Hamiltonian approach to Coulomb gauge Yang-Mills Theory
aa r X i v : . [ h e p - t h ] O c t Hamiltonian approachto Coulomb gauge Yang-Mills Theory
H. Reinhardt ∗ † Institut für Theoretische Physik, Auf der Morgenstelle 14D-72076 Tübingen, GermanyE-mail: [email protected]
W. Schleifenbaum
Institut für Theoretische Physik, Auf der Morgenstelle 14D-72076 Tübingen, Germany
D. Epple
Institut für Theoretische Physik, Auf der Morgenstelle 14D-72076 Tübingen, Germany
C. Feuchter
Institut für Theoretische Physik, Auf der Morgenstelle 14D-72076 Tübingen, Germany
The vacuum wave functional of Coulomb gauge Yang-Mills theory is determined within the vari-ational principle and used to calculate various Green functions and observables. The results showthat heavy quarks are confined by a linearly rising potential and gluons cannot propagate overlarge distances. The ’t Hooft loop shows a perimeter law and thus also indicates confinement.
The XXV International Symposium on Lattice Field TheoryJuly 30-4 August 2007Regensburg, Germany ∗ Speaker. † This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under contract no. Re856/6-1,2. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ang-Mills Theory in Coulomb gauge
H. Reinhardt
1. Introduction
The confinement puzzle has been with us ever since the birth of quantum chromodynamics(QCD). By means of lattice calculations, it has been possible to penetrate the infrared nonpertur-bative sector of QCD and recover a confining potential between (static) quarks [1]. At present,however, available lattice sizes do not suffice to describe the Green functions in the deep infrared[2]. The continuum approach, on the other hand, has the intriguing feature that the infrared limitcan be studied asymptotically. In the last decade, a new understanding of infrared QCD has arisenfrom studying continuum Yang-Mills (YM) theory via Dyson-Schwinger equations. The Landaugauge has the advantage of being covariant and therefore encouraged many to intensive investiga-tion of the infrared properties of YM theory [3, 4]. In Coulomb gauge, non-covariance brings aboutsevere technical difficulties which are only recently on the verge of being overcome [5]. Neverthe-less, the Coulomb gauge might be the more efficient choice to identify the nonabelian degrees offreedom. It is well-known that screening and anti-screening contributions to the interquark poten-tial are neatly separated in Coulomb gauge perturbation theory [6]. As for the infrared domain, theGribov-Zwanziger scenario serves as a transparent confinement mechanism [7, 8].A further advantage of working in the physical Coulomb gauge is that one may pass overto a Hamiltonian description. This opens up direct access to the heavy quark potential via theexpectation value of the Hamiltonian. In recent years, variational methods have been pursued tosolve the Yang-Mills Schrödinger equation with a Gaussian type of wave functional [9, 10, 11,12]. Despite Feynman’s critique [13], it turns out that the wave functional is sensitive to infraredmodes and the variational method a powerful tool, at least for the qualitative description of YMtheory. With careful treatment of the operator ordering in Coulomb gauge [14], it is possible tofind a strictly linearly rising heavy quark potential. We report on the latest results found in theHamiltonian approach to YM theory in Coulomb gauge. This includes the full calculation of gluonand ghost Green functions and a running coupling. Furthermore, the ’t Hooft loop, an (dis-)orderparameter for confinement, will be calculated using the results of the Green functions.This paper is organized as follows. In section 2, the Yang-Mills Hamiltonian in Coulombgauge and the equations of motion are introduced. The latter will be solved variationally and theheavy quark potential and the running coupling are presented in section 3. The ’t Hooft loop isdiscussed in section 4 and conclusions are given in section 5.
2. Yang-Mills Schrödinger equation and Dyson-Schwinger equations
In the canonical quantization approach, we choose A a ( x ) = A ai ( x ) and the conjugate momentum P ai ( x ) to arriveat the Weyl gauge Hamiltonian. Since A originally serves as the Lagrange parameter of the Gausslaw, the choice of Weyl gauge requires a restriction on the Hilbert space,ˆ D P | Y i = g r m | Y i (2.1)where g is the gauge coupling, r am ( x ) the density of external color charges, and ˆ D abi = ¶ i d ab + g ˆ A abi with ˆ A abi = A ci f acb . Fixing the residual time-independent local gauge invariance by the Coulomb2 ang-Mills Theory in Coulomb gauge H. Reinhardt gauge, ¶ i A i =
0, and eliminating the longitudinal part of the momentum operator P by means ofthe Gauss law (2.1), one arrives at the Hamiltonian that depends only on transversal fields, H = Z (cid:0) J − P J P + B + g J − r F J r (cid:1) . (2.2)The appearance of the Faddeev-Popov determinant J [ A ] = det ( − D [ A ] ¶ ) is due to a non-trivialchange of coordinates to the transverse fields and turns out to be crucial to the infrared propertiesof the theory. The latter term in Eq. (2.2) describes the Coulomb interaction of dynamical andexternal charges r = − ˆ A P + r m via F ab ( x , y ) = h x , a | ( − D ¶ ) − ( − ¶ ) ( − D ¶ ) − | y , b i (2.3)and reduces to the familiar Coulomb law in the abelian theory.With the Hamiltonian (2.2) at hand, we may apply the variational principle to find the wavefunctional Y [ A ] = h A | Y i . Inspired by QED, we choose [10] Y [ A ] = N p J Z D A exp (cid:18) − Z A w A (cid:19) (2.4)with a normalization constant N . The factor of J − / is chosen to alleviate the computation ofexpectation values, similar to defining radial states in quantum mechanics. A different power of J in the wave functional does not change the properties of the solution [15]. One may think ofthe variational parameter w as in the inverse of the gluon propagator, D abi j ( x , y ) = h Y | A ai ( x ) A bj ( y ) | Y i = d ab t i j ( x ) w − ( x , y ) (2.5)with t i j being the transverse projector. It is determined by solving the functional Schrödingerequation, i.e. minimizing the energy h Y | H | Y i . This gives rise to a non-linear integral equation in w which we refer to as the gap equation. It was derived to two-loop order in the energy in Ref. [10]and reads in momentum space ( k = | k | ) w ( k ) = k + c ( k ) + I w ( k ) + I w . (2.6)Here, c ( k ) abbreviates the so-called curvature and it is related by c ( k ) = N c Z d q ( p ) (cid:0) − ( ˆ k · ˆ q ) (cid:1) d ( | k − q | ) d ( q )( k − q ) (2.7)to the ghost propagator h Y | ( − D ¶ ) − | Y i = g d ( k ) k . (2.8)A Dyson-Schwinger equation for the ghost form factor d may be derived from the path integral, oralternatively from the following operator identity for G [ A ] = ( − D ¶ ) − , G [ A ] = (cid:0) − ¶ (cid:1) − + (cid:0) − ¶ (cid:1) − g ˆ A ¶ G [ A ] (2.9)which yields d − ( k ) = g − − N c Z d q ( p ) (cid:0) − ( ˆ k · ˆ q ) (cid:1) d ( | k − q | )( k − q ) w ( q ) . (2.10)3 ang-Mills Theory in Coulomb gauge H. Reinhardt
Both in the ghost Dyson-Schwinger equation (2.10) and in the equation for the curvature (2.7), wehave approximated the proper ghost-gluon vertex by its tree-level counterpart G i . This amounts tothe factorization h A i G [ A ] i = D i j (cid:10) G [ A ] G j G [ A ] (cid:11) ≈ D i j h G [ A ] i G j h G [ A ] i . (2.11)The non-renormalization of the ghost-gluon vertex in gauges where the gluon propagator is trans-verse, such as the Coulomb and the Landau gauge [16, 17], suggests that the above approximationis a good one. Dyson-Schwinger studies in both four and three-dimensional Landau gauge as wellas lattice calculations in four-dimensional Landau gauge confirmed that the dressed vertex is closeto tree-level [18]. The case of three-dimensional Landau gauge resembles the Coulomb gauge andtherefore we adopt the approximation (2.11). This vertex’ non-renormalization will have crucialimpact on the IR sector of the solutions.The other momentum dependent term I w ( k ) in the gap equation (2.6) reads I w ( k ) = N C Z d q ( p ) (cid:0) + ( ˆ k · ˆ q ) (cid:1) d ( k − q ) f ( k − q )( k − q ) [ w ( q ) − c ( q ) + c ( k )] − w ( k ) w ( q ) (2.12)and is due to the Coulomb interaction part of the Hamiltonian. Here, the form factor f measuresthe deviation from the factorization of the Coulomb potential, (cid:10) G [ A ]( − ¶ ) G [ A ] (cid:11) = h G [ A ] i ( − ¶ ) f h G [ A ] i . (2.13)In the infrared, we set f ( k ) =
1, factorizing the expectation value for the Coulomb propagator(2.13) equivalently to the one for the ghost-gluon vertex in Eq. (2.11). In the ultraviolet, f ( k ) istreated perturbatively, see [10].In order to fix the Coulomb gauge uniquely, configuration space must be restricted to thecompact fundamental modular region. As suggested in [8], this entails the “horizon condition” forthe ghost form factor, d − ( ) = . (2.14)As we shall see, the horizon condition (2.14) has the consequence that all form factors d , c and w diverge in the infrared.
3. Green functions, heavy quark potential and running coupling
The ultraviolet divergences encountered in the gap equation (2.6) are removed by subtractingthe equations at an arbitrary renormalization scale m . Alternatively, one can eliminate the diver-gences by adding appropriate counter terms to the YM Hamiltonian and to ln J [19]. This elim-inates the UV-divergent constant I w from Eq. (2.6) and involves some renormalization constants,one of them can be chosen as c = lim k → ( w ( k ) − c ( k )) and fixed by the requirement of minimalenergy to be c =
0. For details, see Ref. [24].The solutions for the form factors w ( k ) , d ( k ) and c ( k ) can be seen in Fig. 1. In the asymptoticinfrared, the gluon form factor w ( k ) approaches the curvature c ( k ) , reflecting the dominance ofghost degrees of freedom, cf. Landau gauge [4]. With c ( k ) being infrared enhanced, the ghostcontent of the solution makes propagation of gluons over asymptotically large distances impossible,hence gluons are confined. 4 ang-Mills Theory in Coulomb gauge H. Reinhardt s ] w (k) [ s ]| c (k)| [ s ] 1 10 100 1000 10000 0.001 0.01 0.1 1 10 100 1000k [ s ] d(k) Figure 1:
Left: the gluon form factor w ( k ) and the curvature | c ( k ) | . Right: the ghost form factor d ( k ) . The correlation of the asymptotic infrared power laws is due to the non-renormalization of theghost-gluon vertex, w ( k ) = c ( k ) ∼ d ( k ) ∼ k , f ( k ) = , k → . (3.1)Without imposing the horizon condition (2.14), solutions to d , c and w can be found that approachfinite values in the infrared [19] so that the energy functional is dominated by the ultraviolet modes,as speculated by Feynman [13]. Conversely, the infrared power law solution (3.1) where the horizoncondition is satisfied are not subdominant to ultraviolet modes and turn out independently of thedetails of the wave functional. Even a stochastic vacuum, Y [ A ] =
1, would produce the same resultsfor the infrared [15]. One may therefore be confident using the variational principle.The infrared enhancement of the form factors w ( k ) and d ( k ) is qualitatively reproduced byrecent lattice calculations [20].Equipped with the ghost form factor d ( k ) , the heavy quark potential can be found by choosing r am ( x ) = d a (cid:16) d ( ) ( x − r / ) − d ( ) ( x + r / ) . (cid:17) (3.2)and recalculating the energy h H i with fixed w . There is only one contribution to the energy thatdepends on the distance r between the quarks. Using Eq. (2.13), it reads V c ( r ) = g Z h Y | r m F r m | Y i = Z d q ( p ) d ( q ) f ( q ) q (cid:0) − e i q · r (cid:1) . (3.3)With the infrared behavior of the form factors (3.1), we find that V c ( r ) rises linearly in the infraredand thus confines heavy quarks as shown in Fig. 2. By matching the slope of the linear potential tothe lattice string tension s , one may set the scale.Apart from the solution in Fig. 1 with the asymptotic infrared behavior (3.1) there is one furthersolution with slightly different infrared exponents for the power laws. The latter was discoveredfirst [10], however, it does not have the same attractive features as the one in Fig. 1. In particular,the heavy quark potential is strictly linearly rising only for the solution presented here.A nonperturbative running coupling may be extracted from the ghost-gluon vertex [17, 11], a ( k ) = p k d ( k ) w − ( k ) , (3.4)5 ang-Mills Theory in Coulomb gauge H. Reinhardt -2 0 2 4 6 8 10 12 14 16 0 5 10 15 20 25 30¯r ¯ V(¯r)- ¯ V Linear fit 0.01 0.1 1 0.001 0.01 0.1 1 10 100 1000k [ s ] a (k) [16 p /3N c ] Figure 2:
Right: the Coulomb potential V ( r ) . Right: the running coupling a ( k ) . With a tree-level vertex, it can be shown that one finds a finite value in the infrared, a ( ) = p / ( N c ) [11]. In the ultraviolet, we find the correct 1 / ln ( k / m ) scaling from one-loop pertur-bation theory. However, the first coefficient of the beta function, b , is off by a factor of 8 / a ( k ) in Fig. 2 is amonotonic function.
4. The ’t Hooft loop
A (dis-)order parameter of confinement is the ’t Hooft loop h V ( C ) i [22] whose operator V ( C ) is defined by the relation V ( C ) W ( C ) = Z L ( C , C ) W ( C ) V ( C ) , where W ( C ) is the operator of thespatial Wilson loop, Z is a (non-trivial) center element of the gauge group and L ( C , C ) denotesthe Gaussian linking number. An explicit realization of V ( C ) in continuum Yang-Mills theory wasderived in ref. [23] and is given by V ( C ) = exp (cid:20) i Z d x A ai [ C ]( x ) P ai ( x ) (cid:21) . (4.1)Here A [ C ] denotes the gauge potential of a (spatial) center vortex whose magnetic flux is localizedat the loop C . Since V ( C ) Y ( A ) = Y ( A + A [ C ]) the ’t Hooft loop is a center vortex generator.Using the wave functional found in the variational solution of the Yang-Mills Schrödinger equationin Coulomb gauge, as described above, the expectation value h V ( C ) i ≡ exp ( − S ( C )) was evaluatedfor a planar circular loop C and is was found that the exponential S ( C ) obeys a perimeter lawsignaling confinement [24]. This result is in accord with the linear behavior found for the staticcolor potential.
5. Conclusions
We have solved the Yang-Mills Schrödinger equation approximately and thus determined thevacuum wave functional. Our solutions exhibit the phenomena of confinement of gluons as well6 ang-Mills Theory in Coulomb gauge
H. Reinhardt as heavy quarks. As an improvement on previous results, the heavy quark potential rises strictlylinearly. The nonperturbative running coupling derived from the ghost-gluon vertex was presentedand the ’t Hooft loop was calculated. It is promising that the results have the crucial features ofnonperturbative physics, and that calls for further investigations in the Hamiltonian approach.
6. Acknowledgments
It is a pleasure to thank A. Szczepaniak and A. Weber for continuing correspondence on thesubject. One of us (H.R.) would like to thank the organizers of the conference for the stimulatingevent.
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