Structure of the Vacuum in Deformed Supersymmetric Chiral Models
aa r X i v : . [ h e p - t h ] J u l Structure of the Vacuum in DeformedSupersymmetric Chiral Models
P. Fern´andez a , E. F. Moreno b , F. A. Schaposnik a ∗ a Departamento de F´ısica, Universidad Nacional de La PlataC.C. 67, 1900 La Plata, Argentina b Department of Physics,West Virginia UniversityMorgantown, West Virginia 26506-6315, U.S.A.
January 9, 2019
Abstract
We analyze the vacuum structure of N = 1 / C -deformed superpotentials and canonical and non-canonicaldeformed K¨ahler potentials. We find conditions under which the vacuumconfigurations are affected by the deformations. A C -deformation of the N = 1 superalgebra corresponding to nonanticommu-tative Grassmann coordinates θ α has been shown to arise in string theory in agraviphoton background [1]-[2]. Prompted by this result, nonanticommutativeversions of supersymmetric (SUSY) Yang-Mills theory and Wess-Zumino modelhave been formulated [3]-[4] and their renormalizability established [5]-[7]. Thedeformation preserves the notion of chirality but only half of the N = 1 super-symmetry is preserved as the supercharges Q α , the generators of θ α translations,are conserved while the ¯ Q ˙ α are broken explicitly.In order to analyze the vacuum structure of undeformed SUSY chiral modelswe study the effective potential V for scalar fields since its critical points corre-spond to the possible vacua. Hermiticity of the original theory guarantees thatthe resulting potential is positive definite so that the vanishing of V implies theexistence of a supersymmetric vacuum. But in C -deformed SUSY theories her-miticity is lost, V is not positive definite and the analysis of the critical pointsshould be done at the quantum level using saddle point or steepest descentmethods. ∗ Associated with CICBA N = 1 SQCD that can be seen, in the low-energy effectivetheory, as vacua of an O’Raifeartaigh-type model [9]-[14]. In connection withthis phenomenon, it is the purpose of this work to analyze the structure of thevacuum for C -deformed O’Raifeartaigh-like models, discussing in particular thepossibility of spontaneous breaking of the surviving supersymmetry.As explained in [6] a N = 1 / | vac i state and its dual h vac | to be annihilated by Q α . This is connected tothe fact that the vacuum energy of such state, h vac | E | vac i , vanishes even if theenergy associated with the non-Hermitian deformed Lagrangian is in generalcomplex-valued. Hence, the analysis of the zeroes of the scalar potential stillprovides information about symmetry breaking in deformed models and this isthe route we will follow in this investigation.A discussion of the scalar potential for certain SUSY deformed models hasbeen already presented in refs.[15]-[18] for deformed Wess-Zumino and sigmamodels (with canonical K¨ahler potentials). Here we will consider O’Raifeartaighmodels with more general deformed superpotentials and we will also discuss thecase of deformed non-canonical K¨ahler potentials. The plan of the paper isthe following: In section 2 we establish our conventions for nonanticommuta-tive superspace and present general deformed models containing chiral super-fields. In section 3 we analyze the vacuum structure of rather general deformedO’Raifeartaigh-like models in which the K¨ahler potential is kept canonical, andin section 4 a similar analysis of deformed models with non-canonical K¨ahlerpotential. We summarize and discuss our results in section 5. We consider the deformation of 4 dimensional Euclidean N = 1 superspaceparametrized by superspace bosonic coordinates x µ and chiral and anti-chiralfermionic coordinates θ α , ¯ θ ˙ α as proposed in [3] { θ α , θ β } = C αβ (1) { ¯ θ ˙ α , ¯ θ ˙ β } = 0 , { θ α , ¯ θ ˙ β } = 0 (2)Here C αβ are constant elements of a symmetric matrix. Defining chiral andanti-chiral coordinates according to y µ = x µ + iθσ µ ¯ θ (3)¯ y µ = y µ − iθσ µ ¯ θ (4)we impose [ y µ , y ν ] = [ y µ , θ α ] = [ y µ , ¯ θ ˙ α ] = 0 (5)2nd obtain, as a consequence of (1)-(5),[¯ y µ , ¯ y ν ] = 4¯ θ ¯ θ C µν . (6)where C µν = C αβ ( σ µν ) αβ is antisymmetric and antiselfdual.The non(anti)commutative field theory in such a deformed superspace canbe defined in terms of superfields that are multiplied according to the followingMoyal product [3]Φ( y, θ, ¯ θ ) ∗ Ψ( y, θ, ¯ θ ) = Φ( y, θ, ¯ θ ) exp − C αβ ←− ∂∂θ α −→ ∂∂θ β ! Ψ( y, θ, ¯ θ ) (7)Supercharges and covariant derivatives in chiral coordinates take the form Q α = ∂∂θ α , ¯ Q ˙ α = − ∂∂ ¯ θ ˙ α + 2 iθ α σ µα ˙ α ∂∂y µ , (8) D α = ∂∂θ α + 2 iσ µα ˙ α ¯ θ ˙ α ∂∂y µ , ¯ D ˙ α = − ∂∂ ¯ θ ˙ α (9)The D - D algebra is not modified by the deformation (1) as it also happens forthe Q - D and ¯ Q - D algebra. Concerning the supercharge algebra, it is modifiedaccording to { ¯ Q ˙ α , Q α } = 2 iσ µα ˙ α ∂∂y µ = 2 σ µα ˙ α P µ (10) { Q α , Q β } = 0 (11) { ¯ Q ˙ α , ¯ Q ˙ β } = − C αβ σ µα ˙ α σ νβ ˙ β ∂ ∂y µ ∂y ν = 4 C αβ σ µα ˙ α σ νβ ˙ β P µ P ν (12)Then, only the subalgebra generated by Q α is still preserved and this definesthe chiral N = 1 / In this work we will discuss models containing chiral superfields. In deformedsuperspace, a chiral superfield Φ satisfying ¯ D ˙ α Φ = 0 can be written, as usual,in the form Φ( y, θ ) = φ ( y ) + √ θψ ( y ) + θθF ( y ) (13)Analogously we can define antichiral superfields satisfying D α ¯Φ = 0 (14)which only depend on ¯ θ and ¯ y µ .A general action in terms of chiral and antichiral superfields takes the form S (cid:2) Φ , ¯Φ (cid:3) = Z d y (cid:20)Z d θd ¯ θ K ∗ (cid:16) Φ i , ¯Φ ¯ j (cid:17) + Z d θ W ∗ (cid:0) Φ i (cid:1) + Z d ¯ θ ¯ W ∗ (cid:16) ¯Φ ¯ j (cid:17)(cid:21) (15)3ere we call K ∗ , W ∗ the K¨ahler and superpotential functionals with superfieldsmultiplied using the Moyal product. A very useful formula for handling thesequantities has been derived in [16]-[17]. For example, given the superpotential W ∗ (Φ), we can define a “diffuse superpotential” f W ( φ i , F i ) = Z − dξ W ( φ i + ξcF i ) (16)where fields φ i are multiplied in the r.h.s. with the ordinary product and wehave written c = √− det C . As pointed out in [16], non(anti)commutativityinduces certain fuzziness controlled by auxiliary fields F i .Using eq.(16), we can prove that, in terms of component fields, the scalarpotential can be written V scalar (cid:0) φ i , ¯ φ ¯ i (cid:1) = 12 F i f W , i (cid:12)(cid:12)(cid:12)(cid:12) F i = F i ( φ, ¯ φ ) (17)with all products being ordinary products. Analogously, we can define, startingfrom the K¨ahler potential, the following diffuse quantities [17] Z (cid:0) φ, ¯ φ, F (cid:1) = Z − dξK (cid:0) φ i + ξcF i , ¯ φ ¯ j (cid:1) (18) Y (cid:0) φ, ¯ φ, F, ¯ F (cid:1) = ¯ F ¯ p Z, ¯ p −
12 ( ¯ χ ¯ p ¯ χ ¯ q Z, ¯ p ¯ q )+ c Z − dξξ h ∂ µ ¯ φ ¯ p ∂ µ ¯ φ ¯ q K, ξ ¯ p ¯ q + ∇ ¯ φ ¯ p K ξ ¯ p i (19)Now, calling Z d yL K ≡ Z d y Z d θd ¯ θK ∗ (cid:16) Φ i , ¯Φ ¯ j (cid:17) (20)it can be shown that L K = 12 F i Y, i + 12 ∂ µ ¯ φ ¯ p ∂ µ ¯ φ ¯ q Z, ¯ p ¯ q + 12 ∇ ¯ φ ¯ p Z, ¯ p − (cid:0) χ i χ j (cid:1) Y, ij − i (cid:0) χ i σ µ ¯ χ ¯ p (cid:1) ∂ µ ¯ φ ¯ q Z, i ¯ p ¯ q − i (cid:0) χ i σ µ ∂ µ ¯ χ ¯ p (cid:1) Z, i ¯ p (21) The choice of deforming the anticommutator of θ α (1), without altering that of¯ θ ˙ α implies that ¯ θ ˙ α are not the complex conjugate of θ α , which is only possiblein Euclidean space. Moreover, hermiticity of the theory is lost because of thedeformation and then the usual analysis of the the potential minima shouldbe replaced by a careful analysis of the critical points of the resulting complexexpression. At the quantum level, saddle point or steepest descent methods4hould be applied as usual, but taking into account that trajectories are inprinciple complex and that space is Euclidean.As shown in ref.[6] taking the deformed Wess-Zumino model as a prototypeof N = 1 / h vac | E | vac i = h vac | Q α ¯ Q ˙ α + ¯ Q ˙ α Q α | vac i = 0 (22)Then, in order to have a supersymmetric vacuum Q α , the generator of thesurviving supersymmetry, should annihilate both | vac i and h vac | , Q α | vac i = 0 , h vac | Q α = 0 (23)since, being ¯ Q ˙ α the generator of the explicitly broken supersymmetry, ¯ Q ˙ α | vac i does not vanish in general.Vanishing of the vacuum energy for supersymmetric vacua is not a conse-quence of any specific choice of the deformed superpotential. As explained in([6]), supersymmetric vacua in deformed models with chiral fields impose thecondition ∂ ¯ W ∗ ( ¯Φ) /∂ ¯Φ = 0 which in turn imply the vanishing of the correspond-ing scalar potential. We discuss here how the landscape of extrema of the scalar potential in O’Rai-feartaigh models is affected by the deformation of superspace defined in eq.(1).
Consider three chiral superfields fields Φ i ( i = 1 , ,
3) and a canonical K¨ahlerpotential K = ¯Φ i ∗ Φ i . Concerning the superpotential, we choose W = Φ ∗ (cid:18) h ∗ Φ + f (cid:19) + m Φ ∗ Φ + ST (24)which has the typical O’Raifeartaigh potential form, extended to non(anti)com-mutative space. Here ST includes all necessary symmetrizing terms so thatthe potential is symmetrized with respect to the ∗ product. For simplicity,we take all parameters ( f, m, . . . ) as real numbers. In order to compute thescalar potential for component fields φ we use eq.(17). In view of the form ofthe superpotential, W ( φ i + ξcF i ) as defined in (16) will only have terms withpowers ξ n , n = 0 , , ,
3. Moreover, since integrals with odd powers in ξ vanishwe end with W ( φ i + ξcF i ) = φ (cid:18) h φ ) + ξ c ( F ) + f (cid:19) + ξ c hφ F F
5o the diffuse superpotential f W becomes f W ( φ i , F i ) = 2 φ (cid:18) h φ ) + f (cid:19) + 23 c h ( F ) + hφ F F i leading to a scalar potential V E = F (cid:18) h φ ) + f (cid:19) + m ( F φ + F φ ) + hF φ φ − det C hF ( F ) (25)The subscript E indicates that we are dealing with the Euclidean potentialwhich is minus the Minkowski potential.Using the equations of motion to replace auxiliary fields F i and putting allfermion fields to zero we end with V E = − (cid:18) h φ ) + f (cid:19) (cid:18) h (cid:0) ¯ φ ¯3 (cid:1) + f (cid:19) − ( hφ φ + mφ ) (cid:0) h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 (cid:1) − m φ ¯ φ ¯3 + h C (cid:18) h (cid:0) ¯ φ ¯3 (cid:1) + f (cid:19) (cid:0) h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 (cid:1) (26)For C = 0 we recover the ordinary superspace result with a real potentialprovided φ ∗ = ¯ φ . For det C = 0 the potential becomes complex not only becausethe term proportional to det C is not accompanied by its complex conjugate butalso because in principle ¯ φ is not the complex conjugate of φ .The equations for the extrema of potential (26) read0 = hφ (cid:0) h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 (cid:1) (27)0 = m (cid:0) h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 (cid:1) (28)0 = hφ (cid:18) h (cid:0) ¯ φ ¯3 (cid:1) + f (cid:19) + m ¯ φ ¯3 + hφ (cid:0) h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 (cid:1) (29)0 = ( hφ φ + mφ ) h ¯ φ ¯3 − det Ch (cid:18) h (cid:0) ¯ φ ¯3 (cid:1) + f (cid:19) (cid:0) h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 (cid:1) h ¯ φ ¯3 (30)0 = m ( hφ φ + mφ ) − det Chm (cid:18) h (cid:0) ¯ φ ¯3 (cid:1) + f (cid:19) (cid:0) h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 (cid:1) (31)0 = (cid:18) h φ ) + f (cid:19) h ¯ φ ¯3 + ( hφ φ + mφ ) h ¯ φ ¯1 + m φ − det C h (cid:18) h ¯ φ ¯3 (cid:0) h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 (cid:1) + 2 (cid:18) h φ ¯3 ) + f (cid:19) (cid:0) h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 (cid:1) h ¯ φ ¯1 (cid:19) (32)Let us first consider the case m = 0. In this case, eq.(28) implies h ¯ φ ¯1 ¯ φ ¯3 + m ¯ φ ¯2 = 0 (33)6he l.h.s of this equation appears as a factor in all terms containing det C andhence all dependence on C αβ disappears. Field configurations correspondingto extrema of the potential are not affected by the deformation. Moreover, thevalue of the potential at the extrema is also unaffected by non(anti)commutativitysince terms containing det C are multiplied by the same vanishing factor. Theonly difference with an ordinary superspace theory is that, in principle, ¯ φ ¯ i doesnot necessarily coincide with φ ∗ i . For the particular field configurations where¯ φ ¯ i = φ ∗ i , the results for the undeformed case [10] apply, and we can concludethat there is symmetry breaking, no runaway directions, and a classical pseu-domoduli space with degenerate non supersymmetric vacua (arbitrary φ vac ).Concerning the general case in which ¯ φ ¯ i = φ ∗ i , we find extrema with similarproperties as those with ¯ φ ¯ i = φ ∗ i discussed above except that the pseudomoduliis spanned here by φ and ¯ φ and hence its dimension is doubled. We concludethe discussion of the m = 0 case noting that the theory above corresponds to ageneric supersymmetry breaking potential because the equation V = 0 cannotbe generically solved.We will show that the situation changes when the coefficient m in (24)vanishes. In that case the φ field decouples and the scalar potential takes theform V E = − (cid:18) h φ ) + f (cid:19) (cid:18) h (cid:0) ¯ φ ¯3 (cid:1) + f (cid:19) − h φ φ ¯ φ ¯1 ¯ φ ¯3 + h C (cid:18) h (cid:0) ¯ φ ¯3 (cid:1) + f (cid:19) (cid:0) ¯ φ ¯1 ¯ φ ¯3 (cid:1) (34)In the undeformed case we can easily see that there exist two supersymmetricvacua which correspond to φ vac = 0 and φ = ± p − f /h and a supersymmetrybreaking flat direction for φ vac = 0, φ vac arbitrary, for which V = f (inMinkowski space).In the deformed model there are also six families of supersymmetric config-urations which do not depend on det C . Namely¯ φ ¯1 = 0 , ¯ φ ¯3 = ± r − fh , (35) φ = 0 , ¯ φ ¯3 = ± r − fh (36)¯ φ ¯3 = 0 , φ = ± r − fh (37)All other fields not included in each line are arbitrary.Concerning non-supersymmetric extrema, they are the same for the unde-formed and the deformed case, φ = ¯ φ ¯3 = 0 , φ and ¯ φ ¯1 arbitrary (38)and for these configurations V E = − f .7here are also four solutions for which the fields at the extrema depend ondet C φ = φ = 0 , ¯ φ ¯1 = ± h √− det C , ¯ φ ¯3 = ± r − fh (39)For these configurations V = 0 and hence they correspond to supersymmetricvacua. A remarkable feature of these extrema can be seen by taking det C ∈ R .Indeed, in that case, in the det C → + limit, they correspond to runaway direc-tions which do not satisfy the extrema conditions of the undeformed potential.Hence, they have emerged entirely as a consequence of the deformation.Let us now consider the vacua structure of another potential which resultsfrom the following superpotential W = h Φ ∗ Φ ∗ (Φ − m ) + m Φ ∗ (Φ − m ) + ST (40)In contrast with the superpotential (24), the form of this superpotential allowsfor the existence of critical points ∂ W /∂φ = ∂ W /∂φ = 0.A completely analogous calculation to that presented above leads to thefollowing expression for the scalar potential V E = − ( hφ ( φ − m )) (cid:0) h ¯ φ ¯3 (cid:0) ¯ φ ¯3 − m (cid:1)(cid:1) − m ( φ − m ) m (cid:0) ¯ φ ¯3 − m (cid:1) − [ hφ (2 φ − m ) + mφ ] (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + m ¯ φ ¯2 (cid:3) + det C h (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + m ¯ φ ¯2 (cid:3) (cid:0) h ¯ φ ¯3 (cid:0) ¯ φ ¯3 − m (cid:1)(cid:1) (41)The equations for the extrema of potential (41) read0 = ∂V∂φ = h (2 φ − m ) (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 + m (cid:1) + m ¯ φ ¯2 (cid:3) (42)0 = ∂V∂φ = m (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + m ¯ φ ¯2 (cid:3) (43)0 = ∂V∂φ = (2 hφ − hm ) (cid:0) h ¯ φ ¯3 (cid:0) ¯ φ ¯3 − m (cid:1)(cid:1) + m (cid:0) ¯ φ ¯3 − m (cid:1) (44)+ 2 hφ (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + m ¯ φ ¯2 (cid:3) ∂V∂ ¯ φ ¯1 = h (cid:0) φ ¯3 − m (cid:1) [ hφ (2 φ − m ) + mφ ] − det C h (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + m ¯ φ ¯2 (cid:3) h (cid:0) φ ¯3 − m (cid:1) (cid:0) h ¯ φ ¯3 (cid:0) ¯ φ ¯1 − m (cid:1)(cid:1) (45)0 = ∂V∂ ¯ φ ¯2 = m [ hφ (2 φ − m ) + mφ ] − det C hm (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + m ¯ φ ¯2 (cid:3) (cid:0) h ¯ φ ¯3 (cid:0) ¯ φ ¯3 − m (cid:1)(cid:1) (46)0 = ∂V∂ ¯ φ ¯3 = (2 h ¯ φ ¯3 − hm ) ( hφ ( φ − m )) + ( m ) ( φ − m ) (47)+ 2 h ¯ φ [ hφ (2 φ − m ) + mφ ] − det C h (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + m ¯ φ ¯2 (cid:3) h ¯ φ h ¯ φ ¯3 (cid:0) ¯ φ ¯3 − m (cid:1) − det C h (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + m ¯ φ ¯2 (cid:3) (cid:0) h ¯ φ ¯3 − hm (cid:1)
8s in the previous example, let us first consider the case m = 0. In that case,eq.(43) implies h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + m ¯ φ ¯2 = 0 (48)Again, the l.h.s of this equation appears as a factor in all terms containing det C and hence all dependence on C αβ disappears and field configurations correspond-ing to extrema of the potential are not affected by the deformation. Moreover,the value of the potential is also unaffected by non(anti)commutativity sinceterms containing det C are multiplied by the same vanishing factor. As ex-plained in [10] there are supersymmetric vacua φ Si which corresponds to φ S = m , φ S = − hm m φ S (49)(in the deformed case we should have identical values for fields ¯ φ ¯ i which, in thedeformed case are not automatically related to φ i ).As in the undeformed case, there are also extrema φ M for which V [ φ M ] = 0.In fact, the Euclidean V [ φ M ] is a real negative number which in Minkowskiundeformed superspace would lead to the metastable vacua. The explicit formof the solutions is the same as in the undeformed case.Let us now consider the m = 0 case. In the undeformed (Minkowski) space,the non-supersymmetric (metastable) vacua present for m = 0 are lost but, aswe will see, the situation changes in the deformed case. Indeed for vanishing m the scalar potential takes the form V E = − hφ ( φ − m ) h ¯ φ ¯3 (cid:0) ¯ φ ¯3 − m (cid:1) − hφ (2 φ − m ) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1) + det Ch (cid:2) h ¯ φ ¯1 (cid:0) φ ¯3 − m (cid:1)(cid:3) ¯ φ ¯3 (cid:0) ¯ φ ¯3 − m (cid:1) (50)Let us compare the supersymmetric vacuum states between the undeformedand the deformed case. In the undeformed case, we have four supersymmetricvacuum states: φ = ¯ φ ¯1 = φ = ¯ φ ¯3 = 0 φ = ¯ φ ¯1 = φ = 0 , ¯ φ ¯3 = m φ = ¯ φ ¯1 = ¯ φ ¯3 = 0 , φ = m φ = ¯ φ ¯1 = 0 , φ = ¯ φ ¯3 = m (51)In the deformed case, the vacua (51) are still present. In addition, there areother four supersymmetric vacua: φ = ¯ φ ¯3 = 0 , ¯ φ ¯1 = ± i h √− det C , φ = m / φ = ¯ φ ¯3 = 0 , ¯ φ ¯1 = ± i h √− det C , φ = m (52)9s in the case of the extrema (39) of the previous example, in the limit det C → + these extrema correspond to runaway directions which do not exist in thecase of the undeformed potential det C = 0.Concerning the supersymmetry breaking vacua, there is no difference be-tween the undeformed and deformed case, having in both the pseudomodulispace: φ = ¯ φ = m / V = ( m / h in the undeformed case and V E = − ( m / h in thedeformed one. We end this section discussing conditions on a general cubic superpotential un-der which the vacuum structure remains unaffected by the deformation. Con-sider n chiral superfields Φ i ( i = 1 , , . . . , n ), a canonical K¨ahler potential anda deformed superpotential of the form W (Φ p ) = C + C q Φ q + C qr Φ q ∗ Φ r + C qrs Φ q ∗ Φ r ∗ Φ s (54)with C, C q , C qr , y C qrs arbitrary coefficients, symmetric in all their indices. Asbefore, in view of the form of the superpotential, the functional W ( φ i + ξcF i ),as defined in (16), will just contain terms with powers 0, 1, 2, and 3 of ξ . Onlyeven powers will contribute to ˜ W obtaining W ( φ i + ξcF i ) = C + C q φ q + C qr φ q φ r + C qrs φ q φ r φ s + ξ c (cid:0) C qr F q F r + 3 C qrs ( φ q F r F s ) (cid:1) (55) f W (cid:0) φ i , F i (cid:1) =2 ( C + C q φ q + C qr φ q φ r + C qrs φ q φ r φ s )+ 2 c (cid:0) C qr F q F r + 3 C qrs ( φ q F r F s ) (cid:1) (56)Using the equations of motion for auxiliary fields ¯ F ¯ j we find F i = − ∂ ¯ W /∂ ¯ φ i and then f W , i = 2 ( C i + 2 C ir φ r + 3 C irs φ r φ s ) + 2 c C irs F r F s (57)With this V = F i (cid:2) C i + 2 C ir φ r + 3 C irs φ r φ s + c C irs F r F s (cid:3)(cid:12)(cid:12) F i = F i ( φ j , ¯ φ ¯ j ) (58)The extrema conditions are0 = ∂V∂φ j = 2 F i [ C ij + 3 C ijr φ r ] | F i = F i ( φ j , ¯ φ ¯ j ) (59)0 = ∂V∂ ¯ φ ¯ j = F i , ¯ j (cid:2) C i + 2 C ir φ r + 3 C irs φ r φ s + 3 c C irs F r F s (cid:3)(cid:12)(cid:12) F i = F i ( φ j , ¯ φ ¯ j ) (60)10uppose that the following relations among coefficients C ij and C ijr hold (cid:0) C ij + 3 C ijr φ r (cid:1) = δ ia M j + δ ja M i (61)for some value a ( M i is an arbitrary, field dependent, vector). Such conditionsimply that F a = 0 (unless, for all i , the pairs of coefficients ( C i , C iaa ) areproportional to each other, cf. (59)). If we still impose a more restrictivecondition on C ijr , namely that it vanishes unless it has two indices a , we seethat the extrema conditions (60) are independent of det C and also the potentialat the extrema is unaffected by the deformation.Is easy to see that the above mentioned conditions force the potential to takethe form W = X i = a Φ i ∗ g i (Φ a ) + ST (62)with g i quadratic functions, not all proportional to each other.By the above arguments, the vacuum structure of this superpotential is notdeformed. Note that the explicit examples previously discussed in subsection3.2 belong (for m = 0) to this class of potentials, insensitive to the deformations. As a first simple example of noncanonical K¨ahler potential we consider K = (cid:0) Φ ∗ ¯Φ (cid:1) (63)In this case eqs. (18) and (19) take the form Z (cid:0) φ, ¯ φ, F (cid:1) = Z − dξ (cid:0) φ ¯ φ + ξcF ¯ φ (cid:1) = 2 (cid:0) φ ¯ φ (cid:1) + 23 c (cid:0) ¯ φF (cid:1) Y (cid:0) φ, ¯ φ, F, ¯ F (cid:1) = ¯ F (cid:18) φ ¯ φ + 43 c F ¯ φ (cid:19) −
12 ( ¯ χ ¯ χ ) (cid:18) φ + 43 c F (cid:19) + c Z − dξξ (cid:2) ∂ µ ¯ φ∂ µ ¯ φ (cid:0) φ + 4 ξcφF + 2 ξ c F (cid:1) + 2 (cid:3) ¯ φ (cid:0) φ ¯ φ + ξcF ¯ φ (cid:1) ( φ + ξcF ) (cid:3) = ¯ F (cid:18) φ ¯ φ + 43 c F ¯ φ (cid:19) −
12 ( ¯ χ ¯ χ ) (cid:18) φ + 43 c F ¯ φ (cid:19) + (cid:18) c φF (cid:19) (cid:2) ∂ µ ¯ φ∂ µ ¯ φ + ¯ φ (cid:3) ¯ φ (cid:3)
11o that the kinetic part of the component field Lagrangian reads L K = − F F φ ¯ φ + 2 F ( ¯ χ ¯ χ ) φ + 23 det CF (cid:2) ∂ µ ¯ φ∂ µ ¯ φ + ¯ φ (cid:3) ¯ φ (cid:3) (64) − ∂ µ ¯ φ∂ µ ¯ φ (cid:18) φ −
43 det CF (cid:19) − (cid:3) ¯ φ (cid:18) φ ¯ φ + 43 c F ¯ φ (cid:19) − χχ (cid:0) F ¯ φ − χ ¯ χ (cid:1) + 12 i ( χσ µ ¯ χ ) ∂ µ ¯ φ φ + 12 i ( χσ µ ∂ µ ¯ χ ) 8 φ ¯ φ (65)Since det C only affects kinetic energy terms for ¯ φ , the scalar potential for thisnoncanonical K¨ahler potential could only be deformed by contributions arisingfrom the superpotential.Because L K is a linear functional of the K¨ahler potential the discussion abovealso applies to the case K = Φ ∗ ¯Φ + λ (cid:0) Φ ∗ ¯Φ (cid:1) (66)Such a K¨ahler potential can be though as resulting from the approximation ofa general potential K ∗ (cid:0) Φ , ¯Φ (cid:1) = f (cid:0) Φ ∗ ¯Φ (cid:1) for Φ ≈
0. Then, in the weak-fieldregime we have to expect that only the deformation of the superpotential wouldaffect the vacuum structure.Modifications arise for K¨ahler potentials with higher powers, namely (cid:0) ¯ΦΦ (cid:1) n with n >
2. Consider the simplest case n = 3, K = (cid:0) ¯Φ ∗ Φ (cid:1) (67)Since we are interested in purely bosonic contributions with no derivatives, wewill restrict our analysis to these type of terms which will be indicated with thesubscript “boson”. We have, Z (cid:0) φ, ¯ φ, F (cid:1) = Z − dξ (cid:0) φ ¯ φ + ξcF ¯ φ (cid:1) = 2 h(cid:0) φ ¯ φ (cid:1) − det C ¯ φ φF i Y boson (cid:0) φ, ¯ φ, F, ¯ F (cid:1) = 6 ¯ F (cid:2) φ ¯ φ − det C ¯ φ φF (cid:3) ∂Y boson ∂φ = 6 ¯ F (cid:2) φ ¯ φ − det C ¯ φ F (cid:3) The corresponding contribution to the Lagrangian is, L K | boson = 3 F ¯ F ¯ φ (cid:0) φ − det CF (cid:1) (68)so the relevant parts of the equations of motion for the auxiliary fields are3 F ¯ φ (cid:0) φ − det CF (cid:1) + ∂ ¯ W ∂ ¯ φ = 0 (69)9 ¯ F ¯ φ φ − CF ¯ F ¯ φ + ∂ W ∂φ = 0 (70)12e then conclude that both F and ¯ F will depend on det C independently ofthe choice of the superpotential, so that for a K¨ahler potential cubic in ¯Φ ∗ Φthe scalar potential and, a fortiori, the vacuum structure will be affected by thedeformation.Let us consider a simple example that illustrates the discussion above. Itcorresponds to superpotentials W and ¯ W (recall that in Euclidean space, theyare independent functionals) W = 12 f Φ ∗ Φ , ¯ W = g (71)and the K¨ahler potential defined in (67).Given superpotentials (71) we get for the auxiliary fields, using eqs. ofmotion (69) and (70), F = i √ φ det C (72)¯ F = f φ ¯ φ (73)It can be seen from eq.(17) that, as expected, the scalar potential is affected bythe deformation of the K¨ahler potential through the dependence of F on det C as given by (72).Let us end this section by pointing that a completely analogous behavior canbe found for a K¨ahler potential of the form K n = ( ¯Φ ∗ Φ) n . For example, forodd n we find, instead of eq.(69), that the auxiliary field F obeys the equation n √− det C ¯ φ n − (cid:16) ( φ + √− det CF ) n − ( φ − √− det CF ) n (cid:17) + ∂ ¯ W∂ ¯ φ = 0 (74)This is a degree n polynomial equation for F , with coefficients depending ondet C as a result of the deformation in the K¨ahler potential. The solution for F will be in general det C -dependent (as we have explicitly seen for the particularcase n = 3) and hence the scalar potential as given by (17) will in turn bedeformed. In this work we have discussed the vacuum structure of N = 1 / N = 1 / i are not the complex conjugate of Φ i , scalars ¯ φ ¯ i do not ingeneral coincide with φ ∗ i . This of course complicates the analysis of extrema ofthe potential unless we impose some restrictions on fields and potentials.Restricting the analysis to the case of field configurations such that ¯ φ ¯ i = φ ∗ i ,we have seen in section 3.2 that the vacuum configurations for superpotentials(24) and (40) described in [9]-[10] for undeformed superspace, are also presentin the deformed case when the coefficient m = 0. Hence in both cases thereis symmetry breaking and a classical pseudomoduli space with degenerate nonsupersymmetric vacua. The difference between the two cases is that in the latterthere can be metastable (for an appropriate choice of coefficients) vacua whichare absent in the former.An interesting phenomenon takes place for m = 0: in the limit det C → C = 0. This phenomenon is resemblant ofwhat happens with solitons in θ -deformed noncommutative space: apart fromthose that reproduce the ordinary regular solitons in the θ → C -dependence on the vacuumstructure. The case K = ( ¯Φ ∗ Φ) is a counterexample of this possibility sincewe proved that only the kinetic energy is affected by the deformation. Hence,in a weak-field approximation, the vacuum dependence on the C -deformationwill only enter through the deformed superpotential. We need higher powers( n >
2) of ¯Φ ∗ Φ in order to change the vacuum structure as we have explicitlyshown at the end of section 4.The discussion in this work is valid at tree-level, and should be corrected byincluding leading quantum corrections to the potential. Being the theory non-hermitian, one should resort to complex saddle point or steepest descent meth-ods. We hope to report on this issue in a following investigation. O’Raifeartaigh-type models, as those considered here, can arise naturally and dynamically inthe low-energy limit of simple SUSY gauge theories. In this respect, the exten-sion of the analysis we have presented to the case of deformed super Yang-Millstheory is also a subject we hope to address in the future.Acknowledgements: This work was partially supported by PIP6160-CONICET,BID 1728OC/AR PICT20204-ANPCYT grants and by CIC and UNLP, Ar-gentina. 14 eferences [1] J. de Boer, P. A. Grassi and P. van Nieuwenhuizen, Phys. Lett. B (2003) 98.[2] H. Ooguri and C. Vafa, Adv. Theor. Math. Phys. (2004) 405.[3] N. Seiberg, JHEP (2003) 010.[4] T. Araki, K. Ito and A. Ohtsuka, Phys. Lett. B (2003) 209[5] O. Lunin and S. J. Rey, JHEP (2003) 045.[6] R. Britto, B. Feng and S. J. Rey, JHEP (2003) 067; R. Britto, B. Fengand S. J. Rey, JHEP (2003) 001;[7] R. Britto and B. Feng, Phys. Rev. Lett. (2003) 201601[8] L. O’Raifeartaigh, Nucl. Phys. B (1975) 331.[9] K. Intriligator, N. Seiberg and D. Shih, JHEP , 021 (2006).[10] K. Intriligator and N. Seiberg, Class. Quant. Grav. , S741 (2007).[11] D. Shih, arXiv:hep-th/0703196.[12] S. P. de Alwis, Phys. Rev. D (2007) 086001.[13] K. Intriligator, N. Seiberg and D. Shih, JHEP (2007) 017.[14] S. Ray, Phys. Lett. B (2006) 137.[15] T. Hatanaka, S. Ketov, Y Kobayashi and S. Sasaki, Nucl. Phys. B726 (2005) 481.[16] L. Alvarez-Gaume and M. A. Vazquez-Mozo, JHEP (2005) 007.[17] T. Hatanaka, S. Ketov and S. Sasaki, Phys. Lett.
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