Heavy quarks, gluons and the confinement potential in Coulomb gauge
aa r X i v : . [ h e p - ph ] N ov Heavy quarks, gluons and the confinement potential inCoulomb gauge
Carina Popovici, Peter Watson and Hugo Reinhardt
Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany
Abstract.
We consider the heavy quark limit of Coulomb gauge QCD, with the truncation of the Yang-Mills sector to includeonly (dressed) two-point functions. We find that the rainbow-ladder approximation to the gap and Bethe-Salpeter equations isnonperturbatively exact and moreover, we provide a direct connection between the temporal gluon propagator and the quarkconfinement potential. Further, we show that only bound states of color singlet quark-antiquark (meson) and quark-quark(SU(2) baryon) pairs are physically allowed.
Keywords:
Coulomb gauge, heavy quarks, Bethe-Salpeter equation
PACS:
Coulomb gauge is an ideal choice for investigating theconfinement phenomenon. In this gauge, there is an ap-pealing picture of confinement: the Gribov-Zwanzigerscenario [1], whereby the temporal component of thegluon propagator provides a long-range confining forcewhereas the transverse spatial components are infraredsuppressed (and therefore do not appear as asymptoticstates). Confinement implies the existence of a nonper-turbative scale (the string tension), and this naturallyleads to the question: is there a simple connection be-tween the (gauge fixed) Green’s functions of Yang-Millstheory and this physical scale? To answer this ques-tion, we propose to study here the full (nonperturbative)QCD Bethe-Salpeter equation in Coulomb gauge withina heavy quark mass expansion at leading order. In thistruncated system, we will use the results inspired by thelattice for the explicit Green’s functions of the Yang–Mills sector and in addition we will employ the Slavnov-Taylor identity for the quark-gluon vertex.We begin by considering the explicit quark contribu-tion to the full QCD generating functional (unless oth-erwise specified, the Dirac and color indices in the fun-damental representation are implicit and we follow theconventions from [2]): Z [ c , c ] = Z D F × exp (cid:26) ı Z d xq ( x ) h ı g D + ı ~ g · ~ D − m i q ( x )+ ı Z d x [ c ( x ) q ( x ) + q ( x ) c ( x )] + ı S YM (cid:27) . (1) S YM represents the Yang-Mills part of the action andthe temporal and spatial components of the covariantderivative (in the fundamental color representation) are given by D = ¶ − ıgT a s a ( x ) , ~ D = ~ (cid:209) + ıgT a ~ A a ( x ) , (2)where ~ A and s refer to the spatial and temporal compo-nents of the gluon field, respectively.The full quark field q ( ¯ q is the conjugate field, and c , c are the corresponding sources) is decomposed accordingto the heavy quark transformation: q ( x ) = e − ımx [ h ( x ) + H ( x )] , h ( x ) = e ımx + g q ( x ) , H ( x ) = e ımx − g q ( x ) (3)(similarly for the antiquark field). We now insert this de-composition into the generating functional Eq. (1), in-tegrate out the H -fields, and make an expansion in theheavy quark mass, similar to Heavy Quark Effective The-ory [HQET] [3]. At leading order, we get the followingexpression: Z [ c , c ] = Z D F exp (cid:26) ı Z d xh ( x ) [ ı ¶ x + gT a s a ] h ( x )+ ı Z d x (cid:2) e − ımx c ( x ) h ( x ) + e ımx h ( x ) c ( x ) (cid:3) + ı S YM (cid:27) + O ( / m ) . (4)In the above, we have retained the full quark and anti-quark sources (unlike HQET). This means that we canuse the full gap and Bethe-Salpeter equations of QCDreplacing, however, the kernels, propagators and verticesby their leading order expression in the mass expansion.Also, notice that the spin degrees of freedom have de-coupled from the system.n full Coulomb gauge QCD, the quark gap equationis given by : G qq ( k ) = G ( ) qq ( k ) + ( p ) Z d w × n G ( ) aqq s W qq ( w ) G bqq s ( w , − k , k − w ) W ab ss ( k − w )+ G ( ) aqqAi W qq ( w ) G bqqA j ( w , − k , k − w ) W abAAi j ( k − w ) o (5)( G ’s denote the various proper functions, W denotespropagators, see [4]). The gap equation is supplementedby the Coulomb gauge Slavnov-Taylor identity [2]: k G dqq s ( k , k , k ) = ı k i ~ k G aqqAi ( k , k , k ) G adcc ( − k )+ G qq ( k ) h ˜ G dq ; ccq ( k + q , k − q ; k ) + ıgT d i + h ˜ G dq ; ccq ( k + q , k − q ; k ) − ıgT d i G qq ( − k ) (6)where k + k + k = q is an arbitrary energy injec-tion scale (arising from the noncovariance of Coulombgauge), G cc is the ghost proper two-point function, ˜ G q ; ccq and ˜ G q ; ccq are ghost-quark kernels associated with theGauss-BRST transform (see also [5]).Now, as a consequence of the (Coulomb gauge) de-composition, Eq. (3) and under the assumption thatthe pure Yang-Mills vertices may be neglected (re-taining, however, the dressed gluon propagator), theDyson-Schwinger equation for the nonperturbative spa-tial quark-gluon vertex furnishes the result that G ¯ qqA ∼ O ( / m ) (see [2] for a complete discussion and justi-fication of this truncation). Similarly, the ghost-quarkkernels can be neglected. Thus, under our truncationscheme, the Slavnov-Taylor identity reduces to k G dqq s ( k , k , k ) = ıg h G qq ( k ) T d − T d G qq ( − k ) i + O ( / m ) . (7)This is then inserted into Eq. (5), together with the tree-level quark proper two-point function G ( ) qq ( k ) = ı [ k − m ] + O ( / m ) (8)and the tree level quark gluon vertex G ( ) aqq s ( k , k , k ) = gT a + O ( / m ) (9)that follow from the generating functional Eq. (4). Thegeneral form of the nonperturbative temporal gluon prop-agator is given by [6]: W ab ss ( ~ k ) = d ab ı ~ k D ss ( ~ k ) . (10) This expression is in the second order formalism and can be directlyinferred from the result obtained within the first order formalism [4].
Lattice results [7] motivate that the dressing function D ss is largely independent of energy and likely to be-have as 1 /~ k for vanishing ~ k (the explicit form of D ss will only be needed in the last step of the calculation).Putting all this together, we find the following solutionto Eq. (5) for the heavy quark propagator: W qq ( k ) = − ı [ k − m − I r + ı e ] + O ( / m ) , (11)where the (implicitly regularized, denoted by “ r ”) con-stant is given by I r = ( p ) g C F Z r d ~ w D ss ( ~ w ) ~ w + O ( / m ) , (12)with the Casimir factor C F = ( N c − ) / N c . When solv-ing Eq. (5), the ordering of the integration is set such thatthe temporal integral is performed first, under the con-dition that the spatial integral is regularized and finite.Inserting the solution Eq. (11) into the Slavnov-Tayloridentity, we find that the temporal quark-gluon vertex re-mains nonperturbatively bare: G aqq s ( k , k , k ) = gT a + O ( / m ) . (13)Note that the propagator Eq. (11) has a single pole inthe complex k -plane (due to the mass expansion) andtherefore it is necessary to explicitly define the Feynmanprescription. From Eq. (11) it then follows that the closedquark loops (virtual quark-antiquark pairs) vanish due tothe energy integration, which implies that the theory isquenched in the heavy mass limit: Z dk [ k − m − I r + ı e ] [ k + p − m − I r + ı e ] = . (14)We also emphasize that the position of the pole has nophysical meaning since the quark can never be on-shell.The poles in the quark propagator are situated at infinity(thanks to I r as the regularization is removed) meaningthat either one requires infinite energy to create a quarkfrom the vacuum or, if a hadronic system is considered,only the relative energy (derived from the Bethe-Salpeterequation) is important.The solution for the antiquark propagator reads: W qq ( k ) = − ı [ k + m − I r + ı e ] + O ( / m ) . (15)Notice the assignment of the Feynman prescription, sim-ilar to Eq. (11) – see also the discussion in [2]. This hasthe important consequence that the Bethe-Salpeter equa-tion for the quark-antiquark states has a physical inter-pretation of a bound state equation (see below). The cor-responding vertex reads: G aqq s ( k , k , k ) = − gT a + O ( / m ) . (16)et us now consider the full homogeneous Bethe-Salpeter equation for quark-antiquark bound states: G ( p ; P ) ab = − ( p ) Z dkK ab ; dg ( p , k ; P ) × [ W qq ( k + ) G ( k ; P ) W qq ( k − )] gd (17)where k ± (similarly for p ± ) are the momenta of thequarks (with the notations from [2]), P is the 4-momentum of the bound state (assuming that a solu-tion exists), K represents the Bethe-Salpeter kernel and G is the Bethe-Salpeter vertex function for the particu-lar bound state under consideration. Further, we explic-itly identify the antiquark contribution, i.e. W qq ( k − ) = − W Tqq ( − k − ) (also in the kernel).When constructing the Bethe-Salpeter kernel, weuse the fact that the temporal integration performedover multiple quark propagators with the same relativesign for the Feynman prescription vanishes (similar toEq. (14), but in this case the terms originate from inter-nal quark or antiquark propagators) and hence, the kernelreduces to the ladder truncation [2]: K ab ; dg ( p , k ) = G a ¯ qq sag W ab ss ( ~ k ) G Tbq ¯ q sbd . (18)We now insert the nonperturbative results for the prop-agators and vertices, Eqs. (11,13, 15,16) and take theform, Eq. (10), for the temporal gluon propagator. Fur-ther, we notice that that the Bethe-Salpeter equation is in-dependent of the relative quark energy and hence we canperform the temporal integration over the quark propaga-tors. The energy integration over the quark and antiquarkpropagators now leads to (unlike Eq. (14)): Z ¥ − ¥ dk (cid:2) k + − m − I r + ı e (cid:3) (cid:2) k − − m + I r − ı e (cid:3) = − p ı I r . (19)By using the expression Eq. (12) for I r , and Fouriertransforming to coordinate space, we find the followingsimple energy solution for the pole condition of theBethe-Salpeter equation Eq. (17): P = g ( p ) Z r d ~ w D ss ( ~ w ) ~ w n C F − e ı ~ w · ~ y C M o + O ( / m ) . (20)where C M arises from the color structure and is yet to beidentified ( G is not assumed to be a color singlet): [ T a G ( ~ y ) T a ] ab = C M G ab ( ~ y ) . (21)Because the total color charge of the system is con-served and vanishing [8], a single quark (or antiquark)cannot be prepared in isolation. Thus, the bound stateenergy P can only be either confining for large sepa-rations, i.e. increase linearly with the separation betweenthe quark and antiquark, or be infinite when the hypothet-ical regularization is removed (so that the system cannot be physically created). If the temporal gluon propagatordressing function is more infrared divergent than 1 / | ~ w | ,then C F = C M (22)is required such that the spatial integral is convergent.This gives the condition G ag ( ~ y ) = d ag G ( ~ y ) , (23)which means that the quark-antiquark Bethe-Salpeterequation can only have a finite solution for color singlet states and otherwise the energy of the system is diver-gent. Assuming that in the infrared D ss = X /~ w (as in-dicated by the lattice) where X is some combination ofconstants, then from Eq. (20) with the condition Eq. (22)we find P ≡ s | ~ y | = g C F X p | ~ y | + O ( / m ) . (24)The above result is that there exists a direct connectionbetween the string tension s and the nonperturbativeYang–Mills sector of QCD at least under the truncationscheme considered here.A similar calculation performed for the diquark Bethe-Salpeter equation shows that the diquarks are confinedfor N c = SU ( ) baryon,and otherwise there are no (finite) physical states. ACKNOWLEDGMENTS
C.P. has been supported by the Deutscher Akademis-cher Austausch Dienst (DAAD). P.W. and H.R. havebeen supported by the Deutsche Forschungsgemein-schaft (DFG) under contracts no. DFG-Re856/6-2,3. C.P.thanks the organizers, in particular F. Llanes-Estrada, forthe support.
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