Renormalisation group determination of scalar mass bounds in a simple Yukawa-model

Abstract

The scalar mass is determined in the simplest scalar-fermion Yukawa-model in the whole range of stability of the scalar potential. Two versions of the Functional Renormalisation Group (FRG) equations are solved, where also composite fermionic background is taken into account. The close agreement of the results with previous studies taking into account exclusively the effect of the scalar condensate, supports a rather small systematic truncation error of FRG due to the omission of higher dimensional operators.1 Motivation: from "Moscow-zero" to Higgs-mass boundsThe discovery of the "Moscow-zero" [1] (alias Landau-pole) appears from modern perspective as a true Archimedean "fixed point" on the historical path of renormalized quantum field theories. Its real importance has been recognized after desperate discussions concerning the very sense of quantum field theory. The present conference gives the opportunity to quote from the history notes of A.A. Anselm [2] that researchers of the Theory Division of the Leningrad (at present Petersburg) Nuclear Physics Institute under the "natural scientific leadership" of V.N. Gribov contributed "many important papers during the 50's and the beginning of the 60's to the development of the "null-charge" problem." And the interest of Volodia Gribov did not diminish with time [3].At present two alternative attitudes are taken concerning the exact nature of the field theories which in the framework of (resummed) perturbation theory appear to possess Landau-pole. The concept of asymptotic safety [4] envisages the existence of an UV-attractive fixed point in the renormalisation flow of the model under study. The energy scale of this fixed point should be lower than the scale of the Landau-pole. Assuming the existence of such a fixed point in the joint theory of gravity and the Standard Model at the scale of the Planck-mass a quite accurate prediction(!) of the mass of the Higgs particle was achieved [5]. Some recent papers [6] express very optimistic views concerning the existence of a UV-safe completion of the Standard Model on its own, but to present date no such interactive fixed point could have been found.

Fixed points and the spontaneous breaking of scale invariance

“…Various incarnations of "functional renormalisation" are available [33][34][35][36][37] which, in combination with systematic approximations [38][39][40][41][42][43][44], give access to the relevant physics including at strong coupling. Recent applications of the methodology include models of particle physics [45][46][47], purely fermionic models [48], quantum gravity [49][50][51][52][53][54], and models in fractal or higher dimensions [55][56][57][58][59][60] The primary focus of this work will be on (φ 2 ) 3 d=3 scalar field theories in the limit of infinite N [61,62]. Our interest in this model is twofold: Firstly, the theory offers a rich spectrum of phenomena ranging from interacting ultraviolet fixed points with asymptotic safety to strongly interacting Wilson-Fisher fixed points, phase transitions and dimensional transmutation, and interaction-induced spontaneous breaking of scale invariance [63].…”

Impact of generalized Yukawa interactions on the lower Higgs-mass bound

“…In these studies, based on the functional RG [48][49][50][51] as well as nonperturbative lattice simulations [52][53][54][55][56], a lower Higgs mass bound emerges naturally from the RG flow itself and is primarily connected with a consistency condition on the bare action rather than with the stability of the effective Higgs potential which is a resulting long-range property. Moreover, it can be shown that higher-order operators at the standard-model cutoff scale can have a quantitative impact on the lower Higgs mass bound as well as the stability properties of the bare potential.…”

Renormalization group flow of the Higgs potential

“…Our estimates based on a variety of model studies suggest that even Planck scale operators can induce a relaxation of the conventional lower bound on the order of 1% for the absolute stability bound. While we have considered here the influence of only one φ 6 -type operator as an example, similar features occur, for instance, upon the inclusion of higher-order fermionic operators [30,31] or mixed operators [32] as well as in theories with additional scalar fields, e.g., dark matter candidates [47]. Most of the quantitative studies so far have explored only initial conditions which are already sufficiently close to the Gaußian fixed point.…”

“…As a simple generalization, let us consider the next higher-order operator in the potential ∼ λ 3,Λ φ 6 . In fact, many further operators can be (and have been) studied; see, e.g., [30,31] for higher-order fermionic operators or [32] for mixed operators. The present simple φ 6 example suffices to illustrate an important point here.…”