Practical Look at Non-Covariant Gauges

Absence of Nonlocal Counterterms in the Gauge Boson Propagator in Axial-Type Gauges

“…Moreover, we find, as shown below, that we do not encounter non-local divergence (in this simple example) found with uniform/ML prescription and we suspect this to hold in higher order calculations. Unlike CPV, this treatment has been shown explicitly to preserve a 3 We note that for η 2 > 0, the denominator ( η · k) 2 − k 2 never vanishes and we can simplify by dropping ǫ that went along with it.…”

“…The Lorentz type gauges have the several desirable properties of Lorentz-covariance ,ease of calculations and the availability of a gauge parameter to verify the gaugeindependence. They , however, are burdened with having to include the diagrams involved with the Faddeev-Popov ghosts.As a result of this, another set of gauges ,the axial-type gauges[ these include the light-cone [LCG] and the temporal gauges], have also been employed in the Standard Model [SM] calculations.These gauges purport to have decoupling of ghosts [2,3,4] and consequently require a fewer set of diagrams.The disadvantages these gauges suffer from are the lack of manifest covariance, extra counter-terms arising from an additional vector η { or two additional vectors in case of the LeibbrandtMandelstam/Uniform prescriptions [2,3]} and the problem of how to correctly deal with the spurious singularities of the form 1/(η.q) q .Doubts have also been expressed [5] about the exceptional advantages of the axial-type gauges;nonetheless they have been found useful in practice. The crucial problem of 1/(η.q) q -type singularities has been widely addressed over several decades [2,3,[6][7][8].…”

“…It was found that the PVP could not be used for the light-cone gauges for several reasons [3,4].Further, it was shown that the PVP does not give the correct behavior for the Wilson loop to O[g 4 ] compatible with the calculation in Feynman gauge [9]. To overcome these difficulties, the Leibbrandt-Mandelstam prescription [2,3,4] was proposed and applied to the light-cone gauges.It was further generalized to the other axial gauges through what is called the uniform prescription [2,3]. One of the drawbacks of the L-M/Uniform prescriptions has been the presence of the non-local counterterms in proper vertices to all orders.…”

“…Another drawback is, of course,that the presence of two vectors leads to a larger set of possible divergences. Bassetto and coworkers have developed the techniques for dealing with non-local divergences [2,3]. Sev-eral other ways of dealing with the 1/(η.q) q -type singularities have also been proposed [6][7][8].…”

“…Such a well-defined treatment for defining the path-integral in axial-type gauges [and which is also applicable to host of other gauges [11]]has been established using the FFBRS transformation [12,13,7,8].It is expected that this path-integral so constructed should provide answers to all the questions regarding the various difficulties associated with the variety of the non-covariant gauges [e.g. axial, light-cone, temporal [2,3], Coulomb [14] that are thought to arise from the ill-definedness of the naive treatment. In other words,as we have emphasized, we expect that the way to deal with these difficulties does not require an ad hoc augmentation of rules as to how the diagrams are calculated, but is already contained in the process outlined earlier in references [11,12,13].The above expectation ,based on rigorous formal arguments, derives concrete support from several works [7,8,15].…”