TY - JOUR

T1 - Renormalization in the Coulomb gauge and order parameter for confinement in QCD

AU - Zwanziger, Daniel

N1 - Funding Information:
PACS: 12.38.Aw; 12.38.Bx; ll.15.Bt; ll.10.Gh Keywords: QCD; Coulomb-gauge; Renormalization; Confinement l E-mail: daniel.zwanziger@nyu.edu 2 Research supported in part by the National Science Foundation under grant no. PHY-9520978.

PY - 1998/5/4

Y1 - 1998/5/4

N2 - Renormalization of the Coulomb gauge is studied in the phase space formalism, where one integrates over both the vector potential A, and its canonical momentum II, as well as the usual Faddeev-Popov auxiliary fields. A proof of renormalizability is not attempted. Instead, algebraic identities are derived from BRST invariance which renormalization must satisfy if the Coulomb gauge is renormalizable. In particular, a Ward identity is derived which holds at a fixed time t, and which is an analog of Gauss's law in the BRST formalism, and which we call the Gauss-BRST identity. The familiar Zinn-Justin equation results when this identity is integrated over all t. It is shown that in the Coulomb gauge, g2D0,0 is a renormalization-group invariant, as is its instantaneous part V(R), which we call the color-Coulomb potential. (Here D0,0 is the time-time component of the gluon propagator.) The contribution of V(R) to the Wilson loop exponentiates. It is proposed that the string tension defined by KCoul = limR→∞ CV(R)/R may serve as an order parameter for confinement, where C = (2N)-1(N2 - 1) for SU(N) gauge theory. A remarkable consequence of the above-mentioned Ward identity is that the Fourier transform V(k) of V(R) is of the product form V(k) = [k2Dc,c* (k)]2L(k), where Dc,c* (k) is the ghost propagator, and L(k) is a correlation function of longitudinal gluons. This exact equation combines with a previous analysis of the Gribov problem according to which k2Dc,c* (k) diverges at k = 0, to provide a scenario for confinement.

AB - Renormalization of the Coulomb gauge is studied in the phase space formalism, where one integrates over both the vector potential A, and its canonical momentum II, as well as the usual Faddeev-Popov auxiliary fields. A proof of renormalizability is not attempted. Instead, algebraic identities are derived from BRST invariance which renormalization must satisfy if the Coulomb gauge is renormalizable. In particular, a Ward identity is derived which holds at a fixed time t, and which is an analog of Gauss's law in the BRST formalism, and which we call the Gauss-BRST identity. The familiar Zinn-Justin equation results when this identity is integrated over all t. It is shown that in the Coulomb gauge, g2D0,0 is a renormalization-group invariant, as is its instantaneous part V(R), which we call the color-Coulomb potential. (Here D0,0 is the time-time component of the gluon propagator.) The contribution of V(R) to the Wilson loop exponentiates. It is proposed that the string tension defined by KCoul = limR→∞ CV(R)/R may serve as an order parameter for confinement, where C = (2N)-1(N2 - 1) for SU(N) gauge theory. A remarkable consequence of the above-mentioned Ward identity is that the Fourier transform V(k) of V(R) is of the product form V(k) = [k2Dc,c* (k)]2L(k), where Dc,c* (k) is the ghost propagator, and L(k) is a correlation function of longitudinal gluons. This exact equation combines with a previous analysis of the Gribov problem according to which k2Dc,c* (k) diverges at k = 0, to provide a scenario for confinement.

KW - Confinement

KW - Coulomb-gauge

KW - QCD

KW - Renormalization

UR - http://www.scopus.com/inward/record.url?scp=0032481975&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032481975&partnerID=8YFLogxK

U2 - 10.1016/S0550-3213(98)00031-5

DO - 10.1016/S0550-3213(98)00031-5

M3 - Article

AN - SCOPUS:0032481975

SN - 0550-3213

VL - 518

SP - 237

EP - 272

JO - Nuclear Physics B

JF - Nuclear Physics B

IS - 1-2

ER -