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=v , (20) turns out to be well behaved in the infrared. Note that Eq. (20) is £/(l)invariant and therefore does not depend on the (7(1) gauge-fixing Eq. (7). Let us for the moment assume that a unique non-trivial solution to Eq. (20) exists in some gauge a; we return to this conjecture below. The consequences for the IR-behavior of the model are dramatic. Defining the quantum part a(x) of the auxiliary scalar p by p{x) = v + y/aa(x)
with (a) = 0 ,
(21)
the momentum representation of the Euclidean ghost propagator at tree level becomes
.-*$££•
<*>
Feynman's parameterization of this propagator allows an evaluation of loop integrals using dimensional regularization that is only slightly more complicated than usual. More importantly, the ghost propagator is regular at Euclidean momenta when v ^ 0. Its complex conjugate poles at p2 — ±iv can furthermore not be interpreted as due to asymptotic ghost states 16 . When v / 0 , the W-boson is massless only at tree level and (see Fig. 1) acquires the finite mass m 2 ^ = <72|u|/(167r) at one loop, /
;
;
=ww*-
<23)
s
w£ wwww^wwww Wy Fig. 1. The finite one-loop contribution to the W mass. Technically, the one-loop contribution is finite because the integral in Eq. (23) involves only the <So6-part of the ghost propagator Eq. (22). Since p 2 /(p 4 +u 2 ) = —i; 2 /(p 2 (p 4 + v2)) + 1/p2, the u-dependence of the loop integral is IR- and UV-finite. The quadratic UV-divergence of the 1/p2 subtraction at v = 0 is cancelled by the other, ^-independent, quadratically divergent one-loop contributions - (in dimensional regularization this scale-invariant integral vanishes by itself), m 2 ^ furthermore is positive due to the overall minus sign of the ghost loop. The sign of m ^ is crucial, for it indicates that the model is stable and (as far as the loop expansion is concerned) does not develop tachyonic poles at Euclidean p2 for v ^ 0. Conceptually, the local mass term proportional
265
to 6ltv6ab is finite due to the BRST symmetry Eq. (6), which excludes a mass counter-term. The latter argument implies that contributions to m ^ are finite to all orders of the loop expansion. — -I- ^m
«^
Fig. 2. rCTCT(v,p2) to order g°.
— -< •"
If the model is stable at v ^ 0, the 1PI 2-point function Taa{v,p2) of the scalar must not vanish at Euclidean p2 either. To order g°, Tacr{p2) is given by the 1/g2 term that arises from Eq. (19) upon substitution of Eq. (21) and the one-(ghost)-loop contribution shown in Fig.2. Since a non-trivial solution to the gap equation Eq. (20) relates 1/g2 to a loop integral of zeroth order in the coupling, we may use Eq. (20) to lowest order to obtain a "tree-level" expression for Yaa(v,p2) of order g°. Evaluating the loop integrals, one obtains the real, positive and monotonic function
Ftro-(x
_ »"'•—
„2
_ i
)
—
1
_ 1
+ 2 V 1 ~ 4ircacoth(Vl - 4tar) "1 oo_2„.-l
f
(24)
rVfftp2 > 0) > a/(167r 2 ) to order g° establishes the perturbative stability of a non-trivial solution to Eq. (20) and the fact that this solution is a minimum of the one-loop effective potential. An expansion about a solution v ^ 0 to the gap equation thus has lowest order propagators that are regular at Euclidean momenta for all the elementary fields except the photon A^ (if all the matter fields are massive). The polarization of the photon vanishes at p2 = 0 due to the [/(l)-symmetry regardless of the value of v. Taking into account that the massless Goldstone quartets associated with this symmetry breaking decouple from physical quantities, the situation for v ^ 0 is thus rather similar to QED with an unorthodox massive matter content (extending the notion of "massive matter" to include ghosts and other unphysical fields). 5
The Gap
To complete the argument, we solve Eq. (20) for small coupling. To lowest order in the loop expansion, the relation between the renormalized couplings g, a, the renormalization point \i and an expectation value v ^ 0 implied by
266
Eq. (20) is v2
16TT 2
h = -~T+2
hl
The anomalous dimension j tobed
v
+
°^2)-
(25)
of the expectation value is simultaneously found
^• = - ^ = l ^ - 3 > + 0 < » 4 > -
<*>
2
Using the relation between /z, g and the asymptotic scale parameter A - ^ , we may rewrite Eq. (25) as
where /?o is the lowest order coefficient of the /^-function of this model (fi0 = (22 — 2 n / ) / 3 with n / quark flavors in the fundamental representation as matter). Apart from an anomalous dimension, the non-trivial solution v at sufficiently small coupling is thus proportional to the physical scale A ^ - in the MS
particular gauge a = /3o/2. The anomalous dimension j v in Eq. (26) furthermore is of order gA at a = 3. For rif = 2 quark flavors, the terms of order In g in Eq. (27) thus also vanish in the particular gauge a = /?n/2 = 3 and higher order corrections to the asymptotic value of v at small g2 are analytic in g2. With ri/ = 2 flavors, one can expand the model about i 7 a = e 2 A J L . - ( n / = 2 ) ( l + O(0 2 ))
(28)
in the gauge a = /3o/2, and determine the 0(g2) corrections in Eq. (28) order by order in the loop expansion of the gap equation Eq. (20). Note that this behavior is surprisingly consistent with the previous observation4 for SU(n) in generalized covariant gauges that the lowest order solution to the gap equation remains accurate to order g2 at any finite order of the loop expansion in the gauge a = Po/n when there are n / = n light quark flavors. This does not mean that other gauges are any less physical, but it does single out a = Po/n = 3 as a critical gauge in which the perturbative evaluation of the gap equation Eq. (20) is consistent for sufficiently small values of g2. (In QED the hydrogen spectrum to lowest order is most readily obtained in Coulomb gauge, although it evidently does not depend on the chosen gauge. In the present case asymptotic freedom determines an optimal gauge for solving the gap equation at small coupling.) d
T h i s corrects the error in 17 of ignoring the corrections of order g2 in Z = 1 +
0(g2)
267
At the one-loop level, Eq. (20) has a unique non-trivial solution in any gauge a ^ 0 and Taa of Eq. (24) shows that it corresponds to a minimum of the one-loop action. In the limit a -* 0 at finite coupling, the non-trivial one-loop solution Eq. (25) coincides with the trivial one. On the other hand, some of the couplings in the non-linear gauge-fixing £AG become large in this limit, invalidating any perturbative analysis. The highly singular behavior of the model when a ~ 0 is already apparent in the divergent part of the W self-energy to one loop. The corresponding anomalous dimensions -yw and 7 Q of the vector boson and the gauge parameter are
~~TS£-£(*-!-H+<*^ *=3E?=-£(§"-H+0W,)-
<29»
Gauge dependent interaction terms proportional to g/a at one loop thus lead to a term of order g2/a2 in the longitudinal part of the W self-energy only. The transverse part of the W self-energy is regular in the limit a —• 0. Taking a to vanish thus is rather tricky: Eq. (29) implies that the longitudinal part of the W-propagator at one loop is proportional to 3<72p2 ln(j?) at large momenta and no longer vanishes in this limit. Higher order loop corrections similarly contribute to the longitudinal propagator as a —• 0. j a does not depend on the gauge parameter £ at one loop, due to an Abelian Ward identity that also gives the QED-like relation14 ZA = Zj2 = Z^ between the renormalization constants of the photon, of the coupling g and of the gauge parameter £. The anomalous dimension of the gauge parameter a at sufficiently small g2 is negative for positive values of a when /30 < 6 + 2y/3. With j a < 0, the effective gauge parameter tends to decrease at higher renormalization scales H and direct integration of Eq. (29) gives a vanishing a at a finite value of the coupling g2. As already noted above, the loop expansion, however, is valid only if g2 ^C 1 and g2 ja <£. 1. But Eq. (29) does show that there is no finite UV fixed point for the gauge parameter and that a effectively vanishes at least as fast as g2 as fi -*• oo for any fixed gauge at finite g2. Eq. (28) nevertheless is the asymptotic solution to Eq. (20) in the sense that it is valid at arbitrary small coupling if the gauge at that coupling is chosen to be a(g) = /3o/2. The existence of a (unique) non-trivial solution to the gap equation can be viewed as a consequence of the scale anomaly4. The renormalization point dependence of Eq. (25) and the associated UV-divergence of the loop integral are an indication of this.
268
I would like to thank D. Kabat, D. Zwanziger and R. Alkofer for suggestions, L. Spruch for his continuing support, and L. Baulieu for encouragement and the organizers for this very stimulating conference. 1. J.I. Kapusta, Finite Temperature Field Theory (Cambridge Univ. Press, New York, 1989). 2. R.D. Pisarski, Phys. Rev. Lett. 63,1129; E. Braaten and R.D. Pisarski.i&id. 64, 1338 (1990); Nucl. Phys. B337, 569 (1990);B339,310 (1990);Phys. Rev. D45, R1827 (1992). See I. Zahed and D. Zwanziger, Phys.Rev.D61, 037501 (2000) for a non-perturbative resolution. 3. L. Baulieu, M. Schaden, Int.J.Mod.Phys.A13 985 (1998). 4. M. Schaden, Phys.Rev.D58, 025016 (1998). 5. M. Schaden, Phys.Rev.D59, 014508 (1999). 6. See A. Patrascioiu and E. Seiler, Absence of Asymptotic Freedom in NonAbelian Models, HEP-TH/0002153 and references therein. 7. G. Curci, R. Ferrari, Nuovo Cim. 32, 1 (1976); i6»d.35,151 (1976). 8. A. Blasi, N. Maggiore, Mod. Phys. Lett. A l l , 1665 (1996) includes a lucid analysis of the CF-modef. 9. A. R. Fazio, V. E. Lemes, M. S. Sarandy and S. P. Sorella, The Diagonal Ghost Equation Ward Identity for Yang-Mills Theories in the Maximal Abelian Gauge, HEP-TH/0105060. 10. H. Min, T. Lee and P.Y. Pac, Phys. Rev. D32, 440 (1985). 11. G. 't Hooft, Nucl.Phys. B190[FS3], 455 (1981). 12. K.-I. Kondo, Vacuum Condensate of Mass Dimension 2 as the Origin of Mass Gap and Quark Confinement, Chiba Univ. Preprint CHIBA-EP129, HEP-TH/0105299. 13. C. Becchi, A. Rouet, R. Stora, Ann. Phys. 98, 287 (1976); O. Piguet, A. Rouet, Phys. Rep. 76,1 (1981); L. Baulieu, Phys. Rep. 129,1 (1985); See also F. Delduc, N. Maggiore, 0 . Piguet, S. Wolf, Phys. Lett. B385, 132 (1996). 14. M. Quandt, H. Reinhardt, Phys. Lett. B424, 115 (1998). 15. T. Kugo, I. Ojima, Prog. Theor. Phys. 60, 1869 (1978); ibid.61, 294 (1979). 16. D. Zwanziger, Nucl. Phys. B323, 513 (1989); M. Stingl, Confined Subspaces Univ. of Muenster preprint MS-TPI-99-04 (1999). 17. M. Schaden, Mass Generation in Continuum SU(2) Gauge Theory in Covariant Abelian Gauges, hep-th/9909011; K.-I. Kondo and T. Shinohara, Phys. Lett. B 4 9 1 , 263 (2000).
269
Confinement picture in dual formulation of lattice gauge models O. Borisenko N.N.Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, 03143 Kiev, Ukraine E-mail: [email protected] M. Faber Institute of Nuclear Physics, Vienna University of Technology, Vienna, E-mail: [email protected]
Austria
A dual formulation of three dimensional SU{2) lattice gauge model is constructed. We calculate duals for the partition function and Wilson loop. Using the auxiliary fields we derive a local form of the plaquette representation of the Wilson action and discuss its applicability for the investigation of the confinement properties of the model at weak coupling.
Lattice gauge theory (LGT) provides a non-perturbative approach to quantum field models. The dual transformation (DT) is one of the most powerful method of LGT used to investigate non-perturbative dynamics. Among results obtained using the DT are the Guth theorem on the deconfining transition in 4D U(l) L G T 1 , the Mack-Gopfert result on the confinement in 3D abelian model 2 and many others. The DT is used to study the degrees of freedom responsible for an essential part of the dynamics of the gauge model, such as the confinement mechanism, the phase structure of the theory, etc. 3 . Originally, the DT's have been performed for gauge models with discrete symmetries 4 . DT's for 17(1) LGT's can be found in 3 and 2 . DT's of non-abelian 517(2) gauge model have been carried out in 5 in D •=• 3 and for higher dimensions in 6 . The dual of 3D 5*7(2) LGT can be presented in the form 5
r(p)=0,l/2,l,... \ V
I
Cr(/3) = J dUpXr{UP) exp [f3(TrUp)] = I ( 2 r + l)/ 2r +i(2/3) ,
(2)
where In(x) are the modified Bessel functions and the quantity H(r) is expressed through invariant 6j-symbols S(r) = . I
R,
n Frn n F*orfd. •ven x
odd x
(3)
270
_ f R3 Ri R2 1 J R5 -R3 Ri \ ~ [ ri r2 r3 J \ r 4 r 6 r 5 J R5 Re R 1 f -^6 #2 #4 1 f i?l #2 ^?3 1 ^6 #1 r9 r 8 r 7 / \ n 2 rio r u J \ i? 4 R5 Re J '
/.x W
a similar representation holds for F°dd. The subscripts r, refer to the 12 plaquettes attached to a site x and the subscripts Ri to the 6 links attached to x. To proceed further we divide the lattice into "empty" and "occupied" cubes. One can check that summations over magnetic numbers can be closed only in corners of "occupied" cubes. One then sees that this leads to a decoupling of the summations over rp: all such summations are closed around "occupied" cubes. One can define a dual action as follows. We substitute (3) into (1) and rearrange products over sites to get products over "occupied" cubes. The P F gets the form Z = ^Z?/z(^)exp(5duo') . (5) The "invariant measure" is given by
*WM = IIP* + I> n { £ £ 2 } n { £ £ £ } I
even
x
v
' odd x
v
m
'
and the dual action takes the form -tdual
= 5> £cv1(/3)...£a-6(/3)n&i
(7)
where ^T,c means the sum over all "occupied" cubes. rj,i = 1, ...,6 are representations residing on the 6 plaquettes of a cube. Y[t runs over the 8 corners of the cube c and &j is the 6j-symbol associated with the i-th corner (its exact form can be obtained from (4)). We derive now a dual form for Wilson loop. The Wilson loop in representation J is denned as Wj(C) = (2J+l)~1{Trj Yliec Ui)- The main observation, which we do not prove here and which is similar to before, is that all D-matrices can be combined in such a way to decouple summations over magnetic numbers at each point x. Now, there are two additional external angular momenta on links belonging to the loop C. Summing up over magnetic numbers one sees that the presence of the source J does not destroy the factorization property of the plaquette variables: the sums over rp can again be closed over "occupied" cubes. This results in the following final expression for Wj{C)
Wj(C) = (2j + iy1z-1
J2D^(Ri)^p[sdual(J)] {Ri}
•
(8)
271
The functional form of Sdual(J) coincides with that of Sdual(0) in Eq.(7). Every link I £ C enters in two "occupied" cubes. One such cube includes only i?, ' while the second one - only R\2). D(J,J(RI) is similar to D(iJ(Ri) (6) but includes also 9j-symbols if the site x belongs to the loop C. Formula (8) is valid also for the Polyakov loop if C is closed in the temporal direction. Finally, we want to introduce a plaquette representation for the PF. Substituting the definition (2) of Cr{P) into the expression for the P F (5) one finds
Z = [HdUpexp 0£(TrCy J
v
L
J(UP) ,
(9)
P
where the /3-independent Jacobian is given by J{UP) = J2D^(Rl)l[J2xr1(U1)...J2xr6(U6)l[bi {Ri}
c
n
r6
.
(10)
t=l
The plaquette variables Uv play the role of dynamical degrees of freedom in this representation. Due to the auxiliary link representations Ri, the integrals over plaquette variables are also decoupled over "occupied" cubes. For j3 sufficiently large (the region where one expects the continuum limit of the model), all plaquette matrices fluctuate around the unit matrix. This gives a chance for the construction of a low-temperature expansion which could be applied also for studies of the long-range properties of the model unlike the conventional perturbation theory. The price for this is a rather complicated form of the Jacobian J(U). There seems to be no way to present it in a local form: the summation over Ri makes the Jacobian to a highly non-local quantity. Thus, the task of the construction of the effective model at large f3 reduces to finding the asymptotic expansion for the one-cube expression denned in Eq.(7). This probably can be done making use the uniform asymptotic expansions for the 6j-symbols. Such a possibility is currently under investigation. References 1. 2. 3. 4. 5. 6.
A. Guth, Phys. Rev. D 21, 2291 (1980). M. C'opfert, G. Mack, Com.Math.Phys. 8 1 , 97 (1981). T. Banks, J. Kogut, R. Myerson, Nucl. Phys. B 121, 493 (1977). A. Ukawa, P. Windey, A. Guth, Phys. Rev. D 21, 1013 (1980). R. Anishetty, et.al., Phys. Lett. B 314, 387 (1993). I. Halliday, P. Suranyi, Phys. Lett. B 350, 189 (1995).
272
R E N O R M A L O N S O N T H E LATTICE A N D T H E OPE FOR T H E P L A Q U E T T E : A STATUS R E P O R T
Universitd
F. DI RENZO degli Studi di Parma and I.N.F.N., Gruppo collegato di viale delle Scienze, 43100 Parma, Italy E-mail: [email protected]
Parma,
The first ten coefficients in the perturbative expansion of the plaquette in Lattice SU(3) are computed both on a 8 4 and on a 24 4 lattice. They are shown to be fully consistent with the growth dictated by the first IR Renormalon and with the expected finite size effects on top of that. As already pointed out a few years ago, this leads to a puzzling result on the smaller lattice: when the contribution associated with the Renormalon is subtracted from Monte Carlo measurements of the plaquette, what is left over does not scale (as expected) as a 4 , but as a 2 . While the analysis is not yet complete on the larger lattice, the implications of such a finding is discussed.
1
Introduction: the Gluon Condensate in Lattice Gauge Theory
A longstanding problem in Lattice Gauge Theory (LGT) is that of a first principles determination of the Gluon Condensate (GC). In LGT the GC is given in terms of Wilson Loops, for instance the basic plaquette W, for which an Operator Product Expansion (OPE) can be written in terms of WA
A4
W = W0 + -±r-
+ ...
(1)
Here Q = 1/a is the inverse lattice spacing, acting as the scale needed in every definition of the GC, Wo is the contribution associated with the Identity operator, while W4 is associated with the "genuine" (i.e. dim = 4) condensate. In view of Eq. (1), a standard approach 1 was to exploit the following formula WMC 03)
- J2 c^"n = cZ^)
* (aA)4 + • • •
G
(2)
n
which is to be understood as follows: Wo is computed in Perturbation Theory and is subtracted from the Monte Carlo measurements at various values of /?; the leading contribution that is left on the right is the second term in the OPE, which is the relevant one, whose signature is dictated by Asymptotic Scaling, that is (aA) 4 ~ exp(-/3/(26 0 )). The former procedure is actually ill defined, since the definition of the perturbative contribution to be subtracted is plagued by Renormalons. In Ref.2
273
the first eight coefficients of the perturbative expansion of the plaquette were computed via Numerical Stochastic Perturbation Theory, the Renormalon factorial growth was singled out and shown to introduce an indetermination of order (A/Q) 4 , which is just the order of the term one would be interested in. The situation is in a sense even worse. In Ref.3 the Renormalon contribution was resummed and subtracted: to our astonishment what was left over did not scale as a 4 , but as a 2 , at least on the 8 4 lattice on which the computations were first performed on. Such a result is a challenge to our understanding of power effects: for a lucid discussion see Ref. 4 . 2
Renormalons on the lattice: new results and confirmation
Ne now address the following questions: Was the leading behaviour correctly singled out? Are finite size effects under control (remember that the Renormalon growth impinges more and more on the IR region as higher order orders are computed)? Answers can be got (for details see Ref. s ) by considering the following formula for the Renormalon contribution rQ2
Wr(s,N)
=C /
1.2
JI.2
-J^as{sk2)
= Y,CTnen(C,s,N)p-n
(3)
The integral is simply the expected form (obtained from dimensional and Ren.G. considerations) for the condensate on a finite lattice: the lower limit of integration is the IR cutoff, function of the lattice size N, while the scale s is in charge of matching from a continuum to a lattice scheme. The coefficients in the expansion (computable in terms of Incomplete Gamma functions) are functions of an overall constant, the scale s and the lattice size N. In Ref. 2 the values for s and C were obtained by fitting C™n(C, s,N) to the first eight coefficients on the N = 8 lattice. From C™n(C, s, N) one can now infer higher orders on different lattice sizes (simply changing the parameter N). By doing this and actually computing the first ten coefficients of the expansion both on a 8 4 and on a 244 lattice, one gets for instance Fig (1). While the actual numbers will be published soon elsewhere, an impressive agreement is manifest between the first ten coefficients as inferred as above and as actually computed on the 244 lattice. Finite size effects are of order 3 — 4% at tenth order. 3
Conclusions and perspectives
What about the subtraction? By repeating the procedure the bizarre result is actually stable on the 8 4 lattice, which leaves the puzzle still there. On the larger lattice work is still in progress: first indications are that there could be
274
Figure 1: The first ten coefficients in the expansion of the plaquette on a 24 4 lattice (circles) versus the expected asymptotic behaviour (crosses) as explained in the text.
space for recovering the standard (a 4 ) result, due to a tiny interplay between perturbative and non-perturbative finite size effects. This would of course still pose the question of what to blame for the dependence on finite size. Further study on the sensitivity to boundary conditions is also in progress. Acknowledgments The author is indebted to the other people involved in the NSPT project: G. Burgio, G. Marchesini, E. Onofri, M. Pepe, L. Scorzato. References 1. See for example B. Alles, M. Campostrini, A. Feo and H. Panagopoulos, Phys. Lett. B 324, 443 (1994) and references therein. 2. F. Di Renzo, G. Marchesini and E. Onofri, Nucl. Phys. B 457, 202 (1995). 3. G. Burgio, F. Di Renzo, G. Marchesini and E. Onofri, Phys. Lett. B 422, 98 (1998). 4. M. Beneke, Physics Reports 317, 1 (1999). 5. F. Di Renzo and L. Scorzato, to be issued soon.
275
COLOUR C O N F I N E M E N T IN T H E LATTICE L A N D A U G A U G E QCD SIMULATION Sadataka FURUI School of Science and Engineering, Teikyo University, E-mail: [email protected] Hideo NAKAJIMA Department of Information Science, Utsunomiya University, E-mail: [email protected] The colour confinement criterion proposed by Kugo and Ojima is tested in the lattice Landau gauge QCD simulation. The renormalization effects are studied by measuring the gluon propagator, ghost propagator, three gluon vertex and the ghost-antighost-gluon vertex. The running coupling a, from ghost-antighost-gluon vertex in the infrared region yields A —~—- ~ lGeV, consistent to that from three MOM
gluon vertex in high momentum region.
1
The confinement signal and the definition of the gauge field
Two decades ago, Gribov pointed out a possible mechanism of colour confinement in Coulomb gauge or Landau gauge QCD via infrared divergence of the Faddeev-Popov ghost propagator 3 . At nearly the same time, Kugo and Ojima proposed a criterion for the absence of coloured massless asymptoptic states in Landau gauge QCD using the BRST symmetry1. We study the confinement signal in the gluon propagator, the ghost propagator and the Kugo-Ojima parameter, and their dependence on the gauge field. Usually, the gauge field on lattice A^x) is defined from the link variable Un(x) as Ax>lt = \{Ux,ii - Ultli)\traceUss part, which we call [/-linear version. A more natural definition2 is Ux^ — expAx^, A\ = — AXifl, which we call log U version. We observe that the Kugo-Ojima parameter uab(0) at /? = 5.5 is about -0.7 in the log U and about -0.6 in the [/-linear version. If the configuration is in the core region4, the tensor
divided by N2 — 1, where N is the number of colours, is expected to approach a function E(U) defined by the optimizing function4. They are E(U) = ]T), ^ R e tr[/ ( in [/-linear version, and j ^ j ]£, ) 0 tr (AatS(^4;)A°J, adjAt and S{x) =
t h ffi 2 ),
in log[/ version.
where Ai =
276
0.2
0.2
0.4
0.4
1 li
|i
0.6
1<
0.6
0.8
Figure 1: The dependence of the KugoOjima parameter measured along x,y,z and t axis resp. on the definition of the gauge field. /3 = 6.0,16 4 . The triangles are log U, and diamonds are U— linear version.
Figure 2: T h e ghost propagator as a function of the lattice momentum. j3 = 6.0, 16 4 . Triangles are log U definition and diamonds are {/-linear definition. The dashed curve is 1.162/p 2 5 4 5
In the table below e\ and e-i stand for e = ( \'), in our 164 lattice simulation of the [/-linear and the log U version of the gauge fields, respectively. We define the horizon function h — ( ^ r ^ ) / ( 3 ( i V 2 - 1)) = c - e/d. In general we expect h < 0 and in the continuum limit, we expect h = 0 when the configuration is in the core region. Table 1: /3 dependence of the Kugo-Ojima parameter c, trace e divided by the dimension d, and h = c — e/d. The suffix 1 corresponds to the u-linear and 2 corresponds to the log U version. Data are those of 16 4 , except /3 = 5.5 {/-linear data, which are those of 8 4 .
p
C\
5.5 6.0
0.570(58) 0.576(79)
ei/d 0.780(3) 0.860(1)
h -0.21 -0.28
C2
0.712(18) 0.628(94)
e2/d 0.908(1) 0.943(1)
h -0.20 -0.32
The gluon propagator is infrared finite and the absolute value in log [7 version is about 20% larger than that in [/-linear version The corresponding difference in the ghost propagator is about 10%. 2
T h e Q C D r u n n i n g coupling
The QCD running coupling a(/z) = g2/47r can be measured from the three gluon vertices9 as, g(n2) - G A W ( p .a/c^w'[p f ^)G A (')' (pe=») • S i n c e t h e l a t t i c e d a t a of GA^ is infrared finite, in contrast to the conjecture that the gluon propagator is infrared vanishing4, p(/x2) decreases as /x decreases as 1.5^. This be-
277
f
7.5
15 2.5
\
10 7.5 5
1
^
2.5
0
0.5
1
1.5
2
I
i
2.5
Figure 3: The gluon propagator as a function of the lattice momentum. /? = 6.0, 16 4 . Triangles are logiJ version and diamonds are [/-linear version. 50 samples.
Figure 4: T h e QCD running coupling constant as a function of momentum /i. Triangles and diamonds are g 2 /47r, the same as Fig3, stars and boxes are g2/4n and g2/4n, respectively of log U smeared version.
haviour does not agree with the results of Dyson-Schwinger approach5, which suggest that a s (/x) monotonically increases to a finite constant as /x goes to 0. The running coupling can be obtained also from the ghost-antighost-gluon The coupling g{n2) = %^f)ta"i)^t%)<^Lif!^%)• preliminary results of 2 2 g /4n and g /4:W at symmetric momentum points
2 Ov) = - sin(n„7T/L),
^ nl = 1 0 /*
have a peak which depend on definition of the A^ but at other momentum points g2/4ir and g2/4n are not so different. When the physical scale is fixed and by a-1 = 1.91 ±O.lGeV, h^foM c a l c u l a t e d from t h e 92/^ at J2^nl~8 16 are about lGeV, which are consistent to that obtained from the three gluon vertex in high momentum region6. S.F. thanks Prof. Reinhardt and Prof. Alkofer for helpful discussion and hospitality in Tuebingen in August 2000. This work is supported by JSPS Grant-in-aid No.11640251 and the KEK supercomputer project No.00-57. References 1. T. Kugo and I. Ojima, Prog. Theor. Phys. Supp. 66, 1 (1979). 2. H.Nakajima and S. Furui, Nucl. Phys.B (Proc Suppl.)63A-C,635, 865(1999), idem, B83-84,521 (2000); idem, B94,558 (2001); idem, Nucl. Phys.A680,151c (2000); hep-lat/0006002, 0007001. 3. V.N. Gribov, Nucl. Phys. B 139, 1 (1978).
278
4. D. Zwanziger, Nucl. Phys. B 364, 127 (1991), idem B 412, 657 (1994). 5. L. von Smekal, A. Hauck, R. Alkofer, Ann. Phys.267,1 (1998), hepph/9707327; R. Alkofer and L. von Smekal, hep-ph/0007355. 6. B. Alles et al., Nucl. Phys. B 502, 325 (1997); D. Henty et al.,EPS HEP 1995 239(1996);Ph. Boucaud et al., hep-ph/9810322,9810437
279
H A M I L T O N I A N LATTICE G A U G E THEORY N. E. LIGTERINK ECT*, Villa Tambosi, Strada delle Tabarelle 286, Villazzano (Trento), Italy E-mail: [email protected] I discuss the different aspects of the quark-anti-quark wave functional in Hamiltonian lattice QCD. In this short paper I like to give a physical motivation for the study of Hamiltonian lattice gauge theory, as most of our more technical results on the Hamiltonian lattice gauge theory are recently published.1 In terms of numerical accuracy for comparable calculations, the Hamiltonian approach cannot compete with the Euclidean one. However, different insights could be gained from an approach that requires a wave functional. In this paper I will focus on the mechanism of confinement. The problem of confinement starts with the long-range forces of gauge theories. We have come to terms with the long-range interaction between charges in electrodynamics, since we created a, maybe slighty artificial, handle on the infrared problems of electrodynamics. The strange relation between gauge invariance and covariance already indicates this delicate balance. If one would use a noncovariant gauge, it would seem that the gauge field could change faster than light, however, considered carefully, the gauge field cannot be viewed independently from the charges that lie at its source. Therefore, charges and their surrounding gauge fields, should, in some sense, be viewed as extended objects, similar to the QCD string.2
0
Figure 1. The typical o image of QCD string confinement; the dipolefieldis deformed o into a string-like object which yields a linear potential.
E
—2 A
2N
Figure 2. The permutation of flux-lines due to the action of the electric operator.
Generally, confinement is introduced as a classical concept where the orginal dipole field of two opposite charges get squeezed in the middle such that a flux tube is formed.3 See Fig. 1. The cause of this squeezing is related to the background field, or more appropriately, the vacuum. In a Euclidean approach, or a non-zero temperature approach, the "vacuum is not stable," if we can talk about a vacuum at all. Therefore, non-trivial topology fields, or non-zero
280
background fields, are introduced to explain, physically, the dynamical deformation of the dipole field. In the Hamiltonian approach, on the contrary, we find that the vacuum is always filled with closed flux-lines, or vacuum bubbles. Although it is difficult to determine the vacuum wave functional accurately. Central to a Hamiltonian approach is balance between opposite effects. Due to the quantisation the kinetic and the potential energy are approximately reciprocially related in the wave functional. The potential energy wants to focus the wave function at the bottom of the potential, while the kinetic energy wants to spread the same wave function. Somewhere in the middle a stable balance is found. This principle carries over from quantum mechanics to quantum field theory, to the balancing forces of a dynamical flux tube. In order to analyse the QCD dipole field in a similar manner, we adopted the language of flux-lines. In lattice QCD the only gauge degrees of freedom are the link operators between lattice sites. These, at the classical level, represent both the electric and the magnetic degrees of freedom: a closed loop of links around an elementary space-like plaquette" is the discretised version of the magnetic energy B2, while the two colour charges should be connected via a link operator ng,, = P e x p ( ^ P ^ A - A ^ f r ) } [ * ./x„=x(0)
(1) J
to create a gauge-invariant, global colour singlet, which can be associated with the electric flux from one charge to the other. Prom here on I will refer to this generic path-ordered exponential as the flux-line. Furthermore, since electric and magnetic operators are canonically related, and, their effects cannot be separated from the context, which depends on the residual gauge fixing as well. The proper electric operator follows from quantisation of the gauge field, and is the generalized angular momentum L2 on the gauge-group manifold of SU(3). The action of this operator should be calculated in terms of the degrees of freedom of the wave functional. The wave functional consists of a sum the flux-lines with different paths x(7) between the two charges at x 0 and x&, augmented with a infinite set of closed loops, from the vacuum, which are possibly dependent on the particular path x(7). The effect of the electric operator in this framework follows from the SU(iV) identity for the group generators A": S Kj^ti = ^SuSjk — jfSijSki- Calculating the effect of the electric operator on a single, non-overlapping flux-line, we find it is an eigenstate with an eigenvalue proportional to the length of the flux-line. However, if two flux-lines, U and V, would overlap the electric operator gives an extra contribution in the °In a Hamiltonian approach there are only space-like plaquettes.
281
form of the permutation of the flux-lines: JV2-I
„
5 3 (*aUh(*"V)k, = 2UuVjk - jjUiiVu ,
(2)
a=l
proportional to the length of the overlap (see Fig. 2). If we now would look at the action of the Hamiltonian on a typical dipole wave-functional configuration we find three contributions (see Fig. 3).
o
H^
o
-* < S . o °
o
0
+
,
^
+ o °
O
Q
O
° ^ | °
oo °
Fig. 3. Schematically: diagonal + off-diagonal + magnetic The diagonal part of the electric operator assigns an eigenvalue to each linesegment proportional to its length. The off-diagonal part permutes the overlapping flux-lines of the vacuum and the flux tube, and, in effect, diffuses the flux-lines. The magnetic part adds a fundamental, plaquette-size loops to the "background field." In a sense, the off-diagonal part and the magnetic part conspire to fill the vacuum, and the diagonal and the off-diagonal parts are the typical opposite forces of a quantum theory, which determine the shape of the dipole field. The domination of the diagonal part will lead to a flux-tube, while the domination of the off-diagonal part gives rise to a typical QED dipole. The stationary eigenstate is found when the effects balance each other. The coupling-constant dependence appears in the dipole field only via the vacuum, since the opposite terms of squeezing and percolation are both the result of the action of the electric operator, however, the latter depends on the distribution of vacuum bubbles, and therefore, depends, indirectly, on the coupling constant. For the quantitative analysis of the scenario above a precise knowledge of the vacuum state is necessary. We expect the flux tube, a set of paths x(7), to affect the vacuum only nominally. This work is in progress. References 1. N.E. Ligterink, N.R. Walet, and R.F. Bishop, Ann. Phys. (N.Y.) 284, Aug. (2000)(in print); Nucl. Phys. A 663&664, 983c (2000); Nucl. Phys. (Proc. Suppl.) B 83-84, 956 (2000). 2. N.K. Nielsen and P. Olesen, Nucl. Phys. B 6 1 , 45 (1973); M. Baker, these proceedings; F.V.Gubarev et al, Phys. Lett. B 468, 1999 (134). 3. S. Mandelstam, Phys. Lett. B 53, 476 (1975); Yu.A. Simonov, Phys. Usp. 39, 1996 (313).
282
LATTICE QCD W I T H T H E OVERLAP-DIRAC OPERATOR: ITS A P A R A M E T E R , A N D ONE-LOOP RENORMALIZATION OF F E R M I O N I C C U R R E N T S C. ALEXANDROU, E. FOLLANA, H. PANAG0P0UL0S 0 Department of Physics, University of Cyprus, P.O.Box 20537, Nicosia CY-1678, Cyprus E. VICARI Dipartimento di Fisica dell'Universita and I.N.F.N., Via Buonarroti 2, 1-56127 Pisa, Italy We compute the ratio between the scale A L associated with a lattice formulation of QCD using the overiap-Dirac operator, and AJ^J- To this end, the one-loop relation between the lattice coupling go and the coupling renormalized in the MS scheme is calculated, using the lattice background field technique. We also compute the one-loop renormalization Z^ of the two-quark operators ^Tip, where V denotes a generic Dirac matrix. Furthermore, we study the renormalization of quark bilinears which are more extended and have better chiral properties. Finally, we present improved estimates of Zp, coming from cactus resummation and from mean field perturbation theory.
It has recently been shown that chiral symmetry can be realized in lattice QCD without fermion doubling, circumventing the Nielsen-Ninomiya theorem (for a list of references, see our publications [1,2]; for reviews see, e.g., Refs. [3,4]). This has been achieved by introducing an overlap-Dirac operator derived from the overlap formulation of chiral fermions 5 . The simplest such example, for a massless fermion, is given by the Neuberger-Dirac operator 6 £»N = i p [ l + X ( X t X ) - 1 / 2 ] ,
X = DVf-±p,
(1)
a is the lattice spacing, pG(0,2) a parameter, £>w the Wilson-Dirac operator ^ w - 2 l > ( V M + V P ) - f l V ^ V ^ ] • «V„^(aO = U(x,nW(x
+ aii)-il)(z).
(2)
D N has a number of desirable features: The Ginsparg-Wilson relation: 75 .DN + -ON 75 = a £>N 75 -ON, protects the quark masses from additive renormalization, and implies renormalizability to all orders of perturbation theory 7 . This relation also leads to the existence of an exact chiral symmetry of the lattice action 8 , with chiral Ward identities which ensure the non-renormalization "Presented the talk
283
of vector and flavor non-singlet axial vector currents and the absence of mixing among operators in different chiral representations. Chiral symmetry results in leading scaling corrections to hadron masses which are 0(a2), rather than 0(a). The axial anomaly is correctly reproduced by the fermion integral measure, which is non-invariant under flavour-singlet chiral transformations. £>N avoids fermion doubling at the expense of not being strictly local: Locality is recovered in a more general sense, i.e. allowing an exponential decay of the kernel of £>N at a rate which scales with the lattice spacing and not with the physical quantities 9 . In what follows, we present perturbative calculations, in lattice QCD with the operator £>N> of several quantities which are needed to relate Monte Carlo data to physical observables. Lack of strict locality greatly complicates these calculations, as compared to the Wilson case. Technical details may be found in our publications [1,2]. To evaluate A L / A J ^ - we need to calculate 1 the one-loop relation between go and the renormalized MS coupling g at scale [i: go = Zg(go,a/j,)g. Writing: Zg{go,x)2=l+gl (2&0lnx-Mo) +0(gfc), one has: l0=2b0ln ( A ^ / A ^ g ) . The algebra was performed using a symbolic manipulation package which we have developed in Mathematica. For the present purposes, this package was augmented to include the propagator and vertices of the overlap action. Our results are shown in Figure 1 for different numbers Nf of fermion flavours. Some particular cases of interest are (SU(3), p=l): A A i / M s = 0.034711 (JV)=0), 0.025042 (JV>=1), 0.011273 (JV)=3), (cf. Wilson fermions: AL/Am = 0.029412 (JV>=1), 0.019618 (iV)=3)). 0.10
.04
N f =0
0.06
.03
,.02
'" N =1 L
0.02
f
-0.02
-
-0.06 .00
-0.10 0.0
2.0 1.0 1.5 P Figure 1: Ai/A^^-- in SU(3), as a function of P• 0.5
0.0 0.0
1.0 2.0 P Figure 2: The coefficients 6 L (p), 6Q. (p), as a function of p. 0.5
284
We have furthermore computed 2 , to one loop, the renormalization constants Zo of the local fermionic currents: Oi = rP(x)T^(x),
Tt = 1 (S), 7 5 (P), 7„ (V), j ^
5
(A),
and their extended (non-ultralocal), improved counterparts: O'i = rPTi (1 - a£» N /2) V, O" = $ (1 - a£> N /2) r« (1 - aDN/2) $. (O'i obey Ward identities leading to: Zs< = Zp>, Zy> = Z#. O" are free of 0(a) errors, not only in the spectrum, but also in generic matrix elements.) We have proved that: Zo: = Zoi, and also: Zo:> — Zoi • We calculated Z0i = 1 + g2cF [(c0i - c) In aV + b%f - bm - b0. + bL] (see Ref. [2] for notation). Zoi are independent of the gauge parameter and of the fermion mass. The results for bh and b0. are shown in Figure 2; they do not depend on N or Nf. As an example, Zot at p = 1 is: Zs,p = 1 + g2cF[3(Ina2H2)/16TT2 +0.204977], ZA,v = 1 + 92cF[0.198206], 2 2 ZT = 1 + p C i r [ - (In a V ) / 1 6 T T + 0.204392]. Finally, we have obtained improved estimates for Zo{, coming from a resummation to all orders of "cactus" diagrams 10 . These diagrams are often largely responsible for lattice artifacts. Our method is gauge invariant, and systematic in dressing higher loop contributions; applied to a number of cases of interest, it has yielded results remarkably close to nonperturbative estimates. In particular, for Zy,A we find (at g0 = 1, p = 1): Zy,A — 1-35, as compared to our undressed result: Zy,A = 1.26427. To conclude, some feasible future tasks, alongside with numerical simulation, are: Calculation of the /^-function for the overlap-Dirac operator, running fermion masses, renormalization of 4-fermion operators (ZAS=2, etc.). References 1. C. Alexandrou, H. Panagopoulos, E. Vicari, Nucl. Phys. B 571, 257 (2000). 2. C. Alexandrou, E. Follana, H. Panagopoulos, E. Vicari, Nucl. Phys. B 580, 394 (2000). 3. H. Neuberger, Nucl. Phys. B(PS) 83, 67 (2000). 4. M. Liischer, Nucl. Phys. B(PS) 83, 34 (2000). 5. R. Narayanan and H. Neuberger, Nucl. Phys. B 443, 305 (1995). 6. H. Neuberger, Phys. Lett. B 417, 141 (1998); B 427, 353 (1998). 7. T. Reisz and H. J. Rothe, Nucl. Phys. B 575, 255 (2000). 8. M. Liischer, Phys. Lett. B 428, 342 (1998). 9. P. Hernandez, K. Jansen, M. Liischer, Nucl. Phys. B 552, 363 (1999). 10. H. Panagopoulos and E. Vicari, Phys. Rev. D 58, 114501 (1998); D 59, 057503 (1999).
285
Comments On the Confinement from Dilaton-Gluon Coupling in QCD M. Chabab LPHEA, Physics Department, Faculty of Science, Cadi Ayyad University, P.O. Box 2390, Marrakesh 40001, Morocco E-mail:mchabab@ucam. ac. ma In this talk, I report on a work done in collaboration with R. Markazi and E.H. Saidi [1]. 1
Introduction
Confinement in gauge theories provides one of the most challenging problems in theoretical physics. Various quark confinement models rely on flux tube picture. The latter emerges through the condensation of magnetic monopoles and explain the linear rising potential between color sources. However, a deep understanding of confinement mechanism in still lacking. Recently it has been shown in [2] that a string inspired coupling of a dilaton <j> to the 4d SU(NC) gauge fields yields a phenomenologically interesting interquark potential V(r) with a confining term. Extension of gauge field theories by inclusion of dilatonic degrees of freedom has gained considerable interest. Particularly, Dilatonic Maxwell and Yang Mills theories which, under some assumptions, possess stable, finite energy solutions. 2
Description of Dick Model
The Dick interquark potential [2] was obtained as follow: First start from the following model for the dilaton gluon coupling G((j>): Lit, A) = -^l^F^Fir
~ \{d,f + W(
(1)
Then construct G{(j>) under the requirement that the Coulomb problem still possesses analytical solutions.The coupling G(0) and the potential W{(j>) that emerged are: f2 1 G(4>) = const. + J— , W(<j>) =-m2<j>2 (2) where / is a scale parameter characterizing the strength of the scalar-gluon coupling and m represents the dilaton mass.
286
Next, consider the equations of motion of the fields A^ and (j) and solve them for static point like color source described by the current density Jg = Pcjf®. After some straightforward algebra, Dick shows that the interquark potential Vo(r) is given by (up to a color factor), VD(r) = ^
- gfJ^L—
ln[exp(2mr) - 1]
(3)
Eq.(3) is remarkable since for large values of r it leads to a linear confining potential VD(r) ~ 2gfm^^^r. This derivation provides a challenge to monopole condensations as a new quark confinement scenario. Therefore, it is justified to dedicate more efforts to the investigation of a more general effective coupling function G(
/*2Iffl
(4)
Such form of the potential is very attractive. On the one hand, it extends the usual Coulomb formula Vc ~ 1/r which is recovered from (4) by taking G = 1. Moreover for G ~ r 2 , which by the way corresponds to a coupling G{
n
r
n
(5)
n>0
where Cn are related to the quark and gluon vacuum condensates. Although, the derivation of the formula (4) for the interquark potential from Eq.(l) is by itself an important result, there remain however other steps to overcome before one can exploit (4). A crucial technical step is to determine for what type of couplings G(<j>), one can solve the equation of motion of the scalar field
= J'/
(a)
,
d
^ = ^ - \
F
Z - >
F
r ^ ~
(<>) (6)
287
In trying to explore (4), we have observed that the functional G[
LD = \{v'f
+ ^G{y/r)
(7)
where y — rep, y' — ( ^ ) a n d a = J^N2CJ^1 and where g is the gluon coupling constant. Indeed, one can show that Eq.(6.b) may be interpreted as corresponding to a mechanical system with the action: S = jdrr2[4>2 + W(<j>) + £ < ? ( * ) ] = Jdr[\{y')2
+ r2W(y/r)
+
^G(y/r)]
(8) Consequently the coupling G(<j>) of Eq.(l) appears as a part of interacting potential of Id quantum field theory. 3
Genaralized Dick model
First of all observe that the lagrangian (7) including the Dick model (1) is a particular one dimensional field theory of lagrangian L^\{y')2-U{y,r)
(9)
where U(y, r) is a priori an arbitrary potential. Though simple, this theory is not easy to solve except in some special cases. A class of solvable models is given by potentials of the form : U(y) = AV(" + P > + 7 V ( « - n ) + Syk 2
(10)
2
where n,p,q and k are numbers and A , j and 6 are coupling constants. The next thing to note is that Eq.(10) has no explicit dependence in r and consequently the following identity usually holds : y'2 = U + const.
(11)
Actually Eq.(ll) is just an integral of motion which may be solved under some assumptions. Indeed by making appropriate choices of the coupling A as well as the integral constant, one may linearise y' in Eq.(ll) as follows : y'
= Ui + U2
(12)
and shows that the solutions are classified by the product U1U2 and the ratio UxjV-i.. In [1] we have discussed some interesting examples which are special cases of general models involving interactions classified according to the
288
following constraint: Ui.U2~yk
(13)
This constraint cannot however determine Ui and TJi independently as in general the following realizations are all of them candidates, U1 = \yn+P
,
U2 = 7y«-n
(14)
with k = p + q integer. We have shown that: i) Dick solution is recovered for k — 0. ii) For dilatonic Models with fc = 2 and p = q = l , we get the following solutions: , . .1 ,nmr , ..1 y[r) = [-tan[—^=- + const.)]" A y2
,
, . . 1, , .nmr ..i y(r) = [——tanh{—•=- + const.)]" A v2
.__. (15)
corresponding respectively to A7 = ± m 2 . iii) For Models with general values of fc, one has to know moreover the ratio U1/U2 in order to work out solutions. A simple, but important generalization of Dick Model corresponds to take U\ — Xy and Ui = 72/* _1 . One can check, after some straightforward algebra, that the solution is given by W(r) = [r0c]*Ar
(16)
where
289 ON CONFINEMENT PROVIDED B Y THE SPONTANEOUS SYMMETRY BREAKING
V. E. K U Z M I C H E V , V. V. K U Z M I C H E V Bogolyubov
Institute
for Theoretical Physics, H-B Metrolohichna 03143, Ukraine E-mail: [email protected]
Street,
Kiev
We discuss the model of confinement, in which the decay of hadron into the free quarks and gluons is not strictly forbidden, but the lifetime of hadron with respect to quark-gluon channel of decay is greater or at least of the order of age of the Universe. Our model provides the confinement of a massive quark within the limited region of space by means of constant component of the effective potential which arises as a result of reorganization of vacuum of the scalar field, which effectively describes gluon degrees of freedom, with condensation of corresponding Higgs field. On a concrete example it is shown that the lifetime of hadron equal to age of the Universe leads to the Higgs boson with the mass mn > 40 GeV for realistic coupling and astrophysical constants.
1
Introduction
The experimental data and the predictions of gauge theory of the strong interactions allow to conclude that the effective coupling constant becomes very strong at distances large in comparison with the hadron size and results in quark and gluon confinement. In other words the channels of decay of hadron via quarks and gluons are strictly forbidden. The nature of this prohibition is still unknown and it is introduced into the theory phenomenologically. Since hadron is a local formation in spacetime an cut-off of the confining long-range part of effective potential must be realized at the distances ~ 1 fm. In the present contribution we show that above mentioned cut-off can be provided naturally by the constant component of the effective potential which arises from the reorganization of the vacuum state with condensation of the Higgs field in the system of nonlinearly interacting fields. In such approach from the idea of absolute (strict) confinement (the decay of hadron into the free quarks and gluons is strictly forbidden) we come to the assumption of soft (nonstrict) confinement when the lifetime of hadron with respect to quarkgluon channel of decay is greater or at least of the order of some characteristic time (e.g. age) for our Universe.
290 2
Specification of Model
As it is well known in QCD, if one takes into account spin, colour and flavour degrees of freedom in quark-gluon system, the requirement of gauge invariance of the theory leads to clear structure of the Lagrangian. However, the equations of motion which follow from this Lagrangian prove to be too complicated for analysis and solution. Therefore at study of confinement it is expedient to proceed to some effective action functional, which leads to the equations of motion, which describe this phenomenon, but disregard others insignificant in this case properties of a system. With this purpose we shall consider the action functional for a system of N + 1 fields in the following Lorentz-invariant form
=
d4x I i q^d^q
- rnqq + - d^a
d^ifia - qj^Vrf
\,
(1)
where m is the mass of the fermion (quark) field q and V^ = V^ {ip\,
UM) = {-!)<
2
V, -r
4
f,
(2)
with constants /z? > 0 and A, > 0. In a considered model the free scalar fields ipi are massless. They describe effectively the gluon fields without the account of self-interaction and spin. As a result of interaction between the fields q and
291 3
Effective Potential
Considering the excitation spectrum of massive field q less than the threshold of birth process of particles we proceed to the problem of a massive quark moving in an effective potential V = Vo(
-4
V = ^-sech 2 (£iM8i)tanh 2 0ii|V|*i) + £- s e c h 4 ( ^ | ^ | s 2 ) - 1 Xi 4A2
(3)
where %l> is large component of bispinor q, /2? = ffj/xf, Xi = giXi, Sj = (z — Uit)/yjl — uj, and ut are the velocities of the solitary waves. The part of the potential V in the form of the barrier of the finite magnitude (first term in Eq. (3)) is organized by the solitary wave of sech-type 1 . The attraction (second term) is formed by the kink/antikink wave with the energy density localized near the origin. At large distances the potential V tends to the constant negative value Vc = —(5/I6A) (mjf/m3), where m # = v ^ ^ w and X = m2A2 are Higgs mass and self-constant, g = mg2 is dimensionless coupling constant. The appearance of the constant component in V can be interpreted as a consequence of spontaneous symmetry breaking when the field y>2 comes to the state with stable vacuum. In order to illustrate the role of Vc at first we put Vc = 0 and fix the height of a barrier according to the width T of the ground 15 - state of quark, assuming that V = Ta, where Ta is the decay constant of the nucleus 92sU into a - particle and |o 4 Th 1. Since free quarks are not observed then the lifetime of hadron with respect to quark decay mode is not less then the age of the Universe t0 — 10 - 15 Gyr. The required width T = l/t0 is reached at Vc - - 4 x 106 - 107 GeV for m = 350 MeV, range of barrier 0.4 fm and the height of the barrier fixed above. It leads to the Higgs mass ran = 4 0 - 5 0 GeV x (A/p) 1 / 4 . For (X/g) > 1 this values coincide with current theoretical and experimental estimations. If we fixed the height of the barrier according to decay constant typical for weak interactions, the confinement would be reached at Vc « — 1055 GeV. These estimations show that there exist the values of Vc which provide the quark confinement while remaining finite within the energy scale of high energy physics. References 1. V.E. Kuzmichev and V.V. Kuzmichev, hep-th/0004033; hep-th/0004034.
292 CHIRAL P H Y S I C S I N T H E Q U A R K COMPOSITES A P P R O A C H TO QCD SERGIO CARACCIOLO Scuola Normale Superiore and INFN, Sezione di Pisa, 1-56100 Pisa, FABRIZIO PALUMBO INFN - Laboratori Nazionali di Frascati, P. O. Box 13, 1-00044 Frascati,
ITALIA
ITALIA
ROBERTO SCIMIA Dip. di Fisica dell'Universita and INFN, Sezione di Perugia, Via A. Pascoli, 1-06100 Perugia, ITALIA The standard QCD action is improved by the addition of irrelevant operators built with chiral composites. An effective Lagrangian is derived in terms of auxiliary fields, which has the form of the phenomenological chiral Lagrangians. Our improved QCD action appears promising for numerical simulations as the pion physics is explicitely accounted for by the auxiliary fields.
1
Introduction
We have developed an approach to QCD based on the use of quark composites as fundamental variables, following the idea that a significant part of the binding of the hadrons can be accounted for in this way, so that the "residual interaction" is sufficiently weak for a perturbative treatment. For lack of space we confine ourselves in this contribution to the case of quark composites with the quantum numbers of the chiral mesons, referring t o x for the details. We add to the standard lattice QCD action with Wilson fermions an irrelevant 4-fermions operator Sc, therefore we assume as our starting point the modified partition function Z = j[dV][dXd\]
exp[-SYM
-Sq-
Sc],
(1)
where SYM is the Yang-Mills action, Sq is the action of the quark field, A is the quark field while the gluon field is associated to the link variables V. The chiral composites are the pions and the sigma •ft — i a2fc,rA75TA, CT = a 2 kn AA,
(2)
where a is the lattice spacing and a factor of dimension (length) 2 , necessary to give the composites the dimension of a scalar, has been written in the form a2kv for convenience (see below).
293 The operator Sc has the form
Sc = \(&,C±))-\({±1
+ ±,mq))
(3)
(A) = t r i s o s p i M ,
(4)
1 -a+p?/a?
(5)
where S = a + i T • 7r75, mq is the quark mass and
2
C =
a4
/> is a dimensionless parameter and • the Laplacian on the lattice. The irrelevance by power counting of Sc requires that in the continuum limit p do not to vanish and k„, as well as the product knp, do not diverge. We replace the chiral composites by the auxiliary fields £ = So — 175 T • E by means of the Stratonovich-Hubbard transformation 3 , then we perform a change of variables so as to introduce the pion field by the transformation T, = R U = exp
•J*
u+!+!>&
R2 = Y,l + S 2
GH e s t , < 2 ) -
(6)
The effective action takes the form §
= E
7 < W f l t / t ) > V+(iJC0)> + £{R,R) 2a 2 '
- Tr In (D - Q).
(7)
If the radial field .R acquires a non vanishing expectation value R we set R = fv so that the first term of the effective action 5 is the kinetic part of the chiral action while the radial field should not be dynamical because of its large mass (of order I/a). It can then be shown that the fermionic determinant can be expanded in powers of the derivatives of U, as appropriate to a Goldstone field, and of the explicit symmetry breaking terms. For a first investigation of S we naively forget about the Wilson term in the lattice action of the fermions. In this case we find
294 where H = 24 is the number of quark components. If we neglect the fluctuations of R and put B = l/(2a 2 /„), the first term of S can be identified with the leading term of the chiral models C2 = \il
[
(9)
with the standard relations ml = 2mgB,
^(OlAAlO) = -2f$B
(10)
while the fermionic determinant has an expansion in inverse powers of fn and kn (hopping expansion) which provides corrections to £2- Furthermore the quarks have an effective mass Mq = mg - p2KR
= mq - p2fc7r/jr
(11)
proportional to the inverse of the expansion parametrs and are therefore perturbatively confined. The first to C% is
-^ij^iC
+ v*) [(vJt/'MvJtf) + (v-t/')(v-i0]> (12)
To second order, besides the usual terms of the chiral models, we get a coupling between the pions and the glueballs. But, as it is well known, the Wilson term is necessary to reproduce the abelian anomaly, and this is true also with our effective action 4 . The analysis 1 of the model in presence of the term Wilson term Qw is in progress. From a numerical point of view, the evaluation of the fermionic determinant should be faster in presence of the auxiliary fields, as they already provide the propagation of the pions. Some support to the above can be found in 5 . References 1. S. Caracciolo, F. Palumbo and R. Scimia, hep-lat/9910039 2. S. Caracciolo and F. Palumbo, Nucl. Phys. B512 (1998) 505, hepph/9801389; Erratum, B555 (1999) 656 3. R. L. Stratonovitch, Dokl. Acad. Nauk. S.S.S.R. 115 (1957) 1097 [Sov. Phys. Dokl. 2 (1958) 416 ]; J. Hubbard, Phys. Rev. Lett. 3 (1959) 77 4. S. Caracciolo and F. Palumbo, hep-lat/0004019 5. R. C. Brower, K. Orginos and C. I. Tan, Nucl. Phys. Proc. Suppl. 42 (1995) 42, hep-lat/9501026
PARALLEL SESSION B Light Quarks (and Gluons)
Conveners:
Martin G. OLSSON (Madison) Massimo TESTA (Roma) Nils A. TORNQVIST (Helsinki)
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297 Spectroscopy "windows" of quark-antiquark mesons and glueballs with effective Regge trajectories
Physics Dept,
Maxtina Brisudova University of Florida, FL 32611, USA
Gainesville,
Leonid Burakovsky, Terrance Goldman Theoretical Division, T-16, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Regge trajectories of quark-antiquark mesons can be well approximated for phenomenology purposes by a specific nonlinear form, reflecting that the flux tubes cannot be arbitrarily large, but break due to the effect of pair-production. If confirmed, this would imply that there is only a finite number of bound states on each trajectory, and consequently, an existence of "spectroscopy windows" for each flavor. Here we present our results for these windows.
1 It is a paradigm in QCD that Regge trajectories (roughly speaking, functions that relate angular momentum of a state to its mass squared) of mesons are linear. This picture arises in the Veneziano model for the amplitudes, string theory of hadrons and also, at least asymptotically, from potential models with a linear confining potential. On the other hand, Regge trajectories extracted from data are nonlinear. In addition to being disfavored by experiments, linear trajectories also lead to problems in theory. (For details on both experimental and theoretical aspects, and references, see 1 .) All the theoretical concepts that lead to linear Regge trajectories in QCD overlook one feature of QCD - the production of color singlet pairs. Pair production is only virtual at short distances, but as the energy of the flux tube increases, it is possible (and likely) to create a real pair and the breaking of the flux tube is energetically favorable. One can argue that hadronic Regge trajectories cannot rise indefinitely, and they acquire a curvature due to pair production which screens the confining QCD potential at large distances 2 . Once the nonlinearity of Regge trajectories is an accepted fact, the question of what specific form should be used for phenomenology arises. We have considered a whole class of nonlinear trajectories allowed by dual amplitudes with Mandelstam analyticity (DAMA) f and references therein).
298 DAMA allow for Regge trajectories of a form afi(t)
= a j? (0) + 7 [TX - (Tfi - t)»]
(1)
where v is a constant restricted t o 0 < f < | ; 7 i s a universal constant; T-j is a trajectory threshold, 0^(0) is its intercept, and i,j refer to flavor. For any v ^ 0 in this interval, the trajectory becomes purely imaginary at the trajectory threshold (in other words, its real part terminates); for v — 0 it develops a constant imaginary part. This means that any v ^ 0 trajectory supports only a finite number of bound states, with their value of angular momentum limited by a(T). Beyond the threshold, there are continuum states. This picture seems to have captured the essence of the effect of pair production. The largest deviations of the true trajectory from any of these forms can be expected for the states near the threshold. The sensitivity of the parameters on the specific form (i.e. the value of v) is also interesting. We found that the least sensitive is the threshold (up to few percent), and the most sensitive is the resultant maximum angular momentum (i.e. a(T) for any v ^ 0; for v — 0 it is infinite) 1 . This means, in our opinion, that while the maximum angular momentum for a given flavor cannot be predicted, the values of thresholds extracted from data can be taken seriously, regardless of what DAMA form is assumed. We have argued that both limiting cases [y = 1/2 and v = 0) can be expected to work comparably well for lowest lying states (and thus, any v in between), but the v = 1/2, so-called square-root, form is likely to be more realistic. Therefore, we use the square-root form for phenomenological purposes. Assuming that Regge trajectories are of the form (1) with v = 1/2, we determine thresholds and intercepts of trajectories by using various experimental information. Typically, we use masses of a few known lowest lying states, and in the case of the p trajectory we also use the value of the intercept (which is known and well-established) found from exchange processes. The value of 7 (the universal asymptotic slope) is fit to the p trajectory, and then taken as universal for all other trajectories. The approach has more predictive power than one would naively expect. The number of parameters for any two parity related 5-flavor multiplets" (i.e. 20 trajectories in all) is only 12, in contrast to 40. I would like to concentrate here on just one of the numerous consequences and implications. Specifically, if we are right, there exist "spectroscopy wina An example of a pair of parity partners trajectories is the p and 02 trajectories; the 5-fiavor multiplet containing the p trajectory includes K*, , D*, £>*, J/ip, B*, B*, B* and T trajectories.
299
> M m
a 2
nn
sn
ss
en
cs
cc
bn
bs
be
bb
Figure 1: Mass windows for quark-antiquark states of a listed flavor, together with known masses of states belonging to the vector-, tensor-, pseudoscalar- and axial-vector- trajectories, n refers to light quarks. For example, the nn tower contains 1=1 mesons ir, p, a i , a% etc.
dows" as shown in Figure 1. This may simplify identification of states observed in experiments, and might be beneficial in searches for exotics. Acknowledgments This research is supported in part by the U.S.D.O.E. under contract W-7405ENG-36, and grant DE-FG02-97ER-41029. References 1. M.M. Brisudova, L. Burakovsky, T. Goldman, Phys. Rev. D 6 1 , 054013 (2000). 2. M.M. Brisudova, L. Burakovsky, T. Goldman, Phys. Lett. B 460, 1 (1999). 3. L.L. Jenkovszky, Riv. Nuovo Cim. 10, 1 (1987).
300
T H E TOPOLOGICAL SUSCEPTIBILITY A N D P I O N D E C A Y C O N S T A N T F R O M LATTICE QCD UKQCD Collaboration: A. HART Dept. of Physics and Astronomy, Univ. of Edinburgh, Edinburgh, Scotland and M. T E P E R Theoretical Physics, Univ. of Oxford, 1 Keble Road, Oxford, England We study the topological susceptibility, Xi m t w o flavour lattice QCD. 1 We find clear evidence for the expected suppression of x at small quark mass. The estimate of the pion decay constant, fn = 105 ± 5 j^Jg MeV, is consistent with the experimental value of approximately 93 MeV. We compare \ to the large-Nc prediction and find consistency over a large range of quark masses.
The ability to access the non-perturbative sectors, and to vary parameters fixed in Nature has made lattice Monte Carlo simulation a valuable tool for investigating the role of topological excitations in QCD and related theories. 2 The topological susceptibility is the squared expectation value of the topological charge, normalised by the volume X = ^ ,
Q= ^
j
(1)
Sea quarks induce an instanton-anti-instanton attraction which in the chiral limit becomes stronger, suppressing Q and x3 X = S(m„_1 +md"1)~1,
where
£ = - lim
lim (0|V^|0)
(2)
m,->0 V—>-oo
is the chiral condensate. 4 We assume (0\ipip\0) = (0|mt|0) = (0|dd|0) and neglect contributions of heavier quarks. The Gell-Mann-Oakes-Renner relation, flml
= {mu + md)H + 0{ml)
=>
x=££L+0{mi)
(3)
for Nf light flavours, in a convention where the experimental value of the pion decay constant /„. ~ 93 MeV. Eq. 3 holds in the limit f%m%V > 1, which is satisfied by all our lattices. The higher order terms ensure that x ~* XqUj the quenched value, as mq,m1K —> oo. We find, however, that our measured values are not very much smaller than x q u , so we must consider two possibilities. Firstly, there are phenomenological reasons 5 ' 6 for believing that QCD is 'close' to iVc = oo, and in the case of gluodynamics even SU(2) is demonstrably
301 Figure 1: The measured topological susceptibility, with interpolated quenched points at the same f o and fits independent of the quenched points.
-- *
V
fit to Eqn. 6 fit to Eqn. 7 fit to Eqn. 8 fit to Eqn. 9 physical value
*I------T*
^
_
!
_
Cokf close to SU(oo). 7*2 Fermion effects are non-leading in Nc, so we expect \ ~* Xqu for any fixed value of mq as the number of colours Nc —> oo. For small m , we expect Eq. 4 to hold, with x°°, /oo the quantities at leading order in Nc. 4 Alternatively, our mq ~ J7istrange and perhaps higher order terms are important. In the absence of a QCD prediction, Eq. 5 interpolates between Eq. 3 and the quenched limit. X°°ml X=
+ m*
(4),
X=
J; m > C t ^ ( l ^ml)
(5).
Measurements of x were made on a number of ensembles of Nf = 2 lattice field configurations produced by the UKQCD Collaboration. An SU(3) Wilson gauge action is coupled to clover improved Wilson fermions. 8 The UKQCD ensembles have two notable features. The improvement is fully non-perturbative, with discretisation errors being quadratic rather than linear in the lattice spacing. Second, the couplings are chosen to maintain an approximately constant lattice spacing (as defined by the Sommer scale, r 0 = 0.49 fm 9 ) as the quark mass is varied. This is important, as the susceptibility in gluodynamics varies with the lattice spacing as 2 f^x = 0.072 - 0.208/fo in competition with the variation with mq.l The topological susceptibility is measured from the gauge fields after cooling to remove the UV noise. We plot these data in Fig. 1 along with the interpolated x q u at an equivalent lattice spacing from the above formula for comparison, which vary little owing to the the UKQCD matching. The behaviour with M = {foih„)2 is qualitatively as expected and, more quantita-
302 Table 1: Fits to the Nat most chiral points of x-
Fit Eq. Eq. Eq. Eq. Eq. Eq.
6 6 6 7 7 7
Fit Eq. Eq. Eq. Eq. Eq. Eq.
8 8 8 9 9 9
iVfit 2 3 4 3 4 5
0.0140 0.0112 0.0091 0.0176 0.0170 0.0147
JVfit 3 4 5 3 4 5
0.0208 0.0272 0.0233 0.0186 0.0209 0.0189
Cl
CO
(16) (6) (4) (35) (4) (16) (1) (14) (1)
— -0.0018 (10) (1) -0.0016 (4) (0) -0.0011 (3) (0)
c0 (87) (12) (85) (18) (66) (10) (53) (7) (42) (7) (36) (5)
C3
0.0844 0.0632 0.0717 0.0576 0.0506 0.0550
(427) (35) (114) (6) (147) (3) (175) (6) (55) (5) (69) (6)
X2/d-o.f. 0.805 2.202 9.008 0.964 0.502 2.965
0.265 (27) 0.261 (13) 0.242 (12)
X2/d-o.f. 1.013 0.895 1.847 0.990 0.682 1.929
0.288 0.329 0.305 0.273 0.289 0.275
foU 0.237 (14) 0.212 (6)
TO/TT
(61) (53) (44) (40) (30) (27)
tively, we attempt fits motivated by Eqs. 3, 4, 5: ~ 2»
rp X M
Co
^+«(«'i"^»vta"(5<)
(9)
We include progressively less chiral points until the fit becomes unacceptably bad in Table 1. We note the wide range fitted simply by including an m\ term, and the consistency of our data with large-iVc predictions. The stability and similarity of the fits motivates us to use en from Eq. 7 to estimate fn = 105 ± 5 t\o MeV, with variation between other fits providing the second, systematic error, and in good agreement with the experimental value ~ 93 MeV. References 1. See also A. Hart, M. Teper, Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 476 [hep-lat/9909072]; hep-ph/0004180; UKQCD Collab., in prep. 2. For a recent review, see M. Teper, Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 146 [hep-lat/9909124]. 3. P. Di Vecchia, G. Veneziano, Nucl. Phys. B 171 (1980) 253. 4. H. Leutwyler, A. Smilga, Phys. Rev. D 46 (1992) 5607. 5. G. 't Hooft, Nucl. Phys. B72 (1974) 461. 6. E. Witten, Nucl. Phys. B160 (1979) 57. 7. M. Teper, Phys. Rev. D 59 (1999) 014512 [hep-lat/9804008]; hepth/9812187. 8. J. Garden (UKQCD), Nucl. Phys. B (Proc. Suppl.) 83 (2000) 165 [hep-lat/9909066]. 9. R. Sommer, Nucl. Phys. B411 (1994) 839 [hep-lat/9310022].
303
LIGHT-LIGHT A N D HEAVY-LIGHT M E S O N S S P E C T R A F R O M N O N P E R T U R B A T I V E QCD Alexei Nefediev Institute of Theoretical and Experimental Physics, Moscow 117218, Russia Abstract Properties of light-light mesons axe described by the effective Hamiltonian with spinless quarks derived from QCD. The spectrum is computed by the WKB method and shown to reproduce the celebrated linear Regge trajectories even for the lowest levels. The correct string slope of the trajectories naturally appears in the present approach as the string dynamics is taken into account properly. Similar method is applied to heavy-light mesons and a set of corrections to the Hamiltonian is taken into account including spin-spin and Tomas spin-orbit interactions. The numerical results for the spectrum are compared with the experimental data and with the results of recent lattice calculations. 1
Introduction
One of the most successful models of confinement in QCD is the string picture which exploits the idea of the flux tube formation between the colour constituents in hadrons. The small radius of the string compared to the hadronic size makes it possible to construct quantum mechanical quark models with the interquark interaction described by either non-relativistic 1 or relativistic string (see e.g. 2 ) . The role of the string becomes especially important if light quarks are involved, so that the proper string dynamics should be taken into account together with the quarks one when studying the properties of hadrons. 2
Light-light m e s o n s
Starting from the gauge invariant Green's function of the qq system, neglecting spins and using the Feynman-Schwinger representation for the one-particle propagators as well as the area law asymptotic for the Wilson loop one arrives at the following Lagrangian of the system L(T) = —m\Jx\ — m.i\lx\ ~ a \ d/3^(ww')2
- w2w'2,
(1)
where the interaction is described by the Numbu-Goto term for the minimal string bounded by the quarks trajectories 2 . We use the straight-line anzatz for the minimal string iy^(r,/3) = /3xlM + (1 — 0)x2f, 2 . Introducing einbein
304
fields (i (dynamical mass of the quark) and v (density of the string energy) to get rid of the square roots in (1) and using the standard techniques, one finds the Hamiltonian of the system in the form (we consider equal quark masses) 2 „
p2r + m2
L2/r2
'
^d^(<j2v2
, A
,„ x
Getting rid of the einbein v by taking extremum in Hamiltonian (2) and keeping the other einbein /x as a variational parameter /xo one finds H =f ^ +/x 0 + C/( Mo ,r), (3) Mo where the effective potential U has a rather complicated form; its dependence on fio reflects the nonlocal string-type character of the interaction introduced in (1). Nonrelativistic expansion of (3) gives for the interquark potential V(r) = U(m,r)=ar--^:
+ ...,
(4)
where the correction to the confining potential is known as the string one 3 ' 2 . The spectrum of the Hamiltonian (3) is found by the quasiclassical method and each eigenenergy is minimized tuning /xo. Numerical results 4 reproduce straight-line Regge trajectories in the angular momentum I with the inverse slope 27r<74. The difference between 2ixa and the overestimated value 8c found for the Bethe-Salpeter equation with linear confinement is entirely due to the proper string dynamics missing in the latter case. 3
Heavy-light mesons
Now we apply similar approach to the heavy-light mesons spectrum. The zeroth approximation for the Hamiltonian with the Coulombic potential and the constant term added and the string correction to it read (re = \OLS\. ^+m?
—r
i=l
m\
<J(/X? + /i§ - /ix/xz) L2
__ K
+ Tj+„---Co,V^-
^
- , (5)
v
whereas other corrections are spin-dependent and coincide in form with the Eichten-Feinberg-Gromes results 5 up to the change m» —• /x; in the denominators 6 (note that for a light quark /Xj ~ 500 -=- 800MeV 3> rrij): 8TTK
- -
2
V8d = -z (S1S2) ^(0) ~ 7T ~T 3/xiM2 2 r \ /xf
,S2L\
H
K (S-i.L
S2L\
T + —3 ! /x2, I fir3 \ /xx /x2 /
W
305 2 +
~ 7 1 ^ ( 3 ^ " ^ 2 " ) ~ ( £ & ) ) + 2 ^ 3 (SL) (2-ln(nr)-lE),lE
« 0.57.
Numerical results for the D, Ds, B and Bs mesons spectra calculated in the the given technique with standard parameters 7 ' 8 are in a good agreement with the experimental and lattice data. In conclusion let us briefly discuss the situation with the D*' resonance recently claimed by DELPHI Collaboration 9 . It was reported to have the mass 2637 ± 6MeV that agrees with the predictions of the quark models for the first radial excitation 2 3 S 1 (0~) of the qq pair (our prediction for this state is 2664MeV) but its surprisingly small width of about !5MeV is in a strong contradiction with the theoretical estimates 10 . Meanwhile it was observed 10 that orbitally excited states 1" and 3 _ could have such a small width. Our model predictions for these states are 2663MeV and 2654MeV correspondingly, i.e. they lie even lower than the radially excited one. This resolves the main objection to the identification of the D*' with orbital excitations. Indeed, in quark models 2" and 3~ states lie at least 50—60MeV higher that the experimentally observed value. In our approach extra negative contribution to the masses of orbitally excited states is readily delivered by the string correction discussed above, which comes from the proper string dynamics inside meson. Financial support of INTAS-RFFI grant IR-97-232 and RFFI grants 0002-17836 and 00-15-96786 is gratefully acknowledged. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
N.Isgur and S.Godfrey, Phys.Rev. D32, 189 (1985); A.Yu.Dubin, A.B.Kaidalov, Yu.A.Simonov, Phys.Lett. B323, 41 (1994); J.Merlin and J.Paton, J.Phys. G i l , 439 (1985); V.L.Morgunov,A.V.Nefediev,Yu.A.Simonov,Phys.Lett.B459,653 (1999); E.Eichten and F.L.Feinberg, Phys.Rev. D 2 3 , 2724 (1981); D.Gromes, Z.Phys. C26, 401 (1984); Yu.A.Simonov, preprint ITEP 97-89; J.Pantaleone, S.-H.Henry Tye, Y.J.Ng, Phys.Rev. D 3 3 , 777 (1986); Yu.A.Simonov, Z.Phys. C 5 3 , 419 (1992); Yu.S.Kalashnikova and A.V.Nefediev, in preparation; DELPHI Collaboration, P.Abreu et.al, Phys.Lett. B426, 231 (1998); D.Melikhov and O.Pene, Phys.Lett. B446, 336 (1999); P.R.Page, hep-ph/9809575
306
E L E C T R O M A G N E T I C PROPERTIES OF LIGHT N U C L E O N R E S O N A N C E S IN A CHIRAL C O N S T I T U E N T Q U A R K MODEL * R.F. WAGENBRUNN 0 , M. RADICP' 6 and S. BOFFI"' 6 °Dipartimento di Fisica Nucleare e Teorica, Universita di Pavia, 1-27100, Italy b Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, 1-27100 Pavia, Italy We present results on electromagnetic excitations of nucleon resonances within an extended version of the Goldstone-boson-exchange constituent quark model. The agreement with the data turns out to be not very satisfactory. Relativistic boost effects may change the results drastically, however.
The Goldstone-boson-exchange constituent quark model has been rather successful in reproducing the gross features of the phenomenological spectra of all light and strange baryons 1 , z . An important further test of the model is its performance in predicting electromagnetic properties of baryons. Following the procedure of Ref. 3 a charge-current density operator can be obtained which is consistent with the model Hamiltonian giving a gauge invariant operator without introducing any further parameters. It consists of one- and two-body contributions. When using the extended version of the model of Ref. 2 the experimental magnetic moments of proton and neutron can be reproduced quite well 4 . However, a reasonable description of the dependence of the nucleon form factors on the square of the momentum transfer Q2 is only possible by assuming a quite large size of the constituent quarks themselves, i.e. by introducing constituent quark form factors 5 . After the constituent quark form factors have been fixed by demanding reasonable nucleon form factors one can use them in the calculation for transition form factors to nucleon resonances. For the N -> A transition form factor the result fails to reproduce the data at low Q 2 with helicity amplitudes of about 2/3 of the experimental values at Q2 = 0 (see Ref. 5 ) . On the other hand the overall Q2 dependence is reasonable. As for the transition to the negative-parity states iV(1535) and iV(1520), in Fig. 1 the helicity amplitudes for a proton target are shown. For each resonance, the full result with oneand two-body currents and with just the one-body current only are shown. For each case, constituent quark form factors as in ref. 5 are used as well as the corresponding results for point-like constituent quarks are shown. The upper •This work was partly performed under the contract ERB FMRX-CT-96-0008 within the frame of the Training and Mobility of Researchers Programme of the European Union.
307
Q' [(GeV/cfl
O* [(GeV/cfl
Figure 1: Helicity amplitudes for the electroproduction of JV(1535) and iV(1520). For iV(1520) the helicity asymmetry is also shown. The results obtained with one- and twobody currents with constituent quark form factors are drawn in solid lines, the long-dashed lines show the contribution from one-body current. The corresponding results for point-like constituent quarks are shown by the dashed-dotted and short-dashed lines, respectively.
right plot shows the amplitude for iV"(1535). While at Q2 = 0 the full result is in reasonable agreement with the data it falls off too fast for larger Q 2 . In this case the one-body current alone for point-like constituent quarks yields an almost perfect agreement with the data completely opposite to the situation in case of the elastic form factors. Also for AT(1520) no reasonable description of the data for the helicity amplitudes Ai/2 and A3/2 (upper right plot and lower left plot, respectively) is achieved. However, the helicity asymmetry (-^1/2 _ ^3/2)/(^i/2 + ^3/2) a g r e e s with the data quite well (lower right plot). In this observable deviations which occur in the amplitudes themselves cancel out to a large extent. Results for higher resonances have also been obtained
308
but are not shown because of the inaccurate data available. If the constituent quark model is suitable for calculating the transition form factors, what is responsible for the poor description of the data? In the case of transition form factors the coupling of the resonances to the decay channels (Nn, Nr],...) should be taken into account. A recent study on pion electroproduction within a dynamical model 6 indicates a sizable pion cloud contribution to the N —> A form factor. Moreover, an adequate treatment of relativity should be in order. In the model Hamiltonian the relativistic expression for the kinetic energy of the constituent quarks has been applied in order to take into account that the constituent quarks have quite large individual momenta. This can easily be deduced from the expectation value of their kinetic energy. Even at low Q2 relativistic boosts effects may thus be quite sizable. In the results presented here the boosts have been treated nonrelativistically. In order to take into account the relativistic boosts one can apply the so called point-form formulation of relativistic quantum mechanics 7 . A recent calculation of nucleon form factors within this framework yields indeed tremendously different results as compared to a nonrelativistic approach 8 . In particular it turns out that the nucleon charge radii, magnetic moments, and form factors come out in good agreement with the data assuming point-like constituent quarks. The extension of this kind of calculation to transition amplitudes and form factors is in progress. References 1. L.Ya. Glozman, W. Plessas, K. Varga, and R.F. Wagenbrunn, Phys. Rev. D 58, 094030 (1998). 2. R.F. Wagenbrunn, L.Ya. Glozman, W. Plessas, and K. Varga, Nucl. Phys. A663&664, 703c (2000); ibid. A666&667, 29c (2000). 3. S. BofB, F. Capuzzi, P. Demetriou, and M. Radici, Nucl. Phys. A637, 585 (1998). 4. R.F. Wagenbrunn, M. Radici, S. Boffi, and P. Demetriou, Eur. Phys. J. A 8, (2000). 5. S. Boffi, P. Demetriou, M. Radici, and R.F. Wagenbrunn, in Proceedings of the International Conference on Quark Nuclear Physics, Adelaide, February 2000, to appear in Nucl Phys. A. 6. S.S. Kamalov and S.N. Yang, Phys. Rev. Lett. 83, 4494 (1999). 7. W.H. Klink, Phys. Rev. C 58, 3587 (1998). 8. R.F. Wagenbrunn et al., in preparation.
309
RELATIONSHIP OF P I O N I U M LIFETIME W I T H P I O N SCATTERING LENGTHS IN GENERALIZED CHIRAL P E R T U R B A T I O N THEORY H. SAZDJIAN Groupe de Physique Theorique, Institut de Physique Nucleaire, Universite Paris XI, F-91406 Orsay Cedex, France E-mail: [email protected] The pionium lifetime is calculated in the framework of the quasipotential-constraint theory approach, including the sizable electromagnetic corrections. The framework of generalized chiral perturbation theory allows then an analysis of the lifetime value as a function of the nn S-wave scattering lengths with isospin 7 = 0,2, the latter being dependent on the quark condensate value.
The DIRAC experiment at CERN is expected to measure the pionium lifetime with a 10% accuracy. The pionium is an atom made of 7r+7r~, which decays under the effect of strong interactions into 7r°7r°. The physical interest of the lifetime is that it gives us information about the nir scattering lengths. The nonrelativistic formula of the lifetime was first obtained by Deser et al. 1: 1
_
16TT /2Am 7r (og - al)2 . ,
,rt,l2
where ip^—(0) is the wave function of the pionium at the origin (in z-space) and CLQ, OQ, the 5-wave scattering lengths with isospin 0 and 2, respectively. The evaluation of the relativistic corrections to this formula can be done in a systematic way in the framework of chiral perurbation theory {\PT)2, in the presence of electromagnetism 3 . There arise essentially two types of correction, (i) The pion-photon radiative corrections, which are similar to those met in conventional QED. (ii) The quark-photon radiative corrections, which appear through terms where the photon field is not explicitly present and which are mainly responsible for the pion mass difference at lowest-order. Three different methods of evaluation have been used for the study of the pionium bound state in the framework of x.PX'. The first method uses a threedimensionally reduced form of the Bethe-Salpeter equation within the quasipotential-constraint theory approach 4 . The second method uses the BetheSalpeter equation with the Coulomb gauge 5 . The third one uses the approach of nonrelativistic effective theory 6 . All the above approaches lead to similar estimates, on the order of 6%, for the relativistic corrections to the nonrelativistic formula of the pionium decay width.
310
The theoretical interest of the TTTT scattering lengths is that they allow us to estimate the value of the quark condensate in QCD. The fundamental order parameter of spontaneous chiral symmetry breaking being Fn, the pion weak decay constant, other order parameters may eventually vanish in the chiral limit without contradicting chiral symmetry breaking, as long as Fn remains different from zero in that limit. Such an issue is intimately dependent on the mechanism of chiral symmetry breaking. In standard xPT2, it is assumed that the quark condensate parameter, defined as < 0|<7g|0 > /F%, is on the order of the hadronic mass scale (~ 1 GeV). This hypothesis is verified in the sigmamodel and the Nambu-Jona-Lasiono model. The vacuum state here is similar to a ferromagnetic type medium. On the other hand, in an antiferromagnetic type medium, one would have a vanishing quark condensate and yet chiral symmetry would still be broken 7 , 8 . An intermediate possibility, due to an eventual phase transition in QCD for large values of the light quark flavor number, was also advocated recently 9 . Generalized xPT i s a framework in which the quark condensate value is left as a free parameter subjected to an experimental evaluation 10 . The Goldstone boson scattering amplitudes are sensitive to the quark condensate value and hence their experimental measurment gives us an estimate of the latter quantity. Thus, in the -KIT scattering amplitude relatively small values of the 5-wave isospin-0 scattering length OQ, on the order of, say, 0.21-0.22, correspond to the predictions of standard xPT, while relatively large values of OQ, on the order of, say, 0.28-0.36, correspond to small values of the quark condensate parameter. We have redone the analysis of the pionium lifetime in the framework of generalized xPT11. Eliminating the quark condensate parameter in favor of the combination (a[] — OQ) w e n a v e calculated the pionium lifetime as a function of (OQ — OQ). The corresponding curve is presented in Fig. 1. Values of the lifetime close to 3 fs, lying above 2.9 fs, say, would confirm the scheme of standard xPT. Values of the lifetime lying below 2.4 fs remain outside the domain of predictions of standard %PT and would necessitate an alternative scheme of chiral symmetry breaking. Values of the lifetime lying in the interval 2.4-2.9 fs, because of the possibly existing uncertainties, would be more difficult to interpret and would require a more refined analysis. References 1. S. Deser, M. L. Goldberger, K. Baumann and W. Thirring, Phys. Rev. 96, 774 (1954). 2. J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984).
311
4
"i
i
i
i
i
i
i
i
i
i
r
r(fs) 3.5
1.5 J 1 I L j i i L 1 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 On -
On
Figure 1: The pionium lifetime as a function of the combination (a[j — OQ) of the S-wave scattering lengths (full line). The band delineated by the dotted lines takes into account the estimated uncertainties (2-2.5%).
3. R. Urech, Nucl. Phys. B 433, 234 (1995); M. Knecht and R. Urech, Nucl. Phys. B 519, 329 (1998). 4. H. Jallouli and H. Sazdjian, Phys. Rev. D 58, 014011 (1998)); 099901(E). 5. M. A. Ivanov, V. E. Lyubovitskij, E. Z. Lipartia and A. G. Rusetsky, Phys. Rev. D 58, 094024 (1998). 6. A. Gall, J. Gasser, V. E. Lyubovitskij and A. Rusetsky, Phys. Lett. B 462, 335 (1999); J. Gasser, V. E. Lyubovitskij and A. Rusetsky, Phys. Lett. B 471, 224 (1999). 7. H. Leutwyler, Phys. Rev. D 49, 3033 (1994). 8. J. Stern, preprint hep-ph/9801282. 9. S. Descotes, L. Girlanda and J. Stern, JHEP 01, 041 (2000); S. Descotes and J. Stern, Phys. Rev. D 62, 054011 (2000); preprint hep-ph/0007082. 10. J. Stern, H. Sazdjian and N. H. Fuchs, Phys. Rev. D 47, 3814 (1993). 11. H. Sazdjian, Phys. Lett. B 490, 203 (2000).
312
The Infrared behaviour of the gluon propagator in SU(2) and SU(3) without lattice Gribov copies* C. Alexandrou", Ph. de Forcrand6'0, E. Follana" ° Department of Physics, University of Cyprus, CY-1678 Nicosia, Cyprus b Inst. fur Theoretische Physik, ETH Honggerberg, CH-8093 Zurich, Switzerland c CERN, Theory Division, CH-1211 Geneva 23, Switzerland We present lattice results for the gluon propagator for SU(2) and SU(3) in the Laplacian gauge which avoids lattice Gribov copies. In SU(3) we compare with the most recent lattice calculation in Landau gauge and with various approximate solutions of the Dyson Schwinger equations (DSE). Introduction We first summarize the results obtained within the Landau gauge1: By solving approximately the DSE, Mandelstam found an infrared enhanced gluon propagator of the form D{q2) *~ 4-. Avoiding gauge copies, Gribov obtained D(q2) ~ qi\mi• Using the "pinch technique", Cornwall 2 obtained a solution which fulfills the Ward identities, allows a dynamical mass generation, and also predicts a finite value for D{0) = D(q2 = 0) consistent with our data. Early results for the gluon propagator obtained directly from Lattice QCD on small lattices 4 were interpreted in terms of a massive scalar propagator. Results on larger lattices were accounted for by assuming a positive anomalous dimension 5 : D(q2) ~ 2 ( 1 + ^ 2 . A recent, detailed study of the gluon propagator uses very large lattices 6 . Since we want to compare our results with these, we follow closely their analysis and refer to Refs. 6 ' 7 for details. In the Laplacian gauge, the longitudinal part of the gluon propagator does not vanish; the transverse scalar function D(q2) can be extracted from T>^,(q) as D{q2) = i J E „ i E a ^ ( < 7 ) } " l ^
1
, where F(q*) is determined by
projecting the longitudinal part ofT>f£,(q) using the symmetric tensor q^q". Gauge Fixing Procedure Previous lattice studies all fixed to Landau gauge by using a local iterative maximization algorithm, which converges to any one of many local maxima (lattice Gribov copy), but fails to determine the global one. To overcome this problem, we use a different gauge condition, the Laplacian gauge 3 , which is Lorentz-symmetric and gives a smooth gauge field like the Landau gauge, but which specifies the gauge unambiguously. We consider the maximization ""I^lk given by C. Alexandrou
313
of Q = Re^2XtllTr[g(x)Uli(x)g^(x + p,)-g(x)g(x)^] . If one relaxes the requirement that g € SU(N), maximizing Q is equivalent to minimizing the quadratic form Y^xyfx^xyfy, with A(C/) the covariant Laplacian. Using the (N - 1 ) lowest-lying eigenvectors f*(x) of A(U), one can fix the gauge uniquely by requiring Var,/?(a:) G R , / / ( « ) = 0,j = (• + 1),.., AT7. Results In Fig.l we show the transverse gluon propagator for SU(2) Yang-Mills theory in two different volumes; m 0 = A / D ( 0 ) - 1 for the 164 lattice. Changing the volume has little effect, in particular on D(0). We observe similarly small volume effects in SU(3). This is strikingly different from Landau gauge, where Zwanziger has argued that D(0) should vanish in the infinite lattice volume limit 8 . This prediction is indeed consistent with recent lattice results in SU(2) at finite temperature 9 . In contrast, in the Laplacian gauge, we find that D(0) is finite and independent of the volume V for V larger than about l/2fm ~ £>(0)2. We find £>(0) = 58(2) in lattice units at /? = 6.0, i.e. D(0)-^2 = 248(5) MeV (using a - 1 = 1.885 GeV), corresponding to a length scale of about 0.8 fm. In Fig.2 we compare results for the gluon propagator in SU(3) quenched QCD in Laplacian and Landau gauges, (mo = yjD{Q)~1 in the Laplacian gauge). Scaling is checked on the 163 x 32 lattice for j3 — 5.8 and 6.0. Making a cylindrical cut in the momenta 6 to minimize lattice artifacts, we find that scaling is very well satisfied for the Laplacian gauge, with both sets of data falling on a universal curve 7 . We fit to our data the same models as considered by Leinweber et al. 6 in Landau gauge. Since we have observed scaling, we use our results at the 2.0
1
163x32
/S=6.0
Landau gauge Laplacian gauge
I2 0.5
0.0
Figure 1: T h e SU(2) gluon propagator in two different volumes.
qa Figure 2: The SU(3) gluon propagator in Laplacian and Landau gauges.
314 Model Gribov Stingl Marenzoni Cornwall Model A
Z 2.63(2) 2.63(2) 2.47(3) 7.08(9) 1.96(1)
m 0.203(7) 0.203(13) 0.199(6) 0.281(4) 0.654(17)
A or a
A
0.002 (1.100) 0.237(5) 0.265(8) 2.181(67)
8.91(41)
D(0) 0 0 62 59 43
X 2 /d.o.f 5.7 5.7 4 2.5 1.2
Table 1: best fit of parameter values to our /? = 6.0 data on the 163 x 32 lattice. finer lattice spacing (/? = 6.0) for the fits. Table 1 and Fig. 3 summarize the results of the fits to the various models. We find that Gribov-type models are excluded, whereas Cornwall's model is clearly favored among all analytically motivated models. Model "A" 6 , which gives a better fit, is phenomenological, contains one more parameter, and misses D(0) by 25%. One can then use the fit to Cornwall's model to analytically continue to negative q2 and determine the gluon pole mass. This is carried out in Ref. 7 . In conclusion, we see significant modifications from Landau gauge in the infrared. In particular, we find that D(0) obeys scaling, is finite, and volume independent for large enough volumes. We find support for Cornwall's model which fits the momentum dependence of the propagator rather well, whereas models with infrared enhancement of the type l/(g 2 ) 2 or Gribov-type suppression are excluded.
4
2
4
6
q (GeV)
Figure 3: Fits to various models References 1. J. E. Mandula, Phys. Rep. 315 (1999) 273. 2. J. M. Cornwall, Phys. Rev. D26 (1982) 1453. 3. J. C. Vink and U. Wiese, Phys. Lett. B 289 (1992) 122. 4. J. E. Mandula and M. Ogilvie, Phys. Lett. B185 (1987) 127;R. Gupta et al, Phys. Rev. D36 (1987) 2813. 5. R Marenzoni, G. Martinelli and N. Stella, Nucl. Phys. B455 (1995) 339. 6. D. Leinweber et al., Phys. Rev. D60 (1998) 094507; T. G. Williams, this volume. 7. C. Alexandrou, Ph. de Forcrand and E. Follana, hep-lat/0008012. 8. D. Zwanziger, Phys. Lett. B257 (1991) 168. 9. A. Cucchieri, hep-lat/9908050.
315
T H E 7 ->• 3TT F O R M FACTOR AS A C O N S T R A I N T O N S C H W I N G E R - D Y S O N MODELING OF LIGHT Q U A R K S DUBRAVKO KLABUCAR Physics Department, P.M.F., Zagreb University, Bijenicka c. 32, Zagreb, Croatia BOJAN BISTROVIC Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 The form factor for 77r+ —> 7T+7T0 was calculated in a simple-minded constituent model with a constant quark mass parameter, as well as in the Schwinger-Dyson approach. The comparative discussion of these and various other theoretical results on this anomalous process, as well as the scarce already available data (hopefully to be supplemented by more accurate CEBAF data soon), seem to favor SchwingerDyson modeling which would yield relatively small low-momentum values of the constituent (dynamically dressed) quark mass function.
The form factor for the anomalous process 7(9) -> Tr+{pi)ir°(p2)n~{p3) was calculated as the quark "box"-amplitude in two related approaches 1 ' 2 . In our Ref. *, the intermediate fermion "box" loop is the one of constituent quarks with the constant quark mass parameter M. The predictions of this quark loop model 1 are given in Fig. 1 by the long-dashed curve for M = 330 MeV, by the line of empty boxes for M — 400 MeV, and by the line of crosses for the large value M = 580 MeV. (In the lowest order in pion interactions, they are also the form factors of the cr-model and of the chiral quark model.) Our second Ref. 2 employs the Schwinger-Dyson (SD) approach 3 , where the box loop amplitude is evaluated with the dressed quark propagator s(k)
KJ
1
= 2
=
^(fc2)
itA(k ) + m + B{k*)-ifi+M{k2)
m
K)
containing the momentum-dependent, mostly dynamically generated quark mass function M(k2), while m is the small explicit chiral symmetry breaking. In Fig. 1, the solid curve gives our 737r form factor obtained in the SD approach for the empirical pion mass, m^ = 138.5 MeV, while the dashed curve gives it in the chiral limit, m„ = 0 = m. To understand the relationship between the predictions of these two approaches, one should, besides the curves in Fig. 1, compare also the analytic expressions we derived for the form factors [esp. Eqs. (20)-(21) in Ref. 2 and analogous formulas in Ref. 1 ]. This way, one can see, first, why the constant, momentum-independent term is smaller in the SD case, causing the downward shift of the SD form factors with respect to those
316
si ml] Figure 1: Various predictions for the dependence of the normalized y3n form factor F^w on the Mandelatam variable s = (pi +P2) 2 • The kinematics is as in the Serpukhov measurement: the photon and all three pions are on shell, g2 = 0 and p\ — p\ = p\ = m 2 .
in the constant constituent mass case. Second, this constant term in the both approaches diminishes with the increase of the pertinent mass scales, namely M in the constant-mass case, and the scale which rules the SD-modeling and which is of course closely related to the resulting scale of the dynamically generated constituent mass M(k2 ~ 0). Finally, the momentum-dependent terms are similar in the both approaches; notably, the coefficients of the momentum expansions (in powers of pi • pj) are similarly suppressed by powers of their pertinent scales. This all implies a transparent relationship between M.{k2) at small k2 and the jSir form factor, so that the accurate CEBAF data, which hopefully are to appear soon 4 , should be able to constrain M.(k2) at small k2,
317
and thus the whole infrared SD modeling. Admittedly, we used the Ball-Chiu Ansatz for the dressed quark-photon vertex, but this is adequate since Ref. 5 found that for - 0 . 4 GeV 2 < q2 < 0.2 GeV 2 , the true solution for the dressed vertex is approximated well by this Ansatz, plus the vector-meson resonant contributions which however vanish in our case of the real photon, q2 — 0. Therefore, if the experimental form factor is measured with sufficient precision to judge the present SD model results definitely too low, it will be a clear signal that the SD modeling should be reformulated and refitted so that it is governed by a smaller mass scale and smaller values of Ai{k2 ~ 0). The only already available data, the Serpukhov experimental point 6 (shown in the upper left corner of Fig. 1), is higher than all theoretical predictions and is probably an overestimate. However, the SD predictions are farthest from it. Indeed, in the momentum interval shown in Fig 1, the SD form factors are lower than those of other theoretical approaches (for reasonable values of their parameters) including vector meson dominance 7 (the dotted curve) and of chiral perturbation theory 8 (the dash-dotted curve). Therefore, even the present experimental and theoretical knowledge indicates that the momentum-dependent mass function in the SD model we adopted 2, may already be too large at small k2, where its typical value for light u, d quarks is M{k2 M 0) « 360 MeV. Note that some other (so far very successful, see review 3 ) SD models obtain even significantly higher values, M(k2 w 0 ) w 600 MeV and more, which would lead to even lower JSTT transition form factors. Acknowledgments D. Klabucar thanks the organizers, W. Lucha and N. Brambilla, for their hospitality at the Fourth International Conference on "Quark Confinement and the Hadron Spectrum" in Vienna, 3.-8. July 2000. References 1. 2. 3. 4.
5. 6. 7. 8.
B. Bistrovic and D. Klabucar, Phys. Rev. D 61 (2000) 033006. B. Bistrovic and D. Klabucar, Phys. Lett. B 478 (2000) 127-136. A recent review is: C. D. Roberts, e - p r i n t arXiv:nucl-th/0007054. R. A. Miskimen, K. Wang and A. Yagneswaran (spokesmen), Study of the Axial Anomaly using the ,yir+ -¥ TT+TT° Reaction Near Threshold, Letter of intent, CEBAF-experiment 94-015. P. Maris and P. Tandy, Phys. Rev. C 61 (2000) 045202. Yu. M. Antipov et al., Phys. Rev. D 36 (1987) 21. S. Rudaz, Phys. Lett. B 145 (1984) 281. B. Holstein, Phys. Rev. D 53 (1996) 4099.
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PARALLEL SESSION C Heavy Quarks (and Gluons)
Conveners:
Christine T. H. DAVIES (Glasgow) Dieter GROMES (Heidelberg)
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321
Application of the Shifted-Z Expansion method to B, and D Meson Leptonic Decay Constants in the Semi-Relativistic Wave Equation T. Barakat Near East University, Lefkosa, P.O. BOX 1033 Mersin 10 - Turkey E-mail:[email protected]. edu. tr In this work we extend the shiftd 1-expansion method developped in [1] to the spectra of heavy-light mesons (Qq).
1
Introduction
In a previous work [1], a quickly tractable constituent second-order Schrodingerlike equation with the inclusion of relativistic corrections up to order (v/c)2 in the quarks speeds is established to solve the semi-relativistic wave equation [(p2 + m\ f'2
+ (p2 + m\ Y'2 + V(r) - M ] ^(r) = 0 .
(1)
Therein, it has been demonstrated that the shifted-? expansion (SLET), where 1 is the angular momentum, provides a powerful, systematic, analytic technique for determining the bound states of the semi-relativistic wave equation consisting of two quarks of masses m i , 7712, and total energy M in any spherically symmetric potential, even one which has no small coupling constant parameter. It provide a remarkable accurate and simple analytic expression for coulomb potential [1]. It also handles highly excited states which pose problems for variational methods. Encouraged by the success of SLET in the heavy mesons sector, we feel tempted to extend SLET to solve for the bound states of the above equation in the heavy-light (Qq) mesons sector, and to estimate their leptonic decay constants fs., / s d , / D , > and fod with the different potentials. 2
SLET for the Semi-Relativistic Equation with Spherically Symmetric Potentials
An expansion in the powers of (v/c)2 up to two terms in the semi-relativistic equation Eq.(l) yields
%L ~ h + v{r)} ^(r) = Ent ^ne{r)'
(2)
where Eni = M — mi — m?,, /x = mim2/(m.\ + 7712) is the reduced mass and v — m\m\l(m\ + m\) is a useful parameter. In the physical literature, the
322
second term in Eq.(2) is treated as perturbation by using different trial wave functions . In this work, in order to obtain a Schrodinger-like equation, this term is treated using the reduced Schrodinger equation:
P4 = V M — mi — m 2 — V(r)
(3)
Thus, the second order Schrodinger—like equation to order (v/c)2 becomes fy
~ ^[Eli
+ V2(r) - 2EntV(r)\
+ V ( r ) } v w ( r ) = E^Mr),
(4)
where n = u/fi2. If the angular momentum I is shifted through the relation I = 1 + 0, Eq.(4) becomes 1 d2 2 • 2fxdr — -rz-\ E2 = (^r- + 2r)
[P +1(20+1)+13(0+1)) 2 ^—n 2pr
EnlV{r) H7(r; +
•n
Rnt(r)
Ent)Rnt{r),
(5)
where n in this paper is the radial quantum number. The systematic procedure of SLET can be found in [1], which immediately gives in one batch, an expression for the energy eigen values Ene —
EQ
+
Q
(i)
22)
+
+ 0
(6)
as well as the leading order bound-state wave functions, Rni{r) =
Cnlr^-1)'2e-x^'r^L^-2yG'{2\(rlr0f),
(7)
in which A = l/u, Q — fiu, and a = 21 + 20 + 3. 3 3.1
Applications, Results, and Discussion The Qq Potential and the Meson Spectra
Four different radial forms for V(r) in Eq.(l) are used. All forms have enjoyed some popularity in the literature [2]. The first (I) is V(r) = - ^ + Kr + A, r
(8)
323
whose form is suggested by QCD with ac = 0.47, K = 0.19 GeV2, A=-0.35 GeV, mc = 1.51 GeV, and mi, — 4.95 GeV. The second form (II) is the Powerlaw potential of Martin, V{r) = -7.226 GeV + (6.570 GeV) (r x l G e V ) 0 1 ,
(9)
with m c = 1.49 GeV, and mb = 4.95 GeV. The third form (III) is the Powerlaw potential of Rosner et al., V(r) = -0.772 GeV + 0.801((r x 1 GeV)a - l)/a,
(10)
with a = -0.12, m c = 1.55 GeV, and mb = 4.99 GeV. The last form (IV) is the Logarithmic potential of Quigg and Rosner, V(r) = -0.6635 GeV + (0.733 GeV)ln(r x 1 GeV),
(11)
with m c = 1.49 GeV, and mb = 4.91 GeV. The results of the calculations of the mesons mass spectra are organized in Table (1). 3.2
Leptonic Decay Constants of Pseudo-scalar Mesons Through the Van Royen- Weisskopf Formula
The measurement of leptonic decay constant / p = ( P = £ ) , Ds, B, Bs) is of great interest for weak decays of heavy mesons and for CP violation in B decays. In potential models a relation between fp and the ground state wave function at the origin tpp(0) through the Van Royen-Weisskopf formula (normalized so that fn = 93.3MeV) is given by
fi = ^\m
i2,
d2)
where Mp is the obtained meson mass. On the othe hand the square of wave function at the origin | ^(0) | 2 for S- states can be related to the potential V(r) by the well-known Fermi-Schwinger sum rule, | RS(0) | 2 = 2 M ^ ° ° R*nt(r) l ^ l ^ i r )
2 r
dr.
(13)
Once, the parameters such as I, w, etc., are obtained after determining ro the integral in Eq.(13) can easily be evaluated numerically. Finally, the leptonic decay constants fs,, fB, JD., and fo are calculated from Eq.(12), and are displayed in Table (2) compared with the results of other theoretical investigations and experimental results.
324 Table 1: The mass spectra (in units of GeV), and Decay constants (in units of MeV) of cd, cs, bd, bs, and be systems in the semi-relativistic wave equation with m u ( = mj) = 0.22, and m, = 0.43. Potential MB. MD. MDd fs. fBd ID. fDi MBd
(I) (II) (HI) (IV) (Average)
5.4141 5.4081 5.3827 5.4177 5.4056
5.2679 5.2630 5.2369 5.2722 5.2600
2.0913 2.1020 2.0597 2.1111 2.0910
1.9122 1.9273 1.8801 1.9331 1.9131
199 183 211 197 198
142 132 151 143 142
268 209 251 202 285 225 266 211 268 212
In conclusion, it is observed that, the convergence of the eigen values as well as the wave functions is very rapid as n, and I are increasing. Therefore, we can say, this technique is easy to implement, the results are sufficiently accurate for practical purposes, and the SLET equations are cast in away to be used to the unequal mass cases qQ. Moreover, substantial computation time reduction has been achieved. References 1. T. Barakat, math-ph/0004026, and references therein.
325 T h e M a s s of t h e b Quark from Lattice N R Q C D
The Department
of Physics
S. Collins and Astronomy, The University Scotland, G12 8QQ
of
Glasgow,
We present results for the mass of the 6 quark in the MS scheme obtained by calculating the binding energy of the B meson in the static limit. The self energy of a static quark, .Eg0 needed for this purpose, is now known to 0(a3) in the quenched approximation. We find a preliminary value of m~b(m~b) = 4.34(7) GeV at nf = 0. The error is dominated by the remaining uncertainty in E£f. In addition, using Eff at 0(a2), we estimate that the quark mass is reduced by approximately 70 MeV when two flavours of dynamical quarks are introduced.
1
Introduction
The mass of the 6 quark in the MS scheme (mj) can be extracted on the lattice using NRQCD, via the pole mass, Mpoie, by calculating the binding energy of the B meson, Kund'- Mpoie = Megpt — Aund where M^pt is the spin-average of the experimental B and B* masses and Abind = Esim — EQ. Esim is the energy of the B meson (at rest) in NRQCD and E0 is the 6 quark self energy. The pole mass is then converted to mb at some scale fi using the continuum perturbative factor Zcont: m&(/j) = Zcont(fj)Mpoie. While Mpoie has an 0(AQCD) renormalon ambiguity, this is cancelled by similar effects in Zcont- Tni is well denned. At present, EQ is only known to 0(a) for the b quark. However, in the limit of the b quark mass becoming infinite, EQ is known to 0(a3) if internal quark loops are neglected (the quenched approximation). The tadpole-improved formula can be expressed as E%° = 1.070a, + 0.118a* - 0.3(1.4)a*
:
ap = a£ n / = 0 ) (0.84/a).
(1)
The a3, coefficient has been determined by Lepage et. a l . x . The error on the coefficient is numerical and quite large. However, it provides a realistic estimate of the uncertainty in EQ, compared to using 2 — loop perturbation theory and assuming the contribution of higher order terms is l a , . In Eq. 1, E0 is expressed in terms of a coupling constant denned on the lattice from the plaquette, ap2, evaluated at a characteristic gluon momenta, q* = 0.84/a, calculated using the BLM proceedure 3,4 . In addition, the lattice calculation of ESim has been tadpole improved, whereby all gauge fields on the lattice are divided by a 'mean-field' approximation to the gluon field, UQ,
326
to obtain more continuum-like operators. The corresponding tadpole improvement of E0 leads to the addition of the perturbative series for lnu 0 - These ingredients result in a well behaved perturbative series for EQ. This is certainly not the case if the bare lattice coupling, CKL = 5Q/(47T) is used: £g° = 2.1173a L + 11.152a| + 82.7(1.4)a|
(2)
Di Renzo et. al. 5 have also determined the 0(a3) coefficient. They obtain 86.2(.6), when on, is used. Encouragingly, the two determinations, which have very different systematic errors, agree within 3
:
ap = a p 0) (0.62m^).
(3)
The series is well-behaved and we estimate the uncertainty in Zcont to be 3a*. The error introduced by working in the static limit, i.e. ignoring 0(AQCD/M) contributions to Abind, leads to « 1% uncertainty in m£. The error arising from working in the quenched approximation is also likely to be around 1% (assuming a 10 — 20% shift in EBim when sea quarks are included). These effects are the same size as the error arising from the numerical error in EQ. We obtained E%fm by extrapolating the simulation energy calculated at finite heavy quark mass. The latter was obtained as part of a high statistics study of the B meson spectrum in the quenched approximation at three lattice spacings (a), with a - 1 = 1—2.5 GeV. For details of the simulations see reference 7 . Note that we use the spin-average of the experimental masses for the B and B* mesons in the expression for Mpoie in order to reduce the error in using E%?m. In addition, we performed a study of sea quark effects using results obtained from a simulation including two flavours of sea quarks (n/ = 2) 8 with a - 1 ~ 2 GeV. Only the 0 ( a p ) coefficient for n / = 2 has been computed 9 and hence the comparison with mt at nj — 0 is performed using EQ and Zcont to this order. 2
Results
Tables 1 and 2 summarize our results. Within the combined statistical and systematic errors we see that this is the case and we take the result at {3 = 6.0 as our best determination of rnb(rfTb) = 4.34(7) GeV at n / = 0. Note that the numerical error in EQ dominates the uncertainty in m^. In addition, using the results at /? = 6.0 at rif = 0 from the Bs meson and those obtained at n / = 2 we see that the b quark mass decreases by 70 MeV at 0(ap) when sea quarks are introduced. Assuming the systematic (perturbative) errors for the two simulations are correlated this is ~ 2a in
327 Table 1: Abind and mb(nifc) in GeV from the B meson at n/ = 0. The statistical and main systematic errors are estimated, including those due to determining the inverse lattice spacing ( a - 1 ) and residual discretisation effects in jE„ m ~ 0((AQCDI)2)"•bind
0 5.7 6.0 6.2
stat. .24 .35 .36
(1) (2) (8)
.Eg° l-4a? (11) (6) (5)
a-1
disc.
(1) (1) (2)
(9) (4) (2)
4.43 4.34 4.32
mj,(m(,) Zcont stat EQ° 3a£ 1.4a? (3) (1) (10) (3) (2) (5) (3) (7) (4)
a-1
disc.
(1) (1) (2)
(8) (3) (2)
Table 2: The change in mi,(mi,) from rif = 0 to 2, where 2-loop perturbation theory is used.
Tif ~~0 2
mbijnb) 4~45 4.34
stat. error iol .02
the (remaining) statistical errors and the same size as the error in Tnb(rfTb) at 0(a.p). Further work is necessary to reduce the error in EQ, in order for sea quark and 0(AQCD/M) effects to be significant. Acknowledgements The author acknowledges support from the Royal Society of Edinburgh. References 1. H. Trottier, private communication and G. P. Lepage et. al, Nucl. Phys. Proc. Suppl. 83-84, 866 (2000). 2. C. T. H. Davies et. al, Phys. Lett. B 345, 42 (1995). 3. G. P. Lepage and P. Mackenzie, Phys. Rev. D 48, 2250 (1993). 4. S. J. Brodsky et. al, Phys. Rev. D 28, 228 (1983). 5. F. Di Renzo, private communication and G. Burgio et. al, Nucl. Phys. Proc. Suppl 83-84, 935 (2000). 6. K. Melnikov and T. van Ritbergen, Phys.Lett. B482, 99 (2000). 7. J.Hein et. al, Phys. Rev. D 62, 074503 (2000). 8. S. Collins et. al, Phys. Rev. D 60, 074504 (1999). 9. J. Shigemitsu, private communication.
328
P H E N O M E N O L O G Y OF HEAVY Q U A R K SYSTEMS A N D N R Q C D M A T C H I N G COEFFICIENTS Lewis P. Fulcher Department of Physics and Astronomy, Bowling Green State Bowling Green, Ohio 43403 E-mail: [email protected]
University
The recent lattice calculations of Bali, Schilling and Wachter (BSW) have made possible an unprecedented level of contact with phenomenology since these authors were able to completely determine all of the leading relativistic corrections to the heavy quark potentials. Since this work is based on the effective Lagrangian of NRQCD, it shares with other lattice calculations a need for reliable inputs for the matching coefficients Ci((i,m). We compare calculations based on the BSW determination of the matching coefficients from one-loop expressions with those from the tree-level approximation, namely, Ci(fi,m) = 1. Using the central potential generated from the BSW lattice calculation, we find that the upsilon energies are not sensitive to this difference, but charmonium energies give a preference for the one-loop values.
1
Introduction
The use of the NRQCD Lagrangian,
£=
^{^-S- Cl S- C2 ^- B + C3 8&( D - E - E ' D ^-}^ «
which describes the propagation of a heavy quark of mass m, as the foundation for a new generation of lattice calculations of the properties of heavy quark systems presents an attractive prospect for significant improvements in the accuracy of such calculations. The matching coefficients c, = Ci(fi,m) depend on both the scale /j, and the heavy quark mass and include the effects of radiative corrections. They are chosen so that the local interactions in Eq. (1) represent the theory of QCD to the given order in the velocity. At the tree level all of these coefficients are unity. Since the assumptions underlying the NRQCD Lagrangian of Eq. (1) are compatible with the classic work of Eichten and Feinberg*, Bali, Schilling and Wachter 2 (BSW) were able to use Eichten and Feinberg's espressions relating both the spin-dependent and spin-independent quark-antiquark potentials to the appropriate Wilson loop expectation values. Working in the quenched approximation and integrating out all the gluonic degrees of freedom, BSW calculated the central potential, the leading relativistic corrections to the central potential, the spin-dependent potentials and the momentum-dependent spin-independent potentials studied
329 by Brambilla et al?'*. The relativistic corrections to the central potential and the spin-dependent potentials all have explicit dependence on the matching coefficients. Below we compare calculations based on a one-loop determination of the matching coefficients with those based on a tree-level determination. 2
Potentials and Matching Coefficients
To calculate the energies of charmonium and the upsilon system, BSW showed that the linear plus Coulomb form is a good parametrization of their lattice determination of the central potential. When their results for the 0(v2) corrections 2 are added to the central potential, one has
where we have suppressed the argument /x in the matching coefficients. The string constant A (slightly flavor dependent) is determined by optimizing the fit to the measured charmonium and upsilon energies, and the Coulomb coefficient K = 0.324(0.321) depends on the lattice parameter /? = 6.0(6.2). Since the BSW results for the fine-structure splittings of the P states are not very good, we spin average 5 over the P and D state energies and thus we do not consider the spin-orbit and tensor potentials. However, we do include the spin-spin potentials in order to calculate the S-state hyperfine splittings, that is, C
VSS(T,
. C
S I , S 2 ) = —^ omim2
r
o
-i
C2(mi)c2(m 2 ) - 7C 2 (m2,mi) ( l - C 2 ( m 2 ) ) Vi(r), v L 4 '}
The matching coefficients are determined from 2
m
/25
m e
(3)
/25 5
4
*<* >=(^r • *<* >= (?sr - - <> \as(m)J
\as(m)J
and thus values of the running coupling at 4 different scale and mass arguments are required. These values are taken from the one-loop expression, a
v
(?) = _ 4 7 r
V + ^ i n f-) + o - 1 0 5 8
3 In Ua iUn
STT2
\aqj
(5)
where Un denotes the requisite lattice expectation value for BSW's implementation2 of the Huntley-Michael 6 renormalization prescription. If the lattice parameter /? = 6.0, then the matching coefficient determined from Eqs. (4) are C2{mb) = 1.036 and c2{mc) = 1.224.
330 Table 1: Charmoniura energies (MeV).
State l 3 5i l^o 23Si 21S0 3 3 5! 3% ±*cog \S\avg
3
BSW(rec) £ = 6.0 3132 3014 3684 3593 4102 4020 3467 26.0
BSW(rec) /3 = 6.2 3142 3005 3687 3585 4098 4006 3473 26.4
BSW(tree) 13 = 6.0 3095 3036 3650 3604 4079 4041 3467 32.4
BSW(tree) /3 = 6.2 3105 3046 3651 3604 4081 4048 3473 34.2
EXPT 3097 ± 0.04 2980 ± 2 3686 ± 0.1 3594 ± 5.0
3525 ± 1.0
Results
Our results for charmonium 5 are shown in Table 1. These include recalculations of BSW's results in columns 2 and 3, and the tree-level results in columns 4 and 5. The average deviations for the BSW results from the measured energies are somewhat smaller than those using tree-level values for all of the matching coefficients, an indication that the measured energies support the one-loop determination of the matching coefficients from Eqs. (4). Further, the tree-level results for the IS hyperfine splitting are a factor of about 2 smaller than the one-loop results. Thus, the hyperfine splitting gives a strong preference for the one-loop expressions. Our results for the upsilon system did not show much sensitivity to the procedure used to determine the matching coefficients. However, we did find the hyperfine splitting for the Bc system to be sensitive to this procedure. Finally, we note that the BSW potentials show a surprising degree of agreement with the measured energies of the heavy-light systems when one of the quark masses is extrapolated to a small value. References 1. 2. 3. 4.
E. Eichten and F. Feinberg, Phys. Rev. D 23, 2724 (1981). G. Bali, K. Schilling and A. Wachter, Phys. Rev. D 56, 2566 (1997). N. Brambilla and A. Vairo, Phys. Rev. D 55, 3974 (1997). A. Barchielli, N. Brambilla and G. Prosperi, Nuovo Cimento A 103, 59 (1990). 5. L. Fulcher, Phys. Rev. D 62, 0945XX (2000), accepted for publication. 6. A. Huntley and C. Michael, Nucl. Phys.B286, 211 (1987).
331
T H E R A D I A L W A V E F U N C T I O N S OF A HEAVY-LIGHT M E S O N CALCULATED O N A LATTICE UKQCD Collaboration, A. M. GREEN, J. KOPONEN, P. PENNANEN Department of Physics and Helsinki Institute of Physics P.O. Box 9, FIN-00014 University of Helsinki,Finland E-mail: [email protected], [email protected], [email protected] C. MICHAEL Department of Mathematical Sciences, University of Liverpool, L69 3BX, UK E-mail: [email protected] A brief review is given of attempts to understand the energies of four-quark systems calculated on a lattice in terms of nuclear-physics-inspired many-body techniques involving interquark potentials. Results are given for the next stage of this study where the wavefunctions of heavy-light mesons are also calculated on a lattice.
Over the past few years the authors have been measuring on a lattice the energies of various four-quark systems. In the original papers (see for example Refs. 1 2 3 ) the four quarks involved were all considered to be infinitely heavy. The resultant energies could then be reasonably well understood in terms of a many-body nuclear-physics-inspired approach involving interquark potentials - provided there was introduced a four-quark term similar to a form factor. Neglecting such a factor consistently led to an overestimate of the binding. Later in Ref. 4 a method was developed for treating on a lattice two quark systems, where one of the quarks was a light quark i.e. the case of heavy-light mesons Qq. In that paper the authors concentrated on measuring the S-, P-, D- and F-wave energies. Returning to the four-quark system, in Refs. 5 the energies of the Q2q2 system were calculated using the same techniques that proved successful in Ref. 4 for the basic Qq case. In addition to the presence of light quarks, the works of Refs. 4 5 had two other improvements compared with Refs. 1 2 3 : i) The gauge group used was SU(3) and not SU(2). ii) The lattice configurations were unquenched. In Ref. 6 the earlier nuclear-physics-inspired approach in terms of interquark potentials was extended to the Q2q2 case. This required fitting first the Qq energies of Ref. 4 to extract an effective light-quark mass of about 400 MeV. The main conclusion from this work was that the Q2q2 data could not be understood in terms of purely two-quark potentials and, as in the earlier static case of Refs. 1 2 3 , a four-quark form factor was necessary.
332
Most of the above work has been devoted to the energies of the various quark systems - the exception being Ref. 7 where flux-tube structures were measured. Now we are working on a lattice measurement of the radial wavefunctions of a single heavy-light meson. These wavefunctions consist of the distribution of the light quark and the colour field components around the static quark. The light quark wavefunctions of the ground state and some excited states are being measured. Such wavefunctions have not been measured before and are of relevance to various phenomenological attempts to reproduce meson-decays and scattering of mesons. These include e.g. bag models and semirelativistic Schrodinger and Blankenbecler-Sugar equations. The actual wavefunction measurement is based on the light-quark propagators Gij of Ref. 4 . For a measurement of the Qq energies only one G^ enters in the 2-point correlation as, essentially, C^t) = J2ijGjiUij where Uij is the static quark propagator represented by a straight line of gauge links from point i to point j in a different time slice. However, for the light quark wavefunction measurement two such operators arise giving a 3-point correlation of the form C${t, r) = Yliji GjiOGuUij, where the site I is constrained to be within r spacings from the i,j space coordinates. Here we use the local operators O = 74 and 1, which are probing respectively the light quark charge and matter distributions at a distance r from the heavy quark. The latter are defined as (C3(t,r)/C2(t)). The result of fitting these distributions by F2(0), where F = Aexp(—r/ro), is given in Table 1. There it is seen that the charge distribution has a considerably longer range than that of the matter. Summing over the charge distribution should give the charge of the quark. With the present normalisation this should be PSI on a lattice and, within the expected accuracy, this is indeed the case, when the sum is carried out directly on the lattice - see the column DSum. As discussed in 8 the sumrule for O = 1 has a less direct interpretation. The authors wish to thank the Center for Scientific Computing in Finland for their cooperation in making these studies possible.
Table 1: Parameters for fitting the charge (74) and matter (1) distributions with F2, where F = Aexp(—r/ro). Dsum refers to a direct lattice estimate of the sum of F2. Operator(O) 74 (Charge) 1 (Matter)
r0/a 1.56(2) 1.15(5)
A 0.45(1) 0.46(2)
DSum 1.12(5) 0.25(5)
333
References 1. A.M. Green, C. Michael and J.E. Paton, Nucl.Phys.A554, 701 (1993), hep-lat/9209019. 2. A.M. Green, J. Lukkarinen,P. Pennanen and C. Michael, Phys.Rev. D53, 261 (1996), hep-lat/9508002. 3. A.M. Green and P. Pennanen, Phys. Rev. C57, 3384 (1998), hep-lat/9804003 and references therein. 4. C. Michael and J. Peisa, Phys.Rev. D58, 034506 (1998), hep-lat/9802015. 5. UKQCD Collaboration, C. Michael and P. Pennanen, Phys.Rev.D60, 054012 (1999) hep-lat/9901007; UKQCD Collaboration, C. Michael, P. Pennanen and A.M. Green, Proceedings of the 17th International Symposium on Lattice Field Theory (LATTICE 99), Pisa, Italy, Nucl. Phys. B(Proc. Suppl.) 83-84, 200 (2000), hep-lat/9908032; P. Pennanen and C. Michael, hep-lat/0001015. 6. A.M. Green, J. Koponen and P. Pennanen, Phys.Rev.D61:014014 (2000), hep-ph/9902249 and Proceedings of LATTICE 99, Pisa, Italy, Nucl. Phys. B(Proc. Suppl.) 33-84, 292 (2000), hep-lat/9908016. 7. P. Pennanen, A.M. Green and C. Michael, Phys. Rev. D59, 014504 (1999), hep-lat/9804004; Phys. Rev. D56, 3903 (1997), hep-lat/9705033. 8. UKQCD Collaboration, M. Foster and C. Michael, Phys. Rev. D59, 074503 (1999), h.ep-lat/9810021
Figure 1: The correlation (C${t,r)/C2(i)) b) Matter. Solid(dotted) for t = 8(10).
as a function of r in lattice units: a) Charge and
334
W E A K F O R M FACTORS FOR HEAVY M E S O N D E C A Y S 0
Institut
D. MELIKHOV and B. STECH fur Theoretische Physik, Universitat Heidelberg, D-69120, Heidelberg, Germany
Philosophenweg
16,
We calculate the form factors for weak decays of B(s\ and D/s\ mesons to light pseudoscalar and vector mesons within a relativistic dispersion approach based on the constituent quark picture. This approach gives the form factors as relativistic double spectral representations in terms of the wave functions of the initial and final mesons. The form factors have the correct analytic properties and satisfy general requirements of nonperturbative QCD in the heavy quark limit. The effective quark masses and meson wave functions are determined by fitting the quark model parameters to lattice QCD results for the B —• p transition form factors at large momentum transfers and to the measured D —>• (K, K*)lv decay rates. This allows us to predict numerous form factors for all kinematically accessible q2 values.
The knowledge of the weak transition form factors of heavy mesons is crucial for a proper extraction of the quark mixing parameters, for the analysis of non-leptonic decays and CP violating effects and for a search of New Physics. Theoretical approaches for calculating these form factors are quark models, QCD sum rules, and lattice QCD (a detailed list of references can be found in 1 ). Although in recent years considerable progress has been made, the theoretical uncertainties are still uncomfortably large. An accuracy better than 15% has not been attained. Moreover each of the above methods has only a limited range of applicability: QCD sum rules are suitable for describing the low q2 region of the form factors; lattice QCD gives good predictions for high q2. As a result these methods do not provide for a full picture of the form factors and, more important, for the relations between the various decay channels. Quark models do provide such relations and give the form factors in the full g 2 -range. However, quark models are not closely related to QCD and therefore have input parameters which may not be of fundamental significance. Clearly, a combination of various methods can be fruitful for obtaining reliable predictions for many decay form factors in their full q>2-ranges. To achieve this goal, one needs a general frame for the description of a large variety of processes. This can only be a suitable quark model, since only a quark model connects different processes through the meson soft wave functions and describes the full g2-range of the form factors. This program has been implemented in our recent work * where the predictions of the quark model "Talk presented by D. Melikhov
335
have been considerably improved by incorporating the results from lattice QCD and the available experimental data. 1. The physical picture The constituent quark picture is based on the following phenomena: • the chiral symmetry breaking in the low-energy region which provides for the masses of the constituent quarks; • a strong peaking of the nonperturbative meson wave functions in terms of the quark momenta with a width of the order of the confinement scale; • a qq composition of mesons in terms of the constituent quarks. 2. The formalism For the description of the transition form factors in their full q2 -range and for various initial and final mesons, a fully relativistic treatment is necessary. We make use of the dispersion formulation of the quark model 2 which guarantees the correct spectral and analytic properties of the form factors. The transition form factors in the decay region are given by the relativistic double spectral representations through the wave functions of the initial and final mesons. These spectral representations obey rigorous constraints from QCD on the structure of the long-distance corrections in the heavy quark limit: the form factors of the dispersion quark model have the correct heavyquark expansion at leading and next-to-leading l / m g orders in accordance with QCD for transitions between heavy quarks. For the heavy-to-light transition the dispersion quark model satisfies the relations between the form factors of vector, axial-vector, and tensor currents valid at small recoil. In the limit of the heavy-to-light transitions at small q2 the form factors obey the lowest order 1/TTIQ and 1/E relations of the Large Energy Effective Theory. 3. Parameters of the model A possible way to control quark masses and the meson wave functions is to use the lattice results 3 for the B —> p form factors at large q2 as 'experimental' inputs. In 4 the b and u constituent quark masses and slope parameters of the B, IT, and p wave functions have been obtained through this procedure. I n 1 we have included into consideration also charm and strange mesons and fixed their wave functions and the effective masses mc and ms by fitting the measured rates for the decays D -+ {K,K*)lv. With these few inputs we gave in Ref1 numerous predictions for the form factors for the D^ and B(s) decays into light mesons which nicely agree at places where data are available.
336
4. Results The main results of our analysis are as follows: 1. In spite of the rather different masses and properties of mesons involved in weak transitions, all existing data on the form factors can be understood in the quark picture, i.e. all form factors can be described by the few degrees of freedom of constituent quarks. Details of the soft wave functions are not crucial; only the spatial extention of these wave functions of order of the confinement scale is important. In other words, only the meson radii are essential. 2. The calculated transition form factors are in good agreement with the results from lattice QCD and from sum rules in their regions of validity. The only exception is a disagreement with the sum rules 5 for the Ba —> K* transition. This disagreement is caused by a different way of taking into account the SU(3) violating effects when going from B —> p to Bs —> K* and is not related to specific details of the model. We suspect that the sum rules 5 overestimate the SU(3) breaking in the long-distance region, but this problem deserves further clarification. 3. We have estimated the products of the meson weak and strong coupling constants by extrapolating the form factors to the meson pole. The value of each coupling constant can be obtained independently from the residues of several form factors. In all cases the values extracted from the different form factors agree with each other within the 5-10% accuracy. This gives additional argument in support of the reliability of our estimates for the form factors. Acknowledgments It is a pleasure to thank the Organizers for creating a fruitful working atmosphere during this nice and interesting meeting. D.M. acknowledges the support of the BMBF under project 05 HT 9 HVA3. References 1. 2. 3. 4. 5.
D. Melikhov and B. Stech, Phys. Rev. D 62, 014006 (2000). D. Melikhov, Phys. Rev. D 53, 2460 (1996), 56, 7089 (1997). L. Del Debbio et d, Phys. Lett. B 416, 392 (1998). M. Beyer and D. Melikhov, Phys. Lett. B 436, 344 (1998). P. Ball and V. M. Braun, Phys. Rev. D 58, 094016 (1998).
337
C O M P A R I S O N OF T W O - B O D Y EQUATIONS JOHN H. CONNELL Springfield Technical Community College 1 Armory square, Springfield, Mass. 01105, USA [email protected] A dozen energy levels of two scalar particles bound by a harmonic-oscillator potential are calculated exactly using four different equations: the Bethe-Salpeter equation, the "spinless Salpeter" equation, a new relativistic two-body wave equation local in configuration space, and the non-relativistic SchrSdinger equation. The four equations all disagree with each other, but the Bethe-Salpeter equation and the relativistic two-body wave equation agree most closely.
1
Equations
Because Salpeter's exact reduction of the Bethe-Salpeter equation is so difficult to solve, the so-called "spinless-Salpeter" equation has often been used instead, even though it is also hard to solve. Both equations contain non-local operators. Recently a relativistic two-body wave equation with local operators only which is quite easy to solve has been derived from the Bethe-Salpeter equation. Here we take a single scalar harmonic-oscillator interaction, I(r) = [0.05 GeV3] r 2 - 0.8 GeV
(1)
and work out and compare its exact energy levels for a dozen states using the three equations mentioned, as well as the non-relativistic Schrodinger equation. The constituent particles are scalar and have masses m and M. The total energy E of the bound state is parametrised by b2 using
E = Vm2 + b2 + VM2 + b2
(2)
An offset to the relative energy variable uses the constituents' individual boundstate energies: t = \/m 2 + b2, T = sjM2 + b2, so the BS equation is: TV ^ =
[IJLI-
T( _
^
?m
2M
(Dimensionally the scalar interaction is 4 m M / ( r ) ; the 2m and 2M have been moved so that all dimensions are similar to the spin-1/2 problem.)
338
E X A C T SALPETER R E D U C T I O N . The exact Salpeter reduction, completely equivalent to the BS equation above, is V p 2 +m2 + V p 2 + M2 + Z(p)I(r)
(T)
= E
(3)
in which Z(p) is a correction factor to the interaction I(r): 2 2 2 2 2mM V p + m + v/p + M
Z(p) =
/ v
p 2 + m 2 v / p 2 + M 2 ^ P 2 + m2 + V P 2 + M 2 + E
"SPINLESS SALPETER" EQUATION. When Z{p) is arbitrarily made unity, one obtains an equation which has been considered "simpler" than (3) and studied a great deal, 1 but still contains non-local operators: \ / p 2 + m 2 + y/p2 + M2 + I(r)] (r) = E<j>(r)
(4)
NON-RELATIVISTIC S C H R O D I N G E R . With reduced mass p. [p 2 + 2/iJ(r)] ^(r) = 2;* [£7 - m - M]
{T)
(5)
RELATIVISTIC T W O - B O D Y WAVE EQUATION. Elsewhere 2 we have deduced directly from the BS equation a relativistic two-body wave equation which contains only local operators, as well as energy-dependent constants: {p 2 + 2fiZE [I(r) + REI(r)2}
} 0(r) - b2<J>(r)
(6)
The constant b2 determines the energy E using (2). The other constants are m +M E ' 2
RE
=
mM 1 ~E~ m(m + t)
M(M + T)
Results
With the interaction (1) the energy levels of each equation above can be worked out exactly. At the suggestion of F. Schoberl, equations (3) and (4) are solved as local differential equations in momentum space using r2 = —V2. We earlier worked out a dozen levels for light-light constituents. 3 Here we show results for a heavy-light pair m = 300 MeV and M — 4.7 GeV.
339 State
IS IP ID 2S 2P 2D 3S 3P 3D 4S 4P 4D
Energy Levels in M e V Exact Spls. Wave NR BS Salp. Eqn. Sch. 4947 4711 5136 4832 5232 5005 5301 5253 5445 5278 5477 5674 5472 5245 5522 5674 5654 493 5689 6095 5808 5733 5842 6516 5834 5710 5859 6516 5977 5934 6005 6937 6102 6152 6140 7358 6126 6136 6150 7358 6249 6346 6280 7779 6356 6551 6402 8200
Errors relative to B S Spls. Wave NR Salp. Eqn. Sch. -236 189 -115 -227 69 21 32 -167 229 -227 50 202 -161 35 441
-75 -124
-43 50 10 97 195
34 25 28 38 24 31 46
708 682 960 1256 1232 1530 1844
Except for the ground state, the local relativistic two-body wave equation reproduces the Bethe-Salpeter levels fairly well. Further energy-level corrections for the relativistic two-body wave equation have already been worked out for a Coulomb interaction 2 and will be applied to the present problem later. A complete treatment of the light-light, light-heavy and heavy-heavy systems with further corrections to the relativistic two-body wave equation's levels will be presented in future. 4 Here we just recall that in the light-light case 3 the results are similar except that the uncorrected relativistic two-body wave equation agrees even more closely with the Bethe-Salpeter equation while the "spinless-Salpeter" equation continues to disagree. In the heavy-heavy case all four equations agree fairly well. Therefore the "spinless Salpeter" equation disagrees violently with the Bethe-Salpeter equation when one particle is relativistic or both are. In contrast even the uncorrected local relativistic two-body wave equation's levels are reasonably close to those of the Bethe-Salpeter equation except for the ground state. I am most grateful for the hospitality of the Austrian Academy of Sciences, and to Wolfgang Lucha and the other organisers of this Conference References 1. 2. 3. 4.
W. Lucha & F. Schoberl, Phys. Rev. A60, 5091 (1999) and references. J. Connell, xxx.lanl.gov/abs/hep-th/0006082. Confinement III Proceedings, ed. N. Isgur (World Scientific, 2000). To be posted on xxx.lanl.gov in 2001
340
I N S T A N T A N E O U S B E T H E - S A L P E T E R EQUATION: (SEMI-)ANALYTICAL SOLUTION Wolfgang LUCHA Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18, A-1050 Vienna, Austria E-mail: Wolfgang.luchaQoeaw.ac.at Khin MAUNG MAUNG Department of Physics, Hampton University, Hampton, VA 23668 E-mail: [email protected] Franz F. SCHOBERL Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria E-mail: franz.schoeberWunivie. ac. at The Bethe-Salpeter equation for bound states of a fermion-antifermion pair in the instantaneous approximation for the involved interaction kernel is converted into an equivalent matrix eigenvalue problem with explicitly (algebraically) given matrices.
1
The Instantaneous Bethe— Salpeter Equation (IBSE)
For a system of massless fermion and antifermion forming bound states with the "pion-like" spin, parity, and charge conjugation quantum numbers JPC = 0 _ + , the (homogeneous) Bethe-Salpeter equation, in free-propagator approximation and instantaneous approximation for the involved interaction kernel, reads for a time-component Lorentz vector interaction (i.e., the Dirac structure 7°
/
HA-' k'2
^JL-V0{k,k>)9a(*>)
= MVtf)
2^V1(k,k')*1(k')
= M*2(k).
,
o oo
(1)
0
In this set of coupled equations for the two relevant radial Salpeter amplitudes $ i and *2 in momentum space, with the bound-state masses M as eigenvalues, the interaction potential V (r), usually formulated in configuration space, enters in form of its standard Fourier-Bessel transforms Vr,(/i, k'),L — 0,1. We adopt a linear potential V(r) = Ar (A > 0) as a simple model for quark confinement.
341
2
Efficient M e t h o d of Solution: Expansion in Terms of Basis States
By insertion of the first of Eqs. (1) into the second and by expansion in terms of sets (distinguished by the angular momenta £ = 0,1) of basis states for L 2 (-R + ) —with configuration and momentum-space representations <j>\ '(r) and <j>\ (p), resp.—the solution of the IBSE (1) reduces to the diagonalization of the matrix 3 oo
oo
Mii=^ j Akk^f\kUf\k)
oo
+ 2 j dkk^f\k) f
0
0
oo
^~V0{k,k'Uf\k')
0
oo
+ 2 J dfcfc2# » (*) J ^ o
VX (k, k>) #°> (k')
o
oo
oo
+ J dke^\k) 0
oo
J ^-Vl(k,k')
I ^^V0(k',k")tf(k»)
0
0
. (2)
Allowing these basis functions to depend on a variational parameter fi > 0 gives us more freedom in the search for solutions of the IBSE. All integrations in M.ij are evaluated by (truncated) expansions, with the (//-independent) coefficients3 oo
I^ = ^Ukk^f\k)4>f{k),
i,j = 0,l,2,... ,
o N
°°
btj = - fdkk3 $ 0 ) (fc) 4>f (k) , k $ 0 ) (*) = „ E ^ *T (*)> oo
N
2
0)
ca = fdkk %w (A) $ (k) , #°> (fc) = J2 en 4>f (k) , 0
3=0 N
°°
3
0)
0)
da = j f dk k ?W (k) $ (k) , k $ (fc) = ^ E ^ W W ,
d r r 3 ^ w ( r ) ^ ( r ) , rtfjV) = 7 E V#MV) > * = 0,1 • 0
M
i=o
The explicit algebraic expressions of all these matrices may be found in Ref. 3.° "Let's mention a numerical problem noted for Mathematica 4.0: for, e.g., the matrix element ^49,49 = 101/(2- v / 689) = 1.924 Mathematica finds exactly this value for a working precision of 40 digits but the nonsense value —1.675 x 10 2 2 for the default working precision of 16 digits.
342
In this way, the IBSE (1) is converted to an eigenvalue problem for the matrix 3
MV = V 4 ? + 2A £ bnVff +2A £ £) c:idsiV$ r=0 ,2 p
3
N
N
N
r=0
*=0
t=0
r=0
s=0
Analytical Results (for B o t h Massless and Massive Constituents)
For a matrix size less than or equal to 4, the diagonalization of the matrix Mij may be even performed analytically. In the one-dimensional case, we find, after minimizing w.r.t. the variational parameter /x, for the lowest bound-state mass 3
For 4 A = 0.2 GeV 2 , this expression gives M = 1.696 GeV, only 2.4 % away from the numerical result M = 1.656 GeV, obtained for 15 x 15 matrices and N = 49. For a nonvanishing mass m of the bound-state constituents, we get accordingly 1#2
M
4
D
8896
2
= 8 m +
315^
23 / 128AN 2
x
A +
,
, „. (m
y(45^j
^0)-
Relations Between Matrix Elements and Accuracy of Expansions
Our final question concerns the errors induced by the necessary truncations of the expansion series. The expansion coefficients 6y-, cy, dij are not independent but should satisfy (clearly, only in the limit N —> co exact) relations of the kind N
N c
/ j ri r=0
c
rj
=
"ij
)
N c
/ y ri ^rj r=0
=
/
y
r=0
N " r i °rj
=
"ij
'
/—t r=0
ri
r
i
=
*3
'
For 15x15 matrices and N = 49, these relations are fulfilled with relative errors less than 3 %. For comparison, some integrals in (2) may be evaluated exactly. 3 References 1. 2. 3. 4.
J.-F. Lagae, Phys. Rev. D 45, 305 (1992). M. G. Olsson, S. Veseli, and K. Williams, Phys. Rev. D 52, 5141 (1995). W. Lucha, K. Maung Maung, and F. F. Schoberl, hep-ph/0009185. W. Lucha, F. F. Schoberl, and D. Gromes, Phys. Rep. 200, 127 (1991); W. Lucha and F. F. Schoberl, Int. J. Mod. Phys. A 7, 6431 (1992).
343
GENERALIZED G E L L - M A N N - L O W T H E O R E M A P P L I E D TO A S C A L A R MODEL A. WEBER Institute
de Fisica y Matemdticas, UMSNH, Cd. 58040 Morelia, Michoacdn, Mexico E-mail: [email protected]
Universitaria
The recently established generalization of the Gell-Mann-Low theorem is applied to Wick-Cutkosky-like models, where the exchanged particle can have arbitrary mass. We derive the effective Schrodinger equation to lowest non-trivial order, identify t h e diagrams corresponding to the Bloch-Wilson Hamiltonian, and consider the non-relativistic and one-body limits.
The generalization of the Gell-Mann-Low theorem 1 has been developed with the bound state problem of quantum field theory in mind. In this contribution, we will apply the generalized Gell-Mann-Low theorem (GGL) to lowest non-trivial order to a scalar model containing two charged particles A and B with masses TUA and m j , respectively, and a third uncharged particle of mass fi, coupled via the interaction Hamiltonian Hr = :fd3x
[ ^ ( x ^ C x M x ) + 4 ( x ) ^ ( x ) ^ ( x ) ] :.
(1)
As the ifo-invariant subspace ffc we take the span of all momentum eigenstates I P A J P B ) of one A- and one B-particle, normalized non-covariantly to = (2TT)3<5(PA - p'A)(2n)36(pB
(PA,PB\P'A,PB)
- p'B).
(2)
To second order in the coupling constant, the resulting expression for the matrix elements of the Bloch-Wilson Hamiltonian is (PA,PB\HBW\P'A>PB)
= fa + y/MA+pA
-ig2 f J — OO
+ y/MB+pB^)
(2TT)3S(PA
- p'A)(2n)36(pB
- p'B)
fd3xd3x'
dte~M J
x [ V £ (*, x ' ) ^ (t, x ' ) A F (0 - t, x - x ' ) C (0, X ) ^ B (0, x) + < ; ( 0 , x ) ^ ( 0 , x ) A F ( t - 0, x' - x ) V £ ( t , x'JVg, (t,x')] ,
(3)
and the corresponding diagrams are presented in fig. 1. There are two unlinked
344 t = 0
t = 0
P'A
PA
P'B
PB
P'A
PA PB
P'B
PA
p'A
>
"i >
^—>
PA
+ P'B
PB
P'B
PB
PA
PA
PA
PA
+
+ PB
--• ' \ — "" ""t V.
PB
PB PB
S
PA
PA
P'B
PB
+
PA
7?-*-
PA
+ P'B
PB
Figure 1: The second-order contributions to the Bloch-Wilson Hamiltonian in the t w o particle sector. The propagators are represented by full black and gray lines for the A- and B-particles, respectively, and by dashed lines for the neutral particle. The position of the vertices is integrated over all of 3-space and over time from - c o to 0 for the left vertex, while the right vertex is fixed in time at t = 0. External lines are to be contracted with the corresponding free-particle wave functions.
345
diagrams (containing a part that is not connected to any external line) which merely contribute to the vacuum energy EQ, and four unconnected but linked diagrams that combine to two Feynman diagrams and renormalize the masses rriA and TUB to MA and MB by the usual one-loop contributions (for an onshell renormalization scheme). The two connected diagrams effect the lowestorder interaction between the two particles A and B and have been expressed in (3) through the propagators Ap(x — x') = (ip(x)ip(x')) and the free-particle wave functions
V^(f,x)= e
, u4 = JM\+&,
(4)
and analogously for the B-particles. The corresponding effective Schrodinger equation for the two-particle wave function I/J(PA,PB) = (PA)PslV') c a n be reduced to the c.m.s. and reads ( V ^ 2 + P 2 + \ / M B + P 2 ) V-(P) + / | ^ ( P , P ' M P ' ) = -BV(P) ,
(5)
with p = p ^ = —PB, E' — E — EQ and the (non-local and non-hermitian) potential V(p,p') = -
^ yl2uA2wB2u$2u>*
2w p _p' ^
+ a;p_p- - u}£
u ^ + wp_p-
-v$)
Eq. (5) reduces in the non-relativistic limit to the usual equation for a Yukawa potential, and in the one-body limit to the equation for a particle in the field of a fixed source (as derived by the GGL). The numerical solution of (5) is presently under investigation. Given that there is no relative time dependence in (5), abnormal solutions as in the Wick-Cutkosky model are not expected. Acknowledgments This work was supported by CIC-UMSNH and Conacyt grant 32729-E. References 1. A. Weber, Bloch-Wilson Hamiltonian and a Generalization of the GellMann-Low Theorem, preprint hep-th/9911198, to be published in the proceedings of the VII Mexican Workshop on Particles and Fields held in Merida, Yucatan, Mexico 1999.
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PARALLEL SESSION D Deconfinement Conveners:
Prithjof KARSCH (Bielefeld) Janos POLONYI (Strasbourg)
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349 QCD s u m rules at finite temperature and density S. Mallik Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Calcutta-700064,
India
We discuss the characteristic features of QCD sum rules in the medium, which distinguish them from their vacuum counterparts. Both mesonic and nucleonic correlation functions are considered.
1
Introduction
The original, vacuum QCD sum rules 1 , so successful in dealing with hadron phenomenology in the low energy region, has already been extended to finite temperature and (nuclear) density 2 , 3 . Though the results in the medium are yet to reach the accuracy of those in the vacuum, we believe it could be attained if all the additional features of such sum rules are properly taken into account. Here we do not write any specific sum rule; instead, we discuss these additional features in a general way. A vacuum QCD sum rule may, in principle, be written for any vacuum correlation function, {o\TA{x)B{o)\o), where A(x) and B(x) are any two 'currents', which are local operators composed of quark and gluon fields. To get the corresponding sum rule in the medium we have to consider the ensemble average, (o\TA(x)B(o)\o), where for any operator O, (O) =
Tr[Oe-KH-^]/Tre-KH-»N\
Here H is the QCD hamiltonian, fl is the inverse temperature, and N is the number operator for any species of particle(s) with chemical potential /x. A sum rule in the medium is then obtained by equating (the Borel transform of) its spectral representation to its Operator Product Expansion (OPE). The interaction of the 'current' with the particles in the medium changes the spectral function. Also the set of operators contributing to the OPE is enlarged, as we now have only the spatial 0(3) invariance instead of the 0(3,1) invariance. Alternatively we may restore Lorentz invariance by introducing the 4-velocity vector uM of the heat bath, allowing us to construct additional 0(3,1) invariant operators. The kinematics is also altered. Below we discuss briefly all these three aspects of the sum rules in the medium. 2
Kinematics
We consider the kinematics for both the bosonic vector current, V°{x) = 9(x)7M(r°/2)q{x) and the nucleonic current, 77 ~ q{x)q{x)q(x) 4 . The Fourier
350
tranform of the correlation functions, to be studied below, depend on two independent kinematic variables, q2 and w = q • u, where gM is conjugate to x11. We shall also use q2 = w2 — q2. For the bosonic correlation function T^(q)=ijd4xei"x(T(V^x)Vvb(o))).
(1)
one has two kinematic covariants. Since the dynamical singularity extends here down q2 = 0, it is convenient to choose them as regular at q2 = 0. One such choice is 5 ,
where uM = u^ — toq^lq2. In terms of these the mesonic correlator may be decomposed as Tfi(q) = S^iQ^Tt + P^Tt), (3) However, these covariants (2.2) depend on the direction of
(4)
For the nucleon correlation function, n(q) = i J tPxet-'pTiWr&o)).
(5)
one has the kinematic decomposition,
u(q) = iii + i M + n 3 ^
(6)
giving n x = trH/4, n 3 = {cjtr(0)
n 2 = {wirfoffi) - tr(iU)}/(4q2)
tr(0)}/(4q2), (7)
Thus if we wish to write sum rules for q = 0, it is necessary to extract a factor of q2 from the numerator for II2 and II3. This, however, becomes quite complicted for the spectral contributions from two particle intermediate states. In this limit it is therefore more convenient to replace (2.6) in the rest frame of the medium by
u(q) = u1 + u'lo
(8)
351
3
Spectral representation
The interaction of the current with the medium brings in additional contributions due to the so-called Landau cuts. Thus for the exchange of an intermadiate state of two particles of masses mi and m2, there arise not only the usual unitary cut with q2 > (mi + m-i)2, but also the Landau cut, q2 < (mi — m.2)2 in the low energy region. This point need be kept in mind while looking for intermediate states contributing dominantly to the sum rules. We now enumerate the dominant intermediate states. Consider first the mesonic correlation function in the p meson channel. At finite temperature and zero (nucleon) chemical potential, to order T2, one usually takes the contributions from p and nn intermediate states. But to have a representation extending up to 1 GeV, one must also include intermediate states like nu, 7r0. The temperature dependence of the p meson parameters may be left as unknowns to be determined from the sum rules or be taken as inputs calculated from chiral perturbation theory 6 . In the nuclear medium (p, / 0) at zero temperature, one has, in addition to the contributions from p and NN intermediate states, the contributions from NA states, where A represents a resonance in the nN channel. Next consider the nucleonic correlation function. At finite temperature and zero chemical potential, the contributions proportional to first power in pion density distribution function are from AT, irN and TTA states. The vertices 777^ are obtainable from chiral perturbation theory or equivalently from current algebra and PC AC 6 . Finally for the nucleon correlation function in the nuclear medium at zero temperature, one usually considers the intermediate states N and TTN. However the states pN,ujN are also important. A point to note here is that these contributions are proportional to a fractional power of the nucleon number density. 4
OPE
The expansion at short distance of the product of currents (mesonic or nucleonic) may be obtained by first applying Wick's theorem and then expanding the resulting normal ordered bilocal operators into local ones. 7 , 8 . Although the procedure is simple in principle, it requires quite an amount of algebra to find the (Wilson) coefficients of higher dimension operators. At dimension six, we have besides the possible gluon operators, the following ones, q[D», G^}Yq, qu-Da- Gq,
qHD^G^u'q, q(u • D)3q.
q[u-D,u-G-
7 ]g,
(9)
352
The first two can be converted to 4-quark operators by using the equation of motion for the gluon field. Then by the ground state saturation (similar to vacuum saturation for the vacuum expectation values) their ensemble averages may be further converted to (qq)2 and (q*q)2. The sum rules evaluated so far incorporate only (qq)2 at dimension six, ignoring other operstors — a simplifying assumption, but difficult to justify. 5
Conclusion
We discuss briefly the different characteristic features of the QCD sum rules in the medium, which must be taken into account for reliable results. These involve the particle parameters and the ensemble average of operators, which, to begin with, are unknowns. The dependence of the particle parameters on the medium may be worked out from the chiral perturbation theory, leaving the ensemble average of the higher dimension operators as unknowns. We are presently attempting an evaluation of these operator matrix elements in a framework involving both mesonic and nucleonic correlation functions. 6
Acknowledgement
I wish to thank the Organisers, particularly Professor Wolfgang Lucha, for the local hospitality, which enabled me to participate in the Conference. 1. M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979). For a collection of original papers and comments, see Vacuum Structure and QCD Sum Rules, edited by M.A. Shifman (North Holland, Amsterdam, 1992). 2. A.I. Bochkarev and M.E. Shaposhnikov, Nucl. Phys. B268, 220 (1986). 3. E.G. Drukarev and E.M. Levin, J E T P Lett. 48, 338 (1988). 4. B.L. Ioffe, Nucl. Phys. B 188, 317 (1981). 5. S. Mallik and K. Mukherjee, Phys. Rev. D58, 096011 (1998). 6. H. Leutwyler and A.V. Smilga, Nucl. Phys. 342, 302 (1990). 7. A.V. Smilga, Sov. J. Nucl. Phys. 35, 271 (1982). 8. W. Hubschmid and S. Mallik, Nucl. Phys. B207, 29 (1982).
353
EIGENVALUES OF T H E QCD D I R A C O P E R A T O R AT FINITE TEMPERATURE A N D DENSITY E. BITTNER a , M.-P. LOMBARDO b , H. MARKUM a , R. PULLIRSCH a "'Institut fur Kernphysik, Technische Universitat Wien, A-1040 Vienna, Austria h Istituto Nazionale di Fisica Nucleare, Sezione di Padova, e Gruppo Colhgato di Trento, Italy We investigate the eigenvalue spectrum of the staggered Dirac matrix in two-color QCD at nonzero temperature and at baryon density when the eigenvalues become complex. The quasi-zero modes and their role for chiral symmetry breaking and the deconfinement transition are examined. The bulk of the spectrum and its relation to quantum chaos is considered. Comparison with predictions from random matrix theory is presented.
1
Chiral Condensate
The properties of the eigenvalues of the Dirac operator are of great interest for important features of QCD. The Banks-Casher formula 1 relates the Dirac eigenvalue density p(X) at A = 0 to the chiral condensate, E = | ( ^ ) | = lime_>.o liniy-^oo irp(e)/V. The microscopic spectral density, ps(z) = limy-Hx, p (z/VE) I'VE, should be given by the appropriate result of random matrix theory (RMT), 2 which also generates the Leutwyler-Smilga sum rules. 3 A formulation of the QCD Dirac operator at chemical potential p ^ O o n the lattice in the staggered scheme is given by 4 Mx,y{U,n)
= —
^2
[UAx)vAx)Sy,x^-^.c]
v=x,y,z
+ Ya [Ui^t(*)e"<W- - ul(y)m(y)e'"sy^} ,
(1)
with the link variables U and the staggered phases 77. We report on computations with gauge group SU(2) on a 6 4 lattice at /? — 4/g2 = 1.3 and with Nf = 2 flavors of staggered fermions of mass m = 0.07. For this system the fermion determinant is real and lattice simulations of the full theory with chemical potential become feasible exhibiting a phase transition at \ic « m„/2 w 0.3 where the chiral condensate (nearly) vanishes and a diquark condensate develops. 5 In the left plot of Fig. 1 we compare the densities of the small eigenvalues at p. = 0 to 0.4 on our 6 4 lattice, averaged over at least 160 configurations. Since the eigenvalues move into the complex plane for p > 0, a band of width e = 0.015 parallel to the imaginary axis is considered to construct p(y), i.e.
354
Figure 1: Left plot: Density p(y) of small eigenvalues for two-color QCD on a 6 4 lattice from n = 0 to 0.4. The loss of quasi-zero modes is accompanied by a vanishing of the chiral condensate. Right plot: Chiral condensate extracted by three different methods (see text).
p(y) = J_ dxp(x,y), where p{x,y) is the density of the complex eigenvalues x + iy. The density p(y) is used to estimate a value for the chiral condensate by naively applying the Banks-Casher relation which originally was derived for real eigenvalues appearing in pairs of opposite sign. We further employed the standard definition of the Green's function 3 by inverting the fermionic matrix with a noisy source and by computing its eigenvalues exactly, respectively, to get the condensate. Thus the chiral condensate for two-color QCD with finite chemical potential was extracted by three methods. The preliminary results for < t/>^> > and its variance are shown in the righthand plot of Fig. 1. 2
Quantum Chaos
The fluctuation properties of the eigenvalues in the bulk of the spectrum have also attracted attention. It was shown in Ref. 6 for Hermitian Dirac operators that on the scale of the mean level spacing they are described by RMT. For example, the nearest-neighbor spacing distribution P(s), i.e. the distribution of spacings s between adjacent eigenvalues on the unfolded scale, agrees with the Wigner surmise of RMT. According to the Bohigas-Giannoni-Schmit conjecture, 7 quantum systems whose classical counterparts are chaotic have a nearest-neighbor spacing distribution given by RMT whereas systems whose classical counterparts are integrable obey a Poisson distribution, Pp(s) = e~s. Therefore, the specific form of P(s) is often taken as a criterion for the presence or absence of "quantum chaos". For p, > 0, the Dirac operator loses its Hermiticity properties so that its eigenvalues become complex. The aim of the present analysis is to investigate
355
/x = 3.0
0.4
/x = 0
Figure 2: Nearest-neighbor spacing distribution for two-color QCD with varying fi. The analytic curves are the Wigner distribution, Pw = 262144/(7297r 3 )s 4 exp(-64/(97r)s 2 ) (left), the Ginibre distribution of Eq. (2) (middle) and the Poisson distribution of Eq. (3) (right).
whether non-Hermitian RMT is able to describe the fluctuation properties of the complex eigenvalues of the QCD Dirac operator. We apply a twodimensional unfolding procedure 8 to separate the average eigenvalue density from the fluctuations and construct the nearest-neighbor spacing distribution, P(s), of adjacent eigenvalues in the complex plane. Adjacent eigenvalues are defined to be the pairs for which the Euclidean distance in the complex plane is smallest. The data are then compared to analytical predictions of the Ginibre ensemble 9 of non-Hermitian RMT, which describes the situation where the real and imaginary parts of the strongly correlated eigenvalues have approximately the same average magnitude. In the Ginibre ensemble, the average spectral density is already constant inside a circle and zero outside. In this case, unfolding is not necessary, and P(s) is given by 1 0 W-l
PG (S) = cp(cs),
p(s) = 2s lim AT-»oo
2\ „ - * " J{en{s2)e
l_n=l
N-l
„2n
n!e s2
h »( )'
(2)
where e„(ar) = 5 3 m = o x m / m ' a n ( ^ c = /o°° dssp(s) = 1.1429.... For uncorrec t e d eigenvalues in the complex plane, the Poisson distribution becomes 10
Pp{s) = \se-™ V4.
(3)
This should not be confused with the Wigner distribution for a Hermitian operator. 6 Our results for P(s) are presented in Fig. 2. As a function of fi, we expect to find a transition from Wigner to Ginibre behavior in P(s). This was clearly seen in color-SU(3) with Nf — 3 flavors and quenched chemical potential, 8
356
where differences between both curves are more pronounced. For the symplectic ensemble of color-SU(2) with staggered fermions, the Wigner and Ginibre distributions are very close to each other and thus harder to distinguish. They are reproduced by our preliminary data for /x = 0 and (J. = 0.4, respectively. Even in the deconfined phase, where the effect of the chemical potential might order the system, the gauge fields retain a considerable degree of randomness, which apparently gives rise to quantum chaos. For fi > 1.0, the lattice results for P(s) deviate substantially from the Ginibre distribution and could be interpreted as Poisson behavior, corresponding to uncorrelated eigenvalues. A plausible explanation of the transition to Poisson behavior is provided by the following two (related) observations. First, for large fi the terms containing e^ in Eq. (1) dominate the Dirac matrix giving rise to uncorrelated eigenvalues. Second, for large \x the fermion density on the finite lattice reaches saturation due to the limited box size and the Pauli exclusion principle. Acknowledgments: This study was supported in part by FWF project P14435-TPH. We thank the ECT, Trento, for hospitality during various stages of this work. We further thank B.A. Berg and T. Wettig for collaborations. 11 1. T. Banks and A. Casher, Nucl. Phys. B 169 (1980) 103. 2. E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. A 560 (1992) 306; J.J.M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70 (1993) 3852. 3. H. Leutwyler and A.V. Smilga, Phys. Rev. D 46 (1992) 5607. 4. P. Hasenfratz and F. Karsch, Phys. Lett. B 125 (1983) 308; J. Kogut, H. Matsuoka, M. Stone, H.W. Wyld, S. Shenker, J. Shigemitsu, and D.K. Sinclair, Nucl. Phys. B 225 (1983) 93; I.M. Barbour, Nucl. Phys. B (Proc. Suppl.) 26 (1992) 22. 5. S. Hands, J.B. Kogut, M.-P. Lombardo, and S.E. Morrison, Nucl. Phys. B 558 (1999) 327. 6. M.A. Halasz and J.J.M. Verbaarschot, Phys. Rev. Lett. 74 (1995) 3920; M.A. Halasz, T. Kalkreuter, and J.J.M. Verbaarschot, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 266. 7. 0 . Bohigas, M.-J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52 (1984) 1. 8. H. Markum, R. Pullirsch, and T. Wettig, Phys. Rev. Lett. 83 (1999) 484. 9. J. Ginibre, J. Math. Phys. 6 (1965) 440. 10. R. Grobe, F. Haake, and H.-J. Sommers, Phys. Rev. Lett. 61 (1988) 1899. 11. B.A. Berg, E. Bittner, H. Markum, R. Pullirsch, M.-P. Lombardo, and T. Wettig, hep-lat/0007008.
357
E V I D E N C E FOR D E C O N F I N E M E N T F R O M T H E J/xp S U P P R E S S I O N P A T T E R N I N P b - P b COLLISIONS B Y T H E NA50 EXPERIMENT M.C. Abreu 6 '°, B. Alessandro 10 , C. Alexa 3 , R. Arnaldi 10 , M. Atayan 12 , C. Baglin 1 , A. Baldit 2 , M. Bedjidian 11 , S. Beole 10 , V. Boldea 3 , P. Bordalo 6 ' 6 , A. Bussiere 1 , L. Capelli 11 , L. Casagrande 6 ' c , J. Castor 2 , T. Chambon 2 , B. Chaurand 9 , I. Chevrot 2 , B. Cheynis 11 , E. Chiavassa 10 , C. Cical6 4 , T. Claudino 6 , M.P. Comets 8 , N. Constans 9 , S. Constantinescu 3 , N. De Marco 10 , A. De Falco 4 , G. Dellacasa 10 ' d , A. Devaux 2 , S. Dita 3 , O. Drapier 11 , L. Ducroux 11 , B. Espagnon 2 , J. Fargeix 2 , P. Force 2 , M. Gallio 10 , Y.K. Gavrilov 7 , C. Gerschel 8 , P. Giubellino 10 , M.B. Golubeva 7 , M. Gonin 9 , A.A. Grigorian 12 , J.Y. Grossiord 11 , F.F. Guber 7 , A. Guichard 11 , H. Gulkanyan 12 , R. Hakobyan 12 , R. Haroutunian 1 1 , M. Idzik 10 ' 6 , D. Jouan 8 , T.L. Karavitcheva 7 , L. Kluberg 9 , A.B. Kurepin 7 , Y. Le Bornec 8 , C. Lourengo 5 , P. Macciotta 4 , M. Mac Cormick 8 , A. Marzari-Cliiesa 10 , M. Masera 10 , A. Masoni 4 , S. Mehrabyan 12 , M. Monteno 10 , A. Musso 10 , P. Petiau 9 , A. Piccotti 10 , J.R. Pizzi 11 , F. Prino 1 0 , G. Puddu 4 , C. Quintans 6 , S. Ramos 6 ' 6 , L. Ramello 10ld , P. Rato Mendes 6 , L. Riccati 10 , A. Romana 9 , I. Ropotar 5 , P. Saturnini 2 , E. Scomparin 10 S. Serci 4 , R. Shahoyan 6 -', S. Silva6, M. Sitta 1 0 ' d , C. Soave 10 , P. Sonderegger 5 ' 6 , X. Tarrago 8 , N.S. Topilskaya 7 , G.L. Usai 4 , E. Vercellin 10 , L. Villatte 8 , N. Willis 8 . 1 LAPP, CNRS-IN2P3, Annecy-le- Vieux, France; 2 LPC, Univ. Blaise Pascal and CNRS-IN2P3, Aubiere, France; 3 IFA, Bucharest, Romania; 4 Universita di Cagliari/INFN, Cagliari, Italy; 5 CERN, Geneva, Switzerland; 6 LIP, Lisbon, Portugal; 7 INR, Moscow, Russia; 8 IPN, Univ. de Paris-Sud and CNRS-IN2P3, Orsay, France; 9 LPNHE, Ecole Polytechnique and CNRS-IN2P3, Palaiseau, France; 10 Universita di Torino/INFN, Torino, Italy; u IPN, Univ. Claude Bernard Lyon-I and CNRS-IN2P3, Villeurbanne, France; 12 YerPhI, Yerevan, Armenia. a) also at UCEH, Universidade de Algarve, Faro, Portugal; b) also at 1ST, Universidade Tecnica de Lisboa, Lisbon, Portugal; c) now at CERN; d) Universitd del Piemonte Orientate, Alessandria and INFN-Torino, Italy; e) now at Faculty of Physics and Nuclear Techniques, University of Mining and Metallurgy, Cracow, Poland; f) on leave of absence of YerPhI, Yerevan, Armenia. The NA50 experiment at CERN has found evidence for deconfinement in central Pb-Pb collisions at 158 A GeV/c at the CERN SPS. The anomalous (with respect to ordinary nuclear absorption) suppression of J/ip has been observed in 1995 and 1996 data and has been confirmed with the 1998 data analysis, where a detailed study of the suppression pattern vs. centrality has been performed. NA50 results exclude present conventional hadronic models, while they find a natural explanation in the formation of a deconfined state of quarks and gluons.
358
1
Charmonium production in p-A and light ion interactions
Charmonium suppression was proposed by Matsui and Satz * as a possible signal of colour deconfinement, the mechanism being Debye screening of the cc potential. Experiments NA38 and NA51 at CERN have studied charmonia in proton-nucleus and light ion induced reactions, and experiment NA50 has extended the study to Pb ion induced reactions. Experiments NA38, NA50 and NA51 all share the same muon spectrometer. In the standard analysis the signal is extracted from the fi+fi~ invariant mass spectrum after subtracting the combinatorial background; a fit to the spectrum provides the yields of J/ip, tp', Drell-Yan and open charm. For ion-induced reactions, in experiments NA38 and NA50, the centrality of the reaction is estimated from transverse energy ET and (only for NA50) from zero-degree energy EZDG o r charged multiplicity NchThe systematics of J/tp production from p-p to S-U reactions can be summarized 2 by the simple law OAB = cr0{AB)a (A and B are the projectile and target mass number, resp.) with the exponent a = 0.918 ± 0.015 representing the amount of ordinary suppression. In order to perform the same study as a function of centrality, the DrellYan process is used as a reference (it scales linearly with AB) and the path of nuclear matter L traversed by the resonance is used instead of the product AB. In figure 1 the results for the ratio Bfl,ia(J/,ip)/cT(DY) in p-p, p-d, p-W, p-U and S-U collisions are reported, together with a simple fit to the expression exp(—(TabsPoL), with (Tabs — 6.5±1.0 mb (the Pb-Pb points will be discussed in the next section). Regarding the tp' a quite different behaviour is seen 2 in the S-U data, which give a much higher absorption cross-section a'abs = 25 ± 5 mb. The current interpretation of this so-called ordinary suppression of charmonia relies first of all on a two step formation mechanism: (i) gluon fusion producing cc in both colour singlet and colour octet states (as required by CDF data) and (ii) colour neutralization of the ccg state which, for positive XF in p-A collisions, happens outside the nucleus. The same nuclear absorption is then expected in p-A for both J/ip and ip', since they traverse the nucleus as a not yet fully formed resonance. On the other hand, the larger volume in S-U collisions means that the ip' is more easily suppressed by nucleons and comoving hadrons. 2
N A 5 0 results: anomalous J/xp suppression
Results on J/ip suppression from the 1996 run of NA50 are reported on fig. 1. The anomalous suppression is clearly visible with an onset at L ~ 8 fm. The results reported are coming from the standard analysis which uses the measured
359 >o
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90 80 70
p(450GeV/c)-p,d(NA51)
* p(200-GeV/c)-A (A=W,U) (NA38J •
32
S(32x200*GeV/c)-U(NA38)
208
60
ell
1
Pb(208xl58 GeV/c)-Pb (NA50)
50
J-l
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S >—5
e> -i
zS.
PQ
20
a„h=5.8±
0.6 mb
* reseated to 15S GeV/c
10 2.5
7.5
10
L(fm) Figure 1: The (J/ip)/Drell
- Yan ratio vs. L from pp to Pb-Pb.
Drell-Yan events as a reference. To overcome the statistical limitation coming from the Drell-Yan sample, a new method based on the Minimum Bias trigger has been developed 3 . In fig. 2 results of the new analysis on 1996 and 1998 Pb-Pb data are shown as a function of transverse energy. These results (open symbols) agree with the previous analysis (full circles) and confirm the drop of the J/ip yield in the region 30 < ET < 55 GeV, which is consistent with a sharp threshold due to our resolution of about 10%. Note that 1996 data above ET = 90 GeV have been excluded for reinteraction problems in the 12 mm Pb target. The J/I/J yield exhibits a further decrease at the high ET end, showing no sign of saturation. Hadronic absorption models, based on final state interactions with co-
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Pb-Pb 1996 • -m Pb-Pb 1996 with Minimum Bias O
zm
Pb-Pb 1998 with Minimum Bias
10 5 -
0
20
40
60
80
100
120
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E T (GeV) Figure 2: The (J/ip)/Drell
— Yan ratio vs. ET in Pb-Pb collisions.
movers in addition to ordinary absorption by nucleons, are in general unable to reproduce the observed J/ip suppression pattern 4 . NA50 results exclude such conventional hadronic models, while they find a natural explanation in the formation of a deconfined state of quarks and gluons in central Pb-Pb collisions at the CERN SPS. References 1. 2. 3. 4.
T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986). NA38 Collaboration, M.C. Abreu et al., PLB 466, 408 (1999). NA50 Collaboration, M.C. Abreu et al., Phys. Lett. B 450, 456 (1999). NA50 Collaboration, M.C. Abreu et al., Phys. Lett. B 477, 28 (2000).
P O S T E R SESSION CONTRIBUTIONS
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M O N O P O L E C O N D E N S A T I O N A N D C O N F I N E M E N T IN N O N A B E L I A N LATTICE G A U G E THEORY P. CEA Dipartimento di Fisica and INFN - Sezione di Bari, via Amendola 173, 70126 Bari, Italy E-mail: [email protected] L. COSMAI INFN - Sezione di Bari, via Amendola 173, 70126 Bari, Italy E-mail: [email protected] We give a very short account of our recent investigations * on Abelian monopole condensation in SU(2) and SU(3) lattice gauge theories. Summary To study the vacuum structure of the lattice gauge theories we introduced a gauge invariant effective action, defined by using the lattice Schrodinger functional. The functional integration is extended over links on a lattice with the hypertorus geometry and satisfying a given constraint corresponding to an external background field. To detect monopole condensation at finite temperature we introduced a disorder parameter /x defined in terms of the thermal partition functional. Our thermal partition functional is constructed by means of the lattice Schrodinger functional and is invariant against gauge transformations of the background field. Therefore we do not need to do any gauge fixing to perform the Abelian projection. Our numerical results suggest that the disorder parameter \i ^ 0 in the confined phase and /z —> 0 when approaching the critical coupling in the thermodynamic limit. Moreover our results are consistent with a second order deconfining phase transition in the case of the SU(2) gauge theory and with a first order deconfining phase transition in the case of SU(3). Numerical results also suggest that the SU(3) vacuum reacts moderately stronger in the case of the color hypercharge Abelian monopole. This last result could be useful in the theoretical efforts to understand the pattern of symmetry breaking in the deconfined phase of QCD. References 1. P. Cea and L. Cosmai, hep-lat/006007 to be published on Phys. Rev. DI (1 November 2000), and references therein.
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M A S S OF T H E B O T T O M Q U A R K Wolfgang LUCHA Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18, A-1050 Vienna, Austria E-mail: wolf gang. lucha@oeaw. ac. at Franz F. SCHOBERL Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria E-mail: franz.schoeberWunivie.ac.at If heavy quarkonia may be described by nonrelativistic kinematics and only the perturbative contribution to the quark-antiquark interaction potential (with all nonperturbative effects summarized by Leutwyler-Voloshin-like corrections to the energy) the Hamiltonian describing a system of heavy quark and antiquark, both with constituent mass m, forming a bound state with spin s = 0,1, reads 1
F = 2m-iA-JLA» + ^(r) m 4 m6 +
n
„F ,
s(s+l)
+
^ (m^lrA
+
J ( 3 ) (x) + higher orders ,
^ ^4mr ( C2 t - 2 C 7 A ) r = Ixl ,
where V^ is the perturbative static quark-antiquark interaction potential which is known up to two loops, a(/x) is the strong fine-structure constant (in the MS scheme) at the renormalization scale /x, and CF (CA) are the quadratic Casimirs for the fundamental (adjoint) representation of the gauge group. The (discrete) spectrum of the Hamiltonian H = 2 m — (1 /m) A+VQ (r) may be found exactly by a numerical procedure 2 developed for solving the nonrelativistic Schrodinger equation; the remainder of H has to be treated perturbatively. Comparing the resulting bound-state masses to the observed hadron spectra allows to extract the quark masses. Fitting the ground-state energy, for a QCD scale parameter A(nf = 4) = 0.231§;SI GeV and /x = V6.632 GeV, to the experimental T mass, we find for the b quark pole mass 3 rrn, = 4.983 GeV, with some 100 MeV error. References 1. A. Pineda and F. J. Yndurain, Phys. Rev. D 58, 094022 (1998); 6 1 , 077505 (2000). 2. W. Lucha and F. F. Schoberl, Int. J. Mod. Phys. C 10, 607 (1999). 3. W. Lucha and F. F. Schoberl, Phys. Rev. D 62, 097501 (2000).
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R A N D O M M A T R I X THEORY A N D CHIRAL C O N D E N S A T E FOR QED B.A. BERG Department of Physics, The Florida State University, Tallahassee, FL 32306 H. MARKUM, R. PULLIRSCH Institut fur Kernphysik, Technische Universitat Wien, A-1040 Vienna, Austria T. WETTIG Department of Physics, Yale University, New Haven, CT 06520-8120 and RIKEN BNL Research Center, Upton, NY 11973-5000 We compare the low-lying spectrum of the staggered Dirac operator in the confining phase of compact U(l) gauge theory on the lattice to predictions of chiral random matrix theory. Agreement with the chiral unitary ensemble is observed. The small eigenvalues contribute to chiral symmetry breaking similar as for the SU(2) and SU(3) gauge groups. A value for the chiral condensate of (ijnf>) = 0.352(3) is extracted [hep-lat/0007009].
1 1 1 1 I
1 1 1 1 I
1 1 1 1 I
10
15 z
1 1 1 1
0.004
Figure 1: Microscopic spectral density (left) and distribution of the smallest eigenvalue (right) of the Dirac operator on a lattice of size 8 3 x 6. The histograms represent the lattice data and the solid lines are the R M T predictions.
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S T R O N G DECAYS I N T H E 'T H O O F T MODEL Yulia Kalashnikova and Alexei Nefediev Institute of Theoretical and Experimental Physics, Moscow 117218, Russia Abstract The strong decay amplitudes are studied in the 't Hooft model for the two-dimensional QCD. Special attention is payed to the pions in the final state, and it is demonstrated that any amplitude of the form A —> n + C vanishes identically for any pion momentum. We study the strong decays with pions in the final state in the 't Hooft model for the QCD2 in the limit of infinite number of colours l in the axial gauge A\ = 0. We develop the Hamiltonian approach to the model which allows to define the dressed quark fields 2 and to introduce the non-local operators creating and annihilating bound qq states — mesons 3 . The resulting bound state equation 2 gives two wave functions for each meson which describe forward and backward in time motion of the qq pair inside meson. In the chiral limit there exists a massless solution - chiral pion, and its form can be found explicitly 3 ' 4 . We discuss the two possible ways to consider the hadronic process A —> B + C. The first way is based on the direct calculation of the matrix elements (BC\H\A) for the Hamiltonian of the model. Another one is based on the matrix approach 2 ' 4 and allows to arrive at the same results but in a more elegant way. One can use an effective diagrammatic techniques involving dressed meson-quark-antiquark vertex, quark and mesonic propagators and the full four-quark scattering amplitude (see 5 for similar results in the light-cone gauge). As a result the pionic vertex can be represented as a superposition Tn(p,P) = —iPpVufaP) — iP^a^ipfP), that immediately leads to the conclusion that once the vector and axial-vector currents v^ and o^ are conserved in the weak sense in the chiral limit, then any amplitude of the type M(A -¥ n + C) vanishes identically for any pion momentum. The latter statement is the Adler selfconsistency condition 6 for the 't Hooft model. 1. G.'t Hooft, Nucl.Phys. B75, 461 (1974); 2. I.Bars and M.B.Green, Phys.Rev. D17, 537 (1978); 3. Yu.S.Kalashnikova, A.V.Nefediev and A.V.Volodin, hep-ph/9908226, Phys. Atom.Nucl. in press; 4. Yu.S.Kalashnikova, A.V.Nefediev, Phys.Lett. B487, 371 (2000); 5. M.B.Einhorn, Phys.Rev. D14, 3451 (1976); 6. S.L.Adler, Phys. Rev. 139B, 1638 (1965);
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LIST OF PARTICIPANTS Mohammad Reza ALAGHEBAND (Sharif Univ. of Technology, Iran) David ALBA (INFN, Florence Section, Italy) Constantia ALEXANDROU (Univ. of Cyprus, Cyprus) Andrei ALEXANDRU (Lousiana State Univ., U.S.A.) Federico ANTINORI (CERN, Switzerland) Keith Oliver BAKER (NuHEP, Hampton Univ. and Jefferson Lab, U.S.A.) Marshall BAKER (Seattle, U.S.A.) Gunnar BALI (Glasgow Univ., Scotland) Thabit BARAK AT (Near East Univ., Turkey) Juraj BOHACIK (Bratislava, Slovakia) Nicolai BONDARENKO (Kharkov National Univ., Ukraine) Oleg BORISENKO (Academy of Sciences of Ukraine, Ukraine) Nora BRAMBILLA (Heidelberg, Germany) Martina BRISUDOVA (Univ. of Florida, U.S.A.) Giuseppe BURGIO (Humboldt Univ. Berlin, Germany) Paolo CEA (Univ. of Bari, Italy) Mohamed CHABAB (Cadi Ayyad Univ., Morocco) Gilberto COLANGELO (Zurich, Switzerland) Sara COLLINS (Glasgow Univ., U.K.) John CONNELL (Springfield Technical Community College, U.S.A.) Leonardo COSMAI (INFN Bari, Italy) Stephen COTANCH (Univ. of North Carolina, U.S.A.) Poul Henrik DAMGAARD (Niels Bohr Institute, Denmark) Thomas DEVLIN (Rutgers Univ., U.S.A.) Francesco DI RENZO (Parma Univ., Italy) Michael ENGELHARDT (Univ. Tubingen, Germany)
368
Manfried FABER (Vienna Univ. of Technology, Austria) Lewis FULCHER (Bowling Green State Univ., U.S.A.) Sadataka FURUI (Teikyo Univ., Japan) Howard GEORGI (Harvard Univ., U.S.A.) Leonid GLOZMAN (Univ. Graz, Austria) Anthony GREEN (Univ. of Helsinki, Finland) Dieter GROMES (Univ. Heidelberg, Germany) Alistair HART (Univ. of Edinburgh, Scotland) Thomas HEINZL (Univ. Jena, Germany) Christina HELMINEN (Univ. of Helsinki, Finland) Dubravko KLABUCAR (Zagreb, Croatia) Valentin KUZMICHEV (Bogolyubov Inst, for Theoretical Physics, Ukraine) Kurt LANGFELD (Univ. Tubingen, Germany) Xue-Qian LI (Nankai Univ., China) Zoltan LIGETI (Fermilab, U.S.A.) Norbert LIGTERINK (ECT*, Italy) Wolfgang LUCHA (Austrian Academy of Sciences, Austria) Giuseppe MAIELLA (Naples, INFN, Italy) Samirnath MALLIK (Saha Institute of Nuclear Physics, India) Pieter MARIS (Kent State Univ., U.S.A.) Harald MARKUM (Vienna Univ. of Technology, Austria) Khin MAUNG MAUNG (Hampton Univ., U.S.A.) Dmitri MELIKHOV (Univ. Heidelberg, Germany) Chris MICHAEL (Liverpool, England) Alexei NEFEDIEV (ITEP Moscow, Russia) Alexander NESTERENKO (Moscow State Univ., Russia) Herbert NEUBERGER (Rutgers Univ., U.S.A.)
369 Stefan OLEJNIK (Slovak Academy of Sciences, Slovakia) Martin OLSSON (Univ. of Wisconsin, U.S.A.) Peter ORLAND (City Univ. of New York, U.S.A.) Haralambos PANAGOPOULOS (Univ. of Cyprus, Cyprus) Alessandro PAPA (Univ. della Calabria, INFN Cosenza, Italy) Giovanni M. PROSPERI (Univ. of Milano, Italy) Luciano RAMELLO (Univ. Piemonte Orientale, Italy) Carlos RAMIREZ (Univ. I. de Santander, Colombia) Hugo REINHARDT (Univ. Tubingen, Germany) Jose E. F. RIBEIRO (Lisbon, Portugal) Craig D. ROBERTS (Argonne National Lab, U.S.A.) Hagop SAZDJIAN (Univ. Paris XI, France) Martin SCHADEN (New York Univ., U.S.A.) Franz F. SCHOBERL (Univ. Wien, Austria) Roberto SCIMIA (Univ. of Perugia, INFN Perugia, Italy) Sergei SHABANOV (Univ. of Florida, U.S.A.) Hugh Ph. SHANAHAN (Tsukuba, Japan) Mikhail SHIFMAN (Minnesota, U.S.A.) Lorenz von SMEKAL (Univ. Erlangen, Germany) Peter C. TANDY (Kent State Univ., U.S.A.) Massimo TESTA (Univ. of Rome "La sapienza", Italy) Antonio VAIRO (Univ. Heidelberg, Germany) Robert Ferdinand WAGENBRUNN (Univ. di Pavia, Italy) Katja WAIDELICH (North Carolina State Univ., U.S.A.) Axel WEBER (Univ. Michoacana de San Nicholas de Hidalgo, Mexico) Anthony WILLIAMS (Univ. of Adelaide, Australia) Valentine I. ZAKHAROV (MPI Munich, Germany)
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