Renormalization Group Improvement and Dynamical Breaking of Symmetry in a Supersymmetric Chern-Simons-matter Model

In this work, we investigate the consequences of the Renormalization Group Equation (RGE) in the determination of the effective superpotential and the study of Dynamical Symmetry Breaking (DSB) in an N = 1 supersymmetric theory including an Abelian Chern-Simons superfield coupled to N scalar superfields in (2+1) dimensional spacetime. The classical Lagrangian presents scale invariance, which is broken by radiative corrections to the effective superpotential. We calculate the effective superpotential up to two-loops by using the RGE and the beta functions and anomalous dimensions known in the literature. We then show how the RGE can be used to improve this calculation, by summing up properly defined series of leading logs (LL), next-to-leading logs (NLL) contributions, and so on... We conclude that even if the RGE improvement procedure can indeed be applied in a supersymmetric model, the effects of the consideration of the RGE are not so dramatic as it happens in the non-supersymmetric case.


I. INTRODUCTION
Dynamical Symmetry Breaking (DSB) constitutes a very appealing scenario in field theory, where quantum corrections are entirely responsible for the appearance of nontrivial minima of the effective potential [1]; in the case of a classically scale invariant model, one may additionally say that all mass scales are generated by these quantum corrections and are fixed as functions of the symmetry breaking scale. The absence of a tree level mass term for the scalar field naturally avoids the large quadratic corrections to this mass, thus evading hierarchy problems. This scenario would be particularly interesting in the Standard Model, but earlier calculations pointed to a dead end: quantum corrections turned the effective potential unstable, rendering DSB impossible [2]. It has been shown that this conclusion, based on the effective potential calculated up to the oneloop level, could be modified substantially by more refined techniques [3][4][5][6][7][8][9]; these made use of the Renormalization Group Equation (RGE) to sum up infinite subsets of higher loop contributions to the effective potential. More than a quantitative correction over the one-loop result, these improvements lead to a new scenario, where DSB was operational [3,4]; further improvements were able to include corrections up to five loops in the effective potential [10,11], bringing the prediction for the Higgs mass close to the experimental value indicated by current observations at the LHC [12,13] (for other works regarding conformal symmetry in the Standard Model see for example [14,15]).
Besides being a viable ingredient to the Standard Model phenomenology, DSB also occurs in other contexts, such as lower dimensional theories. Particularly interesting are models involving the Chern-Simons (CS) term [16], which induces a topological mass for the gauge field, and relates for example to exotic statistics and fractional spin, relevant properties for the study of the quantum Hall effect [17]. The basic renormalization properties of models containing a Chern-Simons field have been studied for quite some time [18][19][20][21][22][23][24]. We shall be particularly interested in models with scale invariance at the classical level, that is, with a pure CS field coupled to massless scalars and fermions, with Yukawa quartic interactions and scalar ϕ 6 self-interactions. In these models, the one-loop corrections calculated using the dimensional reduction scheme [25] are rather trivial, since no singularities appear, and no DSB happens either; at the two-loop level, however, one finds renormalizable divergences. Also, the two-loops effective potential V ef f exhibits a nontrivial minima, signalizing the appearance of DSB. Due to the nontrivial β and γ functions at two-loop level, one may obtain an improvement in the calculation of V ef f by imposing the RGE where α i are the coupling constants of the theory, µ is the mass scale introduced when extending the theory to D = 3 − ǫ, γ ϕ is the anomalous dimension of scalar field ϕ, and φ is the vacuum expectation value of ϕ. This improved effective potential was calculated in [26], and it was shown to imply in considerable changes in the properties of DSB in this model, thus providing another context where the consideration of the RGE is essential to a proper analysis of the phase structure of the model.
Our objective is to verify whether in supersymmetric models containing the CS field, one can also use to RGE to improve the calculation of the effective potential in a similar way to what was done in [26], and look for consequences in the DSB. Supersymmetric theories in (2 + 1) dimensions are often considered as useful theoretical laboratories for the study of general properties of supersymmetric models, while still presenting interesting properties by themselves. Recently, for example, extended supersymmetric Chern-Simons models have been studied as duals of specific supergravity models, in the context of AdS/CFT duality [27][28][29][30]. The renormalization properties of supersymmetric models including the CS have also been extensively studied [19,[31][32][33][34]. We shall be particularly interested in calculations done in the superfield language [35,36], in which supersymmetry is manifest in all stages of the calculations.
There are subtleties in the definition of the effective superpotential in three dimensions [33,37], which we will discuss shortly: the outcome is that we will be able to discuss the breaking of scale and gauge symmetries, but supersymmetry will remain unbroken. The first part of our work contains the computation of the renormalization group β and γ functions of a general supersymmetric Chern-Simons theory coupled to matter superfields, up to two-loops, using the superfield formalism. Using this result, we are able to use the RGE in the superfield formalism to find an improved effective superpotential, which will be used to study DSB in our model. We will show that, contrary to what happens in the non supersymmetric case [26], here the RGE improvement lead to slight modifications in the DSB scenario.
This paper is organized as follows: In Sec. II, we present the N = 1 supersymmetric abelian Chern-Simons model coupled to N massless scalar superfields in (2 + 1) dimensions, and we calculate the effective superpotential up to the two-loop level. Section III contains the calculation of the counterterms needed to render the quantum corrections to the vertex functions finite up to two-loop order; these are used in Sec. IV to calculate the β functions and the anomalous dimensions γ. The RGE is then used to calculate the improved effective superpotential, and this result is the base of the study of DSB that is done in Sec. V. Section VI presents our conclusions and perspectives.
Some technical aspects and further clarifications to our calculations are collected in the Appendices.

II. THE GAP EQUATION AND THE EFFECTIVE SUPERPOTENTIAL
Our starting point is the classical action in N = 1 superspace of a Chern-Simons superfield Γ β coupled to N massless scalars superfields Φ, with a quartic self-interaction [34], where N Γ α is the gauge supercovariant derivative and a = 1, ... N , N being the number of flavors of the complex scalar superfield Φ. We follow the conventions for three-dimensional supersymmetry found in [35].
The action in Eq. (3) represents the supersymmetric version of the model considered in [24,26]: indeed, Eq. (3) can be cast in component form as where the superfield Φ a has field components ϕ a , Ψ a and F a that are, respectively, a set of N complex scalar fields, Dirac fields and complex scalar auxiliary fields, while the only component of the CS superfield Γ β that is not fixed by its equations of motion is the CS field A M . Here, M is the usual spacetime index, which assume the values 0, 1, 2, and the sum over repeated flavor indices is implicit (see appendix A for more details).
We shall need to calculate, up to the two-loop level, the effective superpotential and the renormalization group β functions and anomalous dimensions. For the first task, we shall follow a procedure similar to what was described in [32,33,37], that will be briefly summarized in this paragraph. We consider a shift in the N -th component of Φ a , by the background (constant) superfield and we impose Σ = Π = Φ j = 0, for j = 1, 2, ..., (N − 1). On general grounds, the superfield effective potential can be cast as where we call the function K (σ cl ) the Kählerian effective superpotential, inspired by the traditional nomenclature adopted in four-dimensional spacetime. As discussed on [33], the knowledge of K is sufficient for investigating the dynamical breaking of the gauge symmetry, while the study of a hypothetical supersymmetry breaking would need also the calculation of F [38,39]. In this work, we restrict ourselves to the calculation of K, hence we disregard all terms that involve supercovariant derivatives acting on σ cl . With this restriction, the standard superfield formalism can be applied at all steps of the calculation, manifestly preserving supersymmetry.
The shift in Eq. (5) induces in the action a mixing between the scalar and spinor superfields that can be removed using an appropriate gauge fixing and Faddeev-Popov terms, where N σ cl Π is the gauge fixing function, α is the gauge parameter, c andc are the ghosts superfields. This procedure is similar to the choice of the R ξ gauge commonly used in the study of non-superymmetric broken gauge theories [40].
We may write the regularized action for this model as follows, where L ct is the counterterm Lagrangian.
The propagators assume the form where δ 2 12 = δ 2 (θ 1 − θ 2 ) and In the following, we will work in the supersymmetric Landau gauge, α = 0. With this choice, the propagator of the spinor superfield assumes a simpler form, and the ghosts decouple from the theory.
We start by calculating the one point vertex function and imposing that it vanishes, a condition known as the gap equation. In the one-loop level, only four diagrams contribute, which are shown  Fig. 2, omitting an overall factor of i in Fig. 1; the corresponding contributions to the vertex function are 3 . Therefore, the quantum correction to S Imposing S Σ (1)l = 0, one verifies that the one-loop correction is not sufficient to ensure a nontrivial solution to the gap equation, i.e., the only solution is σ cl = 0. Therefore, there is not DSB in the first loop correction. The incorporation of two-loops corrections involve the calculation of the diagrams drawn in Fig. 2, whose results are tabulated in Table I and leads to where and A 4 is the counterterm. Notice that the pole in ǫ = 3 − D is entirely contained in the constant ζ.
In obtaining this result, manipulation of supercovariant derivatives was done with the help of the Mathematica package SusyMath [41] (see also Appendix B for details on the momentum integrals).
The unimproved Kählerian effective superpotential is obtained as in [33], from the integration of the one-point vertex function, Eq. (14) plus the classical contribution from Eq. (9), where with L ≡ ln σ 2 cl /µ .

III. COUNTERTERMS CALCULATION
It is a known fact in the literature that the beta function for the gauge coupling g vanishes [19], and there are no divergent corrections to the two point vertex function of the CS superfield Γ [42,43]. We do have nontrivial contributions for the beta function for the scalar self-coupling λ, as well as the wave function renormalization of the Φ a superfield. To evaluate these quantities, in this Section we calculate in the two-loop approximation the divergent parts of the two point vertex function S ΦΦ 2l and the four-point vertex-function S ΦΦΦΦ 2l , in the symmetric phase. We remind that the regularization method used by us, dimensional reduction to D = 3−ε, ensure that one-loop corrections are finite, while two-loops corrections will be seen to be divergent with one simple pole at ε = 0.
Free propagators for scalar and gauge superfields are given, respectively, by while the elementary vertices are and in the following we will work again in the Landau gauge α = 0.  Table II: Divergent parts of the diagrams appearing in Fig. 3 omitting an overall factor of For the divergent part of the two-point function of the Φ a superfield (Fig. 3) we obtained where the counterterm δ Φ is responsible for the cancellation of the divergence proportional to the constant ζ.

IV. RENORMALIZATION GROUP ANALYSIS
The key point of our procedure is to impose that the effective superpotential obeys the RGE, or To this end, we need to calculate the renormalization group β functions and anomalous dimensions γ.
The β λ function is calculated by differentiating with respect to µ Eqs. (25c) and, and we write it in the form where y = g 2 and The γ Φ function (anomalous dimension) associated to the Φ superfield is These results are in qualitative agreement with the ones in [19], while differences in the numerical coefficients might be due to differences in conventions and/or normalizations.
To calculate γ σ that appears in the RGE, we substitute the β functions, Eqs. (26) and (27), and the Kählerian effective superpotential that was found from the one point vertex function, Eq. (16), into RGE itself, obtaining where A 1 , A 2 and A 3 being the constants defined in Eq. (15). In essence, Eq. (30) implies that For convenience, we solve Eq. (29) for γ σ assuming λ = 0 as well as y λ < 1, in which case λ ′ ∼ λ.
With this approximation, we obtain We stress that the apparent singularity at λ = 0 in γ (2) σ is an artifact of our approximation, which assumed λ = 0, and we will see it will not imply in any singularity in the calculation of the effective potential. If we consider λ → 0 the theory is not singular because from Eq. (31), λ ′ = f y 2 , y 3 , and γ σ could be again obtained from Eq. (29). The case in which λ = 0 at tree level is not the natural one to consider, so we will not discuss it any further.
Now following for example [26], we define and rearrange the contributions to S ef f according to the relation between the aggregate powers of coupling constants and the power of the logarithm L, Hereafter, LL stands for Leading Logarithms, N LL for Next-to-Leading Logarithms, and so on. The RGE, Eq. (36), can also be split in the same fashion, and this is explicit shown in Eq. (35) Comparing Eq. (34) with Eq. (17), one can identify the numerical values of the initial C coefficients, as follows Now, we start by focusing on the terms of order λ n y m L n+m−2 in Eq. (35), i.e., which furnishes the following relation This implies that C LL n,m = 0 for n + m ≥ 2, i.e., Since the coefficient C LL 1,0 is already known from Eq. (37), S LL ef f is completely determined. It is interesting to compare Eq. (38) with its counterpart in the non-supersymmetric model, Eq. (11) in [26]: while both equations determine all higher order coefficients C LL , in the supersymmetric case, these are simply enforced to vanish, meaning the effective superpotential does not receive leading logarithms corrections at higher loop orders. That complies with the general picture that supersymmetric models have less divergences and have simpler renormalization properties.
Having found S LL ef f , we can now consider terms of the order λ n y m L n+m−3 in (35), i.e., The first term has already been determined, and we proceed to find out S N LL ef f which, as before, will be written in the form where and C N LL n,m = 0 for n + m ≥ 4. From these relations, we cannot determine new coefficients but it is possible to reobtain C N LL 3,0 with a value compatible with Eq. (37), which is a consistency check of our procedure.
In the next step, we focus on terms proportional to λ n y m L n+m−4 in (35), from which we find the relation, These relates the 2N LL coefficients with the N LL and LL ones, i.e while other values C 2N LL n,m are zero. Therefore we obtain where C 2N LL

V. DYNAMICAL BREAKING OF SYMMETRY
In this Section we study the dynamical breaking of the conformal symmetry that occurs in the present theory, based on the improved Kählerian effective superpotential that was obtained in the previous Section, ρ being a finite renormalization constant. To this end, we need to find the minima of the component version of the improved Kählerian effective superpotential, V I ef f , obtained as follows, where in the last equality we used the equation of motion for σ 2 , which is the vev of an auxiliary field. The same can be done to find the V ef f , that is the component version of the unimproved Kählerian effective superpotencial, Eq. (16). Our objective is to see whether the scenario of DSB is substantially modified in the improved case, compared with the unimproved one.
The constant ρ is fixed using the normalization condition which follows from the component expansion of the classical action given in Eq. (4), after the shift ϕ → ϕ + √ N σ 1 that parallels Eq. (5). The effective potential V I ef f (σ 1 ) has a minimum at σ 2 1 = µ, i.e., d dσ 1 V I ef f (σ 1 ) This equation is used to determine the value of λ as a function of the free parameters y and N ; upon explicit calculation, Eq. 56 turns out to be a tenth-degree polynomial equation in λ, and among its solutions we will look for those which are real and positive and correspond to a minimum of the potential, i.e., These procedure was implemented in a Mathematica program, and we verified that it can be performed for any value of y and λ (provided that y λ and that both are smaller than unity).
That means DSB is operational in this model for any reasonable value of its parameters. As an example: choosing y = 0.6 and N = 1, we found that λ = 0.0242 for the unimproved effective potential and λ I = 0.0269 for its improved version. We see that the enforcement of the RGE on the calculation of the effective potential provides only a quantitative improvement on the parameters of the DSB. This is rather different from the scenario found in the non-supersymmetric version of the same model, studied in [26], where the RGE improvement provided qualitative changes in the phase structure of the DSB. The incremental aspect of the improvement, in the present case, can also be seen by plotting both the improved and unimproved effective potentials as in Fig. 5. We could also notice that, in our case, for N > 1, the improvement becomes even less important.

VI. CONCLUSIONS
The mechanism of symmetry breaking is central in the formulation of a consistent quantum field model of the known elementary interactions, and the possibility that quantum corrections of a symmetric potential could alone induce such symmetry breaking is a rather interesting one, not only for its mathematical elegance, but also for physical reasons. Recently, for example, a mechanism of dynamical symmetry breaking in a scale-invariant limit of the Standard Model is being discussed as a viable mechanism for generating a mass for the Higgs particle compatible with experimental observations, but protected from hierarchy problems. The idea of using the RGE to improve the calculation of the effective potential, summing up terms arising from higher loop orders organized as leading logarithms, next-to-leading logarithms, and so on, is central to this approach.
This idea is worth extending on other contexts, for example supersymmetric models. This work is a first attempt in that direction. We explicitly show that indeed this program can be applied to a supersymmetric model in the superfield formalism, which is one of the main technical result of this paper.
We discussed a general supersymmetric Abelian Chern-Simons model coupled to arbitrary number of scalar and fermion fields. Matter fields are assumed to be minimally coupled to the CS field, together with quartic self-interaction. For the sake of the RGE improvement, we had to calculate perturbative corrections to the vertex functions up to the two-loop level, finding the renormalization group β functions and anomalous dimensions γ. These, together with the RGE, allowed a calculation of the improved Kählerian effective superpotential in this approximation and, from this, we found improved component effective potential V I ef f . This latter was used to study DSB in our model, and the results compared with the potentials calculated from a simple two-loop study of the tadpole equation.
The end result was that DSB is operational for all reasonable values of the free parameters, and that the RGE improvement produces only a small quantitative change in the properties of the model. In this particular model, the effects of the improvement in the phase structure of the model were not so dramatic as in its non supersymmetric counterpart, however the question remains whether the same might happen in different models.
An important aspect that was left out of this paper because of its technical complexity, its the extension of the present study for the calculation of the auxiliary field effective superpotential F (see Eq. (7)), which would allow us to probe the possibility of spontaneous breaking of supersymmetry. This is currently a work in progress.