Mandelstam-Leibbrandt Prescription Research Articles

Mandelstam-Leibbrandt prescription

The light-cone gauge is used frequently in string theory as well as gauge theories and gravitation. Loop integrals, however, have to be infrared regulated to remove spurious poles. The most popular and consistent of these infrared regulators is the Mandelstam-Leibbrandt prescription (ML). The calculations with the ML are rather cumbersome, though. In this work, we show that the ML can be replaced by a symmetry of the regulator. This symmetry simplifies the calculations, reducing them to conventional dimensional regularization integrals.

Regularization of the light-cone gauge gluon propagator singularities using sub-gauge conditions

Perturbative QCD calculations in the light-cone gauge have long suffered from the ambiguity associated with the regularization of the poles in the gluon propagator. In this work we study sub-gauge conditions within the light-cone gauge corresponding to several known ways of regulating the gluon propagator. Using the functional integral calculation of the gluon propagator, we rederive the known sub-gauge conditions for the theta-function gauges and identify the sub-gauge condition for the principal value (PV) regularization of the gluon propagator's light-cone poles. The obtained sub-gauge condition for the PV case is further verified by a sample calculation of the classical Yang-Mills field of two collinear ultrarelativistic point color charges. Our method does not allow one to construct a sub-gauge condition corresponding to the well-known Mandelstam-Leibbrandt prescription for regulating the gluon propagator poles.

Chern-Simons spinor electrodynamics in the light-cone gauge

The one-loop quantum corrections of Chern-Simons spinor electrodynamics in the light-cone gauge has been investigated. We have calculated the vacuum polarization tensor, fermionic self-energy, and on-shell vertex correction with a hybrid regularization consisting of a higher covariant derivative regularization and dimensional continuation. The Mandelstam-Leibbrandt prescription is used to handle the spurious light-cone singularity in the gauge field propagator. We then perform the finite renormalization to define the quantum theory. The generation of the parity-even Maxwell term and the arising of anomalous magnetic moment from quantum corrections are reproduced as in the case of a covariant gauge choice. The Ward identities in the light-cone gauge are verified to satisfy explicitly. Further, we have found the light-cone vector dependent sector of local quantum effective action for the fermion is explicitly gauge invariant, and hence the Lorentz covariance of $S$-matrix elements of the theory can be achieved.

RENORMALIZATION-GROUP ANATOMY OF TRANSVERSE-MOMENTUM DEPENDENT PARTON DISTRIBUTION FUNCTIONS IN QCD

The ultraviolet and rapidity divergences of transverse-momentum dependent parton distribution functions with lightlike and transverse gauge links is studied, also incorporating a soft eikonal factor. We find that in the light-cone gauge with q--independent pole prescriptions extra divergences appear which amount, at one-loop, to a cusp-like anomalous-dimension. We show that such contributions are absent when the Mandelstam-Leibbrandt prescription is used. In the first case, the soft factor cancels the anomalous dimension defect, while in the second case its ultraviolet-divergent part reduces to unity.

Renormalization-group properties of transverse-momentum dependent parton distribution functions in the light-cone gauge with the Mandelstam-Leibbrandt prescription

The renormalization-group properties of transverse-momentum dependent parton distribution functions in the light-cone gauge with the Mandelstam-Leibbrandt prescription for the gluon propagator are addressed. An expression for the transverse component of the gauge field at light-cone infinity, which plays a crucial role in the description of the final-/initial-state interactions in the light-cone axial gauge, is obtained. The leading-order anomalous dimension is calculated in this gauge and the relation to the results obtained in other gauges is worked out. It is shown that, using the Mandelstam-Leibbrandt prescription, the ensuing anomalous dimension does not receive contributions from extra rapidity divergences related to a cusped junction point of the Wilson lines.

Propagators with the Mandelstam-Leibbrandt prescription in the light-cone gauge

We show that the Feynman propagator in the light-cone gauge with the Mandelstam-Leibbrandt prescription has a logarithmic growth for large $\stackrel{\texttildelow{}}{n}\ifmmode\cdot\else\textperiodcentered\fi{}x$ which is related to the presence of a residual gauge invariance. Furthermore, we show that the retarded propagator for the $\stackrel{\texttildelow{}}{n}\ifmmode\cdot\else\textperiodcentered\fi{}A$ component of the gauge field develops a coordinate dependent mass. We argue that this feature is unphysical and may be eliminated by fixing the residual gauge degrees of freedom.

The calculation of quark–gluon plasma thermodynamic potential in the lightcone gauge

The thermodynamic potential of quark–gluon plasma is investigated in the lightcone gauge. In this gauge, the gluon propagator has unphysical poles of kind 1/(n · k). In order to remove these singularities Mandelstam and Leibbrandt have proposed a new prescription. In this work the thermodynamic potential of quark–gluon plasma is calculated at two-loop approximation using the Mandelstam–Leibbrandt prescription and the obtained expression is shown to be gauge invariant.

Summing planar diagrams for string and gauge theories

Abstract We discuss the problem of summing planar diagrams in both string and field theory using light-cone methods. We advocate the discretization of both x + and p + as a useful tool for attacking this problem. We then discuss some technical aspects of this discretization procedure in non-abelian gauge theories. We describe one-loop calculations in this formalism and explain how the Mandelstam-Leibbrandt prescription might be adapted to it.

Indispensability of residual gauge fields and extended hamiltonian formalism in light-front quantization of gauge fields

We consider introducing residual gauge degrees of freedom into the conventional formulation of the light-front quantization of gauge field theories. For that purpose we construct the canonical formulation of axial gauge fields of the type nμ Aμ = 0 in auxiliary coordinates: xμ = (xτ,xσ,x1,x2), where xτ = x0 sin φ + x3 cos φ and xσ = x0 cos φ − x3 sin φ. We then show that, irrespective of the quantization surface, and irrespective of the gauge fixing condition, residual gauge fields are indispensable in implementing the Mandelstam-Leibbrandt prescription and in regularizing the infrared divergences which are inherent in the canonical quantization of space-like axial gauge fields. With the residual gauge fields in place, we find that the infrared divergences are regulated with the Mandelstam-Leibbrandt prescription in the light-front formulation obtained as the limit φ→π/4 of the n2 = 0 case. In addition we show that, because an explicit quantization surface dependence does not appear in the n2 = 0 case, the light-front temporal gauge limit φ→π/4−0 agrees with the light-front spatial gauge limit φ→π/4+0 and that the perturbative Hamiltonian in the light-front formulation consists of physical degrees of freedom integrated over the hyperplane xl+ = (x0+x3)/√2 = constant and residual degrees of freedom integrated over the hyperplane xl− = (x0 − x3)/√2 = constant.

Extended Hamiltonian Formalism of the Pure Space-Like Axial Gauge Schwinger Model

We demonstrate that pure space-like axial gauge quantizations of gauge fields can be constructed in ways that are free from infrared divergences. To do so, we must extend the Hamiltonian formalism to include residual gauge fields. We construct an operator solution and an extended Hamiltonian of the pure space-like axial gauge Schwinger model. We begin by constructing an axial gauge formulation in auxiliary coordinates, x+=(x+,x−), where x+=x0sin θ+x1cos θ, x−=x0cos θ−x1sin θ, and we take A−=A0cos θ+A1 sin θ=0 as the gauge fixing condition. In the region , we can take x− as the evolution parameter and construct a traditional canonical formulation of the temporal gauge Schwinger model in which residual gauge fields dependent only on x+ are static canonical variables. Then we extrapolate the temporal gauge operator solution into the axial region, , where x+ is taken as the evolution parameter. In the axial region we find that we have to take the representation of the residual gauge fields realizing the Mandelstam-Leibbrandt prescription in order for the infrared divergences resulting from (∂−)−1 to be canceled by corresponding ones resulting from the inverse of the hyperbolic Laplace operator. We overcome the difficulty of constructing the Hamiltonian for the residual gauge fields by employing McCartor and Robertson's method, which gives us a term integrated over x−= constant. Finally, by taking the limit , we obtain an operator solution and the Hamiltonian of the axial gauge (Coulomb gauge) Schwinger model in ordinary coordinates. That solution includes auxiliary fields, and the representation space is of indefinite metric, providing further evidence that “physical” gauges are no more physical than “unphysical” gauges.

Probing negative-dimensional integration: two-loop covariant vertex and one-loop light-cone integrals

The negative-dimensional integration method (NDIM) seems to be a very promising technique for evaluating massless and/or massive Feynman diagrams. It is unique in the sense that the method gives solutions in different regions of external momenta simultaneously. Moreover, it is a technique whereby the difficulties associated with performing parametric integrals in the standard approach are transferred to a simpler solving of a system of linear algebraic equations, thanks to the polynomial character of the relevant integrands. We employ this method to evaluate a scalar integral for a massless two-loop three-point vertex with all the external legs off-shell, and consider several special cases for it, yielding results, even for distinct simpler diagrams. We also consider the possibility of NDIM in non-covariant gauges such as the light-cone gauge and do some illustrative calculations, showing that for one-degree violation of covariance (i.e. one external, gauge-breaking, light-like vector n μ ) the ensuing results are concordant with the ones obtained via either the usual dimensional regularization technique, or the use of the principal value prescription for the gauge-dependent pole, while for two-degree violation of covariance—i.e. two external, light-like vectors n μ , the gauge-breaking one, and (its dual) n μ ∗ —the ensuing results are concordant with the ones obtained via causal constraints or the use of the so-called generalized Mandelstam-Leibbrandt prescription.

Light-cone gauge and the calculation of the two-loop splitting functions

We present calculations of next-to-leading order QCD splitting functions, employing the light-cone gauge method of Curci, Furmanski, and Petronzio (CFP). In contrast to the `principal-value' prescription used in the original CFP paper for dealing with the poles of the light-cone gauge gluon propagator, we adopt the Mandelstam-Leibbrandt prescription which is known to have a solid field-theoretical foundation. We find that indeed the calculation using this prescription is conceptionally clear and avoids the somewhat dubious manipulations of the spurious poles required when the principal-value method is applied. We reproduce the well-known results for the flavour non-singlet splitting function and the N_C^2 part of the gluon-to-gluon singlet splitting function, which are the most complicated ones, and which provide an exhaustive test of the ML prescription. We also discuss in some detail the x=1 endpoint contributions to the splitting functions.

Two-loop anomalous dimension in light-cone gauge with Mandelstam-Leibbrandt prescription

All the next-to-leading order results for Altarelli-Parisi splitting functions have been obtained in the literature either by using the operator product expansion method or by making use of the Curci-Furmanski-Petronzio (CFP) formalism in conjunction with light-like axial gauge, principal value (PV) prescription and dimensional regularization. In this paper we present the calculation of some non-singlet two-loop anomalous dimensions with the CFP formalism using light-cone axial gauge with Mandelstam-Leibbrandt (ML) prescription. We make a detailed comparison between the intermediate results given by the PV versus the ML method. We point out that the ML method is completely consistent and avoids the “phenomenological rules” used in the case of PV regularization.

The eikonal approximation and perturbative QCD in the temporal gauge

The extension from the light-cone gauge to the temporal gauge of the Mandelstam-Leibbrandt prescription for non-covariant poles of the gluon propagator is applied to calculate the infrared behaviour of the Bloch-Nordsieck cross-section for annihilation of a massive quark with its antiparticle, to order α s 2. The eikonal approximation together with unrestricted momenta of virtual gluons is used to isolate the infrared divergent part of the cross-section. The formerly established absence of quark mass singularities in the cross-section is reproduced provided an integral gets a treatment similar to how some Feynman diagrams involving massless quark propagators are dealt with by the use of dimensional regularization scheme in covariant gauges.

The Mandelstam-Leibbrandt prescription in light-cone quantized gauge theories

Quantization of gauge theories on characteristic surfaces and in the light-cone gauge is discussed. Implementation of the Mandelstam-Leibbrandt prescription for the spurious singularity is shown to require two distinct null planes, with independent degrees of freedom initialized on each. The relation of this theory to the usual light-cone formulation of gauge field theory, using a single null plane, is described. A connection is established between this formalism and a recently given operator solution to the Schwinger model in the light-cone gauge.

A note on the extended Mandelstam-Leibbrandt prescription for gauge theories in temporal gauge

The extension to the temporal gauge of the Mandelstam-Leibbrandt prescription for treatment of spurious poles of the gauge- field propagator in the light-cone gauge is applied at lowest order of radiative corrections to the calculation of the vacuum expectation value of the Wilson line integral along a contour with a cusp. The results show that in this definition of the temporal gauge a cusp renormalization constant aries non-locally; to calculate it one has to take into account the interactions of all parts of the contour, in contradistinction to the Feynman gauge, where one needs to consider the Wilson line integral along just the immediate neighbourhood of the cusp.

Gauge invariance and anomalous dimensions of a light-cone Wilson loop in light-like axial gauge

Complete two-loop calculation of a dimensionally regularized Wilson loop with light-like segments is performed in the light-like axial gauge with the Mandelstam-Leibbrandt prescription for the gluon propagator. We find an expression which exactly coincides with the one previously obtained for the same Wilson loop in covariant Feynman gauge. The renormalization of the Wilson loop is performed in the MS-scheme using a general procedure tailored to the light-like axial gauge. We find that the renormalized Wilson loop obeys a renormalization group equation with the same anomalous dimensions as in covariant gauges. Physical implications of our result for the investigation of infrared asymptotics of perturbative QCD are pointed out.

Stochastic interpretation for the propagators in the temporal gauge

We show that the stochastic interpretation based on stochastic mechanics provides a reasonable physical justification for the form of the propagator of a fully-fixed temporal gauge field. The generalized spectral representation of this propagator in euclidean space-time allows it to be linked to the “naive” propagator with the double pole singularity. The time translationally invariant α-prescription also has a stochastic interpretation, though at the expense of the temporal gauge condition. The stochastic formalism does not seem to work for the Mandelstam-Leibbrandt prescription.

Ghost contributions to the LFK kernel in light-cone gauge

Using the Mandelstam-Leibbrandt (ML) prescription for the vector propagator in light-cone axial (LCA) gauge, we show how the infrared singularity (IRS) corresponding to real gluon emission in the Lipatov-Fadin-Kuraev (LFK) equation is cancelled by the axial ghost contribution.

EVALUATING ONE-LOOP LIGHT-CONE INTEGRALS BY "COVARIANTIZING" THE GAUGE-DEPENDENT POLE

Making sure that causality be preserved by means of "covariantizing" the gauge-dependent singularity in the propagator of the vector potential Aμ(x), we show that the evaluation of some basic one-loop light-cone integrals reproduce those results obtained through the Mandelstam-Leibbrandt prescription. Moreover, such a covariantization has the advantage of leading to simpler integrals to be performed in the cone variables (the bonus), although, of course, it introduces an additional α-parameter integral to be performed (the price to pay).