, *R "A K0.6 (still assuming *s"*s "0), as compared to the (constituent)
For consistency note that (5.24) implies, via Eq. (5.21), for the neutron CL (Q)"!0.055$0.014 in agreement with the neutron measurements in Table 2.
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static S;(6) prediction in (5.16), *R K0.6:*R13"1 (5.27) in complete analogy to the #avor nonsinglet sector where, using Eqs. (5.8), (5.9) and (5.12), (5.28) A K1.26:A13", A K0.58:A13"1 . In both cases there are only 30}40% reductions from the static S;(6) expectations which are attributed to helicity nonconservation induced at low-energy scales by "nite quark mass e!ects which break chirality and thus the symmetry in the nonperturbative region which creates a di!erence in the initial values for the perturbative QCD evolution [46,47]. Note that only for conserved quantities one expects, in the absence of chirality breaking e!ects, constituent and parton results to coincide. The reduction of A from their static S;(6) values in (5.28) can be theoretically understood in terms of relativistic binding e!ects within the framework of relativistic quark models [101]. Furthermore, the choice of a Q-independent *R ful"lls the requirement of what might be called an &extended' Adler sum rule or an &extended' baryon (Gross Llewellyn}Smith) sum rule. The usual (unpolarized) Adler [32] and baryon [337] sum rules measure the isospin of the target nucleon and the sum of the baryon number and strangeness of the nucleon, respectively. On the parton level there appear in both cases "rst moment expressions dx[q(x, Q)!q (x, Q)] whose di!erences and sums over di!erent quark #avors are independent of Q. Speci"cally, it follows that the number of quarks of a certain #avor inside the nucleon is &conserved', i.e. Q-independent, to any order in a . The &extended' Adler and baryon sum rules extend this statement to the Q conservation of the number of quarks of a certain helicity, i.e. dx[dq(x, Q)#dq (x, Q)], to be independent of Q. The di!erence in sign of the antiquark contributions has to do with the opposite charge conjugation properties of the vector and axial-vector current. Unfortunately, our present ignorance of all moments of two-loop splitting functions dPL, or GH equivalently of dP(x), calculated in the o!-shell scheme prevents us from a detailed (x, Q) GH dependent analysis of present data, in contrast to the MS regularization/factorization scheme where such an analysis has been performed. In the MS scheme one is confronted with a less plausible large di!erence between the singlet sector *R(Q),A (Q)K0.3;A13"1 , at Q"5 to 10 GeV according to (5.24), and the nonsinglet quantities in (5.28).
(5.29)
5.2. The xrst moment and the anomaly In the OPE approach the possible decomposition, Eq. (5.22), into quark and gluon was not realized because the fundamental quantities of the OPE are not parton densities but matrix elements of certain operators between proton states. 1PS " q c c q " PS2 are the matrix elements I relevant for the total helicities, i.e. for the "rst moment of g [529,530]. More precisely, in the framework of the operator product expansion one obtains the result Eq. (5.23) with A , A and A de"ned by A S "1PS"tM c c j t"PS2 , (5.30) I I
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A S "(31PS"tM c c j t"PS2 , (5.31) I I A S "1PS"tM c c t"PS2 , (5.32) I I where t"(u, d, s)2 and j are the Gell}Mann matrices, a"1,2, 8. Longitudinal polarization ? SN&PN of the proton is assumed throughout this section. In the naive quark parton model one has the identity (*q#*q )S " 1PS"q c c q"PS2 I I
(5.33)
between the "rst moments *q#*q (q"u, d, s) and the matrix elements. However, one should note that j "tM c c t which appears in Eq. (5.32) is the axial vector singlet current which is not I I conserved, RIjO0, but is &anomalous' in the sense of Adler [33] and Bell and Jackiw [108] I f a (5.34) RIj" Q eIJ?@G? G? . ?@ IJ I 2 2p The appearance of the gluon "eld strength G in this equation is the reason why Eqs. (5.22) and ?IJ (5.32) are not in contradiction and instead of Eq. (5.33) one has (in the particular factorization scheme called the o!-shell scheme)
a *q#*q ! Q *g S "1PS"q c c q"PS2 . I I 2p
(5.35)
The anomaly enters in one-loop order through the diagram in Fig. 21, which, in the gluon o!-shell scheme, determines the (Wilson) coe$cient ! of 2(a /2p)*g, and in two-loop order through the Q diagram in Fig. 20 from which Kodaira [413] has obtained his anomalous dimension. *q#*q and !(a /2p)*g appear with the same coe$cient in Eq. (5.35) because there is only one Q operator for the "rst moment singlet contribution. For the gluon there is no direct operator, but only a representation of *g 1PS"KI"PS2"!SI*g
Fig. 21. Triangle anomaly diagram giving rise to Eqs. (5.34) and (5.40).
(5.36)
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in terms of the gauge-dependent current K "eIJMNA? (G? !g f A@ AA ) J MN Q ?@A M N I
(5.37)
whose divergence is RIK "eIJ?@G? G? ?@ IJ I
(5.38)
(so that j !f (a /2p)K is conserved). I Q I An explicit computation by Forte [274,275] of 1PS"(a /2p)KI"PS2 in (5.36) has revealed that Q (a /2p)*g"(a /2p)*g #X where *g corresponds to the perturbatively calculable hard piece of Q Q the anomaly which may be identi"ed with the usual (perturbative) partonic de"nition used thus far; the remaining gluon-topological Chern}Simons (instantonic) piece X is the soft nonperturbative contribution to the triangle anomaly [274,275,100,101] which does not correspond to a hard (large-k ) subprocess and thus should be absorbed into the de"nition of the conserved total quark 2 polarization, i.e. *R !f (a /2p)*g"(*R !f X)!f (a /2p)*g ,*R !f (a /2p)*g . This leaves Q Q Q us formally with our original expression (5.26) with both terms being now separately gauge invariant. The term !fX may be interpreted as an additional sea polarization induced by instantonic e!ects. A "rst attempt [458] to evaluate 1PS"(a /2p)KI"PS2 on a (small) lattice yielded Q the bound 3"(a /2p)*g #X"(0.05. This surprisingly small result can obviously not be used to Q extract some information about *g . It should be noted that Eq. (5.35) is really a representation of (chiral invariant) massless QCD. If fermion masses are introduced, the anomaly Eq. (5.34) is modi"ed according to fa RIj"2itM c Mt# Q eIJ?@G? G? ?@ IJ I 2 2p
(5.39)
where M"diag(m , m , m ) is the fermion mass matrix. The mass term in this equation is able to S B Q conceal the e!ect of the anomalous term under certain circumstances. To examine this e!ect, we write down the anomaly equation, Eq. (5.39), in momentum space for the case of one fermion with mass m (p#q)?CIJ(p, q)"2mCIJ(p, q)!(a /p)eIJ?@p q . ? Q ? @
(5.40)
Here CIJ(p, q) is the triangle diagram (cf. Fig. 21) and ?
\V dy a . CIJ(p, q)"m Q eIJ?@p q dx ? @ m!(p#q)xy p
(5.41)
the contribution from the mass term. If (p#q)4m, the two terms on the right-hand side of Eq. (5.40) (i.e. the anomalous contribution and the mass contribution) cancel each other, up to terms of order (p#q)/m, so that the anomaly is concealed. A similar e!ect occurs in Eq. (5.35) if one includes the quark mass. Namely, the anomalous gluon term &*g disappears and one formally recovers the naive Eq. (5.33). Thus the contribution to g from the anomaly is hidden in the massive theory (as well as in the MS subtraction scheme). This result will be derived in detail in the next section.
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The anomaly induces a point-like interaction between the axial-vector current and the gluons because the amplitude C?IJ(p, q) does not depend on the momentum transfer p!q when m"0. Furthermore, it is known that the anomaly is not a!ected by higher order corrections [34] and it has been argued [363] that this is true even beyond perturbation theory. This leads us to believe that the identi"cation, Eq. (5.35), remains true in higher orders, at least in suitable chirality preserving schemes like the gluon o!-shell scheme. 5.3. Detailed derivation of the gluon contribution The most straightforward way to derive the gluon contribution is the QCD parton model. In the following we shall only consider the #avor singlet piece of g because the gluon contributes only to the singlet. In the QCD parton model the evolution of the "rst moments of the polarized quark and gluon density is given by
*P a (t) *P d *R OO OE b " Q 2p dt *C *P *P! EO EE 4
*R *C
#O(a) . Q
(5.42)
This is an extension of Eq. (5.2) to second order in a , in which the two-loop contribution b to the Q b function arises, cf. Eq. (4.50), because instead of *g we have introduced the product *C"(a (t)/2p)*g (see the discussion in Section 5.1). In Eq. (5.42) we have used quantities *PI Q GH which are the "rst moments of dPI de"ned in Eq. (4.52) GH
*P " GH
a a dx dP (x)"*P# Q *P# Q *P#O(a) . Q GH GH GH 2p 2p GH
(5.43)
They are related to the corresponding anomalous dimensions by the usual factors of ! and ! in the one- and two-loop case (see footnote 1). As is well known, for a complete discussion of "rst-order e!ects the knowledge of second-order (k"1) anomalous dimensions is mandatory. These second-order (two-loop) anomalous dimensions depend on the calculational (i.e. regularization) scheme. For example, it is not true in general that *P"*P"0. However, some speci"c (chirality preserving) schemes have this property, so OO OE that *R is constant in Q and the generalized Adler sum rule holds. Note furthermore that in deriving the evolution equation (5.42) we have used *P"0: *P has to vanish on general OE OE grounds in any scheme [55,407], since the gauge-invariant axial current j is multiplicatively I renormalizable and therefore cannot mix with the gauge-dependent current K in Eq. (5.37). I In the QCD parton model it is well known how to calculate the contribution of *g to C " dx g . Namely, one just has to calculate the diagrams in Fig. 17b and take the "rst 1 1 moment. We call the amplitude squared corresponding to these diagrams X . It has four indices, IJMN k,l for the photon, and o, p for the gluon. To get the contribution proportional to *g in Eqs. (5.22) and (5.23) one has to contract it with eIJ?@p q ;eMNABp q because eMNABp q is proportional to the ? @ A B A B di!erence of products of gluon polarization tensors e > eH> !e \ eH\ of gluons with positive M N M N and negative helicity. This one can see, for example, in the Breit frame where p"E (1, 0, 0, 1) E is the gluon momentum and q"(0, 0, 0,!Q). In this frame the polarization tensors are e "(1/(2)(0, 1,$i, 0). !
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To calculate the diagrams or, more precisely, the coe$cient of the *C contribution to C it is 1 more appropriate to go to the c.m.s. of the photon and gluon in which q"(q , 0, 0,!p ) , p"(p , 0, 0, p ) , k"(k , 0, k sin h, k cos h) , M p "(p , 0,!k sin h, !k cos h) , where q "(p !Q,
p "(p !P,
(pq)!PQ , s
p "
(5.44)
(5.45)
(s k "p "k " . 2
Note that s"(p#q) and the fraction of momentum of the gluon which is carried by the intermediate quark in Fig. 17b is given by z"Q/(s#Q). This is the quantity with respect to which the "rst moment has to be taken. A phase space integration over the production angle h is also necessary. In Eq. (5.45) a gluon o!-shellness P"!p has been introduced. It is needed for regularization purposes because although the "nal result turns out to be "nite, singular expressions arise in intermediate steps of the calculation (from the collinear region hP0). In the o!-shell sheme (PO0) the coe$cient of *C in Eq. (5.22) turns out to be
f
dz(1!2z)
"f
\
1 1 z 1 1 1 # d cos h 2! ! #P u t Q u t 4
Q(1!z) "!f#0(P) , dz(2z!1)ln Pz
(5.46)
where 1 z z , u"!(p!p ) + (1#cos h)#P Q Q 2
(5.47)
z 1 z t"!(k!p) + (1!cos h)#P , Q 2 Q and f"3 is the number of light quarks. In Eq. (5.46) terms like P/u or P/t which do not contribute in the limit PP0 have been left out. However, the terms &P/u and &P/t are important. After integration over h a term &*P ln P arises which drops out because the "rst OE moment of dP"(2z!1) vanishes. There is an overall factor of f because each quark #avor can OE be produced. Instead of P a quark mass m may be introduced to regulate the collinear singularity (&on-shell scheme'). This is the more appropriate procedure for heavy quarks (c and b) but less useful for the
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63
light quarks (u, d and s). In that scheme one has instead of Eq. (5.45) p "p "(pq)/s ,
q "(p !Q,
(5.48)
k "p "(s/2, k "((s/2)(1!4m/s . The coe$cient of *C is now
dz
d cos h
1 1 1 (2z!1) # !1 2 1!b cos h 1#b cos h
\ 1 2m(1!z) 1 # # (1!b cos h) (1#b cos h) s
"O(m) ,
(5.49)
where b"(1!(4m/s). This time the e!ect of the double pole term in Eq. (5.49) &2m/s is such that there is no gluon contribution (for small quark masses mP0). If used for the light quarks, Eq. (5.49) seems to be in contradiction with Eq. (5.18) because they seem to imply, at some conveniently chosen input scale Q"k, two di!erent relations between the axial vector singlet current matrix element and *R and *C namely
*R (k) on-shell scheme , (5.50) A (k)" *R !f*C(k) off-shell scheme . *C is scale independent in "rst order but picks up a scale dependence in higher orders. It is well known, for example, from the calculation of QCD corrections to unpolarized structure functions, that di!erent regularization schemes can lead to di!erent results. Usually, this is interpreted in such a way that the use of di!erent regularization schemes corresponds to di!erent de"nitions of the quark density. In our case this means that one has to deal with two di!erent *R's which are denoted by *R and *R in Eq. (5.50). Going from the second to the "rst relation in Eq. (5.50) means that one absorbs the gluon contribution into a rede"nition of *R . The obtained result, *R , corresponds to the one in the conventional MS scheme in Eq. (5.18) where *C(k) does not explicitly occur due to *C "0 in Eq. (4.92). Although in principle it is a mere matter of convention E to choose a particular factorization scheme the use of the o!-shell scheme (where a potentially large gluon contribution exists) has the advantage that *R is conserved, i.e. scale independent, as will be shown below. This way the Adler sum rule [32] is extended to higher orders of polarized scattering. Furthermore, for the case of the light quarks (u, d, s with m(K ) the massless scheme /!" might be more physical because it could be related to the notion of constituent quarks [46,511]. To really understand why the two de"nitions in Eq. (5.50) do not contradict each other, one must write down the full result for the "rst moment of the singlet component of g in Eq. (4.6), dx g , in 1 both schemes and compare it with Kodaira's original result, cf. Eq. (5.23) [413]. In order to do this it should be noted that dx g is a physical observable and therefore the prediction for it must be 1 scheme independent. The physical prediction is always a product of coe$cient functions, anomalous dimensions and the parton densities/matrix elements, i.e.
*R 1 dx g (x, Q)" C (Q)E (Q, k) 1 9 *C(k)
(5.51)
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*R (k) 1 " C (Q)E (Q, k) 9 *C(k)
(5.52)
1 (Q)E (Q, k)A (k) " C ) 9 )
(5.53)
in the three schemes to be compared, with
a (Q) 3 C "1# Q ! C ) 2p 2 $
(5.54)
and
a (k)!a (Q) 3 Q E "1# Q ! C f (5.55) ) pb 2 $ are taken from the work of Kodaira [413] which was done with massless quarks and o!-shell gluons. For simplicity we work below the charm threshold so that f"3. Furthermore [50],
a (Q) 3 C "(1,!f )# Q ! C , cC 2p 2 $
,
(5.56)
0 a (k)!a (Q) 0 Q E "1# Q (5.57) b . pb *P ! C EE $ 4 Here cC is a second-order correction to the "rst-order gluon photon-fusion process (Fig. 17b). Eq. (5.50) and the entry !C in Eq. (5.56) give rise to the QCD correction factor 1!a (Q)/p in Q $ the coe$cient of A in Eq. (5.23). Both cC and *P can be determined by comparison EE with Kodaira's result, Eqs. (5.54) and (5.55), i.e. by the consistency requirement Eqs. (5.51)}(5.53), which yields [50] *P !b /4"!f*P"!2f. The form of the matrix in Eq. (5.57) is quite EE EO remarkable, in particular the vanishing of the entries in the "rst row. According to Eq. (5.42) this corresponds to *P "0 OO
(5.58)
and *P "0 . (5.59) OE Note that the vanishing of *P and *P is explicit from Eq. (4.26) and the vanishing of *P has OO OE OE been proven by [55]. In [55] the entry *P !b /4 was called c. Eqs. (5.58) and (5.59) go a step EE EE further, saying that in a suitable chirality conserving regularization scheme, *R is a conserved (i.e. Q-independent) quantity beyond the leading order, in fact to any order. It is well known that in massless perturbative QCD the chiral symmetry is unbroken to any order. Eqs. (5.58) and (5.59) show that the o!-shell scheme realizes this fact in the most intuitive way by forbidding chirality #ip interactions to any order of perturbation theory. As we shall see in the following this fact is also intimately related to the reasonable treatment of the ABJ anomaly in the o!-shell scheme. The anomaly term !f in C , Eq. (5.56), will be shown to correspond } via the consistency require ment, Eqs. (5.51)}(5.53) } to a conserved *R and vice versa.
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Eqs. (5.58) and (5.59) can be derived, for example, from the consistency requirement, Eqs. (5.51)}(5.53). To see that let us evaluate the products C E (*R , *C)2 and C E A ) ) appearing in Eqs. (5.51) and (5.53):
*R (k) C (Q)E (Q, k) *C(k)
3 a (k) a (k)!a (Q) 3 b Q " 1! C Q # Q *P !f*P# C OO EO 2 $ 2p pb 2 $2 a (k) a (k)!a (Q) Q # !f#cC Q # Q pb 2p b b ; *P !f *P! !cC *C(k)#O(a) , OE EE Q 2 4
*R (k)
(5.60)
3 a (k) a (k)!a (Q) 3 3 b Q ! C f# C C (Q)E (Q, k)A (k)" 1! C Q # Q ) ) 2 $ 2p pb 2 $ 2 $2 ;[*R (k)!f*C(k)]#O(a) . Q By comparison one obtains (among other things) 3 *P !f*P"! C f . OO EO 2 $
(5.61)
(5.62)
Because of the unambiguously determined leading order *P"3/2C , the right-hand side of EO $ (5.62) ("Kodaira's anomalous dimension) is saturated by the second term on the left-hand side, so that one arrives at Eq. (5.58). Thus *R (k) is scale independent, i.e. conserved. Until now we have discussed only the scheme dependence of the coe$cient of *g in Eq. (5.22). It turns out that other quantities di!er in the on-shell scheme from their corresponding counterparts in the o!-shell scheme as well. Therefore, we make a general ansatz a (Q) C (Q)"(1, 0)# Q (CR , cC ) , 2p
(5.63)
*P a (k)!a (Q) *P OO OE Q E (Q, k)"1# Q (5.64) b pb C *P ! EE $ 4 for the coe$cients and anomalous dimensions in the on-shell scheme. The entry *P"C in EO $ the anomalous dimension matrix is scheme independent. Due to the consistency requirement, Eqs. (5.51)}(5.53), for the physical observable there must be a transformation matrix a (Q) 1# Q m !f 2p (5.65) M(Q)"
such that
0
C (Q)"C (Q)M\(Q) ,
1
(5.66)
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E (Q, k)"M(Q)E (Q, k)M\(k) , *R *R (k) "M(k) . *C(k) *C(k)
(5.67) (5.68)
*C is not rede"ned at the order at which we are working; ergo the two entries 0 and 1 in the second row of M(Q). Eqs. (5.66)}(5.68) can be used to calculate m as well as the complete set of quantities in C and E . One simply has to work out the matrix expressions in Eqs. (5.66)}(5.68). Equivalently, these quantities can be derived by using the transformations, Eqs. (4.68) and (4.69), of Section 4.2 for n"1 (with the transition from *g to *C to be carried out). Here we do not want to bother with all the details of the transition to the on-shell scheme, but concentrate on one particular aspect, the non-conservation of *R (k) which becomes explicit in the result [50] (5.69) *P "!C f OO $ which makes clear that *R (k) is not a conserved but a scale dependent quantity (in contrast to *R ). This result follows from the consistency requirement, Eq. (5.67) (by comparison with the o!-shell scheme) but also from the consistency requirement, Eqs. (5.52) and (5.53) (by comparison with Kodaira's result), in a similar fashion as *P "0 was derived in Eq. (5.58). OO All of the above considerations were carried out in a decent but rather abstract way by imposing the condition of the scheme independence of a physical observable. An explicit check of the truthfulness of the whole approach was made in Ref. [50] where the relation (5.69) was reproven by an explicit two-loop calculation in the on-shell scheme (massive quarks). The main point was to show that the diagrams in Fig. 22 give rise to a nonvanishing anomalous dimension when calculated in the scheme with massive quarks, and a vanishing anomalous dimension in the o!-shell scheme. More precisely, the matrix element corresponding to Fig. 22 is of the form Q ME(Fig. 22)"*P," ln #const . OO k
(5.70)
after integration over the appropriate phase space [50]. Here ND denotes &nondiagonal' transitions between quarks of di!erent #avors q and q (cf. Fig. 22). It was shown in [50] that *P,""!C f for any masses m and m of the quarks q and q , whereas in the o!-shell OO $ scheme *P,""0. In other words, the change in *P when going from the o!-shell to the OO OO on-shell scheme is entirely due to non-diagonal #avor transitions. In this way a completely consistent picture arises, in which results of the o!-shell scheme can be transformed to any other scheme and vice versa. Note that diagram Fig. 22 can be obtained from Fig. 20 by cutting.
Fig. 22. DIS process involving non diagonal transitions between quarks of di!erent #avors.
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Most important among the other schemes is of course the MS scheme, because the two-loop anomalous dimensions are now known for all moments [478,568,569]. The results for the "rst moment were already given in Section 4.3. Here we recollect them for the convenience of the reader. One has, according to Eqs. (4.91)}(4.94) (see footnote 4)
a (Q) 3 ! C , cC C (Q)"(1, 0)# Q 2p 2 $ +1 +1
,
(5.71)
0 a (k)!a (Q) !C$ f Q E "1# Q , (5.72) b pb +1 C *P ! $ 4 EE +1 where cC "C f, which will contribute only in NNLO a and *P !b /4"0 can be also $ Q +1 EE +1 obtained from consistency by comparison with Kodaira's result, namely from
*R (k) 1 1 +1 " C (Q)E (Q, k)A (k) , dx g1 (x, Q)" C (Q)E (Q, k) ) 9 ) 9 +1 +1 *C(k)
(5.73)
in agreement with the results in Eq. (5.19). It should be noted that the entry 0 in Eq. (5.71) corresponds to a vanishing *C in Eq. (4.62) or (4.85), i.e. to the fact that *g decouples from CN L in E 1 the MS scheme, as well as in the on-shell scheme according to Eq. (5.63). It is now instructive, after these many technical details, to summarize the NLO evolution equations for the "rst moments of the #avor singlet quantities in the di!erent schemes. In the MS and on-shell schemes we have
d a (Q) (!2f )*R(Q) " Q , *R(Q) dt 2p +1 +1
(5.74)
a (Q) d 2*R(Q) " Q , *C(Q) 2p dt +1 +1
(5.75)
. In the o!-shell with the singlet contribution A to CN L being given by A (Q)"*R(Q) +1 scheme we have obtained d *R(Q) "0 , dt
(5.76)
a (Q) d [2*R !2f*C(Q) ] *C(Q) " Q 2p dt
(5.77)
with the singlet contribution now being given by A (Q)"*R !f*C(Q) as was already anticipated in Eq. (5.50). Due to the Q-independence of *R , Eq. (5.76), these latter two RG equations can be combined into one simple evolution equation for the singlet combination A : d a (Q) (!2f )[*R!f*C(Q)] . (5.78) [*R!f*C(Q)] " Q dt 2p
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Recalling da a a Q "!b Q !b Q , 4p (4p) dt
(5.79)
the solutions of the above RG equations are straightforward. In the MS and in the on-shell scheme, Eqs. (5.74) and (5.75) give
2f a (k)!a (Q) Q Q #O(a) *R(k) " 1! (5.80) Q b p +1 +1 2f a (k)!a (Q) Q Q "*C(k) ! (5.81) #O(a) *R(k) *C(Q) Q p +1 b +1 +1 with an appropriately chosen input scale Q "k and b "11!2f/3. In the o!-shell scheme, Eqs. (5.76)}(5.78) give *R(Q)
*R(Q) "*R(k) ,*R"const. , (5.82) 2f a (k)!a (Q) 2 a (k)!a (Q) Q Q Q Q #O(a) *C(k) # *R . (5.83) *C(Q) " 1! Q p p b b As an illustrative application of these results, let us estimate dynamically *g(Q) in the o!-shell scheme according to Eq. (5.83) [306,428,307]. The necessary input at Q"k, i.e. the boundary condition, can be deduced from the unpolarized valence-like gluon and sea input densities (i.e. xg(x, k)&x?(1!x)@ with a'0, etc.) which are thus integrable, i.e. their n"1 moments g(k), g(x, k) dx, etc., exist (only) at Q"k with the result g(k)+1 and s(k)+0 at k+0.3 GeV [315,318,319]. This allows for a parameter-free as well as perturbatively stable calculation of structure functions in the small-x region (x410\, at Q'k) which is entirely based on QCD dynamics and agrees with all present measurements obtained at DESY-HERA [319]. Thus, the positivity constraints (4.11) imply
"*g(k)"4g(k)+1,
"*s(k)"4s(k)+0 .
(5.84)
Using therefore *R+A , with A being given in (5.9), Eq. (5.83) yields [307] !2.6:*g(Q) :3.9 (5.85) for Q"10 GeV, which is compatible with Eq. (5.25). Here we have used b "9 for f"3 light quark #avors and a (k)/2p"0.108 and a (10 GeV)/2p"0.0304 in NLO. It is also interesting to Q Q note that on rather general heuristic grounds based on the intrinsic bound-state dynamics of the nucleon, counting rules, consistency constraints for g /g and *g/g as xP1 and xP0, one \ > expects [145] the total gluon helicity to be sizeable, *g+1.2. Note, however, that the scale k remains undetermined in such considerations, in contrast to the RG based result in Eq. (5.85). In the MS as well as in the on-shell scheme where the full "rst moment *g decouples from A (Q) and *R(Q) is not conserved, cf. Eq. (5.80), the experimentally required suppression of A in Eq. (5.7), as required by the constraint (5.24), can be achieved either by *s"*s (0 or by
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*u +*dM (0 in a S;(3) broken scenario. A detailed (x, Q) dependent analysis of present D polarization data will be presented in Section 6. Despite the fact that *g decouples from A , the polarized gluon density dg(x, Q) will in future play an important part in analyzing presently available data for x55;10\ which give rise to a sizeable *g as well. As already mentioned in Section 3.4 and to be discussed in more detail in Section 6, the best information on dg(x, Q) will come from heavy quark production in polarized el pl via cl Hgl Pcc (with no light quark contribution present in LO), or in polarized pl pl collisions at the RHIC facility via the dominant gl gl Pcc subprocess; in the latter case also the production of direct photons should o!er a good possibility to extract information about the magnitude of *g(Q). Alternatively, dg(x, Q) could be determined from the Q-dependence of g if HERA is run with a polarized proton beam. 5.4. The Bjorken sum rule Sum rules are relations for moments of the structure functions. The most important and fundamental sum rule for polarized scattering is the Bjorken sum rule which was derived in 1966 from SU(6) SU(6) current algebra [118,119]. It describes a relationship between spin-dependent DIS and the weak coupling constant de"ned in neutron b-decay
1g dx[gN (x, Q)!gL (x, Q)]" 6g 4 where g and g are the vector and axial vector couplings measured in nuclear b-decay 4 G g L "! $ cos h p c 1! c n [e cI(1!c )l] . @U I g (2 4
(5.86)
(5.87)
Here h is the Cabibbo angle and g /g "1.2573$0.0028 in Eq. (5.8) is known to high precision. 4 It is not di$cult to prove the Bjorken sum rule with the help of the knowledge which was collected in Section 5.1. Qualitatively, it arises as follows: g /g gives the strength at which the 4 axialvector transformation from up to down quark in the proton takes place. In DIS this is measured by *(u#u )!*(d#dM ). Indeed, from Eqs. (5.5), (5.30) and (5.87) one sees that one simply has to show 1PS"tM c c j t"PS2 "(g /g )S . N I N 4 I One can use the isospin symmetry to apply the Wigner}Eckhart theorem
(5.88)
1PS"tM c c j t"PS2 " 1PS"tM c c j t"PS2 , (5.89) N I N N I > L where "2 and "2 denote the proton and neutron states. From Eq. (5.89) one gets N L 1PS"u c c d"PS2 "(g /g )S (5.90) N I L 4 I which completes the proof of the Bjorken sum rule. This sum rule is so very fundamental because it relies only on isospin invariance, i.e. on a S;(2) D symmetry between up- and down-quarks, cf. Eq. (5.89). The Bjorken sum rule is an asymptotic result which relates low- and high-Q quantities. This originates from the fact that, apart from "nite
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O(a ) corrections, the l.h.s. of Eq. (5.86) CN (Q)!CL (Q)"A "(*u#*u !*d!*dM ) is just Q the nonsinglet isospin-3 component in Eq. (5.4) which does not renormalize due to the vanishing of *P and *P in (5.19). Thus it is generally believed that the Bjorken sum rule gets only OO ,1\ moderate QCD corrections. For example, the perturbative QCD corrections are given by the nonsinglet Wilson coe$cient *C in (5.20) and known to be small. In addition, there are probably ,1 nonperturbative higher-twist (HT) corrections, so that instead of (5.86) we have in general
1g c dx[gN (x, Q)!gL (x, Q)]" *C (Q)# &2 ,1 6g Q 4 with [414,415,439]
(5.91)
a (Q) a (Q) a (Q) *C "1! Q !3.5833 Q !20.2153 Q , ,1 p p p
(5.92)
cf. Eq. (5.20). The latter result is obtained for f"3 light #avors, and the range of c has been &2 estimated to be c +!0.025 to #0.03 GeV [84,85,402]. For recent reviews, see [371,464,358] &2 and references therein. Additional renormalon contributions to the perturbative 3-loop result in Eq. (5.92) have been studied as well but their size seems to be small in the relevant perturbative region, Q91 GeV. For a review, see [257] and references therein. Disregarding the small nonperturbative contribution for Q'1 GeV, the Bjorken sum rule (5.91) is in reasonable agreement with experiments: According to Table 2, SLAC(E143) "nds CN (Q)!CL (Q)"0.164$0.017 (5.93) at Q"3 GeV, to be compared with the predicted [r.h.s. of (5.91)] CN (Q)!CL (Q)" 0.187$0.002 at the same Q, using a /p"0.076$0.010. The most recent CERN (SMC) result is Q [28] CN (Q)!CL (Q)"0.195$0.029 (5.94) at Q"10 GeV, where the theoretical prediction is CN (Q)!CL (Q)"0.193$0.002, using a /p"0.061$0.004. A more detailed comparison between theory and experiment can be found, Q for example, in [574,258,48]. Finally, it should be kept in mind that the original small EMC result for CN in Eq. (3.3) implied already dramatic consequences for CL prior to the recent CERN and SLAC measurements of the neutron structure function gL , by assuming the validity of the &safe' Bjorken sum rule: Using Eq. (3.3) in (5.91) one anticipated CL (Q+10 GeV)"!0.067$0.018 (5.95) which, being in agreement with recent measurements in Table 2, is about ten(!) times larger than the naive pre-EMC Ellis}Ja!e expectation CL +!0.008 based on Eq. (5.15) in conjuction with the #( Bjorken sum rules. These predictions for CL can be translated into Bjorken-x space, (5.96) gL (x, Q)"gN (x, Q)![du (x, Q)!dd (x, Q)] T T where dq ,dq!dq . The small NLO contribution due to dC (x) in Eq. (4.47) has been suppressed. T O In order to reproduce the strongly negative x-integrated result Eq. (5.95), it can be anticipated [310] from Eq. (5.96) that gL (x, Q) has to become strongly negative for x:0.1, in contrast to
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gL which gives rise to an almost vanishing CL . It will become transparent from a detailed #( #( (x, Q)-dependent analysis of all present data in Section 6, that such expectations have been con"rmed by the more recent CERN(SMC) and SLAC measurements. 5.5. The Drell}Hearn}Gerasimov sum rule The DHG sum rule [231,291] is a prediction for C at Q"0 and can be considered as giving qualitative information on the magnitude of higher twist e!ects in the region between Q"0 and the present high energy data. It relates the spin-dependent scattering cross section of circularly polarized real photons by longitudinally polarized nucleons N to the anomalous magnetic moment of the nucleon. If we de"ne
M g,(x, Q) dx , I (Q),2 , Q
(5.97)
the statement of the DHG sum rule is I (0)"!i /4 , ,
(5.98)
where M is the mass of the nucleon and i its anomalous magnetic moment, k "(1#i )k and , N N k "i k . i is de"ned by the nucleon}photon coupling L L , c?(iR !e A )!(i k /2)p F?@ ? , ? , ?@
(5.99)
with e "#e and e "0 and from experiment one has i "1.79 and i "!1.91. N L N L The derivation of the DHG sum rule relies on the relation between g and the photoabsorption cross sections, Eqs. (2.10) and (2.14), and on the dispersion relation for forward Compton scattering. In fact, the spin-#ip part of the forward Compton scattering amplitude is proportional to [p !p ]dl/l and, at low Q, is given by !(2pa/M)i which is usually referred to as , the Low-theorem [453]. Fig. 23 compares the result of the DIS data for I (Q) and I (Q), using the recent QN L independent LO results CN "0.146 and CL "!0.064 [309] in Eq. (5.97), with the values at Q"0 from the DHG sum rule. Since perturbative LO and NLO QCD is fully operative for Q91 GeV [309], as will be discussed in more detail in the next section, the most striking feature of this "gure is the necessity of a tremendous Q-dependence (variation) of CN and CN !CL in the low-Q region, Q(1 GeV: In particular, CN must change sign between Q+0 and Q+1 GeV, and this must be due to some strong nonperturbative higher twist e!ect. A parametrization (but not an explanation) of this e!ect was suggested by [64] and later on improved by [151,539]. It turns out that such e!ects seem to be signi"cantly larger than what one obtains on the basis of QCD sum rules. Most recent E143 measurements [6] at small Q con"rm these expectations as well as the trend depicted in Fig. 23. Reviews on this topic can be found in [371,99], for example. Experiments are being planned to test the sum rule directly [139,264]. This is important because, among other things, there has been some criticism towards the argument that connects the spin structure function integral to the Q"0 integral in Eqs. (5.97) and (5.98) [392].
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Fig. 23. DIS predictions for I (Q) according to (5.97) using CN "0.146 and CL "!0.064. These latter numbers are N L Q-independent in LO [309]. The DHG values refer to (5.98).
6. Polarized parton densities and phenomenological applications 6.1. Deep inelastic polarized lepton}nucleon scattering Nowadays sophisticated phenomenological models describing the spin structure functions in LO and NLO QCD are used to interpret the experimental data. Before we discuss these models, we want to give a short historical overview about some simpler (and by now mostly obsolete) models that were discussed in the literature in the past, the Kuti}Weisskopf model [430], the Carlitz}Kaur model [159] and the Cheng}Fischbach [169,170] model. When discussing the "rst moment of g in Section 5.1 we have noted that in the SU(6) constituent picture the nucleon consists of three free quarks and its wave function is completely symmetric in spin and #avor indices. Squaring the SU(6) wave function one obtains the probabilites for an up or down quark with spin up or down and, from that, the SU(6) predictions CN " and CL "0 according to (5.17). This prediction is clearly wrong except in that the proton is expected to have a larger positive asymmetry than the neutron with a very small asymmetry. One of the "rst attempts to describe the nucleon in terms of a relativistic picture was the Kuti}Weisskopf model [430]. In this model the nucleon consists of three valence quarks described by the SU(6) wave function, allowing however for an x-dependent quark-density distribution, plus a core of an inde"nite number of quark}antiquark pairs carrying vacuum quantum numbers and zero angular momentum. The model also postulated the existence of gluons carrying a fraction of the total momentum, the ratio of gluons to the quark}antiquark pairs being the only free
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parameter of the model. At low x the core dominates while at high x only the valence quarks contribute to the scattering. The functional form of g is given by C(c#3(1!a)) 5 x\?(1!x)\>A>\? (6.1) g (x)" 54 C(1!a)C(c#2(1!a)) for the proton while for the neutron it is identically zero. Assuming that the behavior of structure functions near x"1 is related to the elastic form factor one obtains !1#c#2(1!a(0))"3. In their original work, Kuti and Weisskopf [430] considered a(0)K0.5, as suggested by nondi!ractive trajectories, which leads to an increase of the spin asymmetry at small x. This is in contradiction with the present correct view, cf. Section 4.4, and with experimental data. The normalization factor in Eq. (6.1) was chosen to ful"ll the SU(6) predictions for the "rst moment sum rules. Therefore, just as SU(6), the model fails to explain the experimental data. Here, and in the other following pre-QCD approaches, the scale Q at which Eq. (6.1) is expected to hold, cannot be speci"ed except that Q<M, as usual within the framework of the idealized &impulse approximation'. Later attempts [187] tried to improve the model by taking into account the di!erence between constituent and current quarks. The two are related by a Melosh transformation which is shown to be equivalent to a rotation in spin space [187]. The rotation angle h is de"ned so as to "t the modi"ed SU(6) formula g /g " cos 2h to the experimental number, yielding cos 2h+0.75. The 4 predictions for the spin asymmetry are
19 16 ! r(x) cos 2h, AN " 15 15
2 3 AL " ! cos 2h 5r(x) 5
(6.2)
where r(x)"FL (x)/FN (x) is the ratio of the unpolarized structure functions. In this model the prediction for CN is even somewhat larger than that of the Ellis}Ja!e sum rule. In another attempt to satisfy the Bjorken sum rule without completely abandoning the SU(6) picture of the nucleon, Babcock et al. [80] suggested to include perturbative estimates of the qq and gluon sea polarization. Furthermore, for the valence quarks the parametrization du(x)K0.44u (x) T and dd(x)K!0.35d (x) was proposed. Since the authors oriented their results at the prediction of T the Ellis}Ja!e sum rule, they are not in agreement with present data. Most famous among the SU(6) inspired models for the polarized structure functions is certainly the Carlitz}Kaur model [159]. In this model the symmetry is broken by suggesting that the con"guration, in which the non-interacting diquark system is in an isospin-1 state, is suppressed at high x relative to the isospin-0 case. This is quanti"ed in terms of two functions, I (x) and I (x), describing the valence quark distributions, where the subscript refers to the isospin of the non-interacting system and which are obtained from unpolarized DIS data. At low x, the valence quarks are assumed to lose any memory of the parent spin orientation through interaction with the gluon sea. This is described in terms of a factor, called sin h(x), giving the probability that the quark will #ip its spin through interactions with the sea. It is given by 1 H(x)N(x) , sin h(x), 2 H(x)N(x)#1
(6.3)
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where N(x) is the density of the gluon sea relative to the valence quarks and H(x) is the probability of a spin-#ipping interaction between valence quarks and gluons. The behavior of N(x) for xP1 is suggested by dimensional counting rules and the behavior for xP0 by Regge theory, N(x)&(1!x)/(x. Assuming the x-independence of H, a measure of the dilution of the valence quark spin due to this interaction is then given by the spin dilution factor cos 2h(x)" H (1!x)/(x. The polarized quark densities are given by dd(x)"!d (x)cos 2h(x) . (6.4) du(x)"[u (x)!d (x)]cos 2h(x), T T T The only free parameter in the model is H . It is "xed by the Bjorken sum rule, H +0.052. Correspondingly, the spin dilution factor can be calculated. It is almost equal to 1 everywhere except at low values of x. Using Eq. (6.4), CN L can be calculated and turn out to be essentially identical to the predictions of the Ellis}Ja!e sum rule. Furthermore, it should be noted that the d-quark spin density in Eq. (6.4) does not satisfy the perturbative QCD requirement dd(x)& d (x). V T In the Cheng}Fischbach model [169,170], later re"ned by Callaway and Ellis [156] and Cheng and Lai [171], it is simply assumed that the polarized quark distributions are related to the unpolarized ones via du (x)"a(x)u (x), T T
dd (x)"b(x)d (x) . T T
(6.5)
It is further assumed that a(x), b(x)& 1 in order to take account of the argument that the valence V quark at x"1 remembers the spin of the parent nucleon. By contrast, the region near x"0 is expected to be dominated by the sea so that the spin of the nucleon is no longer re#ected by the valence quarks implying a(x), b(x)& 0. Since *d is negative, the boundary condition b(x)& 1 V V implies that dd (x) changes sign as a function of x. Therefore, b(x) was suggested to be of the form T b(x)"((x!x )/(1!x ))xN so that the sign of dd (x) #ips at x"x . Furthermore, Cheng and Lai T [171] have chosen a(x)"x . Roughly, typical values of x and p are of the order 0.5. These valence distributions alone cannot account for the experimental data which require in addition a large polarized sea quark and/or gluon density. In Ref. [171] an ansatz of the form "ds(x)""xAQ s(x), "dg(x)""xAE g(x) was made. Since the data do not really "x ds(x) and dg(x) it was not possible to determine c and c from a "t. Q E Now we turn to more recent developments. According to the discussions and results presented in Sections 4.1, 4.2 and 5.1, it is straightforward to perform LO and NLO (MS) QCD analyses of all presently available polarized DIS data on gN L(x, Q). This a!ords the knowledge of appropriate input parton densities df (x, Q ), f"q, q , g, extracted (as far as possible) from present measure ments at a conveniently chosen Q"Q . The analyses can be performed either directly in Bjorken-x space or, more conveniently, in (Mellin) n-moment space where the RG evolution equations can be solved analytically as in Eqs. (4.60)}(4.64). During the past decade many LO analyses, partly supplemented by the NLO gluon anomaly in Eq. (4.46), (4.65) or (5.22), have been performed, e.g. [56,310,515,493,341,342,344,171,178,172,146,441], and more recently in [52,53,293,313,256,268,97,173,87,138,268]. With the recently completed calculation of all two-loop splitting function (anomalous dimensions) dP(x), i, j"q, g, in the MS factorization scheme GH [478,568,569], also fully consistent NLO (MS) analyses became possible [309,551,294,9,270,
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444,445,559]; alternative factorization schemes have been proposed as well for performing NLO analyses [88,48,9,31,446,367]. In general, the searched for parton densities df (x, Q) have to satisfy the fundamental positivity constraints (4.11) at any value of x and scale Q, as calculated by the unpolarized and polarized evolution equations, within the same factorization scheme. It is thus natural to perform polarized NLO analyzes in the MS scheme where practically all unpolarized NLO parton densities have been analyzed and presented. Furthermore the total helicities, i.e. n"1 moments of df (x, Q) in Eq. (5.1), are constrained by the sum rules (5.8) and(5.9) *u#*u !*d!*dM "F#D"1.2573$0.0028 ,
(6.6)
*u#*u #*d#*dM !2(*s#*s )"3F!D"0.579$0.025 ,
(6.7)
which hold for the #avor SU(3) symmetric &standard' scenario commonly used. It should be D remembered that the #avor nonsinglet combinations A in Eqs. (5.5) and (5.6) which appear in (6.6) and (6.7) are Q independent also in NLO due to *P "0 in (5.19). ,1\ As has been already discussed at the beginning of Section 5.1, there are serious objections to this latter full SU(3) symmetry (mainly due to m ;m ) which results in (6.7), in contrast to the D S B Q unquestioned isospin SU(2) symmetry (m Km ) which gives rise to (6.6). A plausible (but D S B extreme) alternative to the full SU(3) symmetry is a &valence' scenario [450,451,448] where SU(3) D D is (maximally) broken and which is based on the assumption that the #avor-changing hyperon b-decay data "x only the total helicities of valence quarks *q (Q),*q!*q : T *u (Q )!*d (Q )"F#D"1.2573$0.0028 , (6.8) T T *u (Q )#*d (Q )"3F!D"0.579$0.025 (6.9) T T at some appropriately chosen input scale Q"Q . Note that *q (Q) depends (marginally) on T Q in NLO due to *P O0 in (5.19). ,1> In the &standard' SU(3) symmetric scenario we need *s"*s (0 in CN L in Eq. (5.14) or (5.18) in D order to comply with recent experiments (cf. Table 2), i.e. in order to obtain a reduction of the Ellis}Ja!e expectation CN L , Eq. (5.15), based on A and A which are entirely "xed by Eqs. (6.6) #( and (6.7), respectively. [Remember that *g(Q) decouples from CN L in (5.18) in NLO (MS) due to *C "0.] In the &valence' scenario we can do even with *s"*s K0 since here only the valence E contribution to A is "xed (apart from minor Q dependent e!ects in NLO) by Eq. (6.9), with the entire A still being "xed by (6.8) due to the assumption *u "*dM ,*q (which is again violated by minor Q dependent e!ects in NLO). This gives in LO for C in (5.4) (6.10) CN L"$ (F#D)# (3F!D)# (10*q #*s#*s ) and a similar relation holds in NLO [309]. Thus, in contrast to Eq. (5.14), a light polarized sea *q (0 will account for a reduction of CN L even for the extreme SU(3) broken choice *s"*s "0! D Turning to the determination of the polarized LO and NLO parton distributions df (x, Q) it is helpful to consider some reasonable theoretical constraints concerning the sea and gluon densities, in particular in the relevant small-x region where only rather scarce data exist at present (in contrast to unpolarized DIS): apart from the rather general Regge constraints in Eq. (4.102) for xP0, color coherence of the gluon couplings at xK0, i.e. equal partition of the hadron's
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momentum among its partons, implies for the gluon and sea densities [145,144] df (x, Q )/f (x, Q )&x as xP0 , (6.11) and arguments based on helicity retention properties of perturbative QCD of valence densities at large-x imply [145,144,193] dq (x, Q )&q (x, Q ) as xP1 . (6.12) T T The scale Q at which these relations are supposed to hold remains unspeci"ed. Although not strictly compelling, Eqs. (6.11) and (6.12) are expected [145,144] to hold at some &intrinsic' boundstate-like scale (Q :1 GeV, say), but certainly not at much larger purely perturbative scales Q <1 GeV. Despite such &guidelines', presently available scarce polarization data on A (x, Q)Kg (x, Q)/F (x, Q), Eq. (2.14), constrain the polarized parton densities rather little, which holds in particular for dg(x, Q) [309,313,294,88]. The determination of the polarized valence densities is less ambiguous. In order to avoid, as far as possible, pure guesses for the input densities df (x, Q ), it has been suggested in [309,313] to employ the unpolarized valence-like input densities f (x, Q ) at Q "kK0.3 GeV, properly modi"ed so as to comply with polarized DIS data, with the positivity inequalities (4.11) for Q5k and with the constraints (6.11) and (6.12). Subject to these requirements the following general ansatz for the LO and NLO polarized parton densities has been employed [309,9]: dq (x, k)"N T x?OT q (x, k) , T T O dq (x, k)"N x?O (1!x)@O q (x, k) , O (6.13) ds(x, k)"ds (x, k)"N dq (x, k) , Q dg(x, k)"N x?E (1!x)@E g(x, k) , E where the LO and NLO unpolarized input densities f (x, k) at k "0.23 GeV and *k "0.34 GeV, respectively, refer to the recent GRV valence-like densities [319]. It should be ,*noted that employing valence-like gluon and sea input densities [i.e. xg(x, k)&x?, xq (x, k)&x?Y with a, a '0 as xP0] allows for a parameter-free calculation of parton densities and DIS structure functions in the small-x region (x:10\) at Q'k which is entirely based on the QCD dynamics [314,315,319]. The perturbatively stable LO/NLO predictions turned out to be in excellent agreement with all DESY-HERA measurements up to now [11,38,40,220,222,223]. The resulting "t parameters N , a , b for the &standard' and &valence' scenarios in LO and NLO G G G can be found in [309] where appropriate simple parametrizations of the rather complicated QCD evolutions have also been given. The LO and NLO results for the asymmetries AN B(x, Q) measured up to now, as discussed in Section 3, are presented in Fig. 24 for the &standard' scenario. The results for the &valence' scenario are very similar. In both cases the LO and NLO results are perturbatively stable and almost indistinguishable. The expected Q dependence of A (x, Q) is shown in Fig. 25 and compared with recent SLAC-E143 and SMC data. [The di!erence between the LO and NLO results in the small-Q region is mainly due to di!erent LO (k "0.23 GeV) *and NLO (k "0.34 GeV) input scales.] It should be emphasized that A Kg /F is in general ,* expected to be Q dependent as soon as gluon and sea densities become relevant, due to the very di!erent polarized and unpolarized splitting functions dP (x) and P (x), respectively (except GH GH
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Fig. 24. Comparison of LO and NLO results for A (x, Q) as obtained [309] from the "tted inputs at Q"k for *- ,*the &standard' scenario, Eqs. (6.6) and (6.7), with present data [490,77,20,17,18,65,2}4,23}25]. The Q values adopted here correspond to the di!erent values quoted by the experiments for each data point starting at Q91 GeV at the lowest available x-bin.
Fig. 25. The Q dependence of AN B(x, Q) as predicted by LO and NLO QCD evolutions [309] at various "xed values of x, compared with recent SLAC-E143 [2,4] (solid circles) and SMC data [25,17,18,20] (open circles).
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for dP"P which dominates, apart from marginal di!erences in the relevant NLO NS splitting OO OO functions, in the large-x region). Moreover, the smaller x the stronger becomes the dependence of the exactly calculated A (x, Q) on the precise form of the input at Q"k [313]. For practical purposes, however, such ambiguities are irrelevant since A Kg /F K2x g /F P0 as xP0 is already unmeasurably small (of the order 10\) for x:10\. Thus the small-x region is unlikely to be accessible experimentally for g (x, Q), in contrast to the situation for the unpolarized F (x, Q). It is furthermore interesting to note that the approximate asymptotic (xP0) DLL expression (4.110) for A (x, Q) does not even quantitatively reproduce the exact LO results for A (x, Q) for x510\ [313]. It is therefore misleading to use the simple asymptotic DLL formulae (4.107)}(4.110) for quantitative estimates [93,94,263]. The structure functions gN L(x, Q) and gB , given by the relation (3.6), can now be extracted using Eqs. (2.14) and (2.18) where one usually neglects the subleading contributions proportional to c. These results are shown in Figs. 26 and 27. The reason why the LO results are partly larger by more than about 10% than the NLO ones is mainly due to the LO approximation R"0 in Eq. (2.19), as used in [309]. Some of the EMC and E143 asymmetry data [490,77,78,2,4] have been analysed by assuming AN to be independent of Q. This can be partly responsible [309] for these &data' falling somewhat below the NLO predictions in the small-x region, despite the excellent "ts to AN in Fig. 24. The predictions for the NLO parton distributions at the input scale Q"k " ,*0.34 GeV in Eq. (6.13) are shown in Fig. 28 and compared with the reference unpolarized NLO dynamical input densities of [319] which satisfy of course the positivity requirement (4.11) as is obvious from Eq. (6.13). The LO predictions are similar [309]. It should be noted that the strange densities correspond to N "1 in (6.13) for the SU(3) symmetric &standard' scenario, whereas to Q D
Fig. 26. Comparison of the &standard' and &valence' LO and NLO results [309] with the data for gN B(x, Q) [490,77,78,17,18,20,65,2}4,23}25,28]. The SMC data correspond to di!erent Q91 GeV for x50.005, as do the theoretical results.
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Fig. 27. Same as in Fig. 26 but for gL (x, Q). The E142 and E143 data [65,3] correspond to an average 1Q2"2 and 3 GeV, respectively, and the theoretical predictions correspond to a "xed Q"3 GeV.
Fig. 28. Comparison of the "tted &standard' and &valence' input NLO(MS) densities at Q"k "0.34 GeV, accord,*ing to Eq. (6.13) [309], with the unpolarized dynamical input densities of [319].
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Fig. 29. The polarized LO and NLO(MS) densities at Q"4 GeV as obtained from the input densities at Q"k *- ,*as shown in Fig. 28 for the NLO. In the &standard' scenario, ds coincides with the curves shown for dq in LO and NLO due to the SU(3) symmetric input which is only marginally broken in NLO for Q'k . D ,*-
N "0 for the SU(3) broken &valence' scenario [309]. The corresponding polarized densities at Q D Q"4 GeV, as obtained from these inputs at Q"k for the two scenarios in LO and NLO, are shown in Fig. 29. It is interesting to note that, within the radiative approach with its longer Q-evolution &distance' starting at the low input scale k in Eq. (6.13), a xnite (negative) strange polarized sea input ds(x, k) is always required by present data for the &standard' scenario. This holds true even if one uses (somewhat inconsistently in NLO) the &o!-shell' dCI in Eqs. (4.46) or E (4.65) which corresponds to *CI ", giving rise to Eq. (5.22), in contrast to *C+1"0. The shape of E E the polarized gluon densities dg presented in Figs. 30 and 31 is constained rather little by present asymmetry data [309,313,294]: Equally agreeable "ts can be obtained for a fully saturated [inequality (4.11)] gluon input dg(x, k)"g(x, k) as well as for the less saturated dg(x, k)"xg(x, k). A purely dynamical [310] input dg(x, k)"0 is also compatible with present data, but such a choice seems to be unlikely in view of dq (x, k)O0; it furthermore results in an unphysical steep [310] dg(x, Q'k), being mainly concentrated in the very small-x region x(0.01, as in the corresponding case [314,296] for the unpolarized parton distributions in disagreement with experiment. The resulting NLO gluon densities dg(x, Q) at Q"4 GeV which originate from these extreme inputs are compared in Fig. 30 with our &"tted dg' curve of Fig. 29 obtained for the &valence' scenario. Present data allow even for a partly negative dg(x, 4 GeV) [294] and speci"c model calculations can accommodate even a negative *g(4 GeV) [312,376]. It turns out that dg(x, Q) is somewhat less ambiguous if only the more global quantities dq , dR and ,1 dg are used for analyzing present data [88], instead of trying to delineate the individual parton densities. An alternative analysis of polarized structure functions has been performed by Gehrmann and Stirling [294]. This was done in the same spirit as the unpolarized analysis [468], namely the
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Fig. 30. The experimentally allowed range of NLO polarized gluon densities at Q"4 GeV for the &valence' scenario with di!erently chosen dg(x, k ) inputs. The &"tted dg' curve is identical to the one in Fig. 29. Very similar results are ,*obtained if dg(x, k ) is varied accordingly within the &standard' scenario as well as in an LO analysis [309,313]. ,*-
Fig. 31. NLO polarized and unpolarized parton distributions at Q "4 GeV according to [294].
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polarized parton distributions were chosen to be of the general form xdf (x, Q )"g A x?D (1!x)@D (1#c x#o (x) (6.14) D D D D for f"u , d , q , g at the starting scale Q "4 GeV. The normalization factors T T C(a )C(b #1) a C(a #0.5)C(b #1) \ D D D D D (6.15) A " 1#c #o D D a #b #1 C(a #b #1) D C(a #b #1.5) D D D D D D are determined by the condition that the "rst moments of df (x, Q ) are given by g . The parameters D g , a , b , c and o for f"u and d were determined by LO and NLO "ts to recent data. For the D D D D D T T sea, an SU(3) symmetric antiquark polarization was assumed (just as in the &standard' scenario D [309] as discussed above, e.g. Eq. (6.13) with N "1). For dg(x, Q ), which is hardly constrained Q by the data (cf. Fig. 30), three alternative parametrizations were employed, a hard and a soft distribution (A and B) with the spin aligned with that of the parent proton and a distribution (called C) with the spin anti-aligned. All three choices give equally good descriptions of the structure function data, but would be relatively easy to discriminate if data on polarized gluon initiated processes like cHgPcc or qgPcq were available. In all cases the "rst moment g "1.9 was E chosen which is obtained in LO by attributing all the violation of the Ellis}Ja!e sum rule to a large gluon polarization. This number is in the same ballpark as if determined from an NLO analysis of the experimental data (*g(Q )"1.5$0.5 [88] and Eq. (6.16) and Tables 3 and 4 below), but with a large error. A summary of the "tted and chosen parameters together with the s of the "t can be found in Table 2 of [294]. The NLO polarized parton distributions at Q "4 GeV for gluon scenarios A, B and C are shown in Fig. 31. It should be emphasized that present polarization data can be "tted even with a negative dg in the large-x region, as shown in Fig. 31, which refers to a more extreme choice than the ones depicted in Fig. 30. Furthermore, the LO GRV 94 [319] and the NLO MRS-A' [468] parton densities have been chosen as reference distributions, which are necessary for comparing with the positivity constraints (4.11), and are also shown in Fig. 31. The resulting "ts for gN L B(x, Q) are similar in quality [294] as the ones shown in Figs. 26 and 27. Finally let us turn to the "rst moments (total polarizations) *f (Q) in Eq. (5.1) of the polarized parton densities df (x, Q) and the resulting CN L(Q). It should be recalled that, in contrast to the LO, the "rst moments of the NLO (anti)quark densities do renormalize, i.e. are Q dependent, due to the nonvanishing of the 2-loop *P and *P in (5.19). Let us discuss the two scenarios in OO ,1> turn: (i) In the &standard' scenario the input densities in (6.13), being constrained by (6.6) and (6.7), imply in LO [309]
*u "0.9181, *d "!0.3392 , T T *q "*s"*s "!0.0587 ,
(6.16)
*g(k )"0.362, *g(4 GeV)"1.273, *g(10 GeV)"1.570 , *which result in *R"0.227. This gives CN "0.1461,
CL "!0.0635
(6.17)
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Table 3 First moments of polarized NLO parton densities df (x, Q) and of gN L(x, Q) as predicted in the &standard' scenario [309]. Note that the marginal di!erences for *q and *s indicate the typical amount of dynamical SU(3) breaking generated by D the RG Q-evolution to Q'k ,*Q(GeV)
*u T
*d T
*q
*s"*s
*g
*R
CN
CL
k ,*1 4 10
0.9181 0.915 0.914 0.914
!0.3392 !0.338 !0.338 !0.338
!0.0660 !0.067 !0.068 !0.068
!0.0660 !0.068 !0.068 !0.069
0.507 0.961 1.443 1.737
0.183 0.173 0.168 0.166
0.1136 0.124 0.128 0.130
!0.0550 !0.061 !0.064 !0.065
in reasonable agreement with present data (Table 2). The NLO results are shown in Table 3 which are in even better agreement with experiments (Table 2). (ii) In the &valence' scenario the input densities in (6.13), being constrained by (6.8) and (6.9), imply in LO [309] *u "0.9181, T *q "!0.0712,
*d "!0.3392 , T *s"*s "0 ,
(6.18)
*g(k )"0.372, *g(4 GeV)"1.361, *g(10 GeV)"1.684 *which result in *R"0.294. Apart from this maximal SU(3) breaking, these results are similar to D the &standard' ones in (6.16) and yield CN "0.1456, CL "!0.0639 (6.19) on account of (6.10). The NLO results are shown in Table 4. These results for the total helicities ("rst moments) are similar to the ones observed in other recent LO and NLO analyzes [294,88,9]. Due to the similarity of the LO and NLO results in both scenarios, it is obviously impossible to distinguish experimentally between the &standard' (SU(3) symmetric) and &valence' (SU(3) maxiD D mally broken) scenario. In both scenarios the Bjorken sum rule (5.86) manifestly holds due to the constraints (6.6) and (6.8). Furthermore, the observed total helicities carried by the valence quarks, *u and *d , are compatible with the ones obtained very recently from semi-inclusive spin T T asymmetry measurements [27], cf. Section 6.5, which yielded *u "1.01$0.24 and T *d "!0.57$0.25 at 1Q2K10 GeV. It is also very interesting to note that our optimal "t T results shown above favor a sizeable total gluon helicity *g(10 GeV)K1.7, despite the fact that *g(Q) decouples from the full (04x41) "rst moment C (Q) in (5.18) in the MS scheme (since *C+1"0). This implies that for any experimentally relevant analysis (where 0.01:x(1), the E NLO dg(x, Q) in Eq. (4.47), for example, plays an important role almost regardless of the value of the full "rst moment of dC (x). The importance of dg(x, Q) also holds in LO where dg does not E directly appear in g (x, Q), Eq. (4.5), but enters only via the RG evolution equations. This large MS result for *g(Q) is also comparable with the weaker constraint (5.85) obtained in the o!-shell scheme (where *C "!) which is not too surprising since the RG solutions (5.81) E and (5.83) di!er only to O(a ). Furthermore one can be tempted to reinterpret our MS results for Q
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Table 4 The "rst moments (total helicities) as in Table 3, but for the maximally SU(3) broken &valence' scenario [309] D Q(GeV)
*u T
*d T
*q
*s"*s
*g
*R
CN
CL
k ,*1 4 10
0.9181 0.915 0.914 0.914
!0.3392 !0.338 !0.338 !0.338
!0.0778 !0.080 !0.081 !0.081
0 !2.5;10\ !3.5;10\ !3.8;10\
0.496 0.982 1.494 1.807
0.268 0.252 0.245 0.244
0.1142 0.124 0.128 0.130
!0.0544 !0.061 !0.064 !0.065
*R(Q) in terms of *R in Eq. (5.26), by assuming that *g(Q) in the o!-shell scheme is similar to our MS results. This gives *R K0.33 (0.42) (6.20) according to the results for the &standard' (&valence') scenario in Table 3 (Table 4). Thus the sizeable *g(Q) implies a sizeable amount of total helicity of singlet quark densities, *R , which comes close to the naive expectation *R KA K0.6 in Eq. (5.27) in contrast to *R(Q) in the MS scheme. Finally, it is very interesting to observe that at the low input scales Q"k : 0.23, 0.34 GeV *- ,*the nucleon's spin is dominantly carried just by the total helicities of quarks and gluons )#*g(k )K0.5 [0.6] (6.21) *R(k *- ,*-
*- ,*-
according to Eqs. (6.16) and (6.18), and Tables 3 and 4. Thus the helicity sum rule (1.1) implies that )K0 . (6.22) ¸ (k X *- ,*The approximate vanishing of this latter nonperturbative angular momentum, being built up from the intrinsic k carried by partons, is intuitively expected for low bound-state-like scales (but not 2 for Q
a (Q) d ¸O (Q) " Q 2p dt ¸ (Q) E
!C $
f 3
C $
f ! 3
a (Q) !C$ # Q 2p !C $
¸ (Q) O ¸ (Q) E
f 3 !
*R(Q) *g(Q)
,
(6.23)
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where the second inhomogeneous term was "rst studied in [505] with *R and *g evolving according to (5.2). The solution of (6.23) is straightforward:
1 1 3f 1 1 3f ¸ (Q)"! *R# # ¸ (Q )# *R! ¸>D @ , O O 2 2 16#3f 2 2 16#3f
1 16 1 16 # ¸ (Q )#*g(Q )! ¸>D @ ¸ (Q)"!*g(Q)# E E 2 16#3f 2 16#3f
(6.24) (6.25)
with ¸,a (Q)/a (Q ) and *R,*R(Q )"*R(Q) in LO. The last Eq. (6.25) demonstrates Q Q explicitly that the large gluon helicity *g(Q) as obtained above at large Q'k is canceled by an equally large, but negative, gluon orbital momentum. Asymptotically (QPR) the solutions (6.24) and (6.25) become particularly simple ¸ #*R" 3f/(16#3f ), ¸ #*g" 16/(16#3f ) . (6.26) O E Thus the partition of the nucleon spin between quarks and gluons eventually follows the wellknown partition of the quark and gluon momenta in the nucleon [338,339,295]. If the Q evolution is slow, then Eq. (6.26) predicts that quarks carry less than about 50% of the nucleon spin even at low momenta [cf. Eqs. (6.16) and (6.18)]. It should be mentioned that, although ¸ "¸ #¸ can be theoretically formally formulated in X O E a consistent covariant way [381,376,174], there appears to be no direct experimental test of the size as well as of the sign of ¸ (Q), or more ideally of the separate components ¸ (Q) and ¸ (Q). The X O E possible measurement of azimuthal distributions has been proposed [180,477] but these are only sensitive to some average 1k 2 of rotating constituents in a polarized nucleon target. 2 More recently, Ji [395,396] has suggested to use deeply virtual Compton scattering (DVCS) cH(Q)pPcp in the limit of vanishing momentum transfer t"(p !p), in order to get direct information about the gauge invariant combinations J "*R#¸ and J "*g#¸ appearing O E E O in the spin sum rule (1.1): "J (Q)#J (Q). In this limit, J "[A (0)#B (0)] where A (t) O E O E O E O E O E and B (t) are the &Dirac' and &Pauli' form factors of the quark and gluon energy-momentum tensor O E [395,398,498] (which are analogously de"ned as the well known form factors of the electromagnetic vector current). The form factors A (t) and B (t) are then related via sum rules to the n"2 O O moments (Bjorken-x averages) of the structure functions of the nonforward (or ow-forward) DVCS. Although the extrapolation tP0, required to obtain J , is di$cult [397,497,426] (if at all O E possible), rough estimates result in a DVCS cross section at !t(1 GeV above 1 pb at CEBAF and DESY}Hermes kinematics [397,563]. 6.2. Heavy quark production in polarized DIS and in photoproduction In the previous sections we have realized, among other things, the enormous di$culties to extract the polarized gluon distribution and in particular its "rst moment from inclusive deep inelastic data. These problems have been anticipated several years ago by theoretical studies [55,241,158,50], and they are in fact not surprising in view of the well-known subtleties having occurred in all attempts to determine the unpolarized gluon density in unpolarized DIS experiments during the past two decades.
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A popular way out of this dilemma is the study of semi-inclusive cross sections, and in particular of charm production, because the production of heavy quark hadrons is triggered in leading order by the photon}gluon fusion mechanism [cH(c)dgPhhM with h"b, c] and is therefore sensitive to the gluon density inside the proton, whereas the heavy quark content of the proton is usually negligible, if it exists at all, at presently available Q-values. Due to its prominent decay mechanism, J/t production is the most distinct among the charmed events. In contrast to open charm production one faces here, however, the additional model dependence for bound-state production, such as the &duality model' [281,346,403,298,300] (nowadays also called &color evaporation' model) and the color-singlet model [162,111,82,83]. In the duality or &soft color' treatment of color quantum numbers the cross section for bound charm production is given by
dp 1 K" dm AA , p " 9 dm KA
(6.27)
where dp is computed perturbatively in LO and NLO due to cHgPcc , etc. (or due to AA qq Pcc , ggPcc , etc., for hadronically produced quarkonia), since here the color singlet property of the J/t is ignored at the perturbative stage of the calculation. The subsequent hadronization of the color singlet state is then assumed to be characterized by multiple (nonperturbative) soft gluon emissions, i.e. the treatment of color is, on the average, statistical with the factor representing the statistical probability that the 3;3 charm pair is asymptotically in a color singlet state. Alternatively, in the color-singlet model the color singlet property of the produced onium states (J/t, etc.) is enforced already at short distances, *x&m\, by the emission of a perturbative (octet) gluon o! R the produced charm quark. It is this latter assumption which casts doubt on the color-singlet model since it does not seem logical to enforce perturbatively the color-singlet property of the onia at short distances, given that there remains practically an in"nite time for soft gluons to readjust the color of the cc pair before it appears as an asymptotic J/t or, alternatively, DDM state. In other words, it is hard to imagine that a color singlet state formed at a range m\, automatically survives R to form a J/t. Indeed, the duality (&color evaporation') treatment has received renewed attention and appears to be the favorite mechanism of heavy quarkonia production [58,235,526,290,527] (or a variant of it which di!ers in its nonrelativistic treatment of the nonperturbative long-distance part of the cc matrix elements which obey simple &velocity-scaling' laws with respect to the relative velocity b of the cc ; this allows for a systematic expansion in a (2m ) and b [129].) The reason for Q A this revival is that some data on the production of t- and B-states disagree with the simple minded color-singlet model predictions; occasionally by well over one order of magnitude as in the case of t production at the Fermilab Tevatron [140]. Thus it seems to be more appropriate to study and delineate the relevant quarkonia production mechanism using unpolarized reactions "rst, instead of using a particular model to get access to dg(x, Q+m ) via polarized deep inelastic (or photon) ( R production of J/t's [304,305,340,404,325,546]. This statement is even more true for (di!ractive) elastic J/t production where some very speculative models exist [142,516,517,447]. They are based on a two-gluon exchange but it is not clear whether the square of the polarized gluon density dg(x, Q+m ) or some independent two-gluon correlation function appears in the cross section ( R formula. Therefore, from the theoretical point of view a much cleaner signal for the gluon density in heavy quark production is open charm production, although experimentally it has worse statistics due to
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the di$culties in identifying D-mesons. Instead of the deep inelastic process one may as well look at photoproduction, because the mass of the charm quark forces the process to take place in the perturbative regime. The advantage of photoproduction over DIS is its larger cross section. Several "xed target experiments, HERMES at DESY [57], COMPASS at CERN [106] and a new facility at SLAC [73] are being developed to measure the polarized gluon distribution via photoproduction. In leading order the inclusive polarized deep inelastic open charm production cross section is given by, using Eq. (2.7) or (2.41) d 1 4pa d*p A" (p (2!y)gA (x, Q) N !p = )+ dx dy 2 Q dx dy
(6.28)
where [577,311]
dx x a (k ) dg(x , k )dC , Q (6.29) gA (x, Q)" Q $ $ A x x 9p >KA / V is the charm contribution to the polarized structure function g and where dC (z, Q)"(2z!1)ln((1#b)/(1!b))#(3!4z)b (6.30) A is the partonic matrix element due to cHdgPcc . Note that the renormalization scale of a in Q (6.29) has been set equal to the factorization scale k appearing in dg and that one usually takes $ k "2m . Furthermore, b"(1!4m/s( where s( "(p#q)"Q(1!z)/z is the Mandelstam $ A A variable of the subprocess. By combining these formulae with the unpolarized cross section one can obtain the polarization asymmetry AA"d*pA/dpA. If one plugs in the drastically di!ering LO polarized gluon densities of [309] and the oscillating GS}C density of [294] which are for convenience compared to each other in Fig. 32, one obtains the results for gA (x, Q) and the deep-inelastic charm asymmetry AA"gA /FA shown in Fig. 33 at Q"10 GeV [311,553,554]. (Note that the corresponding NLO densities are shown in Figs. 30 and 31 at Q"4 GeV.) It should further be noted that the dashed curve in Fig. 33a corresponds, at x+0.01, to about 10% of the full gN which implies that a fairly accurate high statistics experiment would be required in order
Fig. 32. Polarized gluon densities at Q"10 GeV (+4m) of the four LO sets used in this subsection. The dotted curves A refers to set C of [294] whereas the other densities are taken from [313] as described in Section 6.1.
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Fig. 33. Charm contribution gA to g at Q"10 GeV for the four gluon distributions of Fig. 32, calculated according to Eq. (6.29) using k "2m with m "1.5 GeV [553,554]. $ A A
to extract dg(x, k ) from gA . The size of AA in Fig. 33b in the large-x region, x'0.01, is deceptive $ since here the individual gA and FA "(FA !FA )/2x [308] rapidly decrease as a result of the * threshold condition b50. Realistically, AA is of the order of 5% at xK0.01 where gA is also sizable as shown in Fig. 33. For x:0.005, which could be reached at the HERA el pl collider, the situation is even less favorable, since g(x, k ) becomes much larger than dg(x, k ) as xP0 and AA is $ $ correspondingly small. Furthermore, the contribution dC in (6.30) from the polarized subprocess A cHgPcc changes sign towards the HERA small-x region, so that AA is further suppressed, cf. Fig. 33, and becomes probably unmeasurable below xK0.005. It should be noted that this latter oscillation of dC causes the strong increase of gA in the region of very small x, as shown by the A dotted curve in Fig. 33a, via the convolution with the peculiarly oscillating polarized gluon density GS}C of Fig. 32. The relevant asymmetry AA in Fig. 33b remains negligible due to the enormously increasing unpolarized gluon density for xP0. We now turn to the case of photoproduction of charm. It is straightforward to obtain from the above expressions (6.28)}(6.30) the inclusive open charm photoproduction cross section by taking the simultaneous limits QP0 and zP0 while keeping Q/z+s( "xed:
1#b 4paa (k ) dx Q $ dg(x , k ) 3b!ln *pA (s )" $ AN A 1!b x 9s KA QA A
(6.31)
where b"(1!4m/s( and s( "x s . This integrated cross section depends only on the total A A proton}photon energy s "(P#q) which for a "xed target experiment is given by s "2ME A A A where E is the photon energy. By varying the photon energy it is in principle possible to explore A the x-dependence of dg. Very high photon energies correspond to small values of x. However, as we shall see later, it is not trivial to obtain the "rst moment of dg from the cross section, Eq. (6.31). The expected polarization asymmetry is given as a function of (s in Fig. 34 for various possible forms A
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Fig. 34. Longitudinal spin asymmetry for the total charm photoproduction cross section calculated according to Eq. (6.31), using k "2m with m "1.5 GeV, for the four polarized gluon densities shown in Fig. 32 [553,554]. $ A A
of dg(x, k ). It is clearly seen that it becomes rapidly smaller towards higher energies due to the two $ reasons disussed above [oscillation of the parton matrix element and singular behavior of g(x, 4m)]. Thus the total charm cross section and the asymmetry become unmeasurably small at A HERA energies, (s &200 GeV, to be reached with a possible future doubly polarized el pl collider A [553,554]. On the other hand, it seems more feasible that the total asymmetry in cpPcc can be measured at smaller energies, (s :20 GeV, where future polarized "xed-target experiments like A COMPASS [106] will be performed. It should be noted that the choice of scale k in dg in Eqs. (6.29) and (6.31) is not certain: $ probably 2m is a reasonable choice but it might as well be (s or any number in between. This A A uncertainty re#ects our ignorance about the magnitude of the higher order correction and could be resolved if a higher order calculation of these cross sections will be performed. The same statement holds true for the argument of a . Therefore, in the equations presented below the scale of dg and Q a will be chosen to be more general, k and k , respectively. It turns out that the variation of the Q $ 0 cross sections when one varies k and k is larger (&20%) than that of the asymmetries (45%) $ 0 so that one may speculate that the higher order corrections to AA are small. However, to prove this conjecture a higher order calculation is necessary. Eq. (6.31) was obtained after integration over the charm quark production angle hK (in the gluon}photon cms). If one is interested in the p distribution or wants to introduce a p -cut, it is 2 2 appropriate to keep the hK dependence in the fully di!erential cross section [286,553,554]
ea (k ) tK #u( !2ms( tK #u( d*pA A #4m AN " A Q 0 dg(x , k )b 2 $ A tK u( 16s( tK u( dx d cos hK
(6.32)
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where s( "x s , tK "!(s( /2)(1!b cos hK ) and u( "!(s( /2)(1#b cos hK ). It is possible to make a transA formation to the transverse momentum of the charmed quark by using p "(s( /4!m)sin hK : 2 A ea (k ) dx b *pA (p )" A Q 0 dg(x , k ) AN 2 $ s( /4!m x 16s KA QA A A ( A Q \K dp tK s( tK u( u( 2 ; 2m ! ! !2m # . (6.33) A A u ( tK u ( u ( tK tK N2 (1!(p2 /(s( /4!mA )) There are several good reasons to study the p distribution. First of all and in general, it gives more 2 information than the inclusive cross section. Secondly and in particular, it can be shown that the integrated photoproduction cross section, Eq. (6.31), as well as the corresponding DIS charm production cross section in (6.28) are not sensitive to the "rst moment of dg. The sensitivity is strongly increased, however, if a p -cut of the order of p 51 GeV is introduced, see below. Last 2 2 but not least, it is experimentally reasonable and often necessary to introduce a p -cut. 2 Let us dwell on the "rst moment discussion for a moment. It is true that the "rst moment is only one among an in"nite set of moments and the most interesting quantity to know is the full x-dependence of dg(x, Q) at a given Q. However, as was shown in Section 5, the "rst moment *g certainly has its signi"cance, "rstly because it enters the fundamental spin sum rule (1.1) and secondly because it gives the contribution within the proton to the c anomaly, 1PS"q c c q"PS2"(*q!(a /2p)*g)S , within the proton. In massless DIS it is straightforward to I Q I "nd out what the contribution of the "rst moment to the cross section is. One can apply the convolution theorem (4.23) to see that the contribution of *g is given by the "rst moment of the parton matrix element, i.e. by the n"1 gluonic Wilson coe$cient, cf. Eq. (4.62). If masses are involved, like m , the answer to this question is somewhat more subtle. Since the cross section is not A any more a convolution of the standard form, (4.14), one can formally write the integrals in (6.29) and (6.31) as (dx /x )dg(x , k )H(m/x , Q) where m"(1#(4m/Q))x for DIS charm production K $ A and m"4m/S for photoproduction (Q"0) of charm. Now one can apply the convolution A A theorem by integrating this expression over m and the "rst moment dz H(z, Q) gives essentially the contribution from *g(k ). By integrating Eqs. (6.30) and (6.31) it turns out that both for the $ inclusive charm DIS and photoproduction the corresponding quantities dz H(z, Q) identically vanish [577,311,304,305]. This can be traced back to the small-p behavior of the (perturbative) 2 partonic cross section (Wilson coe$cient) for cHgPcc which cancels the contribution of the large-p region in dz H(z, Q) [462,565,566]. It is not really a surprise in view of the structure of 2 the anomaly in massive QCD [cf. the discussion after Eq. (5.39) and the appendix of [433]]. Since the integrals dz zL\H(z, Q) keep being small in a neighborhood of n"1 one may conclude from this that these cross sections are not suited for determining the "rst moment of dg. Fortunately, the situation changes if one includes a p -cut of greater than 1 GeV. In that case the sensitivity to 2 dg(x, k ) is reestablished because the small-p behavior of the matrix element for cgPcc does not $ 2 cancel the contribution of the large-p region any more [436]. 2 The formulae presented in Eqs. (6.31)}(6.33) were obtained for strictly real photons Q"0. This is a reasonable approximation for the projected "xed target experiment (photoproduction) but may be improved, if one is interested in operating the HERA ep collider also with polarized high-energy protons. In that case the WeizsaK cker}Williams approximation may be introduced [286,553,554] to account for the tail of the photon propagator. The WeizsaK cker}Williams approximation is also
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advantageous because tagging of the electron, needed for the extraction of the cross section at "xed photon energy would reduce the cross section too strongly. On the basis of the WeizsaK cker} Williams approximation one may go on to include a possible resolved photon contribution to the cross section described by polarized photon structure functions df A(x, k ) with f"q, g $ [322,324,320,555]. These quantities are completely unmeasured so far and could, in the &maximal' scenario, contribute up to 20% of the cross section [553,554]. The polarized lowest order cross section for producing a charm quark with transverse momentum p and cms-rapidity g then is 2 d*p( 1 d*pA "2p dx x df C(x , k )x df (x , k ) , (6.34) C C C $ N N $ x !oe\E dtK 2 C dp dg \E E M \M C 2 D D where o,m /(s with m ,(p #m and x ,x oeE/(x !oe\E). The sum runs over all 2 2 2 A N C C relevant parton species. df are e!ective polarized parton densities in the longitudinally polarized electron de"ned by
where *P A C
dy x *P (y)df A x " C , k , A C A y y $ VC is the polarized WeizsaK cker}Williams spectrum
df (x , k )" C $
(6.35)
a 1!(1!y) Q (1!y) *P (y)" ln , (6.36) A C 2p my y C and the same cuts as in the unpolarized case should be used, Q "4 GeV and the y-cuts
0.24y40.85 [221]. The cross section, Eq. (6.34), can be transformed to the more relevant HERA laboratory frame by a simple boost which implies g,g "g !ln(E /E ), where we have * N C counted positive rapidity in the proton forward direction. The spin-dependent di!erential LO subprocess cross sections d*p( /dtK for the resolved processes ggPcc and qq Pcc with m O0 A can be found in [198,409]. The dominant direct (&unresolved') contribution derives from df A(x , k ),d(1!x ) in (6.35) with the corresponding polarized cross sections for the direct A $ A subprocess cgPcc being readily obtained from that for ggPcc by dropping the nonabelian (s-channel) part and multiplying by 2N ea/a (k ) where e "2/3. Note that the resolved photon A A Q $ A contributions are relevant mainly for (real) photoproduction and that they are appreciable only at very high energies (s 5100 GeV. Furthermore, there are experimental techniques to separate the A resolved part from the direct photon contribution, see e.g. [273]. Fig. 35 shows results for the p and g distributions obtained for the four di!erent polarized gluon densities in Fig. 32 for 2 * E "820 GeV and E "27 GeV [553,554]. The curve denoted by &resolved' is an estimated upper N C limit for the resolved photon contribution. It is negligibly small unless p becomes very small. Also 2 shown are the corresponding asymmetries AA which are much larger than for the total cross section if one goes to p of about 10}20 GeV. Furthermore, the asymmetries are sensitive to the size and 2 shape of the polarized gluon distribution used. Included in the asymmetry plots are the expected statistical errors dAA at HERA which can be estimated from dA"1/P P (Lpe , (6.37) C N where P , P are the beam polarizations, L is the integrated luminosity and e the charm detection C N e$ciency, estimated to be P *P "0.5, L"100 pb\ and e"0.15. C N
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Fig. 35. p - and g -dependence of the (negative) polarized charm-photoproduction cross section and asymmetry in 2 * ep-collisions at HERA [553,554]. The considered kinematic regions are !14g 42 for the p -distribution and * 2 p '8 GeV for the g -distribution. The various curves correspond to the various polarized gluon densities in Fig. 32. 2 * For comparison, the &resolved' contribution to the cross section, calculated with the '"tted' dg in Fig. 32 and the &maximally' saturated set of polarized photonic parton densities, is shown by the lower solid curves.
6.3. Heavy quark production in hadronic collisions Hadronic heavy quark production proceeds via the (so far available) LO subprocesses dgdgPhhM , dqdq PhhM ,
(6.38)
and appears to be a very sensitive and presumably the most realistic test of dg(x, k ), since dg enters $ &quadratically' and the dqdq contribution is small [205,229,198,409]. Here the polarized pl pl RHIC collider ((s"50}500 GeV) with high luminosity (L910 cm\ s\) will play a decisive role [148,585]. The di!erential cross sections for the subprocesses in Eq. (6.38) are given by [198]
pa 3 d 1 4 d *p( EE FFM , (p( EE FFM !p( EE FFM )" Q ! >> >\ 8s( s( 3tI u dtK 2 dtK
2ms( tI #u ! F (tI #u ) , tI u
d d pa 4 tI #u #2ms( F , *p( OO FFM "! p( OO FFM "! Q s( dtK dtK 3s( 3
(6.39) (6.40)
where $ refers to the helicity of the incoming partons, a "a (k ) and tI ,tK !m, u ,u( !m, i.e. Q Q $ F F s( #tI #u "0. By integrating with respect to tI the total cross sections are then easily obtained:
pa 17 1#b *p( EE FFM (s( , k )" Q 2 3b! ln #5b(5!b) , $ 16s( 3 1!b
(6.41)
pa 4 *p( OO FFM (s( , k )"!p( OO FFM (s( , k )"! Q b(3!b) $ $ 9s( 3
(6.42)
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with b"1!4m/s( . The unpolarized di!erential and total cross sections for ggPhhM are well F known [298], with the latter being given by 1 p( EE FFM (s( , k ), (p( EE FFM #p( EE FFM ) >\ $ 2 >> pa " Q 16s(
1 1#b 1 11!6b# b ln # b(31b!59) . 3 1!b 3
(6.43)
A NLO analysis of the polarized cross section *p( GH for heavy quark production is unfortunately still missing. For not too small polarized gluon densities, the total polarized cross section for heavy quark production (4m4s( 4s) is dominated by the gg-initiated subprocess, because the F contribution from the subprocess qq PhhM is marginal for large energies ((s950 GeV). Although the latter can be easily implemented [198], we give here for simplicity only the result for the former cross section:
*pF (s)" NN
"
Q
F
K
ds( dx dx dg(x , k )dg(x , k )*p( EE FFM (s( , k )d(s( !x x s) $ $ $
F
dx
F
dx dg(x , k )dg(x , k )*p( EE FFM (x x s, k ) $ $ $
K Q K QV dq " *p( EE FFM (qs, k )U (q, k ) , $ EE $ q KF Q where the polarized gluon luminosity (#ux) is given by
dx q dg(x , k )dg , k . $ $ x x O The relevant spin}spin asymmetry is de"ned by U (q, k )"q EE $
(6.44)
(6.45)
AF (s)"*pF (s)/pF (s) , (6.46) NN NN NN where the unpolarized cross section is analogous to the polarized cross section, Eq. (6.44), with g(x, k ) appearing instead of dg(x, k ) and the unpolarized partonic cross section (6.43) has to be $ $ used instead of *p( . The typical scale to be used in (6.44) is again k +2m . For the production of $ F heavy quarkonia (J/t, B, etc.) one proceeds in the same way except that in Eq. (6.44) the region of integration has to be taken as 4m4s( 44m or 4m4s( 44m , instead of 4m4s( 4s. However, A " @ F one faces here the additional bound-state model dependence as discussed in the previous section [205,229]. The expected asymmetries (6.46) for total charm and bottom production at typical RHIC energies are shown in Fig. 36 for the various possible forms of dg(x, k ) shown in Fig. 32. $ Although the asymmetries are very sensitive to the polarized gluon density dg(x, 4m), they become F relatively small at top RHIC energies, with A@ being almost an order of magnitude larger than NN AA . It seems that realistic measurements of dg will be possible preferably at medium RHIC NN
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Fig. 36. Asymmetries for the total charm and bottom hadroproduction, according to Eq. (6.46), at RHIC energies using m "1.5 GeV and m "4.5 GeV, for the four polarized gluon densities shown in Fig. 32. A @
energies, (s:100 GeV, or at future HERA-No energies, (s+40 GeV (for a recent review see [423], and references therein). It should be noted, however, that the introduction of a p -cut, as 2 discussed in the previous section for charm photoproduction [553,554,436], could result in sizeably larger asymmetries at high energies. 6.4. High p jets in high energy lepton}nucleon collisions 2 A possible way to distinguish the spin-dependent parton distributions is to study jet production at HERA operated in the polarized mode [158,56,311]. The idea is based on the fact that the parton subprocesses cHqPgq and cHgPqq lead to a di!erent large-p behavior so that the contributions 2 from dq(x, k ) and from dg(x, k ) might be distinguished. The undetermined QCD renormaliz$ $ ation/factorization scale k is taken to be k "Q, although a choice like k "p is equally $ $ $ 2 feasible. Unfortunately, at CERN (SMC), SLAC and DESY (Hermes) jet cross sections are di$cult to measure because the signatures are minute at small (s. Furthermore, it turns out that the polarization asymmetries for jet production even at HERA ((s+300 GeV) are not too large, and in view of the statistical error obtained for luminosities L&100 pb\, cf. Eq. (6.37) with e&1, one should not be too optimistic that dg(x, k ) can be determined from such measurements. This $ situation will obviously improve for integrated luminosities L&500 pb\ as may be optimistically expected at a future fully polarized HERA el pl collider [219,224,124]. An overview of the physics issues will be given in the following. The result for 1gN 2 when written di!erentially in j"4p /Q is 2
a (k ) d dx gN (x, Q, j)" Q 0 e[C M (j)(*q#*q )(k )#¹ M (j)*g(k )] O $ O $ 0 E $ 4p dj O
(6.47)
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with C "4/3 and ¹ "1/2, and where the functions M (j) and M (j) are given by $ 0 O E j## (1#j#1 12#11j H ln M (j)" ! , O (1#j) (1#j!1 4j(1#j) j!2 j(4#j) (1#j#1 M (j)" ! ln . E 2(1#j) 4(1#j) (1#j!1
95
(6.48) (6.49)
As usual, one takes k "k "Q. The important point to observe is [56] that M (j) gives 0 $ E a negative contribution which, for a large positive *g&3 would be the dominant contribution in Eq. (6.47). However, this derivation is strongly idealized, because the negative M gets a large E contribution from the small-x region where measurements become di$cult due to the large unpolarized cross section. Imposing a cut x5x on the x-integration [311], typically x " Q/2ME with x 4x4(1#j)\, the negative signal from M is drastically reduced so that it is J E doubtful that this e!ect can be used to determine *g. Instead of using a cut one can directly verify this "nding by a study of the distribution dgN (x, Q, j)/dj as a function of x and j [311]. The result is shown in Fig. 37. It can be seen that only for x40.001 a clear negative signal in the perturbatively safe region p 94 GeV develops. 2 This feature has nothing to do with the particular dg(x, Q) and dq(x, Q) chosen for this plot, but is a consequence of the structure of the perturbative parton matrix element. So far in this subsection we have discussed the determination of the "rst moment *g. Now we turn to the directly measurable x-dependent densities like dg(x, k ). In general, the cross section for $ polarized deep inelastic electron proton scattering with several partons in the "nal state e\ (l)#p (P)Pe\(l )#remnant(p )#parton 1(p )#2#parton n(p ) v v P L
Fig. 37. p depencence of gN (x, p , Q) for various values of x at Q"10 GeV [311]. 2 2
(6.50)
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is generically given by
(6.51) d*p " dx df (x , k ) d*p( D(p"x P, a (k ), k , k ) , D D $ D Q 0 0 $ LU D where the sum runs over incident partons f"q, q , g which carry a fraction x of the proton D momentum. *p( D denotes the partonic cross section from which collinear initial state singularities are factorized out (in a next-to-leading order calculation) and included in the scale-dependent parton densities df. A natural scale for multi-jet production in DIS may be k " 0 k "1/4 ( p ( j)), with the average p ( j) being de"ned in the Breit frame by 2E(1!cos h ), $ H 2 2 H H. where the subscripts j and P denote the jet and proton. Results about this cross section including HO e!ects can be found in [479,265]. If the jets are de"ned in a modi"ed JADE scheme, the theoretical uncertainties of the two-jet cross section can be very large due to higher order e!ects. These uncertainties are smaller for the cone algorithm, which is de"ned in the laboratory frame. In this algorithm the distance *R"((*g)#(* ) between two partons decides whether they should be recombined to a single jet. Here the variables are the pseudo-rapidity g and the azimuthal angle . Partons with *R(1 are recombined. Furthermore, a cut on the jet transverse momenta of p ( j)'5 GeV in the lab frame and in the Breit frame was imposed. Using the 2 polarized parton densities (set A) of [293] it was found [265] that the LO polarized dijet cross section *p(2-jet) is !45 pb at HERA energies ((s+300 GeV). This negative result for the polarized dijet cross section is entirely due to the boson}gluon fusion process, which is negative for x:0.025 and its contribution to the total polarized dijet cross section is !53 pb. The contribution from the quark initiated subprocess is positive over the whole kinematical range and contributes with 8 pb to the resulting dijet cross section. With these numbers one obtains a rather small average asymmetry of +!0.015. To obtain these numbers, polarizations of 70% for both the electron and the proton beams have been assumed. Note, however, that the shape of the polarized gluon density is hardly (or even not at all) constrained by currently available DIS data, in particular for small x. Alternative parametrizations of the polarized gluon distributions in the small-x region, which are still consistent with all present data [293,309], can lead to asymmetries which are a factor two larger. Although the dijet events are in principle sensitive to this lower x range, the resulting numbers for the asymmetries are in general small because of the dominance of the unpolarized gluon density in the small-x region. Reducing the proton beam energy to 410 GeV, instead of the nominal 820 GeV, does not improve the signal, although the mean value of y is higher. The asymmetry signal increases only for a few points around x'0.1, since a lower incident energy probes slightly higher values of x [479,265]. One may also have a look at large-p jet photoproduction [553,554]. In that case the 2 generic cross section formula for the production of a single jet with transverse momentum p and 2 rapidity g is similar to that in (6.34), the sum now running over all properly symmetrized 2P2 subprocesses for the direct (cbPcd) and resolved (abPcd) cases. The corresponding di!erential helicity-dependent LO subprocess cross sections can be found in [80] for the case that only light #avors are involved. One may neglect the charm content of the nucleon and consider charm only contributing as a "nal state via cgPcc (for the direct part) and ggPcc , qq Pcc (for the resolved part).
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Just as for the case of heavy quark photoproduction it turns out that the resolved photon contributions are at most subdominant (in the &maximal' scenario studied by [553,554]), if not negligible. As these authors have found, the &direct' cross sections are fairly large over the whole rapidity range, and also as a funtion of p , and sensitive to the shape and size of dg(x, p ) with, 2 2 unfortunately, not too sizeable asymmetries as compared to the statistical errors for L"100 pb\. A measurement of dg thus appears to be possible under the imposed conditions only if luminosities exceeding 100 pb\ can be reached. 6.5. Semi-inclusive polarization asymmetries A straightforward idea to study polarized DIS in more detail is to analyze semi-inclusive cross sections. Heavy quark production as discussed in Sections 6.2 and 6.3 is an example for this. Another possibility is to tag for light quark hadrons, like the pions, kaons or protons, in order to obtain information on the spin #avor content of the nucleon, i.e. consider the process eo #po Pe #H#X .
(6.52)
The charged hadrons H most suited for this analysis are mainly p! and K! but also p and p may be interesting [27,30,13]. Recently, p production has been used as well for extracting ds(x, Q) [79]. In particular, one can compare cross sections of hadrons with positive and negative electric charge to obtain additional information. In the unpolarized case and within the framework of the LO-QCD quark parton model the cross section is given by 1 dp! eq(x, Q)D&(z, Q) O " O & O , eq(x, Q) p dz O O
(6.53)
where p is the inclusive cross section and one is considering the production of hadrons H!. The fragmentation function D&(z, Q) represents the probability that a struck quark with #avor O q fragments into a hadron H carrying fractional momentum z of the parent ("struck) quark q (see, for example, [45,509]). In NLO the cross section in (6.53) does not factorize in x and z anymore, but instead one has more complicated double convolution integrals over parton densities, fragmentation functions and Wilson coe$cients [49,288]. In the polarized case one may de"ne asymmetries A!, similar to the inclusive asymmetry, Eq. (2.10), namely A!"(p! !p! )/(p! #p! ) .
(6.54)
In analogy to (6.53) these semi-inclusive asymmetries in LO-QCD are given by edq(x, Q)D&(z, Q) O . A!(x, z, Q)" O & O eq(x, Q)D&(z, Q) O & O O
(6.55)
This result holds under the (to some extent questionable) assumption that the fragmentation functions do not depend on the quark helicity [278,190]. The idea then is to take the q(x, Q) and the D&(z, Q) from other (unpolarized) experimental data and use the measurement of A! to O
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determine the dq(x, Q). By using di!erent targets (proton, deuteron and He) one measures di!erent linear combinations of the dq(x, Q) according to Eq. (6.55). For example, the deuteron cross section is considered the sum of the proton and neutron cross section, corrected by the D-state factor 1!w , as will be discussed in Section 6.10. Because of isospin symmetry the " asymmetries for proton and neutron are related by exchange of up and down quarks and antiquarks. To increase the statistics, the LO relation (6.55) is sometimes integrated over the measured z-domain [27], i.e. D&(Q)" dz D&(z, Q) is used in (6.55) instead of D&(z, Q). O O O One may take higher order QCD corrections to Eq. (6.55) into account. The NLO corrections to the denominator of the asymmetry are well known [49,288], and the NLO corrections to the numerator have been calculated too [177,269]. In the MS scheme, where the anomalous gluon contribution is hidden in the de"nition of the singlet quarks *R, the NLO corrections turn out to be much smaller than the present experimental accuracy. Furthermore, present measurements [27] do not involve the small-x region x:0.005. If one considers the production of the 6 charged hadrons p!, K!, p and p from 3 quarks and 3 antiquarks, there are in principle 36 independent fragmentation functions. Among these the fragmentation functions of strange quarks into pions can be neglected. The fragmentation functions of nonstrange quarks into pions can be obtained, for example, from EMC measurements [70] by using charge conjugation and isospin symmetry. The number of independent fragmentation \ \ \ \ SQ and D) S Q"Dp S B functions can be further reduced by assumptions like D) S Q"D) Q B B BM for unfavored and favored fragmentations, respectively, so that "nally the number of independent quark fragmentation functions is 6. In addition, there are the three gluon fragmentation functions D&(z, Q) which in LO enter only via the evolution equations, whereas in NLO they enter the cross E sections and asymmetries (6.55) directly. The quark fragmentation functions together with the unpolarized parton densities and a measurement of the spin asymmetries of proton and deuteron and/or He serve as input to determine all the valence and sea quark densities dq (x, Q)"dq(x, Q)!dq (x, Q) and T dq (x, Q)"2dq (x, Q) (q"u, d, s). If one measures the spin asymmetries in certain x-bins, there is for each x-bin a system of linear equations for the six unknown spin distributions. The weight of the strange quark distributions in these equations is marginal, so that they cannot be determined. Recently, however, asymmetries for p"(p>#p\) production o! a He target have been measured [79] which allow the extraction of ds/s according to a suggestion of [278]. Although the statistics is still inferior, there is an indication that ds(x, Q) turns negative for x:0.1 and vanishes at larger values of x as theoretically anticipated. The statistical errors of present polarized experiments are so large that the power of the method can be increased by the additional assumption du (x, Q)"ddM (x, Q),dq (x, Q). Finally, one is left with 3 unknown functions du (x, Q), dd (x, Q) and dq (x, Q). These have been determined in a recent analysis by T T the SMC collaboration [27] for 12 x-bins between 0.005 and 0.48 and assuming Q-independence of the asymmetries. Their results show that at the present stage this method is not really accurate enough for a quantitative analysis. Typical errors are about 50% or larger. However, in future this method will be certainly very fruitful to discriminate between the polarized up and down and sea quark contribution. A check on the consistency of the procedure is possible by using the relation gN (x, Q)!gL (x, Q)"[du (x, Q)!dd (x, Q)]#O(du !ddM )#(HO!QCD) . T T
(6.56)
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The l.h.s. of this equation is obtained from inclusive data while the r.h.s. can be obtained from the semi-inclusive asymmetries. In addition, the assumptions du (x, Q)"ddM (x, Q) and the smallness of the HO corrections can be checked. An alternative procedure, in case of pions, has been suggested by [278]. These authors consider the asymmetry (p> !p\ )!(p> !p\ ) , A " (6.57)
(p> !p\ )#(p> !p\ ) where &mixed' refers to the combination p>!p\ of p! production, for example, and which has a simple expression in terms of up- and down-quark densities, 4du !gdd T . (6.58) A " T
4u !gd T T For the combination p>!p\ the fragmentation functions cancel and thus g"1. If, however, the hadrons are not identi"ed, as in the SMC experiment [27], the factor g has to be evaluated by averaging the relevant fragmentation functions and is found to be about 0.5 for z'0.2. This method, however, is not as simple as it looks, because the spectrometer acceptance is di!erent for positive and negative hadrons and the ratio of these acceptances does not cancel in the asymmetries, Eq. (6.57). Still one can measure this ratio and correct for the acceptance di!erence. Finally, the quark distributions du and dd obtained this way should be in agreement with the distributions T T obtained from A!. There is the possibility to obtain information on ds by looking at processes with fast kaons (K\"u s) in the "nal state [190]. These have a high probability to contain the initial struck quark, i.e. a fast kaon can be a signal for a s or u quark struck by the photon. Observation of the corresponding polarization asymmetry A)\ is a signature of the existence of a polarized s resp. u sea. However, there are strong systematic uncertainties in such an experiment because some contribution will arise from ss pairs created in the photon}gluon fusion process. Polarization asymmetries for semi-inclusive pion production have been measured from doubly longitudinally polarized (anti)proton}proton collisions at the Fermilab SPF [14] resulting in Ap +0 for 1:pp:4 GeV at EN "200 GeV, i.e. (s"20 GeV. It should be pointed out that ** 2 this result does not necessarily imply a vanishing [14,500,501] gluon polarization *g, but is equally consistent with a large *g+3}6 [572]. A clean distinction between a large and a small *g scenario could be achieved, if it were possible to perform such a semi-inclusive experiment at, say, (s+100 GeV with pp95 GeV [572]. 2 6.6. Information from elastic neutrino}proton scattering (Quasi)-elastic neutrino}proton scattering (lpPlp) is mediated by the exchange of the Z boson. Since the parity violating Z-quark coupling involves c c , the unpolarized cross section will depend I on the proton matrix element of the axial vector current. As will be shown below this o!ers in particular the possibility to measure a combination of the strange quark matrix 1p"s c c s"p2 and I the anomalous gluon component. Theoretical aspects of this process have been reviewed in [407] and more recently in [42]. A very readable experimental paper on the subject is [39]. New neutrino
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experiments like CHORUS, NOMAD, ICARUS, MINOS, COSMOS, etc., [481] aimed to search for neutrino oscillations are taking data or are under preparation and could also be used to get information on the NC neutrino (and antineutrino) elastic scattering on protons. The one-nucleon matrix element of the hadronic neutral current has the form
i p qJF8 (Q)#c c G8 (Q) u(p) . 1p "J8"p2"u (p ) c F8 (Q)# I 4 + I I 2M IJ
(6.59)
As will be explained now, F8 , F8 and G8 can be written as linear combinations of normalized 4 + matrix elements (&form factors') of the following ; ;; quark currents:
ip qJ 1N(p )"< "N(p)2"u (p ) F (Q)c #F (Q) IJ u(p) , I I 2M
ip qJ 1N(p )"<"N(p)2"u (p ) F (Q)c #F (Q) IJ q u(p) , I I 2M
(6.62) (6.63)
1N(p )"A "N(p)2"G (Q)u (p )c c u(p) , (6.64) I I 1N(p )"<"N(p)2"G (Q)u (p )c c q u(p) . (6.65) I I The four vector form factors F can be measured in electromagnetic scattering. At Q"0 they are given in terms of the anomalous magnetic moments i of the nucleons by , (6.66) F (0)", F (0)"(i !i ) , N L (6.67) F (0)"(3, F (0)"(3(i #i ) . N L The QP0 limit of F is the baryon number, F (0)"1 . (6.68) The nonsinglet axial vector form factors G at zero momentum transfer are related to the constants F and D introduced in hyperon semileptonic decays, cf. Eqs. (5.10) and (5.11), 1g 1 G (0)" (F#D)" , 2g 2 4
1 G (0)" (3F!D) . (12
(6.69)
One is left with two singlet form factors undetermined at QP0, F (0) and G (0). F (0) is the &anomalous baryon number magnetic moment'. G (0) is the singlet axial current form factor
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relevant to spin physics. Both F and G can be measured in elastic neutral current scattering. In the naive parton model G (0) can be expressed by the singlet polarized quark density *R introduced in Section 5. As discussed in detail in Section 5, higher order QCD corrections may introduce, within certain schemes, an anomalous gluon contribution. Since the hadronic neutral current is a linear combination of the ; ;; currents
(6.72)
(6.73) a ", a "!1, a "!1/(3 . The impact of these results, and in particular of the last relation in Eq. (6.71), on spin physics is as follows: In the naive (i.e. non-QCD) parton model there is no Q dependence and one can identify the axial vector form factors with combinations of "rst moments of polarized parton densities, for f"3 active #avors, G "[*(u#u )#*(d#dM )#*(s#s )] , (6.74) G "[*(u#u )!*(d#dM )] , G "(1/(12)[*(u#u )#*(d#dM )!2*(s#s )] . If these results are combined to G8 in Eq. (6.71), one "nds that G8 measures the following combination of "rst moments: (6.75) G8 "![*(u#u )!*(d#dM )!*(s#s )] . This is an important result because it shows that (quasi)-elastic neutral current processes allow to determine a linear combination of "rst moments of polarized parton densities, which is independent and di!erent of what is measured in polarized deep inelastic electroproduction. Higher order QCD, QED and quark mass corrections to this result have been estimated by [407] to be small. There is some e!ect from the renormalization group running of a between the scale (1 GeV at which the hadronic matrix elements are de"ned and the scale m 8 at which the couplings are given. The e!ect of this running can be summarized as e!ectively replacing a " by a +0.48. The running of the other couplings can be neglected to a good approximation.
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The discussion so far seems to imply that elastic neutrino scattering is an extremely elegant method to determine the spin of the proton. Unfortunately, the above discussion is not the whole story. The point is that to "t neutrino-hadron scattering data one needs the form factors at QO0 and there is a signi"cant Q-dependence due to the "nite extension of the proton. The exact Q-dependence of the form factors being unknown, a dipole approximation of the form &(1#Q/M )\ is usually applied. In these expressions new phenomenological parameters " M (di!erent for each form factor) appear, which have to be determined experimentally. This " uncertainty in the Q-dependence reduces the potential of the method very much [39,188]. Recently, a method has been suggested by [42] to reduce this sensitivity on the poorly known (non-strange) axial form factor and increase the accuracy and sensitivity to the strange axial form factor (*s) by considering a certain combination of neutral and charged current neutrino and antineutrino cross sections, namely the asymmetry (dp/dQ),!!(dp/dQ),! J, J , . A (Q)" , (dp/dQ)!!!(dp/dQ)!! JL J N
(6.76)
In principle, it should be possible to obtain information about the proton spin from the investigation of the quasi-elastic CC processes l #nPk\#p and l #pPk>#n. However, the existing I I CC data are not accurate enough to compete with the NC elastic scattering data [289]. These and other suggestions are awaiting the experimental tests. For example, a measurement of G in elastic scattering of unpolarized electrons o! unpolarized protons might be feasible, because the forward backward asymmetry of the electrons in the cms is determined by the c-Z interference terms and these in turn are proportional to G . 6.7. The OPE and QCD parton model for g and g > In Section 2.4 the &kinematics' of polarized charged and neutral current processes has been introduced and the structure functions g have been de"ned. In this section we want to discuss the parton model and the phenomenological consequences of polarization e!ects involving charged and neutral currents. Further information can be found in the literature [211,432,571,388,508,472}474,393,62]. The "rst reference [211] is an old review and summarizes the theoretical articles [489,218,37,410,154] from before the startup of the CERN experiments. Just as for g the naive parton model can be applied as a "rst approximation, whenever longitudinally polarized leptons probe longitudinally polarized protons (P "MS ) at high energies (much larger I I than K ). Under this condition the relevant structure functions in the hadronic tensor Eq. (2.31) /!" are g and g " : g #g . > Let us "rst consider neutrino nucleon scattering lpPl\X which proceeds via => exchange. (Analogously, for antineutrino scattering, lNPl>X refers to the charged =\ current.) The results can also be taken over rather directly to the charged current reactions l>pPlX. Neutrinos couple to d-type quarks and to u -type antiquarks so that the densities of these partons will appear in the structure functions. In principle, one has also to take into account the proper CKM mixing occuring at the charged current vertex. However, if one considers the contributions of 4 #avors (u, d, s and c), one always encounters the factor cos h #sinh "1. One can get the parton model expressions for g and g by an explicit calculation of the lowest order processes and by >
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comparing it to the general form of the hadron tensor (2.28) gJ,(x, Q)"dd(x, Q)#ds(x, Q)#du (x, Q)#dc (x, Q) , (6.77) gJ,(x, Q)"![dd(x, Q)#ds(x, Q)!du (x, Q)!dc (x, Q)] , (6.78) gJ, (x, Q)"2xgJ,(x, Q) , (6.79) > where the index lN always implies => exchange. One obtains the corresponding formulae for lN (=\ exchange) by the #avor interchanges d u and s c. The contribution of gJ, to the cross \ section vanishes in the framework of the QCD improved parton model. Some remarks are in order. Due to the charge conjugation property of c the antiquarks always appear with an opposite sign in g and g as compared to g . As discussed in Section 5, the sum > a (6.80) dx(gJ,#gJ ,)(x, Q)"*R!f Q *g(Q) 2p is a measure of the axial vector singlet current matrix element and its measurement would be a nice way to verify the value of *R!f (a /2p)*g without recurrence to the low energy determination of Q the matrix elements A and A . In contrast to g , the structure functions g and g measure > solely nonsinglet combinations of parton densities (just as F in unpolarized DIS), so that they do not get a contribution from the gluon densitiy. They can yield informations of the following type: For example, by scattering on an isoscalar target one could "nd out, how large the polarized strange quark sea is
(6.81) (gJ,!gJ ,) (x, Q)#(gJ,!gJ ,) (x, Q)"2(dc#dc !ds!ds )(x, Q) . N L Other combinations have been studied in [472}474,62]. Furthermore, it should be mentioned that the treatment of the charm quark contributions to the above structure functions in terms of a massless intrinsic density dc(x, Q)"dc (x, Q) is controversial. It is more appropriate to calculate the heavy quark contributions to neutral and charged current processes perturbatively via the subprocess =>gPcs [571], for example. Just as F and F in unpolarized DIS, g and g are related by a Callan}Gross-like relation, > Eq. (6.79), originally derived by [228]. This relation will be violated beyond the leading order of QCD, except in the case of neutral currents where g and g do not receive a gluonic > contribution to O(a ) [432,571]. Further new relations analogous to the Wandzura}Wilczek Q relation for g (cf. [576] and Section 8.1) have been recently derived from a study of the twist-2 and twist-3 OPE [121,122]. The close relationship between unpolarized and polarized DIS in the limit QPR, in which only longitudinal polarization survives, becomes very transparent within the OPE. In the high energy limit one can stick to the leading twist-2 operators, and the OPE for the hadronic tensor reads
2 L q 2q L\ (RNI 2IL\ EL#QNI 2IL\ CL) =J,"ie qH G G G G I I IJ IJHN Q G L 2 L q q # !g # I J q q 2q L\ (RNI 2IL\
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q q # !g # I N IN q
q q !g # J I JI q
2 L\ (6.82) ;2 q 2q L\ (RNI 2IL\ =L#QNI 2IL\ >L) , I G G G G I Q L G where R are the operators relevant for polarized scattering (Eqs. (4.72) and (4.73)) and Q are the G G corresponding operators for unpolarized scattering (essentially the same as Eqs. (4.72) and (4.73) but without a c ). EL, CL, L are the corresponding Wilson coe$cients. For example, G G G G G G EL is related to the structure function g , CL to the structure function F , etc. More precisely, one G G has [508]
dx xL\F (x, Q)" bG XL(Q/k, a ), n"1, 3, 5,2 , (6.83) L G Q G dx xL\F (x, Q)"2 bG >L(Q/k, a ), n"1, 3, 5,2 , (6.84) L G Q G 1 (6.85) dx xL\F (x, Q)" bG CL(Q/k, a ), n"1, 3, 5,2 , L G Q 2 G 1 dx xL\g (x, Q)" aG EL(Q/k, a ), n"1, 3, 5,2 , (6.86) L G Q 2 G dx xL\g (x, Q)" aG L G Q G where the aG (bG ) are the matrix elements for a (un)polarized proton, cf. Eq. (4.74). From these L L equations the Callan}Gross relations are apparent, because in leading order all Wilson coe$cients are equal to 1. For more details we refer the reader to [508,121,122]. Let us now turn to neutral current exchange where the description is similar, although somewhat complicated by the cZ-interference. In LO the parton model predictions for the structure functions are gA8(x, Q)" e v (dq#dq )(x, Q) , O O O
(6.89)
gA8(x, Q)" e a (dq!dq )(x, Q) , O O O 1 g88(x, Q)" (v#a)(dq#dq )(x, Q) , O O 2 O
(6.90)
g88(x, Q)" v a (dq!dq )(x, Q) , O O O
(6.91) (6.92)
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where v and a are the vector and axialvector coupling of the quark #avor q to the O O Z, v "! sinh , a ", v "!# sin h and a "!, cf. Section 2.4. Furthermore, 5 S B 5 B S one has gA8 88"2xgA8 88 and gA as before, cf. Eq. (4.5). NLO contributions can be found in > [271,432,472}474,571,557]. For comparison let us include here the corresponding LO relations for the unpolarized structure functions, namely FA8 88" (e v , v#a)(q#q ), FA8 88" O O O O O 2 (e a , v a )(q!q ) and FA8 88"2xFA8 88. For the charged current processes l>NPlX one can O O O O O take over the expressions for gJ,, etc., from above, Eqs. (6.77)}(6.81). 6.8. Single-spin asymmetries and handedness Single spin asymmetries A "(pN !p= )/(pN #p= ) are asymmetries that arise if only one of * the external particles is longitudinally (or transversally) polarized. In deep inelastic lepton}nucleon scattering at Q;m all single-spin asymmetries vanish } at least within the framework of the 8 (twist-2) parton model } and the same holds true for many processes at RHIC, like direct photon, heavy quark or jet production. Still, there are circumstances under which nonvanishing single-spin asymmetries arise. For instance, many authors have tried to derive single-spin asymmetries from higher twist e!ects like the intrinsic transverse momentum of the partons within hadrons [534,535,195,196,132,240,494,495,435,237,560,76,59}61,423]. Such considerations have been triggered by several experiments which have shown that single-spin asymmetries can indeed be large both in semi-inclusive pptPpX [15,16,22] and exclusive pptPpp [67,157,209] reactions. These ideas go beyond the perturbative QCD (twist-2) parton model and are more di$cult to test than the parton model inspired predictions which usually lead to double-spin asymmetries. Another example of single-spin asymmetries is the possibility to determine the spin of "nal state particles, like the handedness of jets. All these ideas and approaches will be discussed at the end of this section. We shall start the discussion with a leading, twist-2 source of single-spin asymmetries which is due to weak interactions, i.e. the presence of parity violating couplings in processes in which =! and Z are involved. In such processes the c from the polarized particle and the c from the axial vector coupling combine in the matrix element to give a nonvanishing single-spin asymmetry even within the ordinary parton model. This point will be discussed "rst in this section because it can lead to drastic e!ects and, at RHIC, is an independent way to determine the polarized parton densities. Let us recall that in unpolarized proton scattering the production of W's via ppP=!X is sensitive to the form of the antiquark distributions, because the dominant contribution to the cross section comes from the quark-antiquark fusion reactions udM P=> and u dP=\. The same holds true for polarized scattering where at least one of the protons is polarized. The corresponding single-spin asymmetry is given by pN !p= du(x , m )dM (x , m )!(u dM ) 5 5 " . (6.93) A.4" * pN #p= u(x , m )dM (x , m )#(u dM ) 5 5 Note that for kinematical reasons A.4 depends only on y"ln x /x because one has * x "(m /(S)eW and x "(m /(S)e\W. Eq. (6.93) is a crude LO approximation but it shows how 5 5 sensitive one is to the polarized antiquark densities. A phenomenological analysis on the basis of this formula has been presented in [137,541] for the RHIC machine parameters chosen to be
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Fig. 38. The parity-violating spin asymmetries for =! and Z production as a function of y under RHIC conditions [137]. The curves correspond to a suggested choice of the sea quark polarizations, cf. Ref. [135]. Fig. 39. The parity-violating single-spin asymmetries in ppP=!#jet at y"0 as a function of p under RHIC 2 conditions. The results are taken from Ref. [136].
(s"0.5 TeV, ¸"2;10 cm\ s\ and a polarization of 70%. The resulting single-spin asymmetries are shown in Fig. 38 as a function of y. In Ref. [136] the analysis has been extended to processes like ppP=!#jet#X. The corresponding single-spin asymmetries are shown in Fig. 39 as a function of p . Later on, a Monte Carlo study including collinear gluon bremsstrahlung 2 has been carried out in [519]. The full NLO-QCD calculation has been performed in [579]. In principle, there are two other parity-violating asymmetries which can be measured if both protons are polarized, namely N = N A.4 (y)"(p= = !pN )/(p= #pN ) **
(6.94)
and N = N A.4 (y)"(p= (6.95) N !p= )/(pN #p= ) ** Note that if parity is conserved one has p "p and p "p so that all these asymmetries, ? @ \? \@ ? \? including A.4, vanish. In the parity violating =! production process they are given in leading * order by [137]
[du(x , m )dM (x , m )!u(x , m )ddM (x , m )]![u dM ] 5 5 5 5 A.4 (y)" ** [u(x , m )dM (x , m )!du(x , m )ddM (x , m )]#[u dM ] 5 5 5 5
(6.96)
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and [u(x , m )ddM (x , m )#du(x , m )dM (x , m )]![u dM ] 5 5 5 5 A.4 (y)" (6.97) ** [u(x , m )dM (x , m )#du(x , m )ddM (x , m )]#[u dM ] 5 5 5 5 with the property A.4 (y)"A.4 (!y) and A.4 (y)"!A.4 (!y) and again sensitive to ** ** ** ** the polarized sea. Results for these asymmetries are included in Fig. 38, obtained under the assumptions of Ref. [137]. These authors have suggested that one should make the approximation " du(x )ddM (x )";u(x )dM (x ). In that case the three asymmetries are not independent quantities any more but A.4 (y) and A.4 (y) can be expressed in terms of A.4(y) and A.4(!y). Whether this ** ** * * approximation is reasonable or not will be shown in future. In any case, among the three, A.4 is * certainly the most interesting one because it will have the smallest statistical error. This is true despite the fact that the single-spin asymmetry will be smaller in absolute magnitude than the double-spin asymmetries by roughly a factor of 2. Similar considerations as for =-production apply for Z-production at RHIC. One may also consider the Z-contribution to the Drell}Yan process ppPcH, ZHPk>k\ which is relevant for large invariant masses of the k>k\ pair. There have been studies of large-p = [136] (and also 2 large-p ZH [442,443] and large-p jet production) induced by the higher order processes 2 2 q q P=g and q gPq =, where iOj denotes the quark #avor. Let us discuss now qualitatively G H G H the signi"cance of these processes for the determination of the polarized gluon density dg(x, k ) $ where k may be roughly chosen as k +(p #m )/4. If the beams are polarized and the initial $ $ 2 5 partons are carrying a given helicity, one has for the Compton-type scattering process q (h)g(j)Pq = the parton level cross section G H paa dp( Q " (h!1)(c j#c (1!j)) , (6.98) 12 sin h s( u( dtK 5 where c "(s( !m )#(u( !m ) and c "2(u( !m ). For q (h)g(j)Pq = the same formula 5 5 5 G H holds but with hP!h and jP!j. The resulting parton level single-spin asymmetries are a( .4"1 for polarized quarks and a( .4"1!(c /c ) for polarized gluons. Since the last quantity is * * rather small on average [136], the single hadron helicity asymmetry will be dominated by polarized quarks and is not very sensitive to dg. If it were possible to determine the polarization of the outgoing photon or jet, one would have nonvanishing single-spin asymmetries in processes without parity violation. Namely, one could study processes like q#go Pq#co or q#go Pqo #g, etc., either at RHIC or one could use the energetic unpolarized proton beams of the Tevatron p p or HERA ep colliders (E +1 TeV) to be N scattered o! a polarized "xed proton target. In the former case one would have to measure the circular polarization of the "nal state photon. This is quite di$cult. It could be done either by selective absorbtion using a polarized detector [210] or by making use of the fact that in high energy photon induced showers the longitudinal polarization is conserved to a good degree of accuracy [471,323]. In the latter case an unpolarized quark collides with a polarized gluon and one would have to measure the polarization of the outgoing quark jet. It has been speculated [485,236,214,238,133] that information about the polarization of the initiating parton, i.e. the outgoing quark, can be obtained from the &handedness' of a jet. As will be discussed below, present
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experimental data show unfortunately so far no evidence for this concept to be relevant. Here handedness is de"ned as follows: consider the triple product of vectors S " : t ) (k ;k ) where t is a unit vector along the jet axis, and k and k are the momenta of two particles in the jet chosen by some de"nite prescription, e.g. the two fastest particles. The jet is de"ned as left(right)-handed if S is negative(positive). For an ensemble of jets the handedness is de"ned as the asymmetry in the number of left- and right-handed jets, H " : (N !N )/(N #N ). It can then be 1 1 1 1 asserted that H"aP where P is the average polarization of the underlying partons in the ensemble of jets and a is the &analyzing power' of the handedness method. In order to obtain information about dg(x, k ) from processes like p#pl Pjel t#X one needs to $ know the value of a. It has been attempted [1] to determine a from jet production at SLD where a polarized Z-boson decays into two jets originating from a qq pair. The quark and antiquark in a Z decay have opposite helicities. The SM predicts P +0.67 and P +0.94 S A B Q @ for the average polarizations of quark #avors so that the quarks are produced predominantly left-handed and the antiquarks predominantly right-handed. In order to observe a net polarization in an ensemble of jets from Z decays it is necessary to distinguish quark jets from antiquark jets. This separation can be achieved at SLC where Z bosons are produced in collisions of highly longitudinally polarized electrons with unpolarized positrons. In this case the SM predicts a large di!erence in polar angle distributions between quarks and antiquarks. Unfortunately, no evidence for handedness was found by the SLD collaboration [1]. This negative result seems to imply that the connection between the polarization of the hard partons before the fragmentation and the orientation of the "nal state hadrons is very loose and washed out by con"nement e!ects. Instead of producing and delineating a polarized photon, Drell}Yan dilepton production has been suggested as a probe for polarized densities: here either the polarization of one of the "nal leptons in ppo PcHXPk>ko \X has to be measured [199], or the angular distribution of the produced lepton pair [160] as a polarimeter for the virtual intermediate photon. Similarly, instead of producing a polarized jet (quark), semi-inclusive polarized singleparticle (e.g. Ko ) production has been suggested and analyzed as well. Either e>eo \PKo X [152] or l po PlKo X [377,454,455,412], where in the latter case a longitudinally polarized lepton beam instead of a polarized nucleon target could do as well as a sensitive probe of dg(x, Q). Here the (time-like) polarized fragmentation functions dDK (z, Q), f"q, q , g also enter, which are D de"ned in analogy to the space-like polarized parton densities. Theoretically, these processes have been studied rather thoroughly during the past few years, not only in LO, but in NLO as well [272,556]. Finally, we want to come back to higher twist contributions as a source of single (transverse) spin asymmetries A . Let us "rst discuss semi-inclusive reactions such as pptPpX and pptPcX. At , present, there are basically three sources for a nonvanishing A : , (i) dynamical contributions, i.e. &hard' partonic twist-3 scattering e!ects, which result from a short distance part calculable in perturbative QCD combined with a long distance part related to quark}gluon correlations [240,494,495,560]; (ii) intrinsic k e!ects in parton distribution functions which, being nonperturbative universal 2 nucleon properties, give rise to twist-3 contributions when convoluted with the hard partonic cross sections. Such contributions are usually referred to as &Sivers e!ect' [534,535,59,483, 484,424];
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Fig. 40. Sample twist-3 quark-gluon correlation diagrams corresponding to fermionic (a) and gluonic (b) pole contributions which give rise to a nonvanishing single-spin asymmetry in hadronic direct-photon production.
(iii) intrinsic k e!ects in the parton fragmentation functions which is known as &Collins' or 2 &sheared jet' e!ect [195,196,483,484,76]. A typical example for dynamical quark}gluon correlation contributions (i) to direct-c production is depicted in Fig. 40. They give rise to AA O0 and correspond to fermionic (quark) pole [240] and/or , gluonic pole [494,495,247] dominance in the calculation of the twist-3 partonic scattering cross sections. Clearly, single-spin asymmetries for direct-c production are not 'contaminated' by possible intrinsic k e!ects in fragmentation functions (iii), but k e!ects in parton densities (ii) 2 2 could also be relevant besides (i). In the latter case (ii) one expects that the number of quarks with longitudinal momentum fraction x and transverse intrinsic motion k depends, on account of 2 soft gluon interactions between initial state partons, on the transverse spin direction of the parent nucleon, so that the &quark distribution analysing power' N (k ),( f t (x, k )! 2 O 2 O , f s (x, k ))/( f t (x, k )#f s (x, k )) can be di!erent from zero. Similar soft interactions in the 2 O , 2 O , 2 O , "nal (fragmentation) state may give rise to the &Collins e!ect' (iii); it simply amounts to say that the number of hadrons h (say, pions) resulting from the fragmentation of a transversely polarized quark, with longitudinal momentum fraction z and transverse momentum k , 2 depends on the quark spin orientation. That is, one expects the &quark fragmentation analyzing power' A (k ),(D t (z, k )!D s (z, k ))/(D t (z, k )#D s (z, k )) to be di!erent from 2 F O 2 F O 2 F O 2 O 2 F O zero, where, by parity invariance, the quark spin should be orthogonal to the q!h plane. Notice also that time-reversal invariance does not forbid such a quantity to be nonvanishing because of the (necessary) soft interactions of the fragmenting quark with external strong "elds, i.e. because of "nal state interactions. This idea has been applied [76] to the computation of the single-spin asymmetries observed in pptPpX [15,16,22]. As mentioned above, both A (k ) and N (k ) are leading twist quantities which, when convoluted with the O 2 O 2 elementary cross sections and integrated over, give twist-3 contributions to the single-spin asymmetries. Each of the above mechanisms might be present and important in understanding twist-3 contributions. It is then important to study possible ways of disentangling these di!erent contributions in order to be able to assess the importance of each of them. We discuss now single-spin asymmetries for various processes ABtPCX. To obtain a complete picture one needs to consider nucleon}nucleon interactions together with other processes, like lepton}nucleon scattering which might add valuable information too. For each of them one should discuss the possible sources of higher twist contributions, distinguishing, according to the above discussion, between those
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originating from the hard scattering and those originating either from the quark fragmentation or distribution analyzing power [61,60,423]: E pNtPhX: all kinds of higher twist contributions may be present; this asymmetry alone could not help in evaluating the relative importance of the di!erent terms; E pNtPcX, pNtPk>k\X, pNtPjets#X: no fragmentation process is involved and one remains with possible sources of nonzero single-spin asymmetries in the hard scattering (i) or the quark distribution analyzing power due to an intrinsic k (ii); 2 E lNtPhX: a single-spin asymmetry can originate either from hard scattering (i) or from k e!ects 2 in the fragmentation function (iii), but not in the distribution functions, as soft initial state interactions are suppressed by powers of a ; Q E lNtPcX, cNtPcX, lNtPk>k\X, lNtPjets#X: a single-spin asymmetry in any of these processes may only be due to higher-twist dynamical hard scattering e!ects (i). In the following we want to concentrate on some speci"c processes, in order to see to what extent valuable information can be obtained from measuring single-spin asymmetries at RHIC and at the future HERA-No (where we speci"cally refer to the &phase I' program, i.e. to polarized "xed target experiments with an unpolarized beam; &phase II' refers to a polarized proton beam as well [423]). One example is semi-inclusive p! production by ptp collisions which is known to exhibit surprisingly large single-spin asymmetries at 200 GeV. This was measured a few years ago by the E704 Collaboration using a transversely polarized beam [15,16,22]. For any kind of pions the asymmetry shows a considerable rise above x '0.3, i.e. in the fragmentation region of the $ polarized nucleon. It is positive ('20% ) for both p> and p mesons, while it has the opposite sign for p\ mesons. The charged pion data were taken in the 0.2(p (2 GeV range and it was found 2 that the asymmetry is larger for p '1 GeV than for p (1 GeV. Theoretical approaches as to the 2 2 interpretation of these data have been put forward in [534,535,132,237,76,59]. The authors of Refs. [60,423] have examined to what accuracy this asymmetry could be measured at HERA-No . They have shown that it would be possible to quantitatively determine the p dependence of the 2 asymmetry up to p values of about 10 GeV. 2 Another example is to measure the single transverse spin asymmetry in inclusive direct photon production, pptPcX. Since this process proceeds without fragmentation, i.e. the photon carries directly the information from the hard scattering process, it measures a combination of initial k e!ects and hard scattering twist-3 processes [240,494,495,247]. The "rst and only results up to 2 now were obtained by the E704 Collaboration [21] showing an asymmetry compatible with zero within large errors for 2.5(p (3.1 GeV in the central region "x "(0.15. Again, the authors of 2 $ Refs. [60,423] have examined to what accuracy this asymmetry could be measured at HERA-No . The contributions of the gluon-Compton scattering (qgPcq) and quark-antiquark annihilation (qq Pcg) were compared to the background photons that originate mainly from p and g decays. It turns out that a good sensitivity dAA of about 0.05 can be maintained up to p ( 8 GeV. For , 2 increasing transverse momentum the annihilation subprocess and the background photons are becoming less essential. Thirdly, the single-spin asymmetry in Drell}Yan production, pptPllM X, at small transverse momenta was calculated [347] in the framework of twist-3 perturbative QCD at HERA-No energies. The resulting asymmetry does not exceed 2% and depends strongly on the kinematical domain. It should be noted that asymmetries of the size of a few percent represent the canonical
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order of magnitude for single-spin asymmetries induced by twist-3 perturbative QCD e!ects, and that one may expect larger values only under very special circumstances. Note also, that such asymmetries on the few percent level are di$cult to measure, even with su$ciently small statistical errors, since the systematic errors originating mainly from beam and target polarization measurements constitute a severe limit. Recently, it has been suggested [435] that deep inelastic Compton scattering lpPlcX, where either the incoming lepton or proton are polarized, might lead to appreciable and well-measurable single-spin asymmetries in the small-x region. The idea is that intrinsic k and absorptive e!ects of 2 the strong interactions lead to a complex phase in the vertex between the incoming quark and the prompt photon. This phase, in turn, induces a single-spin asymmetry. Finally, it should be remembered that large spin e!ects in proton}proton elastic scattering, pptPpp, have been discovered many years ago [67,157,209] with proton beam energies of 24 and 28 GeV. The single-spin asymmetry was found signi"cantly di!erent from zero for transverse momenta of the outgoing protons larger than 2 GeV. The transverse single-spin asymmetry in elastic pp scattering at HERA-No and RHIC energies has been calculated in a dynamical model that leads to spin-dependent pomeron couplings [328]. The predicted asymmetry is about 0.1 for p "4}5 GeV with a projected statistical error of 0.01}0.02 for HERA-No , i.e. a signi"cant 2 measurement of the asymmetry can be performed to test the spin dependence of elastic pp scattering at high energies. Although we do not have any detailed quantitative theoretical understanding of elastic single-spin asymmetries for the time being [133], it should be emphasized that, apart from possible helicity nonconserving e!ects occurring in the hadronic wave function, one expects at least qualitatively a nonvanishing elastic single-spin asymmetry due to degenerate multiple Regge exchanges which give rise to di!erent phases in di!erent helicity amplitudes (see, e.g., [537] for a recent review). 6.9. Structure functions in DIS from polarized hadrons and nuclei of arbitrary spin Shortly after the measurement of gN at CERN there were several proposals to investigate spin e!ects in other nuclear targets. Most prominent among them are of course the deuteron (with spin J"1) and Helium-3 (J"), because they are used to determine gL , but there are other potentially polarizable nuclei as well, like Li(J"), B(J"3) and Al(J"). In those higher spin con"g urations additional structure functions arise which are in principle measureable, although they are typically smaller than the unpolarized structure functions, because polarization dependent e!ects arise only from the unpaired nucleons in the nucleus and are therefore suppressed as 1/A. Furthermore, it should be noted that all these new structure functions would vanish for free nucleons. Thus, such experiments o!er the possibility to con"rm the spin structure of nucleons, and at the same time to obtain new information on nuclear binding e!ects. Among the new structure functions originally studied and de"ned by [380,361,507,508] there are some which have a simple description within the parton model (like g for the nucleons) and there are others which vanish in the parton model (i.e., appear only in higher orders of QCD like F or * are of higher twist like g for the nucleons). Those with a parton model interpretation can be described in terms of parton densities q(+(x, Q) which give the probability to "nd a quark ! with momentum fraction x and helicity $ in a spin-J target with spin M along the z-axis, at some scale Q. Measuring the cross section for an unpolarized target determines the averaged quark
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distribution 1 ( q((x, Q)" q(+(x, Q) . (6.99) 2J#1 +\( Because of QCD parity invariance one has q(+"q( \+ so that only 2J#2 (2J) of the 4J#2 (4J) ! 8 polarized parton densities are independent. (The numbers in brackets hold for half integer J.) It is appropriate to introduce combinations F(+"(q(+#q(+) , (6.100) > \ g(+"(q(+!q(+) , (6.101) > \ because in the Bjorken limit the cross sections on targets with spin (JM) probed with helicity $ electrons can be written as d(p(+#p(+) 8paME > \ " [yx#2x(1!y)]F(+(x, Q) (6.102) dx dy Q and d(p(+!p(+) 8paME > \ " y(2!y)xg(+(x, Q) dx dy Q
(6.103)
with M denoting the target mass. Up to now we have introduced 2J#2 (2J) quark densities for (half)integer J corresponding to 2J#2 (2J) naive parton model structure functions. In addition, there are 4J (4J#1) functions which get contributions either from higher twist operators or from HO QCD so that one has altogether 6J#2 (6J#1) independent structure functions for integer (half integer) spin, as will be shown after Eq. (6.106). For example, for J" there are F , F !2xF , g and g , among them F and g of leading and F !2xF of higher order in QCD, and g being a higher twist contribtuion. For J"1 one has F , F !2xF , b , b !2xb , g , g , b and b . In the naive quark parton approximation the only nonzero structure functions are F , g , b and b . For example, b can be measured by scattering an unpolarized electron beam from a polarized deuteron target with the target spin directed parallel to the direction of the incident electron beam and arranged in each of its m "#1, 0,!1 substates, ' b (x, Q)" e[F >(x, Q)#F \(x, Q)!2F (x, Q)] . O O
(6.104)
Instead of Eqs. (6.100) and (6.101) it is sometimes more convenient to de"ne &multipole' structure functions [J¸] " : ( (!1)(\+(JMJ!M"¸0)q(+ where (JMJ!M"¸0) are the Clebsch}Gordan coe$cients. These function +\( > can be measured using (un)polarized leptons for odd (even) ¸. They are particularly useful if the relative motion of nucleons inside the nucleus is to be considered. Furthermore, it should be noted that the "rst ¸!1 moments of these functions vanish.
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In general, all b can be measured using unpolarized lepton beams. Furthermore, they all arise G from nuclear binding e!ects and are not present for a system of free nucleons. One has b "0 if the spin-1 target consists of two spin constituents at rest, or in relative s-wave, and thus one expects b (deuteron)+0, because the relative motion of the nucleons in the deuteron is nonrelativistic. In contrast one expects a large bM because the quarks in the o-meson move relativistically [380,361]. Furthermore, an interesting sum rule for b has been derived which is related to the electric quadrupole moment of the spin-1 target and to its polarized sea-quark content [189]. It is possible to write down the general hadron tensor for a spin-1 target b b b P P = "!F g #F I J !b r # (s #t #u )# (s !u )# (s !t ) IJ IJ IJ IJ IJ IJ IJ P)q 6 IJ 2 IJ 2 IJ ig ig # e qHsN# e qH(P ) qsN!s ) qPN) IJHN P)q (P ) q) IJHN
(6.105)
and to obtain the cross section from it. In Eq. (6.105) we have de"ned r "(g / IJ IJ (P ) q))(q ) EHq ) E!i(P ) q)) with i"1#4xM/Q, s "(2P P /(P ) q))(q ) EHq ) E!i; IJ I J (P ) q)), t "(1/2(P ) q))(q ) EHP E #q ) EHP E #q ) EP EH#q ) EP EH!P ) qP P ) and I J IJ I J J I I J J I u "(1/P ) q)(EHE #EHE #Mg !P P ). E is the polarization 4-vector of the spin-1 IJ I J J I IJ I J I target, i.e. it ful"lls P ) E"0 and E ) E"!M and sN,(!i/M)eN?@OEHE P . ? @ O Instead of this cumbersome approach one may gain more physical insight by studying the various amplitudes appearing in the forward Compton helicity matrix elements. Namely, just as for the spin- target the optical theorem relates the hadron tensor for arbitrary spin to the imaginary part of the forward Compton scattering amplitude which in turn can be expressed in terms of helicity amplitudes. Let A denote the imaginary part of the forward Compton helicity K+ KY+Y amplitude for c #target Pc #target , K + KY +Y A( "eHI=(+Y+eJ , K+ KY+Y KY IJ K
(6.106)
where eJ , m"$1, 0, are the photon polarization vectors. All the structure functions like b , K b , etc., discussed before can be expressed as linear combinations of these amplitudes. The A( are easily enumerated. Angular momentum conservation requires that the K+ KY+Y total helicity is conserved, m#M"m #M , which leaves 18J#1 independent helicity amplitudes. Time reversal invariance requires A( "A( which leaves 12J#2 indepenK+ KY+Y KY+Y K+ dent amplitudes. Parity invariance requires A( "A( so that one "nally has the K+ KY+Y \K\+ \KY\+Y 6J#2 (6J#1) independent amplitudes, as mentioned before. Among them are diagonal transverse amplitudes A( which in the Bjorken limit correspond to the quark densities !+ !+ q(+ introduced above: A( "q(+. ! !+ !+ ! The amplitudes naturally arise when the hadron tensors =(+Y+ are multiplied with the lepton IJ tensor to form the cross section. To work that out in detail one has to expand the lepton tensor, Eq. (2.29), on the basis of virtual photon helicity eigenstates
1 j 2Q jeI eJ # (j#iy)(eIHeJ #eIHeJ )! (eIHeJ e (#eIHeJ e\ () ¸ " 2 > > \ \ \ > IJ iy 2 > \
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y #j 1! (eIHeJ e\ (#eI eJ e (!eIHeJ e (!eI eJ e\ () \ > > 2 \ 2Q # y(i
y j 1! (eIHeJ !eIHeJ )! (eIHeJ e\ (#eI eJ e (#eIHeJ e (#eI eJ e\ () , \ \ \ > > 2 > > 2 \ (6.107)
where j"2(1!y)!(y/2)(i!1) and is the azimuthal angle (measured with respect to the x-axis in the xy-plane) of the "nal lepton. The photon momentum is used as the spin quantization axis. Note that the "rst term (&2Q/iy) in Eq. (6.107) is spin-independent and the last term (&2Q/y(i) is the spin-dependent term. The resulting expressions for the cross sections in terms of the helicity amplitudes are
dR(HYH 8paMEx 1 1 " j A( ! A( e\ (! A( e ( + +Y 2 >+ \+Y dx dy d
2pQi 2 \+ >+Y 1 # (j#iy)(A( #A( ) >+ >+Y \+ \+Y 2
y # 1! j(A( e (#A( e\ (!A( e\ (!A( e (), , \+ +Y + \+Y >+ +Y + >+Y 2 (6.108)
8paMEx y 1 d*R(HYH " y 1! (A( !A( )! yj(A( e (#A( e\ ( >+ >+Y \+ \+Y \+ +Y + \+Y 2 2 dx dy d
2pQ(i
#A( e\ (#A( e () . >+ +Y + >+Y
(6.109)
These cross sections are not yet in a useful form because the helicities M and M are de"ned w.r.t. the virtual photon direction which changes event by event. It is better to de"ne cross sections for targets with de"nite helicites j and j w.r.t. the incident beam direction. To transform between the 2 frames one has to perform an Euler rotation. The state "Jj2 can be written in terms of "JM2 as "Jj2" e H\+(d( (b)"JM2 , (6.110) +H + where d( (b) is the Wigner rotation matrix and (b) the angle between q and the incoming lepton +H direction k. The cross sections for targets with de"nite polarizations in the lab frame can therefore be obtained from Eqs. (6.108) and (6.109) as dp(HYH dR(HYH " d( (b)d( (b)e H\HY>+Y\+( , +YHY dx dy d
dx dy d +H + +Y d*R(HYH d*p(HYH " d( (b)d( (b)e H\HY>+Y\+( . +YHY dxdyd +H dxdyd
+ +Y
(6.111) (6.112)
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For more details see [361,380]. In those references models have been studied, like the bag model, to predict some of the &new' nuclear structure functions F(+ and g(+. 6.10. Nuclear bound state ewects In the previous section only the appearance of new structure functions in higher spin nuclei has been discussed. Another, probably even more important question is how the &old' functions like g are modi"ed by nuclear e!ects, i.e. how much e.g. g" deviates from gN #gL . This is a very important question because gL cannot be determined directly but only through a measurement of g" or g (He). Let us start with the discussion of deuterium, for which a number of studies exists. The simplest and "rmest theoretical approach to the deuteron is to approximate it as a free neutron and proton with polarization 1!w , where w is the deuteron d-state probability. " " All other nuclear corrections are essentially small because they are suppressed by powers of p/M where p is the 3-momentum of the nucleon in the restframe of the nucleus. w can be calculated " in NN potential models to be w +0.050$0.010 [147,72]. There is a relatively large theo" retical error in the prediction of w , because depending on which of the pheno" menological potentials one uses one gets di!erent answers. This theoretical error is by far the dominant source of uncertainty in the deuteron analysis, much larger than the O(p/M) e!ects mentioned above. This is true except for the large-x region (x50.8) where large nuclear e!ects are known to be present for unpolarized scattering and expected for polarized scattering as well. It is not clear whether these e!ects are spin independent, in the sense that they drop out in the asymmetries &g /F , or not. The fact that w is not precisely known may become a problem for future precision measure" ments of gL because a variation of 2% in w corresponds to an error of roughly 10% in gL . This is " also the order of magnitude which other nuclear e!ects may have on the determination of gL . To incorporate these e!ects there has been a sequence of papers ([475,476] and references therein, but see also the work of [582] to be discussed below) working in the impulse approximation, i.e. neglecting "nal state interactions. There may be "nal state interactions and related e!ects like "nal state pion exchange or nuclear shadowing, but these e!ects are usually assumed to be small for light nuclei and to some extent spin independent. At "rst the so-called convolution model was applied to include nuclear binding and relativistic e!ects, and afterwards the e!ects of o!-mass shellness of bound nucleons were studied. In the convolution model the free nucleon structure function is convoluted with the light cone momentum distribution of nucleons in the nucleus [512]
dy x df (y)g, , Q . (6.113) g"(x, Q)" y y , " ,L N V This formula holds in the Bjorken limit (in"nite Q and l). For "nite Q there is a spectral representation which generalizes Eq. (6.113) [475,476,561]. It was found [562] that the convolution approach gives a result very close to the description with a constant factor 1!w , at least for " x40.7. Above 0.7 there are appreciable corrections to this factor which can become as large as 10%. The simple behavior for x40.7 is not modi"ed when o!-mass-shell corrections [475,476] are included but there are additional contributions at x50.7 (of up to 5%) which should be taken into
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account. Although the nuclear e!ects are within the error bars of presently available data, they will be required in the high statistics E154 and HERMES experiments. Now coming to He, as a "rst approximation one may say that g (He) directly gives gL because the spins of the protons compensate each other due to the Pauli principle. This relies on the assumption that all nucleons are in a S wave. However, such a cancellation does not occur if other components of the three body wave function are considered. In Refs. [582,182,183] the question has been quantitatively discussed as to whether and to what extent the extraction of gL from the asymmetry of the process Ho e(el , e )X could be hindered by nuclear e!ects arising from small wave function components of He, as well as from Fermi motion and binding e!ects on DIS. The basic nuclear ingredient used in [182,183] is the spin-dependent spectral function of He, which allows one to take into account at the same time Fermi motion and binding corrections. Of particular relevance are the up and down quark spectral functions P, , N ,P, and N ,P, , \ \\ NNY+ > because the integral of their di!erence determines the nucleon polarizations P!, N"p, n , [182,183]. These functions enter the so-called spectral representation of g (He) as a function of g,. Note that in the Bjorken limit this spectral representation goes over into Eq. (6.113). In Ref. [582] the Ho e asymmetry has been calculated taking into account S and D waves but considering only Fermi motion and omitting Q dependent terms. Woloshyn used He wave functions [35] calculated using the momentum space Faddeev equation, which are known to give a good description of the spin-averaged quasi-elastic data at medium energies. The extent to which the proton contribution to the spin asymmetry of He can be neglected, can be envisaged by a look at the proton momentum distributions for spins parallel and antiparallel to the target, cf. Fig. 41 where they are plotted against the light cone momentum fraction. The spin dependence is seen to be quite small. Also shown in Fig. 41 are the corresponding curves for the neutron. The neutron curves answer the important question, how large the probability is of having a neutron with spin antiparallel to the nuclear spin. One sees that n is very small as compared to n but somewhat \ > broader since it comes from components in the He wave function with a larger high momentum tail than the dominant S-wave. Fig. 42 shows typical results for g (He)(x, Q) as compared to gL (x, Q) at some unknown (small) value of Q. The curves are given only up to x+0.9 because above this value the calculation becomes unreliable. In fact, nuclear e!ects are expected to become much larger as xP1. It is possible to summarize the e!ects of higher wave functions S and D in He by means of the following procedure: In a pure S wave state the nucleon polarizations are given by P>"1, L P\"0 and P>"P\", whereas for a three-body wave function containing S, S and L N N D waves, one has (6.114) P!"$GD , L P!"GD , (6.115) N where D"[P #2P ] and D "[P !P ]. From world calculations on the three-body " " 1Y 1Y system one obtains, in correspondence with the experimental value of the binding energy of He, D"0.07$0.01 and D "0.014$0.002 [280]. Thus if the S and D waves are considered and Fermi motion and binding e!ects are disregarded, one can write g (He)(x, Q)"2p gN (x, Q)#p gL (x, Q) , N L A o "2f p A o #f p A o , & N N N L L L
(6.116) (6.117)
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Fig. 41. Nucleon momentum distributions for spins parallel and antiparallel to a He target, with z being the momentum fraction of the nucleon in the target, according to [582]. Fig. 42. g (x) for neutron (dashed) and He (solid curve) according to [582].
where f (x, Q)"FNL(x, Q)/[2FN (x, Q)#FL (x, Q)] is the proton (neutron) dilution factor, NL are the e!ective A o o (x, Q)"gNL(x, Q)/FNL(x, Q) is the proton (neutron) asymmetry and p NL NL nucleon polarizations p "P>!P\"!0.028$0.004 , (6.118) N N N p "P>!P\"0.86$0.02 . (6.119) L L L The above values correspond to Eqs. (6.114) and(6.115), while the spin-dependent spectral functions of [582,182,183] yield p "!0.030 and p "0.88. N L For future precision studies one will have to go even beyond those approximations, because there may be modi"cations due to the presence of the nuclear medium, like nuclear swelling and binding e!ects on the parton densities. These e!ects have recently been studied by [368] by depleting the parton densities of [309] at small Q and evolving them according to the AP evolution equations. It turns out that these e!ects are far below the present experimental accuracy, but may be of some relevance for future precision measurements, in particular at small x40.03. 6.11. Direct photons and related processes in proton collisions using polarized beams The most interesting prospect for polarized high energetic proton experiments is the possibility to determine dg(x, k ) from the process pl pl PcX with a high energetic photon in the "nal state $ [248,135,112,341,342,171,201}203,331,297,136,148,471,345,541]. In unpolarized scattering hard photons are known to be a clean probe of the gluon distribution, because they can be directly detected, without undergoing fragmentation. Note that for physics beyond the standard model direct photon production is claimed to be a signal to probe compositeness. The observation of spin asymmetries in this reaction could help to disentangle the structure of new contact interactions [135].
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On the parton level the process is induced in lowest order by the annihilation of light quarks qq Pcg and by the Compton scattering gqPcq, and is strongly dependent on the magnitude of dg(x, k ) with k &pA . It may also be possible to determine dg(x, k ) from high-p jet production, $ $ $ 2 2 but this process has a large background of quark-initiated events qqPpartons, and is therefore less sensitive to dg(x, k ). In Table 5 and Fig. 43 the parton level asymmetries for all the possible $ partonic 2P2 processes including direct photon production are shown as a function of the scattering angle in the parton-cms. The "gure clearly shows that the photon processes are expected to give potentially large e!ects. From a theoretical point of view it would also be interesting to study the production of high-p 2 muon pairs in Drell}Yan processes involving o!-shell photons, because they o!er the possibility to obtain additional information, namely about the polarization of the "nal state. This information
Table 5 Tree-level partonic cross sections and asymmetries a( GH "d*p( /dp( , including direct photon production processes ** GH GH [80,134]. A factor of pa/s( has been factored out of the jet cross sections and pe aa /s( has been factored out of the Q / Q single-photon cross sections a( GH **
dp( GH dtK
qqPqq
4 s( #u( s( #tK 2 s( # ! 9 tK u( 3 tK u(
qq Pqq
4 s( #u( 9 tK 4 tK #u( 9 s(
qq Pq q
(s( !u( )/tK #(s( !tK )/u( ! s( /tK u( (s( #u( )/tK #(s( #tK )/u( ! s( /tK u( s( !u( s( #u( !1
qq Pqq
4 tK #u( s( #u( 2 u( # ! 9 s( tK 3 s( tK
qq Pgg
32 tK #u( 8 tK #u( ! 27 tK u( 3 s(
qgPqg
s( #u( 4 s( #u( ! tK 9 s( u(
s( !u( s( #u(
ggPqq
1 tK #u( 3 tK #u( ! 6 tK u( 8 s(
!1
ggPgg
9 s( u( s( tK tK u( 3! ! ! 2 tK u( s(
qgPqc
1 s( u( ! # 3 u( s(
qq Pcg
8 tK u( # 9 u( tK
(s( !u( )/tK !(tK #u( )/s( # u( /s( tK (s( #u( )/tK #(tK #u( )/s( ! u( /s( tK !1
!3#2s( /tK u( #tK u( /s( 3!s( u( /tK !s( tK /u( !tK u( /s( s( !u( s( #u( !1
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Fig. 43. Parton level asymmetries, including direct photon production processes as a function of the scattering angle in the parton-cms [134].
can be obtained, for example, from the angular distribution of the leptons [171,443,442]. For an interesting application of HO QCD to this process see [160]. The Drell}Yan process will be discussed later on in this subsection. The observable to be measured in direct-c production is the inclusive double-spin asymmetry N = N p= = #pN !pN !p= AA " = = N ** p= #pN N #pN #p=
(6.120)
which reduces to = p= = !pN AA " = ** p= #p= N
(6.121)
if parity is conserved. In Eqs. (6.120) and (6.121) it has been assumed that both protons are longitudinally polarized. More explicitly, this asymmetry is given by
1 dx dx +df (x, k )df (x , k )a( GH dp( #(i j), (6.122) AA dp" G $ H $ ** GH 1#d ** GH G HO O E where dp( and dp are the parton and hadron level cross sections for unpolarized direct-c GH production,
1 dx dx + f (x, k ) f (x , k )dp( #(i j), . (6.123) dp" G $ H $ GH 1#d GH G HO O E The prime refers to the second proton and f and f are the parton densities in the two protons. G H a( GH are the subprocess asymmetries, which along with the dp( can be calculated in perturbative ** GH QCD, cf. Table 5 and Fig. 43. The product d*p( "a( GH dp( is much larger for the Compton GH ** GH subprocess than for the annihilation subprocess. This can be deduced from the explicit form of the
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parton level cross sections and asymmetries in Table 5 and Fig. 43. The Compton subprocess has a positive a( GE HO and leads to a positive contribution to AA which is proportional to dg. For the ** ** annihilation subprocess one has a negative a( GH . Since the Compton subprocess dominates on the ** parton level, one has usually a much smaller asymmetry AA (positive or negative) if dg(x, k ) is $ ** encounters large positive values small, in contrast to a large dg(x, k ). In the latter case one of $ AA up to 50%. In Fig. 44 AA is shown as a function of the photon-p for the two scenarios of large ** 2 ** small dg. It is clearly seen and that a measurement of AA provides a valuable probe of dg(x, k ). $ ** Higher order QCD corrections to the process pl pl PcX involving polarized proton beams have been calculated by [201}204,331]. A particular feature is the appearance of the process ggPcqq whose contribution to the spin asymmetry is of the order of [dg]. The calculation has been performed in the framework of dimensional regularization, in which case special care is needed for the treatment of c . In Refs. [201}203] the method of dimensional reduction was used and it was shown how to circumvent the problems with c . In Ref. [331] the 't Hooft}Veltman scheme was used and the dependence of the cross section on isolation cuts has been examined carefully. Isolation cuts are needed in order to single out isolated direct hard photons from the background [332,330].
Fig. 44. AA at 300 GeV pp beam energy with a large dg(x, k ) as a function of p at h "453 (upper solid curve) and ** $ 2 h "903 (dashed curve). The lower short-dashed curve corresponds to a small dg(x, k ) at h "903. The factorization $ scale was chosen to be k "p . Analogous curves for (s" 600 GeV are also included [136]. $ 2 Fig. 45. K-factor for direct c-production in pl pl -collisions as a function of x [201}203]. The following parameters have 2 been used: K "200 MeV with 4 #avors and the parton distributions of Ref. [56] (set 1) with k "p . The three curves $ 2 +1 correspond to (s"38(dashed), 100(dotted) and 500(solid) GeV.
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The results of the calculation of Ref. [201}203] are shown in Fig. 45 in the form of the K-factor for the polarized cross section de"ned as the ratio of K"(LO#HO)/LO. The K-factors are shown as a function of x "2p /(s for various values of the rapidity g and (s. Note that for the 2 2 p distribution one has the simple formula 2
1 d(D)p d(D)p( GH A6 " dx(d)f (x, p ) dx (d)f (x , p ) G 2 H 2 1#d dp dp 2 2 1 V1 GH N N 2 2 G HO O E
(6.124)
where d(D)p( /dp "(d(D)p( /dtK )s( /(tK !u( ) and d(D)p( /dtK was given in Table 5. One "nds in Fig. 45 2 that in all cases the HO corrections are positive and quite large, in particular at large x . Due 2 to the quite large K-factors there is some ambiguity of the results concerning the unknown higher orders. However, this ambiguity is much smaller than the dependence on the input parton densities as has been discussed in [201}203] with a rather moderate gluon density as input. It should be kept in mind, however, that strictly speaking, there is a very subtle interplay between the magnitude of the K-factor in various schemes and the correct choice of the HO parton densities, the value of K, etc. Similar results have been obtained, using more recent polarized parton densities, for prompt photon production at a future "xed target HERA-No (phase II) experiment ((s"39 GeV) [332] and at the RHIC collider ((s"50}500 GeV) [330]. Instead of direct photon production one may consider production of ='s or Z's balanced by a jet. These processes are very closely related because they involve analogous diagrams. The heavy vector bosons show the additional feature that spin asymmetries arise even if only one of the proton beams is polarized (&single-spin asymmetries'). This is due to the axialvector couplings involved and essentially a parity violating e!ect, as already discussed in Section 6.8. The corresponding single-spin asymmetry A5 8"(pN !p= )/(pN #p= ) *
(6.125)
involves products of polarized and unpolarized parton densities and would in principle be sensitive to dg(x, m ). It turns out, however, that the QCD matrix element for &Compton' scattering of 5 a polarized gluon on an unpolarized quark and associated production of a =! is very small, so that the single-spin asymmetries are not very suitable for the determination of dg(x, m ). They are, 5 however, suitable for the determination of the various #avor contributions du(x, m ) and dd(x, m ) 5 5 to the proton spin. Recent analyses of these e!ects can be found in [519,135,137,541] where it is shown that the process ppo P=!X at RHIC can lead to asymmetries around 50% with a large e!ect in the asymmetry for ppo P=\X if du (x, m ) is small. 5 We now come to the Drell}Yan process [503,341,342,171,136,471,578,405,292]. We shall discuss here only the case of longitudinal polarization. The NLO corrections to the Drell}Yan process in transversely polarized hadron}hadron collisions have been analyzed in [200,573]. The H longitudinal spin asymmetry AA "d*p/dm /dp/dm is de"ned in analogy to AA but depends on ** JJ JJ ** the invariant mass of the produced lepton pair m . The polarized and unpolarized cross sections JJ are given by
d(D)p 4pa dx dx "(!) e(d)q(x , m )(d)q (x , m )[d(1!z) O JJ JJ dm 9sm x x JJ OS B Q JJ
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a (m ) a (m ) # Q JJ h(1!z)(d)c ]# Q JJ h(1!z)(d)c (z)(d)g(x , m ) OO OE JJ 2p 2p
(6.126) ; e[(d)q(x , m )#(d)q (x , m )] #(1 2) O JJ JJ OS B Q where z"m /x x s and the coe$cient functions (d)c (z) and (d)c (z) for the relevant subprocesses JJ OO OE are in the MS-scheme given by [288,503,578,472}474,405,292] c (z)"C OO $
2 ln(1!z) p!8 d(1!z)#4(1#z) 3 1!z
c (z)"¹ (2z!2z#1)ln OE 0
(1!z) 7 1 ! z#3z# z 2 2
dc (z)"!c (z) , OO OO
dc (z)"!¹ (2z!1)ln OE 0
!2
> ,
1#z ln z , 1!z
(6.127) (6.128) (6.129)
(1!z) 3 5 ! z!z# z 2 2
.
(6.130)
It should be noted that the dominant O(a ) contribution in Eq. (6.126) comes from the d-function Q term in Eq. (6.127). It gives a large e!ect for the cross sections but not in the double-spin asymmetry H AA . The asymmetry is shown in Fig. 46 as a function of q"m /s for di!erent parametrizations of ** JJ dg(x, m ) and ds(x, m ). The asymmetries in Fig. 46 are of the same order of magnitude as for the JJ JJ direct photon process. In principle, one could expect that both the dg and the ds terms contribute about the same to the asymmetry. However, due to the smallness of dc (z) it turns out that the sea OE polarization plays a stronger role in AAH than the gluon polarization. This is in contrast to the ** direct photon process which depends strongly on the polarization of the gluons. Fig. 46 also shows that the Drell}Yan process could be very valuable to determine the strange sea polarization in the proton. A measurement of the sign of AAH would already give information **
Fig. 46. Predicted Drell}Yan spin asymmetry as a function of q"Q/s at (s"100 GeV for large positive and large negative ds [171]. Dashed curves are the asymmetries without gluon contributions.
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about the sign of ds. Unfortunately, the Drell}Yan process has a larger background than the direct photon production so that the expected statistical errors at RHIC are larger. Finally, we want to add some remarks about the prospects of determining the polarized parton densities from jet production in hadronic collisions, in particular at RHIC [80,502,248,135,136,429, 178,500,501,230,541]. Quite in general many features of unpolarized jet production can be applied directly to the polarized case. For example, at low p the main contribution comes from 2 gluon}gluon scattering so that this region is particularly suited for attempts to determine the (polarized) gluon density. Secondly, event rates and statistics are much larger than in direct photon production. A drawback is that the signatures are less clear as will be discussed below. One has in principle the possibility to analyze single-jet production pl pl PJX as well as di-jet production pl pl PJ J X. The latter is more di$cult but not impossible to analyze in a high luminosity machine like RHIC. The single jet spin asymmetry can be calculated as a function of rapidity and transverse momentum of the jet from the ratio of the polarized and the unpolarized cross sections. These are given by
1 2x x d(D)p( d(D)p 1 GH #(i j) , (6.131) " dx (d)f (x , k )(d)f (x , k ) E G $ H $ p 1#d 2x !x eW dtK dp
V GH 2 G H where x "2p /(s, x "x eW/(2!x e\W) and x "x x e\W/(2x !x eW). The explicit expres 2 2 2 2 2 2 sions for the parton cross sections can be found in Table 5 [80,134] and the corresponding parton level spin asymmetries are shown in Fig. 43. For all the dominant subprocesses the corresponding asymmetries a( GH are positive, except for qq annihilation. This leads to positive values of the single ** jet double spin asymmetry A "d*p/dp /dp/dp in the low p -range where the gluons dominate. 2 2 2 ** The parton level spin asymmetries combine with suitable parton densities to give the proton asymmetries. These are shown in Fig. 47 as a function of p at two values of the beam energy for 2 a dg(x, k ), k &p , with a small and with a large "rst moment *g(k ) [136]. The signi"cance of the $ $ 2 $ small and intermediate p regime for the determination of dg is clearly exhibited. If *g(p ) is large 2 2
Fig. 47. Predicted spin asymmetries for single jet production as a function of the jet-p for a large (solid curve) and small 2 (dashed curve) "rst moment *g(p ) and two values of the RHIC beam energy [136]. 2
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one can easily have asymmetries of about 20% in the small p range. The relatively large RHIC 2 luminosity L&10 cm\ s\ in combination with the large jet cross sections make the signal in principle large and will lead to small statistical errors [541]. The drawback of this method to determine dg(x, p ) is the large background of soft events in the small p region. 2 2 Let us now discuss 2-jet production. An observable to study is the distribution in the 2-jet invariant mass, M "2p cosh(yH), where yH"(y !y ). It is given by (( 2 d(D)p M >7 W+ (d) f (x , k )(d) f (x , k ) d(D)p( GH , G $ H $ dy " (( dy (6.132) dM dtK 2s cosh(yH) WK (( G H \7 where x "qe! W >W , y "max(!>, log q!y ) and y "min(!>,!log q!y ). >&1 K + is a cuto! on the jet rapidity and q is de"ned as q"((4p /s)cosh(yH). Numerical results for these 2 distributions have been presented in [178,136]. They are shown in Fig. 48 as a function of the jet pair mass for two values of the proton energy and two choices of the polarized gluon density, a small dg(x, M ) (dashed curve) and a large one (solid curve). At intermediate values of the jet pair (( mass M &100 GeV, they are even more sensitive to the magnitude of the polarized gluon density (( than the single jet p distribution but somewhat more di$cult to measure. One may also examine 2 di-jet production and higher order e!ects [230].
6.12. Spin-dependent structure functions and parton densities of the polarized photon Structure functions FA(x, Q) and parton densities f A(x, Q), with f"q, q , g, of unpolarized, i.e. G helicity averaged, photons are theoretically well known (see, e.g. [303,316] and references therein) and experimentally rather well studied (see, e.g., [110,262] and references therein). In contrast to the (un)polarized hadronic parton densities studied so far, these densities obey inhomogeneous evolution equations where the inhomogeneous LO and NLO terms k (x) derive from the GO E point-like splitting of the photon into quarks and gluons which can be calculated from "rst QED principles. In LO, cPqq gives rise to k, and k"0. They are the so-called driving terms which O E
Fig. 48. Predicted asymmetries A for dijet production as a function of the jet pair mass for two values of the proton ** energy and two choices of the polarized gluon density, a small dg(x, M ) (dashed curve) and a large one (solid curve) (( [136].
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uniquely "x the &point-like' (inhomogeneous) solution of the evolution equations for the parton densities of the photon, once a speci"c input scale Q for a (Q ) has been chosen. This is in contrast Q to the conventional &hadronic' (homogeneous) part of the general solution, which derives from the common homogeneous evolution equations and which requires some nonperturbative hadronic (vector-meson-dominance oriented) input f A (x, Q ). In complete analogy to the helicity-averaged case f A"f A #f A , the spin-dependent parton > \ densities of a longitudinally (more precisely, circularly) polarized photon are de"ned as [373,349,583,465] df A(x, Q)"f A (x, Q)!f A (x, Q) (6.133) > \ with f A (f A ) denoting the parton densities in the photon aligned (anti-aligned) with its helicity. As in > \ Eq. (4.11), they satisfy the general positivity constraints "df A(x, Q)"4f A(x, Q) .
(6.134)
Similarly to Eq. (4.5), the polarized photon structure function is given by 1 (6.135) gA (x, Q)" e[dqA(x, Q)#dq A(x, Q)]#O(a , a) O Q 2 O where the NLO (two-loop) contributions [555] have been suppressed. Note that dqA"dq A and dqA"O(a/a ) in LO. In Bjorken x-space, these photonic parton densitites obey the following Q inhomogeneous LO evolution equations: a a (Q) d A dq (x, Q)" dk(x)# Q dP dqA , ,1 ,1 2p ,1 2p dt ,1
(6.136)
dP 2fdP dRA d dRA(x, Q) a dk O #aQ (Q) OO OE " , (6.137) dt dgA(x, Q) 2p 0 2p dP dP dgA EO EE which are a straightforward generalization of Eqs. (4.12) and (4.19), taking into account the &point-like' photon splitting cPq which gives rise to the inhomogeneous terms: dk"dP can be O OA obtained, apart from obvious charge factors, from dP in (4.21) by multiplying it with 2fN /¹ : OE A 0 f (e!1e2)\dk"1e2\dk"6f (2x!1) , (6.138) O ,1 O where 1e2,f \ e. Furthermore, dk"dP"0 has been used in (6.137). The LO equations O O E EA (6.135)}(6.137) can be straightforwardly extended to NLO [555] where the O(aa ) terms dk and Q O dk derive from the C ¹ pieces of 2fdP and dP in (A.4) and (A.6) in the Appendix, E $ D OE EE respectively, multiplied by fN /¹ . A D The evolution equations (6.136) and (6.137) are most conveniently solved in Mellin n-moment space (cf. Eqs. (4.22) and (4.23)) where the solutions can be given analytically and one can easily keep track of the contributions stemming form di!erent powers of a in order to avoid terms Q beyond the order considered. The inversion to Bjorken x-space is again straightforward with the help of (4.37). The general solution decomposes into df A L(Q)"df A L(Q)#df A L (Q) .*
(6.139)
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with the &point-like' (inhomogeneous) solution being given by a 1 4p L [1!¸\ @ B.,1 ] dkL , dqA L (Q)" ,1 .* ,1 1!(2/b )dPL 2pb a (Q) Q ,1 dRA L(Q) a dkL 4p 1 L O .* " [1!¸\ @ B.K ] a (Q) 1!(2/b )dPK L 2pb dgA L(Q) 0 Q .* and the &hadronic' (homogeneous) solution given by
(6.140)
(6.141)
dqA L (Q)"¸\ @ B.,1 dqA L (Q ) (6.142) ,1 ,1 dRA L (Q) dR A L (Q ) , "¸\ @ B.K L (6.143) dgA L (Q) dgA L (Q ) where ¸(Q),a (Q)/a (Q ). Note that the &hadronic' solutions are formally identical to the usual Q Q ones in Eqs. (4.27) and (4.28) since they are derived from the homogeneous part of the evolution equations (6.136) and (6.137). The structure of these LO solutions can be straightforwardly extended to NLO [316,555] by using the techniques discussed in Section 4.2 being based on the (two-loop) evolution matrix in Eq. (4.56). The new ingredient of these solutions is the &point-like' component which is driven by the inhomogeneous (photon splitting) terms dk in Eqs. (6.140) and (6.141), i.e. they uniquely G determine the &point-like' parton densities in the photon once an appropriate input scale Q has been speci"ed. This is in contrast to the hadronic components in (6.142) and (6.143) which require, as usual, also the speci"cation of the input densities df A (x, Q ). In general, one expects the &point-like' df A (x, Q) to be dominant in the large-x region since the dk in (6.140) and (6.141) .* G increase as xP1 according to (6.138). This is in contrast to the &hadronic' df A (x, Q) where the (VMD oriented) input df A (x, Q )&(1!x)? as xP1. Such expectations have been well estab lished in the case of unpolarized photons [316,317,110,262]. For polarized photons, several model calculations have been performed for estimating df A(x, Q) in LO [373,349,583]. More recently, in order to obtain a somewhat more realistic estimate for the theoretical uncertainties in the experimentally entirely unknown df A, two very di!erent scenarios were considered in [322,324]: The &minimal scenario' is characterized by the input L
df A (x, k)"0 , whereas the &maximal scenario' is de"ned by the other extreme input
(6.144)
df A (x, k)"f A (x, k) , (6.145) with Q ,k"k "0.25 GeV and the unpolarized LO GRV photon densities f A (x, k) have * been taken from [317]. It should be mentioned that the range of such VMD inspired inputs can be further restricted [324] by using the sum rule
dqA(x, Q) dx"0
(6.146)
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Fig. 49. LO and NLO expectations for gA according to the &minimal' and &maximal' inputs in Eqs. (6.144) and (6.145), respectively [555]. The results shown correspond to f"3 #avors.
which derives from the vanishing of the "rst moment of gA due to the conservation of the electromagnetic current (gauge invariance) [243,98,487,279]. Note that the &minimal' hadronic input (6.144) satis"es this sum rule automatically. Such inputs have also been implemented in NLO [555] and some typical expectations for gA are shown in Fig. 49. The strong increase in the large-x region is typical for the &point-like' dqA as discussed above. The &minimal' scenario is &point-like' .* throughout the entire x-region due to the vanishing hadronic input in (6.144). The possible importance of the hadronic components in Eqs. (6.142) and (6.143) is illustrated by the &maximal' scenario results in Fig. 49 which involve an additional hadronic component due to the input (6.145). Several suggestions have been proposed to measure df A(x, Q) at polarized ep [321,322] and e>e\ colliders [320]. It has been discussed already in Section 6.2 that these resolved photon contributions could amount up to 20% of the total charm production cross section at future el pl collisions [553,554]. The situation improves for photoproduction of jets at future el pl colliders ((s +200 GeV) where the much larger size of the resolved photon contributions to singleAN inclusive jet and, in particular, dijet production [553,554] could give rise to experimentally testable signatures of the parton densities of a polarized photon in the not too distant future.
7. Nonperturbative approaches to the proton spin In Section 5 it was shown that the apparent smallness of the #avor singlet axial vector current matrix element is due either to a negative sea polarization or to a positive gluon polarization. However, no theoretical prediction for the value of this matrix element was presented. There are several nonperturbative approaches which try to remedy this situation [564,531,174,116] and lattice calculations have been attempted as well [459,326,213,287]. The well-established isotriplet Goldberger}Treiman (GT) relation (2f pg , g (0)" 2M p ,,
f "132 MeV , p
(7.1)
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which "xes the axial coupling g (0)"g "1.2573$0.0028 in terms of the strong coupling constant g , has inspired [564,531] to generalize it to the isosinglet ; (1) case to see if one can p ,, learn something about the magnitude of g S ,1PS"tM c c t"PS2. I I Many discussions on the isosinglet GT relation were mainly motivated by the desire to understand why the g inferred from the EMC experiment is so small (g &0.15, cf. Table 6) [167,282}285,523,96,564,531,194,350,389,117,164,242,260}262,575,452,532,488,245]. At "rst sight, the ; (1) GT relation seems not to be in the right ballpark, as the naive SU(6) quark-model prediction yields far too large a value g +0.80, because g "((6/5)g . It was then p ,, E ,, and suggested that the &physical' g -nucleon coupling is composed of a bare coupling g E ,, a coupling between the ghost "eld and the nucleon. In QCD the ghost "eld G,R KI, cf. (5.38), is I necessary to solve the ; (1) problem. It mixes with the bare g and is allowed to have a direct ; (1) invariant interaction g with the proton. Unfortunately, the de"nition of this coupling is %,, not free of ambiguities. For example, it is sometimes assumed in the literature to be the coupling between the glueball and the nucleon. Furthermore, unlike in the case of g (0), the predictive power of the ; (1) GT relation is partly lost by the mixing. In chirally invariant schemes (like the o!-shell regularization discussed in Section 5) one may at least identify the term ((n f /2M)g with the E ,, Dp total quark spin inside the proton *R , cf. Section 5 [531]. It is interesting to obtain a scheme-independent relation. For the isotriplet GT relation this is possible because it holds irrespective of the light quark masses. For mO0 it may be derived from p PCAC, while in the chiral limit g (q) can be related to an isotriplet form factor which in turn receives nonvanishing pion pole contributions. By the same token it can be shown in the OZI limit that the isosinglet ; (1) GT relation [531,532] (7.2) g (0)"((3f /2M)g p E ,, is a bare unphysical coupling remains totally scheme and mass independent. In Eq. (7.2) g E ,, between g and the nucleon. The g is not a physical meson but constructed from the mass eigenstates via [523,96,174]
p 1 h cos h #h cos h h sin h !h cos h p (7.3) g " !h cos h sin h g , g h ! sin h cos h g where h "!0.016, h "!0.0085 and h "!18.53 are the mixing angles of the p}g system [245]. Consequently, the complete ("isotriplet#octet#singlet) GT relation in terms of physical coupling constants reads (2f (2f pg p [g $g " g (0)" (h sin h !h cos h ) EY,, 2M p ,, 2M p,, $g (h cos h #h sin h )] , E,,
(7.4)
Note that in Section 5 a di!erent notation was used, namely A S "*R S "1PS"tM c c t"PS2 and g (0)"g /g . I I I 4 Further, in Section 6.6 an isotriplet form factor G (Q) has been introduced which is normalized di!erently than g (Q), namely G (Q)"g (Q).
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(6f (6f pg p (g " g (0)" cos h #g sin h Gg h ) , EY,, p,, 2M E ,, 2M E,,
(7.5)
(3f (3f p g " p (g g (0)" cos h !g sin h $g h )#2 , E,, p,, 2M E ,, 2M EY,,
(7.6)
where the upper sign is for the proton and the lower sign for the neutron and the ellipses in the last relation are related to the ghost coupling to be discussed below. Although the isosinglet GT relation is scheme invariant, the interpretation of the g "eld is the predicted scheme dependent. When the S;(6) quark model is applied to the coupling g E ,, g is too large, g "0.80. This can be resolved by applying the S;(6) model to the physical . One receives a two-component expression [531] coupling g rather than to g E ,, E ,, (3f 1 p g ! g (0)" m f g (7.7) 2M E ,, 2(3 E p %,,
where g is the coupling de"ned in the e!ective QCD nucleon Lagrangian %,, g (3 L"2# %,, RIG¹r(NM c c N)# (RK)g #2 , (7.8) I f 4M p where RK"(1/(3)m f g # g m f RI¹r(NM c c N) is the anomaly with matrix element E p %,, E p I Mg (0). " 1N"RK"N2"1N"RIJ"N2"(1/(3)f g I p E ,, In order to arrive at a small EMC-like g one may now argue that there is a cancellation between the two terms of Eq. (7.7). This is reminiscent to the situation in the perturbative framework where a cancellation between *R and *g was proposed to explain the spin EMC e!ect. However, there is no scheme independent identi"cation between the various terms of the perturbative and nonperturbative approach. Some authors have questioned [245] the ;(1) GT relation (7.7) but it seems that it survives as long one takes proper care of all the mixing e!ects. Still it should be stressed that until now no direct experimental con"rmation exists, and in this sense it is one of the speculations from the realms of nonperturbative QCD. The point is that both sides of the relation are di$cult to grab. The right-hand side (i.e. the low-energy couplings) cannot really be determined from low energy data, because g is not known. Even the gNN couplings are only known within large errors, %,, cf. the discussion in [174]. For example, a determination of g using F, D and h values gives EY,, g "3.4 [175] whereas an estimate of the g P2c decay rate through the baryon triangle EY,, contributions yields g "6.3 [81]. The analysis of the NN potential gives g "7.3 [233] EY,, EY,, while the forward NN scattering analyzed by dispersion relations gives g (3 [335]. On the EY,, other hand, the only chance to determine g is from the polarized DIS data via the ;(1) GT %,,
There is a conjecture, though no proof, that g is given by the quark sea contribution (&disconnected insertion'), %,, !(m f /4M)g "*(u #u )#*(d #dM )#*(s#s )+3*(s#s ) [174]. Furthermore, g could in principle be obE p %,, Q Q %,, tained from low-energy baryon}baryon scattering in which an additional S;(3) singlet contact interaction arises from the ghost interaction but this is very di$cult to measure [523].
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relation. However, there is no direct identi"cation between g and an observable de"ned in %,, high-energy scattering. Therefore, the whole issue remains somehow speculative and undecided. Another attempt to obtain information on the proton spin matrix elements from low energy meson properties was recently made by Birkel and Fritzsch [116]. They used the masses and properties of the axial vector mesons with quantum numbers J.!"1>>. The spectrum consists of the isoscalar mesons f (1285), f (1420) and f (1510). After correcting for the mixing with the isovector a (1260) and the strange isodoublet K , the f mesons can be written as a linear combination of the three axial vector states "N2"(1/(2)"u u#dM d2, "S2""s s2 and an exotic gluonic meson state "G2. The latter interpretation arises because within the S;(3) multiplets one expects only two isoscalar mesons f . A relatively large mixing is found between the N, S and G G state, and this is due largely to the e!ect of the anomaly and reminiscent of what happens in the case of pseudoscalar mesons [116]. On the basis of this analysis one may use the idea of &axial vector' dominance to get information on the proton spin matrix element. The basic relation is 1P"q c c q"P2" I
10"q c c q"A21AP"P2 I m !k I
(7.9)
where 10"q c c q"A2 denotes the transition element of the axial vector current between the vacuum I and the axial vector meson A, and 1AP"P2 describes the coupling of the axial vector meson to the proton. The 4-momentum transfer is k. Using Eq. (7.9) one can relate *(u#u ), *(d#dM ) and *(s#s ) to the corresponding couplings of axial vector mesons. Without the gluonic state, a relatively large value of the #avor singlet quark spin contribution *R+0.52 is obtained. Inclusion of the gluonic state 1GP"P2O0 leads to numbers of the order *R+0.25 in accordance with DIS data. However, just as in the U(1) Goldberger Treiman model there is a free parameter which has to be "xed, namely the coupling g to the proton de"ned by %. 1GP"P2"ig u (P)c c u(P)eJ where eJ is the polarization vector of the gluonic state. It is hard to %. J determine g experimentally. The above number *R+0.25 corresponds to the choice g "19. %. %. A more general method to obtain nonperturbative results on the proton spin matrix elements is lattice gauge theory. After the 1987 EMC spin surprise, several attempts were made to compute *g and 1PS"tM c c t"PS2 using lattice QCD. A "rst direct calculation of *R was made in [459] but I without "nal results. Successful lattice computations in the quenched approximation were published recently [213,287,326]. A more ambitious program of computing the polarized structure functions g and g is also feasible and encouraging early results were reported in [326]. In Refs. [213,287], the scheme and gauge invariant matrix elements 1PS"tM c c t"PS2 were calculated. The I results are shown in Table 6 and compared to the experimental data, although one should keep in mind that the computation of sea quark contributions might be questionable in a quenched calculation. Varying the quark masses it was found in [287] that a considerable amount of the sea contributions is mass independent and therefore must be induced by gluons through the ABJ anomaly. This is in accord with arguments based on perturbative QCD and presented in Section 5. However, for a direct lattice computation of *g one would need gauge con"gurations on a sizeable lattice not available so far. It is hoped that lattice results for *g will be available in the near future. The most recent improvement in this "eld is the implementation of an improved action by the DESY/HLRZ collaboration [115], i.e. of a systematic procedure for the removal of all terms linear in the lattice spacing a from the lattice observables [558,456,457] which reduces the cuto! errors
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Table 6 Axial couplings and quark spin content of the proton from lattice calculations according to [174]. Note that the &experimental' singlet results are model and scheme dependent and the stated numbers refer to an average of recent LO and NLO(MS) analyses. In the o!-shell scheme one obtains [48], for example, g "0.45(9); see, however, Ref. [9]
g g g *(u#u ) *(d#dM ) *(s#s ) F D
[213]
[287]
Experiment
0.25(12) 1.20(10) 0.61(13) 0.79(11) !0.42(11) !0.12(1) 0.45(6) 0.75(11)
0.18(10) 0.985(25) * 0.638(54) !0.347(46) !0.109(30) 0.382(18) 0.607(14)
0.22(6) 1.2573(28) 0.579(25) 0.80(3) !0.46(3) !0.12(3) 0.459(8) 0.798(8)
order by order in a, yielding a better extrapolation towards the continuum limit [115]. This O(a) improved lattice theory yields, for example, *u "0.841(52) and *d "!0.245(15), in reasonable T T agreement with DIS experiments (cf. Section 6.1 and Ref. [9]). There are also attempts to explain the smallness of g , i.e. of *R, by invoking the Skyrme model [143,255,518,184,185,114]. Here one argues [143,255] that the quark singlet contribution *R is suppressed by 1/N while *(q#q )"O(1) for each separate #avor, and a similar suppression A should hold for *g(Q). Besides the fact that the precise relation between the Skyrme model and QCD is somewhat unclear, in particular in connection with *g, this explanation has been questioned within the Skyrme model itself [518,184,185,114,381]. Alternatively, QCD sum rules [109,179] result in a similarly small total singlet spin contribution g "0.1}0.2 [343,354]. Again, a distinction between the individual *R and *g contributions cannot be obtained. Very promising appears to be the chiral soliton approach towards the structure of the nucleon within the e!ective chiral theory [226,227,580] which allows for the calculation of the full x-dependence of the structure functions and the parton densities from "rst principles, in contrast to just their nth moments as obtained in the nonperturbative approaches discussed so far. The relevance and in#uence of the instanton vacuum on low-energy QCD observables has been emphasized in particular by Diakonov et al. [226,227] who calculated unpolarized (spin-averaged) and polarized valence and sea input densities from "rst principles at the typical scale which is set by the inverse average instanton size o, i.e. Q +o\+0.36 GeV. The instanton size remains the only free parameter in the calculation, and its inverse serves as an UV cuto! of the nonrenormalizable e!ective chiral "eld theory. What makes this approach quite promising is the fact that it predicts [226,227], besides the valence densities, a valence-like input (unpolarized) sea density in the small-x region at Q "0.3}0.4 GeV, which forms the basic ingredient for understanding and predicting all small-x unpolarized DIS HERA-data [11,38,40,220,222,223] from "rst principles, i.e. pure (parameter-free) QCD dynamics [316,318,319]. So far, the polarized sea and the (un)polarized gluon input densities have not been calculated. It is in particular the valence-like gluon densities [319,309], being 1/N &suppressed', which have to come out sizeable at Q "0.3}0.5 GeV. A
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Otherwise the chiral soliton approach does not refer to a perturbative (twist-2) input scale reachable by perturbative RG evolutions, but instead would refer to some nonperturbative input quark scale which cannot be reached by perturbative evolutions. Nevertheless, for the time being, the chiral soliton approach appears to be a realistic model of nonperturbative QCD which might eventually link, from "rst principles, the con"ning regime to the perturbative sector. It should be stressed that only future (dedicated) experiments can ultimately decide about the physical reality of the various theoretical ideas and scenarios discussed so far. It should be kept in mind, however, that all realistic experiments are of course sensitive to the explicit x-dependence of the polarized parton densities df (x, Q) and in most cases are not directly related to their "rst moments *f (Q). 8. Transverse polarization 8.1. The structure function g For pure photon exchange the complete polarization part of the hadron tensor is antisymmetric and given by, cf. (2.1),
Sq M qM SNg (x, Q)# SN! PN g (x, Q) . = "i e IJ Pq Pq IJMN
(8.1)
For longitudinal polarization and at Q much larger than M the g -piece gives the dominant contribution, with g being suppressed by a factor xM/Q according to (2.7). However, for a nucleon transversely polarized with respect to the beam direction, = is proportional to IJ (xM/Q)(g #g ), so that g and g enter with equal coe$cients but the whole contribution is down by a factor xM/Q with respect to g in the longitudinally polarized case. This can again be derived from the inclusive, fully di!erential cross section Eq. (2.7). Therefore g being measured at SLAC and DESY will have much less accuracy than g . Furthermore, it is really the combination g ,g #g which is the &transverse spin structure function' although, for obvious reasons, one 2 usually refers just to g [375]. Since g is related to a transverse polarization, it is not easy to "nd a partonic interpretation [45,383,348]: in a transversely polarized nucleon, the quark spin operator projected along the nucleon spin, R "c c S. with S. &c , does not commute with the free quark Hamiltonian 2 2 2 H "a p and thus there exists no energy eigenstate "p 2 such that R "p 2"j "p 2. Therefore, X X X 2 X 2 X R is a &bad' operator and depends on the dynamics. Nevertheless, a transverse-spin average for 2 quarks can still be de"ned in the nucleon and it is just g ,g #g which is sensitive to the 2 quark}gluon interactions } a clear sign that no simple parton interpretation can be made for it [375,382]. This is in contrast to the longitudinally polarized nucleon, where the quark helicity operator R "c c S. , with S. &c , commutes with H and thus g (x) measures directly the quark helicity distribution. It should be further noted, that g vanishes [45,382] for a free (massless or massive) quark, i.e. for a pointlike nucleon, and thus g cannot be expressed as an incoherent sum over free on-shell partons. The partons must be interacting and/or virtual in order to contribute to g . Therefore, g will serve as a unique probe of &higher twist' (twist,dimension spin"3) as well.
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Regardless of the di$culties with a partonic interpretation, g itself consists of a twist-2 (g55) and a twist-3 (g ) contribution, g "g55#g (8.2) both of which can a priori contribute the same order of magnitude. The twist-2 contribution g55 is the so-called &Wandzura}Wilczek' piece [576] which will be discussed "rst: In leading twist-2, the same operators in Eqs. (4.72) and (4.73) contribute to g and g ; therefore one has, through the optical theorem [375,382],
1 n!1 dx xL\g55(x, Q)"! MLEL(Q/k, a ), n"1, 3, 5,2 G G Q 2 n G By comparing this with the corresponding equation for g , (4.75), one obtains [576] n!1 U "0 . g (x, Q)#g55(x, Q) dx xL\ n This can be inverted to Bjorken-x space to give
(8.3)
(8.4)
dy g (y, Q) , (8.5) y V which is the so-called Wandzura}Wilczek relation [576]. This twist-2 expectation for g and the present experimental results on g are shown in Fig. 50. Note that the twist-2 Wandzura}Wilczek piece g55 of g obeys automatically the so-called Burkhardt}Cottingham sum rule [153], g55(x, Q) dx"0, which follows from Eqs. (8.4) or (8.5) and to which we shall return below. It is a unique feature of g that the higher twist term g in (8.2) is not suppressed by inverse powers of Q and thus could in principle be equally important as the twist-2 contributions g55 discussed so far. It should, however, be kept in mind that g vanishes for ultrarelativistic on-shell quarks where SN&PN, i.e. in this case there are not enough four vectors to form the g55(x, Q)"!g (x, Q)#
Fig. 50. Twist-2 expectations for the structure function g55(x, Q"10 GeV), using a parametrization of the EMC data (solid curve) [77,78] on gN (x, Q"10 GeV), according to the Wandzura}Wilczek relation (8.5).
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required antisymmetric combination in (8.1). Furthermore, one might naively expect g to be small because nonrelativistic corrections, being of the order m /M or m /K, are small for light quarks. O O Future experiments will prove to what extent this is actually the case. A considerable twist-3 contribution, however, might be due to the o!-shellness of interacting quarks with virtuality k where k/K is not small [533,46,47,375,382]. On the other hand, within the covariant relativistic parton model approach it has been argued [374] that the quark virtuality k is expected to be not sizeable and therefore deviations from the Wandzura}Wilczek relation (8.5) should be small. In the following we shall explain this argument, because it allows to get some physical insight into the properties of g . As a straightforward generalization of the unpolarized case [301], the covariant parton model expression for the antisymmetric part (8.1) of the hadron tensor is given by a convolution over the struck parton's momentum in the &hand bag' diagram,
(8.6) = (q, P, S)" dk f (P, k, S)w (q, k)d[(k#q)!m] , Q IJ Q IJ Q! where f is some Lorentz-invariant hadronic wave function. The antisymmetric tensor for the Q parton}photon interaction is given by q?n@ w " tr[(1Gc n. )(k. #m)c (k. #q. #m)c ]!(k l)"$ime I J IJ?@ IJ ! with n@ being the o!-shell parton spin vector, k ) n"0, n"!1. De"ning
(8.7)
m M ( f !f ) , fI (P ) k, k)"! \ k ) S (k!Mk/(P ) k) >
(8.8)
a comparison of Eqs. (8.6) and (8.1) yields [374]
px k#k 2 , k g (x)" dk dk fI x# 2 8 xM
k#k 2 1! xM
2k 1! xM#k#k 2 k px k#k 2 2 , k g (x),g (x)#g (x)" dk dk fI x# 2 2 xM 8 xM
,
(8.9) (8.10)
for a longitudinally and transversely polarized photon, respectively. The last equation, (8.10), immediately proves that a "nite transverse polarization result g can arise only for a nonvanishing 2 parton transverse momentum k O0. Therefore, the EMC observation of a nonvanishing value of 2 g (x) dx is direct evidence for k O0 (via the Burkhardt}Cottingham sum rule g (x) dx"0, to 2 be discussed below). Furthermore, the second factor in parentheses in the integrand of Eq. (8.9) proportional to 2k violates the Wandzura}Wilczek sum rule and explicitly describes higher-twist corrections (g ) to it. In the absence of this term one easily derives from (8.9) and (8.10) 1 d [g (x)#g (x)]"! g (x) (8.11) x dx and hence the Wandzura}Wilczek sum rule (8.5). There seems to be not much room for a sizeable kO0 since the "rst term in parentheses in the integrand of Eq. (8.9) gives rise to a zero for g (x) if k#k +xM. The lack of experimental evidence for such a zero suggests that k#k ;xM. 2 2
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Ignoring the possibility that k(0 contributes appreciably, the second factor in parentheses in (8.9) is essentially unity, giving thus only small corrections to the Wandzura}Wilczek sum rule. It should be noted that g (x) in Eq. (8.10) gets no additional o!-mass-shell correction, besides 2 from the one entailed in fI , and that a measurement of g gives a direct estimate [374] of the mean intrinsic k of partons as a function of x, in complete analogy with the unpolarized case [301]. 2 Unfortunately, it must be noted that the scale Q, at which these results are supposed to hold, remains undetermined within the covariant parton model [301]. Therefore, these results have only a qualitative rather than a quantitative character. Furthermore, and more importantly, the virtuality k alone is not a reliable measure for the importance of all possible twist-3 contributions to g , because of the appearance of additional twist-3 operators which describe quark}gluon correlations and explicitly quark-mass-dependent operators. These operators will now be discussed in some detail: In the OPE approach to g the higher twist terms are determined by a tower of operators whose number increases with increasing moments. Therefore, Eqs. (8.3)}(8.5) are incomplete due to the neglect of g in (8.2), and in principle signi"cant modi"cations of Fig. 50 might be expected. The modi"cations g due to higher twist can be described in terms of matrix elements of the following twist-3 operators [414,533,382,417,418]:
L\ iL\ (n!1)tM c cND+I 2DIL\ ,t! tM c cIJ D+NDI 2DIJ\ DIJ> 2DIL\ ,t , RNI 2IL\" $ n J (8.12) RNIO 2IL\ "iL\tM m c cND+I 2DIL\ cIL\ ,t , K O 1 RNI 2IL\ " (< !< #; #; ), I L\\I I L\\I 2n I
(8.13) (8.14)
where m in (8.13) represents the quark mass (matrix) and + , means symmetrization over the O Lorentz indices; furthermore, the #avor structure (j ) for the quark "elds t has been suppressed for ? simplicity and the appropriate subtraction of trace terms is always implied in order to render the resulting operators traceless, i.e. of de"nite spin. The operators in (8.14) contain explicitly the gluon G?@ and are given by "eld strength G and its dual tensor GI "e IJ IJ IJ?@ < "iLgStM c DI 2GNII 2DIL\ cIL\ t , I ; "iL\gStM DI 2GI NII 2DIL\ cIL\ t , I where S means symmetrization over k and g is the QCD coupling constant. It is a well-known fact G [533,46,47,375,382,417,418] that these operators (8.12)}(8.14) are not independent and related through the &equation of motion' operator n!1 S[tM c cNDI 2DIL\ cIL\ (iD. !m )t RNI 2IL\ "iL\ O 2n (8.15) # tM (iD. !m )c cNDI 2DIL\ cIL\ t] . O Making use of the identities D "+c , D. , and [D , D ]"gG one can obtain the following I J IJ I I relation for the twist-3 operators:
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L\ n!1 RNI 2IL\ " RNIO 2IL\ # (n!1!k)RNI 2IL\ #RNI 2IL\ . (8.16) $ K I n I Leaving aside the Q dependence for the time being, the moments of g are given by 1 n!1 (d !a ), n"3, 5, 7,2 (8.17) dx xL\g (x, Q)" L L 2 n which represents the generalization of the pure twist-2 relation (8.3) and where the contribution from the twist-2 operators is summarized in a " MLEL and d is the matrix element of the sum L G G G L of all twist-3 operators contributing to the nth moment of g . Note that this formula is only true if one formally keeps R in the operator basis because the matrix element d is de"ned by $ L n!1 1P, S"RNI 2IL\ "P, S2"! d (SNPI !SI PN)PI 2PIL\ . (8.18) $ n L
If one eliminates R from the basis via Eq. (8.16), the matrix elements of the operators R O , R $ K I and R will appear. Note further that there is no relation (8.17) for the "rst moment n"1, because there is no twist-3 operator for n"1. This is in contrast to g whose "rst moment is "xed in the operator product expansion by the matrix element of the axial vector singlet current (cf. Section 5). Combining Eq. (8.17) with the analogous pure twist-2 relation (4.75) for g , 1 (8.19) dx xL\g (x, Q)" a , 2 L it has become customary to extract the pure twist-3 matrix element d : L n n d (Q)"2 dx xL\ g (x, Q)# dx xL\g (x, Q) , (8.20) g (x, Q) "2 L n!1 n!1 where the latter equality follows from Eqs. (8.2) and (8.4). Being pure twist-3, d (Q) is a direct probe L of nonpartonic e!ects such as quark}gluon correlations. In other words, it is a direct measure of deviations from the (twist-2) Wandzura}Wilczek relation (8.5). Several experimental attempts at CERN (SMC [19]) and SLAC (E143 [5], E154 [8]) to observe such deviations by measuring gN L B(x, Q) via A in (2.13) did not result in any statistically relevant twist-3 contribution (g ) to g , 2 i.e. present data are in agreement with the twist-2 Wandzura}Wilczek prediction derived from (8.5), g (x, Q)+g55(x, Q), at presently attainable values of Q. Qualitatively, the observed tendency is that gN (x, Q) is positive in the region of smaller x and negative in the region of larger x values, in agreement with the twist-2 Wandzura}Wilczek expectations [5] (cf. Fig. 50). More speci"cally, present measurements imply, for example, for the third n"3 moment d (Q+5 GeV) in Eq. (8.20) the following results [5,8]:
dN "(5.4$5.0);10\ (E143) , dB "(3.9$9.2);10\ (E143) , (8.21) dL "(!10$15);10\ (E154, SLAC average) . Comparing these results with bag model expectations [550,402,542}544], dN +(6 to 18);10\, dB +(3 to 7);10\ and dL +(!2.5 to 0.3);10\ or with those obtained from QCD sum rules
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[84,85,548], dN +(!9 to 0);10\, dB +(!22 to !8);10\ and dL +(!40 to !15); 10\, it becomes clear that for the time being there is no possibility to distinguish between di!erent models. As a side remark, it should be noted that the matrix elements d appear also in higher twist L corrections to g . For example, the "rst moment of g has an expansion [246,402]
M M 1 (a #4d #4f )#O . dx g (x, Q)" a # Q 9Q 2
(8.22)
The higher twist corrections in this result are not completely "xed by a and d , but there is another matrix element f of a twist-4 operator involved, de"ned by e1PS"q gGI c@q"PS2"2Mf S . O ?@ ? G
(8.23)
This twist-4 matrix element f has been estimated [84,85,246,400], partly from a QCD sum rule approach, with similar uncertainties as the above estimates of d . There have been attempts to calculate also g (x, Q) in the bag model [382,550,542}544]. The bag model does not contain gluon "elds explicitly, but the boundary of the bag-con"ned quarks simulates the binding e!ect coming from quark}gluon and gluon}gluon interactions. Hence, the structure function g calculated in the bag model includes higher twist e!ects and measures possibly large twist-3 matrix elements comparable in size to the twist-2 ones. Indeed, sizeable departures from the Wandzura}Wilczek relation [550] have been noted in an extended version of the MIT bag model [525]. They are mainly induced by the noncovariance of the relativistic bag model, which originates from the implementation of the bag boundary in the equation of motion. This is in spite of the fact that the average parton virtuality in the bag is small, from which one might erroneously conclude that g +0 according to the covariant parton model discussed above. The predictions for g [550] are shown in Fig. 51. It should, however, be emphasized that the results of such bound-state models are expected to hold at some nonperturbative bound-state scale, typically Q&K. Strictly speaking, it is therefore not even possible to use these predictions as an input for an evolution to a larger scale Q91 GeV, unless one arbitrarily chooses the bag bound-state scale Q
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Fig. 51. Predictions for g [550] of an extended (one-gluon exchange) version of the MIT bag model [525] using a bag radius R"0.7 fm and spin-singlet and -triplet intermediate diquark masses of 0.56 and 0.8 GeV. The EMC prediction for g55 of Fig. 50 is also shown.
We have already noted that for the "rst moment of g there is no twist-3 operator within the light-cone OPE. This is a hint, though no proof, of the so-called &Burkhardt}Cottingham (BC) sum rule' [153]
dx g (x, Q)"0 . (8.24) This relation can be derived as a superconvergence relation based on Regge asymptotics (see [375] for a review). Recently, it has been shown by an explicit calculation that it is ful"lled in higher-order QCD [51,416]. The (more general but also more questionable) Regge proof was derived by considering the asymptotic behavior of the virtual Compton helicity amplitude A related to g , l Im A (q, l) , (8.25) g (x, Q)" 2pM where l,P ) q. The "rst moment of g is given in terms of A as Q dx g (x, Q)" dl Im A (q, l ) . (8.26) 2pM \O Combining Cauchy's theorem with crossing symmetry one "nds the following dispersion relation for A : dl 2 Im A (q, l ) . (8.27) A (q, l)" l l !l p \O Burkhardt and Cottingham argue that all known Regge singularities contributing to A have an intercept a(0) less than zero. This is equivalent to the statement that A falls o! to zero stronger than 1/l as lPR (more precisely &l?\ modulo logarithms). This implies that one can take the limit lPR under the l dispersion integral and obtains
2 A (q, l)+! dl Im A (q, l ) pl \O
(8.28)
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which is compatible with the rapid Regge fall-o! only if the integral vanishes. Thus (8.26) implies the BC sum rule (8.24). The derivation raises some questions. For example, Heimann [353] has argued that there might be contributions from multi-pomeron cuts which invalidate the assumption of intercept less than zero and would imply a highly singular behavior g (x)Px\ as xP0 so that the "rst moment integral would not even converge. On the other hand, Regge cuts with branch points at a(0)50 or speci"c nonpolynomial residue J"0 "xed poles in Compton amplitudes [166,375] could invalidate the BC sum rule by terms of order at most 1/Q. It is, therefore, possible that the BC sum rule might be restored asymptotically, as QPR, because the residues of the Pomeron cuts fall o! at QPR. Apart from such an &exotic' situation, the BC sum rule (8.24) appears to be very robust and is most probably true [375,382]. Furthermore, in the framework of perturbative QCD there is no indication of a violation of the BC sum rule [51,416] so that we believe that it is probably valid at su$ciently high Q. It should be emphasized again that the light-cone OPE per se is noncommittal about the BC sum rule since the twist-3 operators with mixed symmetry in (8.12)}(8.14) need at least two (antisymmetrized) indices (n'1) which implies the validity of Eq. (8.17) only for n'1. We would need the nP1 continuation of the twist-3 matrix element d : if d is less singular L L than (n!1)\ as nP1, the BC sum rule (8.24) would hold, as one might naively expect; it would not hold if d &(n!1)\ for nP1 (but on the other hand we know that the twist-2 matrix element L a in (8.19) is not singular because the n"1 moment of g is "nite). It is clearly important to check the validity of the BC sum rule experimentally } as far as possible. Present measurements at an average Q of 3}5 GeV imply [5,8]
dx gN (x, Q)"!0.013$0.028,
dx gL (x, Q)"0.06$0.15 , (8.29) which are consistent with (8.24) but certainly not conclusive. It would nevertheless be very interesting if the BC sum rule were not true, because this would imply a nonconventional behavior of twist-3 operators (d &(n!1)\) or the importance of long-range e!ects [382,463,461,521]. L Another interesting, but less problematic sum rule concerns the valence content of g and g [239,244] dx x[g (x, Q)#2g (x, Q)] "0
(8.30)
which amounts to the vanishing of all twist-3 contributions to the second (n"2) moment of g . Apart from the fact that this sum rule can be independently derived from the OPE [122], there is also a wealth of similar sum rules between structure functions in the electroweak sector (see [121,122] and references therein) which are unfortunately beyond the experimental reach for the time being. Let us "nally come to the logarithmic scaling violations to Eq. (8.17), in particular the Q-evolution of g (x, Q) is theoretically unknown and in general an unsolved intricate problem, because the number of independent twist-3 operators increases with n [68,149,150,504]. Thus, the dimension of the anomalous dimension matrix becomes larger with increasing n and therefore there exist no Altarelli}Parisi-type evolution equations. Anomalous dimensions and coe$cient functions for g have been calculated, see [399,418] and earlier references quoted therein, but for
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principal reasons a satisfactory physical model to predict g (x, Q) does not exist. There have been attempts [44], to which we shall return below, to construct practical approximations to the Q evolution in the large-n limit (xP1) as well as in the limit N PR. Unfortunately, the very A large-x region and probably also the large N limit are experimentally not very relevant. The latter A just gives an indication that the e!ect of the Q evolution of g (x, Q) might be large throughout the whole x-region [550]. Furthermore, a parton inspired picture involves two-parton &correlation functions' C(x , x ), where two partons (with Bjorken x and x ) split from the proton at the same time and interact with the photon (see, e.g., [461,560]). However, these functions are as undetermined as the matrix elements of the higher twist operators. At the end of this section we want to discuss the operator mixing and its e!ects on the Q-evolution and the coe$cient functions for the simple case of n"3. The coe$cient functions at the tree level and the anomalous dimensions for the operators depend separately upon the choice of the independent operator basis. But the Q evolution of the moments of g does of course not depend on it. In the case n"3 there are four operators with the constraint (8.31) R "R #R #R , $ K where the Lorentz indices of operators are omitted. One may choose the operators R , R , and $ K R as independent operators and eliminate the &EOM' operator R . One then gets the following renormalization matrix for the composite operators:
R Z Z Z Z R $ $ R Z Z Z Z R " , (8.32) R 0 0 Z 0 R K K R 0 0 0 Z R 0 where the Z can be calculated in dimensional regularization and are of form Z "d # GH GH GH (1/e)(g/16p)z with D"4!2e. A straightforward but tedious calculation gives [417,418] GH z "C #N , z "!C #N , $ A $ A z "!N , z "3C !N , A $ A (8.33) z "!C #N , z "C !N , $ A $ A z "!C # N , z "N , A $ A z "6C , z "0 . $ R is a gauge non-invariant &equation of motion' operator RNI I "iS[tM c cNRI cI (iD. !m )t#tM (iD. !m )c cNRI cI t] (8.34) O O which comes into play when renormalizing the operators in (8.31). Although this operator is gauge noninvariant, it may be chosen to appear in the operator basis because it vanishes by the equation of motion. The above results in Eq. (8.34) satisfy the equalities [417] z #z "z #z , z #z "z #z #z , 2 z !z "z ! z #z . 3
(8.35)
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What happens if one chooses R , R , R and R , and eliminates R ? Using (8.31) and the K $ relations (8.35) one gets the renormalization matrix
R Z #Z Z #Z Z Z R R 0 Z 0 0 R K " K . (8.36) 0 0 Z !Z Z !Z R R 0 0 0 Z R R 0 This choice of basis was adopted, for example, in [44] in their approximate calculation of anomalous dimensions in the large N limit: A L 1 1 1 cL "4N ! ! (8.37) ,1 A j 4 2n H valid for large n, i.e. for large values of x close to 1. It should be noted that this anomalous dimension di!ers substantially from the one naively obtained by ignoring the operator mixing as was done in the early days [356,520,36,37,414,415,68]. In that approximation the evolution equation for the moments of the twist-3 contribution g to g in Eq. (8.2) reads L a (Q) A,1 @ dx xL\g (x, Q)" Q dx xL\g (x, Q ) (8.38) a (Q ) Q with b given in (4.13). This Q evolution has been quantitatively studied in [550] and used by [5] to compare bound-state model predictions, e.g. in Fig. 51, with experimental results, such as the ones in (8.21), at larger values of Q.
8.2. Transverse chiral-odd (&transversity') structure functions In analogy to the unpolarized and polarized structure functions F and g , the &transversity' structure function [499,414,150,75,206,383,384,506] is, in LO, formally given by (cf. Eq. (4.5)) 1 h (x, Q)" e[d q(x, Q)#d q (x, Q)] O 2 2 2 O where, similarly to Eqs. (4.1) and (4.2), d \ q (x, Q),\ q t(x, Q)!\ q s(x, Q) 2
(8.39)
(8.40)
describes the (anti)quark &transversity' distribution with \ q t (\ q s) being the probability of "nding a (anti)quark in a transversely polarized proton with spin parallel (antiparallel) to the proton spin. The transverse polarization of a quark entering an interaction kernel is obtained by using u(p, s)u (p, s)"!p. s. c for its spinor u(p, s) with s ) p"0. The d \ q are related to the tensor current 2 q ipIJc q which is chiral (and charge conjugation) odd, i.e. measure correltions between left- and
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right-handed quarks, q q , induced for example by nonperturbative condensates 1q q 2 in the * 0 * 0 nucleon. Thus, unlike in the familiar case of the q(x, Q) and dq(x, Q), there is no gluonic transversity density at leading twist [379,391,540]. Together, q, dq and d q provide a complete 2 description of quark momentum and spin at leading twist as can be seen generically from a spin-density matrix representation in the quark- and nucleon-helicity basis, F(x, Q)"q(x, Q)I I#dq(x, Q)p p #d q(x, Q)(p p #p p ) . 2 > \ \ >
(8.41)
Thus the d \ q (x, Q) are leading twist-2 densities and complete the twist-2 sector of nucleonic 2 parton distributions, and are therefore in principle as interesting as the familiar f (x, Q) and d f (x, Q). Unfortunately, the d \ q densities are experimentally more di$cult to access and 2 entirely unknown so far. (Although the name &transversity' is fairly universal, the notation is not. In addition to d q, notations such as hO , h , D q, D q, dq, etc., are common as well [499,206,383, 2 2 2 384,75]). It should be remembered that the common unpolarized and longitudinally polarized quark densities q and dq, respectively, considered up to now are related to the matrix elements of vector and axial}vector currents, q c q and q c c q"q c c q #q c c q , respectively, which preserve I I * I * 0 I 0 chirality, i.e. are chiral-even (q Pq , q Pq ) in contrast to the chiral-odd d q. Thus, the * * 0 0 2 transverse spin structure function g "g #g (or g ) of the previous Section 8.1, which preserves 2 chirality, must not be confused with h which #ips chirality. We have seen that for g , arising from transversally polarized nucleons, the cross section picks up a factor of M/(Q because in these processes the nucleon helicity changes but the quark chirality does not change in the hard scattering subprocess [383,384,148]. Therefore it is obviously not possible to measure the chiralodd transversity structure function h (x, Q) in usual ep inclusive DIS which, apart from small quark mass corrections, gives always rise to chiral-even transitions (cHq Pq , etc.), i.e. to F and * * g . The chiral-even transitions are illustrated in Fig. 52 where the quark lines leaving and entering the nucleon are of a single chirality. This is so because the photons and gluons participating in the hard scattering process have a vector-like interaction with the massless quark. Thus in lowest order only two independent quark-nucleon amplitudes enter the description of DIS: the sum in Fig. 53 over chirality gives the unpolarized structure function F (or F ), whereas the di!erence gives the structure function g of longitudinal polarization. On the other hand, in hadronic collisions, the chirality of the quark lines leaving and entering a given nucleon need not be the same, as illustrated in Fig. 54. This is due to nonperturbative condensates 1q q 2O0 which break the chiral symmetry of the QCD vacuum as well as of the * 0
Fig. 52. Chiral (even) structure of deep inelastic scattering.
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Fig. 53. Left- and right-handed chiral even quark distributions as measured in DIS whose sum and di!erence give F (x, Q) and g (x, Q), respectively. Fig. 54. Chiral (odd) #ip distribution which gives the &transversity' structure function h (x, Q).
nucleon structure due to bound state e!ects [383,384,148]. The corresponding new leading twist-2 structure function is referred to as h (x, Q) in (8.39). Its interpretation is obscure in the chiral basis, but is revealed by using the transverse projection operators POP "(1$c S. ) instead of 2 P "(1$c ), cf. Eq. (1.3). Here the transversely polarized nucleon and quarks are classi"ed as 0 * eigenstates of the transversely projected Pauli}Lubanski spin operator c S. (which is the more 2 JJMPN), i.e. c S. u(P , S )"$u(P , S ). In contrast familiar form proportional to = "!e 2 X 2 X 2 I IJMN to the transverse quark spin operator c c S. relevant for g in the previous Section 8.1, the 2 Pauli}Lubanski operator c S. commutes with the free quark Hamiltonian H "a p , and 2 X X therefore h (x, Q) can be interpreted in terms of the quark-parton model as done in (8.39). There are further subleading (twist-3) transversity distributions [383,384,483] such as h which is the chiral-odd analog of g . There is, furthermore, a twist-3 chiral-odd scalar distribution e(x, Q) related to the unit operator, i.e. measuring, via its "rst moment, the nucleon p-term. More explicitly, the complete set of structure functions are de"ned by the light cone Fourier transform of nucleon matrix elements of quark bilocal operators as follows:
dj e HV1PS"q (0)c q(jn)"PS2"2[f O(x, Q)p #Mf O(x, Q)n ] , I I I 2p
(8.42)
dj e HV1PS"q (0)c c q(jn)"PS2"2M[gO (x, Q)p S ) n#gO (x, Q)S I I 2I 2 2p
# MgO (x, Q)n S ) n] , I
dj e HV1PS"q (0)q(jn)"PS2"2MeO(x, Q) , 2p
(8.43) (8.44)
dj e HV1PS"q (0)ip c q(jn)"PS2"2[hO (x, Q)(S p !S p ) IJ 2I J 2J I 2p
# hO (x, Q)M(p n !p n )S ) n# hO (x, Q)M(S n !S n )] , (8.45) I J J I 2I J 2J I * which holds at some factorization scale Q, hO , hO #hO , and in order to follow more closely the * original notation [383,384] we have used f O,q, gO ,dq, hO ,d q, etc. Two light-like null-vectors 2 (p"n"0) have been introduced via the relation P "p #(M/2)n with p ) n"1, and the I I I nucleon spin vector is decomposed as S "S ) np #S ) pn #S . The twist-4 contributions f O, I I I 2I gO and hO will not be considered in the following.
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The d \ q and the longitudinal d\ q densities are not entirely independent of each other since 2 one clearly has \ q #\ q "\ q t#\ q s by rotational invariance, which implies the general > \ positivity constraints [381,384] "d \ q (x, Q)"4\ q (x, Q) 2
(8.46)
in complete analogy to (4.11). A second inequality has been derived by So!er [538], q (x, Q)#d\ q (x, Q)]"\ q (x, Q) , "d \ q (x, Q)"4[\ > 2
(8.47)
which has its origin in the positivity properties of helicity amplitudes. This latter inequality can also be derived in the context of the parton model [538,536,327]. Being a twist-2 quantity, where only fermions contribute, h (x, Q), i.e. d\ q (x, Q) obey 2 a simple nonsinglet-type Altarelli}Parisi evolution equation. In LO it is similar to (4.12) and reads [75] a (Q) d \ d q (x, Q)" Q d P d \ q 2 2 2p 2 OO dt where t"ln Q/Q and
(8.48)
2x 3 # d(1!x) (8.49) d P(x)"C 2 OO $ (1!x) 2 > with C ". Alternatively, in Mellin n-moment space this RG evolution equation becomes similar $ to (4.24) and reads d \ a (Q) d q L(Q)" Q d PL d \ q L(Q) , 2 dt 2p 2 OO 2
(8.50)
where the nth moment is de"ned by Eq. (4.22) and the nth moment of (8.49) is given by (8.51) d PL"C [!2S (n)] 2 OO $ with S (n) being de"ned right after Eq. (4.26). The solution of (8.50) is straightforward and is formally identical to the one in (4.27). Recently, the calculation of the NLO two-loop splitting functions d P (x) has been completed in the MS factorization scheme [427,351,570] for the 2 OO! #avor combinations d q ,d q$d q . The LO evolutions of the twist-3 distributions e(x, Q) and 2 ! 2 2 h (x, Q) in (8.44) and (8.45) have been studied as well recently [421,422,86,107]. * Since the corresponding evolution kernels are entirely di!erent for the \ q (x, Q), d\ q (x, Q) and d \ q (x, Q) densities, the question immediately arises whether the So!er inequality (8.47) is 2 maintained when QCD is applied [327,406,90]. It turns out, however, that this inequality is preserved by LO [90] and NLO [570,470] QCD evolutions at any Q'Q provided it is valid at the input scale Q"Q .
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As stated above, the transversity distributions are experimentally entirely unknown so far. Being chiral odd, they cannot be directly determined from the common fully inclusive DIS process, i.e. h (x, Q) is not directly measurable despite its formal de"nition in (8.39). One would need, for example, to measure a (single) transverse asymmetry in single transversely polarized semi-inclusive DIS eptsPehtX with h"K, jet, 2 [74,75,195}197,394]. This requires, however, a (di$cult) measurement of the transverse polarization of the "nal state ht. Old ideas [485,241] have been revived [238,195}197,394] for determining the polarization of an outgoing quark (or gluon), in particular chiral-odd fragmentation functions, via the hadron distribution in the jet, for example. Perhaps a more feasible possibility has been suggested recently [385,378] to extract d q from DIS 2 two-meson production, e.g. eptsPep>p\X via a Collins-angle-like distribution [195,196] by measuring the observable p>;p\ ) S , i.e. the correlation of the normal to the plane formed by the 2 three-momenta p! with the nucleon's transverse spin. This, however, requires the cross section to be held fully di!erential to avoid averaging the meson}meson "nal state interaction phase to zero. It is conceivable that HERMES and (in the not too distant future) COMPASS can perform such measurements. A more natural way to search for transversity densities is in doubly transversely polarized hadron}hadron initiated reactions like ptptsPk>k\, c, jj, cc ,2X [499,75,383,384,148,206, 390,573,200,405,406,570,386,469,470,91,513]. Here the chirality changing densities (cf. Fig. 54) appear automatically in the initial states without extra suppressions as illustrated for Drell}Yan dilepton production in Fig. 55. The expected double transverse asymmetries A turn out, 22 however, to be prohibitively small, A ;A (typically, smaller by about an order of magnitude), 22 ** i.e. much smaller than the doubly longitudinally polarized asymmetries (inlcuding the DIS A and A ) considered thus far. Single transverse asymmetries A measured in reactions like pptsPjtX 2 might be sizeable (about 10%) [552], but require again a delicate measurement of the polarizations of the outgoing quarks and gluon via the hadron distribution in the "nal jet jt. Such experiments could be performed at HERA-No (phase I) and RHIC. Whereas these purely transverse asymmetries A and A measure solely d \ q , a mixed longitudinal-transverse asymmetry A , down by 22 2 2 *2 a factor M/Q, gives access to h (x, Q),h #h provided g and g "g #g are known 2 * [383,384,390]. It should be noted that most of these processes have been analyzed in LO-QCD, with the exception of transversely polarized Drell}Yan dimuon production which has been already extended to the NLO within di!erent factorization schemes [573,200,405,570,470].
Fig. 55. Chiral structure of hadronic Drell}Yan production of lepton pairs with invariant mass Q,q'0 produced by the virtual photons. Chirality #ip (R¸) occurs without suppression due to the two initial hadrons (nonperturbative bound states) involved in the process.
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Due to the lack of any experimental information, all these studies and expectations for transverse asymmetries are based on theoretical model estimates for d \ q and the other subleading transver2 sity densities. In the nonrelativistic quark model, d q(x, Q)"dq(x, Q) due to rotational invari2 ance, whereas in the bag model [383,384,550,528] d q(x, Q)9dq(x, Q) in the medium to large 2 x-region } both expectations hold presumably at some nonperturbative bound-state-like scale Q"O(K). Somewhat smaller results, d q(x, Q):dq(x, Q), are obtained by a chiral chromodi2 electric con"nement model [92] and by the chiral soliton approach [491] at QK0.3 GeV, and with QCD sum rules [372] at Q91 GeV. It seems that transversity densities are sizeable and not too di!erent from the (longitudinal) helicity distributions. Anyway, d qOdq (h Og ) directly 2 probes relativistic e!ects in the wave function. In particular, h could develop a very di!erent small-x behavior as compared to g [92,359,411,528]. There have also been attempts [384,352,538,524,92] to estimate the nucleon's &tensor charge', i.e. the total transversity D q(Q) carried by quarks, 2 [d q(x, Q)!d q (x, Q)] dx,* q(Q) , (8.52) 2 2 2 which is a #avor nonsinglet valence quantity (quarks minus antiquarks, since the tensor current is C-odd) and measures a simple local operator analogous to the axial charge (cf. Eq. (5.33))
1PS"q ip c q"PS2"(1/M)(S P !S P )* q . (8.53) IJ I J J I 2 Unlike its vector and (NS) axial}vector equivalents, the tensor charge is not conserved in QCD, so it has an anomalous dimension at one loop. To understand the physical meaning of the tensor charge it is helpful to make a comparison between the chirally odd and even matrix elements in the rest frame of the nucleon, PI"(M, 0) and SI"(0, S) 1PS"q ip c q"PS2"1PS"q R q"PS2"S * q , (8.54) G G G 2 1PS"q c c q"PS2"1PS"q c R q"PS2"S *(q#q ) , (8.55) G G G i.e. the spin operator related to axial current di!ers by an additional c from the expectation value of the Pauli}Lubanski &transversity' operator. The above estimates imply [352,524,92] * R(Q),* (u#d#s)"0.6 to 1 at Q of about 5}10 GeV which compares well with a recent 2 2 lattice calculation [69], * RK0.6$0.1. These results are intriguing since they seem to imply that 2 the tensor charge behaves more like the &naive' quark model expectation (5.16), * R"*R13"1, 2 in contrast to the experimental result *R(Q)K0.2 (cf. Table 3, for example) that quarks carry much less of the nucleon's spin than naively expected. Furthermore, one of the outstanding puzzles is how to obtain an independent measure of * q(Q) and thereby formulate a &transversity sum 2 rule' analogous to those that have been so helpful in the study of *(q#q ) in connection with the spin of the proton. Keeping in mind that we need the transversity densities d \ q (x, Q), besides the more common 2 f (x, Q) and df (x, Q), for a complete understanding of the leading twist-2 sector of the nucleon's parton structure, we face the curious situation of having reached a remarkably advanced theoretical sophistication without having any experimental &transversity' measurements whatsoever!
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Note added. Very recently the calculation of the NLO contributions to polarized photoproduction of heavy quarks, co po Pcc X, as discussed in Section 6.2, has been completed [130,131]. As expected, among other things, the dependence of *pA in (6.31) on k is considerably reduced in AN $ NLO.
Acknowledgements One of us (E.R.) expresses his warmest thanks to M. GluK ck for numerous discussions and for a fruitful collaboration on various topics presented here during the past decade. He also thanks M. Stratmann and W. Vogelsang for their collaboration during the past years and for their help in preparing some of the "gures presented. We are also grateful to G. Altarelli for some helpful and clarifying comments and discussions. Furthermore, B.L. would like to thank J. Bartels, H. Fritzsch, S. Forte, P. Gambino, M. GoK ckeler, Y. Koike, J. Kodaira, G. Piller, G. Ridol", P. Ratcli!e, M.G. Ryskin, A. Schaefer, J. So!er and E. Ste!ens for comments and discussions in connection with this work. Finally we thank Mrs. H. Heininger, Mrs. R. Jurgeleit and Mrs. S. Laurent for their help in typing the manuscript and in preparing some of the "gures. This work has been supported in part by the &Bundesministerium fuK r Bildung, Wissenschaft, Forschung und Technologie', Bonn, as well as by the &Deutsche Forschungsgemeinschaft'.
Appendix. Two-loop splitting functions and anomalous dimensions In addition to the well-known LO (one-loop) polarized splitting functions [54] which are given in Eqs. (4.16) and (4.21), with their nth moment, as de"ned in (4.22), given in (4.26). For completeness we list here the results for the NLO (two-loop) splitting functions dP as recently calculated in GH [478,568,569]. As discussed in the text (Section 4, etc.), these calculations were done in the MS scheme. To deal with c and the e-tensor, the HVBM scheme [due to its not fully anticommuting c , additional "nite renormalizations are required in order to guarantee the conservation of the #avor nonsinglet axial vector current, Eq. (4.93)] or equivalently the reading point method has been used. The #avor nonsinglet splitting functions in (4.49) are the same as for the unpolarized case [212] dP (x)"P (x)"P(x)$P (x) ,1! ,1! OO OO "C [!(2 ln x ln(1!x)# ln x)dp (x)!(#x)ln x OO $ ! (1#x) ln x!5(1!x)#(!p#6f(3))d(1!x)] # C C [( ln x# ln x#!p)dp (x)#(1#x) ln x# (1!x) OO $ #(#p!3f(3))d(1!x)] # C ¹ [!( ln x#)dp (x)!(1!x)!(#p)d(1!x)] OO $ D $(C !C C )[2(1#x) ln x#4(1!x)#2S (x)dp (!x)] . OO $ $
(A.1)
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Note that the &$' meaning of dP has been interchanged in [568,569]. The #avor-singlet ,1! splitting functions in (4.53) are given by [478,568,569] (see footnotes 3 and 5) dP(x)"dP (x)#dP (x) OO ,1\ .1 OO
(A.2)
dP (x)"2C ¹ [1!x!(1!3x) ln x!(1#x) ln x] , .1 OO $ D
(A.3)
with
and 2fdP(x)"C ¹ [!22#27x!9 ln x#8(1!x) ln (1!x) OE $ D # dp (x)(2 ln (1!x)!4 ln(1!x) ln x#ln x!p)] OE # C ¹ [2(12!11x)!8(1!x) ln(1!x)#2(1#8x) ln x D !2(ln (1!x)!p)dp (x)!(2S (x)!3 lnx)dp (!x)] , OE OE
(A.4)
dP(x)"C ¹ [!(x#4)!dp (x) ln(1!x)] EO EO $ D # C [!!(4!x) ln x!dp (!x) ln(1!x) EO $ #(!4!ln (1!x)# ln x)dp (x)] EO # C C [(4!13x) ln x#(10#x) ln(1!x)#(41#35x) $ #(!2S (x)#3 lnx)dp (!x) EO #(ln (1!x)!2 ln(1!x) ln x!p)dp (x)] , EO
(A.5)
dP(x)"!C ¹ [10(1!x)#2(5!x) ln x#2(1#x) ln x#d(1!x)] EE $ D ! C ¹ [4(1!x)#(1#x) ln x#dp (x)#d(1!x)] D EE #C [(29!67x) ln x!(1!x)#4(1#x) lnx!2S (x)dp (!x) EE #(!4 ln(1!x) ln x#ln x! p)dp (x)#(3f(3)#)d(1!x)] , EE
(A.6)
where C "4/3, C "3, ¹ "f ¹ "f/2, f(3)+1.202057 and $ D 0 2 dp (x)" !x!1 , OO (1!x) > dp (x)"2x!1, dp (x)"2!x , OE EO 1 dp (x)" !2x#1 , EE (1!x) >
(A.7)
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and the &#' description has obviously to be omitted for dp (!x). Furthermore, GG >V dz 1!z S (x)" ln . (A.8) z z V >V For relating the above results to those of [478] the following expression for S (x) is needed: (A.9) S (x)"!2Li (!x)!2 ln x ln(1#x)# ln x!p , where Li (x) is the usual Dilogarithm [225]. The coe$cient functions relevant for a consistent NLO(MS) analysis of g (x, Q) in (4.47) are given by Eqs. (4.41) and (4.42). It should be recalled that convolutions involving the ( ) distributions can be conveniently calculated numerically with the > help of Eq. (4.18). The nth moments of these splitting functions, de"ned in (4.22), which are needed for the evolution equations (4.54) and (4.55) in Mellin-moment space, are as follows. The moments of the LO splitting functions, dPL, are given in (4.26). The moments of the dP (x) in (A.1), being the same GH ,1! as for the unpolarized case [212,267] (see footnote 4), are
2n#1 1 S (n)#2 2S (n)! !dPL "C 2 ,1! $ n(n#1) n(n#1)
n S (n)!S 2
n 3n#n!1 3 2n#2n#1 # 3S (n)#8SI (n)!S ! ! G2 2 n(n#1) 8 n(n#1)
67 1 #C C S (n)! 2S (n)! $ 9 n(n#1)
n 2S (n)!S 2
n 11 1 ! S (n)!4SI (n)# S 3 2 2 !
1 151n#236n#88n#3n#18 17 2n#2n#1 ! $ 18 24 n(n#1) n(n#1)
20 4 2 11n#5n!3 1 # C ¹ ! S (n)# S (n)# # . $ D 9 3 9 n(n#1) 6
(A.10)
The moments of the #avor singlet splitting functions in (A.2)}(A.6) are given by [478,309] (see footnote 5) with
dPL"dPL #dPL OO ,1\ .1 OO
!dPL "2C ¹ .1 OO $ D
(A.11)
n#2n#2n#5n#2 n(n#1)
(A.12)
and
n!1 n!1 (S (n)!S (n))#4 S (n) !2fdPL"C ¹ 2 OE $ D n(n#1) n(n#1) !
5n#5n!10n!n#3n!2 n(n#1)
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n!1 n 4 # 2C ¹ !S (n)#S #S (n) ! S (n) D n(n#1) 2 n(n#1) !
n#n!4n#3n!7n!2 n(n#1)
,
n#2 5n#12n#4 !dPL"4C ¹ ! S (n)# EO $ D 3n(n#1) 9n(n#1)
(A.13)
n#2 3n#7n#2 # C (S (n)#S (n))! S (n) $ n(n#1) n(n#1) #
9n#30n#24n!7n!16n!4 2n(n#1)
n#2 n 11n#22n#12 #C C !S (n)#S !S (n) # S (n) $ n(n#1) 2 3n(n#1) !
76n#271n#254n#41n#72n#36 , 9n(n#1)
!dPL"C ¹ EE $ D
(A.14)
n#3n#5n#n!8n#2n#4 n(n#1)
5 3n#6n#16n#13n!3 # 4C ¹ ! S (n)# D 9 9n(n#1)
n n n 1 4 # C ! S !2S (n)S #4SI (n)# S 2 2 2 n(n#1) 2 #
67n#134n#67n#144n#72 S (n) 9n(n#1)
!
48n#144n#469n#698n#7n#258n#144 , 18n(n#1)
(A.15)
where L 1 S (n), , I jI H n n n!1 L 1#(!)H 1 1 S ,2I\ " (1#g)S # (1!g)S , I 2 I I jI 2 2 2 2 H S (n) p Li (x) L (!)H 5 SI (n), S ( j)"! f(3)#g # G(n)# dx xL\ j n 1#x 8 12 H
(A.16)
(A.17)
(A.18)
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with G(n),t((n#1)/2)!t(n/2), t(z)"d ln C(z)/dz and g"$1 for dPL and g"!1 for the ,1! #avor singlet anomalous dimensions. The analytic continuations in n, required for the Mellin inversion of these sums to Bjorken-x space [cf. Eq. (4.37)], are well known [314,316] (see also [581]). The conventional anomalous dimensions dcL are related to the dPL via dcL" GH GH GH !8dPL (cf. footnotes 1 and 5). The moments of the relevant MS coe$cient functions for gL (Q) GH are given in Eqs. (4.63) and (4.64). Finally, we list for completeness the "rst moments *P,dPL" dx dP(x) and the ones of the coe$cient functions *C ,dCL: GH GH GH G G *P "0, *P "(C !C C )(!p#4f(3)) , ,1\ ,1> $ $ *P"!3C ¹ , *P"0 OO $ D OE (A.19) *P"!C #C C !C ¹ EO $ $ $ D b *P"C !C ¹ !C ¹ , EE $ D D 4 and *C "!C , O $
*C "0 . E
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THE PHOTON STRUCTURE FROM DEEP INELASTIC ELECTRON}PHOTON SCATTERING
Richard NISIUS CERN, CH-1211 Gene` ve 23, Switzerland
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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The photon structure from deep inelastic electron}photon scattering Richard Nisius CERN, CH-1211 Gene% ve 23, Switzerland Received October 1999; editor: J.V. Allaby Contents 1. Introduction 1.1. Theoretical description of photon interactions 2. Deep inelastic electron}photon scattering (DIS) 2.1. Kinematics 2.2. Experimental considerations 3. Theoretical framework 3.1. Individual cross-sections 3.2. Equivalent photon approximation 3.3. QED structure functions 3.4. Hadronic structure function FA 3.5. Vector meson dominance and the hadronlike part of FA 3.6. Alternative predictions for FA 4. Parton distribution functions 5. Tools to extract the structure functions 5.1. Event generators 5.2. Unfolding methods 6. Measurements of the QED structure of the photon 7. Measurements of the hadronic structure of the photon
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7.1. Description of the hadronic "nal state 7.2. Hadronic structure function FA 7.3. Hadronic structure of virtual photons 8. Future of structure function measurements 8.1. LEP2 programme 8.2. A future linear collider 9. Probing the structure of the photon apart from DIS 9.1. Photon}photon scattering at e>e\ colliders 9.2. Photon structure from HERA Acknowledgements Appendix A. Connecting the cross-section and the structure function picture Appendix B. General concepts for deriving the parton distribution functions Appendix C. Collection of results on the QED structure of the photon Appendix D. Collection of results on the hadronic structure of the photon References
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Abstract The present knowledge of the structure of the photon is presented with emphasis on measurements of the photon structure obtained from deep inelastic electron}photon scattering at e>e\ colliders. This review covers the leptonic and hadronic structure of quasi-real and also of highly virtual photons, based on
E-mail address: [email protected] (R. Nisius) 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 1 5 - 5
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measurements of structure functions and di!erential cross-sections. Future prospects of the investigation of the photon structure in view of the ongoing LEP2 programme and of a possible linear collider are addressed. The most relevant results in the context of measurements of the photon structure from photon}photon scattering at LEP and from photon}proton and electron}proton scattering at HERA are summarised. 2000 Elsevier Science B.V. All rights reserved. PACS: 13.60.!r; 12.20.Fv; 12.38.Qk Keywords: Photon; Deep inelastic scattering; QED; QCD
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1. Introduction The photon is a fundamental ingredient of our present understanding of the interactions of quarks and leptons. These interactions are successfully described in the framework of the standard model, a theory which consists of a combination of gauge theories. Being the gauge boson of the theory of quantum electro dynamics (QED) the photon mediates the electromagnetic force between charged objects. In these interactions the photon can be regarded as a structureless object, called the direct, or the bare photon. Since QED is an abelian gauge theory, the photon has no self-couplings and to the best of our knowledge the photon is a massless particle. Due to the Heisenberg uncertainty principle, written as *E*t'1, the photon, denoted with c, is allowed to violate the rule of conservation of energy by an amount of energy *E for a short period of time *t and to #uctuate into a charged fermion anti-fermion, ffM , system carrying the same quantum numbers as the photon, cPffM Pc. If, during such a #uctuation, one of the fermions interacts via a gauge boson with another object, then the parton content of the photon is resolved and the photon reveals its structure. In such interactions the photon can be regarded as an extended object consisting of charged fermions and also gluons, the so-called resolved photon. This possibility for the photon to interact either directly or in a resolved manner is another dual nature of the photon, which is the cause of a variety of phenomena and makes the photon a very interesting object to investigate. One possible description of the structure of the photon is given by the concept of photon structure functions, which is the main subject of this review. Today the main results on the structure of the photon are obtained from the electron}positron collider LEP and the electron}proton collider HERA. The largest part of this review is devoted to the discussion of deep inelastic electron}photon scattering and to the measurements of QED and hadronic structure functions of the photon. Other LEP results on the structure of the photon apart from those obtained from deep inelastic electron}photon scattering, as well as the measurements in photoproduction and deep inelastic electron}proton scattering at HERA, are summarised brie#y. The review is organised in the following way. In Section 2.1 the kinematical quantities are introduced and in Section 2.2 the capabilities of the detectors to measure the deep inelastic electron}photon scattering process are discussed. The theoretical formalism needed to measure the photon structure is outlined in Section 3, with special emphasis on the QED and hadronic structure functions of the photon in Sections 3.3 and 3.4, respectively. A review of the available parametrisations of parton distribution functions of the photon is given in Section 4. The most important tools used to measure the photon structure are described in Section 5, concentrating on event generators and unfolding methods. The measurements of the QED and hadronic structure of the photon obtained from leptonic and hadronic "nal states are discussed in Sections 6 and 7, respectively. The prospects of future determinations of the structure of the photon are outlined. The measurements expected to be performed at LEP, using the high statistics, high-energy data still expected within the ongoing LEP2 programme, are discussed in Section 8.1, followed by the discussion of measurements to be performed at a possible future linear collider in Section 8.2.
The units used are c" "1. Fermions and anti-fermions are not distinguished, for example, electrons and positrons are referred to as electrons.
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Fig. 1. Probing the structure of quasi-real photons, c, by highly virtual photons, c夹.
Fig. 2. The di!erent appearances of the photon. Shown are (a) the direct or bare photon, and (b, c) the resolved photon, which can be either point-like, (b), or hadron-like, (c).
Complementary investigations of the photon structure from LEP and selected HERA results are addressed in Section 9. Additional information is presented in the appendices. The relation between the cross-section picture and the structure function picture is outlined in Appendix A, followed by a discussion of the general relation between the photon structure function, the parton distribution functions and the evolution equations given in Appendix B. Appendices C and D contain a collection of numerical results on measurements of the photon structure. 1.1. Theoretical description of photon interactions In deep inelastic electron}photon scattering the structure of a quasi-real photon, c, is probed by a highly virtual photon, c夹, emitted by a deeply inelastically scattered electron, as sketched in Fig. 1. The photon, as the mediator of the electromagnetic force, couples to charged objects. The fundamental coupling of the photon as described in the framework of QED is the coupling to charged fermions, f, which can be either quarks, q, or leptons, l, with l"ekq. For the case of lepton pair production, the process can be calculated in QED. The relevant formulae are listed in Section 3 and the results on the QED structure of the photon are discussed in Section 6. For the production of quark pairs the situation is more complex, since the spectrum of #uctuations is richer, and QCD corrections have to be taken into account. Therefore, the photon interactions receive several contributions shown in Fig. 2. The leading-order contributions are discussed in detail. The reactions of the photon are usually classi"ed depending on the object which takes part in the hard interaction. If the photon directly, as a whole, takes part in the hard interaction, as shown in Fig. 2(a), then it does not reveal a structure. These reactions are called
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direct interactions and the photon is named the direct, or the bare photon. If the photon "rst #uctuates into a hadronic state which subsequently interacts, the processes are called resolved photon processes and structure functions of the photon can be de"ned. The resolved photon processes are further subdivided into two parts. The "rst part, shown in Fig. 2(b), is perturbatively calculable, as explained in Section 3.4, and called the contribution of the point-like, or the anomalous photon. Here the photon perturbatively splits into a quark pair of a certain relative transverse momentum and subsequently one of the quarks takes part in the hard interaction, which for deep inelastic electron}photon scattering in leading order is the process c夹qPq. The second part, where the photon #uctuates into a hadronic state with the same quantum numbers as the photon, as shown in Fig. 2(c), is usually called the hadron-like, or hadronic contribution. The photon behaves like a hadron, and the hadron-like part of the hadronic photon structure function FA can successfully be described by the vector meson dominance model (VMD) considering the low mass vector mesons o, u and , as outlined in Section 3.5. The leading-order contributions are subject to QCD corrections due to the coupling of quarks to gluons. The hadronic photon structure function FA receives contributions both from the point-like part and from the hadron-like part of the photon structure, discussed in detail in Section 3.4.
2. Deep inelastic electron}photon scattering (DIS) The classical way to investigate the structure of the photon at e>e\ colliders is the measurement of photon structure functions using the process eePeec夹cPeeX .
(1)
In this section the kinematical variables used to describe the reaction are introduced in Section 2.1 and experimental aspects are discussed in Section 2.2. 2.1. Kinematics Fig. 3 shows a diagram of the scattering of two electrons, proceeding via the exchange of two photons of arbitrary virtualities, in the case of leading-order fermion pair production, X"!. The reaction is described in the following notation: e(p )e(p )Pe(p )e(p )c夹(q)c夹(p)!e(p )e(p )f(p )f(p ) . (2) The terms in brackets denote the four vectors of the respective particles, and E is the energy of the electrons of the beams. In addition the energies, momentum vectors and polar scattering angles of the outgoing particles are introduced in Fig. 3. The symbol (夹) indicates that the photons can be either quasi-real, c, or virtual, c夹. The virtual photons have negative virtualities q, p40. For simplicity, the de"nitions Q"!q50 and P"!p50 are used, and the particles are
In this review the two parts of the resolved photon will be called point-like and hadron-like to avoid confusion with the term hadronic structure function of the photon which is used for the full FA .
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Fig. 3. A diagram of the reaction eePee!, proceeding via the exchange of two photons.
ordered such that Q'P. A list of commonly used variables, which are valid for arbitrary virtualities and for any "nal state X, is given below: s ,(p #p )"2p ) p "4E , s ,(p #p) , A 1!x s ,=,(q#p)"Q !P , AA x Q Q " , x, 2p ) q =#Q#P
(3) (4) (5) (6)
y,p ) q/p ) p , (7) r,p ) q/p ) q , (8) Q"xy(s #P)"2EE (1!cos h ) , (9) A P"2EE (1!cos h ) . (10) Here s is the invariant mass squared of the electron}electron system, s the invariant mass A squared of the electron}photon system, s the invariant mass squared of the photon}photon AA system, and the mass of the electron has been neglected. Deep inelastic electron}photon scattering is characterised in the limit where one photon is highly virtual, Q<0, and the other is quasi-real, P+0. In this case, P is neglected in the equations above and some simpli"ed expressions are found: s "(p #p)"2p ) p"4EE , (11) A A y"1!(E /2E)(1#cos h ) , (12) r"E /E,z . (13) A Here E is the energy of the quasi-real photon. The reaction receives contributions in leading order A from the di!erent Feynman diagrams shown in Fig. 4. The relative sizes of the contributions of the
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Fig. 4. The di!erent contributions to the reactions eePeec夹c夹Pee!. Shown are (a) the multiperipheral diagram, (b) the t-channel bremsstrahlung diagram and (c) the s-channel bremsstrahlung diagram. In all cases only one possible leading-order diagram is shown.
di!erent Feynman diagrams depend on the kinematical situation. In the region of deep inelastic scattering, Q
M is de"ned as the angle between the scattering planes of the two electrons in the photon}photon centre-of-mass frame, as shown in Fig. 5(a). The polar angle h夹 is de"ned as the scattering angle of the fermion or anti-fermion with respect to the photon}photon axis in the photon}photon centre-of-mass frame, as shown in Fig. 5(b). In this report, the azimuthal angle s is de"ned, as in Ref. [2], as the angle between the observed electron and the fermion which, in the photon}photon centre-of-mass frame, is scattered at cos h夹(0, as shown in Fig. 5(b). There exist slightly di!erent de"nitions of s in the literature. The di!erent de"nitions are due to the di!erent choices made to accommodate the fact that the unintegrated structure function FI A is antisymmetric in cos h夹 if the
The contributions to the s-channel bremsstrahlung diagrams are sometimes called the annihilation and the conversion diagram, reserving the term bremsstrahlung only for the t-channel bremsstrahlung diagram.
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Fig. 5. Illustrations of the scattering angles M , h夹 and s in the photon}photon centre-of-mass system. Shown are (a) the angle between the scattering planes of the two scattered electrons, M , and (b) the scattering angle h夹 of the fermion or anti-fermion with respect to the photon}photon axis, as well as the azimuthal angle s, de"ned as the angle between the observed electron and the fermion which, in the photon}photon centre-of-mass frame, is scattered at cos h夹(0.
angle s is chosen to be the angle between the electron and the fermion or anti-fermion. There are several ways to rede"ne s in such a way that the integration of FI A with respect to cos h夹 does not vanish, see Section 3.3 for details. 2.2. Experimental considerations The measurement of structure functions involves the determination of x, Q, P and s. The capabilities of the di!erent LEP detectors are very similar and they have only slightly di!erent acceptances for the scattered electrons and the "nal state X. As an example, the acceptance of the OPAL detector, shown in Fig. 6, is discussed. The scattered electrons are detected by various electromagnetic calorimeters. The "nal state X is measured with tracking devices and calorimeters which are sensitive to electromagnetic as well as hadronic energy deposits, supplemented by muon detectors. The acceptance ranges of the various components of the OPAL detector are listed in Table 1. For two values of the energy of the beam electrons E"45.6 and 100 GeV the covered phase space in terms of x and Q, for P"0 is schematically shown in Fig. 7. The values of x and Q are obtained from the kinematical relations listed above, using a range in photon}photon invariant mass of 2.5(=(40/60 GeV, for E"45.6/100 GeV and assuming that the observed electrons carry at least 50% of the energy of the beam electrons. The kinematical coverage in principle extends from 10\ to about 1 in x and from 10\ to 3;10 GeV in Q, but measurements of the photon structure cover only the approximate ranges of 10\(x(1 and 1(Q(10 GeV. This is because at large Q the statistics are small, and at very low Q the background conditions are severe. Therefore the present measurements of the photon structure are limited to h '33 mrad, which means Q'Q +1.1/5.5 GeV, for E"45.6/100 GeV, as shown in Fig. 7.
Here Q , calculated from Eq. (9) for E "0.5E and h "33 mrad, is the minimum photon
virtuality at which an electron can be observed. The considerations for the Q acceptance also apply to the acceptance in P for the second photon. Due to the limited coverage of the detector close to the beam direction the scattered electrons radiating the quasi-real photons cannot be detected up to h "33 mrad with the
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Fig. 6. A schematic view of the OPAL detector.
Table 1 The main parameters of the OPAL detector relevant for measurements of the photon structure. Shown are the acceptance ranges in polar angle h for the scattered electrons and the "nal state X. The number 33 mrad re#ects the electron acceptance for beam energies within the LEP2 programme, at LEP1 energies the clean acceptance already started at approximately 27 mrad Scattered electrons Electromagnetic cluster
4}8, 33}55, 60}120, '200 (mrad) Final state X
Charged particles Electromagnetic cluster Hadronic cluster Muons
"cos h"(0.96 "cos h"(0.98 "cos h"(0.99 "cos h"(0.98
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Fig. 7. The kinematical coverage of the OPAL detector. Shown are the accepted ranges in x and Q, for P"0 and for two values of the energy of the beam electrons, 45.6 and 100 GeV. The numbers are obtained for speci"c ranges in = and a minimum energy required for the scattered electron, explained in the text.
exception of a small region of 4}8 mrad. Consequently, for deep inelastic electron}photon scattering the experiments e!ectively integrate over the invisible part of the P range up to a value P .
Because P depends on energy and angle of the electron, P is not a "xed number, but depends on
the minimum angle and energy required to observe an electron. Approximations of P are
Q and the P value corresponding to an electron carrying the energy of the beam electrons and
escaping at h , which is two times Q in the example from above.
As a consequence of the limited acceptance the measured structure functions depend on the P distribution of the not observed quasi-real photon, and this dependence increases for increasing energy of the beam electrons.
3. Theoretical framework In this section the formalism needed for the interpretation of the measurements performed by the various experiments is outlined. The discussion is not complete, but focuses on general considerations and on the formulae relevant for the understanding of the experimental results. These results concern measurements of structure functions and of di!erential cross-sections, related to the QED and hadronic structure of quasi-real and virtual photons. The structure of quasi-real photons is investigated for Q
In principle, one has to specify the number of #avours to which K corresponds, and in next-to-leading order also the factorisation scheme in which K is expressed. For example, K+1 means four active #avours and the MS factorisation scheme, see Ref. [3] for details. However, when constructing parton distribution functions of the photon, in some cases K is taken as a "xed number independent of the number of #avours used, because, given the number of free parameters, there is no sensitivity to K. For simplicity, unless explicitly stated otherwise, here K is used either to denote a "xed number or as a shorthand for K . Numerically, in leading order, K "0.2 GeV corresponds to K "0.232 GeV.
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3.1. Individual cross-sections The general form of the di!erential cross-section for the scattering of two electrons via the exchange of two photons, Eq. (2), using the multipheripheral diagram integrated over all angles except M is given as
a (p ) q)!QP dp dp dp"dp(eePeeX)" E E 16pQP (p ) p )!mm ;(4o>>o>>p #2"o>\o>\"q cos 2 M #2o>>op 22 22 2* #2oo>>p #oop !8"o>o>"q cos M ) , (14) *2 ** 2* taken from Ref. [4, Eq. 5.12]. The four vectors and kinematic variables are de"ned in Section 2.1. The total cross-sections p , p , p and p and the interference terms q and q correspond to 22 2* *2 ** 22 2* speci"c helicity states of the photons (T"transverse and L"longitudinal). Since a real photon can have only transverse polarisation, the terms where at least one photon has longitudinal polarisation have to vanish in the corresponding limit QP0 or PP0. These terms have the following functional form: p JQ, p JP, p JQP and q J(QP. The terms oHI *2 2* ** 2* and oHI, where j, k3(#,!,0) denote the photon helicities, are elements of the photon density matrix which depend only on the four vectors q, p, p and p and on m . They are taken from Ref. [4, Eq. 5.13] and have the following form: m (2p ) p!p ) q) #1!4 , 2o>>" Q (p ) q)!QP m o"2o>>!2#4 , Q
(2p ) q!p ) q) m 2o>>" #1!4 , (p ) q)!QP P
m o"2o>>!2#4 , P
(15) "o>\""o>>!1 , "o>""((o#1)"o>\" . G G G G G Experimentally two kinematical limits are studied for leptonic and hadronic "nal states. Firstly the situation where both photons are highly virtual and secondly the situation where one photon is quasi-real and the other highly virtual: the situation of deep inelastic electron}photon scattering. The corresponding limits of Eq. (14) are discussed next. If both photons are highly virtual the di!erential cross-section reduces to a much more compact form, because Eq. (14) can be evaluated in the limit Q, P<m. In this limit the following relations can be obtained between the oHI given in Eq. (15): G o o"2(o>>!1) , "o>\"" G , G G G 2
o#1 "o>\" 2(o!o/2#o>>) o "o>" G G " G G G G G " o>> o>> 2o>> 2o>> o>> G G G G G o o G #1 G " . 2o>> 2o>> G G
(16)
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177
De"ning o/2o>>,e Eq. (14) reads G G G
a (p ) q)!QP dp dp 4o>>o>> dp" E E 16pQP (p ) p )!mm ;(p #e p #e p #e e p #e e q cos 2 M 22 2* *2 ** 22 !(2(e #1)e (2(e #1)e q cos M ) . (17) 2* Finally for e +1, which is ful"lled by selecting events at low values of y and r, the di!erential G cross-section can be written as
a (p ) q)!QP dp dp 4o>>o>> dp" E E 16pQP ( p ) p )!mm ;(p #p #p #p #q cos 2 M !4q cos M ) . 2* 22 2* *2 ** 22
(18)
This equation can be used to de"ne an e!ective structure function FA Jp #p #p # 22 2* *2 p #q cos 2 M !4q cos M . This e!ective structure function can be measured by experiments. 2* ** 22 However, to relate FA to the structure functions FA and FA discussed below, further assumptions * are needed. By assuming that the interference terms do not contribute, that p is negligible and ** also using p "p , the e!ective structure function can be expressed by means of Eq. (20), as 2* *2 FA "FA #3/2FA . * If the interference terms q and q are independent of M , the integration over M of the terms 22 2* containing cos M and cos 2 M vanishes, and the cross-section is proportional to p #p # 22 2* p #p . The total cross-sections and interference terms can be expressed using Q, P, =, and *2 ** the mass of the produced fermion, m . However, there is a kinematical correlation between these variables and M , which leads to the fact that in several kinematical regions q and q are not 22 2* independent of M . Consequently, the terms proportional to cos M and cos 2 M do not vanish, even when integrated over the full range in M , as explained in Ref. [5]. The resulting contributions can be very large, depending on the ratios Q/P, Q/= and P/=. Due to the large interference terms in some regions of phase space, cancellations occur in Eq. (18) between the cross-section and interference terms, and therefore no clear relation between a structure function and the crosssection terms can be found. In this situation the cleanest experimentally accessible measurement is the di!erential cross-section as de"ned by Eq. (18). For the case of leading-order QED fermion pair production the relevance of the individual terms for di!erent kinematical regions can be studied. For example, Fig. 8 shows the di!erential cross-section dp/dx for muon pair production in the kinematical acceptance range of the PLUTO experiment [6], and for two di!erent lower limits on =. The kinematical requirements are, E , E '0.35E for E"17.3 GeV, 100(h (250 mrad, 31(h (55 mrad, and in addition ='2m in Fig. 8(a), and ='20m in Fig. 8(b). This leads to average values of P and Q of I I 1P2+0.44 GeV and 1Q2+5.3 GeV. The individual contributions are listed in Table 2. Shown in Fig. 8 are the di!erential cross sections dp/dx for three di!erent scenarios: dp/dx using all terms of Eq. (14), using only p , or neglecting the interference terms q and q , all as predicted by 22 22 2* the GALUGA program [7], which is described in Section 5.1. The di!erence between dp/dx using
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Fig. 8. The predicted di!erential cross-section dp/dx for the reaction eePeek>k\ for the acceptance of the PLUTO experiment, and for two di!erent lower limits on =. Shown are the di!erential cross-sections for ='2m in (a), and for I ='20m in (b). See text for further details. The three di!erent histograms correspond to the di!erential cross-section I dp/dx using all terms of Eq. (14) (full), using only p (dash) or neglecting only the interference terms q and q 22 22 2* (dot-dash).
Table 2 The individual contributions to the di!erential cross-section for muon pair production. The numbers given are for several kinematical situations explained in the text, and correspond to the integrals of the distributions shown in Figs. 8 and 9 p (pb) Used terms in Eq. (14)
8(a)
8(b)
9(a)
9(b)
p 22 p 22 p 22 p 22 p 22 p 22
2.20 3.24 4.20 4.30 4.17 3.35
1.26 1.68 2.07 2.12 2.11 1.93
2.29 2.88 3.36 3.38 3.37 3.24
285 349 360 362 360 350
p 2* p 2* p 2* p 2* p 2*
p *2 p *2 p *2 p *2
p ** p ** p **
q 22 q 22
q
2*
only p and dp/dx by neglecting only the interference terms, shows that there are large contribu22 tions from the cross-sections containing at least one longitudinal photon, p , p and p . But 2* *2 ** also the interference terms themselves give large negative contributions, as shown by the di!erence between the dp/dx using all terms and dp/dx by neglecting the interference terms. The importance of the interference terms decreases for increasing =, as shown in Fig. 8(b). However, this comes at the expense of a signi"cant reduction in the acceptance at high values of x.
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Fig. 9. The predicted di!erential cross-section dp/dx for the reaction eePeek>k\ for some typical acceptances of a LEP experiment. Shown are the di!erential cross sections for ='2m . In (a) dp/dx is shown for a typical acceptance for the I exchange of two virtual photons with two detected electrons and for E"94.5 GeV, and in (b) dp/dx is shown for a typical acceptance for deep inelastic electron}photon scattering for E"45.6 GeV. See text for further details. The three di!erent histograms correspond to the di!erential cross-section dp/dx using all terms of Eq. (14) (full), using only p (dash) or 22 neglecting only the interference terms q and q (dot-dash). 22 2*
Fig. 9(a) shows the same quantities for the typical acceptance of a LEP detector at E"94.5 GeV, when using the very low angle electromagnetic calorimeters and the calorimeters used for the high-precision luminosity measurement. In this case, the kinematical requirements are E , E ' 0.5E for E"94.5 GeV, 33(h (120 mrad, 4(h (8 mrad, and ='2m . The increase in the I energy of the beam electrons is compensated by a smaller value of h , resulting in an average value of 1P2"0.3 GeV, similar to the PLUTO acceptance. However, the average value of Q is increased to 1Q2"24.5 GeV, which results in increased ratios Q/P and Q/=. The result is that the total contribution of the interference terms decreases the di!erential cross section by less than 4%, compared to 28(10)% in the case of the PLUTO acceptance for ='2(20)m . This shows I that the importance of the interference terms varies strongly as a function of the kinematical range. In the kinematical region of the LEP high-energy programme the importance of the interference terms is smaller than for the PLUTO region. Unfortunately, no general statement of the importance of these terms can be made for the case of quark pair production in the framework of QCD. However, in the regions of phase space where the leading-order point-like qq production process dominates, the cross-section for quark pair production, in the quark parton model, is exactly the same as for muon pair production, except for the di!erent masses of muons and quarks, and the above considerations can be applied. For deep inelastic electron}photon scattering, Q
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longitudinal quasi-real photons can be neglected because the longitudinal polarisation state vanishes for P"0. Also the term proportional to q vanishes, although q itself does not vanish, 22 22 because for P"0 the angle M is unde"ned. Consequently, for deep inelastic electron}photon scattering, the di!erential cross-section, Eq. (14), reduces to
a (p ) q)!QP o dp dp 4o>>o>> p # p . dp" 22 2o>> *2 E E 16pQP (p ) p )!mm
(19)
This means that only the terms p and p contribute. They correspond to the situation where the 22 *2 structure of a transverse target photon, p, is probed by a transverse or longitudinal virtual photon, q, respectively. Experimentally, due to the limited acceptance discussed in Section 2.2, P can only be kept small, but it is not exactly zero. The numerical e!ect of the various contributions due to the "nite P are shown in Fig. 9(b) for a typical acceptance of a LEP detector for E"45.6 GeV. The kinematical requirements are E '0.5E, 27(h (120 mrad, h (27 mrad, and ='2m . The importance of I the reduction of the cross-section by the interference terms is further decreased to around 3%, and the contribution of p and p to the cross-section is also around 3% and positive, such that the 2* ** two almost cancel each other. In this situation the total cross-section is accurately described by p and p only. 22 *2 In the case of muon pair production, !"k>k\, the cross-section is determined by QED. Eq. (14) and consequently also the limits discussed above, Eqs. (18), (19), contain the full information, and it is su$cient to describe the reaction in terms of cross-sections. However, most of the experimental results are expressed in terms of structure functions, since in the case of quark pair production the cross-section cannot be calculated in QCD and has to be parametrised by structure functions. The relations between the cross-sections and the structure functions are de"ned as Q ((p ) q)!QP [p (x, Q, P)!p (x, Q, P)] , 2xFA (x, Q, P)" 22 2* 2 p)q 4pa p)q Q FA (x, Q, P)" 4pa ((p ) q)!QP ;[p (x, Q, P)#p (x, Q, P)!p (x, Q, P)!p (x, Q, P)] , 22 *2 ** 2* FA (x, Q, P)"FA (x, Q, P)!2xFA (x, Q, P) , * 2
(20)
as given, for example, in Ref. [8]. These equations can be used for the de"nition of both QED and hadronic structure functions. In the limit P"0 the relations 2xFA (x, Q)"(Q/4pa)p (x, Q) , 2 22 FA (x, Q)"(Q/4pa)[p (x, Q)#p (x, Q)] , 22 *2 FA (x, Q)"(Q/4pa)p (x, Q) * *2
(21)
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are obtained. In the QED case the structure functions can be calculated as discussed in Section 3.3, whereas for the hadronic structure functions model assumptions have to be made which are discussed in detail in Section 3.4. 3.2. Equivalent photon approximation In many experimental analyses of deep inelastic electron}photon scattering the di!erential cross-section for the reaction is not described in terms of cross-sections corresponding to speci"c helicity states of the photons, as outlined in Section 3.1, but in terms of structure functions of the transverse quasi-real photon times a #ux factor for the incoming quasi-real photons of transverse polarisation. In this notation the di!erential cross-section, Eq. (19), can be written in a factorised form as
2(1!y) dN2 2pa dp A " [1#(1!y)] 2xFA (x, Q)# FA (x, Q) . 2 1#(1!y) * dx dQ dz dP dz dP xQ
(22)
In Appendix A this equation is derived from Eq. (19) using the limit PP0. By using in addition FA "2xFA #FA the widely used formula * 2 dp dN2 2pa A " [(1#(1!y))FA (x, Q)!yFA (x, Q)] (23) * dx dQ dz dP dz dP xQ is obtained. Sometimes this formula is also used to study the P dependence of FA by using FA (x, Q, P) instead of FA (x, Q). It should be kept in mind that the main approximation made in calculating Eq. (23) is (p ) q)!QP+(p ) q) and that only results in the same limit of FA are meaningful, see Appendix A for details. To avoid this complication, Eq. (19) should be used instead. The factor dN2/dz dP describing the #ux of incoming transversely polarised quasi-real A photons of "nite virtuality is the equivalent photon approximation (EPA) which was "rst derived in Ref. [9]. The EPA is given by
dN2 a 1#(1!z) 1 2 m z A " ! , dz dP 2p z P P
(24)
where the "rst term is dominant. The #ux of longitudinal photons is
dN* a 2(1!z) 1 A " dz dP 2p z P
,
(25)
such that the ratio is given by
dN* dN2 2(1!z) A A + ,e(z) . dz dP dz dP 1#(1!z)
(26)
Comparing this functional form to Eq. (22) shows that the term in front of FA corresponds to the * ratio of the transverse and longitudinal #ux of virtual photons. For the experimental situation where the electron which radiates the quasi-real photon is not detected, the EPA is often used integrated over the invisible part of the P range. The integration
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boundary P is given by four-vector conservation and P is determined by the experimental
acceptance. The experimental values of P strongly depend on the detector acceptance and the
energy of the beam electrons, as has been discussed in Section 2.2. The integration of the EPA leads to the WeizsaK cker}Williams approximation [10,11], which is a formula for the #ux of collinear real photons:
. 1 dN2 a 1#(1!z) P 1 dN2 A " A" dP ln !2mz ! z P dz dP 2p P P dz .
,
(27)
where P "mz/(1!z) and P "(1!z)Eh .
The strong dependence of the EPA on the virtuality of the quasi-real photon is demonstrated in Fig. 10, where the EPA is shown for three values of P. Shown are, "rstly P"P the smallest
value possible, secondly P"P (z"0)"1.4 GeV, a typical value for a LEP detector for an
e>e\ centre-of-mass energy of the mass of the Z boson, (s "m , and thirdly a typical value of 8 an average P observed in an analysis of the QED structure of the photon, P"0.05 GeV. In the range P to P the EPA is reduced by about six orders of magnitude. In addition, the
EPA is compared to the WeizsaK cker}Williams approximation, Eq. (27), using P and the same
value of P . In this speci"c case the result of the integration is rather close to the EPA at the
average P. It is clear that for di!erent levels of accuracy di!erent formulae have to be chosen for adequate comparisons to the theoretical predictions, and special care has to be taken when the P dependence is studied. Several improvements of the EPA have been suggested in the literature for di!erent applications in electron}positron and electron}proton collisions. The discussion of these improvements is beyond the scope of this review and the reader is referred to the original publications [12}15]. 3.3. QED structure functions Two topics concerning QED structure functions have been experimentally addressed by studying the deep inelastic electron}photon scattering reaction, shown in Fig. 11. First the s distribution has been measured, leading to the determination of the structure functions FA and FA , /#" /#" which are obtained for real photons, P"0. Second the structure function FA , and its /#" dependence on P has been measured. The theoretical framework of these two topics is discussed here in turn. The starting point for the measurement of FA and FA is the full di!erential cross-section /#" /#" for deep inelastic electron}photon scattering for real photons at P"0, 2pa dp A ) #e(y)FI A " [1#(1!y)]+(2xFI A * /#" 2 /#" xQ dx dQ dz ds/2p !o(y)FI A cos s#e(y)FI A cos 2s, , /#" /#"
(28)
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Fig. 10. Comparison of the equivalent photon and the WeizsaK cker}Williams approximations. The EPA is shown for three choices of P: P , P with P (z"0)"1.4 GeV and a "xed value of P"0.05 GeV. The EPA is compared
to the WeizsaK cker}Williams approximation (WW), obtained by integrating the EPA, using the same values of P
and P .
Fig. 11. A diagram of the reaction ecPec夹cPe!.
where the functions e(y) and o(y) are both of the form 1!O(y) (2!y)(1!y 2(1!y) , o(y)" "(2[e(y)#1]e(y) . e(y)" 1#(1!y) 1#(1!y)
(29)
The function e(y), already de"ned in Eq. (26), is obtained from e in the limit P"0, see Appendix A. The function o(y) stems from (2"o>"/o>> evaluated in the same limit, as can be seen from Eq. (16). In leading-order QED, the di!erential cross-section depends on four non-zero unintegrated structure functions, namely FI A , FI A , FI A and FI A . They are functions 2 /#" * /#" /#" /#" only of x, b and z , but do not depend on s. The kinematic variables are de"ned from the four
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vectors in Fig. 3 and listed in Section 2.1. The variable z is related to the fermion scattering angle h夹 in the photon}photon centre-of-mass frame, via z "(1#b cos h夹), with b"(1!4m/=, where m denotes the mass of the fermion. and , FI A , FI A For real photons, P"0, the unintegrated structure functions, FI A 2 /#" * /#" /#" FI A have been calculated in the leading logarithmic approximation and can be found, for /#" example, in Ref. [16]. Only recently, in Ref. [2], the calculation has been extended beyond the leading logarithmic approximation, for all four unintegrated structure functions, by retaining the full dependence on the mass of the produced fermion up to terms of the order of O(m/=). But the limitation to real photons, P"0, is still retained. These structure functions are proportional to the cross-section for the transverse real photon to interact with di!erent polarisation states of the virtual photon: transverse (T), longitudinal (L), transverse}longitudinal interference (A) and interference between the two transverse polarisations (B). They are connected to the unintegrated forms of p , p , q and q , respectively. The structure function FI A ,2xFI A #FI A is 22 *2 2* 22 /#" 2 /#" * /#" a combination of these structure functions. Using this relation and the limit e(y)"o(y)"1, Eq. (28) reduces to dp 2pa A " [1#(1!y)][FI A !FI A cos s#FI A cos 2s] . /#" /#" /#" dx dQ dz ds/2p xQ
(30)
In this equation, z and s always refer to the produced fermion. However, to achieve a structure function FI A , which does not vanish when integrated over z , the angle s is de"ned slightly /#" di!erently, as the azimuth of whichever produced particle (fermion or anti-fermion) has the smaller value of z , or cos h夹, as shown in Fig. 5(b). This de"nition leaves all the structure functions unchanged except that FI A now is symmetric in z , thereby allowing for an integration over the /#" full kinematically allowed range in z , namely (1!b)/2 to (1#b)/2. The integration with respect to z leads to dp 2pa A " [1#(1!y)][FA !FA cos s#FA cos 2s] . /#" /#" /#" dx dQ ds/2p xQ
(31)
This formula is used in the experimental determinations of FA and FA . The full set of /#" /#" functions can be found in Ref. [2]; here only the functions used for the determination of FA and /#" are listed: FA /#"
ea 1#b FA (x, b)" x [x#(1!x)] ln ! b#8bx(1!x)!b(1!b)(1!x) /#" p 1!b #(1!b)(1!x)
1 1#b (1!x)(1#b)!2x ln 2 1!b
4ea 1!x FA (x, b)" x(x(1!x)(1!2x) b 1#(1!b) /#" p 1!2x #
3x!2 (1!b arccos((1!b) , 1!2x
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FA
/#"
185
4ea 1!x 1 (x, b)" x(1!x) b 1!(1!b) # (1!b) p 2x 2
;
1!2x 1!x 1#b ! (1!b) ln 2x x 1!b
.
(32)
Here e is the charge (in units of the electron charge) of the produced fermion. The structure can be obtained functions FA and FA are new. In contrast, the structure function FA /#" /#" /#" from Eq. (21), together with the cross-sections listed in Ref. [4], taking the appropriate limit. The corrections compared to the leading logarithmic approximation are of order O(m/=) for FA and FA . For FA they are already of order O(m /=). The structure functions in the /#" /#" /#" leading logarithmic approximation can be obtained from Eqs. (32) in the limit bP1. They are listed, for example, in Ref. [16], and have the following form:
= ea !1#8x(1!x) , FA (x, b"1)" x [x#(1!x)] ln /#" m p 4ea FA (x, b"1)" +x(1!2x)(x(1!x), , /#" p FA
4ea (x, b"1)" +x(1!x), . /#" p
(33)
So far, the structure functions FA and FA have only been measured for the k>k\ "nal state /#" /#" using the Q range from 1.5 to 30 GeV, as discussed in Section 6. The inclusion of the massdependent terms signi"cantly changes the structure functions in the present experimentally accessible range in Q. The numerical e!ect is most prominent at low values of Q and gets less important as Q increases, as demonstrated for the case of k>k\ production. In Fig. 12 for Q"1 GeV, the mass corrections are extremely important, especially at large values of x, while in Fig. 13 for Q"100 GeV, they are small. The second measurement of QED structure functions performed by the experiments is the measurement of FA for Q
= ea FA (x, P)" x [x#(1!x)] ln m#x(1!x)P p x(1!x)P !1#8x(1!x)! , (34) m#x(1!x)P obtained in the limit m;Q, =, which is rather accurate for small values of P. However, for P'0.01 GeV the approximation starts to deviate signi"cantly from the exact formula and should not be used anymore. The structure function FA is strongly suppressed as a function of /#"
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Fig. 12. The structure functions FA , FA and FA for k>k\ "nal states at Q"1 GeV. The structure functions /#" /#" /#" are shown with the full mass dependence (full) and in the leading logarithmic approximation (dash). Shown are (a) FA , /#" (b) FA , and (c) FA . /#" /#" Fig. 13. The structure functions FA , FA and FA for k>k\ "nal states at Q"100 GeV. The same quantities /#" /#" /#" as in Fig. 12 are shown.
Fig. 14. The P dependence of the structure function FA . The structure function FA (full) and the approximation /#" /#" FA (dash) are shown for Q"5.4 GeV, and for various P values, 0.001, 0.01, 0.05, 0.1 and 1.0 GeV.
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P for increasing P, for example, for x"0.5 and Q"5.4 GeV the ratio of FA for P"0 and /#" P"0.05 GeV is 1.4. This suppression is clearly observed in the data, as discussed in Section 6. The QED structure functions de"ned above can only be used for the analysis of leptonic "nal states. For hadronic "nal states the leading-order QED diagrams are not su$cient and QCD corrections are important. Therefore, the cross-sections and consequently also the structure functions cannot be calculated and parametrisations are used instead. This is the subject of the next section. 3.4. Hadronic structure function FA After the "rst suggestions that the structure functions of the photon may be obtained from deep inelastic electron}photon scattering at e>e\ colliders in Refs. [17,18], much theoretical work has been devoted to the investigation of the hadronic structure function FA . The striking di!erence between the photon structure function FA and the structure function of a hadron, for example, the proton structure function F , is due to the point-like coupling of the photon to quarks, as shown in Fig. 2(b). This point-like coupling leads to the fact that FA rises towards large values of x, whereas the structure function of a hadron decreases. Furthermore, due to the point-like coupling, the logarithmic evolution of the photon structure function FA with Q has a positive slope for all values of x, or in other words, the photon structure function FA exhibits positive scaling violations for all values of x, even without accounting for QCD corrections. This is in contrast to the scaling violations observed for the proton structure function F , which exhibits positive scaling violations at small values of x, and negative scaling violations at large values of x, caused by pair production of quarks from gluons and by gluon radiation, respectively. In the case of the photon, the &loss' of quarks at large values of x due to gluon radiation is overcompensated by the &creation' of quarks at large values of x due to the point-like coupling of the photon to quarks. The quark parton model (QPM) already predicts a logarithmic evolution of the photon structure function FA with Q. This was "rst realised in Refs. [19,20] based on the calculation of the Q dependence of the so-called box diagram, for the reaction c夹cPqq , shown in Fig. 3. The QPM result for quarks of mass m I is O
= L e a !1#8x(1!x) , (35) FA (x, Q)"N OI x [x#(1!x)] ln /.+ p mI O I where N is the number of colours and the sum runs over all active #avours i"1,2, n . The QPM formula is equivalent to the leading logarithmic approximation of FA given in Eq. (33). This /#" result, shown in Fig. 15 for three light quark species, is referred to as the calculation of FA based on the Born term, the box diagram FA , the QPM result for FA , or as the QED structure function FA . In Fig. 15 the contributions from the di!erent quark species are added up for the smallest and largest value of Q for which measurements of FA at LEP exist. In this Q range the photon structure function rises by about a factor of two at large values of x. Due to the dependence on the quark charge, the photon structure function FA for light quarks is dominated by the contribution from up quarks. The pioneering investigation of the photon structure function in the framework of QCD was performed by Witten [21], using the technique of operator product expansion. The calculation
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Fig. 15. The QPM prediction for the structure function FA for light quarks. Shown are the predictions of Eq. (35) adding up the contributions for light quarks using a mass of m I "m"0.2 GeV for all quark species k"u, d, s. The di!erent O curves correspond to FA from the down quarks alone (dot-dash), FA from the down and up quarks (dash) and also adding strange quarks (full).
showed that by including the leading logarithmic QCD corrections in the limit of large values of Q, the behaviour of FA is logarithmic and similar to the QPM prediction. Schematically the result reads
Q a (x) "a a(x) ln . FA (x, Q)"a K a
(36)
The term ln(=/mI ) of Eq. (35) is replaced by ln(Q/K), which means the mass is replaced by the O QCD scale K, and = is replaced by Q, which in the leading logarithmic approximation is equivalent, because = and Q are related by a term which depends only on x, as can be seen from Eq. (5) for P"0. However, the x dependence of FA , as predicted by the QPM, which treats the quarks as free, is altered by including the QCD corrections. The result from Witten is called the leading-order asymptotic solution for the photon structure function FA , since it is a calculation of FA using the leading-order logarithmic terms, but summing all orders in the strong coupling constant a , and for the limit of asymptotically large values of Q. The photon structure function FA in the leading-order asymptotic solution is inversely proportional to a , and the Q evolution, as well as the normalisation, are predicted by perturbative QCD at large values of Q. Therefore, there was hope that the measurement of the photon structure function would lead to a precise measurement of a . However, the asymptotic calculation simpli"es the full equations by retaining only the asymptotic terms, which means the terms which dominate for QPR. The nonasymptotic terms are connected to the contribution from the hadron-like part of the structure function, shown in Fig. 2(c). The asymptotic solution is well behaved for xP1 and removes the
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divergence of the QPM result for vanishing quark masses, but in the limit xP0 it diverges like x\ , as was already realised in Ref. [21]. The asymptotic solution has also been re-derived in a diagrammatic approach in Refs. [22}24], and in Ref. [25], by using the Altarelli}Parisi splitting technique from Refs. [26,27]. No closed analytic form of the x dependence of the asymptotic solution can be obtained, since the asymptotic solution is given in moment space using Mellin moments. Consequently, only parametrisations of the x dependence of the parton distribution functions of the photon, based on the "ndings of the asymptotic solution of FA , have been derived. The "rst parametrisation, given in Ref. [28], has been obtained by factoring out the singular behaviour at xP0 and expanding the remaining x dependence by Jacobi polynomials. Another parametrisation has been obained in Ref. [29]. The most recent available parametrisation has been derived in Ref. [30] based on the technique of solving the evolution equations directly in x space. In Ref. [30], it is compared to the two parametrisations discussed above and it is found to be the most accurate parametrisation of the asymptotic solution. The predictions of the parametrisations of the asymptotic solution are compared in Fig. 16 for Q"10 GeV, K"0.2 GeV, and assuming three quark #avours. The three parametrisations are rather close to each other in the range 0.2(x(0.8, where they agree to better than 10%, but at larger and smaller values of x the di!erences are much larger. The asymptotic solution, Eq. (36), factorises the x and Q dependence of FA , which is not the case when solving the evolution equations as discussed in Appendix B. Fig. 17 shows the di!erence between the asymptotic solution and the result from the GRV parametrisation of the photon structure function FA from Refs. [31,32]. The GRV parametrisation is obtained by solving the full evolution equations. In this "gure the logarithmic Q behaviour is factored out and the asymptotic solution is compared to the leading-order GRV parametrisation of FA for several values of Q. The asymptotic solution is consistently lower than the GRV parametrisation in the range 0.2(x(0.8, and for all values of Q. However, the agreement improves with increasing x and Q. For example at Q"100 GeV the agreement is better than 20%, for the whole range 0.2(x(0.8. The asymptotic solution has been extended to next-to-leading order in QCD in Ref. [33], leading to
Q FA (x, Q)"a a(x) ln #b(x) . K
(37)
It was found in Ref. [34] that the next-to-leading order corrections to the asymptotic solution are large for large x, and that the structure function FA is negative for x smaller than about 0.2. In addition, the divergence at low values of x gets more and more severe in higher orders in QCD, and also extends to larger values in x, as discussed in Refs. [35,36]. The divergence at small x of the perturbative, but asymptotic, result, which is cancelled by including the non-asymptotic contribution to the photon structure function, has attracted an extensive theoretical debate. For the real photon, the hadron-like part of the photon structure function FA cannot be calculated in perturbative QCD, and only its Q evolution is predicted, as in the case of the proton structure function. Given this, the predictive power of QCD for the calculation of the photon structure function is reduced, and the scope for the determining a from the photon structure function FA is obscured.
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Fig. 16. Comparison of the parametrisations of the x dependence of the leading-order asymptotic solution of FA . The parametrisations are compared for Q"10 GeV, K"0.2 GeV, and for three quark #avours. Shown are in (a) the structure function FA obtained from the Gordon Storrow parametrisation (full), the Duke and Owens parametrisation (dash), and the Nicolaidis parametrisation (dot-dash). In (b) the di!erences are explored by dividing the older parametrisations by the Gordon Storrow parametrisation. The result for the Duke and Owens parametrisation is shown as a dashed line and the result for the Nicolaidis parametrisation as a dot-dashed line. Fig. 17. Comparison of the asymptotic solution and the leading-order GRV parametrisation of FA . The logarithmic Q dependence is factored out for K"0.2 GeV, and three quark #avours are assumed. Shown are in (a) the asymptotic solution of FA using the Gordon Storrow parametrisation (full), and the result of the GRV parametrisation (dot-dash) obtained by solving the full evolution equations. The GRV parametrisation is shown for several values of Q, 10, 100, 1000 and 10 000 GeV. In (b) the di!erences are explored by dividing the Gordon Storrow parametrisation by the GRV result.
Several strategies have been taken to deal with this problem. It is clear from the singularities of the asymptotic, point-like, contribution that describing FA as a simple superposition of the asymptotic solution and a regular hadron-like contribution, as derived, for example, based on VMD arguments, cannot solve the problem, because a hadron-like part, which is chosen to be regular, will never remove the singularity. Therefore, either a singular part has to be added by hand, to remove the singularities of the asymptotic solution, or the singularities have to be dealt with by including the non-asymptotic contribution as a supplement to the point-like part of the photon structure function FA . The various approaches attempted along these lines will be discussed brie#y. The "rst approach to deal with the singularities was suggested and outlined in Ref. [37]. This method tries to retain as much as possible of the predictive power of the point-like contribution to the structure function, and the possibility to extract a from the photon structure function FA . The solution chosen to remove the divergent behaviour consists of a reformulation of the structure function by isolating the singular structure of the asymptotic, point-like part at low values of x, based on the analysis of the singular structure of FA in moment space. Then, an ad hoc term is
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introduced, which removes the singularity and regularises FA , but depends on an additional para meter, which has to be obtained by experiments, for example, by performing a "t to the low-x behaviour of FA . Several data analyses using this approach have been performed, as summarised in Ref. [8]. A second way to separate the perturbative and the non-perturbative part of the photon structure function, known as the FKP approach, was developed in Refs. [39}41]. Here, the separation into the perturbative and the non-perturbative parts of FA is done on the basis of the transverse momentum squared p of the quarks in the splitting cPqq , motivated by the experimental observation that for transverse momenta above a certain minimum value, the data can be described by a purely perturbative ansatz. The minimum transverse momentum squared p was found to be of the order of 1}2 GeV. Given this large scale, no signi"cant sensitivity of the point-like part of the photon structure function to a remains. It has been argued in Ref. [42] that these values are too high, and that still some sensitivity to a is left, even when using the FKP ansatz. The FKP approach has several weaknesses, which are discussed, for example, in Refs. [43,44]. The main shortcomings are that terms are included which formally are of higher order, and that the parametrisation is based only on &valence quark' contributions, which means that FA vanishes in the limit xP0, whereas the &sea quarks' result in a rising FA at small values of x. This ansatz is therefore currently not widely used. The last approach discussed here is outlined in Refs. [45,46], and is driven by the observation that by using the full evolution equations, the solution to FA is regular both in leading and in next-to-leading order for all values of x. The method is analogous to the proton case and the starting point is the de"nition of input parton distribution functions for the photon at a virtuality scale Q . The relation between the quark parton distribution functions qA and the structure function FA in I leading order is given by the following relation: L (38) FA (x, Q)"x eI [qA (x, Q)#q A (x, Q)] . O I I I The #avour singlet quark part RA(x, Q) and the #avour non-singlet part qA (x, Q) of the photon ,1 structure function are de"ned by FA (x, Q)"x[qA (x, Q)#1e2RA(x, Q)] ,1 such that L RA(x, Q)" [qA (x, Q)#q A (x, Q)] , I I I L qA (x, Q)" [eI !1e2][qA (x, Q)#q A (x, Q)] , I I O ,1 I where 1e2"1/n L eI is the average charge squared of the quarks. I O
(39)
(40)
The regularisation term of Ref. [37] is based on the photon}parton splitting functions of Ref. [33]. Unfortunately, the photon}gluon splitting function in next-to-leading order erroneously contained a factor d(1!x), which was removed later in Refs. [31,38]. As discussed in Ref. [31], this weakens the next-to-leading order singularity at low values of x, and therefore, it will also a!ect the exact form of the proposed regularisation term of Ref. [37].
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The input distribution functions are evolved in Q using the QCD evolution equations. With this, the x dependence at an input scale Q has to be obtained either from theoretical consider ations, which are usually based on VMD arguments if Q is chosen as a low scale, or "xed by a measurement of the structure function FA . This approach gives up the predictive power of QCD for the normalisation of the photon structure function and retains only, as in the proton case, the a sensitivity of QCD to the Q evolution. Because the evolution with Q is only logarithmic, the length of the lever arm in Q is very important, and consequently the sensitivity to a crucially depends on the range of Q where measurements of FA can be obtained. There are several groups using this approach. They di!er however in the choice of Q , the factorisation scheme, and the assumptions concerning the input parton distribution functions at the starting scales. The mathematical framework is outlined in Appendix B, following the discussion given in Ref. [47], and the available parton distribution functions are reviewed in Section 4. Using this framework the predictions of perturbative QCD on the evolution of FA can be experimentally tested by "rst "xing the non-perturbative input by measuring FA at some value of Q and then exploring the evolution of FA for "xed values of x as function of Q. Given the large lever arm in Q from 1 GeV to about 1000 GeV when exploiting the full statistics from LEP at all e>e\ centre-of-mass energies, there is some sensitivity left for measuring a from the photon structure function, especially at large values of x, as discussed in detail in Refs. [16,48]. This completes the discussion of the quasi-real photons, and virtual photons are discussed in the following. For virtual photons the point-like contribution to the photon structure function FA has been derived in the limit Q
Fig. 19. The leading-order contributions to FA . Shown are examples of leading order diagrams contributing to (a) the point-like, and (b) the hadron-like part of the heavy quark structure function FA , with Q"c, b, t.
treatment of the point-like contribution of heavy quarks to the structure function is given by the prediction of the lowest order Bethe-Heitler formula. Due to the large mass scale QCD e!ects are small and this QED result is in general su$cient. The structure function can be obtained from Eq. (20), together with the cross-sections listed in Ref. [4]. The resulting formula is very long and the approximation made in Ref. [51], which is valid for 2xP/Q<1, is su$ciently accurate in
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most cases and is used, for example, when constructing parton distribution functions. This approximation for virtual photons P'0 is given by
e a 1#bc FA "N O x [x#(1!x)] ln !b#6bx(1!x) p 1!bc
bc(1!b) 1!c !(1!b)(1!x) # 2x(1!x)! 1!bc 1!b #(1!b)(1!x)
1 1#bc (1!x)(1#b)!2x ln 2 1!bc
with
4xP 4m c" 1! , and b" 1! . Q =
(41)
For real photons, P"0 and c"1, this reduces to FA as given in Eq. (32). For real photons the /#" next-to-leading order predictions have also been calculated in Ref. [52]. For the hadron-like contribution the photon-quark coupling must be replaced by the gluon}quark coupling, e aPe a /6, and the Bethe-Heitler formula has to be integrated over the O O allowed range in fractional momentum of the gluon. The hadron-like contribution, discussed in Section 4, is only important at small values of x. The dominant point-like contribution to the structure function FA for charm and bottom quarks, using m "1.5 GeV and m "4.5 GeV, is shown in Fig. 20 for three values of Q, 10, 100 and 1000 GeV, and three values of P, 0, 1 and 5 GeV. Several observations can be made. The structure functions rises with Q and also, due to Eq. (6), the large x part is more and more populated. Due to their small charge and large mass, the contribution from bottom quarks is very small. The suppression with P is stronger for the charm quarks since they are lighter than the bottom quarks. At large Q and large invariant masses, =<2m , the mass of the heavy quarks can be neglected in the evolution of FA , provided that the usual continuity relations are respected and the appropriate number of #avours are taken into account in K. This concludes the discussion on the hadronic structure function FA , and the remaining part of this section is devoted to the electron structure function and to radiative corrections to the deep inelastic scattering process. Recently, as described, for example, in Refs. [14,53}55] it has been proposed not to measure the photon structure function, but to measure the electron structure function instead. In measuring the electron structure function the situation is similar to the measurement of the proton structure function in the sense that the energy of the incoming particle, the electron in this case, is known. Therefore there is probably no need for an unfolding of x, explained in Section 5, which is needed for the measurement of the photon structure function. This, on "rst sight, is an appealing feature since it promises greater precision in the measurement of the electron structure function than in the measurement of the photon structure function. But, as already discussed in Ref. [56], the advantage of greater measurement precision is negated by uncertainties which arise in interpreting the results in terms of the photon structure, because the di!erences in the predictions of the photon structure functions are integrated out. The region of low values of x "xz receives contributions from the
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Fig. 20. The point-like heavy quark contribution to FA for various values of Q. Shown are in (a) the contribution of charm quarks, FA , and in (b) the contribution of bottom quarks, FA , to the photon structure function. Both contributions are calculated for three values of Q, 10 GeV (dot-dash), 100 GeV (dash), and 1000 GeV (full). At each Q, three curves are shown, which correspond to P"0, 1 and 5 GeV, where the suppression of FA gets stronger for increasing P. To put both contributions on the same scale the bottom part has been multiplied by 30 as indicated in the "gure.
regions of large momentum fraction x and low scaled photon energy z, and small momentum fraction x and large scaled photon energy z. Due to this, largely di!erent photon structure functions lead to very similar electron structure functions, as was demonstrated in Ref. [56]. Given this, further pursuit of this method does not seem very promising, since it does not give more insight into the structure of the photon. The last topic discussed in this section is the size of radiative corrections. Radiative corrections to the process eePeec夹cPeeX have been calculated for a (pseudo) scalar particle X in Refs. [57}60] and for the k>k\ "nal state in Refs. [60}62]. It has been found that they are very small, on the per cent level, for the case where both photons have small virtualities and the scattered electrons are not observed. Consequently, the equivalent photon approximation has only small QED corrections. For the case of deep inelastic electron}photon scattering a detailed analysis has been presented in Refs. [63,64]. The theoretical calculation is performed in the leading logarithmic approximation which means that the corrections are dominated by radiation from the deeply inelastically scattered electron. Only photon exchange is taken into account, since Z boson exchange can be safely neglected at presently accessible values of Q. The calculation is analogous to the experimental determination of the kinematical variables. The momentum transfer squared Q is determined from the scattered electron, whereas x is based on mixed variables, which means Q is obtained from the scattered electron and = is taken from the hadronic variables. The radiative corrections are dominated by initial state radiation, whereas "nal state radiation and the
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Compton process are found to contribute very little. Final state radiation is usually not resolved experimentally due to the limited granularity of the electromagnetic calorimeters used. The Compton process contributes less than 0.5% to the cross-section for the range 3.2;10\(x(1 and 3.2(Q(10 GeV. The contribution of initial state radiation is usually negative and for a given Q its absolute value is largest at the smallest accessible x and decreases with increasing x. For most of the phase space covered by the presently available experimental data the radiative corrections amount to less than 5%. Due to the capabilities of the Monte Carlo programs used in the experimental analyses of photon structure functions discussed in Section 5.1, the radiative corrections are usually neglected in the determination of FA . They are, however, accounted for when measuring FA . /#" 3.5. Vector meson dominance and the hadron-like part of FA In this section the parametrisations of the hadron-like part FA of the photon structure function, which are constructed based on VMD arguments, are brie#y reviewed. Only the main arguments needed to construct FA are given; for details the reader is referred to the original publications. There have been several attempts to derive the hadron-like part of the photon structure function FA based on VMD arguments motivated by the fact that the photon can #uctuate into a o meson. No precise data on the structure function of the o meson exist, and the structure function of the o is approximated by the structure function of the pion, Fp . In the "rst attempts to measure the photon structure function FA , it was approximated by the sum of the point-like and the hadron-like part, where FA was constructed as a function of x alone, and its Q evolution was ignored. In the context of the evolution of the parton distribution functions, the Q dependence given by perturbative QCD is also taken into account, and only the x dependence at the scale Q is obtained from VMD arguments. These two issues will be discussed in the following. The most widely used approximation for an Q-independent hadron-like component of the photon structure function FA was obtained in Refs. [65,66]. The quark distribution functions of the o meson are taken to be xqM(x)"1/2(1!x) and the photon is modelled as an incoherent sum of G o and u, leading to 8 4pa xqM(x)"a[0.2(1!x)] , FA " 9 f G M
(42)
where f is the o decay constant with 4p/f "1/2.2, as taken from Ref. [8]. This approximation was M M used in several measurements of the photon structure function FA given in Refs. [67}71]. A similar parametrisation has been proposed by Duke and Owens [34]. This parametrisation, which is assumed to be valid at Q"3 GeV, is given by FA "(4pa/f )[0.417(x(1!x)#0.133(1!x)] . M
(43)
Parametrisations of FA have been obtained experimentally from a measurement of the photon structure function FA by the TPC/2c experiment and from measurements of the pion structure function Fp , for example, by the NA3 experiment. The parametrisation obtained in Ref. [72] by
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Fig. 21. Comparison of parametrisations of the hadron-like contribution to the photon structure function FA . Shown are in (a) the theoretically motivated parametrisations obtained by Peterson Walsh and Zerwas (PWZ, full), and Duke and Owens (DO, dash), together with the experimentally determined parametrisations from the TPC/2c (TPC/2c, dot-dash), and the NA3 (NA3, dot), experiments. The evolution of the hadron-like part of FA is shown in (b), for the leading-order parametrisations from GluK ck, Reya and Vogt (GRV, full), for several values of Q. In addition, shown is the hadron-like input distribution from Gordon and Storrow (GS, dash), which is valid for Q"5.3 GeV.
the TPC/2c experiment, is based on a measurement of FA in the range 0.3(Q(1.6 GeV, with an average value of 1Q2"0.7 GeV. The "t to the data yields FA "a[(0.22$0.01)x ! (1!x) #(0.06$0.01)(1!x) ! ] . (44) The pion structure function Fp has been measured from the Drell-Yan process by the NA3 experiment for an average invariant mass squared of the k>k\ system of 25 GeV, as detailed in Ref. [73]. The NA3 data have been re"tted by the TPC/2c experiment and the best "t to the data, as listed in Ref. [72], is given by Fp "a[0.22x (1!x) #0.26(1!x) ] , (45) where the "rst part describes the contribution from valence quarks and the second part is the result for the sea quark contribution. In Fig. 21(a), the theoretically motivated parametrisations, Eqs. (42) and (43), are shown, together with the experimentally determined parametrisations, Eqs. (44) and (45). In the region of large values of x the various parametrisations are rather similar. In contrast, for small values of x, where there was no precise data, the di!erent parametrisations show a large spread. However, the Q dependence has not been taken into account in these parametrisations and the parametrisations are determined for di!erent values of Q. The inclusion of the Q dependence of FA has been performed by several groups when constructing the parton distribution functions as discussed in Section 4. As examples, the leading-order parton distribution functions of Gordon and Storrow [30], and GluK ck, Reya and
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Vogt [31,32], are discussed, which use VMD motivated input distribution functions based on measurements of Fp . In deriving the input distribution functions several assumptions are made. 1. The photon is assumed to behave like a o meson, which means that FA can be expressed as 4pa L eI xqM(x) , FA "i O I f M I where f has been de"ned above and i is a proportionality factor to take into account higher M mass mesons using an incoherent sum. 2. The structure function of the o meson is assumed to be the same as the structure function of the p, which is expressed as half the sum of the n> and n\ structure functions. 3. The constituent quarks of the pions have a valence, v, and a sea, m, contribution, and the other quarks have only a sea contribution. For example, in the n> the up quark has valence and sea contributions, whereas the dM has only a sea contribution. In addition, the valence quark distributions are assumed to be the same for all quark species, as are the sea quark contributions. Then by using Eq. (38), for example, for three light quark species, FA "a2.0/2.2(5v!9m)/9 is obtained. In the case of the leading-order GRV parton distribution functions of the photon the valence and the sea parts are expressed at Q "0.25 GeV by the published parton distribution functions of the pion, as given by Ref. [74]. In the case of the Gordon}Storrow parton distribution functions the VMD contribution is derived using basically the same assumptions. The parametrisation at the input scale Q "5.3 GeV, and for three light quark species, taken from Ref. [30], is given by FA "a[1.3360(x(1!x)#0.641(1!x)# 0.0742(x(1!x)] .
(46)
In Fig. 21(b) the two parametrisations are compared for three #avours, and in addition the Q evolution of the GRV prediction is studied. The parametrisation from GRV is shown at the scale where the evolution starts, Q "0.25 GeV, at the scale where the parametrisation from GS is derived, Q"5.3 GeV, and for two large scales Q"100 and 1000 GeV. The evolution slowly reduces FA at large values of x with increasing Q, and also creates a steep rise of FA at low values of x, as in the case of the proton structure function F . At Q"5.3 GeV the two parametrisations are similar for x'0.2, but at smaller values of x the GRV parametrisation has already evolved a steep rise, which is purely driven by the evolution equations and not based on data. This rise cannot be obtained in the case of the GS parametrisation, because this parametrisation is obtained from a "t to data for Q'5.3 GeV, which do not cover the region of small x. The importance of the hadron-like contribution to the structure function FA decreases for increasing Q, as can be seen from Fig. 22, where the hadron-like contribution is shown together with the full structure function FA as predicted by the leading-order GRV parametrisation, both using n "4, for increasing values of Q. At the input scale the two functions coincide by construction. However, as Q increases there is a strong rise of FA and a slow decrease of FA at large values of x.
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Fig. 22. The Q dependence of the photon structure function FA in comparison to the hadron-like contribution. The GRV parametrisation of the structure function FA in leading-order is compared to the hadron-like part of FA taken as predicted by the evolved hadron-like input distribution function of the GRV parametrisation of the photon structure function FA . Both functions are shown for four active #avours.
3.6. Alternative predictions for FA There have been several attempts to construct the photon structure function FA di!erently from the leading twist procedure to derive FA from the evolution equations, as described in Appendix B. These attempts, which include power corrections, will be summarised brie#y below. The model for FA from Ref. [75] is an extension of the model constructed for the proton case in Refs. [76,77]. It describes FA as a superposition of a hadron-like part based on a VMD estimate and a point-like part given by the perturbative QCD solution of FA , suppressed however by 1/Q at low values of Q: FA (=, Q)"FA (=, Q)#FA (=, Q)
M C4> \ p (=) Q#Q Q 3Q 4 A4 , Q#Q . # FA " (Q#M ) Q#Q Q#= 4pa 4 M S (
(47)
Here M is the mass and C4> \ the leptonic width of the vector meson <, and Q "1.2 GeV, as in 4 Ref. [76]. The total cross-sections p are represented by the sum of pomeron and reggeon A4 contributions with parameters given in Ref. [78]. For moderate values of Q the structure function FA is given by this ad hoc superposition and in the limit of high Q the perturbative QCD solution of FA is recovered, but with 1/Q corrections from the hadron-like part. The model has been shown to describe the results of the measured p 夹 cross-sections from Ref. [79] for the ranges A A 0.2(Q(7 GeV and 2(=(10 GeV.
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The model for FA from Ref. [80] relies on the Gribov factorisation described in Ref. [81]. This factorisation is based on the assumption that at high energies the total cross-section of two interacting particles can be described by a universal pomeron exchange. In the model for FA it is assumed that this factorisation also holds for virtual photon exchange at low values of x, as explained in Ref. [82]. Using this, the Gribov factorisation relates the ratio of the photon}proton and proton}proton cross-sections to the ratio of the photon and proton structure functions FA (x, Q)"F (x, Q) p (=)/p (=) . A
(48)
In this framework a prediction for the photon structure function at low values of x can be obtained from the measurement of the proton structure function F at low values of x. This extends the knowledge of FA to lower values of x because the results on F reach down to x+10\, whereas the data on FA probe only the photon structure down to x+10\. However, this information can never replace a real measurement of FA . The parton distribution functions are constructed using a phenomenological ansatz similar to the LAC case described in Section 4 for four massless quark #avours. All quark distribution functions have the same functional form and the strange and charm quarks are suppressed with respect to the up and down quarks simply by constant factors. The parametrisation of FA is obtained for Q "4 GeV from a "t to the data of the photon structure function FA from Refs. [67,69}72,83}87] and the proton structure function data for x(0.01. Unfortunately, the starting scale of the evolution is too high so that no valid comparisons with the low Q measurements of FA can be made. The model for FA from Ref. [88] is based on the assumption that for =
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functions, are purely phenomenological "ts to the data, starting from an x-dependent ansatz for the parton distribution functions. The second class of parametrisations base their input distribution functions on theoretical prejudice and obtain them from the measured pion structure function, using VMD arguments and the additive quark model, as done in the case of GRV, GRSc and AFG, or on VMD plus the quark parton model result mentioned above, as done in the GS parametrisation. The third class consists of the SaS distributions which use ideas of the two classes above, and in addition relate the input distribution functions to the measured photon}proton cross-section. The main features of the di!erent sets are described below, concentrating on the predictions for FA derived from the parametrisations. The individual parton distribution functions, for example, the gluon distribution functions are not addressed, only their impact on FA is discussed. For more details the reader is referred to the original publications. 1. DG [93]: The "rst parton distribution functions were obtained by Drees and Grassie. This approach uses the evolution equations in leading order with K"0.4 GeV. The x-dependent ansatz for the input distributions at Q "1 GeV is parametrised by 13 parameters and "tted to the only data available at that time, the preliminary PLUTO data at Q"5.3 GeV from Ref. [94]. Due to the limited amount of data available, further assumptions had to be made. The quark distribution functions for quarks carrying the same charge are assumed to be equal, qA "qA and qA "qA , and the gluon distribution function is generated purely dynamically, which means the gluon input distribution function is set to zero. Three independent sets are constructed for n "3, 4, 5, which means that they are not necessarily smooth at the #avour thresholds. The charm and bottom quarks are treated as massless and enter only via the number of #avours used in the evolution equations. They are included for Q'20 and 200 GeV respectively. The parametrisations clearly su!er from limited experimental input and they are not widely used today for measurements of FA . 2. LAC [95]: The parametrisations from Levy, Abramowicz and Charchula use essentially the same procedure as the ones from Drees and Grassie, but are based on much more data, and therefore no assumptions on the relative sizes of the quark input distribution functions are made. An x-dependent ansatz, similar to the DG ansatz, using 12 parameters is evolved using the leading-order evolution equations for four massless quarks, where K is "xed to 0.2 GeV. The charm quark contributes only for ='2m , otherwise the charm quark is treated as massless. No parton distribution function for bottom quarks is available. Three sets are constructed which di!er from each other in the starting scale Q and in the assumptions made concerning the gluon distribution. The sets LAC1 and LAC2 start from Q "4 GeV, whereas LAC3 uses Q "1 GeV. In addition, the sets LAC1 and LAC2 di!er in the parametrisation of the gluon distribution. In the set LAC1 the gluon distribution is assumed to be xg(x)&x@(1!x)A, where b and c are "tted to the data, while the set LAC2 "xed b"0. The data used in the "ts are from Refs. [67}70,72,83,96}100]. The structure function FA obtained from the LAC parametrisations is shown in Fig. 23 for two typical values of Q where data are available from the LEP
The parton distribution functions are usually abbreviated with the "rst letters of the names of the corresponding authors, which will be mentioned below.
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Fig. 23. The structure function FA from the LAC parton distribution functions. Shown is the predicted structure function FA for the three sets LAC1-3 for two values of Q. In (a) the prediction is shown on a logarithmic scale in x and for Q"5 GeV, whereas in (b) a linear scale is used for Q"135 GeV.
experiments, Q"5 and 135 GeV. The sets LAC1 and LAC2 are almost identical for x'0.2 for both values of Q, and although the gluon distribution function of the set LAC3 is very di!erent from the ones used in the sets LAC1 and LAC2, as can be seen from Ref. [95], the structure function FA di!ers by less than 15% for x'0.2. For x(0.2 and at low values of Q however the di!erences in the predictions are larger than the experimental errors. 3. WHIT [101]: The parametrisations of parton distribution functions of the photon from Watanabe, Hagiwara, Izubuchi and Tanaka use a leading-order approach, with three light #avours and a starting scale of Q "4 GeV. The charm contribution, with m "1.5 GeV, is added according to the Bethe-Heitler formula in the region 4(Q(100 GeV, while for higher values, Q'100 GeV, the massive evolution equations from Ref. [102] are used. No parton distribution function for bottom quarks is available. The distributions of the light quarks are separated into distributions for valence quarks and distributions for sea quarks, which are linear combinations of the #avour singlet and non-singlet contributions to FA , introduced in Eq. (40). The valence quark distributions describe the quarks which directly stem from the photon and they are parametrised as functions of x at Q . The sea quark distributions, account for the quarks produced in the process c夹gPqq , and at Q"Q they are approximated by the Bethe-Heitler formula using 0.5 GeV for the mass of the three light quark species. The QCD scale is taken to be K"0.4 GeV. The gluon distribution function is parametrised as xg(x)"a(c#1)(1!x)A and six sets with a"0.5, 1 and c"3, 9, 15 are constructed, all being consistent with the data of the structure function FA used in the "ts. The data used are published data from Refs. [67}70,72,83,103] and preliminary data from Refs. [104}106]. They are subject to an additional requirement of x 'Q/(Q#= ), which is introduced to remove the part
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Fig. 24. The structure function FA from the WHIT parton distribution functions. The structure function FA is shown in (a) for the individual sets WHIT1-6 and in (b) the sets WHIT2-6 are all divided by the set WHIT1. The individual sets in (a) fall into two groups containing three curves each, which coincide at large values of x. The sets WHIT1-3 predict a higher structure function at large values of x than the sets WHIT4-6 and at low values of x the sets WHIT4-6 start rising earlier for decreasing values of x than the sets WHIT1-3.
of the data that was taken at the upper acceptance boundary in =, which means at low values of x. The di!erent predictions for FA of the various sets are shown in Fig. 24 for Q"100 GeV. Fig. 24(a) shows the individual sets and in Fig. 24(b) they are all normalised to the set WHIT1. The kink in the distributions in Fig. 24(a) at x+0.9 is typical for all four #avour parametrisations of FA using massive charm quarks, and is due to the charm quark mass threshold. Only for x values to the left of the threshold is charm production possible, and the threshold varies with Q, as can be seen from Eq. (6) and Fig. 20. The sets fall into two groups depending on the value of the parameter a, with WHIT1-3 having a"0.5, and WHIT4-6 using a"1. The larger value of a makes the sets WHIT4-6 start rising earlier for decreasing values of x. The two groups agree with each other to better than 5% for x'0.3, and for small x, where the gluon part becomes important, they di!er by more than a factor of two. For most of the data on FA the di!erence between the individual sets is much smaller than the experimental accuracy. But at small values of x the data are precise enough to disentangle the very di!erent predictions of the various sets. 4. GRV [31,32]: The parton distribution functions from GluK ck, Reya and Vogt are constructed using basically the same strategy which is also successfully used for the description of the proton and pion structure functions. The parton distribution functions are available in leading order and next-to-leading order. They are evolved from Q "0.25 GeV in leading order and from Q "0.30 GeV in next-to-leading order. The starting distribution is a hadron-like contribution based on VMD arguments, by using the parton distribution functions of the pion from Ref. [74], the similarity of the o and p mesons, and a proportionality factor, i, to account for the sum of
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Fig. 25. Comparison of the GRV leading-order and next-to-leading-order parametrisations of the photon structure function FA . In (a) the photon structure function FA is shown in leading order (dash) and next-to-leading order (full), for four active #avours and for several values of Q, 0.8, 1.9, 15 and 100 GeV, and in (b) the ratio of the next-to-leading order and the leading-order parametrisations is explored for the same values of Q.
o, u and mesons, as explained in detail in Section 3.5. The functional form of the starting distribution is qA"q A"gA"i(4pa/f ) fp(x, Q ), where xf (x, Q )&x@(1!x)A with b'0. The p M parameter 1/f "2.2 is taken from Ref. [8], leaving i as the only free parameter, which is M obtained from a "t to the data in the region 0.71(Q(100 GeV, for ='2 GeV, to avoid resonance production. The point-like contribution is chosen to vanish at Q"Q and for Q'Q , it is generated dynamically using the full evolution equations, as is also done for the evolution of the hadron-like component. The full evolution equations for massless quarks, with K"0.2 GeV, are used in the DIS factorisation scheme, while removing all spurious higher A order terms. The charm and bottom quarks are included via the Bethe-Heitler formula for m "1.5 GeV and m "4.5 GeV, and at high values of = they are treated as massless quarks in the evolution. The data used in the "ts are published data from Refs. [67}70,72,83,97}100] and preliminary data from Ref. [96], all subject to the additional requirement ='2 GeV mentioned above. The leading-order and next-to-leading-order predictions are shown in Fig. 25 for several values of Q. The values chosen are: a very low scale, the lowest Q value where a measurement of FA from LEP is available, and two typical values of Q for structure function analyses at LEP, Q"0.8, 1.9, 15 and 100 GeV. The behaviour of the leading-order and next-to-leading-order predictions are rather di!erent at very low and at high values of x. In the central part 0.1(x(0.9, and for Q"1.9 GeV they di!er by no more than 20%. Because none of the predictions is consistently higher in this region, and since the experiments integrate over rather large ranges in x when measuring the photon structure function FA , it will be very hard to disentangle the two in this region in the near future. At lower values of x however the data start to be precise enough.
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Fig. 26. Comparison of the higher order structure function FA from AFG and GRV. The predicted higher order structure function FA from AFG (dash) is compared to the prediction from GRV (full) for the three values of Q, 2, 15 and 100 GeV.
5. AFG [107]: The strategy used in constructing these parametrisations by Aurenche, Fontannaz and Guillet is very similar to the one used for the GRV parametrisations. The starting scale for the evolution is very low, Q "0.5 GeV. This value is obtained from the requirement that the point-like contribution to the photon structure function vanishes at Q"Q . Consequently, the input is taken as purely hadron-like, based on VMD arguments, where a coherent sum of low mass vector mesons o, u and is used. The AFG distributions are obtained in the MS factorisation scheme. Therefore, the input distributions contain an additional technical input, as shown in Eq. (B.17), which was derived from a study of the factorisation scheme dependence and the momentum integration of the box diagram. With this choice of the factorisation scheme and the technical input, the parton distribution functions are universal and process independent. In contrast, the DIS scheme introduces a process dependence, because the C as given by A A Eq. (B.14), which is absorbed into the quark distribution functions when using the DIS scheme, A contains process-dependent terms, as explained in Ref. [107]. The evolution is performed in the massless scheme for three #avours for Q(m"2 GeV and for four #avours for Q'm, always using K"0.2 GeV. No parton distribution function for bottom quarks is available. An additional scale factor, K, is provided to adjust the VMD contribution. In the standard set this parameter is "xed to K"1. Otherwise K is obtained from a "t to published data taken from Refs. [67}70,83]. In Fig. 26 the higher order prediction of FA from AFG is compared to the GRV prediction for three values of Q, 2, 15 and 100 GeV. At low Q there are large di!erences between the two predictions which tend to get smaller as Q increases. 6. GS [30,108]: The parton distribution functions from Gordon and Storrow are available in leading and in next-to-leading order. They were "rst constructed in Ref. [30], starting the evolution at Q "5.3 GeV, and later updated in Ref. [108] by including more data, and reducing the starting scale to Q "3.0 GeV. Since the data on the photon structure function
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Fig. 27. Comparison of the leading-order GRV and GRSc parametrisations of FA . In (a) the photon structure function FA is shown in leading order for the GRV (dash) and the GRSc (full) parametrisations, for four active #avours and for several values of Q, 0.8, 1.9, 15 and 100 GeV, and in (b) the ratio of the GRSc and the GRV parametrisations is explored for the same values of Q.
FA only indirectly constrain the gluon distribution of the photon, a "rst attempt was made to "t jet production data from TOPAZ [71,109], and AMY [110], which show some sensitivity to the gluon distribution via the contributions of resolved photon processes to the jet production. However, the data are not precise enough to considerably constrain the gluon distribution function. Due to the large starting scale, the input distributions cannot be based only on VMD arguments. The authors choose a VMD input similar to the one used in the GRV ansatz, but supplement it with an ansatz of the point-like component, based on the lowest order BetheHeitler formula for three light quarks. The quark masses are constrained to ful"ll 0.25(m "m (0.4 GeV and 0.35(m (0.55 GeV and are "tted to the data, resulting in masses of 0.29 GeV for up and down quarks and 0.41 GeV for strange quarks, as explained in Ref. [30]. As both contributions, the hadron-like and the point-like, vanish as xP1, the GS quark distribution functions are greatly suppressed at high values of x compared to, for example, the quark distribution functions from GRV. The contribution from charm quarks is added via the Bethe-Heitler formula with a charm quark mass of m "1.5 GeV. The evolution is performed for three light #avours using K"0.2 GeV and this is supplemented with the Bethe-Heitler charm contribution up to Q"50 GeV. At Q"50 GeV this result is matched to a four #avour evolution ansatz, which was started at Q"10 GeV, in such a way that FA is continuous. For Q'50 GeV a four #avour massless approach is chosen, which is known to overestimate the charm contribution. To remove the negative structure function FA obtained at large x when working in next-to-leading order in the MS scheme, the quark distributions are supplemented by a technical input, de"ned in Eq. (B.14), which removes the divergence. The leading-order and next-to-leading-order parton distribution functions are connected to each
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other by the requirements that they are identical at Q , and that the gluon distribution is the same for the leading-order and the next-to-leading-order parametrisations. The data used in the "ts are published data from Refs. [67}72,83,84,99,103,109}112] and preliminary data from Refs. [96,104,113]. 7. GRSc [114]: The parton distribution functions from GluK ck, Reya and Schienbein are constructed using basically the same strategy as the GRV parametrisations. However, in addition to the use of the new pion input from Ref. [115], some conceptual changes have been made. As a result, as in the case of AFG, no parameters have to be obtained from a "t to the FA data. Like the AFG parametrisations, the GRSc parametrisations use a coherent sum of vector mesons, and therefore there is no free parameter i, which was used in the case of the GRV parametrisations when using an incoherent sum. The treatment of a has been changed from the approximate next-to-leading-order formula to an exact solution of the renormalisation group equation for a in next-to-leading order, using K"0.204/0.299 GeV in leading/next-to-leading order. The contribution of charm quarks was changed from 1.5 to 1.4 GeV. The numerical di!erences between F A as predicted by the leading-order GRV and GRSc parametrisations is shown in Fig. 27 for several values of Q. At low values of Q the two parametrisations are very di!erent especially at low values of x. For increasing Q they get closer, and for Q'15 GeV the di!erences are smaller than 10%. 8. SaS [43,116,117]: Two sets are constructed by Schuler and SjoK strand in leading order, SaS1 using Q "0.36 GeV as starting scale and SaS2 for Q "4 GeV. Both sets use K"0.2 GeV, the massless evolution equations for light quarks, and the Bethe-Heitler formula, Eq. (32), for contributions of charm and bottom quarks with masses m "1.3 GeV and m "4.6 GeV. The @ leading order parton distribution functions are derived both in the MS and the DIS scheme. A The parameters are "tted to data for Q'Q and the dependence on the photon virtuality P is kept to allow for an extension to virtual photons, discussed below. The motivation for the choice of the two sets SaS1 and SaS2 is an investigation of the correlation between the size of the hadron-like input function and the starting scale Q . Consequently, the main di!erence between the two sets is that the set SaS2 contains a larger VMD contribution compared to the set SaS1, which is needed to still "t the data, while starting at a much larger scale Q . For the set SaS1 the normalisation of the VMD contribution, as well as the starting scale is determined from the analysis of cp scattering data, only the shape of the VMD distribution is "tted to the data of the photon structure function. In contrast, for the set SaS2, the starting scale is "xed to Q "4 GeV, the functional form of the distribution functions is changed, and an additional proportionality factor K is introduced for the VMD contribution, and "tted to the data, resulting in K"2.422. This factor corresponds to an inclusion of higher mass vector mesons to compensate for the fact that no point-like contribution is allowed to evolve from 0.36(Q(4 GeV. The subdivision into point-like and hadron-like parton distribution functions is made explicit in the SaS distribution functions, allowing for an independent treatment of the two, for example,
The scheme dependence is introduced by hand into the leading-order parton distribution functions, by including the universal part of C (x)"3+[x#(1!x)] ln(1/x)!1#6x(1!x),, which formally is of next-to-leading order. This A choice is motivated by the fact that although formally C (x) is of higher order, numerically it is important. A
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Fig. 28. The structure function FA from the SaS parton distribution functions for several values of Q. The values chosen are the starting scale of the evolution of the sets SaS2, Q"4 GeV and Q"100 GeV. The structure function FA is shown in (a) for the individual sets 1D, 1M, 2D and 2M and in (b) the sets 1M, 2D and 2M are divided by the prediction of the set SaS1D.
in the simulation of the properties of hadronic "nal states, originating from the hadron-like and the point-like part of the photon structure function. The point-like part is further factorised into a term which describes the probability of the photon to split into a qq state at a perturbatively large scale, and a so-called state distribution which describes the parton distribution functions within this qq state. This subdivision is made to facilitate the proper use of the parton distribution functions in Monte Carlo programs, when using the parton shower concept. The data used in the "ts are published data from Refs. [67,69}71,83,98,103] and preliminary data from Ref. [96]. Fig. 28(a) shows the FA prediction of the individual sets for Q"4 and 100 GeV, and in Fig. 28(b) they are normalised to the set SaS1D. Some general trends can be seen from the "gure. The SaS2 sets predict a larger hadron-like part and therefore they are larger at small values of x, than the SaS1 sets. This di!erence decreases with increasing Q, as can be seen from the ratios displayed in Fig. 28(b). At large values of x the hadron-like part is small and the di!erence mainly comes from the di!erent treatment of C (x), which makes the sets 1D and 2D agree A with each other for 100 GeV and also the sets 1M and 2M. At large values of x the structure function FA is more strongly suppressed when using the MS scheme, which makes the sets 1M and 2M vanish faster as x approaches unity. The main features of the parton distribution functions for real photons described above, are listed in Table 3. In general, the parametrisations of the parton distribution functions do not di!er much in the quark distribution functions at medium values of x, because they are constrained by the FA data used in the "ts. In contrast, in the region of high and low values of x the available parton
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Table 3 The parton distribution functions for real photons. The table contains a compilation of the most recent versions of the available parton distribution functions for real photons. The abbreviations evol means that the charm contribution is included as massless or massive quark in the evolution equations, whereas BH denotes the inclusion of massive charm quarks via the Bethe-Heitler formula Authors
Set
DG
Q (GeV)
Scheme
K (GeV)
Gluon
1.0
0.400
1 2 3
4.0 4.0 1.0
0.200 0.200 0.200
b"0
WHIT
1 2 3 4 5 6
4.0 4.0 4.0 4.0 4.0 4.0
0.400 0.400 0.400 0.400 0.400 0.400
a"0.5, a"0.5, a"0.5, a"1.0, a"1.0, a"1.0,
GRV
LO HO
0.25 0.30
LAC
0.5
DIS A MS
LO HO
3.0 3.0
MS
LO HO
0.5 0.5
1D 2D 1M 2M
0.36 4.0 0.36 4.0
AFG GS GRSc SaS
DIS A &DIS ' A &DIS ' A &MS' &MS'
c"3 c"9 c"15 c"3 c"9 c"15
Charm
Ref.
evol
[93]
evol evol evol
[95]
BH, BH, BH, BH, BH, BH,
evol evol evol evol evol evol
[101]
0.200 0.200
BH, evol BH, evol
[32]
0.200
BH, evol
[107]
0.200 0.200
BH, evol BH, evol
[108]
0.204 0.299
BH, evol BH, evol
[114]
0.200 0.200 0.200 0.200
BH, BH, BH, BH,
[43]
evol evol evol evol
distribution functions are not well constrained. The highest value of x reached in the measurements of FA is restricted by the minimum value of invariant mass required to be well above the region of resonance production. Therefore, for example, the very di!erent quark distribution functions of GS and GRV are still consistent with the data on FA . For low values of x, measurements became available only recently, and they are not yet incorporated into the presently available parton distribution functions. Consequently, there was considerable freedom in the gluon distribution function, which is important at low-x. This freedom has been exploited in the di!erences of the various sets constructed by several groups. This results in very di!erent gluon distribution functions but also, driven by the gluons, in di!erent quark distribution functions at low values of x. The new data on FA now start to constrain the parton distribution functions also at low values of x, and, although in leading order the photon only couples to quarks, the measurements of FA give an indirect constraint on the gluon distribution function as well.
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Fig. 29. Predictions for the Q evolution of the photon structure function FA for various x ranges. Predictions of the GRV, SaS and WHIT1 parametrisations are compared to the evolution of the purely point-like part for three light #avours for K "0.232 GeV, combined with FA as predicted by the Bethe-Heitler formula, denoted with PL(uds)#BH(c), and to the hadron-like part of the GRV parametrisation, labelled VMD(GRV).
Also promising is the inclusion of jet production data, either from the reaction ccP jets or cpP jets, which are studied at LEP and HERA. These data are directly sensitive to the gluon distribution function at low values of x, for example, via the boson}gluon fusion diagram or gluon}gluon scattering. In addition, jet production can be used to explore the region of high values of x and to further constrain the quark distribution functions in this region. Because the main subject here is the photon structure function FA , and the HERA results have not yet been incorporated into the construction of parton distribution functions, this interesting topic is not discussed in further detail here. However, the results from the HERA experiment on jet production cross-sections, and what can be learned about the parton distribution function of the photon will be brie#y discussed in Section 9.2.1. As there is the freedom to choose both, the input distribution functions, and the value of K, di!erently for the leading-order and next-to-leading-order parton distribution functions, the predicted structure functions FA in leading and next-to-leading order are similar. However, to perform a meaningful investigation of the sensitivity of the photon structure function FA to the running coupling constant a , by studying the Q evolution of FA , it is mandatory to use the next-to-leading-order approach, in order for the scale K to be "xed. For illustration, the predicted Q evolution of FA for four active #avours, and for various leading-order parametrisations of FA is shown in Fig. 29. The predictions of the GRV, SaS and WHIT1 parametrisations are compared to the evolution of the purely point-like part for three light #avours for K "0.232 GeV, combined with FA as predicted by the Bethe-Heitler formula, denoted with PL(uds)#BH(c), and to the hadron-like part of the GRV parametrisation, labelled VMD(GRV). The bins in x used, correspond to the experimental analyses of Ref. [90]. The hadron-like part of FA dominates at low values of
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x for all values of Q, but for x'0.1 and Q'10 GeV the point-like part is more important, and for x'0.6 the hadron-like part is negligible for Q'10 GeV. As the importance of the hadronlike part of FA decreases for increasing x, the predictions of the GRV, SaS and WHIT1 paramet risations get closer to each other. All parametrisations predict a strongly increasing slope for increasing x, driven by the point-like contribution, with WHIT1 showing the #attest behaviour. Several parton distribution functions for transverse virtual photons have been constructed. They can be applied to any process which is dominated by the contribution of transverse virtual photons. There is no unique prescription on how to extend the parton distribution functions for P'0, and di!erent approaches have been performed. All parton distribution functions are constructed such that they reproduce the correct limits for small and large values of P. For P"0 the parton distribution functions for real photons are recovered, and the limit K;P;Q is given by the perturbative QCD results of Refs. [49,50]. There exists one simple approach by Drees and Godbole, which is independent of the speci"c choice of parton distribution functions used for P"0, and therefore can be applied on top of any of the existing parton distribution functions listed above. In addition, three parton distribution functions for real photons, GRV, GRSc and SaS, have been extended to also incorporate the region of P'0. The extension of the GRV parton distribution functions is called GRS. 1. DG [14]: In the simple model by Drees and Godbole the parton distribution functions for virtual photons are obtained by simple, P-dependent, multiplicative factors from the parton distribution functions for real photons, where by construction the gluon distribution function is more suppressed than the quark distribution functions, as suggested in Ref. [118]:
Q#P P#P ,qA (x, Q)¸ , qA (x, Q, P)"qA (x, Q) I I I Q#P ln P gA(x, Q, P)"gA(x, Q)¸ , ln
(49)
where P should be chosen to be a typical hadronic scale, which means, to be in the range K4P41 GeV. With this, the parton distribution functions are globally suppressed, which means the x dependence of the parton distribution functions for real photons is not altered for P'0. 2. GRS [119]: In the parton distribution functions from GluK ck, Reya and Stratmann a boundary condition, similar to the one used for real photons, is applied at Q"max(P, Q ). This condition allows to smoothly interpolate between P"0 and P
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Fig. 30. Comparison of leading-order parametrisations of the photon structure function FA for virtual photons. The photon structure function FA is shown for the SaS1D (full) and the GRS (dash) prediction for Q"30 GeV and for several values of P, 0, 0.05, 0.5, 1 and 5 GeV, always for three light #avours, n "3. The predictions decrease with increasing P. For the SaS1D prediction the choice of P suppression is made such to have the result which is closest to the GRS prediction, resulting in IP2"2. Fig. 31. The variation of the P suppression of FA in the SaS1D parametrisations. The di!erent choices of the P suppression of FA are shown by varying the parameter IP2 for Q"30 GeV and for two values of P. The curves shown in (a) and (b) are for P"0.05 and 0.5 GeV, respectively. The predictions for the di!erent choices of IP2 are all divided by the result obtained for IP2"2.
photons the photon virtuality should entirely be taken care of by the #ux factors, which are valid for Q
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Fig. 32. The predicted P suppression of the photon structure function FA (uds). Shown are the prediction of FA from the GRS parton distribution functions (dash), together with the prediction from the SaS1D parton distribution functions for two modes of the P suppression. The two modes chosen are the recommended suppression (IP2"0, full) and the one which is most similar to the GRS suppression (IP2"2, dot-dash). The values of FA are normalised to the prediction for real photons, P"0. The curves are calculated for Q"30 GeV, for three values of x, 0.1, 0.5 and 0.9, and for three #avours. Fig. 33. Comparison of the photon structure function FA for virtual photons with the purely perturbative point-like part. The predictions of the structure function FA are shown for three active #avours, for two values of Q, 10 and 100 GeV, and for two values of P, 0.5 and 1.0 GeV, and for K "0.232 GeV. The prediction from the GRS parametrisations is compared to the purely perturbative point-like part which is equivalent to the prediction from Uematsu and Walsh (UW). In (a) the individual predictions are shown and in (b) the GRS predictions are divided by the contribution of the purely point-like part.
prediction is about 8% higher and for smaller values of x the rise is much faster than in the case of the SaS1D prediction. As soon as P'0.5 GeV they perfectly agree with each other for x'0.1. The theoretical uncertainty on how the structure function FA is suppressed for increasing P is explored in the SaS distribution functions. In Fig. 31 the various choices are compared to the choice which is closest to the GRS prediction. The larger the value of P the more the various choices di!er, as can be seen from Fig. 31(a), where at P"0.05 GeV the predictions are close together, whereas at P"0.5 GeV, Fig. 31(b), sizeable di!erences are seen. Taking the variations as an estimate of the theoretical uncertainty, it amounts to about 20% at P"0.5 GeV. Although the absolute predictions for FA from the GRS and SaS1D parametrisations agree quite well for P'0.5 GeV, they di!er in the relative suppression as a function of P, shown in Fig. 32 for Q"30 GeV, and for several values of x. The GRS predictions are compared to the ones from SaS1D using IP2"0 and 2. The suppression decreases with increasing x and the suppression as predicted by SaS, using the recommended scheme, IP2"0, is always stronger than the one predicted by GRS. The kinks in the distributions at P"0.36 and 0.25 GeV for the SaS and GRS predictions are due to the boundary conditions applied.
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Fig. 34. The point-like and hadron-like contributions to FA . The predictions of the SaS1D (full) and the GRS (dash) parametrisations are shown for Q"30 GeV and for two values of P. (a) is for P"0 and (b) uses P"1 GeV. For the GRS prediction in addition the contribution from the point-like process alone (dot-dash) is shown.
Based on the GRS parametrisations the sensitivity to the non-perturbative input distribution functions can be studied. In Fig. 33 the full solution for three light quark species is compared to the purely point-like part which is equivalent to the prediction from Uematsu and Walsh from Ref. [49]. The comparison is done for two values of Q, 10 and 100 GeV, and for two values of P, 0.1 and 1.0 GeV, and all predictions are for K "0.232 GeV. The GRS parametrisations predict a signi"cant hadron-like contribution at small values of x for all values of P. The important decreases for increasing Q and even stronger for increasing P, as can be seen from Fig. 33(b), where the ratio of the GRS prediction and the purely point-like part is shown. At large values of x the hadron-like contribution is less important. For example, the hadron-like contribution amounts to less than 10% for x'0.3 for Q"100 GeV and P"1.0 GeV. This is a region which is still accessible within the LEP2 programme, however only with very limited statistics. The last issue discussed in the comparison of the SaS1D and the GRS parametrisations is the contribution to FA from the point-like and hadron-like production of charm quark pairs. The two predictions are shown in Fig. 34 for Q"30 GeV and for two values of P, 0 and 1 GeV. The mass of the charm quark is m "1.5 GeV for the GRS parametrisation, whereas SaS uses m "1.3 GeV. For the GRS prediction in addition the contribution from the point-like part alone is shown. The point-like contribution is found to dominate for x'0.1, whereas at smaller values of x the hadron-like component gives a signi"cant contribution and dominates as x approaches zero. The di!erence between the two predictions for the point-like part is entirely due to the di!erent choice for the mass, which means when changing the mass in GRS to m "1.3 GeV they are identical. However, GRS and SaS di!er in the contribution from the hadron-like part, with GRS predicting a faster rise for small values of x. This di!erence is due to the di!erent gluon distribution functions. For increasing P the hadron-like part gets less important.
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Experimentally, the measurement of the heavy quark contributions to FA is very di$cult, mainly because of the low statistics available. Firstly, the heavy quark production is suppressed by the large quark masses and secondly, to establish a heavy quark contribution, the quark #avour has to be identi"ed, which can only be done with small e$ciencies. With the available data the measurement of FA is hopeless, because, due to the large bottom mass and the small electric charge the number of events is too small. However, the measurement of the charm contribution to FA is likely to be performed soon for the "rst time, because experimentally about 30 events with positively identi"ed charm quarks are available, as has been reported in Ref. [122].
5. Tools to extract the structure functions The general experimental procedure to measure structure functions is the following. The data are divided into ranges in Q and the structure functions are obtained as functions of x from the distributions of measured values of x, usually denoted by x . If the energies of both incoming particles are known, like in the case of deep inelastic charged lepton}nucleon scattering as, for example, in electron}proton scattering at HERA, the values of Q and x can be obtained from measuring the energy and angle of the scattered electron. Consequently, in regions of acceptable resolution in Q and x as measured from the scattered electron, the proton structure function F can be derived without the measurement of the hadronic "nal state. In addition, the known energy of the proton can be used to replace some less well-measured quantities and to obtain Q and x from the hadronic "nal state. For deep inelastic electron}photon scattering the energy of the incoming quasi-real photon is not known. It could only be obtained from the measurement of the energy of the corresponding electron. For most of the phase space of quasi-real photons the corresponding electrons are not observed in the detectors and no measurement of the photon energy can be performed. Only in the situation of the exchange of two highly virtual photons both electrons are observed in the detectors and the invariant mass of the photon}photon system, as well as x, can be obtained from the two scattered electrons. Consequently, for the measurement of the structure functions of the quasi-real photon, x has to be derived using Eq. (6) from measuring the invariant mass of the "nal state X. In the case of lepton pair production it is required that both leptons are measured in the tracking devices of the detectors, and an accurate measurement of = can be performed. In contrast, the hadronic "nal state is usually only partly observed in the detectors and the measurement of = is much less precise. Due to this, a good description of the observed hadronic "nal state by the Monte Carlo models is much more important for the measurement of the photon structure function, than for the measurement of the proton structure function at HERA. The value of x is obtained from the measurement of the visible hadronic mass = , together with the well measured Q, and therefore the uncertainty of x is completely dominated by the uncertainty of = . The uncertain ty of = receives two contributions. Firstly, = is a!ected by the uncertainty of the measurement of the seen hadrons, which are observed by tracking devices and electromagnetic as well as hadronic calorimeters, and secondly the measurement of = su!ers from the fact that some of the hadrons are scattered outside of the acceptance of the detectors. To account for these de"ciencies, in most of the analyses the structure function FA is obtained from an unfolding procedure which relies on the correlation between the measured x and the underlying value of x, as predicted by
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the Monte Carlo programs. Therefore, the Monte Carlo programs and the unfolding programs are the most important tools used in the measurement of photon structure functions. They are discussed in Sections 5.1 and 5.2, respectively. 5.1. Event generators Event generators are extensively used in the determination of photon structure functions. In this section the most commonly used programs are discussed. Only the main features of the Monte Carlo programs relevant for deep inelastic electron}photon scattering are described, details can be found in the individual program manuals, and a general overview is given in Ref. [123]. There exist two groups of programs relevant for the measurement of photon structure functions. The "rst group deals with low multiplicity "nal states, like resonances, charm quark pairs, or lepton pairs. The programs used for the measurement of the QED structure functions are BDK, GALUGA and Vermaseren, where by now the Vermaseren program can be regarded as the standard Monte Carlo for lepton pair production. 1. GALUGA [7]: The GALUGA Monte Carlo is a more recent program, which contains an implementation of the full cross-section formula from Ref. [4]. It is generally not used as an event generator, but as a useful tool to investigate the importance of the individual terms to the di!erential cross-section, as listed in Eq. (14). 2. Vermaseren [1,124}126]: In most applications the Vermaseren program is based only on the cross-section for the multiperipheral diagram, shown in Fig. 4(a). The full dependence on the mass of the muon and on P is kept. The program generally is used to generate large size event samples, which are compared to the data. Sometimes this program is also abbreviated with JAMVG by using the initials of the author. 3. BDK [62,127}129]: The BDK program is similar to the Vermaseren program, and in addition QED radiative corrections to the process are contained. This program is mainly used for the determination of the radiative corrections to be applied to the data which, after correction, are compared to the predictions of the Vermaseren Monte Carlo. The QED predictions of the three programs are very similar and they nicely agree with the data, as discussed in Section 6. The second group of programs is used for the determination of the hadronic structure function FA . The situation for the multi-particle hadronic "nal state is more complex than for the case of the leptonic "nal state, as it involves QCD. Because the multi-particle hadronic "nal state cannot be predicted by perturbative QCD, there is some freedom on how to model it, and the available programs follow di!erent philosophies to predict the properties of the multi-particle hadronic "nal state. The programs can be further subdivided into two classes. The "rst class consists of the special-purpose Monte Carlo programs TWOGAM [123] and TWOGEN [130], which contain only electron}photon scattering reactions, and are therefore very hard to test thoroughly, except by No detailed description of this Monte Carlo program is publically available.
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using the electron}photon scattering data themselves. This is dangerous, as the measurement of the hadronic structure function and the modelling of the hadronic "nal state are intimately related. For this reason, and also because the programs do not contain parton showers, their importance is gradually decreasing. The second class consists of the general-purpose Monte Carlo programs HERWIG [131}137], PHOJET [138,139] and PYTHIA [140]. These general-purpose Monte Carlo programs are also successfully used to describe electron}proton and proton}proton interactions. Even more important for electron}photon scattering is the fact that some of the parameters are constrained by electron}proton and proton}proton scattering data, therefore leaving less freedom for adjustments to the electron}photon scattering data. The general procedure for the event generation by Monte Carlo methods splits the reaction into di!erent phases, and in each of these phases, speci"c choices are made. For deep inelastic electron}photon scattering "rst a photon is radiated from one of the electrons using an approximation for the photon #ux, discussed in Section 3.2. Then a parton inside the photon is selected according to the prediction of one the various parton distribution functions, discussed in Section 4. The selected parton takes part in the hard sub-process, which is generated using "xed-order matrix elements. A con"guration for the photon remnant is chosen. The emission of further partons is generated from the initial partons, using a prescription of the backward evolution of the initial state parton shower, and from the outgoing partons, modelled by the "nal state parton shower, which is identical to the parton shower used in the e>e\ annihilation events. Finally, all partons are converted into hadrons by means of some hadronisation model, and these hadrons are allowed to decay, using decay tables. The special-purpose Monte Carlo programs used are: 1. TWOGAM [123]: The special-purpose Monte Carlo program TWOGAM was developed within the DELPHI collaboration. The events are separated into three event classes, point-like events, hadron-like events and the so-called resolved photon component. The simulation of point-like events is based on a full implementation of Eq. (14) using the QED cross-sections, with free values chosen for the light quark masses. The hadron-like events are generated according to some VMD prescription. The resolved photon component is added for the scattering of two real, or virtual, photons with transverse polarisation in the following way. The probability to "nd a parton in a photon is given by a set of parton distribution functions for real photons, suppressed by a factor which depends on the virtuality of the photon. The generated partons then undergo a hard 2P2 scattering process. No parton showers are included, and the hadronisation is based on the Lund string model. By using this concept also the virtual photon is allowed to #uctuate into a hadronic state. 2. TWOGEN [130]: The special-purpose Monte Carlo program TWOGEN was developed within the OPAL collaboration. The version used for structure function analyses is called F2GEN. This program is in principle based on Eq. (14), but neglects all but the term proportional to p . 22 Then the di!erential cross-section is expressed as a product of the transverse}transverse luminosity function for real and virtual photons, and the cross-section p . The cross-section 22 p is implemented only for real photons, and as given in Eq. (20), it is proportional to FA . The 22 program generates only the multiperipheral diagram. The angular distribution of the quark anti-quark "nal state is chosen to be like in the case of leptons for point-like events and according to a limited transverse momentum model, called peripheral, for hadron-like events.
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A mixture of the point-like and peripheral events can be generated based on a hit and miss method. This combination is called perimiss. No parton showers are included, and the hadronisation is based on the Lund string model. The general-purpose Monte Carlo programs used are: 1. HERWIG [131}137]: The general-purpose Monte Carlo program HERWIG has been extended to electron}photon processes in the LEP2 workshop [123]. The "rst available version was HERWIG5.8d. The next version used in experimental analyses is HERWIG5.9, which improved as detailed in Section 7.1. The improvements are called HERWIG5.9#k and HERWIG5.9# k (dyn), re#ecting the changes applied to the intrinsic transverse momentum of the quarks in the photon, k , either with "xed or dynamically (dyn) adjusted upper limit. In the HERWIG model the photon #ux is based on the EPA, Eq. (24), and the hard interaction is simulated as eqPeq scattering, where the incoming quark is generated according to a set of parton distribution functions. The incoming quark is subject to an initial state parton shower which contains the cPqq vertex. The initial-state parton shower is designed in such a way that the hardest emission is matched to the sum of the matrix elements for the higher order resolved processes, gPqq and qPqg and the point-like cPqq process. The parton shower uses the transverse momentum as evolution parameter and obeys angular ordering. This procedure dynamically separates the events into point-like and hadron-like events, and this separation will be di!erent from the choice made in the parton distribution functions. For hadron-like events the photon remnant gets a transverse momentum k with respect to the direction of the incoming photon discussed above, where originally the transverse momentum was generated from a gaussian distribution. The outgoing partons undergo "nal-state parton showers as in the case of e>e\ annihilations. The hadronisation is based on the cluster model. 2. PHOJET [138,139]: The general-purpose Monte Carlo program PHOJET is based on the dual parton model from Ref. [141]. It was designed for photon}photon collisions, where originally only real or quasi-real photons were considered. It has recently been extended to match the deep inelastic electron}photon scattering case, if one of the photons is highly virtual. It can also be used for the scattering of two highly virtual photons. Both photons are allowed to #uctuate into a hadronic state before they interact. For the case of deep inelastic scattering the program is not based on the DIS formula, but rather the c夹c cross-section is calculated from the cc cross-section by extrapolating in Q on the basis of the Generalised Vector Dominance model using Ref. [142]. The events are generated from soft and hard partonic processes, where a cut-o! on the transverse momentum of the scattered partons in the photon}photon centre-of-mass system is used to separate the two classes of events. The present value of this cut-o! is 2.5 GeV, which means that for =(5 GeV only soft processes are generated. This results in a strange behaviour of the = distribution for =(5 GeV, which has to be treated with special care. The sum of the processes is matched to the deep inelastic scattering cross-section, or in other words to FA . However, in the present version this matching is imperfect, which results in the fact that the actual distribution in x generated is not the same as one would expect from the input photon structure function FA used in the simulation. This makes it di$cult to use PHOJET for a direct unfolding procedure, but rather it should only be used to determine the transformation matrix relating the generated value of x to the observed value x . Initial-state parton showers are simulated with a backward evolution algorithm using the transverse momentum as evolution
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scale. Final-state parton showers are generated with the Lund scheme. Both satisfy angular ordering implied by coherence e!ects. The hadronisation is based on the Lund string model. 3. PYTHIA [140]: In the general-purpose Monte Carlo program PYTHIA the process is implemented rather similar than in the HERWIG program. PYTHIA includes the reaction as deep inelastic electron}quark scattering, where the quarks are generated according to parton distribution functions of the quasi-real photon. The #ux of the quasi-real photon has to be externally provided, and the corresponding electron is only modelled in the collinear approximation. The program relies on the leading-order matrix element for the eqPeq scattering process. Higher order QCD processes are subsequently generated via parton showers, without a matching prescription to the exact matrix elements. Initial- and "nal-state parton showers are implemented, using the parton virtuality as the evolution parameter. The separation into point-like and hadron-like events is taken from the parton distribution functions, if available. In this way, a consistent subdivision into point-like and hadron-like events can be achieved in the event generation and the parton distribution functions, by using the SaS parton distribution functions together with PYTHIA. For hadron-like events the photon remnant gets a k generated from a Gaussian distribution, and for point-like events the transverse momentum follows a powerlike behaviour, dk/k, with k "Q as the upper limit. The hadronisation is based on the Lund string model. Due to the di!erent choices made in the various steps of the event generation the predictions of the Monte Carlo programs di!er signi"cantly. The quality of the description of the data by the various programs is an active "eld of research. The results of the investigations are discussed in Section 7.1. 5.2. Unfolding methods The determination of the structure function FA (x, Q) involves the measurement of x and Q. For the hadronic "nal state the resolution in =, and therefore the resolution in x, is not very good, due to mismeasurements of the hadrons and losses of particles outside of the acceptance of the detectors. Therefore, unfolding programs are used to relate the observed hadronic "nal state to the underlying value of x. The unfolding problem as well as the programs used for the unfolding are described below. The principle problem which is solved by the unfolding is the following. The distribution g of a quantity u (e.g. x ) directly measured by the detector is related to the distribution f of a partonic variable u (e.g. x) by an integral equation which expresses the convolution of the true distribution with all e!ects that occur between the creation of the hard process and the measurement
g(u)" A(u, u) f (u) du#B(u) ,
(50)
A new version of the PYTHIA program exists, PYTHIA6.0. The description given here is still based on the capabilities of the version PYTHIA5.7, because this is the version which was used in the experimental analyses.
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where B(u) represents an additional contribution from background events. The task of the unfolding procedure is to obtain the underlying distribution f , from the measured distribution g using the transformation A and the background contribution, usually obtained from Monte Carlo simulations. This is done either by discretising and inverting the equation, or by using Bayes' theorem. The relevant programs used, are based on di!erent statistical methods and have slightly di!erent capabilities. They are discussed below. 1. RUN [143}145]: The RUN program by Blobel is used since long in structure function analyses. It is based on a regularised unfolding technique and allows for an unfolding in one dimension. The integral equation is transformed into a matrix equation, and solved numerically, leading to the histogram f (u). This simple method can produce spurious oscillating components in the result due to limited detector resolution and statistical #uctuations. Therefore the method is improved by a regularisation procedure which reduces these oscillations. The regularisation is implemented in the program using the assumption that the resulting underlying distribution has minimum curvature. Technically, the unfolding program RUN works as follows. A set of Monte Carlo events is used as an input to the unfolding program. These events are based on an input FA (x, Q) and implicitly carry the information about the response function A(x , x). A continu ous weight function f (x) is de"ned which depends only on x. This function is used to calculate
an individual weight factor for each Monte Carlo event. The weight function is obtained by a "t of the x distribution of the Monte Carlo sample to the measured x distribution of the data, such that the reweighted Monte Carlo events describe as well as possible the x distribution of the data. After the unfolding both distributions agree with each other on a statistical basis. The unfolded FA (x, Q) from the data is then obtained by multiplying the input FA (x, Q) of the Monte Carlo with the weight function f (x).
2. GURU [146]: The GURU program by HoK cker and Kartvelishvili is a more recent, slightly di!erent, implementation of an regularised unfolding technique based on the method of single vector decomposition (SVD). In this method the matrix A is decomposed into the product A";S<2, where ; and < are orthogonal matrices and S is a diagonal matrix with nonnegative diagonal elements, the so-called singular values. The regularisation procedure of the GURU program is very similar to the one used in the RUN program. The problem is regularised by adding a regularisation term proportional to the regularisation parameter q. In contrast to the automated procedure to determine the value of q implemented in the RUN program, in the GURU program the value of q has to be adjusted to the problem under study, by determining the number of terms of the decomposition which are statistically signi"cant, as explained in detail in Ref. [146]. However, there is one practical advantage of the GURU program, it allows for an unfolding in several dimensions. This is a very interesting feature, as two-dimensional unfolding is a promising candidate to improve on the error of FA stemming from the dependence of the unfolded result of FA on the underlying Monte Carlo program used to simulate the hadronic "nal state. 3. BAYES [147]: The BAYES program by D'Agostini is based on Bayes' theorem. This method is completely di!erent from the two above, because the matrix inversion is avoided by using Bayes' theorem. Starting point is the existence of a number of independent causes C , i"1, 2,2, n , G which can produce one e!ect, E, for example, an observed event. Then if one knows the initial
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probability of the cause, P(C ), and the conditional probability, P(E"C ) of the cause C to G G G produce the e!ect E, then Bayes' theorem can be formulated as P(E"C )P(C ) G G P(C "E)" . (51) G L P(E"C )P(C ) I I I This formula can be used for multidimensional unfolding. In the one-dimensional case the following identi"cations can be made: P(C )"f , P(E"C )"A and the distribution of the G G e!ects E is equivalent to g. The best results are obtained if one uses some a priori knowledge on P(C ), then after some iterations the "nal result is obtained. A careful study of the possible G bias due to the choice of the initial distribution has to be performed. In general, when using the one-dimensional unfolding, there is not much di!erence in the results obtained with the various methods. Clearly, for all programs the dependence on the transformation between the generated variables and the measured ones as given by P(E"C ), or A has to be carefully G investigated by using the predictions of several Monte Carlo models. Traditionally, the unfolding was performed only in the variable x. Motivated by the limited quality of the description of the observed hadronic "nal state by the Monte Carlo programs, discussed in Section 7.1, there have been investigations to study the unfolding in two dimensions to accommodate this shortcoming, as discussed, for example, in Ref. [148]. The main idea is the following. In the one-dimensional unfolding using the variable x, the result is independent of the actual shape of the input distribution function f (x) used in the unfolding and depends only on the transformation A(x , x), which partly depends on the Monte Carlo model used, but also to a large extent on the detector capabilities which are independent of the chosen model. By using a second variable, v, the same argument applies to this variable. Now the result is independent of the joint input distribution function f (x, v) of x and v and only the transformation A(x , v , x, v) matters, which now also depends on the transformation of v. Because only the transformation of v but not its actual distribution a!ects the unfolding result, a part of the dependence on the Monte Carlo model is removed. There are some indications, for example, shown in Ref. [148], that the unfolding in two dimensions may reduce the systematical error of the structure function results, and this technique has been used in recent structure function analyses, as explained in Section 7.2. Although some improvements of the unfolding procedure has been achieved, the main emphasis should be on the understanding of the reasons for the shortcomings of the Monte Carlo programs and on the improvement of their description of the data. This work has been started by the ALEPH, L3 and OPAL collaborations and the LEP Working Group for Two-Photon Physics, reported in Ref. [149], but meanwhile, better tools to cope with the situation are certainly useful.
6. Measurements of the QED structure of the photon The QED structure of the photon has been investigated for all leptonic "nal states e>e\, k>k\ and q>q\. Most results are obtained for k>k\ "nal states for various reasons. For e>e\ "nal states This is of course only true if the same detector parts are populated with particles by the di!erent Monte Carlo models.
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more Feynman diagrams contribute, which makes the analysis in terms of the photon structure more di$cult. The q>q\ "nal states are rare, as the q is heavy, and also q>q\ "nal states are more di$cult to identify, because only the decay products of the q can be observed. The hadronic decays of the q su!er from large background from qq production and the muonic decays of the q from the k>k\ production process. The most promising channel is the one where one q decays to a muon and the other to an electron, but these are very rare. In contrast, the k>k\ "nal states has a clear signature, a large cross section and is almost background free, which makes it ideal for the measurements of the QED structure of the photon. Several measurements of QED structure functions have been performed by various experiments. Prior to LEP, mainly the structure function FA was measured. The LEP experiments re"ned /#" the analysis of the k>k\ "nal state, to derive more information on the QED structure of the photon. The k>k\ "nal state is such a clean environment that it allows for much more subtle measurements to be performed, than in the case of hadronic "nal states. Examples are, the measurement of the dependence of FA on the small, but "nite, virtuality of the quasi-real /#" photon, P, which is often referred to as the target mass e!ect, and the measurement of the structure functions FA and FA , which are deduced from the distribution of the azimuthal angle s, as /#" /#" outlined in Section 3.3. The interest in the investigation of QED structure functions is threefold. Firstly the investigations serve as tests of QED to order O(a), secondly, and also very important, the investigations are used to re"ne the experimentalists tools in a real but clean situation to investigate the possibilities of extracting similar information from the much more complex hadronic "nal state, and thirdly, the measurement of the QED structure of the photon can give some information on the hadronic structure of the photon as well, because at large values of x the quark parton model, which is nothing but QED, gives a fair approximation of the hadronic structure of the photon. The various results are discussed below, starting with the measurements of FA , followed by /#" the measurements of the structure functions FA and FA of quasi-real photons. The "nal /#" /#" topic discussed is the investigation of the structure of highly virtual photons by a measurement of the di!erential cross-section for the exchange of two highly virtual photons, which only recently has been performed quantitatively for the "rst time. The structure function FA has been measured for average virtualities in the range /#" 0.45(1Q2(130 GeV. The results from the CELLO [150], DELPHI [86], L3 [151], OPAL [152], PLUTO [153] and TPC/2c [154] experiments can be found in Tables 9}15. In addition there exist preliminary results from the ALEPH and DELPHI experiment presented in Refs. [155,156]. The ALEPH results are preliminary since more than two years and therefore they are not considered here. The DELPHI results are listed in Table 16. The result at 1Q2"12.5 GeV is going to replace the published measurement at Q"12 GeV, which will still be used here. Special care has to be taken when comparing the experimental results to the QED predictions, because slightly di!erent quantities are derived by the experiments. Some of the experiments express their result as an average structure function, 1FA (x, Q, 1P2)2, measured within their /#" experimental Q acceptance, whereas the other experiments unfold their result as a structure function for an average Q value, FA (x, 1Q2, 1P2). The second choice is much more appropri /#" ate for comparisons to theory, because in this case all experimental dependence is removed, whereas in the "rst case the measured average structure function still depends on the experimental acceptance, which can only approximately be modelled by theory. Fortunately, for not too large
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Fig. 35. The measured average structure function 1FA 2 from the CELLO, DELPHI, L3 and the TPC/2c experi /#" ments, compared to QED predictions. The points represent the data with their statistical (inner error bars) and total errors (outer error bars). The tic marks at the top of the "gures indicate the bin boundaries. The results of the four experiments are compared to four di!erent QED predictions, namely the structure function FA at the lower /#" and upper limits of the Q range studied (dash), and the two quantities 1FA (x, Q, 1P2)2 (dot-dash) and /#" FA (x, 1Q2, 1P2) (full) explained in the text. /#"
bins in Q, and assuming a constant experimental acceptance as a function of Q, the two quantities 1FA (x, Q, 1P2)2 and FA (x, 1Q2, 1P2) are very similar, as can be seen from Fig. 35, /#" /#" where the experimental results on the average structure function 1FA 2 from the CELLO, /#" DELPHI, L3 and TPC/2c experiments are shown together with several QED predictions. The measurements are compared to FA at the lower and upper limits of the Q range studied, and to /#" the two quantities 1FA (x, Q, 1P2)2 and FA (x, 1Q2, 1P2), using the values of 1P2 listed /#" /#" below. Here the average structure function 1FA (x, Q, 1P2)2 is calculated as the average of /#" FA within the Q range used by the experiments, but without taking into account the /#" Q dependence of the cross section. The Q range is divided into 100 bins on a linear scale in Q and for each point in x the average is calculated from all non-zero values of FA . The di!erence /#" between 1FA (x, Q, 1P2)2 and FA (x, 1Q2, 1P2) is small compared to the experimental /#" /#" errors of the CELLO, DELPHI and TPC/2c measurements. However, for the measurement of L3, the size of the di!erence is comparable to the experimental uncertainty, especially at large values of x. Fig. 36 shows the world summary of the FA measurements, where the experimental results /#" are compared either to the predicted 1FA (x, Q, 1P2)2 or to FA (x, 1Q2, 1P2). For the /#" /#"
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Fig. 36. The world summary of FA measurements. The data are compared to FA (x, 1Q2, 1P2), or /#" /#" 1FA (x, Q,1P2)2, with numbers as given in the text. The points represent the data with their statistical (inner error /#" bars) and total errors (outer error bars). The quoted errors for (h) are statistical only. The tic marks at the top of the "gures indicate the bin boundaries.
measurements which quote an average P for their dataset, where 1P2 is either obtained from the Monte Carlo prediction, or from a best "t of the QED prediction to the data, this value is chosen in the comparison. For the comparisons of the other results P"0 is used. The curves shown
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Fig. 37. The measured Q evolution of FA . The measurements of FA as a function of Q for various x ranges /#" /#" compared to QED. The points represent the data with their total errors. The data from Fig. 36 are shown after correcting for the e!ect of non-zero P in the data. The curves correspond to the QED prediction for P"0. The upper two PLUTO points at Q"40 GeV belong to N"8 and 10.
correspond to (a) 1FA (x, 0.14}1.28, 0)2, (b) FA (x, 2.2, 0.05), (c) 1FA (x, 1.4}7.6, 0.033)2, /#" /#" /#" (d) FA (x, 4.2, 0.05), (e) FA (x, 5.5, 0), (f) FA (x, 8.4, 0.05), (g) 1FA (x, 1.4}35, 0)2, /#" /#" /#" /#" (h) 1FA (x, 4}30, 0.04)2, (i) FA (x, 12.4, 0.05), (j) FA (x, 21, 0.05), (k) FA (x, 40, 0), /#" /#" /#" /#" (l) FA (x, 120, 0.066), and (m) FA (x, 130, 0.05), where all numbers are given in GeV. /#" /#" There is agreement between the data and the QED expectations to order O(a) for three orders of magnitude in Q. Some di!erences are seen for the TPC/2c result, but at these low values of Q this could also be due to the simple averaging procedure used for the theoretical prediction. Another way to compare data and theory is exploited in Fig. 37, where the same data is displayed as a function of Q in bins of x with bin sizes of 0.1 if possible, and with central values of x as indicated in the "gure. To separate the measurements from each other an integer value, N, counting the bin number is added to the measured FA . To be able to compare all results to the same /#" QED prediction all data which quote an average P for their measurement are corrected for this e!ect by multiplying the quoted result by the ratio of FA calculated at P"0 and at /#" P"1P2. The measurements which were obtained for di!erent bin sizes in x than the ones used in the "gure are displayed at the closest central value. All curves represent the predicted average FA in the x bin under study, for P"0. In general, the agreement between the data and the /#"
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Fig. 38. The dependence of FA on P and on the mass of the muon. The OPAL data for 1Q2"3 GeV are /#" compared to several QED predictions of FA (x, 1Q2, 1P2, m ), where in (a) 1P2 is varied for a "xed mass of the /#" I muon of m "0.106 GeV and in (b) the mass of the muon is varied for "xed 1P2"0.05 GeV. The variations shown in I (a) are 1P2"0 (dash), 0.05 (full) and 0.1 GeV (dot-dash), and the chosen masses in (b) are m "0.056 (dash), I 0.113> (full), and 0.156 GeV (dot-dash). The points represent the data with their statistical (inner error bars) and \ total errors (outer error bars). The tic marks at the top of the "gures indicate the bin boundaries.
predictions is acceptable and the prediction clearly follows the increasing slope for increasing x observed in the data. The LEP data are precise enough that the e!ect of the small virtuality P of the quasi-real photon can be investigated in detail. As an illustration the comparison is made for the most precise data coming from the OPAL experiment in Fig. 38. The data consist of the dataset at 1Q2"3.0 GeV listed in Table 13. The dependence of FA on P can be clearly established, /#" and the experimental result shown in Fig 38(a) is consistent with the QED expectation for the average value of P predicted by the QED Monte Carlo program Vermaseren, 1P2"0.05 GeV. The dependence of FA on the mass squared of the muon and on P is similar, as can be seen /#" from Eq. (41) and from its approximation Eq. (34). Consequently, the data can also be used to measure the mass of the muon, by assuming the 1P2 value predicted by QED. A precision of about 14% on the mass of the muon can be derived from Fig. 38(b) using the following procedure. A "t to the data using the QED prediction for Q"3.0 GeV and P"0.05 GeV yields as a best "t result m "0.113 GeV, for a s of 12.2 for nine degrees of freedom. The shape of the s distribuI tion is close to a parabola, and by varying the mass in each direction until the minimum s increases by one unit, the error on m is determined. The "nal result is m "0.113> GeV. I I \ Although this is not a very precise measurement of the mass of the muon it can serve as an indication on the precision possible for the determination of K, if it only were for the point-like contribution to the hadronic structure function FA .
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Fig. 39. The measurements of FA /FA and 1/2FA /FA . In (a) the OPAL, the L3 and the preliminary /#" /#" /#" /#" DELPHI results are compared to the theoretical prediction of FA /FA (x, 1Q2) and in (b) to 1/2FA /FA /#" /#" /#" /#" (x,1Q2), always using the structure functions given in Eq. (32). The points represent the data with their statistical (inner error bars) and total errors (outer error bars). The tic marks at the top of the "gures indicate the bin boundaries of the OPAL and L3 analyses. The di!erent curves correspond to the di!erent values of 1Q2, 3.25 (dash), 5.4 (full) and 12.5 GeV (dot-dash). Fig. 40. The measurements of FA and FA . In (a) the OPAL and the L3 results are compared to the theoretical /#" /#" prediction of FA (x, 1Q2) and in (b) to FA (x, 1Q2), always using the structure functions given in Eq. (32). The /#" /#" points represent the data with their statistical (inner error bars) and total errors (outer error bars). The tic marks at the top of the "gures indicate the bin boundaries of the analyses. The di!erent curves correspond to the di!erent values of 1Q2, 3.25 (dash) and 5.4 (full).
The structure functions FA and FA are obtained from the measured shape of the /#" /#" distribution of the azimuthal angle s, which can be written as dN/ds&(1!A cos s#B cos 2s) .
(52)
For small values of y, the two parameters A and B can be identi"ed with FA /FA and /#" /#" /FA , by comparing to Eq. (31), which is valid in the limit o(y)"e(y)"1. The two FA /#" /#" parameters A and B are "tted to obtain the structure function ratios. By multiplying the measured structure function ratios with the measured FA , the structure functions FA and FA are /#" /#" /#" obtained. The error of this measurement is completely dominated by the error on the "tted values of A and B, and the main contribution to this error is of statistical nature. The structure functions FA and FA were measured by L3 in Ref. [151] and by OPAL in Ref. [152], and they are /#" /#" listed in Tables 17 and 18, respectively. In addition, preliminary results on FA /FA and /#" /#" 1/2FA /FA from ALEPH and DELPHI are available in Refs. [155,156]. For the same /#" /#" reasons as mentioned above for FA the ALEPH results are not considered here, the DELPHI results are listed in Table 19.
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The measurements of FA /FA and 1/2FA /FA are compared in Fig. 39. They all /#" /#" /#" /#" agree with each other and with the QED prediction from Ref. [2]. The measurements of FA and /#" FA from the L3 and OPAL experiments are compared in Fig. 40. The measurements from L3 /#" and OPAL are performed in slightly di!erent ways. The strength of the s dependence varies with the scattering angle cos h夹 of the muons in the photon}photon centre-of-mass system. Reducing the and FA which acceptance of cos h夹 enhances the s dependence but, to obtain a result for FA /#" /#" is valid for the full range of cos h夹, the measurement has to be extrapolated using the predictions of QED. The measurements from L3 are obtained in the range "cos h夹"(0.7, and extrapolated to the full range in cos h夹, whereas the measurements presented by OPAL are valid for the full angular range "cos h夹"(1. There are two other di!erences in the analyses. The OPAL measurements uses the predictions including the mass corrections, Eq. (32), whereas the result from the L3 experiment is obtained based on the leading logarithmic approximation, Eq. (33). As the predictions for FA and FA are only valid for P"0, the OPAL measurement of FA is corrected for the /#" /#" /#" e!ect of non-zero P in the data by multiplying the result of the unfolding for 1P2"0.05 GeV by the ratio of FA for P"0 and FA for 1P2"0.05 GeV, both as predicted by QED. In the /#" /#" case of L3 the measured ratios FA /FA and 1/2FA /FA are multiplied with the /#" /#" /#" /#" measured, and therefore P-dependent, FA , without correcting for the e!ect of non-zero P in /#" the data. Despite the di!erences in the analyses strategies, the measurements are consistent with each other, and they are nicely described by the QED prediction. With these measurements it can be experimentally established that both FA and FA are /#" /#" A di!erent from zero. The shape of F cannot be accurately determined, however it is signi" /#" cantly di!erent from a constant. The best "t to a constant value leads to FA /a"0.032 and /#" 0.042 with s/dof of 8.9 and 3.1 for the L3 and OPAL results, respectively. Because the precision of the measurements is limited mainly by the statistical error, and the luminosities used for the results by the experiments amount to about 100 pb\, taken at LEP1 energies, a signi"cant improvement is expected from exploiting the full expected statistics of 500 pb\ of the LEP2 programme. Several investigations to also measure FA and FA for hadronic "nal states are underway by the LEP experiments, but no results are available yet. This concludes the discussion on the QED structure of the quasi-real photon and the remaining part of this section deals with the structure of highly virtual photons. Following the discussion of Section 3.1 the experimentally extracted quantity is the di!erential QED cross section dp/dx for highly virtual photons, given in Eq. (14). The measurement from OPAL is listed in Table 20 and shown in Fig. 41 for two ranges in photon virtualities. In Fig. 41(a) the ranges 1.5(Q(6 GeV and 1.5(P(6 GeV are used, and Fig. 41(b) is for 5(Q( 30 GeV and 1.5(P(20 GeV. The data are compared to various QED predictions. The full line denotes the di!erential cross-sections as predicted by the Vermaseren Monte Carlo using the same bins as for the data. The additional three histograms represent the cross-section calculations from the GALUGA Monte Carlo for three di!erent scenarios: the full cross-section (full), the cross-section obtained for vanishing q (dot-dash) and the cross-section obtained for vanishing 22 q and q (dash), all as de"ned in Eq. (18). There is good agreement between the data and the 22 2* QED predictions from the Vermaseren and the GALUGA Monte Carlo programs, provided all terms of the di!erential cross-section, Eq. (18), are used. However, if either q or both q and 22 22 q are neglected in the QED prediction as implemented in the GALUGA Monte Carlo, there is 2* a clear disagreement between the data and the QED prediction. This measurement clearly
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Fig. 41. The measurement of the di!erential QED cross section dp/dx for highly virtual photons. The di!erential cross-sections dp/dx, for the reaction eePeec夹c夹Peek>k\, unfolded from the data for (a) 1.5(Q(6 GeV and 1.5(P(6 GeV and (b) 5(Q(30 GeV and 1.5(P(20 GeV. The points represent the data with their statistical (inner error bars) and total errors (outer error bars). The tic marks at the top of the "gures indicate the bin boundaries. The data are compared to various QED predictions explained in the text.
establishes the contributions of the interference terms q and q , described in Section 3.1, to the 2* 22 cross-section. Both terms, q and q , are present in the data, mainly at x'0.1, and the cor22 2* responding contributions to the cross-section are negative. Especially, the contribution from q is 2* very large in the speci"c kinematical region of the OPAL analysis. Since the kinematically accessible range in terms of Q and P for the measurement of the leptonic and the hadronic structure of the photon is the same, and given the size of the interference terms in the leptonic case, special care has to be taken when the measurements on the hadronic structure are interpreted in terms of structure functions of virtual photons.
7. Measurements of the hadronic structure of the photon One of the most powerful methods to investigate the hadronic structure of quasi-real photons is the measurement of photon structure functions in deep inelastic electron}photon scattering at e>e\ colliders. These measurements have by now a tradition of almost 20 years, since the "rst FA was obtained in 1981 by the PLUTO experiment in Ref. [157]. The main idea of this measurement is given by Eq. (23), which means that by measuring the di!erential cross-section, and accounting for the kinematical factors, the photon structure function FA is obtained. The photon structure function FA in leading order is proportional to the quark content of the photon, Eq. (38), and therefore the measurement of FA reveals the structure of the photon. The discussion of the experimental results is divided into three parts. The description of the experimentally observed distributions of the hadronic "nal state by the Monte Carlo models is reviewed "rst, followed by
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Fig. 42. The general behaviour of the hadronic energy #ow. The hadronic energy #ow per event based on the HERWIG5.8d generator is shown as a function of the pseudorapidity g for 1Q2"13 GeV. The observed electron is always at negative rapidities, !3.5(g(!2.8, and is not shown. The dark shaded histogram represents the energy reconstructed by the OPAL detector after the simulation of the detector response to the HERWIG5.8d events. The generated energy distribution for these events is represented by the lightly shaded histogram. The vertical lines show the acceptance regions of the OPAL detector components.
the discussion of the measurements of the hadronic structure function FA , and the description of the measurements concerning the hadronic structure for the exchange of two virtual photons. 7.1. Description of the hadronic xnal state As has been explained in Section 5 the adequate description of the hadronic "nal state by the Monte Carlo models is very important for measurements of the photon structure. With the advent of the LEP2 workshop general-purpose Monte Carlo programs, for the "rst time also containing the deep inelastic electron}photon scattering reaction, became available. The "rst serious attempt to confront these models with the experimental data has been performed by the OPAL experiment in Ref. [87]. None of the Monte Carlo programs available at that time was able to satisfactorily reproduce the data distributions. Therefore, the full spread of the predictions was included in the systematic error of the FA measurement, which consequently su!ered from large systematic errors. This observation initiated an intensive work on the understanding of the shortcomings of the Monte Carlo models. The results of these studies and the attempts to improve on the Monte Carlo models are summarised in this section. The #ow of hadronic energy as a function of the pseudorapidity, for an average event, 1/N dE/dg, is shown in Fig. 42, taken from Ref. [87]. The generated hadronic energy #ow as predicted by the HERWIG5.8d Monte Carlo is compared to the visible #ow of the hadronic energy as observed after simulating the response of the OPAL detector. The hadronic energy #ow sums
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over all charged and neutral particles. The pseudorapidity is de"ned as g"!ln(tan(h /2)), where h is the polar angle of the particle measured from the direction of the beam that has produced the quasi-real photon, so the observed electron is at !3.5(g(!2.8, but is not shown. This "gure demonstrates that a signi"cant fraction of the energy #ow in events from the HERWIG5.8d generator goes into the forward region of the detector. About two-thirds of the energy is deposited in the central region of the detector, and 30% goes into the forward region. As little as 5% of the total hadronic energy is lost in the beampipe. The reconstructed energy #ow is rather similar to the generated energy #ow in the central region, but signi"cantly di!erent in the forward region. The small ine$ciency in the central detector region is mostly due to the fact that some hadrons in this region carry low energy, and therefore fail quality cuts applied to the events in the experimental analyses. The forward regions of the LEP detectors are only equipped with electromagnetic calorimeters and the hadronic energy in the forward region can only be sampled by using these electromagnetic calorimeters. Consequently, for example in the case of the OPAL detector, only about 42% of the total hadronic energy in the forward region can be recovered, with an energy resolution of *E/E"30% of the seen energy, as explained in Ref. [87]. Given this, the detectors are precise enough to disentangle various predictions in the central part of the detector. However, they are not able to distinguish well between models which produce di!erent energy #ow distributions in the forward region. The di!erent Monte Carlo models produce rather di!erent hadronic energy #ows also within the clear acceptance of the detectors, which leads to the fact that for a given value of =, the visible invariant mass = is rather di!erent when using di!erent Monte Carlo models. The correlation between the generated and visible invariant masses is shown in Fig. 43, taken from Ref. [87], for two Monte Carlo models described in Section 5.1. The level of correlation achieved between = and =, strongly depends on the acceptance region for the hadronic "nal state and also on the model chosen. A Monte Carlo model like F2GEN predicts a much stronger correlation than, for example, the HERWIG5.8d Monte Carlo model. This strongly e!ects the acceptance of the events and therefore the x distributions, especially at low values of x and correspondingly low values of x. The di!erences of the predictions can most clearly be seen in variables like the energy transverse to the plane de"ned by the beam axis and the momentum vector of the observed electron, E , which is shown in Fig. 44, taken from Ref. [87], in bins of x . The value of E is obtained by summing up the absolute values of the energy transverse to the tag plane for all objects. The F2GEN model predicts the hardest spectrum and lies above the data, whereas the HERWIG5.8d model lies below the data and the PYTHIA prediction does not even populate the tail of the E distribution. It is apparent from this "gure that the largest di!erences occur at low values of x . Taking the di!erences of the models into account in a bin-by-bin correction procedure the observed hadronic energy #ow can be corrected to the hadron level and compared to various model predictions. Examples of this are shown in Fig. 45 for 1Q2"13 GeV, and in Fig. 46 for
The term hadron level means that all cuts are applied to generated quantities and that the observable shown is calculated from generated stable particles, which are usually de"ned with lifetimes of more than 1 ns. In contrast, the detector level distributions are obtained by applying cuts to the measured quantities and also calculating the observable under study from measured objects after applying quality cuts to observed tracks and calorimetric clusters.
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Fig. 43. The correlation between the measured and generated hadronic invariant mass. The correlation between the generated hadronic invariant mass = and the visible mass = with and without the hadronic energy sampled in the forward region (FR) of the OPAL detector. In (a) the correlation is shown for HERWIG5.8d and (b) for F2GEN; in each case for two cuts on the minimum polar angle of the acceptance region. The symbols represent the average = in each bin, and the vertical error bar its standard deviation. The dashed line corresponds to a perfect correlation = "=.
1Q2"135 GeV, both taken from Ref. [87]. The errors take into account the dependence of the correction on the Monte Carlo model chosen for correcting the data. A detailed discussion of the various models used can be found in Section 5.1. The shape of the hadronic energy #ow drastically changes from a two peak structure at low values of Q to a one peak structure for increasing Q, which also means increasing values of x. The di!erences between the models shrink considerably, and in addition the predictions lie much closer to the data. This shows "rstly that the problem is located in the region of low values of x and large values of =, and secondly that the data are certainly precise enough to further constrain the models. None of the models shown is able to describe the data at 1Q2"13 GeV, but the agreement improves for 1Q2"135 GeV. After these "ndings were reported, several methods were investigated to reduce the dependence of the measured FA on the Monte Carlo models. A "rst attempt to improve on the HERWIG5.9 model was made in Refs. [148,158] by altering the distribution of the transverse momentum, k , of the quarks inside the photon from the program default. The default Gaussian behaviour was replaced by a power-law function of the form dk/(k#k ) with k "0.66 GeV, motivated by the
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Fig. 44. The measured E distribution for 1Q2"13 GeV in bins of x. The energy transverse to the plane de"ned by the beam axis and the momentum vector of the observed electron, E , is shown at the detector level for three ranges in x .
observation made in photoproduction studies at HERA that a better description of the data is achieved if the intrinsic transverse momentum distribution is changed to the power-law behaviour, as explained in Ref. [159]. The upper limit of k in HERWIG5.9#k was "xed at k "25 GeV, which is almost the upper limit of Q for the OPAL analysis from Ref. [87]. This led to some improvements in the description of the OPAL data by the HERWIG5.9#k Monte Carlo, as reported, for example, in Ref. [158]. A second attempt to improve on the situation is based on a purely kinematic consideration already explained in Ref. [123]. The longitudinal momentum of the photon}photon system is unknown, but the transverse momentum is well constrained by measuring the transverse momentum of the scattered electron. In addition, when the #z axis is chosen in the hemisphere of the observed electron, the unseen electron which radiated the quasi-real photon escapes with E !"p "+0 along the beam line. Here E and p denote the energy and longitudinal X X momentum of the unseen electron. This fact can be used to replace E #p , the sum of the X energy and longitudinal momentum of the total hadronic system, by quantities obtained from the scattered electron. If in addition the transverse momentum squared of the hadronic system, p , is replaced by that of the scattered electron, p , a part of the uncertainty of the measurement of
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Fig. 45. The corrected hadronic energy #ow for 1Q2"13 Gev. The measured energy #ow per event is corrected for the detector ine$ciencies, as a function of pseudorapidity g, and compared to the generated energy #ow of the HERWIG5.8d and PYTHIA Monte Carlo models and the energy #ow of samples of point-like and perimiss events from the F2GEN model. The vertical error bars on the data points are the sum of the statistical and systematic errors, and the horizontal bars indicate the bin widths. Fig. 46. The corrected hadronic energy #ow for 1Q2"135 Gev. Same as Fig. 45 but for 1Q2"135 Gev.
the hadronic "nal state can be eliminated. The value of = reconstructed in this scheme is called = and has the following form: = "(E #p )(E !p )!p X X "[2E!(E #p )](E !p )!p . (53) X X Because the quantities which are replaced depend on the properties of the hadronic "nal state, the improvement is expected to show some dependence on the Monte Carlo programs used. For example, for the PHOJET Monte Carlo, the improvement on the resolution in = can be seen from Fig. 47, taken from Ref. [89]. The generated values of ="= are compared to = and AA = using the PHOJET Monte Carlo model in the Q range 1.2}9 GeV. The improvement is expected to be largest for L3, because this detector, on top of the general problems discussed above, su!ers from a dead region in acceptance, as can be seen from Fig. 48, taken from Ref. [89]. The distribution of = is closer to the = distribution than the = distribution, but still the agreement with = is not very good. Recently, in Ref. [89], the PHOJET Monte Carlo model has been used for the "rst time in a structure function analysis. Also for the "rst time in this analysis the TWOGAM Monte Carlo program was used outside the DELPHI collaboration. In Fig. 48, taken from Ref. [89], the prediction of the hadronic energy #ow for these two models are compared to the L3 data for the
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Fig. 47. Comparison of di!erent methods for the reconstruction of the invariant mass of the hadronic "nal state. Shown are the generated mass ="= , the visible mass obtained from the observed hadrons in the L3 detector, = and = , AA de"ned in Eq. (53). All distributions are for the PHOJET Monte Carlo model in the Q range 1.2}9 GeV.
Q range 1.2}9 GeV. Again, these two models, although closer to the data than the HERWIG5.8d and PYTHIA predictions in the case of the OPAL analysis, do not accurately account for the features observed in the data distributions. In the case of L3 the HERWIG5.9#k model, which was tuned for the Q region of 6}30 GeV, does not provide a satisfactory description of the data taken in the region 1.2(Q(9 GeV, and therefore the L3 analysis of the photon structure function FA is only based on the PHOJET and TWOGAM models. The above information is valuable in understanding the discrepancies; however, the investigations su!er from three main shortcomings. Firstly, always slightly di!erent cuts are applied to the experimental data, and therefore, although the data are rather indicative, it is not clear, whether a consistent picture emerges from the results of the di!erent experiments. Secondly, in the present form, the data cannot be directly compared to the generated quantities obtained without simulating the detector response. This is because either the data are not corrected for detector e!ects, as in the case of L3, Fig. 48, or they still depend on cuts applied to the data, which is the case for the OPAL distributions (Figs. 45 and 46), which are only obtained for events ful"lling the experimental cuts applied at the detector level. However, in order for the authors to improve on their models it is mandatory that they can compare to corrected distributions provided by the experiments, without the need of simulating the detector response. Thirdly, it is hard to get a reliable estimate of the systematic error of the experimental result within one experiment, because in this case it can only be obtained from varying the Monte Carlo predictions using models which do not accurately describe the data, certainly not a very reliable method. To overcome these shortcomings a combined e!ort by the ALEPH, L3 and OPAL collaborations and the LEP Working Group for Two-Photon Physics has been undertaken, and preliminary results of this work have been reported in Ref. [149]. The data of the experiments have been
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Fig. 48. The measured hadronic energy #ow from L3. The measured hadronic energy #ow is compared to the Monte Carlo predictions in bins of x , obtained from = and Q, and in bins of Q. The models used are HERWIG5.9#k , PHOJET and TWOGAM.
analysed in two regions of Q, 1.2}6.3 and 6}30 GeV, using identical cuts and also identical Monte Carlo events passed through the respective programs of the individual experiments to simulate the detector response. The data are corrected to the hadron level in a phase space region which is purely determined by cuts at the hadron level, and the systematic error is estimated by the spread of the corrected distributions of the three experiments. Then the data are compared to predictions from the HERWIG5.9#k and the PHOJET Monte Carlo models. Several distributions are studied, namely, the reconstructed invariant hadronic mass, de"ned within a restricted range in polar angles, the transverse energy out of the plane de"ned by the beam direction and the direction of the observed electron, the number of tracks, the transverse momenta of tracks with respect to the beam axis, and the hadronic energy #ow as a function of the pseudorapidity. Preliminary results of this investigation have been reported in Ref. [149]. It is found that for large regions in most of the distributions studied, the results of the di!erent experiments are closer to each other than the sizeable di!erences which are observed between the data and the models. Since the data distributions are corrected to the hadron level, they can be directly compared to the predictions of the Monte Carlo models. Therefore the combined LEP data serve as an important input to improve on the Monte Carlo models. The investigation already led to an improved version of the
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HERWIG5.9#k program obtained by again altering the modelling of the intrinsic transverse momentum of the quarks within the photon. While the "xed limit of k "25 GeV was adequate for the region 6(Q(30 GeV, for lower values of Q, it introduces too much transverse momentum. This has been overcome by dynamically adjusting the upper limit of k by the event kinematic on an event by event basis. In this scheme, called HERWIG5.9#k (dyn) the maximum transverse momentum is set to k +Q. This change leads to an improved descrip tion of the data also for the region 1.2(Q(6.3 GeV. Another way of reducing the model dependence of the measured FA is to perform the unfolding in two dimensions, as described in Section 5.2. Recent preliminary results from the ALEPH and OPAL experiments, presented in Refs. [160,161] respectively, show that this indeed reduces the systematic uncertainty on the structure function measurements. From the discussion above it is clear that the error on the measurement of FA will vary strongly with the selection of Monte Carlo models chosen to obtain the size of the systematic uncertainty. However, given the improved understanding of the shortcomings and the combined e!ort in improving on the Monte Carlo description of the data, it is likely that the error on FA will shrink considerably in future measurements. This closes the discussion about the description of the hadronic "nal state by the Monte Carlo models, and the measurements of FA will be discussed next. 7.2. Hadronic structure function FA Many measurements of the hadronic structure function FA have been performed at several e>e\ colliders. Because in some cases it is not easy to correctly derive the errors of several of the measurements, a detailed survey of the available results has been performed, the outcome of which is presented in Appendix D. The measurements and what can be learned from them about the structure of the photon and on its description by perturbative QCD is discussed in the following. The interpretation of the data will only be based on published results, and on preliminary results from the LEP experiments, which are likely to be published soon. In contrast the preliminary results from Refs. [96,104}106,113], which were used in the "t procedures of several of the parton distribution functions of the photon, but which never got published, are regarded as obsolete, and will not be considered here. In all summary "gures presented below only those preliminary results from the LEP experiments are included which are based on data which have not yet been published. For the preliminary results which are meant to replace a previous measurement in the near future the previously published result will be shown until the new result is "nalised. The range in 1Q2 covered by the various experiments is 0.24(1Q2(400 GeV, which is impressive given the small cross-section of the process. The published results from the ALEPH [162], AMY [84,85], DELPHI [86], JADE [67], L3 [89,163], OPAL [87,90,91], PLUTO [68,69], TASSO [70], TPC/2c [72] and TOPAZ [71] experiments can be found in Tables 21}30. The additional preliminary results from the ALEPH [160], L3 [164] and DELPHI [165,166] experiments are listed in Tables 31}33. In the present investigations of the photon structure function FA two distinct features of the photon structure are investigated. Firstly, the shape of FA is measured as a function of x at "xed Q. Particular emphasis is put on measuring the low-x behaviour of FA in comparison to F as obtained at HERA. Secondly, the evolution of FA with Q is investigated. As explained in Section 3.4, this evolution is predicted by QCD to be logarithmic. These two issues are discussed.
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Fig. 49. Measurements of the hadronic structure function FA except from LEP. The points represent the data with their statistical (inner error bars) and total errors (outer error bars), if available, otherwise only the full errors are shown. The measured data are shown in comparison to the prediction of FA obtained from the GRV higher order parton distribution function, using the Q values given in brackets.
The collection of measurements on FA which have been performed at e>e\ centre-of-mass energies below the mass of the Z boson is shown in Fig. 49. Their precision is mainly limited by the statistical error and, due to the simple assumptions made on the hadronic "nal state, the systematic errors are small, but in light of the discussion above, they may be underestimated. The global behaviour of the data is roughly described, for example, by the FA obtained from the GRV higher order parton distribution function. However, some of the data show quite unexpected features. For example, the structure function as obtained from the TPC/2c experiment shows an unexpected shape at low values of x, and also the results from TOPAZ rise very fast towards low values of x. In addition there is a clear disagreement between the TASSO and JADE data at 1Q2"23}24 GeV. Certainly at this stage much more data were needed to clarify the situation. The measurement of FA has attracted much interest at LEP over the last years. The LEP Collaborations have measured the photon structure function FA in the range 0.002(x:1 and 1.86(1Q2(400 GeV. The "rst published result of the low-x behaviour of FA at low values of Q performed on a logarithmic scale in x is shown in Fig. 50. The data have been unfolded based on the HERWIG5.8d Monte Carlo model. Only a weak indication of a possible rise at low values of x for Q(4 GeV is observed. More important, the data seem to be consistently higher than what is predicted by the GRV and SaS1D parametrisations. In addition, there emerges an inconsistency
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Fig. 50. Comparison of measurements of FA at low values of Q. The OPAL data at 1Q2 of 1.86 and 3.76 GeV are compared to previous results from the PLUTO and TPC/2c experiments. The points represent the OPAL data with their statistical (inner error bars) and total errors (outer error bars). For the previous data only the total errors are shown. The tic marks at the top of the "gures indicate the bin boundaries of the OPAL analysis.
at Q+4 GeV, between the OPAL and PLUTO data of Refs. [91,68] on the one hand, and the TPC/2c data of Ref. [72] on the other. Recently, a preliminary update of the OPAL measurement at low values of Q, shown in Fig. 51, has been presented in Ref. [161]. The new OPAL analysis is based on the same data as the published results, but uses the PHOJET and the improved HERWIG5.9#k Monte Carlo models explained above, which much better describe the experimentally observed distributions than the HERWIG5.8d model used for the results in Fig. 50. In addition, the method of two-dimensional unfolding based on the GURU program has been explored. With these improvements the systematic errors could be considerably reduced and now, this new measurement is consistent with the GRV leading-order prediction. By repeating the analysis with the HERWIG5.8d Monte Carlo model, but using two-dimensional unfolding, the new analysis leads to results consistent with the published results from Ref. [91], however with reduced errors. From this it can be concluded that "rstly, the high values of the published results are due to the bad description of the data by the HERWIG5.8d model, and secondly that the two-dimensional unfolding reduces the error of the measurement, even when using an inaccurate model for the unfolding of FA . A similar analysis at low values of x and Q, shown in Fig. 52, was performed by the L3 experiment in Ref. [89]. Two results for FA were presented which di!er in the model chosen for the
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Fig. 51. Preliminary update of FA at low values of Q from OPAL. The preliminary OPAL data at 1Q2 of 1.9 and 3.8 GeV are compared to the results from the PLUTO and L3 experiments. The points represent the data with their statistical (inner error bars) and total errors (outer error bars). For the PLUTO result only the total errors are shown. For the L3 result the errors are obtained as explained in Table 25.
unfolding of FA from the data. The published results are for PHOJET (set1) and TWOGAM (set2). From the measurement it is clear that although some improved description of the hadronic "nal state can be achieved by the PHOJET and TWOGAM models, the model uncertainty still is the dominant error source at low values of x. At Q"5 GeV the LAC1 and LAC2 predictions are consistent with the L3 result. However, the L3 result is consistently higher than the SaS1D and the leading-order GRV parametrisations of FA for both values of Q. Given the quoted errors of the L3 result the GRV and SaS parametrisations need to be revisited. In addition, as can be seen from Fig. 51, the preliminary OPAL and the L3 measurements are consistent with each other. The measurements discussed above are based on the entire data of the LEP1 running period at e>e\ centre-of-mass energies around the mass of the Z boson. The "rst published result based on data for (s 'm is shown in Fig. 53, taken from Ref. [90]. Due to the higher energy of the beam 8 electrons the Q acceptance of the detectors is changed, see Fig. 7 and Eq. (9). As a rule of thumb the values of Q accepted at LEP2 energies is about a factor of four higher than those accepted at LEP1 energies, when using the same detector device to measure the scattered electron. The results in Fig. 53 cover the 1Q2 range from 9 to 59 GeV. The measured FA as a function of x is almost #at within this region and the absolute normalisation of FA is well described by the predictions of the
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Fig. 52. The measurements of FA at low values of Q from L3. The points represent the L3 data with their total experimental errors, but excluding the large error stemming from the choice of Monte Carlo model. The models chosen for the unfolding of FA from the data are PHOJET (set1) and TWOGAM (set2).
leading-order GRV (solid) and the SaS1D (dot-dash) parametrisations of FA evaluated at the corresponding values of 1Q2. Fig. 53 also shows an augmented asymptotic prediction for FA (ASYM). The contribution to FA from the three light #avours is approximated by the leading-order asymptotic form from Ref. [21], using the parametrisation given in Ref. [30]. This has been augmented by adding a point-like charm contribution and a prediction for the hadron-like part of FA for K "0.232 GeV. The point-like charm contribution has been evaluated from the leading-order Bethe-Heitler formula, Eq. (41) for P"0 and m "1.5 GeV. The estimate of the hadron-like part of FA is given by the hadron-like part of the GRV leading order parametrisation of FA for four active #avours, and evolved to the corresponding values of 1Q2. It is known that the asymptotic solution has de"cits in the region of low-x, because of the divergences in the solution which do not occur in the solution of the full evolution equations, as explained in Section 3.4. However, the asymptotic solution has the appealing feature that it is calculable in QCD, even at higher order and for medium x and with increasing Q it should be more reliable. In addition, at high values of x and Q the hadron-like contribution is expected to be small. In the region of medium values of x studied in Fig. 53 this asymptotic prediction in general lies higher than the GRV and SaS predictions but it is still consistent with the data. The importance of the hadron-like contribution to FA (HAD), which is shown separately at the bottom of the "gure, decreases with increasing x and Q, and it accounts for only 15% of FA at Q"59 GeV and x"0.5. The asymptotic solution increases with decreasing K. For Q"59 GeV and x"0.5 the change in FA is #24% and !16% if K is changed from K "0.232 GeV to 0.1 and 0.4 GeV, respectively.
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Fig. 53. The "rst measurement of FA for (s 'm . The structure function FA is measured for four active #avours in 8 four bins in Q with mean values of (a) 1Q2"9 GeV, (b) 1Q2"30 GeV, (c) 1Q2"14.5 GeV, and (d) 1Q2"59 GeV. In (e) the measurement for the combined data sets of (a) and (c), and in (f ) the measurement for the combined data sets of (b) and (d) is shown. The points represent the OPAL data with their statistical (inner error bars) and total errors (outer error bars). The tic marks at the top of the "gures indicate the bin boundaries. The data are compared to several predictions described in the text.
By now, many more measurements for (s 'm using much higher data luminosities have 8 been performed by the LEP experiments. All LEP measurements are displayed in Fig. 54. The measurements obtained at LEP1 energies are shown with open symbols, whereas those obtained at LEP2 energies are shown with closed symbols. The varying energies of the beam electrons give the opportunity to compare data at similar Q but using di!erent detector devices to measure the scattered electron. The results are consistent with each other, which gives con"dence that the systematic errors are well enough controlled. A summary of all measurements of FA is shown in Fig. 55. The comparison to the GRV higher order prediction of FA shows an overall agreement, but also some regions where the prediction does not so well coincide with the data. This large amount of data, which partly is rather precise, gives the possibility to study the consistency of the predictions with the data. The quality of the agreement is evaluated by a simple s method based on
FA !1FA (x, 1Q2, 0)2 G , s" p G G
(54)
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Fig. 54. Measurements of the hadronic structure function FH from LEP. Same as Fig. 49 but showing the results from the LEP experiments.
where the sum runs over all experiments, all 1Q2 values, and all bins in x. The term FA denotes G the measured value of FA in the ith bin and p is its total error. The theoretical expectation is G approximated by the average FA in that bin in x at Q"1Q2 and at P"0, abbreviated with 1FA (x, 1Q2, 0)2. If, as in the case of the OPAL, the experiments quote asymmetric errors, this is taken into account by choosing the appropriate error depending on whether the prediction lies above or below the measured value. The procedure is not very precise, as it does not take into account the correlation of the errors between the data points and the experiments. However, because there are common sources of errors, it is most likely that by using this method the experimental error is overestimated. Given this, the predictions which are not compatible with the data are probably even worse approximations of the data when the comparison is done more
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Fig. 55. Summary of measurement of the hadronic structure function FH . Same as Fig. 49 but showing all available results on FH .
precisely. A more accurate analysis would require to study in detail the correlation between the results within one experiment, but even more di$cult, the correlation between the results from di!erent experiments. This is a major task which is beyond the scope of the comparison presented here. The predictions used in the comparison are the WHIT parametrisations, which are the most recent parametrisations based on purely phenomenological "ts to the data, and the GRV, GRSc and SaS predictions, which use some theoretical prejudice to construct FA as detailed in Section 4. The theoretical expectation is approximated by 1FA (x, 1Q2, 0)2. If FA (1x2, 1Q2, 0), the structure function at the average value of x, is taken instead, the results only slightly di!er, which means that the comparison is not very sensitive to the shape of FA within the bins chosen. The results of this comparison are listed in Tables 4 and 5. None of the parametrisations has di$culties to describe the AMY, JADE, PLUTO and TASSO data, and they all disfavour the TPC/2c results, which show an unexpected shape as function of x.
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Table 4 Comparison of the GRV, GRSc and SaS1 predictions with measurements of FA . The values are calculated using all data shown in Fig. 55, apart from TPC/2c at 1Q2"0.24 GeV which has a 1Q2 below the lowest value for which a parametrisation of FA exists. Listed are the number of data points (dof) as well as the values of s/dof as calculated from Eq. (54) for the individual experiments and for all data points (All). The minimum Q for the GRSc parametrisation is larger than the value for TPC/2c at 1Q2"0.38 GeV, therefore for GRSc the number of points is only 15 and 161 for TPC/2c and for all data, compared to the total number of 19 and 165. For experiments which have measured FA for 1Q2 values below and above 4 GeV in addition the s/dof values for the comparison based only on data for 1Q2'4 GeV are shown in a second row GRV LO Exp.
dof
AMY JADE PLUTO 1Q2'4 GeV TASSO TPC/2c 1Q2'4 GeV TOPAZ ALEPH DELPHI L3 1Q2'4 GeV OPAL 1Q2'4 GeV All 1Q2'4 GeV
8 8 13 10 5 19/15 3 8 20 24 28 22 32 24 165/161 132
HO
GRSc LO
SaS 1D
1M
0.99 1.09 0.60 0.71 1.03 2.00 0.40 2.40 1.74 1.12 3.93 3.43 1.22 0.79 1.81 1.56
0.86 1.00 0.51 0.61 0.85 3.98 0.58 1.89 1.01 0.74 2.36 1.95 0.85 0.50 1.50 1.02
s/dof 0.75 1.01 0.50 0.55 0.97 4.52 0.54 1.89 0.96 0.69 2.40 1.91 0.84 0.45 1.55 0.98
1.03 1.16 0.46 0.38 0.77 4.11 0.67 2.15 1.41 1.47 2.10 2.21 0.80 0.55 1.64 1.29
0.71 1.03 0.53 0.60 1.03 7.34 0.92 1.67 0.90 0.99 1.82 1.20 0.41 0.37 1.58 0.90
The WHIT parametrisations predict a faster rise at low-x than the GRV, GRSc and the SaS parametrisations. Therefore, the agreement with the TOPAZ data is satisfactory for the WHIT parametrisations, whereas the GRV, GRSc and the SaS1 parametrisations yield values of s/dof of around 2, and the SaS2 parametrisations lie somewhere between these extremes. For the same reason the WHIT parametrisations fail to describe the ALEPH and DELPHI data which tend to be low at low values of x, thereby leading to large s/dof for the WHIT parametrisations, especially for the sets WHIT4-6 which use a"1, as explained in Section 4. The only acceptable agreement is achieved by using the set WHIT1. The OPAL results tend to be high at low values of x and also they have larger errors, therefore only the extreme cases WHIT5-6 lead to unacceptable values of s/dof. The L3 experiment quotes the smallest uncertainties on their results which tend to be high at low values of Q. Consequently, none of the parametrisations which are valid below Q"4 GeV is able to describe the L3 data and all lead to large values of s/dof. For Q'4 GeV the agreement improves but the values of s/dof are still too large, except for GRSc. For the parametrisations valid for Q'4 GeV the best agreement with the L3 data is obtained for
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Table 5 Comparison of the SaS2 and WHIT predictions with measurements of FA . Same as Table 4, but including only data with 1Q2'Q "4 GeV SaS 2D Exp.
dof
AMY JADE PLUTO TASSO TPC/2c TOPAZ ALEPH DELPHI L3 OPAL All
8 8 10 5 3 8 20 24 22 24 132
WHIT 2M
1
2
3
4
5
6
0.76 1.27 0.81 0.83 3.02 0.63 14.04 13.69 3.78 1.14 5.78
0.63 1.11 0.65 0.80 2.23 0.89 52.58 46.39 9.61 1.99 18.66
0.58 1.08 0.63 0.74 1.55 1.07 81.73 73.32 11.40 2.18 28.29
s/dof 1.01 1.52 0.58 0.52 2.38 1.29 1.09 0.95 1.22 0.45 0.97
1.09 1.51 0.60 0.45 2.50 1.34 1.18 0.88 1.37 0.48 1.01
0.74 1.25 0.62 0.64 1.83 1.10 1.76 1.72 0.86 0.40 1.10
0.67 1.09 0.57 0.66 1.53 1.05 8.39 7.90 1.26 0.45 3.27
0.71 1.05 0.56 0.66 1.26 1.19 14.43 13.76 1.88 0.53 5.37
WHIT1. This comparison shows that already at the present level of accuracy the measurements of FA are precise enough to constrain the parametrisations and to discard those which predict a fast rise at low-x driven by large gluon distribution functions. In conclusion, for the parametrisations valid for Q'4 GeV satisfactory agreement is found with the SaS2 and the WHIT1 parametrisations, except for the measurements of TPC/2c. For the parametrisations evolved from lower scales, agreement is found for Q'4 GeV with the exception of the L3 and TOPAZ data, and at lower values of Q they are not able to account for the TPC/2c and L3 results. The second topic which is extensively studied using the large lever arm in Q, is the evolution of FA with Q. The "rst measurements of this type were performed for FA for three light #avours and the contribution to FA from charm quarks was subtracted from the data based on the QPM prediction. This was motivated by the fact that at low values of Q the charm contribution is small and that the main aim of the analyses was to compare to the perturbative predictions for light quarks based on the asymptotic solution. At the time of most of the measurements no parton distribution functions of the photon were available. A summary of the published measurements of the Q evolution of FA (Q, uds) is given in Table 34 and shown in Fig. 56, where the point for TASSO has been obtained from combining the three middle bins listed in Table 28. The data are nicely described by the predictions from the SaS1D and GRV parametrisations of FA . The purely asymptotic prediction from Ref. [21], using the parametrisation given in Ref. [30] for K "0.232 GeV (Witten), predicts a slightly lower FA than is seen in the data, whereas the augmented asymptotic solution (ASYM) is somewhat high compared to the data, but both are still consistent with the measured FA . At higher values of Q the charm quark contribution to FA gets larger and today also parametrisations of FA for four active #avours are available. Consequently, more recent analyses of
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Fig. 56. The measured Q evolution of FA for three active #avours. The data with their full errors are compared to leading-order predictions of FA for the region 0.3(x(0.8. Shown are the GRV and SaS1D parametrisations of FA , the augmented asymptotic FA (ASYM), and the purely asymptotic prediction (Witten), described in the text. The asymptotic predictions are evaluated for K "0.232 GeV.
the evolution of FA with Q are based on measurements of FA for four active #avours. In addition, due to the larger statistics available, the experiments start to look into the evolution using several ranges in x for the same value of 1Q2. The "rst LEP measurement of this type is shown in Fig. 57, taken from Ref. [87]. In Fig. 57(a) the result for the range 0.1(x(0.6 is compared to several parametrisations of FA . Shown are the leading-order (LO) predictions of the GRV and the SaS1D parametrisations, both including the contribution to FA from massive charm quarks, and a higher order (HO) calculation provided by E. Laenen, based on the GRV higher order parametrisation for three light quarks, complemented by the contribution of charm quarks to FA based on the higher order calculation using massive charm quarks of Ref. [52]. The di!erences between the three predictions are small compared to the experimental errors, and all predictions nicely agree with the data. In addition, the data are compared to the augmented asymptotic prediction as detailed above. This approximation lies higher than the data at low Q and approaches the data at the highest Q reached. The evolution of FA with Q is measured by "tting a linear function of the form a#b ln(Q/GeV) to the data for the region 0.1(x(0.6. Here a and b are parameters which are taken to be independent of x within the bin in x chosen. The "t to the OPAL data in the Q range of 7.5}135 GeV yields FA (Q)/a"(0.16$0.05> )#(0.10$0.02> )ln(Q) , \ \
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Fig. 57. The measured Q evolution of FA from OPAL. The measurement of FA is shown for four active #avours as a function of Q, in (a) for the range 0.1(x(0.6, and in (b) subdivided into 0.02(x(0.10, 0.10(x(0.25 and 0.25(x(0.60. In addition shown in (a) are the FA of the GRV leading-order (LO) and the SaS1D parametrisations, the FA of the augmented asymptotic prediction (ASYM) and the result of a higher order calculation (HO), where the last two predictions are only shown for Q'4 GeV. In (b) the data are only compared to the higher order prediction. The points represent the OPAL data with their statistical (inner error bars) and total errors (outer error bars). In some of the cases the statistical errors are not visible because they are smaller than the size of the symbols.
where Q is in GeV, with s/dof"0.77 for the central value, as quoted in Ref. [90]. The slope d(FA /a)/d ln Q is signi"cantly di!erent from zero but not precisely measured yet. The photon structure function FA is expected to increase with Q for all values of x, but the size of the scaling violation is expected to depend on x, as shown in Fig. 29. To examine whether the data exhibit the predicted variation in d(FA /a)/d ln Q, the Q range 1.86}135 GeV is analysed using common x ranges. Fig. 57(b) shows the measurement in comparison to the higher order calculation explained above. The points of in#ection of FA for Q below 15 GeV are caused by the charm threshold. The data show a slightly steeper rise with Q for increasing values of x, which is reproduced by the prediction of the higher order parametrisation of FA . However, to experi mentally observe the variation of d(FA /a)/d ln Q with x the inclusion of more data and a reduction of the systematic error are needed. A similar analysis performed by the L3 experiment is shown in Fig. 58, taken from Ref. [163]. Unfortunately, the x ranges are slightly di!erent and the data cannot easily be combined. For x'0.2 the L3 data are described by all shown leading order parametrisations of FA , from LAC1,
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Fig. 58. The measured Q evolution of FA from L3. The measurement of FA is shown for four active #avours as a function of Q, for four ranges in x, 0.01(x(0.1, 0.1(x(0.2, 0.2(x(0.3 and 0.3(x(0.5. The data are compared to FA from the leading-order GRV, SaS1D, and LAC1 parametrisations. In addition shown is a "t to the data explained in the text. The points represent the L3 data with their total errors.
SaS1D and GRV. For smaller values of x some di!erences are seen. In the range 0.1(x(0.2 the data show a steeper behaviour than what is predicted by the three parametrisations of FA , and for 0.01(x(0.1 they are higher than the SaS1D and GRV predictions, but show a similar slope, whereas the FA based on the LAC1 parametrisation predicts a much too fast rise with Q. The L3 data were "tted, as explained for the OPAL result above, in two regions of x, 0.01(x(0.1 and 0.1(x(0.2, using the Q range of 1.2}30 GeV. The results for the two regions are FA (Q)/a"(0.13$0.01$0.02)#(0.080$0.009$0.009) ln(Q/GeV) , FA (Q)/a"(0.04$0.08$0.08)#(0.13$0.03$0.03) ln(Q/GeV) , with s/dof of 0.69 and 0.13 for the central values. The results obtained by the L3 experiment are consistent with the OPAL result, which is valid for 0.1(x(0.6. A collection of all available measurements of the evolution of FA for four active #avours and at medium values of x is listed in Table 35 and shown in Fig. 59. For the PLUTO result the average
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Fig. 59. The measured Q evolution of FA at medium x. The data with their full errors are compared to the predictions based on the leading-order GRV and SaS1D parametrisations and to a higher order prediction (HO), as well as to an augmented asymptotic FA (ASYM), both described in the text, all for the range 0.1(x(0.6. In addition shown are the leading-order GRV predictions for two other ranges in x, 0.2(x(0.9 and 0.3(x(0.8.
FA in the range 0.3(x(0.8 for the 1Q2 values of the analyses has been added to the published A three #avour result. The charm contribution has been obtained from Eq. (41), for P"0 and m "1.5 GeV. The only signi"cant contribution from charm quarks is for 1Q2"45 GeV, where FA increases from 0.55 to 0.73. As above for the three #avour result, the point for TASSO has been obtained from combining the three middle bins listed in Table 28. Unfortunately, the di!erent experiments quote their results for di!erent ranges in x which makes the comparison more di$cult because the predictions for the various ranges in x start to be signi"cantly di!erent for Q'100 GeV, as can be seen from the GRV predictions for three di!erent ranges of x shown in Fig. 59. The measurements are consistent with each other and a clear rise of FA with Q is observed. Again, this rise can be described reasonably well by the leading order augmented asymptotic prediction for K "0.232 GeV. In Fig. 59 only the medium x region is studied. The large amount of data shown in Fig. 55 enables to investigate the variation of the scaling violation as a function of x in more detail. For this purpose the data from Fig. 55 are displayed di!erently in Fig. 60. The data are shown as a function of Q, divided in bins of x, with bin boundaries of 0.001, 0.01, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 0.99 and central values as shown in the "gure. Each individual measurement is attributed to the bin with the closest central value in x used. To separate the measurements from each other an integer value, N, counting the bin number is added to the measured FA . The theoretical predictions are taken as the
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Fig. 60. Summary of the measurements of the Q evolution of FA . The data with their full errors are compared to the predictions based on the leading-order GRV and SaS1D parametrisations and to an augmented asymptotic FA , described in the text. The bins used have boundaries of 0.001, 0.01, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 0.99, with central values as shown in the "gure. To separate the measurements from each other an integer value, N, counting the bin number is added to the measured FA .
average FA in the bin. The Q ranges used for the predictions are the maximum ranges possible for 1(=(250 GeV and Q (Q(1000 GeV, where Q is the starting scale of the evolution for the respective parametrisation of FA . The general trend of the data is followed by the predictions of the augmented asymptotic solution, and the GRV and SaS1D leading-order parametrisations of FA , however, di!erences are seen in speci"c ranges in x which were discussed above in connection with Fig. 55. To quantify the increasing slope as function of Q for increasing values of x, the data are "tted, in bins of x by a linear function of the form a#b ln(Q/K), with K"0.2 GeV. The results of the "t are displayed in Fig. 61 and listed in Table 6. Because some of the data contain asymmetric errors, the central values and errors of the "t parameters are not obtained from analytically solving the problem, but rather the MINUIT program from Ref. [167] has been used to perform the "t. The "tted values for the parameters a and b, as well as their errors are given. The errors are calculated as the one p parameter errors de"ned by the MINUIT program, as explained in Ref. [167]. They re#ect the change of a given parameter, when the s is changed from s
to s #1. The parameters a and b are almost 100% anticorrelated. The errors of the
"tted functions are indicated in Fig. 61 using the full error matrix. For comparison the GRV leading-order predictions are shown as well. Although the prediction, for example, from the
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Fig. 61. Measurements of the Q evolution of FA compared to a linear "t. The same data as in Fig. 60 are compared to the results of linear "ts using the function a#b ln(Q/K), with K"0.2 GeV (dash) together with the errors of the "t (dot). In addition, the predictions of the leading-order GRV parametrisations of FA are shown as full lines.
GRV leading-order parametrisation is not compatible with such a linear approximation, the data, at the present level of accuracy, can be "tted with linear functions with acceptable values of s/dof. In some cases s/dof is much smaller than unity indicating that using the full error is overestimating the errors. Consequently, the data is more precise and a combined "t with a careful estimation of the correlation of the errors should be performed soon. The results of the "ts listed in Table 6, show a signi"cant increase in slope for increasing x in accordance with the theoretical expectation. To compare the data more directly to the asymptotic solution of FA , without the complication due to the heavy quark contribution, the charm quark contribution is subtracted from the measurements based on the point-like QPM prediction, Eq. (41), for P"0 and for a charm quark mass of m "1.5 GeV. In Fig. 62 the three #avour result is compared to the leading-order asymptotic prediction from Ref. [21], using the parametrisation given in Ref. [30], for K " 0.232 GeV, and to the QPM prediction for the three light quarks assuming m I "0.2 GeV. The O values of s/dof as calculated from Eq. (54) are given in Table 7. At low values of x both predictions undershoot the data, and the agreement improves with increasing values of x. For x'0.3 the asymptotic prediction gives a slightly better description than the QPM prediction resulting in s/dof values around 1. It is a very interesting observation that the perturbative prediction is able to describe the behaviour for large x for a reasonable value of the only free parameter K .
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Table 6 Fit results of the Q evolution of FA . Listed are the results for the parameters a and b of the "t using the function a#b ln(Q/K), with K"0.2 GeV. The errors are the one p parameter errors as de"ned by the MINUIT program. The number of degrees of freedom is denoted with dof, and the correlation of the two parameters with cor x 0.001}0.01 0.01}0.1 0.1}0.2 0.2}0.3 0.3}0.4 0.4}0.6 0.6}0.8 0.8}0.98
a$p
?
!0.02$0.01 !0.04$0.01 !0.08$0.02 !0.18$0.04 !0.28$0.08 !0.44$0.08 !0.21$0.17 !1.4$1.1
b$p
dof
s/dof
cor
0.052$0.004 0.062$0.003 0.078$0.004 0.102$0.009 0.12$0.02 0.15$0.01 0.12$0.03 0.30$0.15
18 36 33 15 13 20 15 2
1.69 1.36 0.51 1.12 0.65 0.28 0.87 0.35
!0.95 !0.94 !0.92 !0.95 !0.97 !0.98 !0.98 !0.99
@
Fig. 62. The Q evolution of FA for three active #avours compared to the asymptotic solution and the QPM prediction. The same data as in Fig. 60 are used and the charm contribution is subtracted as explained in the text. The three #avour results are compared to the prediction of the asymptotic solution with K "0.232 GeV (Witten, full) and to the QPM prediction assuming m I "0.2 GeV (QPM, dash). O
To make a more quantitative statement on the description of the measured FA by the per turbative prediction, an x-dependent parametrisation of the next-to-leading-order asymptotic prediction must be available. Then, the data should be compared to the next-to-leading-order asymptotic prediction to "x the QCD scale K, with a proper de"nition of the region of validity of
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Table 7 Comparison of the Q evolution of FA to the asymptotic solution and to the QPM prediction. Listed are the ranges in x used, the number of data points in each range, dof, and the s/dof values for the predictions of FA from the asymptotic solution (Witten), for K "0.232 GeV and the quark parton model (QPM), for m I "0.2 GeV O Witten
QPM
x
dof
s/dof
s/dof
0.001}0.01 0.01}0.1 0.1}0.2 0.2}0.3 0.3}0.4 0.4}0.6 0.6}0.8 0.8}0.98
20 38 35 17 15 22 17 4
14.7 16.2 3.76 2.84 1.01 1.02 0.84 0.76
56.9 18.8 2.85 1.69 1.86 1.27 0.99 1.94
this approximation to avoid the singularities. In addition, then the charm subtraction could also be based on the next-to-leading-order calculation from Ref. [52]. Finally, the contribution of the hadron-like component of the charm production should be investigated, especially at low values of x. 7.3. Hadronic structure of virtual photons The structure functions of virtual photons can be determined in the region Q
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Fig. 63. The e!ective photon structure function FA from PLUTO. In (a) FA as a function of x is compared to the next-to-leading-order (NLO) result of the GRS parametrisation of the parton distribution functions of virtual photons with, and without, including a hadron-like component at the starting scale of the evolution, denoted with NLO (full) and NLO (g"0) respectively. In (b) the P evolution is shown in comparison to the full next-to-leading-order result for three values of x, 0.3, 0.2 and 0.15, and in comparison to the prediction for g"0 at x"0.15.
Recently, preliminary results of a similar measurement, using the full data taken at LEP1 energies, has been presented by the L3 experiment in Ref. [164]. The average virtualities for the L3 result are 1Q2"120 GeV and 1P2"3.7 GeV. In Fig. 64, taken from Ref. [164], the measurement of FA for quasi-real photons and the e!ective structure function, both as functions of x, are compared to several theoretical predictions. As in the case of PLUTO the QPM result is too low compared to the data. Taking only FA as calculated from the GRS parametrisation of the parton distribution functions of the photon, labelled as GRS F2, gets closer to the data, and the best
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Fig. 64. The e!ective photon structure function FA from L3. In (a) the preliminary measurement of the structure function FA for quasi-real photons is compared to the quark parton model prediction (QPM), to FA as predicted by the GRV and AFG parametrisation of the parton distribution functions of real photons, and to a higher order calculation based on Ref. [52], denoted with LRSN. In (b) the e!ective structure function FA is compared to the QPM prediction, to FA predicted from the GRS parametrisation of the parton distribution functions of virtual photons (GRS F2), and to the full GRS prediction obtained from the GRS FA together with the contribution of FA as given by the quark parton model. * Fig. 65. The virtuality dependence of FA . In (a) the P dependence of the L3 result of FA is compared to the quark parton model result (QPM), and in (b) the Q dependence of the results from PLUTO and L3 are shown in comparison to the QPM prediction, for the average value of ln(1Q2/1P2), which is around three.
description is found if the contribution of FA , based on the prediction from the QPM, is added to * this, denoted with GRS. The data show a faster rise with x than any of the predictions, however with large errors for increasing x, which are mainly due to the low statistics available. The P evolution of the L3 result of FA is shown in Fig. 65(a). The QPM prediction is consistent in shape with the data, but the predicted FA is too low. However, the main di!erence comes from FA for P"0, which is not described by the quark parton model for x(0.4. This is expected, because in this region the hadron-like part is predicted to be largest, as can be seen from Fig. 64(a). But in this region the data are even higher than the predictions of all parametrisations of FA which do contain a hadron-like contribution. The measurement for P'0 cannot rule out the quark parton model prediction, although it is consistently higher and does not favour the QPM prediction. The ratio of 1Q2/1P2 is similar for the PLUTO and the L3 measurements, leading to values for ln(1Q2/1P2) of 2.6 and 3.5, respectively. This enables to compare the Q evolution of the two measurements, as shown in Fig. 65(b), for 0.05(x(0.75.
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The evolution is consistent with the expectation of the quark parton model for ln(Q/P)"3, and using the range 0.05(x(0.98. In summary, a consistent picture is found for the e!ective structure function FA of the virtual photon between the PLUTO and preliminary L3 data and the general features of both measurements are described by the next-to-leading-order predictions. However, the data do not constrain the predictions strongly and for detailed comparisons to be made the full statistics of the LEP2 programme has to be explored. If both photons have similar virtualities the photon structure function picture can no longer be applied and the data are interpreted in terms of the di!erential cross-section. Due to the large virtualities the cross-section is small and large integrated luminosities are needed to precisely measure it. The main interest is the investigation of the hadronic structure of the interaction of two virtual photons. However, the interest in performing these measurements increased considerably in the last years, because calculations in the framework of the leading-order BFKL evolution equation, which sums ln(1/x) contributions, predicted a large cross-section for this kinematical range, see Refs. [168}171]. The predicted cross-section is so large that already measurements with low statistics are able to decide whether the BFKL picture is in agreement with the experimental observations. Recently, theoretical progress has been made in Refs. [172,173] to also include next-to-leadingorder pieces in the calculations in the BFKL picture, as explained in Ref. [174]. Large negative corrections to the leading-order results were found, for example, shown in Refs. [175]. Consequently, there is some doubt about the perturbative stability of the BFKL calculation. The theoretical development is underway and this should be kept in mind in all comparisons to the BFKL predictions. The most suitable region for the comparison is ="ln
2p ) q (QP
=#P#Q "ln
(QP
= +ln
(QP
.
(55)
where the approximation is only valid for ="ln(=/(QP) is shown in Fig. 66, taken from Ref. [177]. The data are described by the TWOGAM Monte Carlo for (s "91 GeV and (s "183 GeV. The PHOJET model gives an adequate description at (s "183 GeV and (s"189 GeV, whereas it fails to describe the data
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Fig. 66. The hadronic cross-section for the exchange of two virtual photons from L3. The data of the L3 experiment are compared to the predictions of the PHOJET and TWOGAM Monte Carlo models and to an estimation in the framework of the quark parton model obtained from the Vermaseren Monte Carlo (JAMVG). Fig. 67. The hadronic cross-section for the exchange of two virtual photons from OPAL. The data of the OPAL experiment are compared to the prediction of the PHOJET Monte Carlo model for data taken at (s "189 GeV. In (a)}(c) the corrected cross-sections are shown as functions of =, x and Q.
(s "91 GeV, probably due to the low cut in = applied for this data. For low values of = the PHOJET Monte Carlo is known to be not very reliable, as explained in Section 5.1. A prediction in the framework of the quark parton model obtained from the Vermaseren Monte Carlo is found to be too low at all energies. The presently predicted cross-sections by the BFKL calculation (not shown) are much higher than the measurements and are ruled out by the data. A similar analysis has been performed by the OPAL experiment. Preliminary results were presented in Ref. [178] based on data at (s "189 GeV, for an integrated luminosity of about 170 pb\, with average photon virtualities of about 10 GeV, and for ='5 GeV. The di!erential cross-section as functions of =, x and Q, corrected to the phase space de"ned by E , E '65 GeV, 34(h ,h (55 mrad, and ='5 GeV, are shown in Fig. 67, taken from Ref. [178]. Due to the larger electron energies required in the OPAL analysis compared to the L3 result, the reach in = for OPAL is only about ="35 GeV. This means the smallest value of x reached is only about 8;10\. The measured cross-section in the selected phase space is 0.32$0.05(stat)> (sys) pb, compared to the predicted cross-sections of 0.17 pb for PHOJET \ and 2.2/0.26 pb based on the BFKL calculation in leading/higher order. Also for the OPAL analysis, the data at (s "189 GeV are perfectly described by the PHOJET model and there is no room for large additional contributions. The precision of the results on the di!erential crosssections are limited by the low statistics and they can considerably be improved by using the full statistics of the LEP2 programme.
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8. Future of structure function measurements As discussed in the previous sections, the QED and the hadronic structure of the photon have been measured up to average photon virtualities of 1Q2"130 and 1Q2"400 GeV, respectively. In the future, the analysis of the photon structure can be extended to higher photon virtualities "rstly, by exploring the high luminosity of the complete LEP2 programme and secondly, by using the full potential of the planned linear collider project. The prospects of these two parts of the future of structure function measurements are discussed brie#y in this section. 8.1. LEP2 programme So far the measurements of the QED structure of the photon are based on data taken at LEP1 energies for integrated luminosities of about 100 pb\ per experiment and the results are mainly limited by statistics. Therefore, exploring the full integrated luminosity of 500 pb\ expected at LEP2 energies, a reduction of the error by about a factor of two can be expected. For the case of the hadronic structure of the photon the situation is more di$cult. Certainly the measurement can be extended to higher values of Q. The preliminary data from DELPHI already reaches 1Q2"400 GeV and, using the full data sample at LEP2 energies, decent statistics up to 1Q2"1000 GeV can be reached. For the measurement of the hadronic structure at low values of Q the situation is di!erent. Also in future this measurement su!ers from theoretical uncertainties and considerable improvement in the description of the hadronic "nal state by the Monte Carlo models is needed "rst, before the experimental measurements can get more precise. The measurement of the hadronic structure for the exchange of two virtual photons su!ers from low statistics, therefore using the expected 500 pb\ at LEP2 energies will help to bring down the statistical errors, but it should be kept in mind that the results for the region Q+P e\ colliders LEP and SLC. Table 8 shows the improvements on several machine parameters which have to be achieved to arrive at a luminosity of the order of 10/cm s, which would lead to an integrated luminosity of about 100 fb\ per year of operation. The linear collider, even when running under optimal conditions will produce a huge amount of background where many particles are produced especially in the forward regions of the detector. Detailed background studies for the linear collider were performed. The background sources will lead mainly to e> e\ pair creation and to hadronic background. For the e> e\ collision mode the background simulation of Ref. [180] showed that the amount of background expected per bunch crossing for the TESLA design is about 10 e> e\ pairs with a total energy of 1.5;10 GeV and
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Table 8 Parameters for a future linear collider of the TESLA design. Some approximate values of parameters of the present LEP and SLC colliders are shown together with goals for a future linear collider of the TESLA design
Total length Gradient Beam size p /p V W Electron energy Luminosity L
(km) (MV/m) (lm/lm) (GeV) (10/cm s) (1/pb yr)
LEP
SLC
TESLA
26.7 6 110/5 100 7.4 200
4 10 1.4/0.5 50 0.1 15
33 25 0.845/0.019 250 5000}10 000 20 000
about 0.13 events of the type ccPhadrons for hadronic masses ='5 GeV with an average visible energy of 10 GeV. To accommodate this background the main detector has to be shielded with a massive mask as shown, for example for the TESLA design, in Fig. 68, taken from Ref. [180]. In addition, the photon radiation from the beam electrons will also lead to a signi"cant energy smearing for the electrons of the beams. The prospects of structure function measurements have to be discussed in the context of this expected machine parameter dependent &soft' underlying background, and the energy spread of the beam electrons. There is also a strong interest in the construction of a photon linear collider which would in several aspects be complementary to an electron linear collider. For several reactions the cross sections for incoming photons are larger than for incoming electrons of the same energy. In addition, some reactions, for example, the very important process ccPH only have su$ciently large event rates, when using the large #ux of high energetic incoming photons from a photon linear collider. The linear collider, when operated in the electron}photon mode, would also be an ideal source of high energetic photons for structure function measurements, because the energy of the incoming real photons would be known rather precisely. The construction of a photon linear collider is very demanding and only general concepts are available so far. The method to produce a beam of high energetic photons from an electron beam by means of the Compton backscattering process is shown in Fig. 69, taken from Ref. [181]. The photons are produced by a high-intensity laser and brought into collision with the electron beams at distances of about 0.1}1 cm from the interaction point. The photons are scattered into a small cone around the initial electron direction and receive a large fraction of the electron energy. By properly adjusting the machine parameters, like the distance along the beam line between the production of the backscattered photons and the interaction region, by selecting the polarisations of the laser and the electron beams, and by magnetic re#ection of the spent beam, the energy spectrum of the photons can be selected. As a result of this a typical distribution of ec luminosity as a function of the invariant mass of the electron photon system, (s , is expected to peak at the maximum reachable invariant mass of around 0.8(s with a width of 5%, as described in Ref. [182]. The two main questions concerning the photon structure function FA addressed at LEP, namely the low-x behaviour of FA and the Q evolution of FA can be studied at a future linear collider but stringent requirements have to be imposed on the detector design. The region of high values of
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Fig. 68. The forward region of a future linear collider detector. A sketch of the proposed mask to protect the detector from the background for the TESLA design.
Fig. 69. The production mechanism for high energetic photons. A sketch of the creation of the photon beam by Compton backscattering of laser photons o! the beam electrons.
Q and x can already be studied with an electromagnetic calorimeter, located outside the shielding mask, and covering polar angles of the observed electrons of h '175 mrad, which is able to detect electrons with energies above 50% of the energy of the beam electrons, as shown in Fig. 70(a,b). The errors assumed in Fig. 70, taken from Ref. [183], are the quadratic sum of the statistical and the systematic components. The statistical error is calculated based on the leading-order GRV structure function FA for an integrated luminosity of 10 fb\. The systematic error is assumed to be equal to the statistical error but amounts to at least 5%. Therefore, the precision indicated in Fig. 70 has to be taken with care, as the systematic errors shown do not re#ect the present level of
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Fig. 70. Prospects for structure function measurements at a future linear collider. Shown are hypothetical LEP data and linear collider data for di!erent minimal detection angles of the deeply inelastically scattered electrons, h . In (a,b) h '175 mrad is assumed and (c,d) are based on h '40 mrad.
precision of the LEP data, as detailed in Section 7. To achieve overlap in Q with the LEP data the electron detection has to be possible down to h '40 mrad, shown in Fig. 70(c,d), which means the mask has to be instrumented, and the calorimeter has to be able to detect electrons which carry 50% of the energy of the beam electrons in the huge but #at background of electron pairs discussed above, certainly a non-trivial task.
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Fig. 71. Prospects for the measurements of the Q evolution of FA at a future linear collider. The measured Q evolution of FA is shown, together with the possible extensions at a future lilnear collider, denoted with LCI and LC2. For the hypothetical data, the inner error bars indicate the statistical and the outer error bars the quadratic sum of the assumed statistical and systematic errors.
The measurement of the Q evolution of the structure function FA constitutes a fundamental test of QCD. In Fig. 71 the prospects of the extension of this measurement at a future linear collider with (s "500 GeV are shown for two scenarios. This "gure has been taken from Ref. [179] and the measurements from PLUTO and TASSO have been added. As above, the scenarios assume that electrons can be observed for energies above 50% of the energy of the beam electrons, and for angles of h '40 mrad (LC1) and h '175 mrad (LC2). It is further assumed that the measured structure function is equal to the prediction of the leading-order GRV photon structure function FA in the respective range in x. The statistical errors of the hypothetical measurements are calculated from the number of events for an integrated luminosity of 10 fb\, predicted by the HERWIG Monte Carlo for the leading-order GRV photon structure function FA in bins of Q using the ranges in x as indicated in Fig. 71. The systematic error is taken to be 6.7% and to be independent of Q. This assumption is based on the systematic error of the published OPAL [90] result at Q"135 GeV. The symmetrised value of the published systematic error is 13.4%. It is assumed that this error can be improved by a factor of two. With these assumptions the error on the measurement is dominated by the systematic error up to the highest values of Q. It is clear from Fig. 71 that overlap in Q with the existing data can only be achieved if electron detection with h '40 mrad is possible. For h '175 mrad su$cient statistics is only available for Q above around 1000 GeV.
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In summary with the data from the linear collider the measurement of the Q evolution of the structure function FA , can be extended to about Q"10 000 GeV, and the low-x behaviour can be investigated down to x+5;10\ (x+5;10\) for an electron acceptance of h '175 mrad (h '40 mrad). At the largest values in Q also the contributions from Z exchange in deep inelastic electron} photon scattering could be measured and the charged-current process ecPlX could be used to study the weak structure of the photon. Together these measurements would allow for a separate measurement of the parton distribution functions for up and down-type quark species, as has been discussed in Ref. [184]. For a photon linear collider operating in the electron}photon mode a completely new scenario for photon structure function measurements would be opened. For the "rst time measurements could be performed with beams of high energetic photons of known energy with a rather small energy spread, instead of measurements using the broad bremsstrahlungs spectrum of photons radiated by electrons. With this the measurements of the photon structure function would be on a similar ground than the measurement of the proton structure function at HERA, which would probably also result in a strong reduction of the systematic error. Another very important improvement for structure function measurements would be the detection of the electron that radiates the quasi-real photon and is scattered under almost zero angle. The possibility of such very low angle tagging is presently under study, but it is not yet clear whether it can be realised. If this could be achieved the precision of structure function measurements may signi"cantly be improved, because x could be calculated from the two detected electrons. However, it should be kept in mind that this requires a very good resolution on the measured electron energy, despite the large background discussed above, a very demanding requirement. If this could be reached the measured electron energy would be used to determine the much smaller photon energy, which would allow for a measurement of = independently of the hadronic "nal state. Given that the dominant systematic error of the structure function measurement comes from the imperfect description of the hadronic "nal state by the Monte Carlo models, this would be an important step to reduce the systematic error of structure function measurements. From the above it is clear that the investigation of the structure of the photon would greatly pro"t from the measurements performed at a future linear collider. However, it has to be kept in mind that at the moment it is not clear whether several of the desired features of the detector, like zero-angle tagging and excellent calorimetry inside the shielding mask, can be achieved.
9. Probing the structure of the photon apart from DIS In addition to the results on the structure of the photon from deep inelastic electron}photon scattering the photon structure has been studied in the scattering of two quasi-real photons at e>e\ colliders, and in photoproduction and deep inelastic electron}proton scattering at HERA. These two rich "elds of investigations of the photon structure cannot be covered in all details here. Only the most important topics in the context of this review will be discussed below, focusing on the general ideas and the main results. For the important details, which are not given here the reader is referred to the most recent publications and to summaries of the LEP and HERA results, which can be found in Refs. [185}188].
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9.1. Photon}photon scattering at e>e\ colliders The scattering of two quasi-real photons has been studied in detail at LEP. The photon}photon scattering reaction has the largest hadronic cross-section at LEP2 energies and therefore, in most cases, the results are mainly limited by systematic uncertainties. Results have been derived on general properties of the hadronic "nal states in Refs. [189}191], on the total hadronic photon}photon cross-section in Refs. [176,192,193], on hadron production in Ref. [194], on jet cross-sections in Refs. [195,196], on heavy quark production in Refs. [122,176,197}200], on lepton pair production in Ref. [201] and on resonances in Refs. [202}208]. The selected topics discussed below are the total hadronic cross-section for photon}photon scattering, p , and more exclusively, AA hadron production, jet cross-sections and the production of heavy quarks. 9.1.1. Total hadronic cross-section for photon}photon scattering The measurement of p is both, interesting and challenging. It is interesting, because in the AA framework of Regge theory p can be related to the total hadronic cross-sections for photon} AA proton and hadron}hadron scattering, p and p , and a slow rise with the photon}photon A centre-of-mass energy squared, =, is predicted. It is challenging, "rstly because experimentally the determination of = is very di$cult due to limited acceptance and resolution for the hadrons created in the reaction and secondly, because the composition of di!erent event classes, for example, di!ractive and quasi-elastic processes, is rather uncertain, which a!ects the overall acceptance of the events. The "rst problem is dealt with by determining = from the visible hadronic invariant mass using unfolding programs, similarly to the measurements of the hadronic structure function. The second uncertainty is taken into account by using two models, namely PHOJET and PYTHIA, for the description of the hadronic "nal state and for the correction from the accepted cross-section to p , leading to the largest uncertainty of the result. AA The published measurements of p by L3 in Ref. [191] and by OPAL in Ref. [192] are shown AA in Fig. 72, and preliminary measurements by L3 presented in Ref. [193] are shown in Fig. 73. All results show a clear rise as a function of =. The cross-section p is interpreted within the framework of Regge theory, motivated by the fact AA that p and p are well described by Regge parametrisations using terms to account for pomeron A and reggeon exchanges. The originally proposed form of the Regge parametrisations for p is AA (56) p (=)"X (=)C #> (=)\E , AA AA AA where = is taken in units of GeV. The "rst term in the equation is due to soft pomeron exchange and the second term is due to reggeon exchange. The exponents e and g are assumed to be universal. The presently used values of e "0.095$0.002 and g "0.034$0.02 are taken from Ref. [3]. The parameters were obtained by a "t to the total hadronic cross-sections of pp, pp , n!p, K!p, cp and cc scattering reactions. The coe$cients X and > have to be extracted from the cc AA AA data. The values obtained in Ref. [3] by a "t to previous cc data, including those of L3 from Ref. [191], are X "(156$18) nb and > "(320$130) nb. Recently, an additional hard pomeron AA AA component has been suggested in Ref. [209] leading to p (=)"X (=)C #X (=)C #> (=)\E , AA AA AA AA with a proposed value of e "0.418 and an expected uncertainty of e of about $0.05.
(57)
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Fig. 72. Published results on p
AA
as a function of =.
Fig. 73. Preliminary results on p as a function of =. The total hadronic cross-section for photon}photon scattering as AA a function of = is shown for L3 data at (s "189 GeV and using two di!erent Monte Carlo models for correcting the data.
Di!erent "ts to the data have been performed by the experiments. The interpretation of the results is very di$cult, because, "rstly, the parameters are highly correlated, secondly, the main region of sensitivity to the reggeon term is not covered by the OPAL measurement and thirdly, di!erent assumptions have been made when performing the "ts. The correlation of the parameters of Eq. (57) can be clearly seen in Fig. 74(a) and (b), where the theoretical predictions are shown, exploring the uncertainties for the soft pomeron term in (a) and for the reggeon as well as for the hard pomeron term in (b), using the central values and errors quoted in Ref. [3]. It is clear from Fig. 74(a) and (b) that by changing di!erent parameters in (a) and (b) a very similar e!ect on the rise of the total cross-section can be achieved. Fig. 74(c) shows the spread of the best "t curves for various data and various "t assumptions, explained below. In Fig. 74(a)}(c) in addition the results from Ref. [192] are shown to illustrate the size of the experimental uncertainties. The di!erent "ts performed by the experiments yield the following results: 1. OPAL: The OPAL data taken at (s "161}183 GeV, within the present range of =, can be accounted for without the presence of the hard pomeron term. When "xing all exponents and > to the values listed above the "t yields X "(0.5$0.2> ) nb, which is not signi"cantly AA AA \ di!erent from zero, and X "(182$3$22) nb, which is consistent with the values from AA Ref. [3]. Using X "0 and leaving only e and X as free parameters results in AA AA e "0.101$0.004> and X "(180$5> ) nb, Fig. 74(c, full), again consistent with \ AA \ Ref. [3].
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Fig. 74. Fits to p using various data and "t assumptions. In (a) and (b) the present theoretical predictions are shown AA using the central values and errors quoted in Ref. [3]. In (c) results for "ts to various data as explained in the text are shown. In addition, the measurement from Ref. [192] is shown to illustrate the size of the experimental uncertainties.
2. L3: In all "ts performed by L3 the hard pomeron term is set to zero. The L3 data from Ref. [191] can be "tted using the old values for the exponents of e "0.0790$0.0011 and g " 0.4678$0.0059 from Ref. [210] leading to X "(173$7) nb and > "(519$125) nb, AA AA Fig. 74(c, dash). The L3 data at (s "189 GeV indicate a faster rise with energy. Using e "0.95 and g "0.34, and the PHOJET Monte Carlo for correcting the data, leads to X "(172$3) nb and > "(325$65) nb, but the con"dence level of the "t is only AA AA 0.000034. Fixing only the reggeon exponent to g "0.34 leads to e "0.222$0.019/0.206$0.013, X "(50$9)/(78$10) nb and > "(1153$114)/(753$116) nb, when using PHOJET/ AA AA PYTHIA, Fig. 74(c, dot/dot-dash). In summary, the situation is unclear at the moment with OPAL being consistent with the universal Regge prediction, whereas L3 indicating a faster rise with = in connection with a very large reggeon component for the data at (s "189 GeV. In addition, the L3 data taken at di!erent centre-of-mass energies show a di!erent behaviour of the measured cross-section, with the data taken at (s "133}161 GeV being lower, especially for =(30 GeV. 9.1.2. Production of charged hadrons The production of charged hadrons is sensitive to the structure of the photon}photon interactions without theoretical and experimental problems related to the de"nition and reconstruction of jets. The two main results from the study of hadron production at LEP are shown in Figs. 75 and 76.
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Fig. 75. Transverse momentum distribution dp/dp for hadron production in photon}photon scattering compared to 2 other experiments. The photon}photon scattering data taken at (s "161}172 GeV are compared to other experi ments for 10(=(30 GeV. Fig. 76. Transverse momentum distribution dp/dp for hadron production in photon}photon scattering compared to 2 next-to-leading-order calculations. The photon}photon scattering data taken at (s "161}172 GeV are compared to next-to-leading-order calculations for 10(=(125 GeV.
In Fig. 75 the di!erential single particle inclusive cross-section dp/dp for charged hadrons for cc 2 scattering from Ref. [194], with 10(=(30 GeV, is shown, together with results from cp, np and Kp scattering from WA69 with a hadronic invariant mass of 16 GeV from Ref. [211]. The WA69 data are normalised to the cc data at p +0.2 GeV. In addition, ZEUS data on charged particle 2 production in cp scattering with a di!ractively dissociated photon from Ref. [212] are shown. These data have an average invariant mass of the di!ractive system of 10 GeV, and again they are normalised to the OPAL data. In Fig. 76 the di!erential single particle inclusive cross-section for 10(=(125 GeV is compared to next-to-leading-order QCD predictions. The main "ndings are: 1. The spectrum of transverse momentum of charged hadrons in photon}photon scattering is much harder than in the case of photon}proton, hadron}proton and &photon}Pomeron' interactions. This can be attributed to the direct component of the photon}photon interactions. 2. The production of charged hadrons is found to be described by the next-to-leading-order QCD predictions from Ref. [213] over a wide range of =. These next-to-leading-order calculations are based on the QCD partonic cross-sections, the next-to-leading-order GRV parametrisation of the parton distribution functions for the photon and on fragmentation functions "tted to e>e\ data. The renormalisation and factorisation scales are set equal to p . 2
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9.1.3. Jet production Jet production is the classical way to study the partonic structure of particle interactions. At LEP the di-jet cross-section in cc scattering was studied in Ref. [196] at (s "161}172 GeV using the cone jet "nding algorithm with R"1. Three event classes are de"ned, direct, single-resolved and double-resolved interactions. As explained in Section 1.1, direct means that the photon as a whole takes part in the hard interaction, as shown in Fig. 2(a), whereas resolved means that a parton of a hadronic #uctuation of the photon participates in the hard scattering reaction, as shown in Fig. 2(b) and (c). Experimentally, direct and double-resolved interactions can be clearly separated using the quantity (E$p ) X , (58) x!" A (E$p ) X whereas a selection of single-resolved events cannot be achieved with high purity. Here E and p are X the energy and longitudinal momentum of a hadron, and the sum either runs over all hadrons in the two hard jets or over all observed hadrons. Ideally, in leading order, direct interactions have x!"1. However, due to resolution and higher order corrections the measured values of x! are A A smaller. Experimentally, samples containing large fractions of direct events can be selected by requiring x!'0.8, and samples containing large fractions of double-resolved events by using A x!(0.8. A The measurement of the distribution of cos h夹 the cosine of the scattering angle in the photon} photon centre-of-mass system, allows for a test of the di!erent matrix elements contributing to the reaction. The scattering angle is calculated from the jet rapidities in the laboratory frame using cos h夹"tanh
g !g . 2
(59)
In leading order the direct contribution ccPqq leads to an angular dependence of the form (1!cos h夹)\, whereas double-resolved events, which are dominated by gluon-induced reactions, are expected to behave approximately as (1!cos h夹)\. The steeper angular dependence of the double-resolved interactions can be clearly seen in Fig. 77(a), where the shape of the di-jet cross-section, for events with di-jet masses above 12 GeV and average rapidities of "(g #g )/2"(1, is compared to leading-order predictions. In addition, the shape of the angular distribution observed in the data is roughly described by the next-to-leading-order prediction from Refs. [214], as shown in Fig. 77(b). In both cases the theoretical predictions are normalised to the data in the "rst three bins. These next-to-leading-order calculations well account for the observed inclusive di!erential di-jet cross-section, dp/dE , as a function of jet transverse energy, E , for di-jet events 2 2 with pseudorapidities "g "(2. As expected, the direct component can account for most of the cross-section at large E , whereas the region of low E is dominated by the double-resolved 2 2 contribution, shown in Fig. 78. The calculation for three di!erent next-to-leading-order parametrisations of the parton distribution functions of the photon are in good agreement with the data shown in Fig. 79, except in the "rst bin, where theoretical as well as experimental uncertainties are large. Unfortunately, this is the region which shows the largest sensitivity to the di!erences of the parton distribution functions of the photon.
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Fig. 77. Angular dependence of di-jet production in photon}photon scattering. The data at (s "161}172 GeV are compared to leading-order matrix elements in (a) and to next-to-leading-order (NLO) predictions in (b).
Fig. 78. Transverse energy distribution dp/dE for jet production in photon}photon scattering compared to next-to2 leading-order calculations. The measured di-jet production at (s "161}172 GeV is compared to next-to-leading order (NLO) predictions for di!erent event classes. Fig. 79. Transverse energy distribution dp/dE for jet production in photon}photon scattering compared to predictions 2 for di!erent parton distribution functions. The measured di-jet production at (s "161}172 GeV is compared to next-to-leading-order (NLO) predictions for the GRV, GRS and GS parton distribution functions of the photon.
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9.1.4. Heavy quark production Similarly to the case of deep inelastic electron}photon scattering discussed in Section 3.4, in photon}photon scattering the production of heavy quarks is dominated by charm quark production, because the bottom quarks are much heavier and have a smaller electric charge. Due to the large scale of the process provided by the charm quark mass, the production of charm quarks can be predicted in next-to-leading-order perturbative QCD. In QCD the production of charm quarks at LEP2 energies receives sizeable contributions from the direct and the single-resolved process. In contrast, the double-resolved contribution is expected to be very small, as discussed in Ref. [215]. The direct contribution allows to test a pure QCD prediction and the single-resolved contribution is sensitive to the gluon distribution function of the photon. In photon}photon collisions the charm quarks have been tagged using standard techniques, either based on the observation of semileptonic decays of charm quarks using identi"ed electrons and muons in Ref. [198], or by the measurement of D夹 production in Refs. [122,197,199], using the decay D夹PDn, where the pion has very low energy, followed by the D decay observed in one of the decay channels, DPKn, Knn, Knnn. The leptons as well as the D夹 can be clearly separated from background processes. However, due to the small branching ratios and selection ine$ciencies the selected event samples are small and the measurements are limited mainly by the statistical error. Based on these tagging methods di!erential cross-sections for charm quark production and D夹 production in restricted kinematical regions have been obtained, examples of which are shown in Figs. 80 and 81. Fig. 80, taken from Ref. [199], shows the di!erential cross-section for charm quark production, with semileptonic decays into electrons ful"lling "cos h "(0.9 and E '0.6 GeV and for ='3 GeV. The data are compared to the leading-order prediction from PYTHIA, normalised to the number of data events observed. The shape of the distribution is well reproduced by the leading-order prediction. Fig. 81, taken from Ref. [122], shows the di!erential cross-sections 夹 for D夹 production as a function of the transverse momentum of the D夹, for "g" "(1.5 compared to the next-to-leading-order predictions from Ref. [216] calculated in the massless approach. The di!erential cross-sections as functions of the transverse momentum and rapidity of the D夹 are well reproduced by the next-to-leading-order perturbative QCD predictions, both for the OPAL results presented in Ref. [122] and for the L3 results from Ref. [199]. The shape of the OPAL data can be reproduced by the NLO calculations from Ref. [215]; however, the theoretical predictions are somewhat lower than the data, especially at low values of transverse momentum of the D夹. Based on the observed cross-sections in the restricted ranges in phase space the total charm quark production cross-section is derived, very much relying on the Monte Carlo predictions for the unseen part of the cross-section. Two issues are addressed, "rstly the relative contribution of the direct and single-resolved processes, and secondly the total charm quark production cross-section. The direct and single-resolved events, for example, as predicted by the PYTHIA Monte Carlo, 夹 show a di!erent distribution as a function of the transverse momentum of the D夹 meson, p" , 2 normalised to the visible hadronic invariant mass, = , as can be seen in Fig. 82 from Ref. [122]. This feature has been used to experimentally determine the relative contribution of direct and single-resolved events, which are found to contribute about equally to the cross-section. The total cross-section for the production of charm quarks is shown in Fig. 83 together with previous results summarised in Ref. [16]. The "gure, taken from Ref. [122], has been extended by additional L3 measurements presented in Ref. [200], by the author of Ref. [122]. The LEP results are consistent with each other and the theoretical predictions are in agreement with the data, both
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Fig. 80. Di!erential cross-sections for charm quark production with semileptonic decays into electrons. The data with electrons ful"lling "cos h "(0.9 and E '0.6 GeV and for invariant masses ='3 GeV are compared to the prediction of the PYTHIA Monte Carlo model, normalised to the number of data events observed. Fig. 81. Di!erential cross-sections for charm quark production with tagged D夹 mesons. The measured di!erential 夹 cross-sections for D夹 mesons with "g" "(1.5 is compared to a next-to-leading-order perturbative QCD predictions using the massless scheme.
for the NLO prediction of the full cross-section based on the GRV parametrisation and for the leading-order prediction of the direct component. The results su!er from additional errors due to the assumptions made in the extrapolation from the accepted to the total cross-section, which are avoided by measuring only cross-sections in restricted ranges in phase space. It has been shown in Ref. [215] that the NLO calculations are #exible enough to account for the phase space restrictions of the experimental analyses and that the predicted cross-sections in restricted ranges in phase space are less sensitive to variations of the charm quark mass and to alterations of the renormalisation as well as the factorisation scale. Given this, more insight into several aspects of charm quark production may be gained by comparing experimental results and theoretical predictions for cross-sections in restricted ranges in phase space. In addition to the measurements of the charm quark production cross-sections, a preliminary measurement of the cross-section for bottom quark production has been reported in Ref. [200]. 9.2. Photon structure from HERA At HERA the photon structure is investigated mainly by measurements of the total photon} proton cross-section in Refs. [217}219] and by measurements of the production of charged particles and jets in Refs. [220}228]. The most important results in the context of this review are the
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Fig. 82. The separation of the D夹 production into direct and single-resolved contributions. The measured distribution of 夹 the transverse momentum of the D夹 meson, p" , normalised to the visible hadronic invariant mass, = , is "tted by 2 a superposition of the predicted distributions for direct and resolved events based on the PYTHIA Monte Carlo. Fig. 83. Total cross-section for charm quark production. The measured cross-sections from LEP using lepton tags and D夹 tags are compared to previous measurements.
ones which try to extract information on the partonic content of quasi-real and also of virtual photons from photon}proton scattering and from deep inelastic electron}proton scattering. At e>e\ colliders the partonic structure of the quasi-real or virtual target photon, c(P, z), is probed by the highly virtual photon c夹(Q), in the region Q
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The photon}proton scattering reaction as well as deep inelastic electron}proton scattering also depends on the structure of the proton. Therefore, the investigations of the photon structure are restricted to phase space regions where the parton distribution functions of the proton are well constrained such that the dependence on the proton structure is removed as much as possible. There is one conceptual di!erence between the results obtained by ZEUS and those derived by H1. In the case of ZEUS all results are given at the hadron level, which means, the data are corrected for detector e!ects only. The phase-space regions are selected such that hadronisation corrections, as predicted by Monte Carlo models, are expected to be small. However, the data are not corrected for these e!ects. The results at the hadron level are then compared to NLO calculations which are valid at the partonic level and do not contain hadronisation corrections. H1 also measures cross-sections corrected for detector e!ects. However, based on these crosssections leading-order partonic quantities are reconstructed, which can directly be compared to perturbative calculations at the parton level. This approach certainly reconstructs more fundamental quantities. However, they have additional uncertainties compared to the hadron level crosssections stemming from the di!erences in the hadronisation procedures as assumed in the Monte Carlo programs used for the unfolding. In some cases these additional uncertainties even contribute the dominant error, as explained, for example, in Ref. [224]. Based on measurements of jet production, and charged particle production, information on the partonic structure of quasi-real and also of virtual photons have been derived. These results will be discussed in Sections 9.2.1 and 9.2.2, respectively. 9.2.1. Structure of quasi-real photons Jet cross-sections for photoproduction reactions have been measured by H1 and ZEUS. Fig. 84, taken from Ref. [228], shows an example of a measured di-jet cross-section in electron}proton scattering from the ZEUS experiment. In this "gure, additional theoretical predictions presented in Ref. [226] have been added by the author of Ref. [228]. The di-jet cross-section is corrected for detector e!ects and displayed for di!erent values for the minimum required E . The jets are found 2 using the k -clustering algorithm in the inclusive mode from Refs. [229,230], and the minimum required transverse energies of the jets are E '14 GeV and E '11 GeV for jets with 2 2 !1(g (2. This ensures good stability of the next-to-leading-order QCD predictions due to the asymmetric cuts on E . The di-jet cross-section is corrected for detector e!ects using 2 a bin-by-bin correction procedure. In addition, the photon virtuality is restricted to be smaller than 1 GeV and the scaled photon energy is required to be in the range 0.5(z(0.85. The restriction in the photon energy enhances the sensitivity to the parton distribution functions of the photon. The in#uence of the hadronisation on the di-jet cross-section has been studied based on the Monte Carlo programs HERWIG and PYTHIA. It was found that the di-jet cross-section at the parton level is 10}50% higher than the di-jet cross-section at the hadron level and the largest corrections are predicted for the con"guration where both jets have g (0, which means the jets go in the same hemisphere as the incoming electron. In addition, based on the investigation of the transverse energy #ow around the jets, it has been concluded that no inclusion of soft interactions in addition to the primary hard parton}parton scattering reaction is needed to describe the observed jet pro"les at this large values of E . This means, the jet energy pro"les can be described 2 without inclusion of the so-called soft-underlying event, a method to describe additional soft interactions between the photon and proton remnants. However, it should be noted that the
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Fig. 84. Di-jet cross-section in photon}proton scattering compared to next-to-leading-order QCD predictions. In (a)}(c) the data are shown as a function of g in bins of g , for 0.50(z"y(0.85, for all selected events and for events with an observed value of x '0.75. The rows correspond to di!erent values for the minimum required E . The inner error A 2 bar indicates the statistical and the outer error bar the full error. The shaded band shows the uncertainty related to the energy scale. The curves denote the theoretical predictions based on the AFG (full), GRV (dash) and GS (dot) parametrisations of the parton distribution functions of the photon.
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predicted cross-sections of the HERWIG and PYTHIA Monte Carlo programs had to be scaled by large factors to account for the measured x distribution. The numbers used are 1.28/1.27 and A 1.83/1.72 for the direct/resolved reactions, when using the PYTHIA and HERWIG Monte Carlo programs, respectively. The measured jet cross-section is higher than the predictions from several groups of authors, especially for the region g '0. It has been shown in Ref. [226] that the various predictions are in good agreement with each other. The cross-section shows some sensitivity to the choice of the parton distribution functions of the photon, given by the spread of the predictions seen in the "rst row of Fig. 84. The di!erent regions in g correspond to di!erent regions in x : A E e\E #E e\E 2 . x " 2 A 2zE
(60)
The region of large x is easiest identi"ed by the region in g where the curves for x '0.75 A A approach those for the full range in x . It is in this part where the parton distribution functions A have the largest spread because they are only mildly constraint by the measurements of FA shown in Fig. 55. In this region the cross-section can be described by the perturbative calculations. However, for smaller values of x , shown in Fig. 84(b) and (c) for increasing values of g , none of A the used sets of the parton distribution functions of the photon is able to describe the data, which is taken as an indication that they may underestimate the parton content of the photon. This region corresponds to about 0.1 to 0.6 in x , and here the constraints from the measurements of FA are A rather tight, as indicated by the small spread of the curves. It remains an interesting, but still open question, whether the parton distribution functions of the photon can be changed to describe the jet data and still being consistent with the measured FA . However, it has to be taken into account that the data are only corrected for detector e!ects, and thus a hadron level quantity is compared to theoretical predictions at the parton level. Similarly, in the case of H1, the measured di-jet cross-section as a function of the average transverse energy squared of the jets in bins of x , is the starting point to investigate the partonic A structure of the photon in Ref. [222]. Again jets are found using the k -clustering algorithm in the inclusive mode from Refs. [229,230]. The average of the transverse energies of the two jets with the highest transverse energies is required to be above 10 GeV and the di!erence of the transverse energies should be less than 50% of their average. The average jet rapidity is constrained in the region 0}2 and the absolute di!erence to be smaller than unity. This ensures E '7.5 GeV and, as above for the ZEUS measurement, good stability of the next-to-leading2 order QCD predictions. The cross-section is integrated for P(4 GeV and 0.2(z(0.83. The measured di!erential electron}proton di-jet cross-section is shown in Fig. 85, taken from Ref. [222]. The di-jet cross-section is corrected for detector e!ects only, and compared to the predictions of the leading order PYTHIA Monte Carlo and to the next-to-leading-order parton level predictions using the GRV and the GS parton distribution functions of the photon. The predictions well account for the observed jet cross-section with the exceptions of low and large values of x . For x (0.2 the next-to-leading-order prediction based on the GRV parametrisations A A is lower than the data, whereas the GS parametrisations are able to describe the observed cross-section. As in the case of ZEUS, for large values of x the GRV and GS predictions tend to lie A
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Fig. 85. Inclusive di-jet cross-section for photon}proton scattering from H1. The data is shown as a function of the average transverse energy squared of the jets for several bins in x . The inner error bars represent the statistical errors, the A outer error bars the full errors. The data are compared to the leading-order prediction from the PYTHIA generator (dash), and to analytical next-to-leading calculations using the GRV (full) and the GS (dot) parton distribution functions of the photon.
below the data, with GS predicting a smaller cross-section due to the strongly suppressed quark distribution functions at large values of x discussed in Section 4. However, for H1 the disagreement seems to be less pronounced and also to be more concentrated at larger values of x , 0.6(x (0.75. In the highest bin in x and at low values of E both parametrisations predict A A A 2 too large cross-sections. This cross-section is then used to determine an e!ective parton distribution function of the photon. The reaction is factorised into the radiation of the photon o! the incoming electron, followed by a subsequent photon}proton scattering reaction. The #ux of transverse photons is described using the WeizsaK cker}Williams approximation, Eq. (27), discussed in Section 3.2. The photon}proton cross-section is approximated using the concept of the single e!ective subprocess matrix element, M , from Ref. [231]. This leading-order approach relies on the fact that the 1#1 angular dependence of the matrix elements of the most important contributions to the process is very similar, as can be seen from Fig. 77. Therefore, the contributions to the photon}proton crosssection can be approximated by the product of the e!ective parton distribution functions and the
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M , leading to 1#1 dp 1 dN2 fI (x , Q) fI (x , Q) A A A "M (cos h夹)" , 夹J 1#1 dz dx dx d cos h x x z dz A A with e!ective parton distribution functions de"ned as
(61)
L fI (x , Q), [qA (x , Q)#q A (x , Q)]#gA(x , Q) , (62) A A I A I A A I L fI (x , Q), [q (x , Q)#q (x , Q)]#g(x , Q) . (63) I I I Similarly to deep inelastic electron}photon scattering, the factorisation and renormalisation scales are taken to be equal in the analysis by H1. The factorisation scale Q is identi"ed with p, the transverse momentum squared of the partons with respect to the photon}proton axis in the photon}proton centre-of-mass system. It has been veri"ed that the factorisation of the process into the photon #ux and the two e!ective parton distribution functions is meaningful. This has been done by showing that within the experimental uncertainty, the observed x distribution is independent of the measured values of z and x for a "xed product zx , which means a "xed energy A A entering the hard parton}parton scattering from the photon side. In addition, in the region of the H1 analysis it is found that the relative amount of quark and gluon initiated di-jet events agree with the weight 9/4 to better than 5%. The evolution of the extracted leading-order e!ective parton distribution function x fI /a as AA a function of the factorisation scale Q is shown in Fig. 86 taken from Ref. [222] for two regions of x , 0.2(x (0.4 and 0.4(x (0.7. The data are compared to three predictions based on the A A A GRV parametrisation of the parton distribution functions of the photon. The predictions shown are the e!ective parton distribution function (full), the quark component of fI (dot), and the VMD A contribution to fI (dash), based on the VMD prediction of FA , explained in Section 3.5. The full A prediction describes the measurement. As expected, the VMD contribution is not able to account for the data, and the importance of the gluon part increases for decreasing value of x . Under the A assumption that the factorisation and renormalisation scales can be identi"ed with the transverse momentum of the partons, and within the uncertainties of the concept of the single e!ective subprocess matrix element and the e!ective parton distribution functions, the measurement shows the universality of the parton distribution functions of the photon, which are able to describe both photon}proton scattering and deep inelastic electron}photon scattering reactions. In addition, this analysis extends the measurement of the photon structure to factorisation scales of the order of 1Q2"1p( 2+900 GeV. The sensitivity to the gluon distribution function of the photon seen in Fig. 86 can be explored in the measurement of the di-jet cross-section and by using the production of charged particles to obtain the gluon distribution function of the photon. Both methods have been used by H1 in Refs. [220,225] and [223], respectively. The two methods are complementary. The jet cross-section receives error contributions, for example, from the accuracy of the knowledge of the energy scales of the calorimeters and from the jet de"nition, which are absent when using charged particles. In addition, for su$ciently large transverse momenta of the particles, the dependence on the soft
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Fig. 86. Leading-order e!ective parton distribution function of the photon from H1. The data are shown as a function of the factorisation scale Q"p( , averaged over x in the ranges (a) 0.2(x (0.4 and (b) 0.4(x (0.7. The inner error A A A bar indicates the statistical and the outer error bar the full error. The data are compared to several theoretical predictions explained in the text.
underlying event is also reduced, since most of the particles produced in the soft underlying event have momenta of the order of 0.3 GeV. In contrast, the distribution of charged particles are more sensitive to details of the hadronisation, which are integrated over when using jets. The variable x is either obtained from the jets using Eq. (58), or similarly, from the sum over all charged A particles with transverse momentum with respect to the beam axis of more than 2 GeV using the relation x +1/E p e\E. Since the incoming partons of the hard scattering process cannot be A A identi"ed, no distinction can be made between quark and gluon initiated processes. Therefore, both methods yield only an indirect measurement of the gluon distribution function, because the quark initiated contribution has to be subtracted based on existing parton distribution functions obtained from measurements of FA at e>e\ colliders. The results from the two methods as a function of x are shown in Fig. 87 (left, right) taken from A Refs. [223] and [225], respectively. Both "gures are shown because the older "gure contains more comparisons to theoretical predictions. The two results, obtained from single particles and jets respectively, are consistent and the gluon distribution function is found to be small at large values of x and to rise towards small values of x. The measured leading order gluon distribution function is consistent with the existing parametrisation from GRV. As in the case of the structure function measurements discussed in Section 7.2, the measurements disfavour the strongly rising gluon distribution functions of the photon, for example, the LAC1 gluon distribution function. In addition, the recent H1 measurements are above the SaS1D prediction which is slightly disfavoured. This is similar to the measurements of FA shown in Figs. 50 and 52 which tend to be above the SaS1D prediction. However, the preliminary update of the FA measurement at low values of Q from OPAL (Fig. 52) seems to be consistent with the SaS1D prediction. Certainly, more precise data is needed to draw de"nite conclusions, but, again these measurements show the universality of the parton distribution functions of the photon. This concludes the discussion of measurements of the structure of quasi-real photons and the remaining part is devoted to virtual photons.
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Fig. 87. Measurements of the gluon distribution function of the photon from H1. The leading-order gluon distribution function as a function of x , as obtained from the charged particle cross-section is shown twice (left full circles and right A open squares) for an average factorisation scale 1Q2"1p( 2"38 GeV, together measurements obtained from di-jet cross-sections (left open circles from Ref. [220] and right closed circles from Ref. [225]) for 1Q2"75 GeV. The inner error bar indicates the statistical and the outer error bar the full error. The data are compared to several theoretical predictions explained in the text.
9.2.2. Structure of virtual photons As has been discussed in Section 7.3 the structure of the virtual photon can be studied via the measurement of photon structure functions in the kinematical regime Qe\ data is very limited due to low statistics and further information on the structure of the virtual photon is certainly needed. At HERA both the evolution with Q and the suppression with P can be investigated in deep inelastic electron}proton scattering. The largest part of the cross-section is due to the direct coupling of the virtual photon to the partons in the proton, but there is a small region of phase space where p
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Fig. 88. Triple di!erential jet cross-section dp/dPdE dx for virtual photons from H1. The data are shown as 2 A a function of the photon virtuality P"Q in the ranges of x and using several bins in the factorisation scale Q"E . A 2 The inner error bar indicates the statistical and the outer error bar the full error. The data are compared to several theoretical predictions explained in the text.
much higher than the prediction from the direct coupling, and this di!erence is attributed to resolved interaction due to the hadronic structure of the virtual photon. A much better description of the data by the RAPGAP prediction is achieved when also the partonic structure of the virtual photon is taken into account by using the GRV parton distribution functions of the quasi-real photon, suppressed by the Drees Godbole scheme (full), Eq. (49), with P"u"0.04 GeV. Based on this observation the leading-order e!ective parton distribution function of virtual photons is extracted very similar to the case of quasi-real photons, discussed above. Again the photon}proton cross-section is approximated by the product of the e!ective parton distribution function and the M . Due to the non-zero virtuality P of the photon the situation 1#1
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Fig. 89. Leading-order e!ective parton distribution function of virtual photons from H1. The e!ective parton distribution function is shown as a function of x for several bins for the factorisation scale Q"p and for the photon virtuality A P"Q. The inner error bar indicates the statistical and the outer error bar the full error. The data are compared to several theoretical predictions explained in the text.
is more complex. Firstly, also the #ux of longitudinal photons has to be taken into account and secondly, parton distribution functions of longitudinal virtual photons are needed, which have not yet been determined. Given the known ratio of the #ux of transverse and longitudinal photons, e(z), de"ned in Eq. (26), only the #ux of transverse photons, Eq. (24), is needed and the cross-section can be expressed in a factorised form by dp 1 dN2 fI (x , Q, P) fI (x , Q) A A A J "M (cos h夹)" , 夹 1#1 x x dz dx dx d cos h dP z dz dP A A
(64)
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283
Fig. 90. The P suppression of the leading-order e!ective parton distribution function of virtual photon from H1. The e!ective parton distribution function is shown as a function of the photon virtuality P"Q for several bins in x and A using several bins for the factorisation scale Q"p. The inner error bar indicates the statistical and the outer error bar the full error. The data are compared to several theoretical predictions explained in the text.
with e!ective parton distribution functions de"ned as L fI (x , Q, P), [qA (x , Q, P)#q A (x , Q, P]#gA(x , Q, P) , I A I A A A A I
(65)
and fI (x , Q) as above. Here fI is to be understood as the polarisation averaged e!ective parton A distribution function fI "fI 2#e(z) fI *. According to Refs. [13,118,233] fI * is expected to be small in A A A A most of the kinematical range of the H1 analysis. If this is the case fI reduces to the purely A transverse e!ective parton distribution function for virtual photons.
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In Fig. 89 the measured leading-order e!ective parton distribution function x fI /a is shown as A A a function of x in the region 0.2(x (0.7 and in bins of the factorisation scale Q and the photon A A virtuality P. A slow increase for increasing x is observed for all values of the factorisation scale A and the photon virtuality. The lever arm in the factorisation scale Q is too small to observe the predicted logarithmic growth. The observed decrease with the photon virtuality P is much stronger at low values than at large values of x . For example at x "0.275, x fI /a decreases from A A AA 0.55 to 0.16 when P changes from 2.4 to 12.7 GeV, whereas at x "0.6 and for the same range in A P the decrease is only from 0.95 to 0.54. In most of the phase space the data presented in Fig. 89 can be described by the predictions based on the SaS1D (dash) and SaS2D (dot) parton distribution functions of the virtual photon using the default suppression with the photon virtuality IP2"0, see Section 4 for details. In addition, also the GRV parton distribution functions of the quasi-real photon suppressed by the Drees Godbole scheme (full), Eq. (49) with P"u"0.01 GeV, is in agreement with the data. The only exception is the region of large values of x and large photon virtualities, A where the predictions tend to lie below the data, however, in this region the data su!er from large errors. In Fig. 90 the e!ective parton distribution function is shown as a function of the photon virtuality P in bins of x and of the factorisation scale Q. Again the observed decrease with P is A described for the region Q
Acknowledgements I very much enjoy to participate in this active "eld of research dealing with the structure of the photon. It is clear that the achievements in this area of research are due to the e!ort of many people. Given this, it is only natural that in writing this review I greatly pro"ted from the discussions with my colleagues about experimental and theoretical aspects of the investigations of the photon structure. I also received strong support from various people who provided me with their software and valuable advice on how to use it. Without this I could not have performed the comparisons between the data and the theoretical predictions. The excellent working conditions within the CERN OPAL group were very helpful in various aspects of this research.
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I would like to thank very much the following persons: V.P. Andreev, P. Aurenche, C. Berger, V. Blobel, A. BoK hrer, C. Brew, A. Buijs, J. Chyla, A. De Roeck, R. Engel, M. Erdmann, F.C. ErneH , A. Finch, M. Fontannaz, R. Freudenreich, L. Gordon, K. Hagiwara, R.D. Heuer, R.G. Kellogg, E. Laenen, J.A. Lauber, C.H. Lin, D.J. Miller, J. Patt, D.E. Plane, I. Schienbein, G.A. Schuler, M.H. Seymour, T. SjoK strand, S. SoK ldner-Rembold, H. Spiesberger, M. Stratmann, B. Surrow, G. Susinno, I. Tyapkin, A. Vogt, J.H. Vossebeld, M. Wadhwa, A. Wagner, P.M. Zerwas and V. Zintchenko. Even more important was the continuous encouragement and patience of my wife. Without her help, I would have never been able to write this review during a long period of illness, vielen Dank.
Appendix A. Connecting the cross-section and the structure function picture In this section the connection between the cross-section and the structure function picture for deep inelastic scattering, Q
(A.1)
with p "p !p. This leads to the simpli"ed equations (p !p ) ) q E!E E p)q +1! A " A "z " r" p )q E E p )q
(A.2)
(p ) q)!QP+(p ) q) .
(A.3)
and
In addition, p ) p"2EE A
(A.4)
will be used. Inserting the relations between the structure functions and the cross-sections in the limit of Eq. (A.3) as given in Eq. (21) into Eq. (19) yields
dp dp a (p ) q)!QP dp" 4o>>o>> E E 16pQP (p ) p )!mm
4pa o ; 2xFA (x, Q)# FA (x, Q) . 2 Q 2o>> * The limits for the individual terms are derived below.
(A.5)
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The term in the "rst square brackets reduces to
(p ) q)!QP p ) q y " " p ) p"yz , (A.6) (p ) p )!mm 2E 2E where Eqs. (A.3), (A.4) and (3) have been used. The photon density matrix element o>> de"ned in Eq. (15) can be written as
(2p ) q!p ) q) m 2p ) q m !1 #1!4 2o>>" #1!4 " (p ) q)!QP P p)q P
"
2 m 2 1#(1!z) 2mz !1 #1!4 " ! P r P z z
,
where Eqs. (A.3), (7) and (13) have been used. Similarly, o>> leads to 2 1#(1!y) 2o>>" , y y
(A.7)
(A.8)
where, due to Q<m, the mass term can be neglected. Using this and the relation o"2(o>>!1), Eq. (16), the term in front of FA simpli"es to * o o>>!1 y 2(1!y) e " " "1! " "e(y) . (A.9) 2o>> o>> 1#(1!y) 1#(1!y) This shows that all terms factorise into quantities which depend only on the quasi-real or on the virtual photon, but not on a combination of them. This is certainly not obvious from the original form of Eq. (19). Next the change of variables is performed. Di!erentiating Eq. (10) with respect to cos h yields dP"!2EE d cos h (A.10) and by in addition using !dE "dE "Edz one derives A dp "E dE d d cos h "!p dz dP . (A.11) E Combining Eqs. (12) and (9) for P"0, a relation between E and x is obtained Q QE E " #E! , (A.12) 4E xs A and therefore QE dE " dx . s x A Using in addition dQ"!2EE d cos h ,
(A.13)
(A.14)
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derived from Eq. (9) as above for P, the other di!erential reads dp y "E dE d d cos h "!p dx dQ . E x
(A.15)
Inserting all pieces into Eq. (A.5) recovers Eq. (22) 4pay dp " xQ dx dQ dz dP
a 1#(1!z) 1 2mz a 1#(1!y) 1 ; ! 2p P 2p z P y Q
2(1!y) ; 2xFA (x, Q)# FA (x, Q) . 2 1#(1!y) *
(A.16)
In this form the individual pieces can be nicely identi"ed. The second line is the #ux of the transverse quasi-real target photons. The third line is the #ux of the transverse virtual photons, where the mass term has been neglected. The term in front of FA is the ratio of the #ux of the * transverse and longitudinal virtual photons. Finally the structure functions FA and FA contain the 2 * information on the structure of the transverse quasi-real target photons when probed by transverse and longitudinal virtual photons, respectively. The most important approximation made in deriving Eq. (22) from Eq. (19) is Eq. (A.3). As can be seen from the functional form of p and p , listed in the appendix of Ref. [4], exactly this term 22 *2 also appears in these cross-sections and therefore, for example, in FA . Given this, Eq. (22) should /#" not be used when studying the P dependence of FA , and Eq. (19) should be used instead. /#" Appendix B. General concepts for deriving the parton distribution functions In this section the procedure to derive the parton distribution functions by solving the full evolution equations is discussed. There are several groups using this approach. However, they di!er in the choices made for Q , for the factorisation scheme and for the assumptions on the input parton distribution functions at the starting scales. The general strategy is outlined following the discussion given in Ref. [47]. The individual sets of parton distribution functions have been discussed in Section 4. The parton distribution functions of the photon obey the following inhomogeneous evolution equations:
a L a dqA G " P CA# [P qA #P q A ]#P gA , I O G OI I OG E OG OI 2p d ln Q 2p OG A I a L a dq A G " P CA# [P qA #P q A ]#P gA , I O G OI I OG E OG OI d ln Q 2p OG A 2p I
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dgA L a a " P CA# [P I qA #P I q A ]#P gA , EA EO I EO I EE d ln Q 2p 2p I a L dCA " P CA# [P I qA #P I q A ]#P gA . (B.1) AA AO I AO I AE d ln Q 2p I The parton distribution functions for the quarks and antiquarks are denoted with qA(x, Q) and G q A(x, Q), the gluon distribution function with gA(x, Q), and CA(x, Q) is the photon distribution G function. The symbol represents the convolution integral, de"ned as
dy x P ) qA(y, Q) . (B.2) y y V The sum runs over all active quark #avours k"1,2, n , and the P are the usual Altarelli}Parisi ?@ splitting kernels P qA(x, Q)"
aJaK PJ K(z) . (B.3) P (z, a, a )" ?@ (2p)J>K ?@ J K Since a+1/137 is very small, the expansion of Eq. (B.1) in powers of a is cut at O(a). To this order the terms P I , P I and P do not contribute and the evolution equation for the photon inside the AO AE AO photon can be solved directly. Since photon radiation from photons starts at order a one can use P Jd(1!x) to all orders in a , and with this, the photon distribution function in leading order is AA given by
a CA (x, Q)"d(1!x) 1! *p
Q L eI ln #c , (B.4) O Q I where Q is the starting scale of the evolution and c is an unknown parameter. Since the photon coupling to quarks and anti-quarks is the same the quark distribution functions ful"ll qA"q A and G G one of the "rst two equations from Eq. (B.1) can be removed. Since qA and gA are already of order G a only the O(1) contribution from CA has to be used in the remaining evolution equations which thereby reduce to
dqA L a a G " P # (P #P ) qA #P gA , G O OG O I OG OI I OG E A d ln Q 2p 2p I L a a dgA " P # (P I #P I ) qA #P gA . (B5) I EE EO EO d ln Q 2p EA 2p I The Altarelli}Parisi splitting kernels describe the parton branchings, as illustrated in Fig. 91. In leading order they have the following form:
4 1#z (z)"d #2d(1!z) , GI 3 (1!z) > P G (z)"[z#(1!z)] , P (z)"0 , OE EA
P G (z)"3eG [z#(1!z)] , O OA P G I (z)"0 , OO
P
OG OI
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Fig. 91. Feynman diagrams illustrating the Altarelli Parisi splitting kernels. Shown are (a) the branching of a photon into a quark pair, (b) the branching of a quark into a quark and a gluon, (c) the branching of a gluon into a quark pair, and (d) the branching of a gluon into two gluons.
4 1#(1!z) P I (z)" , EO z 3
P I (z)"P I (z) , EO EO
1!z 11 n z # #z(1!z)# ! d(1!z) . (B.6) z 12 18 (1!z) > The evolution equations are inhomogeneous, because of the occurrence of the term P G describing OA the coupling of the photon to quarks. If it were not for this term, the evolution equations would be identical to the evolution equations for parton distribution functions of hadrons like the proton. This is why the solution of the homogeneous evolution equations can be identi"ed with the hadron-like part of the photon structure function, and its x and Q behaviour is just as in the hadron case. A particular solution to the inhomogeneous evolution equations can be identi"ed with the point-like part of the photon structure function. The parton distribution functions are subject to a momentum sum rule, which can be expressed as P (z)"6 EE
dx+x[RA(x, Q)#gA(x, Q)#CA(x, Q)],"1 ,
(B.7)
where RA is de"ned in Eq. (40). This momentum sum rule holds order by order in a, thus by using Eq. (B.4) to order O(a) the momentum sum rule reads a dx+x[RA (x, Q)#gA (x, Q)]," **p
Q L eI ln #c . (B.8) O Q I This means that the quark and gluon distribution functions of the photon do not obey a momentum sum rule which is independent of Q as, for example, the parton distribution functions of the proton. In contrast, the momentum carried by the partons of the photon rises logarithmically with Q, with an unknown parameter c , which has to be obtained from somewhere else, as
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performed, for example, in Refs. [43,234,235]. This has been done by relating Eq. (B.4) to the hadronic e>e\ annihilation cross-section p(e>e\Phadrons) by means of a dispersion relation in the photon virtuality. For example, the value of c /p obtained in Ref. [43] is about 0.55 with an uncertainty of about 20% for Q "0.36 GeV. The main consequence of this is that for the photon the constraint on the gluon distribution function from the momentum sum rule together with the measurement of FA is not as powerful as in the case of the proton. In addition, there is a theoretical debate on this issue, and the applicability of this sum rule has been questioned in Ref. [236]. The general solution of the inhomogeneous evolution equations, Eqs. (B.5), for the #avour singlet part, which is given by
qA(x, Q)"
RA(x, Q) gA(x, Q)
"qA (x, Q)#qA (x, Q) , .*
(B.9)
can be expressed in terms of the point-like part, qA (x, Q), and the hadron-like part, qA (x, Q), .* taken as the solution of the homogeneous evolution equation. At next-to-leading order these solutions can be written as
qA (x, Q)"
a BK V a a BK V a BK V # ;K (x) ! ;K (x) a 2p a a
qA(Q )
(B.10)
and
1 2p a >BK V #;K (x) 1! a(x) qA (x, Q)" .* 1#dK (x) a a a BK V 1 b(x) , (B.11) # 1! a dK (x) where a "a (Q) and a "a (Q ). The quantities a(x), b(x), dK (x) and ;K (x) abbreviate com binations of the splitting functions and the QCD b-function. The solutions to the leading-order evolution equations are contained in Eqs. (B.10) and (B.11), and are obtained for ;K (x)"0 and b(x)"0. The asymptotic point-like solution is obtained from Eq. (B.11) in next-to-leading order for ;K (x)"0 and by dropping the terms proportional to a /a , which vanish for QPR leading to 1 1 2p a(x)# b(x) . (B.12) qA (x, Q)" dK (x) a (Q) 1#dK (x) The leading-order result is obtained by in addition setting b(x)"0. The pole at dK (x)"!1 is responsible for the divergence of the asymptotic solution. The most recent parametrisation of the leading-order asymptotic solution is given in Ref. [30]. The next-to-leading-order structure function FA for light quarks is given by a a , (B.13) FA "2x eI qA # (C qA #C gA)# eI C I E O I 2p O 2p O A I
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where the C are the next-to-leading order coe$cient functions. In next-to-leading order there G exist the factorisation scheme ambiguity, which means a freedom in the de"nition of the terms belonging to the parton density functions and the terms which are included in the hard scattering matrix elements. The di!erent choices are known as factorisation schemes. A physics quantity like FA , which is a combination of parton density functions and hard scattering matrix elements, is invariant under this choice, if calculated to all orders in perturbation theory, but in "xed order the results from di!erent factorisation schemes can di!er by "nite terms, as discussed, for example, in Ref. [107]. Commonly used factorisation schemes for the proton structure function are the MS scheme and the DIS scheme. For the photon structure function the DIS scheme, as introduced A in Ref. [31], is motivated by the DIS scheme. The photonic, non-universal part, C , in A next-to-leading order is given by
C+1 (x)"3 [x#(1!x)] ln A
1!x !1#8x(1!x) . x
(B.14)
In the DIS scheme this term is absorbed into the quark distribution functions by using the A de"nitions a (B.15) qA A "qA # eI C+1 , C"'1A "0 . A I "'1 I +1 2p O A If the calculation is performed in the DIS factorisation scheme, then there is a good stability of the A perturbative prediction when comparing the leading-order and next-to-leading-order results, as can be seen from Ref. [31]. By applying the MS factorisation scheme, there are much larger di!erences between the leading-order and next-to-leading-order results of FA , stemming from the large negative contribution to the point-like part of the purely photonic part C A in next-to-leading order at large values of x, as can be seen from Eq. (B.14) in the limit xP1. This results even in a negative structure function FA at large values of x, if it is not compensated for by carefully choosing also the point-like input distribution functions, as done in Refs. [107,108]. The negative structure function FA at large values of x can by avoided by using a technical next-to-leading-order input distribution function for the point-like part of the following form: a qA (x, Q )"! eI C (x), gA (x, Q )"0 . 2p O A .* I .* Suitable expression for the term C are either C+1 (x) as de"ned in Eq. (B.14) or A A C (x)"3+[x#(1!x)] ln(1!x)#2x(1!x), . A
(B.16)
(B.17)
The DIS scheme absorbs all higher order terms into the de"nition of the quark distribution functions, such that F is proportional solely to the quark distribution functions in all orders in a . In contrast to this, the DIS scheme in A next-to-leading order absorbs only the purely photonic part, C , into the quark distribution functions. A
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The "rst solution is chosen in Ref. [108] for the construction of the parametrisations from Gordon and Storrow, and Eq. (B.17) was developed for the parametrisations from Aurenche et al. [107], based on an analysis of the momentum integration of the box diagram. As stated above these are purely technical questions on how to deal with the factorisation scheme ambiguity. The predictions for the photon structure function FA calculated in the di!erent schemes are di!erent. However, the di!erences in FA are much smaller than the di!erences in the parton distribution functions, as can be seen from Ref. [47]. This also shows that although FA is a well de"ned quantity its interpretation in terms of parton distribution functions of the photon is a delicate issue.
Appendix C. Collection of results on the QED structure of the photon This section contains a summary of the available results on the QED structure of the photon obtained either from measurements of deep inelastic electron}photon scattering or by exploring the exchange of two highly virtual photons. The numbers listed in Tables 9}20 are the basis of the summary plots shown in Section 6. The sources of information used to obtain the numbers are always given in the caption of the respective table. If the total error p has been obtained in this review, it was calculated from the statistical error p and the systematic error p using the relation p "(p #p .
(C.1)
First the measurements of FA are listed, followed by results on FA and FA , and by the /#" /#" /#" results for the exchange of two highly virtual photons.
Table 9 Results on the average photon structure function 1FA 2 from the CELLO experiment. The numbers are read o! the /#" published "gure, which probably contains only the statistical error. The additional quoted systematic error of 8% is added in quadrature. The measured FA is averaged over the Q range 1.2}39 GeV, with an average value of /#" 1Q2"9.5 GeV. No information is available to which value of 1P2 the result corresponds CELLO Q (GeV)
x
1FA 2 /#"
p
p
Ref.
1.2}39
0.00}0.10 0.10}0.20 0.20}0.30 0.30}0.40 0.40}0.50 0.50}0.60 0.60}0.70 0.70}0.80 0.80}0.90 0.90}1.00
0.222 0.426 0.562 0.511 0.597 0.571 0.545 1.202 1.057 1.185
0.077 0.128 0.162 0.153 0.170 0.170 0.170 0.256 0.273 0.528
0.079 0.132 0.168 0.158 0.177 0.176 0.176 0.273 0.286 0.536
[150]
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293
Table 10 Results on the average photon structure function 1FA 2 from the DELPHI experiment. The numbers are provided by /#" V. Podznyakov. Only statistical errors are available. The measured FA is averaged over the Q range 4}30 GeV, with /#" an average of 1Q2"12 GeV. The best "t of the QED prediction to the data is obtained for 1P2"0.04 GeV. In addition, there exist preliminary results, which are listed in Table 16 DELPHI Q (GeV)
x
1FA 2 /#"
p
4}30
0.00}0.08 0.08}0.18 0.18}0.31 0.31}0.48 0.48}0.70 0.70}1.00
0.077 0.193 0.327 0.513 0.719 0.969
0.013 0.016 0.026 0.032 0.051 0.109
p
Ref. [86]
Table 11 Results on the average photon structure function 1FA 2 from the L3 experiment. The numbers for 1FA 2 are read /#" /#" o! the published "gure, and the average value of Q is provided by G. Susinno. The measured FA is averaged over the /#" Q range 1.4}7.6 GeV, with an average of 1Q2"3.25 GeV. The best "t of the QED prediction to the data is obtained for 1P2"0.033 GeV L3 Q (GeV)
x
1FA 2 /#"
p
p
Ref.
1.4}7.6
0.00}0.10 0.10}0.20 0.20}0.30 0.30}0.40 0.40}0.50 0.50}0.60 0.60}0.70 0.70}0.80 0.80}0.90 0.90}1.00
0.062 0.216 0.326 0.391 0.477 0.534 0.654 0.709 0.775 0.549
0.060 0.015 0.022 0.026 0.028 0.029 0.035 0.037 0.046 0.069
0.063 0.018 0.025 0.030 0.032 0.035 0.039 0.044 0.052 0.072
[151]
Appendix D. Collection of results on the hadronic structure of the photon Because in some cases it is not easy to correctly derive the errors of several of the measurements, a detailed survey of the available results has been performed, the outcome of which is presented below. In some of the cases it is rather di$cult to obtain the central values and the errors of the measurements, because, especially in older publications, it is not always clear which errors
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Table 12 Results on the photon structure function FA from the OPAL experiment. For explanations see Table 13 /#" OPAL 1Q2 (GeV)
x
FA /#"
p
p
Ref.
2.2
0.00}0.10 0.10}0.20 0.20}0.30 0.30}0.40 0.40}0.50 0.50}0.60 0.60}0.70 0.70}0.80 0.80}0.90 0.90}0.97
0.115 0.219 0.282 0.347 0.356 0.400 0.483 0.491 0.532 0.308
0.007 0.010 0.012 0.015 0.017 0.020 0.025 0.031 0.034 0.032
0.009 0.013 0.016 0.019 0.020 0.023 0.030 0.033 0.036 0.078
[152]
4.2
0.00}0.10 0.10}0.20 0.20}0.30 0.30}0.40 0.40}0.50 0.50}0.60 0.60}0.70 0.70}0.80 0.80}0.90 0.90}0.97
0.108 0.237 0.320 0.378 0.373 0.421 0.519 0.556 0.601 0.470
0.010 0.014 0.018 0.020 0.020 0.025 0.029 0.034 0.040 0.041
0.019 0.017 0.022 0.023 0.022 0.028 0.032 0.036 0.042 0.065
[152]
8.4
0.00}0.10 0.10}0.20 0.20}0.30 0.30}0.40 0.40}0.50 0.50}0.60 0.60}0.70 0.70}0.80 0.80}0.90 0.90}0.97
0.090 0.271 0.334 0.409 0.496 0.563 0.596 0.687 0.891 0.761
0.012 0.022 0.029 0.033 0.038 0.043 0.049 0.056 0.072 0.074
0.014 0.029 0.035 0.040 0.046 0.050 0.054 0.061 0.084 0.089
[152]
12.4
0.00}0.15 0.15}0.30 0.30}0.45 0.45}0.60 0.60}0.75 0.75}0.90 0.90}0.97
0.151 0.297 0.402 0.434 0.758 0.723 0.714
0.022 0.033 0.041 0.044 0.062 0.072 0.085
0.025 0.037 0.048 0.056 0.074 0.077 0.090
[152]
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295
Table 13 Results on the photon structure function FA from the OPAL experiment continued. The values are taken from the /#" published tables of Ref. [152]. These results on FA supersede the results published in Ref. [237]. The results given at /#" 1Q2"2.2, 4.2, 8.4, 12.4, 21.0 and 130 GeV are all statistically independent and are unfolded from data in the Q ranges 1.5}3, 3}7, 6}10, 10}15, 15}30, and 70}400 GeV. The results given at 1Q2"3.0 GeV are unfolded from data in the Q ranges 1.5}7 GeV, which means they are not independent results, but contain the data at 1Q2"2.2 and 4.2 GeV. This data is used for the comparisons made in Fig. 38. All results are unfolded for 1P2"0.05 GeV OPAL 1Q2 (GeV)
x
FA /#"
p
p
Ref.
21.0
0.00}0.15 0.15}0.30 0.30}0.45 0.45}0.60 0.60}0.75 0.75}0.90 0.90}0.97
0.117 0.302 0.403 0.559 0.782 0.907 0.802
0.028 0.039 0.051 0.058 0.070 0.080 0.103
0.030 0.044 0.059 0.065 0.078 0.087 0.108
[152]
130
0.10}0.40 0.40}0.60 0.60}0.80 0.80}0.90
0.343 0.578 0.936 1.125
0.094 0.079 0.109 0.130
0.100 0.095 0.126 0.142
[152]
3.0
0.00}0.10 0.10}0.20 0.20}0.30 0.30}0.40 0.40}0.50 0.50}0.60 0.60}0.70 0.70}0.80 0.80}0.90 0.90}0.97
0.113 0.230 0.300 0.363 0.364 0.409 0.507 0.516 0.574 0.397
0.006 0.009 0.011 0.013 0.014 0.017 0.021 0.025 0.029 0.029
0.011 0.012 0.015 0.018 0.016 0.020 0.029 0.018 0.045 0.066
[152]
are contained in the "gures. The strategy taken to obtain the results is the following. If possible the values are taken from the published numbers. If no numbers are published, the values are obtained from inspecting the "gures, either based on the RAL database or by the author himself. In some cases it is unclear whether the errors shown in the "gures are only statistical, or whether they include also the systematic error. If only statistical errors are given in the "gures, the total error is obtained using Eq. (C.1), which means by adding in quadrature the statistical errors and the global systematic errors given in the publications. The prescription on how the central values and the errors are evaluated can always be found in the corresponding tables. The numbers listed in Tables 21}36 are the basis of the summary plots shown in Section 7.
296
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Table 14 Results on the photon structure function FA from the PLUTO experiment. The numbers are read o! the published /#" "gure, which probably contains the full error. The results given at 1Q2"5.5 and 40 GeV are unfolded from data in the Q ranges 1}16 and 10}160 GeV. No information is available to which value of 1P2 the result corresponds PLUTO 1Q2 (GeV)
x
FA /#"
5.5
0.00}0.10 0.10}0.20 0.20}0.30 0.30}0.40 0.40}0.50 0.50}0.60 0.60}0.70 0.70}0.80
40
0.00}0.20 0.20}0.40 0.40}0.60 0.60}0.80 0.80}1.00
p
p
Ref.
0.081 0.177 0.532 0.403 0.532 0.597 0.952 0.887
0.040 0.048 0.089 0.105 0.113 0.161 0.322 0.444
[153]
0.177 0.565 0.532 1.532 0.807
0.113 0.210 0.241 0.468 0.581
[153]
Table 15 Results on the average photon structure function 1FA 2 from the TPC/2c experiment. The numbers are read o! the /#" published "gure. The measured FA is averaged over the approximate range in Q of 0.14}1.28 GeV, with an average /#" of about 1Q2"0.45 GeV, which was estimated from the GALUGA Monte Carlo using the experimental requirements of the TPC/2c analysis. No information is available to which value of 1P2 the result corresponds TPC/2c 1Q2 (GeV)
x
1FA 2 /#"
p
p
Ref.
0.14}1.28
0.00}0.05 0.05}0.10 0.10}0.15 0.15}0.20 0.20}0.25 0.25}0.30 0.30}0.35 0.35}0.40 0.40}0.50 0.50}0.60 0.60}0.70 0.70}0.80 0.80}0.90 0.90}1.00
0.038 0.104 0.135 0.172 0.219 0.281 0.320 0.344 0.370 0.373 0.357 0.354 0.291 0.323
0.010 0.010 0.017 0.021 0.031 0.042 0.042 0.037 0.042 0.037 0.037 0.037 0.031 0.068
0.012 0.020 0.027 0.035 0.047 0.061 0.066 0.066 0.072 0.070 0.068 0.068 0.056 0.085
[154]
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297
Table 16 Preliminary results on the photon structure function FA from the DELPHI experiment. The numbers are provided /#" by A. Zintchenko. The results given at 1Q2"12.5 and 120 GeV are unfolded from data in the Q ranges 2.4}51.2 and 45.9}752.8 GeV. The best "t of the QED prediction to the data is obtained for 1P2"0.025 and 0.066 GeV for 1Q2"12.5 and 120 GeV DELPHI preliminary 1Q2 (GeV)
x
FA /#"
p
p
Ref.
12.5
0.00}0.10 0.10}0.20 0.20}0.30 0.30}0.40 0.40}0.50 0.50}0.60 0.60}0.70 0.70}0.80 0.80}1.00
0.106 0.273 0.426 0.515 0.573 0.645 0.743 0.942 1.152
0.008 0.012 0.017 0.021 0.024 0.029 0.038 0.060 0.112
0.024 0.017 0.021 0.024 0.024 0.029 0.043 0.080 0.146
[156]
120
0.00}0.20 0.20}0.40 0.40}0.60 0.60}0.80 0.80}1.00
0.426 0.436 0.678 1.039 1.524
0.291 0.134 0.143 0.170 0.247
0.291 0.144 0.153 0.176 0.257
[156]
Table 17 Results on the photon structure functions FA and FA from the L3 experiment. The numbers are taken from the /#" /#" published table of Ref. [151]. The original numbers for the measurement of FA are multiplied by !1/2 to account /#" for the di!erent de"nitions of FA as detailed in Section 2.1. For the measurements of FA /FA and /#" /#" /#" 1/2FA /FA the "rst error is statistical and the second systematic. The measured structure functions are averaged /#" /#" over the Q range 1.4}7.6 GeV, with an average of 1Q2"3.25 GeV. The values of FA are not corrected for the /#" e!ect of non-zero P in the data 1Q2"3.25 GeV, Ref. [151]
L3 x
FA /FA /#" /#"
1/2FA
0.00}0.25 0.25}0.50 0.50}0.75 0.75}1.00
0.159$0.040$0.034 0.087$0.071$0.056 !0.210$0.102$0.057 !0.236$0.091$0.079
0.046$0.012$0.012 0.111$0.019$0.038 0.141$0.026$0.048 0.061$0.019$0.030
x
FA /#"
FA /#"
FA
0.014$0.024 0.036$0.032 !0.126$0.052 !0.174$0.062
0.008$0.010 0.090$0.021 0.168$0.040 0.089$0.045
0.00}0.25 0.25}0.50 0.50}0.75 0.75}1.00
0.090$0.008 0.404$0.016 0.597$0.020 0.731$0.032
/#"
/FA /#"
/#"
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Table 18 Results on the photon structure functions FA and FA from the OPAL experiment. The numbers are taken from /#" /#" the published table of Ref. [152]. The "rst error is statistical and the second systematic. The results given at 1Q2"5.4 GeV are unfolded from data in the Q range 1.5}30 GeV. The values of FA are corrected for the P e!ect /#" and correspond to FA for P"0 /#" 1Q2"5.4 GeV, Ref. [152]
OPAL x
FA /FA /#" /#"
1/2FA
x(0.25 0.25}0.50 0.50}0.75 x'0.75
0.176$0.031$0.010 0.018$0.028$0.008 !0.171$0.029$0.007 !0.228$0.037$0.014
0.075$0.025$0.008 0.099$0.024$0.010 0.081$0.027$0.011 0.037$0.033$0.011
x
FA /#"
FA /#"
FA
x(0.25 0.25}0.50 0.50}0.75 x'0.75
0.249$0.006$0.008 0.523$0.011$0.014 0.738$0.017$0.019 0.871$0.027$0.021
0.039$0.007$0.003 0.011$0.016$0.004 !0.122$0.021$0.006 !0.201$0.033$0.013
0.029$0.010$0.003 0.101$0.025$0.011 0.121$0.041$0.017 0.063$0.056$0.018
/#"
/FA /#"
/#"
Table 19 Preliminary results on the photon structure function ratios FA /FA and 1/2FA /FA from the DELPHI /#" /#" /#" /#" experiment. The numbers are provided by A. Zintchenko. The original numbers for the measurement of FA /FA /#" /#" are multiplied by !1 to account for the di!erent de"nitions of FA as detailed in Section 2.1. The "rst error is /#" statistical and the second systematic. The results given at 1Q2"12.5 GeV are unfolded from data in the Q range 2.4}51.2 GeV 1Q2"12.5 GeV, Ref. [156]
DELPHI preliminary x
FA /FA /#" /#"
1/2FA
0.0}0.2 0.2}0.4 0.6}0.6 0.6}1.0
0.135$0.037$0.016 0.140$0.034$0.012 0.038$0.034$0.023 !0.263$0.049$0.035
0.004$0.026$0.009 0.077$0.024$0.009 0.099$0.028$0.015 0.182$0.035$0.022
/#"
/FA /#"
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299
Table 20 Di!erential QED cross-section dp/dx for two exchanged virtual photons. The results on the di!erential cross-section dp/dx from the OPAL experiment are taken from the published Table of Ref. [152]. The results given at 1Q2"3.6 GeV and 1P2"2.3 GeV are unfolded from data in the Q and P range 1.5}6 GeV, and the results given at 1Q2"14.0 GeV and 1P2"5.0 GeV are unfolded from data in the Q range 6}30 GeV and the P range 1.5}20 GeV OPAL 1Q2 (GeV)
1P2 (GeV)
x
dp/dx
p
p
Ref.
3.6
2.3
0.00}0.20 0.20}0.40 0.40}0.65
9.77 10.45 4.34
1.62 1.26 1.07
1.80 1.39 1.09
[152]
14.0
5.0
0.00}0.25 0.25}0.50 0.50}0.75
5.26 6.87 2.75
0.82 0.78 0.60
1.29 1.08 0.63
[152]
Table 21 Results on the photon structure function FA from the ALEPH experiment. The results at 1Q2"9.9, 20.7 and 284 GeV are unfolded from data in the Q ranges 6}13, 13}44 and 35}3000 GeV. All numbers are taken from the published tables. In addition, there exist preliminary results, which are listed in Table 31 ALEPH 1Q2 (GeV)
n
x
FA
p
p
Ref.
9.9
4
0.005}0.080 0.080}0.200 0.200}0.400 0.400}0.800
0.30 0.40 0.41 0.27
0.02 0.03 0.05 0.13
0.03 0.07 0.10 0.16
[162]
20.7
4
0.009}0.120 0.120}0.270 0.270}0.500 0.500}0.890
0.36 0.34 0.56 0.45
0.02 0.03 0.05 0.11
0.05 0.12 0.11 0.12
[162]
4
0.03}0.35 0.35}0.65 0.65}0.97
0.65 0.70 1.28
0.10 0.16 0.26
0.14 0.25 0.37
[162]
284
300
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Table 22 Results on the photon structure function FA from the AMY experiment. The result from Ref. [84] at 1Q2"73 GeV is an update of the measurement from Ref. [83] at the same value of 1Q2, where the previous measurement is no more included in this review. The results at 1Q2"6.8, 73 and 390 GeV are unfolded from data in the Q ranges 3.5}12, 25}220 GeV and for Q'110 GeV. The total error has been calculated in this review from the quadratic sum of the statistical error and the systematic error using the errors given in the published tables AMY 1Q2 (GeV)
n
x
FA
p
p
Ref.
4
0.015}0.125 0.125}0.375 0.373}0.620
0.337 0.302 0.322
0.030 0.040 0.049
0.053 0.049 0.097
[85]
73
4
0.125}0.375 0.375}0.625 0.625}0.875
0.65 0.60 0.65
0.08 0.16 0.11
0.10 0.16 0.14
[84]
390
4
0.120}0.500 0.500}0.800
0.94 0.82
0.23 0.16
0.25 0.19
[84]
6.8
Table 23 Results on the photon structure function FA from the DELPHI experiment. The two results are not independent, but use the same data, which is unfolded for four bins on a linear scale in x, and also for three bins on a logarithmic x scale for x(0.35. The results at 1Q2"12 GeV are unfolded from data in the Q range 4}30 GeV. The total error has been calculated in this review from the quadratic sum of the statistical error and the systematic error using the errors given in the published table of Ref. [86]. In addition, there exist preliminary results, which are listed in Table 32 DELPHI 1Q2 (GeV)
n
x
FA
p
p
Ref.
12
4
0.001}0.080 0.080}0.213 0.213}0.428 0.428}0.847
0.21 0.41 0.45 0.45
0.03 0.04 0.05 0.11
0.05 0.06 0.07 0.15
[86]
12
4
0.001}0.046 0.046}0.117 0.117}0.350
0.24 0.41 0.46
0.03 0.05 0.17
0.06 0.09 0.19
[86]
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301
Table 24 Results on the photon structure function FA from the JADE experiment. The results at 1Q2"24 and 100 GeV are unfolded from data in the Q ranges 10}55 and 30}220 GeV. The full errors are obtained by the RAL database from the "gures of Ref. [67] which contain only the full errors, and the statistical errors are not available JADE 1Q2 (GeV)
p
Ref.
0.51 0.29 0.34 0.59 0.23
0.15 0.12 0.10 0.12 0.12
[67]
0.52 0.75 0.90
0.23 0.22 0.27
[67]
n
x
FA
24
4
0.000}0.100 0.100}0.200 0.200}0.400 0.400}0.600 0.600}0.900
100
4
0.100}0.300 0.300}0.600 0.600}0.900
p
Table 25 Results on the photon structure function FA from the L3 experiment. The results given at 1Q2"1.9 and 5.0 GeV are unfolded from data in the Q ranges 1.2}3.0, 3.0}9.0 GeV. For these measurements two results are given in Ref. [89], one is obtained by an unfolding based on the PHOJET Monte Carlo, the other is based on the TWOGAM Monte Carlo. The numbers given here use the results based on PHOJET as the central values with the corresponding statistical error. The systematic error is calculated from the quadratic sum of the systematic error for the result based on PHOJET and the di!erence between the results obtained from PHOJET and TWOGAM. The results given at 1Q2"10.8,15.3 and 23.1 GeV are unfolded from data in the Q ranges 9}13, 13}18 and 13}30 GeV. For these results the central values with the corresponding statistical errors are taken from the published table in Ref. [163]. The systematic error is calculated from the quadratic sum of the systematic error and the additional systematic error due to the dependence on the Monte Carlo model. In addition, there exist preliminary results, which are listed in Table 33 L3 1Q2 (GeV)
n
x
FA
p
p
Ref.
1.9
4
0.002}0.005 0.005}0.010 0.010}0.020 0.020}0.030 0.030}0.050 0.050}0.100
0.184 0.179 0.176 0.191 0.193 0.185
0.009 0.007 0.006 0.008 0.008 0.007
0.050 0.023 0.017 0.009 0.012 0.027
[89]
5.0
4
0.005}0.010 0.010}0.020 0.020}0.040 0.040}0.060 0.060}0.100 0.100}0.200
0.307 0.282 0.263 0.278 0.270 0.252
0.021 0.014 0.011 0.013 0.012 0.011
0.096 0.047 0.023 0.015 0.023 0.047
[89]
10.8
4
0.01}0.10 0.10}0.20 0.20}0.30
0.30 0.35 0.30
0.02 0.03 0.04
0.04 0.04 0.11
[163]
302
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Table 25 Continued L3 1Q2 (GeV)
n
x
FA
p
p
Ref.
15.3
4
0.01}0.10 0.10}0.20 0.20}0.30 0.30}0.50
0.37 0.42 0.42 0.35
0.02 0.04 0.05 0.05
0.04 0.05 0.09 0.16
[163]
23.1
4
0.01}0.10 0.10}0.20 0.20}0.30 0.30}0.50
0.40 0.44 0.47 0.44
0.03 0.04 0.05 0.05
0.05 0.06 0.06 0.13
[163]
Table 26 Results on the photon structure function FA from the OPAL experiment. The results at 1Q2" 7.5, 14.7, 135, 9.0, 14.5, 30.0, 59.0, 1.86 and 3.76 GeV are unfolded from data in the Q ranges 6}8, 8}30, 60}400, 6}11, 11}20, 20}40, 40}100, 1.1}2.5 and 2.5}6.6 GeV. The numbers are taken from the published tables in Refs. [87,90,91]. Only the statistically independent results are listed here, the original publication also contains results on combined Q ranges. The asymmetric errors are listed using the positive/negative values OPAL 1Q2 (GeV)
n
x
FA
p
p
Ref.
7.5
4
0.001}0.091 0.091}0.283 0.283}0.649
0.28 0.32 0.38
0.02 0.02 0.04
0.04/0.10 0.08/0.14 0.07/0.22
[87]
14.7
4
0.006}0.137 0.137}0.324 0.324}0.522 0.522}0.836
0.38 0.41 0.41 0.54
0.01 0.02 0.03 0.05
0.06/0.13 0.06/0.04 0.09/0.12 0.31/0.14
[87]
4
0.100}0.300 0.300}0.600 0.600}0.800
0.65 0.73 0.72
0.09 0.08 0.10
0.35/0.11 0.09/0.11 0.81/0.12
[87]
9.0
4
0.020}0.100 0.100}0.250 0.250}0.600
0.33 0.29 0.39
0.03 0.04 0.08
0.07/0.07 0.06/0.06 0.13/0.31
[90]
14.5
4
0.020}0.100 0.100}0.250 0.250}0.600
0.37 0.42 0.39
0.03 0.05 0.06
0.16/0.03 0.06/0.15 0.12/0.13
[90]
30.0
4
0.050}0.100 0.100}0.230 0.230}0.600 0.600}0.800
0.32 0.52 0.41 0.46
0.04 0.05 0.09 0.15
0.12/0.05 0.08/0.14 0.22/0.10 0.42/0.21
[90]
135
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303
Table 26 Continued OPAL 1Q2 (GeV) 59.0
n 4
p 0.06 0.07 0.09 0.14
p 0.29/0.09 0.11/0.10 0.19/0.14 0.50/0.14
Ref.
0.050}0.100 0.100}0.230 0.230}0.600 0.600}0.800
FA 0.37 0.44 0.48 0.51
x
[90]
1.86
4
0.0025}0.0063 0.0063}0.0200 0.0200}0.0400 0.0400}0.1000
0.27 0.22 0.20 0.23
0.03 0.02 0.02 0.02
0.06/0.08 0.03/0.05 0.09/0.03 0.04/0.05
[91]
3.76
4
0.0063}0.0200 0.0200}0.0400 0.0400}0.1000 0.1000}0.2000
0.35 0.29 0.32 0.32
0.03 0.03 0.02 0.03
0.09/0.09 0.07/0.07 0.07/0.05 0.09/0.05
[91]
Table 27 Results on the photon structure function FA from the PLUTO experiment. The results from Ref. [68] at 1Q2"2.4, 4.3 and 9.2 GeV are unfolded from data in the Q ranges 1.5}3, 3}6 and 6}16 GeV. The measurement at 1Q2"5.3 GeV contains all data and is not an independent measurement. The total error has been calculated in this review from the quadratic sum of the statistical error and the systematic error using the statistical error and the systematic error of 15/25% for data above/below x"0.2 of the measured values for n "4, as given in Ref. [68]. The data from Ref. [69] at 1Q2"45 GeV are unfolded from data in the Q range 18}100 GeV. The statistical errors are obtained by the RAL database from Fig. 5 of Ref. [69] which contains only statistical errors, and the systematic error has been added according to the quoted systematic error of 10%. For both publications, the charm subtraction has been performed by PLUTO PLUTO 1Q2 (GeV)
p 0.014 0.026 0.064
p 0.053 0.049 0.072
Ref.
0.016}0.110 0.110}0.370 0.370}0.700
FA 0.183/0.204 0.263/0.272 0.222/0.222
2.4
n 3/4
4.3
3/4
0.030}0.170 0.170}0.440 0.440}0.800
0.218/0.256 0.273/0.295 0.336/0.336
0.014 0.020 0.044
0.066 0.048 0.067
[68]
9.2
3/4
0.060}0.230 0.230}0.540 0.540}0.900
0.300/0.354 0.340/0.402 0.492/0.492
0.027 0.029 0.069
0.093 0.067 0.101
[68]
3/4
0.100}0.250 0.250}0.500 0.500}0.750 0.750}0.900
0.360/0.480 0.400/0.550 0.770/0.890 0.840/0.870
0.170 0.120 0.160 0.260
0.177 0.132 0.183 0.274
[69]
3/4
0.035}0.072 0.072}0.174 0.174}0.319 0.319}0.490 0.490}0.650 0.650}0.840
0.216/0.245 0.258/0.307 0.222/0.277 0.329/0.329 0.439/0.439 0.361/0.361
0.015 0.010 0.025 0.037 0.052 0.076
0.063 0.078 0.049 0.061 0.084 0.093
[68]
45
5.3
x
[68]
304
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Table 28 Results on the photon structure function FA from the TASSO experiment. The results at 1Q2"23 GeV are unfolded from data in the Q range 7}70 GeV. The statistical errors are obtained by the RAL database from Fig. 7 of Ref. [70], which probably contains only statistical errors, and the systematic error has been added according to the quoted systematic error of 19% on the x distribution, which in several cases is larger than the full error given in Fig. 7 of Ref. [70] TASSO 1Q2 (GeV)
n
x
FA
p
p
Ref.
23
4
0.020}0.200 0.200}0.400 0.400}0.600 0.600}0.800 0.800}0.980
0.366 0.670 0.722 0.693 0.407
0.089 0.086 0.104 0.116 0.222
0.112 0.153 0.172 0.176 0.235
[70]
3
0.020}0.200 0.200}0.400 0.400}0.600 0.600}0.800 0.800}0.980
0.281 0.441 0.469 0.549 0.422
0.087 0.085 0.104 0.115 0.224
0.111 0.153 0.172 0.175 0.237
Table 29 Results on the photon structure function FA from the TPC/2c experiment. The results at 1Q2"0.24, 0.38, 0.71, 1.3, 2.8 and 5.1 GeV are unfolded from data in the Q ranges 0.2}0.3, 0.3}0.5, 0.5}1.0, 1.0}1.6, 1.8}4.0 and 4.0}6.6 GeV. The statistical errors are obtained by the RAL database. The quoted systematic errors from Ref. [72] have been added. They amount to 11% for the regions 0.2(Q(1 GeV with x(0.1, and 1(Q(7 GeV with x(0.2, and to 14% for the regions 0.2(Q(1 GeV with x'0.1, and 1(Q(7 GeV with x'0.2 TPC/2c 1Q2 (GeV)
n
x
FA
p
p
Ref.
0.24
4
0.000}0.020 0.020}0.060 0.060}0.180
0.084 0.074 0.062
0.005 0.008 0.013
0.011 0.012 0.015
[72]
0.38
4
0.000}0.020 0.020}0.055 0.055}0.111 0.111}0.243
0.113 0.118 0.171 0.151
0.007 0.011 0.021 0.028
0.014 0.017 0.023 0.044
[72]
0.71
4
0.000}0.028 0.028}0.065 0.065}0.121 0.121}0.340
0.117 0.130 0.170 0.133
0.006 0.010 0.017 0.013
0.014 0.018 0.025 0.023
[72]
1.3
4
0.000}0.050 0.050}0.126 0.126}0.215 0.215}0.507
0.107 0.184 0.215 0.102
0.013 0.021 0.034 0.031
0.017 0.029 0.041 0.034
[72]
R. Nisius / Physics Reports 332 (2000) 165}317
305
Table 29 Continued TPC/2c 1Q2 (GeV)
n
x
FA
p
p
Ref.
2.8
4
0.000}0.080 0.080}0.156 0.156}0.303 0.303}0.600
0.134 0.234 0.198 0.160
0.018 0.031 0.042 0.033
0.023 0.040 0.050 0.040
[72]
5.1
4
0.021}0.199 0.199}0.359 0.359}0.740
0.224 0.373 0.300
0.034 0.057 0.044
0.042 0.077 0.061
[72]
Table 30 Results on the photon structure function FA from the TOPAZ experiment. The results at 1Q2"5.1, 16, 80 GeV are unfolded from data in the Q ranges 3}10, 10}30 and 45}130 GeV. The total error has been calculated in this review from the quadratic sum of the statistical error and the systematic error using the errors given in the published table TOPAZ 1Q2 (GeV)
n
x
FA
p
p
Ref.
5.1
4
0.010}0.076 0.076}0.200
0.33 0.29
0.02 0.03
0.05 0.04
[71]
16
4
0.020}0.150 0.150}0.330 0.330}0.780
0.60 0.56 0.46
0.08 0.09 0.15
0.10 0.10 0.16
[71]
80
4
0.060}0.320 0.320}0.590 0.590}0.980
0.68 0.83 0.53
0.26 0.22 0.21
0.27 0.23 0.22
[71]
Table 31 Additional preliminary results on the photon structure function FA from the ALEPH experiment. The results at 1Q2"13.7 and 56.6 GeV are unfolded from data in the Q ranges 7}24, and 17}200 GeV. The numbers for 1Q2"13.7 GeV are taken from the published table and the numbers for 1Q2"56.6 GeV are read o! the "gures presented in Ref. [160] ALEPH preliminary 1Q2 (GeV)
n
x
FA
p
p
Ref.
13.7
4
0.002}0.065 0.130}0.343 0.343}0.560 0.560}0.900
0.32 0.41 0.53 0.37
0.02 0.03 0.04 0.07
0.04 0.03 0.06 0.12
[160]
56.5
4
0.003}0.05 0.05}0.25 0.25}0.48 0.48}0.98
0.48 0.41 0.38 0.54
0.04 0.04 0.06 0.07
0.05 0.10 0.09 0.16
[160]
306
R. Nisius / Physics Reports 332 (2000) 165}317
Table 32 Additional preliminary results on the photon structure function FA from the DELPHI experiment. For the results reported in Ref. [165] the numbers have been provided by I. Tiapkin, whereas the results from Ref. [166] have been taken from the published tables. No information is available which ranges of Q have been used for the results DELPHI preliminary 1Q2 (GeV) 6.3
p
Ref.
0.204 0.261 0.303 0.377
0.03 0.03 0.04 0.11
[165]
0.03 0.03 0.04 0.05
n
x
FA
4
0.002}0.020 0.020}0.070 0.070}0.200 0.200}0.700
p
[165]
13
4
0.002}0.020 0.023}0.140 0.140}0.280 0.280}0.750
0.266 0.316 0.366 0.424
21
4
0.01}0.10 0.10}0.30 0.30}0.80
0.33 0.41 0.51
0.01 0.03 0.05
0.03 0.04 0.06
[166]
42
4
0.01}0.10 0.10}0.30 0.30}0.80
0.41 0.48 0.59
0.01 0.02 0.03
0.03 0.03 0.05
[166]
99
4
0.01}0.10 0.10}0.30 0.30}0.80
0.45 0.52 0.73
0.06 0.05 0.05
0.06 0.06 0.06
[166]
400
4
0.01}0.10 0.10}0.30 0.30}0.80
0.5 0.7 1.0
0.3 0.2 0.1
0.3 0.3 0.3
[166]
Table 33 Additional preliminary results on the photon structure function FA from the L3 experiment. The results at 1Q2"120 GeV are unfolded from data in the Q range 40}500 GeV. The numbers have been provided by F.C. ErneH L3 preliminary 1Q2 (GeV)
n
x
FA
p
p
Ref.
120
4
0.05}0.20 0.20}0.40 0.40}0.60 0.60}0.80 0.80}0.98
0.66 0.81 0.76 0.85 0.91
0.06 0.04 0.10 0.12 0.18
0.08 0.08 0.12 0.14 0.19
[164]
R. Nisius / Physics Reports 332 (2000) 165}317
307
Table 34 Results on the Q evolution of FA for three active #avours. The values for the AMY and the TOPAZ experiments are the published numbers from Refs. [84,71]. The values for the PLUTO experiment are obtained by the RAL database from the published "gures of Ref. [69] The TASSO result is the sum of the three middle bins of Table 28 Exp.
1Q2 (GeV)
x
FA $p
Ref.
AMY
73 390
0.3}0.8
0.42$0.08 0.50$0.18
[84]
PLUTO
2.4 4.3 9.2 45
0.3}0.8
0.24$0.08 0.30$0.06 0.36$0.07 0.55$0.12
[69]
TASSO
23
0.2}0.8
0.48$0.10
[70]
TOPAZ
16 80 338
0.3}0.8
0.38$0.08 0.49$0.15 0.72$0.37
[71]
Table 35 Results on the Q evolution of FA for four active #avours. The values for the AMY, ALEPH, DELPHI, OPAL and TOPAZ experiments are the published numbers from Refs. [84,162,86,90,71], respectively. The values for the JADE and PLUTO experiments are obtained by the RAL database from the published "gures of Refs. [67,69]. For the PLUTO result the contribution from charm quarks has been added as explained in Section 7.2. The TASSO result is the sum of the three middle bins of Table 28. For the TPC/2c the statistical errors are obtained by the RAL database. The quoted systematic error from Ref. [72] has been added. In addition, there exist preliminary results from the DELPHI and L3 experiments, which are listed in Table 36 Exp.
1Q2 (GeV)
x
FA $p
Ref.
ALEPH
9.9 20.7 284
0.1}0.6
0.38$0.05 0.50$0.05 0.68$0.12
[162]
AMY
73 390
0.3}0.8
0.63$0.07 0.85$0.18
[84]
12.0
0.3}0.8
0.45$0.08
[86]
JADE
13.4 21.2 28.3 35.8 46.9 100
'0.1
0.28$0.06 0.37$0.04 0.46$0.09 0.67$0.10 0.66$0.16 0.73$0.13
[67]
OPAL
7.5 9 13.5 14.7 30 59 135
0.1}0.6
0.36> \ 0.36> \ 0.41> \ 0.41> \ 0.48> \ 0.46> \ 0.71> \
[90]
DELPHI
308
R. Nisius / Physics Reports 332 (2000) 165}317
Table 35 Continued Exp.
1Q2 (GeV)
x
FA $p
Ref.
PLUTO
2.4 4.3 9.2 45
0.3}0.8
0.24$0.08 0.30$0.06 0.39$0.07 0.73$0.12
[69]
TASSO
23
0.2}0.8
0.69$0.10
[70]
0.3}0.6
0.31$0.07
[72]
0.3}0.8
0.47$0.08 0.70$0.15 1.07$0.37
[71]
TPC/2c TOPAZ
5.1 16 80 338
Table 36 Preliminary results on the Q evolution of FA . For the DELPHI results reported in Ref. [165] the numbers have been provided by I. Tiapkin, whereas the results from Ref. [166] have been taken from the published tables. The numbers from the L3 experiment have been provided by F.C. ErneH Exp.
1Q2 (GeV)
x
FA $p
Ref.
DELPHI prel.
6.6 11.2 17.4 20 35.5 63.0 102.0
0.3}0.8
0.38$0.08 0.43$0.05 0.52$0.07 0.49$0.06 0.64$0.06 0.77$0.08 0.84$0.11
[165]
21 42 99 400
0.3}0.8
0.51$0.06 0.59$0.05 0.73$0.06 1.00$0.32
[166]
50 80 125 225
0.3}0.8
0.62$0.19 0.75$0.10 0.88$0.13 1.18$0.23
[164]
L3 prel.
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309
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CONTENTS VOLUME 332 B. Lampe, E. Reya. Spin physics and polarized structure functions R. Nisius. The photon structure from deep inelastic electron}photon scattering
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