Pinch technique at two loops: The case of massless Yang-Mills theories

Abstract

The generalization of the pinch technique beyond one loop is presented. It is shown that the crucial physical principles of gauge-invariance, unitarity, and gauge-fixing-parameter independence are instrumental for accomplishing this task, and is explained how the aforementioned requirements single out at two loops exactly the same algorithm which has been used to define the pinch technique at one loop, without any additional assumptions. The two-loop construction of the pinch technique gluon self-energy, and quark-gluon vertex are carried out in detail for the case of mass-less YangMills theories, such as perturbative QCD. We present two different but complementary derivations. First we carry out the construction by directly rearranging two-loop diagrams. The analysis reveals that, quite interestingly, the well-known one-loop correspondence between the pinch technique and the background field method in the Feynman gauge persists also at two-loops. Since we use dimensional regularization the entire construction does not depend on the value of the space-time dimension d. The renormalization (when d = 4) is discussed in detail, and is shown to respect the aforementioned correspondence. Second, we present an absorptive derivation, exploiting the unitarity of the S-matrix and the underlying BRS symmetry; at this stage we deal only with tree-level and one-loop physical amplitudes. The gauge-invariant sub-amplitudes defined by means of this absorptive construction correspond precisely to the imaginary parts of the n-point functions defined in the full two-loop derivation, thus furnishing a highly nontrivial self-consistency check for the entire method. Various future applications are briefly discussed.

Pinch technique: Theory and applications

“…Operationally the situation is significantly more involved compared to the one-loop case; the full details have been presented in [161]. Here we will give a brief qualitative description of how this construction proceeds.…”

“…s-t cancellation of Section 2; but when they hit on gg| T [4] |qq * they will trigger the one-loop version of the same s-t cancellation [43,161]. To demonstrate this latter cancellation one needs to know the result of the action of the longitudinal momenta k…”

“…In this subsection we present the PT construction at two loops [43,161]. Specifically, we will show that the PT can be generalized at two loops by resorting exactly to the same physical and field-theoretical principles as at one-loop.…”

“…Finally, in order to make contact with the notation of [43,161], we introduce the dimensionful projection operator t µν (q) ≡ q 2 P µν (q) . (6.2)…”

The Pinch Technique at Two Loops

“…It is only after all terms have been considered that the aforementioned crucial cancellations become possible. Finally, the counterterms of Γ (2) α (Q, Q ′ ) must be correctly accounted for [14]. Γ…”