Advances in High Energy Physics

Volume 2018, Article ID 5294394, 8 pages

https://doi.org/10.1155/2018/5294394

## Standard Model Effective Potential from Trace Anomalies

National Institute of Physics and Nuclear Engineering, P.O. Box MG-6, Bucharest-Magurele, Romania

Correspondence should be addressed to Renata Jora; moc.oohay@arojrc

Received 15 November 2017; Revised 18 January 2018; Accepted 24 January 2018; Published 12 March 2018

Academic Editor: Andrzej Okniński

Copyright © 2018 Renata Jora. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

By analogy with the low energy QCD effective linear sigma model, we construct a standard model effective potential based entirely on the requirement that the tree level and quantum level trace anomalies must be satisfied. We discuss a particular realization of this potential in connection with the Higgs boson mass and Higgs boson effective couplings to two photons and two gluons. We find that this kind of potential may describe well the known phenomenology of the Higgs boson.

#### 1. Introduction

With the discovery of the electroweak Higgs boson by the Atlas [1] and CMS [2] experiments, the standard model has entered an era of unprecedented experimental confirmation with few hints with regard to its possible extensions to accommodate other particles, interactions, or symmetries. Even in its early years, the standard model Higgs boson has been the subject of a flurry of theoretical papers that dealt with its properties [3–9], the effective one or two loops’ potential [10–12], naturalness of the electroweak scale [13], or the vacuum stability of the standard model [14–16].

It is relatively straightforward to compute the effective potential for a theory with spontaneous symmetry breaking [11, 12]. This potential then may be renormalization group improved [14, 15], constrained to be scale invariant, and the associated vacuum expectation value or effective mass computed.

Historically, the electroweak model with spontaneous symmetry breaking and the linear sigma model for low energy QCD have been strongly related. The latter also displays spontaneous symmetry breaking associated with the formation of quark condensates and possessed also three Goldstone bosons, the pions. However the QCD linear sigma model was not straightforwardly derived in some loop order from the more basic QCD as the hadron detailed structure is as of yet unknown but rather based on the specific properties and symmetries already observed in the hadron spectrum. It is worth mentioning that linear sigma models have long been a basic tool for some effective description for low energy QCD and were generalized to the more comprehensive global group [17–20] and also to include four quark states [21–25] that can accommodate two scalar and two pseudoscalar nonets. Moreover without the specific knowledge of the detailed interaction, one can add phenomenological terms that mock up the axial [26] and the trace anomalies [27] with significant role in the hadron properties and good agreement with the experimental data.

In this work, we will construct an effective potential based entirely on the trace anomaly terms at tree and quantum level. First in Section 2, we will propose a general version where the parameters are constrained only by the requirement of mocking up exactly the trace anomaly. Then the model will contain a number of unknown parameters which should be determined from the phenomenological data. Further on, in Section 3 based on the analogy with low energy QCD, we introduce a particular version of the same potential where all the parameters are specified. We will study in this context the minimum equations and the mass of the Higgs boson. In Section 4 we analyze in the same framework the Higgs effective couplings to two photons and two gluons, relevant for the associated decay. Section 5 is dedicated to conclusions.

#### 2. Trace Anomaly Induced Potential

We start by considering the relevant part of the standard model Lagrangian apart from the kinetic terms for the fermions and for the Higgs doublet. Our choice is motivated by the fact that these terms are scale invariant at the tree level and for the quantum renormalized Lagrangian there is no contribution to the trace anomaly since there is no coupling constant in front of these terms. The gauge fields however behave differently; we can always transform the gauge field as , where is generic arbitrary gauge field and is its coupling constant. Then the corresponding kinetic term in the Lagrangian will appear with a factor which will contribute to the trace anomaly through its beta function. The relevant part of the Lagrangian is thenwhere is the Higgs doublet andand is the field tensor and is the one. Moreover since all fermions except for the top quark have small masses compared to the electroweak scale, we considered only the term pertaining to the top quark and its associated left handed doublet. By definition, all the terms in (1) are gauge invariant.

The next step is to take into account all trace anomalies known at both tree and quantum levels. It is known that the mass terms break scale invariance at tree level. However the quantum breaking of the scale transformation deserves a more detailed discussion. The trace anomaly refers to the renormalized Lagrangian. Then for a general Lagrangian depending on the fields (fermions or bosons) and coupling constants the quantum corrections to the trace anomalies are given bywhere and is the scale associated with the scale transformation. One may writeNext we observe that in functional sense as it is the case:which is zero by the equation of motion (the subscript indicates the functional sense). Since the trace anomaly calculations implicitly assume that the equation of motion for the renormalized field is satisfied, then clearly .

Consequently only the terms that contain coupling constants contribute to the trace anomaly whereas the contribution from the dependence of the renormalized fields with the scale is cancelled by the equation of motion. For a general gauge theory with fermions and scalars, one can make from the beginning the change of variable such that the coupling constant is eliminated from all gauge covariant derivatives. Thus the gauge invariant kinetic terms of the matter fermions or bosons bring no contribution to the trace anomaly.

We shall start with the gauge group that we will analyze in detail and just write down the results for and that can be easily obtained by applying the same procedure. All our calculations and definitions are inspired by the work in [17–20] by analogy with low energy QCD. Thus for a generic Lagrangian of the type,the new improved energy momentum tensor is defined asThis leads upon applying the equation of motion for the field to the following trace of the energy momentum tensor:We will apply (8) consistently in all our subsequent calculations of course adjusted to the specific Lagrangian.

The trace anomaly for reads [28, 29]We rescale the gauge fields back to their original form which leads toThen we consider a slowly varying background Higgs field and introduce the termwhere is some arbitrary scale and , , , and are arbitrary dimensionless coefficients. We compute the trace of the energy momentum tensor for the potential in (11) asto determine that it iswhich leads to the constraint . Then we apply the equation of motionand extract the field from the potential :We introduce the result in (15) into the expression for the potential in (11) to obtainA similar expression can be determined for :where , and are arbitrary dimensionless coefficients, and is the weak coupling constant. Then the potential induce by is justwith , and arbitrary dimensionless coefficients, and the strong coupling constant.

One can associate with the trace anomaly corresponding to the top Yukawa term in the Lagrangian the potentialwhere the anomaly requires that , where is the top Yukawa coupling. Here again and are arbitrary dimensionless coefficients.

The most interesting and complicated term to evaluate is however that of the mass of the Higgs bosons in conjunction with that of the quadrilinear coupling . The mass term is not scale invariant already at tree level and the trace of the energy momentum tensor will receive corrections also at the quantum level. A suitable potential is thenHere is the regular Higgs doublet and and are arbitrary dimensionless coefficients. First term gives the correct mass anomaly and the second and third terms give the correct anomaly provided that . The equation of motion leads toEquation (21) is a transcendental equation. To solve it, we first make the notations:Then (21) may be rewritten asWe denote to determinewhere is the Lambert function. We can assume to be small (making the final choice of the coefficients as such), in which case . This leads toWe introduce (25) into (20) to determineor by using (22)The full potential is thenNote that we could safely introduce the term because it is scale invariant.

#### 3. Trace Anomaly Inspired Particular Potential

The effective potential in (28) is constructed by analogy with low energy QCD effective models and contains in its most general form 20 parameters and 5 constraints. In order to substantiate that nevertheless this is a good phenomenological model, we need to stress out three important points: (1) The effective potential built here is an all orders potential and even if it contains a proliferation of parameters, these parameters encapsulate the intrinsic dependence on higher order loops without making any explicit calculations. Since computing beta functions and anomalous dimensions is far more amenable than calculating an effective potential or other processes at the same loop order, the apparent complexity leads in essence to an effective simplification. (2) The potential in (28) is completely independent of the nature elementary or composite of the Higgs boson and it is a reliable description also for the case when some strong dynamics are at play in the electroweak symmetry breaking. Note that the composite scenario is not completely excluded by the LHC or other experimental data [30]. (3) The number of parameters may be greatly reduced by making an educated guess of some of the parameters by analogy with low energy QCD [28] or even with the standard construction of the Higgs one-loop effective potential as described in the literature [14, 15]. Thus one can choose from physical arguments related to the relative renormalization of the wave function of the Higgs field the following expression for the constrained parameters:Here , , , , and are the beta functions for the coupling constants of the , , and groups, the quadrilinear term in the Higgs potential, and the top Yukawa coupling, respectively (we use the results in [11]). Moreover is the anomalous dimension of the Higgs field.

Moreover the parameters and are redundant because they are already associated with a factor in front of the respective terms so they may be chosen asHere we took into account as arguments of the logarithm the natural expressions of the scalar polynomials as they appear in the Lagrangian.

Next we will set constant and apply the standard approach for constructing effective potentials. We denote [14, 15]where is the running parameter andNote that, for , . We consider all the parameters computed at this scale and apply the minimum equation:In (33), all couplings are known except for . We further require that the minimum is obtained for GeV and solve the minimum equation for the parameter . Here we make the underlying assumption that if the potential is phenomenologically viable as an effective potential, then it should lead to a mass of the Higgs boson very close to the actual mass (this is actually exact in the on-shell subtraction scheme where the renormalized mass is equal to the pole one).

In Figure 1, we plot in terms of the parameter to determine . We use this value to further calculateThen the resulting mass of GeV is very close to the actual experimental mass of the Higgs boson GeV. By varying the top Yukawa coupling (here we took with the mass of the top quark GeV [30]), one can reproduce the exact pole mass of the Higgs boson.