### Abstract

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, g^{2}D_{0,0} is a renormalization-group invariant, as is its instantaneous part V(R), which we call the color-Coulomb potential. (Here D_{0,0} 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 K_{Coul} = lim_{R→∞} CV(R)/R may serve as an order parameter for confinement, where C = (2N)^{-1}(N^{2} - 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) = [k^{2}D^{c,c*} (k)]^{2}L(k), where D^{c,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 k^{2}D^{c,c*} (k) diverges at k = 0, to provide a scenario for confinement.

Original language | English (US) |
---|---|

Pages (from-to) | 237-272 |

Number of pages | 36 |

Journal | Nuclear Physics, Section B |

Volume | 518 |

Issue number | 1-2 |

State | Published - May 4 1998 |

### Fingerprint

### Keywords

- Confinement
- Coulomb-gauge
- QCD
- Renormalization

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Nuclear Physics, Section B*,

*518*(1-2), 237-272.

**Renormalization in the Coulomb gauge and order parameter for confinement in QCD.** / Zwanziger, Daniel.

Research output: Contribution to journal › Article

*Nuclear Physics, Section B*, vol. 518, no. 1-2, pp. 237-272.

}

TY - JOUR

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

AU - Zwanziger, Daniel

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

M3 - Article

VL - 518

SP - 237

EP - 272

JO - Nuclear Physics B

JF - Nuclear Physics B

SN - 0550-3213

IS - 1-2

ER -