### Abstract

We derive the equations of time-independent stochastic quantization, without reference to an unphysical fifth time, from the principle of gauge equivalence. It asserts that probability distributions P that give the same expectation values for gauge-invariant observables 〈W〉 = ∫dAWP are physically indistinguishable. This method escapes the Gribov critique. We derive an exact system of equations that closely resembles the Dyson-Schwinger equations of Faddeev-Popov theory. The system is truncated and solved nonperturbatively, by means of a power law ansatz, for the critical exponents that characterize the asymptotic form at k≈0 of the gluon propagator in Landau gauge. For the transverse and longitudinal parts, we find, respectively, D ^{T} ∼(k ^{2}) ^{-1-α} _{T}≈(k ^{2}) ^{0.043}, suppressed and in fact vanishing, though weakly, and D ^{L}∼a(k ^{2}) ^{-1-α} _{L} ≈ a(k ^{2}) ^{-1.521}, enhanced, with α _{T}= -2α _{L}. Although the longitudinal part vanishes with the gauge parameter a in the Landau-gauge limit a→0 there are vertices of order a ^{-1} so, counterintuitively, the longitudinal part of the gluon propagator does contribute in internal lines in the Landau gauge, replacing the ghost that occurs in Faddeev-Popov theory. We compare our results with the corresponding results in Faddeev-Popov theory.

Original language | English (US) |
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Article number | 105001 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 67 |

Issue number | 10 |

DOIs | |

State | Published - May 15 2003 |

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### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

**Time-independent stochastic quantization, Dyson-Schwinger equations, and infrared critical exponents in QCD.** / Zwanziger, Daniel.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Time-independent stochastic quantization, Dyson-Schwinger equations, and infrared critical exponents in QCD

AU - Zwanziger, Daniel

PY - 2003/5/15

Y1 - 2003/5/15

N2 - We derive the equations of time-independent stochastic quantization, without reference to an unphysical fifth time, from the principle of gauge equivalence. It asserts that probability distributions P that give the same expectation values for gauge-invariant observables 〈W〉 = ∫dAWP are physically indistinguishable. This method escapes the Gribov critique. We derive an exact system of equations that closely resembles the Dyson-Schwinger equations of Faddeev-Popov theory. The system is truncated and solved nonperturbatively, by means of a power law ansatz, for the critical exponents that characterize the asymptotic form at k≈0 of the gluon propagator in Landau gauge. For the transverse and longitudinal parts, we find, respectively, D T ∼(k 2) -1-α T≈(k 2) 0.043, suppressed and in fact vanishing, though weakly, and D L∼a(k 2) -1-α L ≈ a(k 2) -1.521, enhanced, with α T= -2α L. Although the longitudinal part vanishes with the gauge parameter a in the Landau-gauge limit a→0 there are vertices of order a -1 so, counterintuitively, the longitudinal part of the gluon propagator does contribute in internal lines in the Landau gauge, replacing the ghost that occurs in Faddeev-Popov theory. We compare our results with the corresponding results in Faddeev-Popov theory.

AB - We derive the equations of time-independent stochastic quantization, without reference to an unphysical fifth time, from the principle of gauge equivalence. It asserts that probability distributions P that give the same expectation values for gauge-invariant observables 〈W〉 = ∫dAWP are physically indistinguishable. This method escapes the Gribov critique. We derive an exact system of equations that closely resembles the Dyson-Schwinger equations of Faddeev-Popov theory. The system is truncated and solved nonperturbatively, by means of a power law ansatz, for the critical exponents that characterize the asymptotic form at k≈0 of the gluon propagator in Landau gauge. For the transverse and longitudinal parts, we find, respectively, D T ∼(k 2) -1-α T≈(k 2) 0.043, suppressed and in fact vanishing, though weakly, and D L∼a(k 2) -1-α L ≈ a(k 2) -1.521, enhanced, with α T= -2α L. Although the longitudinal part vanishes with the gauge parameter a in the Landau-gauge limit a→0 there are vertices of order a -1 so, counterintuitively, the longitudinal part of the gluon propagator does contribute in internal lines in the Landau gauge, replacing the ghost that occurs in Faddeev-Popov theory. We compare our results with the corresponding results in Faddeev-Popov theory.

UR - http://www.scopus.com/inward/record.url?scp=84865036630&partnerID=8YFLogxK

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U2 - 10.1103/PhysRevD.67.105001

DO - 10.1103/PhysRevD.67.105001

M3 - Article

AN - SCOPUS:84865036630

VL - 67

JO - Physical review D: Particles and fields

JF - Physical review D: Particles and fields

SN - 1550-7998

IS - 10

M1 - 105001

ER -