Leibbrandt-Mandelstam Prescription Research Articles

Computing noncommutative Chern–Simons theory radiative corrections on the back of an envelope

We show that the renormalized U(N) noncommutative Chern-Simons theory can be defined in perturbation theory so that there are no loop corrections to the 1PI functional of the theory in an arbitrary homogeneous axial (time-like, light-like or space-like) gauge. We define the free propagators of the fields of the theory by using the Leibbrandt-Mandelstam prescription --which allows Wick rotation and is consistent with power-counting-- and regularize its Green functions with the help of a family of regulators which explicitly preserve the infinitesimal vector Grassmann symmetry of the theory. We also show that in perturbation theory the nonvanishing Green functions of the elementary fields of the theory are products of the free propagators.

Obtaining gluon propagator with Leibbrandt–Mandelstam prescription

The quantization problems of the gluon fields are investigated in the lightcone gauge, n μ A μ a ( x)=0, where n 2=0. In this gauge, the gluon propagator has unphysical poles of kind 1/( nk). In order to remove these singularities, Leibbrandt and Mandelstam have proposed a new prescription. In this work we introduced a new modified lightcone gauge, which is n μ+ iε (n ∗k)k 2 k μ A μ a=0 where n μ ∗ is the dual vector to n μ . We proved that Leibbrandt–Mandelstam prescription can be obtained on the basis of consistent quantization. The Ward identity in the modified lightcone gauge was also investigated.

Evolution kernels of twist-3 light-ray operators in polarized deep inelastic scattering

The twist three contributions to the Q 2 -evolution of the spin-dependent structure function g 2 ( x ) are considered in the non-local operator product approach. Starting from the perturbative expansion of the T-product of two electromagnetic currents, we introduce the nonlocal light-cone expansion proved by Anikin and Zavialov and determine the physical relevant set of light-ray operators of twist three. Using the equations of motion we show the equivalence of these operators to the Shuryak-Vainshtein operators plus the mass operator, and we determine their evolution kernels using the light-cone gauge with the Leibbrandt-Mandelstam prescription. The result of Balitsky and Braun for the twist three evolution kernel (nonsinglet case) is confirmed.

Pole prescription in axial gauge at finite temperature

>We establish Yang-Mills theory at finite temperature in axial gauge. We encounter the breakdown of the Hata-Kugo expression for the partition function in the special choice of temporal axial gauge. For n ≠ 0 the finite temperature propagator is calculated in both the imaginary time formalism and the real time formalism. In the latter we recover the Leibbrandt-Mandelstam prescription in the lijit T→0. As a check we calculate the mean energy to lowest order.

Axial gauge Chern-Simons theory in the Leibbrandt-Mandelstam prescription for a certain class of UV regulators

We show that the complete perturbative effective action of an SU( N) Chern-Simons theory in an homogenous axial-type gauge ( nA a =0, n 2>0, n 2<0, n 2=0) with the generalized Leibbrandt-Mandelstam prescription is the tree-level action for a certain class of regulators. The Gauss linking number is expressed in terms of the axial propagators for the gauge fields.

The three-gluon vertex in the temporal-planar gauge with the Leibbrandt-Vienna prescription: The one-loop effective action

The UV-divergent part of the one-loop contribution to the three-gluon vertex, Γ μνϱ abc , is computed in the temporal-planar gauge. The generalized Leibbrandt-Mandelstam (or Leibbrandt-Vienna) prescription - which is free from the inconsistencies that are known to affect the principal value prescription - is used to define the unphysical poles 1/( k· n) j . We compute the (one-loop) UV-divergent part of the gauge-field effective action up to order g 3. We show that all the awkward non-local divergences can be written as the BRS variation of some functional X whereas the β-function coincides with the usual (covariant) result. In the homogeneous case we show that a non-local renormalization of the gauge field A μ a of the form A 0μ a = Z g − 1 2 A μ a − h ̷ ƒ μ a(A), A 0μ a and A μ a having identical gauge transformations, and the usual coupling constant renormalization, g 0 = Z g μ 2 ϵ g, suffice to get rid of all the UV-divergences in Γ μνϱ abc and the vacuum polarization tensor gP μν ab .

THE DEPENDENCE OF THE TEMPORAL GAUGE GREEN FUNCTIONS ON THE GAUGE-FIXING VECTORS THROUGH BRS SYMMETRY

The UV-divergent part of the one-loop contribution to the vacuum polarization tensor [Formula: see text] is shown to satisfy the BRS identities governing its dependence on the time-like vector n and the auxiliary space-like vector n′ introduced by the generalized Leibbrandt-Mandelstam (LM) prescription. As far as we know the BRS identity for the n′ dependence cannot be obtained within the pragmatic approach to the LM prescription. In order to circumvent the problem the LM prescription is incorporated into a BRS-invariant action. The BRS identity for n′ provides an example of a situation where terms of order η combine with loop integrals that are singular as η→0 yielding a finite contribution.

Checking the S matrix in QCD in axial gauges within the generalized Leibbrandt-Mandelstam prescription.

We apply the Leibbrandt-Mandelstam prescription to QCD within a family of gauges comprising the homogeneous axial gauge and the planar gauge. We calculate to one-loop order the UV-divergent parts of the quark self-energy, the quark-quark-gluon vertex, and the four-quark one-particle-irreducible vertex. The results are found to satisfy the Slavnov-Taylor-Lee identity. Summing up the contributions to the $S$-matrix element for quark-quark scattering, we show that the result is indeed independent of the gauge parameters ${n}_{\ensuremath{\mu}}$ and $\ensuremath{\alpha}$ and of the auxiliary vector ${n}_{\ensuremath{\mu}}^{*}$. The nonlocal counterterms persisting in Green's functions for ${n}^{2}\ensuremath{\ne}0$ are thus compatible with gauge independence of physical quantities. The Lorentz-noninvariant divergent structures (local as well as nonlocal ones) turn out to be proportional to the classical equations of motion.

Canonical formalism and the Leibbrandt-Mandelstam prescription for noncovariant gauges.

A very simple and elegant approach to the Leibbrandt-Mandelstam regularization is given within the canonical formalism. For any value of {ital n}{sup 2} with {ital n}{sub 0} {ital and} {bold n} different from zero, it consists of introducing the hyperbolic operator {ital n}{sup *}{center dot}{partial derivative}n{center dot}{partial derivative}=(n{sub 0}{partial derivative}{sub 0}){sup 2} {minus}({ital Rn}{center dot}{partial derivative}){sup 2} inside the field equations. The Cauchy problem is solved in the free field theory leading to a propagator regularized by means of the Leibbrandt-Mandelstam prescription. In the non-Abelian theory, these gauges are now on the same footing as relativistic gauges but the contribution of ghost loops vanishes.

Gauge (in)dependence, nonlocal counterterms and extended BRST-symmetry. A pedagogical example

For n 2≠0 a generalized Leibbrandt-Mandelstam (LM) prescription is used to analyze the one-loop counterterms of ordinary QED. Using a recently proposed form of the gauge fixing where n μ and its dual vector n ∗μ are present already at the classical level allows to formulate an extended BRST symmetry involving also the variations of the gauge parameters n μ and n ∗μ into Grassmann variables χ μ and χ ∗μ .

Are axial gauges useful in perturbative QCD?

The general source of the problems with the time-like and space-like axial gauges is analysed. These problems are avoided in the light-like case, using the Leibbrandt-Mandelstam prescription. However, this does not satisfy graph-by-graph unitarity, and so its usefulness in typical perturbative QCD arguments is open to question.

Renormalization of Yang-Mills theories in axial gauges within a uniform prescription.

We investigate the renormalization properties of Yang-Mills theories in axial gauges within the generalized Leibbrandt-Mandelstam prescription which unifies all axial gauges, including the light-cone gauge, within one uniform prescription. We calculate the divergent part of the gluon self-energy in a planar-type gauge fixing. All known results on the planar gauge and on the light-cone gauge are recovered as special cases. At variance to the light-cone gauge for ${n}^{2}$\ensuremath{\ne}0 only the S matrix, but not the Green's functions, can be renormalized by local counterterms.

Limit properties of Feynman integrals in noncovariant algebraic gauges.

We show that the Leibbrandt-Mandelstam prescription for the spurious singularity (nxk)/sup -1/ in the algebraic noncovariant gauges tends indeed to the Cauchy principal value when n/sub 0/..-->..0, at variance with a claim that recently appeared in the literature. The possible relation with renormalization in the light-cone case is discussed

The gluon self-energy in the planar gauge with the Leibbrandt-Mandelstam prescription

We develop an explicit calculation of the divergent part of the gluon self-energy in the planar gauge with the Leibbrandt-Mandelstam prescription for the spurious poles; the Lee-Ward identity for the self-energy tensor is also checked.

Symmetries of axial gauge pole prescriptions

We investigate various pole prescriptions for the 1 qn singularity in axial and light-cone gauges according to their symmetry content. To go beyond the principal value technique it is natural to generalize the Leibbrandt-Mandelstam prescription, which is less symmetric, but in a sense more regular. We analyse the analytical properties of the corresponding one-loop Feynman integrals.

The Leibbrandt–Mandelstam prescription for general axial gauge one-loop integrals

It is well known that, in doing light-cone gauge calculations, it is mandatory to regularize the unphysical (q n)−β poles by use of the Leibbrandt–Mandelstam prescription. This technique is also applied to general axial gauges and it is proved that it is a suitable regularization procedure for these gauges as well. In order to find the relation between the Leibbrandt prescription and the more familiar principal value prescription with its simpler Lorentz structure the temporal gauge limit n→0 is performed (within dimensional regularization). Although this limit is found to be singular for multiple poles, the analytically regularized one-loop integrals agree with the results obtained within the principal value technique for the temporal gauge.

Nonlocal counterterms in QED?

The Leibbrandt-Mandelstam prescription is used to investigate ordinary QED in a general axial gauge. It is shown that the usual renormalization program is applicable at one-loop order without difficulties. For n 2≠0 the individual four-fermion 1PI graphs develop non-polynomial divergent structures. The sum of these contributions, however, vanishes, yielding a finite four-point vertex. For n 2 = 0 we recover the well-known results, whereas the limits n 0→0 or n→0 show an unexpected behaviour.