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On the effective superpotential in the supersymmetric Chern-Simons theory with matter

We develop a superfield approach to the effective potential in the supersymmetric Chern-Simons theory coupled to matter, and, in Landau gauge, calculate the one-loop Kahlerian effective potential and give some qualitative prescriptions for the two-loop one.

It is well known that the low-energy effective dynamics of a quantum field theory is characterized by the effective potential [1]. Its study is especially relevant in the context of supersymmetric theories since the effective (super)potential allows to obtain important information about supersymmetry and/or gauge symmetry breaking. In four space-time dimensions, the methodology of studying the effective superpotential has been well developed in [2,3] and then successfully applied in many examples, including the Wess-Zumino model, both in commutative [3] and noncommutative [4] cases, general chiral superfield model [5], super-Yang-Mills theory [6] and higher-derivative theories [7].
Recently, a great deal of attention has been devoted to studies of the supersymmetric Chern-Simons theory with matter. The reason is the fact that N = 6 and N = 8 supersymmetric Chern-Simons theories are finite and conformal invariant, which allows for studying the AdS 4 /CF T 3 correspondence [8]. Earlier, different aspects of the supersymmetric Chern-Simons theories have been studied in [9].
In three space-time dimensions, the first attempt to formulate a superfield approach to the study of the supersymmetric effective potential has been carried out in [10,11] (some preliminary discussions of this approach were presented in [12,13], see also [14] for some issues related to extended supersymmetry in these theories). Additional studies of the effective potential including a component analysis of the Coleman-Weinberg and Wess-Zumino models were performed in [15].
In this work we consider the superfield version of the supersymmetric Coleman-Weinberg model, that is, the Chern-Simons theory coupled to a self-interacting massless scalar matter, which is the simplest example of a three-dimensional supersymmetric gauge theory with matter, where all couplings are dimensionless. Employing an adequate method, by us devised, we will calculate the one-and discuss the two-loop superpotentials. Throughout the paper, we follow the notations and conventions adopted in [16].
The action of the Coleman-Weinberg theory looks like where Φ is a (complex) scalar superfield and V (Φ) = λ 2 (ΦΦ) 2 is a classical potential. The A α is a gauge superfield, with W α = 1 2 D β D α A β the corresponding gauge-invariant superfield strength. The last term of the expression (1) is the gauge-fixing action. Since the theory is Abelian, the ghosts completely decouple, and their action will be omitted.
We will start by evaluating the superfield effective action, within the methodology of the loop expansion [17]. To do it, we make a shift Φ → Φ + φ in the superfield Φ (the shift for theΦ will be similar), where now Φ is a background (super)field, and φ is a quantum one. Through this study, the gauge field A α is taken to be purely quantum since our aim consists in studying quantum corrections depending only on the scalar fields. As a result, the classical action (1) takes the form Here we eliminated the linear terms in quantum fields since they produce only irrelevant, one-particle-reducible contributions. The effective action Γ[Φ] is defined by the expression where the subscript 1P I stands for one-particle-irreducible supergraphs. The general structure of the effective action can be cast in a form similar to the four-dimensional case [2,3]: where the K(Φ) is the Kählerian effective potential which depends only on the superfield Φ but not on its derivatives. The F term is called auxiliary fields' effective potential whose key property is its vanishing in the case when all derivatives of the superfields are equal to zero (within this paper we will not discuss it). We restrict ourselves to the Kählerian effective potential.
We will work within a loop expansion for the effective action Γ, and for the Kählerian potential K, that is, the tree-order Kählerian effective potential is For the background fields equal to zero, the free propagators of the scalar and gauge superfields corresponding to the action (1) are The key point of this paper consists in the development of a methodology for the background dependent propagators and their use for the calculation of the effective action. While in the four-dimensional superfield theories this formalism has been well developed and successfully applied in a number of papers, see f.e. [3,[5][6][7], in the three-dimensional superfield theories it has been used only in [10] for the purely scalar superfield model. Also, in the works [11] a simplified form of this methodology based on the imposition of restrictions on the structure of the background scalar field (Φ = φ 1 −θ 2 φ 2 ) was used, whereas in the present paper the general form Φ = φ 1 + θ α ψ α − θ 2 φ 2 is used, with no restrictions except for the condition D α Φ = 0, leading to the Kählerian effective potential; actually, we do not employ the component expansion of Φ. Moreover, we generalize the powerful technique of summation over one-loop diagrams (which has been intensively applied in the four dimensions [5,7]) for the three-dimensional case, specially, for the supersymmetric gauge theories, which is done here for the first time. We expect that this methodology can be efficiently applied to more sophisticated theories.
It will be convenient to obtain the effective propagators on the base of summation of different sequences of free propagators. First of all, we can take into account the situation with the triple gauge-scalar vertices. Including such a vertex into a diagram, with one background scalar leg, produces a fragment of the diagram as shown in Fig.1. In that figure the bold line is for the background field, and the factor D β D α + ξD α D β originates from the gauge propagator. The indices α and β are the indices of gauge fields contracted into this propagator. We see that if we move the derivative D α , originated from the interaction vertex, to the propagator of the gauge fields proportional to D β D α + ξD α D β , we will annihilate its gauge-independent part, and the gauge-dependent part is proportional to ξ, by imposing the gauge ξ = 0 (Landau gauge [16]), the calculations are simplified, removing from considerations all diagrams with the vertices (ΦA α D αφ −ΦA α D α φ).
Let us find the one-loop Kählerian effective potential. Taking into account the conclusion made in the previous paragraph, we can find that in the Landau gauge there are two types of contributions to the Kählerian effective potential: in the first of them, all supergraphs involve only the gauge propagators, and in the second of them, all supergraphs involve only matter propagators. It is equivalent to write down two contributions to the one-loop effective action as two traces of the logarithms: where m = 2λΦΦ,M = λΦ 2 , M = λΦ 2 . The plus sign in Γ 1 arose due to the fermionic statistics of A α . The Γ 1 , at the ξ = 0 limit, can be explicitly obtained via the expansion of the trace of the logarithm: since the operator inverse to 1 , at ξ = 0, as Here we took into account that each relevant scalar-gauge vertex looks like − 1 2 g 2 ΦΦA α A α , and < A α (z 1 )A β (z 2 ) >= i 4 D β D α δ 12 . Using the commutator for the supercovariant derivatives [D γ , D β ] = 2δ γ β D 2 together with the property D α D β D α = 0, one finds that D α 2 D α 1 D α 3 D α 2 . . . D α 1 D αn δ 12 | θ 1 =θ 2 = 2 n (n−1)/2 , for n = 2l + 1; instead, if n = 2l is even, this expression vanishes. Hence we have Then, by carrying out the Fourier transform ( → −k 2 ) we arrive at the following contribution to the Kählerian effective potential (as usual, the corresponding effective action can be restored from the relation Γ where µ 2 = g 2 ΦΦ. To do the summation, we can consider This expression can be rearranged, summed and integrated with use of the dimensional regularization as Here we performed a Wick rotation which yields k 2 E = k 2 . All this allows us to integrate the equation (12) This expression, in its functional structure, is similar to the results of [10]. Indeed, it is finite, polynomial and does not involve any logarithm-like dependence.
At the same time, we can find the contribution from the purely matter sector: This sum can be evaluated in the same way as above yielding Hence, the complete one-loop Kählerian effective potential can be read off from the sum of (14) and (18): As it was already noted, it is finite and polynomial.
Now, let us turn to the two-loop approximation which we restrict to some qualitative remarks.
First we obtain the effective propagator of A α field via the sum depicted at Fig. 2.

FIG. 2: Calculation of the "dressed" propagator.
To do it, we start with the quadratic action of the A α field: which can be read off from (2). As we already noted, after the calculation, we must impose ξ = 0. Then, we employ the identity: which straightforwardly follows from the well-known relation . To obtain the background-dependent propagator from (20), we must invert the operator It is easy to verify that if O αβ = AD β D α + BD α D β , then the corresponding propagator, that is, G βγ defined such that O αβ G βγ = −iδ α γ δ 5 (z − z ′ ), looks like where Replacing the values A = 1 + µ 2 2D 2 and B = 1 ξ − µ 2 2D 2 , see (22), we find Imposing the Landau gauge ξ = 0, we get The next step consists in obtaining the background dependent propagator for the superfields φ,φ. To do it, we consider the quadratic action of only these quantum fields. As we have already noted before, it looks like Therefore, the matrix propagator is Thus, we have found all background-dependent propagators.
The interaction part of the action (2), after absorbing some terms into the propagator and removing irrelevant terms mentioned above looks like Using the vertices from this expression, we must calculate the contributions from the diagrams depicted in Fig. 3. We will not calculate these diagrams exactly, but the dimensional analysis and a straightforward inspection of their contributions show that the two-loop Kählerian effective potential has the following form: Here c 1 is a function of couplings involving the factor 1 ǫ , ǫ = d − 3, where d is a spacetime dimension within the dimensional regularization prescription, and c 2 is a function of couplings, which, however, is finite. The µ r is a renormalization scale.
To obtain the renormalized effective Kählerian potential, after adding the corresponding counterterms, we impose the following normalization condition: where v is a mass scale, and K = K (0) + K (1) + K (2) , where K (0) = λ 2 (ΦΦ) 2 , and K (1) and K (2) given by eqs. (19, 30) respectively, is the complete Kählerian effective potential up to two-loop order. Using this condition to eliminate the dependence on µ r , we arrive at the following renormalized Kählerian effective potential: We see that after the two-loop calculations, a mass scale has been generated, breaking thus the scale invariance just as it occurs in the Coleman-Weinberg model [1].
In summary, in this paper we presented a superfield method for the calculation of the effective potential in three-dimensional supersymmetric gauge field theories. We succeeded to obtain explicit expressions for the Kählerian effective potential (which depends on superfield Φ but not on its derivatives) up to two loops. We found that, in the two-loop order, a mass scale is generated, thus, the scale invariance and superconformal symmetry are broken. One must emphasize the difference of our methodology from the one used in [11]. While in that paper, the calculations were performed for a strongly restricted background field, here we have done the calculations without any restriction on its structure, and also did not use any component expansion. Nevertheless, the functional dependence of the effective action on the background fields is the same as in [11] being given by (32). To carry out an exact calculation of the two-loop effective potential for this arbitrary background, however, one must calculate the contributions of the supergraphs depicted at Fig. 3, where the background dependent propagators are given by (26) and (28). Such a calculation is extremely complicated from the technical viewpoint. In principle, our approach can be directly generalized for higher loops, and, also for theories with extended supersymmetry and noncommutativity.
Also, due to the similarity between the structure of two-dimensional and three-dimensional supersymmetry algebras, we expect that this approach can be also applied for the study of the two-dimensional superfield theories.