Masslesslike minimal subtraction for massive scalar field theory

Abstract

We introduce the simplest minimal subtraction method for massive λφ 4 field theory with O(N ) internal symmetry, which resembles the same method applied to massless fields by using two steps. First, the utilization of the partial-p operation in every diagram of the two-point vertex part in order to separate it into a sum of squared mass and external momentum, respectively, with different coefficients. Then, the loop integral which is the coefficient of the quadratic mass can be solved entirely in terms of the mass, no longer depending upon the external momentum, using the parametric dissociation transform. It consists in the choice of a certain set of fixed values of Feynman parameters replaced inside the remaining loop integral after solving the internal subdiagrams. We check the results in the diagrammatic computation of critical exponents at least up to two-loop order using a flat metric with Euclidean signature.

Masslesslike minimal subtraction for massive scalar field theory

“…This has the standard form since all the terms in the bare Lagrangian density are negative and the weight factor is an exponential of a negative quantity.With this new notation for the bare Lagrangian density, Γ (4) (p i , µ 0 , λ, Λ) in the present paper corresponds to −Γ (4) (p i , µ 0 , λ, Λ) in ref. [17]. In addition, odd powers of the coupling constant receive a minus sign in all primitively divergent vertex parts (without the factor of i present in the Minkowski case).…”

“…To begin with we promote a slight change of notation in connection with our previous work [17]. The reason for this change of notation is that we want to address the problem of a scalar quantum field on Minkowski spacetime in complete analogy with the order parameter (scalar field) defined on Euclidean space in the ambit of static critical phenomena.…”

“…Although the arguments are entirely analogous for Minkowski and Euclidean metric signatures, we now outline these different versions separately in order to avoid any possibility of misunderstanding in the unified description of this technique. We will be a little more explicit in the discussion of the situation with the Euclidean metric signature as a follow-up discussion to reference [17].…”

“…The implementation of the parametric dissociation transform (P DT ) [17] is described in Appendix A. The idea is just highlighted here as follows.…”

“…This work is devoted to provide the details of the method briefly described in [17]. Namely, we present the modification of the diagrammatic expansions of the primitively divergent vertex parts as well as their renormalization resulting in the simplest minimal subtraction method for massive scalar field to date, with a minimal number of diagrams.…”