Radiatively Induced Breaking of Conformal Symmetry in a Superpotential

Radiatively induced symmetry breaking is considered for a toy model with one scalar and one fermion field unified in a superfield. It is shown that the classical quartic self-interaction of the superfield possesses a quantum infrared singularity. Application of the Coleman-Weinberg mechanism for effective potential leads to the appearance of condensates and masses for both scalar and fermion components. That induces a spontaneous breaking of the initial classical symmetries: the supersymmetry and the conformal one. The energy scales for the scalar and fermion condensates appear to be of the same order, while the renormalization scale is many orders of magnitude higher. A possibility to relate the considered toy model to conformal symmetry breaking in the Standard Model is discussed.


I. INTRODUCTION
It was shown [1] that infrared divergences in quantum loop contributions to effective potentials of various models in quantum field theory (QFT) can lead to the necessity to introduce a non-zero renormalization scale and thus generate a spontaneous breaking of the conformal symmetry. In particular, the φ 4 model as well as Abelian and Yang-Mills gauge models were considered in [1]. Here we will apply the Coleman-Weinberg (CW) mechanism to a simple QFT model for a superfield which joins scalar and fermion physical fields, see for example [2,3] for application of the CW mechanism in different physical scenarios. energy scale of about 100 GeV is seen both in the Higgs (and electroweak sector) and the top quark mass. The relation 4m 2 H ≈ 2m 2 t ≈ v 2 between the observed Higgs boson mass m H , the top quark mass m t , and the Higgs boson vacuum expectation value v holds with a high accuracy [4]. In any case, the coincidence of the scales is an intriguing puzzle. Another face of the electroweak scale puzzle is the hierarchy problem of the SM due to quadratic divergences in the running of the Higgs boson mass within the SM. In fact, renormalization of m H suffers from fine tuning between the (loop) contributions due to the top quark, the Higgs boson, and EW bosons (note that only longitudinal components of W and Z bosons, i.e. the scalar Goldstones, contribute). It is well known that resolution of the fine tuning problem can be done by a supersymmetric extension of the SM. In any case, a certain (symmetry?) relation between fermionic and bosonic contribution is required to solve the problem.
On the other hand there are many indirect indications the the Conformal Symmetry (CS) might be the proper symmetry of the underlying fundamental theory, while the SM is just an effective model emerged after a spontaneous breaking of the CS, see e.g. Refs. [5,6].
In Ref. [7], the possibility to generate a soft breaking of the conformal symmetry in the sector of the SM which joins the Higgs boson and the top quark was discussed. In fact, the infrared singularity is present in this system and the Coleman-Weinberg mechanism can be applied. Nevertheless, the question about the relation between renormalization conditions for the scalar and the fermion field remains unsolved. As discussed, a certain bootstrap should happen in the SM between the Higgs boson and the top quark. In this paper we suggest to look what comes out if the two fields are joined into a superfield. Of course, it is just one of many other possibilities but it provides a certain feeling of the bootstrap.
It is frequent to find in the current research proposals that suggest (supersymmetric) extensions of the Standard Model with introduction of a hidden sector that shows conformal invariance above a certain high energy scale (see in a different context Ref. The first aim of this paper is to investigate the supersymmetric extension of the model based on the superconformal field theory by means of introduction of a scalar superfield, as we will show in Section II. It is well known that the four dimensional superconformal field theory is powerful enough to obtain the crucial dynamical information about the physics of particle interaction due to the fact that the interaction itself is hidden inside quadratic and dynamical terms in the Lagrangian of the theory. For example, the relation between the R-charge and the conformal dimension determines the conformal dimensions of the chiral operators beyond the perturbation theory. We also have more severe inequalities for conformal dimensions that are not available in non-supersymmetric theories. In this sense, the introduction of the SUSY is theoretically well motivated. Physically the interplay between conformal symmetry and supersymmetry and their breaking (of both or of any of them, total or partial) introduces automatically extra constraints on particle physics.
As we pointed out before, previous investigations on the particle physics within the context of (super) conformal models assume that a certain particle sector remains conformal at least down to the electroweak scale, at which any experimental evidence is expected. The problem is claimed usually of a partial breaking of SUSY or conformal symmetry and how to conciliate both. It is usually solved by means of a gauge mediation or by tuning the Kähler potential. However there are no such problems through our paper, consequently this particular point not will be analyzed here.

II. CONFORMAL MODELS AND SUPERSYMMETRY
The hints in order to introduce the fermionic interactions in any classical bosonic action endowed by conformal symmetry were presented for the first time in the seminal papers of Akulov and Pashnev [9] where the starting point was the well know the AFF (de Alfaro, Fubini and Furlan) conformal model [10]. Without going into details (see [9]), the idea was to introduce a superfield having the form into the following general n-dimensional action with the standard super-kinetic term where Φ is the chiral superfield (with standard anti-chiral counterpart Φ ) and the super- The interaction part was defined in the general form as Without loss of generality the simplest 4-dimensional case will be treated. Remind now the effective potential for a scalar field with a ϕ 4 interaction, which was derived by S. Coleman and E. Weinberg [1] in the one-loop approximation The presence of the renormalization scale M indicates the radiatively induced breaking of the conformal symmetry in this model.
We can pass from the bosonic effective potential to the supersymmetric one by introducing the superfield. Note that due that the standard version of the four-dimensional supersymmetry, the simplest superfields contain a complex Lorentz scalar and a chiral (lefthanded or right-handed) fermion. To avoid confusion henceforth we define ϕ 2 ≡ ϕϕ. Then we obtain the following expression where the Grassman integration was performed under the (physical) measure where a and b are constants related to the group manifold structure (volume), that must be included into the above measure in order to recover the original Coleman-Weinberg potential when all fermions vanish.
Let us look for a minimum of the potential. The conditions lead to the following solution for the scalar and fermion condensate values: We assumed that the coupling constant λ < ∼ 1 so that the perturbative solution is reliable.
We would like to make a parallel to the sector of the Standard Model, which joins the Higgs self-interaction and the Yukawa term of the top quark, see [7] for details. In fact the structure of this sector is exactly the same as the one of our toy model. The spontaneous breaking of the conformal symmetry in the system leads to generation of masses both for the scalar and fermion fields in the standard way after the shift of the Note that the energy scale as for the masses as well as for the condensates of both fields is the same: Remind that the coincidence of scales of the Higgs boson mass and of the top quark one is one of the puzzles in the SM.
In the case if we take the model with a Kähler structure, the potential is slightly modified as: with |c| < 1 that for sign (c) = +1 we have compact manifold, and for sign (c) = −1 we have a non-compact one. The modified potential leads to a similar solution for the minimum position.

III. DISCUSSION
In general, even without any (super)symmetry between the scalar and fermion fields we should look for the minimum of the effective potential. The symmetry condition just helps to derive a relation between the condensate values. Certainly, the top quark and the Higgs boson in the SM are not related by SUSY. On the other hand, we clearly see that they are somehow tightly linked to each other. So, we take SUSY just as a toy model to test the connection between t and H. We suppose that some of the features which appear in the SUSY relation might be relevant for the true (still unclear) picture. Because, the physical states, that we are interested in, become to be part of a same supersymmetric multiplet, they are not independent. This fact reduces the number of independent coupling constants.
Notice or not [11]. Note that instead of introducing multiple Higgs to provide partners for the remaining quarks, we can introduce only one Higgs with quaternionic symmetry [12] (for example, in the Weinberg-Salaam model the main idea in order to increase the number of fields is based on the observation that there exists the following underlying quaternionic symmetry, namely 1 2 g ′ B µ + gσ i A i µ ≡ Q µ , this issue is now under research [12]). It is worth to note that there is a close relation between breaking of the conformal symmetry and SUSY. It was was pointed out in Ref. [13] that quantization of theories within the Hamiltonian formulation suffers from difficulties associated with the ordering of operators. Moreover, the presence of fermionic operators creates additional difficulties that are translated to the breaking of symmetries of the physical system under consideration.
These ordering difficulties are present as in the definition of SUSY charges as well as in the corresponding SUSY generators. Generally, they appear as operators with arbitrary factors that take into account all ways of ordering where in the two solutions are non-normed which corresponds to the case of spontaneously broken supersymmetry. Thus, spontaneous breaking of supersymmetry at the quantum level is possible due the indefiniteness in ordering of operators.
Since normalization is not inherent to the conformal symmetry condition, it seems ap-parently that the breaking of SUSY would not affect the conformal symmetry. But at the quantum level, the ambiguity in the ordering of operators establishes a connection between both symmetry breaking. Precisely this condition seems to be related to what happens in the breaking of supersymmetry at finite temperature, even in the microcanonical picture [? ]. This allows one to make a conjecture on triality between supersymmetry breaking, breaking of conformal symmetry, and non-zero temperature from the quantum level. Note also that since LHC started looking for superpartners, the task becomes extremely hard. Probably the difficulties to interpret the absence of hints for supersymmetry at LHC implies that there exist a supersymmetric Λ value that can be greater than expected, consequently higher values of Λ can be justified in such a case.
Searching for similar effects in other quantum mechanical models is of considerable interest and will allow further studies of the phenomenon of spontaneous symmetry breaking in physics.