PLB34759S0370-2693(19)30463-010.1016/j.physletb.2019.02.052The Author(s)TheoryEffective potential in the 3D massive 2-form gauge superfield theoryF.S.Gamaafisicofabricio@yahoo.com.brJ.R.Nascimentoajroberto@fisica.ufpb.brA.Yu.Petrova⁎petrov@fisica.ufpb.brP.J.Porfíriobpporfirio89@gmail.comaDepartamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970, João Pessoa, Paraíba, BrazilDepartamento de FísicaUniversidade Federal da ParaíbaCaixa Postal 5008João PessoaParaíba58051-970BrazilDepartamento de Física, Universidade Federal da Paraíba Caixa Postal 5008, 58051-970, João Pessoa, Paraíba, BrazilbDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USADepartment of Physics and AstronomyUniversity of PennsylvaniaPhiladelphiaPA19104USADepartament of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA⁎Corresponding author.Editor: M. CvetičAbstractIn the N=1, d=3 superspace, we propose a massive superfield theory formulated in terms of a spinor gauge superfield, whose component content includes a two-form field, and a real scalar matter superfield. For this model, we explicitly calculate the one-loop correction to the superfield effective potential. In particular, we show that the one-loop effective potential is independent the gauge-fixing parameters.1IntroductionAs it is well known, the most studied supersymmetric models are based on a gauge multiplet, describing gauge fields and their superpartners, and a scalar multiplet describing the usual matter. Many issues related to these models in different cases were studied, both in classical and quantum contexts. Nevertheless, other supersymmetry multiplets, including those ones presented in [1], also deserve to be considered. One of the important examples is the tensor multiplet whose component content includes an antisymmetric tensor field [2], which, as is well known, plays an important role since it emerges in string theory [3], and it has been studied in many other contexts, such as Lorentz symmetry violation [4], quantum equivalence [5–7], paramagnetism-ferromagnetism phase transition [8], and cosmological inflation [9]. The quantum impacts of the tensor multiplet were studied in the four-dimensional space-time, where it is described by the chiral spinor superfield, in [10], where the one-loop effective potential was calculated in the model including this superfield some further development of this model has been carried out in [11]. Therefore, the natural problem consists in generalization of this study to three dimensions through treating a theory of the three-dimensional tensor multiplet which is known to be described by the gauge spinor superfield. The corresponding superfield description of a theory on the tree level has been developed already in [12]. Therefore, it is natural to promote this study to the quantum level, introducing a coupling of the gauge spinor superfield to some matter, and calculating the one-loop quantum corrections in this theory. This is the aim we pursue in this paper.Our calculations are based on the methodology of calculating the superfield effective potential developed for the three-dimensional case originally in [13] and then used for various three-dimensional superfield theories in a number of papers, f.e. [14,15]. We calculate the effective potential in the one-loop approximation.The structure of our paper looks like follows. In the section 2 we consider the classical actions of a theory involving the real spinor superfield. In the section 3, we explicitly calculate the one-loop effective potential for this theory, and in the section 4, the results are summarized.2The modelBy imposing some constraints on the field strength for the three-dimensional 2-form gauge superfield ΓAB, it is possible to show that ΓAB can be completely expressed in terms of a prepotential Bα, which is an unconstrained real spinor gauge superfield [1]. Having this in mind, we start with the following definition(1)Sk[Bα,Φ]=−12∫d5z[(DαG)2+(DαΦ)2], where G≡−DαBα is a gauge invariant field strength and Φ is the usual real scalar matter superfield. The identity DαDβDα=0 ensures that Sk is invariant under the gauge transformation(2)Φ→ΦΛ=Φ;Bα→BαΛ=Bα+12DβDαΛβ, with a spinor gauge parameter Λα. The model (1) is an example of first-stage reducible theory. Indeed, the parameter Λα in (2) is not unique, but it is defined up to the transformation Λβ′=Λβ+DβL, where L is an arbitrary scalar superfield, in other words, there are gauge transformations for gauge parameters. The methodology for studying reducible theories has been developed in [16], and the general discussion of such theories can be found in [17]. The four-dimensional analogue of the theory (1), within supergravity context, has been considered in [6].Now, we want to introduce mass terms for the theory (1). These terms are defined as(3)Sm[Bα,Φ]=∫d5z[mΦG+mB2BαBα+12mΦΦ2]. The term mΦG corresponds to the supersymmetric extension of the topological BF model [12]. It is worth to note that mB2BαBα explicitly breaks the gauge invariance of Sm under the transformation (2).Let us check that Sk+Sm indeed describes a massive gauge theory. For this, we need to obtain the free superfield equations for Bα and Φ, which are derived from the principle of stationary action. Thus, we get from Sk+Sm:(4)δ(Sk+Sm)δBα=DαD2G+mDαΦ+2mB2Bα=0;(5)δ(Sk+Sm)δΦ=D2Φ+mG+mΦΦ=0. On the one hand, if mB=mΦ=0 and m≠0, then we can multiply Eq. (4) by Dα/2 and use (D2)2=□ to obtain(6)(□−m2)G=0. We can carry out a similar calculation to show that DαΦ satisfies a Klein-Gordon equation.On the other hand, if mB≠0 and m=0, then we can multiply Eq. (4) by D2DαDγ and use DαDγDα=0 to obtain(7)D2DαDγBα=0. Substituting this back into the equation (4) and using D2[Dα,Dβ]=2Cβα□, we get(8)(□−mB2)Bα=0. It is trivial to show that Φ also satisfies a Klein-Gordon equation for mΦ≠0 and m=0. Therefore, we demonstrated that Sk+Sm describes a massive gauge theory.Since the model under investigation Sk+Sm is a free superfield theory, and the main purpose of this paper is to calculate the one-loop effective potential, then we must extend Sk+Sm to include interactions. Here, we define the interaction between Bα and Φ as(9)Sint[Bα,Φ]=∫d5z[V0(Φ)+V1(Φ)G+12V2(Φ)G2+V3(Φ)BαBα], where Vi(Φ)'s are analytical functions of their arguments. Note that we have ignored in (9) terms higher than quadratic in Bα due to the fact that these terms will not contribute at the one-loop level to the effective potential. Moreover, in addition to (3), Sint also lacks gauge invariance.The lack of gauge invariance of Sm and Sint is inconvenient for quantum calculations. In order to improve the situation, we will restore the gauge symmetry by introducing a Stückelberg superfield Ωα [18]. Thus, instead of the theory Sk+Sm+Sint, we will study in this work the following gauge-invariant theory, obtained from the previous one through adding some new terms, whose action is(10)S[Bα,Ωα,Φ]=12∫d5z{GD2G+ΦD2Φ+mΦΦ2+2V0(Φ)+2[mΦ+V1(Φ)]G+V2(Φ)G2+2[mB2+V3(Φ)](Bα−WαmB)(Bα−WαmB)}, with Wα≡12DβDαΩβ. The new action (10) is invariant under the following transformations(11)Φ→ΦΛ=Φ;Bα→BαΛ=Bα+12DβDαΛβ;Ωα→ΩαΛ=Ωα+mBΛα, with spinor gauge parameter Λα. Moreover, S is also invariant under the gauge transformation(12)Φ→ΦK=Φ;Bα→BαK=Bα;Ωα→ΩαK=Ωα+DαK, with an arbitrary scalar gauge parameter K.Since (10) is gauge invariant, it follows that a gauge fixing is necessary for the calculation of quantum corrections to the effective potential. Thus, the gauge-fixed action is defined as the sum S+Sgf, where S is given in Eq. (10) and the gauge-fixing term Sgf is given by(13)Sgf[Bα,Ωα]=12∫d5z(BαΩα)×(−1αD2DβDαmBDβDαmBDβDα−2αmB2δαβ−1ξD2DαDβ)×(BβΩβ). In particular, if we choose m=0 and the supersymmetric Fermi-Feynman gauge α=ξ=1, the kinetic terms take the particularly simple forms ∼Bα(□−mB2)Bα and ∼Ωα(□−mB2)Ωα.Of course, there be also ghosts in the gauge-fixed action. Indeed, besides the usual ghosts, there are also ghosts for ghosts due to the fact that (10) describes a first-stage reducible theory. However, since the ghosts do not interact with the scalar superfield Φ, it follows that the ghost terms do not contribute to the one-loop effective potential. For this reason, we can omit such terms. We note that the similar situation takes place in four dimensions [11].3One-loop calculationsIn this section, we calculate the one-loop effective potential for the theory (10). To do this, we employ the background field method [19]. Within this approach, we perform the calculations by making a linear split of the superfields into background superfields (Bα,Ωα,Φ) and quantum fluctuations (bα,ωα,ϕ):(14)Bα→Bα+bα;Ωα→Ωα+ωα;Φ→Φ+ϕ. By definition, the effective potential depends only on the matter superfield Φ. Thus, we assume a trivial background for the gauge superfields Bα, Ωα, and the derivatives of Φ:(15)Bα=Ωα=0;DαΦ=0;∂αα˙Φ=0, while a background Φ differs from zero.For the sake of simplicity, before we consider the general problem, we first study the particular case where mB2=V3(Φ)=0. We denote the effective potential calculated in this case by K(1)A. The importance of this choice is based by the fact that in this case the superfield Ωα completely decouples from theory (10). Therefore, expanding S+Sgf around the background superfields and keeping only the quadratic terms in the quantum fluctuations, one finds(16)S2[Φ;ϕ,bα]=SK+SINT;(17)SK=12∫d5z{bα[D2(DαDβ−1αDβDα)]bβ+ϕD2ϕ};(18)SINT=12∫d5z[(mΦ+V0″)ϕ2+2(m+V1′)ϕg+V2g2], where g≡−Dαbα, V1′≡dV1/dΦ, and V0″≡d2V0/dΦ2.The interaction vertices can be read off directly from SINT, and the propagators are obtained by inverting the differential operators in SK, being given by(19)〈bα(1)bβ(2)〉=−14k4D12(D1αD1β−αD1βD1α)δ2(θ1−θ2);(20)〈ϕ(1)ϕ(2)〉=D12k2δ2(θ1−θ2). Notice in Eq. (18) that the quantum superfield bα interacts with the background one Φ through its field strength g. Thus, instead of the propagator 〈bα(1)bβ(2)〉, it is sufficient to use the propagator with no spinor indices 〈g(1)g(2)〉, which is given by:(21)〈g(1)g(2)〉=D1αD2β〈bα(1)bβ(2)〉=D12k2δ2(θ1−θ2). It is clear that (21) does not depend on the gauge parameter α introduced in the gauge-fixing procedure. Therefore, before we start the calculation of the one-loop effective potential KA(1)(Φ), we can already conclude that KA(1)(Φ) is gauge independent as it occurs in some other three-dimensional supergauge theories, see f.e. [14].The propagators (20), (21), and the vertices (18) can be written in a matrix form. In order to do this, we make the definitions(22)χi≡(gϕ);χj≡(gϕ);Mij≡(V2m+V1′m+V1′mΦ+V0″), so that we can show that(23)〈χi(1)χj(2)〉=D12k2δijδ5(θ1−θ2);SINT=12∫d5zχiMijχj. These propagators and vertices are quite similar to ones used in our previous work [14], where we have calculated K(1)(Φ) for a generic superfield higher-derivative gauge theory. Due to this similarity, we simply quote the result here:(24)KA(1)(Φ)=12∫d3k(2π)31|k|[arctan(λ+|k|)+arctan(λ−|k|)], where the λ's are the eigenvalues of the matrix Mij, and |k|=k2.Substituting the eigenvalues into (24) and calculating the integral over the momenta, we obtain(25)KA(1)(Φ)=−116π[(mΦ+V0″)2+2(m+V1′)2+V22]. Just as in the usual three-dimensional field theories, this one-loop contribution to the effective potential is UV finite, and its functional structure is given by a polynomial function of V0″,V1′, and V2. Indeed, in contrast to four-dimensional theories, logarithmic functions begin to occur only at the two-loop level due to the divergences of the Feynman integrals [13,15]. Additionally, as we already said before, (25) is independent of the gauge-fixing parameter α. This result was expected because the theory (10) with mB2=V3(Φ)=0 is classically equivalent to a theory with two massive real scalar superfields, even though G is by definition a field strength. However, it is not clear whether K(1)(Φ) is independent of α when mB2≠V3(Φ)≠0. Thus, let us move on and calculate K(1)(Φ) in the general case mB2≠V3(Φ)≠0. We denote the effective potential calculated in this case by K(1)B.Again, in order to evaluate the KB(1)(Φ) one should expand (10) around the background superfields (14) and keep the terms quadratic in the fluctuations:(26)S2[Φ;ϕ,bα,ωα]=12∫d5z{bα[D2(DαDβ−1αDβDα)+2mB2δαβ]bβ+ωα[D2(DβDα−1ξDαDβ)−2αmB2δαβ]ωβ+ϕD2ϕ+(mΦ+V0″)ϕ2+2(m+V1′)ϕDαbα−V2bαDαDβbβ+2V3bαbα−2V3mBωβDαDβbα+V3mB2ωαD2DβDαωβ}, where we have now taken into account the contributions of ωα.The quadratic mixing terms between the quantum superfields make the calculations troublesome. Fortunately, we can overcome this complication by a non-local change of variables in the path integral, as was done in [20]. Thus, we can diagonalize (26) with the choice(27)ϕ(z)⟶ϕ(z)−∫d5wG(z,w)[m+V1′(Φ(w))]Dwαbα(w);(28)ωα(z)⟶ωα(z)+∫d5wGαβ(z,w)V3(Φ(w))mBDwγDwβbγ(w), where G(z,w) and Gαβ(z,w) are Green's functions, which are defined as solutions of the equations(29)(D2+mΦ+V0″)G(z,w)=δ5(z−w);(30)D2[(1−αmB2□+V3mB)DγDα+(αmB2□−1ξ)DαDγ]Gγβ(z,w)=δαβδ5(z−w). It is possible to show that these functions can be expressed in the form(31)G(z,w)=D2−(mΦ+V0″)□−(mΦ+V0″)2δ5(z−w);(32)Gγβ(z,w)=D24□[1□−αmB2+V3mB2□DβDγ−ξ□−αξmB2DγDβ]δ5(z−w). It is worth to point out that we assume that the quantum variable bα does not change under the transformations (27) and (28). For this reason, these transformations correspond to translations on the field space, so that the corresponding Jacobian is equal to unity.Therefore, after the change of variables (27) and (28), the functional S2 can be rewritten as:(33)S2=12∫d5z{bα[(D2(1−mB2+V3□)−V2)DαDβ+D2(mB2+V3□−1α)DβDα]bβ+ωαD2[(1−αmB2□+V3mB2)DβDα+(αmB2□−1ξ)DαDβ]ωβ+ϕ(D2+mΦ+V0″)ϕ}−12∫d5zd5wbα(z)bβ(w)[(m+V1′)2×D2+mΦ+V0″□−(mΦ+V0″)2DαDβ+(V3mB)2D2DβDα□−αmB2+V3mB2□]×δ5(z−w). In principle, we could derive the Feynman rules for the functional (33) and calculate the one-loop supergraphs which contribute to the effective potential. However, it is much easier to perform the calculation using the well-known formula for the one-loop Euclidean effective action [21,22](34)ΓB(1)[Φ]=−12sTrlnOˆ, where sTr denotes the supertrace over the discrete and continuous indices of Oˆ.It follows from (33) that Oˆ is a block diagonal matrix. Thus, Eq. (34) can be split into three contributions:(35)ΓB(1)[Φ]=Γω[Φ]+Γb[Φ]+Γϕ[Φ], where(36)Γω[Φ]=12Trln[(1−αmB2□+V3mB2)D2DβDα+(αmB2□−1ξ)D2DαDβ];(37)Γb[Φ]=12Trln{[D2(1−mB2+V3□)−V2]DαDβ+(mB2+V3□−1α)D2DβDα−(m+V1′)2D2+mΦ+V0″□−(mΦ+V0″)2DαDβ−(V3mB)2D2DβDα□−αmB2+V3mB2□};(38)Γϕ[Φ]=−12Trln(D2+mΦ+V0″). Notice that ωα and bα are fermionic variables, so that Γω and Γb got an overall plus sign.Now, let us start with the first contribution Γω. First, we factor out the inverse of the ωα-propagator from (36). Thus, Eq. (36) can be rewritten as(39)Γω=12Trln(D2DγDα−1ξD2DαDγ)+12Trln{δγβ+12□[(−αmB2□+V3m2B)D2DβDγ+ξαmB2□D2DγDβ]}. Note that the first trace does not depend on the background superfield, then it can be disregarded. The second trace can be split into two parts with the help of the identity DαDβDα=0. Therefore,(40)Γω=12Trln{δγλ+12□(−αmB2□+V3mB2)D2DλDγ}+12Trln{δλβ+ξαmB22□2D2DλDβ}. Again, the second trace is a constant independent of the background superfield and it can be dropped. To solve the first trace, we have to perform a series expansion of the logarithm. Therefore,(41)Γω=−12∫d5z∫d3k(2π)3∑n=1∞1n[12k2(αmB2k2+V3mB2)]n×(D2)nDα2Dα1Dα3Dα2⋯DαnDαn−1×Dα1Dαnδ2(θ−θ′)|θ=θ′. Each term of the expansion can be evaluated using the D-algebra and the following identities:(42)δ2(θ−θ′)|θ=θ′=0;Dαδ2(θ−θ′)|θ=θ′=0;D2δ2(θ−θ′)|θ=θ′=1. Thus, it is possible to show that each term in the expansion (41) vanishes. Therefore, we obtain(43)Γω[Φ]=0. In the context of three-dimensional super-QED, a vanishing contribution to K(1)(Φ) was also found in Refs. [14,23], where was shown that the contribution of the gauge superfield to K(1)(Φ) vanishes in the Landau gauge. In contrast to [14,23], we have shown that the contribution of the Stückelberg superfield vanishes for any values of the gauge parameters α and ξ.Now, let us consider the contribution of the quantum prepotential bα to ΓB(1)[Φ]. By repeating the same reasoning that led from (36) to (40), we can prove that (37) can be rewritten as(44)Γb=12Trln(D2DαDγ−1αD2DγDα)++12Trln{δγλ+12□[D2(mB2+V3□+(m+V1′)2□−(mΦ+V0″)2)++V2+(m+V1′)2(mΦ+V0″)□−(mΦ+V0″)2]DγDλ}++12Trln{δλβ−α2□2[mB2+V3−−(V3mB)2□□−αmB2+V3mB2□]D2DβDλ}. Notice that only the second trace is nonvanishing and independent of α. In order to make progress, we need the identity(45)δγλ+AD2DγDλ+BDγDλ=(δγα+AD2DγDα)(δαλ+B1−2□ADαDλ). Thus, by applying this identity to (44), we find(46)Γb=12Trln{δγα+12□[mB2+V3□+(m+V1′)2□−(mΦ+V0″)2]D2DγDα}+12Trln{δαλ+12[□−(mΦ+V0″)2]V2+(m+V1′)2(mΦ+V0″)[□−(mΦ+V0″)2](□−mB2−V3)−(m+V1′)2□DαDλ}. In order to evaluate the second trace (the first one is equal to zero), we shall make the simplifying assumption that m=V1=0. Therefore, under such a simplifying assumption, we find(47)Γb=−12∫d5z∫d3k(2π)3∑n=1∞12nnV2n(k2+mB2+V3)n×Dα1Dα2Dα2Dα3⋯Dαn−1DαnDαnDα1δ2(θ−θ′)|θ=θ′. Again, with the help of the D-algebra and the identities (42), we are able to formally show that(48)Dα1Dα2Dα2Dα3⋯Dαn−1DαnDαnDα1δ2(θ−θ′)|θ=θ′={−2n(−k2)n−1,ifn=2ℓ+10,ifn=2ℓ. Substituting this formula into (47), we obtain(49)Γb=12∫d5z∑ℓ=0∞(−1)ℓV22ℓ+12ℓ+1×∫d3k(2π)3(k2)ℓ(k2+mB2+V3)2ℓ+1. We can evaluate this well-known integral over the momenta and sum the results over ℓ to get(50)Γb[Φ]=−116π∫d5zV24(mB2+V3)+V22. The last (and easiest) contribution which is needed to be calculated is (38). We can simply repeat the same reasoning that led us to Eqs. (43) and (50), but we will not calculate explicitly Γϕ. Therefore, the final result is given by(51)Γϕ[Φ]=−116π∫d5z(mΦ+V0″)2. Finally, substituting (43), (50), and (51) into (35) and using the relation ΓB(1)=∫d5zKB(1), we find(52)KB(1)(Φ)=−116π[(mΦ+V0″)2+V24(mB2+V3)+V22]. Similarly to KA(1) [see Eq. (25)], KB(1) is UV finite and, therefore, no additional renormalization is needed. Moreover, KB(1) is also independent of the gauge-fixing parameters α and ξ. In contrast to KA(1), the functional structure of KB(1) is not given by a polynomial function of V0″,V2, and V3. In the N=1, d=3 superspace, such non-polynomial structure is also found in one-loop effective potentials in the context of higher-derivative theories (see, for example, [14]). We conclude this section with the remark that the results (25) and (52), which were obtained by different methods, coincide with each other when m=mB=V1=V3=0. This shows that KB(1) obtained through evaluation of the matrix trace is consistent with KA(1) obtained with use of eigenvalues of the mass matrix.4SummaryWe formulated a supersymmetric theory of three-dimensional two-form field. In the superfield language, this theory is described by a spinor prepotential Bα. We started with a gauge invariant strength G defined in terms of Bα, and further introduced a mass term for this field, a coupling of this field to an usual scalar superfield Φ and a Stückelberg superfield in order to implement gauge symmetry in the presence of the mass term. Afterwards, we calculated the one-loop effective potential of Φ in a resulting theory, using a functional approach. The effective potential turns out to be finite as it must occur in three-dimensional theories. We explicitly demonstrated that our results are rather analogous to the one-loop results in supergauge theories constructed on the base of the usual vector supermultiplet.Essentially, the main result of our paper is a first example of successful formulation of a consistent coupling of three-dimensional spinor superfield to a scalar matter, with the theory turns out to possess gauge symmetry under transformations different from those one in usual supersymmetric QED, and successful calculation of quantum corrections in this theory. Effectively, the main conclusion is that we developed a new supergauge theory with a consistent coupling.Further development of our study could consist in development of non-Abelian generalization of our theory and in study of higher loop corrections. We expect to do these studies in forthcoming papers.AcknowledgementsAuthors are grateful to R.V. Maluf for valuable discussions. P.J. 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