Consistency Condition for the Pinch Technique Self - Energies at Two Loops

A simple and testable necessary condition for the gauge independence of the Pinch Technique self-energies at two loops is discussed. It is then shown that, in the case of the Z and W self-energies, the condition is indeed satisfied by the Papavassiliou-Pilaftsis formulation. The Pinch Technique (PT) is a convenient algorithm that automatically rearranges S-matrix elements of gauge theories into modified, gauge-independent self-energies, vertex, and box diagrams. In turn, the new corrections exhibit very desirable theoretical properties. For these reasons, the PT has been frequently employed in recent discussions of QCD and Electroweak Physics [1]. A temporary drawback is that the approach has been fully developed only at the one-loop level. Very recently, however, Papavassiliou and Pilaftsis (P-P) proposed a method to construct PT self energies at higher orders [2]. Calling Π̂ and Π the PT and Rξ transverse self-energies, respectively, and focusing on the Z case, one has Π̂ ZZ (s) = Π(s) + (Π(s)) , (1) where the “pinch part” (Π(s)) has the structure (Π(s)) = c1(s−M 2 0 )V P (s) + c2(s−M 2 0 )B (s)− R(s) . (2) In Eq.(2) the bare mass M0 is assumed to be defined in a gauge invariant manner, tadpole contributions are included in Π(s), V P (s) and B (s) are the pinch parts from vertex and box diagrams, respectively, and R(s) is a residual amplitude of O(g) proposed in Ref.[2]. It is discussed in detail later on at the O(g) level. Because of the limited knowledge currently available concerning multi-loop amplitudes in gauge theories, a general proof that Π̂ ZZ (s) is gauge invariant in higher orders is not presently available. One of the aims of this report is to note that by judiciously restricting the domain of s to lie in the neighborhood of s̄, the complex-valued position of the propagator’s pole, one can obtain an expression for which the gauge independence can be tested on the basis of current knowledge. Specifically, we consider the neighborhood |s− s̄| ≤ O(g|s̄|), which roughly includes the resonance region. Recalling that s̄ − M 0 = O(g), through O(g) Eqs.(1,2) become Π̂ ZZ (s) = Π 1 (s) + c1(s−M 2 0 )V P 1 (s) + Π 2 (s̄)−R 2 (s̄) +O(g) , (3) where the indices i = 1, 2 denote O(g) and O(g) contributions. Through O(g) the first two terms in the r.h.s. of Eq.(3) equal the one-loop PT self-energy Π̂ ZZ