Abstract
The present paper continues the series [V. V. Vereshagin, True self-energy function and reducibility in effective scalar theories, Phys. Rev. D 89, 125022 (2014); A. Vereshagin and V. Vereshagin, Resultant parameters of effective theory, Phys. Rev. D 69, 025002 (2004); K. Semenov-Tian-Shansky, A. Vereshagin, and V. Vereshagin, S-matrix renormalization in effective theories, Phys. Rev. D 73, 025020 (2006)] devoted to the systematic study of effective scattering theories. We consider matrix elements of the effective Lagrangian monomials (in the interaction picture) of arbitrary high dimension D and show that the full set of corresponding coupling constants contains parameters of both kinds: essential and redundant. Since it would be pointless to formulate renormalization prescriptions for redundant parameters, it is necessary to select the full set of the essential ones. This is done in the present paper for the case of the single scalar field.
Highlights
AND PRELIMINARIESIn this paper we continue constructing the renormalization scheme suitable for the single-scalar effective scattering theory
Let us recall that the theory is called effective if the interaction Lagrangian in the interaction picture contains all the monomials consistent with the given algebraic symmetry
In this paper we study the effective scattering theory (EST) that is just the effective field theory (EFT) only designed for perturbative calculations of the S-matrix elements on the mass shell
Summary
In this paper we continue constructing the renormalization scheme suitable for the single-scalar effective scattering theory. We need to suggest the more suitable form (without loss of generality) and, to suggest the complete set of essential coupling constants (see [3]) needed to fix the four-leg minimal effective vertices.4 To put it another way: it is necessary to point out the complete set of basic four-leg Lagrangian monomials (basis) with certain highest dimension D (see footnote 55). The coefficients at the individual basic monomials present the essential coupling constants It is the problem of constructing the basic set of monomials (more correctly, of the corresponding matrix elements in the momentum representation) which we solve in the given paper. This issue is important in the renormalization theory, because it would make no sense to formulate the renormalization prescriptions (RPs) that correspond to redundant parameters. (Latin letters show the number of Lorentz indices which are denoted by Greek letters.)
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