^{1}

^{3}.

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.

With the discovery of the electroweak Higgs boson by the Atlas [

It is relatively straightforward to compute the effective potential for a theory with spontaneous symmetry breaking [

Historically, the electroweak model with spontaneous symmetry breaking and the

In this work, we will construct an effective potential based entirely on the trace anomaly terms at tree and quantum level. First in Section

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

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

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

We shall start with the

The trace anomaly for

One can associate with the trace anomaly corresponding to the top Yukawa term in the Lagrangian the potential

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 effective potential in (

Moreover the parameters

Next we will set

In Figure

Plot of

An unusual feature of the potential obtained in Section

To illustrate this, we first consider the decay of the Higgs boson to two photons discussed in detail in the literature [

For comparison, we will determine the Higgs couplings in our potential before integrating out the gauge fields. We will explain in detail how this works for the decay to two photons of the Higgs boson and apply briefly our results to the two-gluon decay of the Higgs boson because the results are very similar. The relevant term in the Lagrangian is

The logarithms of the gauge fields are then expanded around the scale where the coupling constant is strong which is

Consequently, the kinetic terms for the electromagnetic field receive a factor

The same method can be applied to the two-gluon decay of the Higgs boson at the scale of reference

In this section, we computed the decay widths to two photons and two gluons of the Higgs boson in the context of an effective model. These decay widths as stated here depend on four parameters

In this work, we proposed an effective Higgs model constructed not by integrating out at one or two loops the gauge, fermion, and scalar degrees of freedom but by analogy with the low energy QCD linear sigma models. Thus this kind of model may be suitable for both when the Higgs boson is elementary and also when it is the result of some unknown strong dynamics. All terms in this potential apart from the

Next we consider a particular case of the potential again inspired by the construction of low energy QCD effective models [

In the same framework but before integrating out the gauge degrees of freedom, we determined the effective Higgs couplings to two photons and two gluons again in good accordance with what we know from one-loop calculations. This shows that the particular case of the model we proposed already describes very well at least a few phenomenological quantities. The potential derived here can be used to extract other possible couplings along the same lines.

The presence of a large number of parameters in the most general version of the model constructed here should not be regarded as a lack of predictability as compared to standard calculations at some loop order of the effective potential but as a way of encapsulating our lack of knowledge with regard to higher loop corrections to the phenomenological parameters. Since in general it is easier to compute beta functions and anomalous dimensions than intricate processes, this kind of model, especially if one uses physical arguments to further constrain or determine some of the parameters, may have important applications.

One potential application of our model would be to study the vacuum stability of the standard model. This topic was thoroughly studied in the framework of regular one-loop or two-loop renormalization improved effective potentials [

Other possible aspects and applications of our method will be investigated in further work.

The author declares that they have no conflicts of interest.