Supersymmetric Nambu Jona-Lasinio Model on = 1/2 four-dimensional − N Non(anti)commutative Superspace Tadafumi Ohsaku Institut fu¨r Theoretische Physik, Universit¨at zu K¨oln, 50937 K¨oln, Deutschland (Dated: February 1, 2008) We construct the Lagrangian of the N = 1 four-dimensional generalized supersymmetric Nambu−Jona-Lasinio ( SNJL ) model, which has N = 1/2 supersymmetry ( SUSY ) on non(anti)commutativesuperspace. Aspecialattentionispaidtotheexaminationonthenonpertur- bativequantum dynamics: The phenomenon of dynamical-symmetry-breaking/mass-generation on thedeformedsuperspaceisinvestigated. ThemodelLagrangian andthemethodofSUSYauxiliary fields of composites are examined in terms of component fields. We derive the effective action, ex- amineit,andsolvethegapequationforself-consistent massparameters. (Keywords: Superspaces, Non-CommutativeGeometry, SupersymmetricEffective Theories. ) 8 0 PACSnumbers: 11.10.Nx,11.10.Lm,11.25.-w,11.30.Pb,11.30.Qc 0 2 n I. INTRODUCTION a J Field theories on noncommutative spacetime defined by the Heisenberg algebra [xµ,xν] = iΘµν = 0 have various 1 6 interesting properties [1,2], and considered that they might capture important aspects of quantum gravity. Recently, 3 it was shown that, low-energy effective descriptions of four-dimensional superstrings in the self-dual graviphoton 3 background fields Fαβ ( with setting anti-self-dual part Fα˙β˙ = 0 ) give the deformations of superspaces [3,4] ( in a v ten-dimensional case, see Ref. [5,6] ) as =1 =1/2, with giving 5 N →N 3 θα,θβ = 2α′Fαβ. (1) 1 { } 3 Here, α′ is the inverse of a string tension. Motivated by these examinations, various physical and mathematical 0 aspects of the deformation of superspace obtained much attentions [7-22]. Under this context, supersymmetric ( 6 SUSY ) Euclidean field theories on non(anti)commutative superspace were first considered in Ref. [7]. The =1/2 0 N / deformed Wess-Zumino ( WZ ) model [7,11,12,21] and deformed SUSY gauge models with(out) matter [7,10,13,14] h wereexaminedbyseveralworks,andthe preservationofthe locality,Lorentzsymmetryandthe renormalizabilitiesof t - allordersinperturbativeexpansionsinthesemodelswereproved[13]. ItwasalsoshowninRefs.[11,12]thatthechiral p superpotential of the WZ model obtains quantum radiative corrections while the antichiral superpotential part of it e obeysthenonrenormalizationtheorembytheperturbativeanalyses. Bythequantumdynamicsofitself,theWZmodel h : gets a term which is not included in the classical deformed action of it [11,12,21]. The methods to construct generic v sigma models both on = 1 four-dimensional and = 2 two-dimensional deformed superspaces were proposed, i N N X and the component field expressions for them were derived [15-19]. Until now, various examinations were performed under the classical or perturbative quantum levels. Hence, it is interesting for us to examine nonperturbative effects r a of quantum dynamics of a model on a deformed superspace. In this paper, we investigate a nonperturbative quantum dynamics in a SUSY model on the = 1/2 four- N dimensional non(anti)commutative superspace. Of particular interest in here is the phenomenon of dynamical sym- metry breaking of the mass generation of the BCS-NJL mechanism ( Bardeen, Cooper, Schrieffer, Nambu and Jona- Lasinio ) [23,24,25]. Until now, there is no attempt for an investigation of the nonperturbative dynamical mass generation in literature. We introduce an = 1 four-dimensional supersymmetric Nambu Jona-Lasinio ( SNJL N − ) model [26-31], regarded as the simplest example which might realize the dynamical symmetry-breaking/mass- generation,andexaminethecharacteristicaspectsofthemodelonthedeformedsuperspace. Thenonrenormalization theorem in the = 1 case forbids the dynamical break down of chiral symmetry by the mass generation, while N it can take place when = 1 SUSY is completely broken by introducing a non-holomorphic soft-SUSY-breaking N mass [26-31]. Because the nonrenormalization theorem is partly broken in the = 1/2 superspace, it is interesting N foruswhethertheSNJLmodelgetsadynamicalmassradiativelyornot. InSec. II,wesummarizethedefinitionsand algebraic properties of the deformed superspace. The SNJL model in the deformed superspace will be examined in detail in Sec.III. The method of SUSY auxiliaryfields of composites onthe deformedsuperspace is investigated. The one-loopeffective actioninthe leadingorderofthe large-N expansionwillbe evaluatedby the superspaceformalism, and the gap equation for a self-consistent mass will be derived and solved. The comments on further investigations for our method are given in Sec. IV. 2 II. SUPERFIELDS ON NON(ANTI)COMMUTATIVE SUPERSPACE First, we summarize several algebraic definitions of = 1/2 four-dimensional non(anti)commutative superspace N for the preparation of our calculations given later. The notations and spinor conventions follow both Ref. [7] and the textbook of Wess and Bagger [32]. The (anti)commutation relations for bosonic coordinates xµ and fermionic coordinates θα, θ¯α˙ ( α,α˙ =1,2 ) are required to satisfy the following relations: θα,θβ Cαβ =Cβα =0, (2) { } ≡ 6 θ¯α˙,θ¯β˙ = θ¯α˙,θα =[θ¯α˙,xµ]=0. (3) { } { } Cαβ arethe deformationparametersof superspace. Hence (θα)∗ =θ¯α˙, they are independent with eachother. At this stage, the Lorentz symmetry SO(4) = SU(2) SU(2) is exp6licitly broken to SU(2) at Cαβ = 0. Due to this L R R fact, we have to work in the Euclidean spacetime×R4, though the Lorentzian signature ηµν =diag(6 1,1,1,1)is used throughout this paper. Any functions of θ have to be ordered, and we use Weyl ordering: (θαθβ−) 1[θα,θβ] = W ≡ 2 1ǫαβθθ. For example, θθ is Weyl ordered. The anticommutator (2) of the Clifford algebra gives nontrivial results −2 comparedwith the ordinary =1 SUSY case. Clearly,the Hermiticity of a theorywas lost,and the unitarity might N be violated in a field theory on non(anti)commutative superspace. Thus, a potential energy could become complex and it would give an unstable vacuum of a theory. However, it was proved that the vacuum energy vanishes in the deformed WZ model because the notion of anti-holomorphicity remains in the part of its antichiral superpotential, and this fact makes its vacuum stable and vanishes the vacuum energy [11,12]. This situation might be true in a model which has an antichiral superpotential. From these exotic characters, it is particularly interesting for us to generalize several methods of quantum field theory to cases on the deformed superspace. The chiral coordinates yµ are defined as yµ xµ+iθασµ θ¯α˙, (4) ≡ αα˙ while, the antichiral coordinates y¯µ are given as follows: y¯µ yµ 2iθασµ θ¯α˙ =xµ iθασµ θ¯α˙. (5) ≡ − αα˙ − αα˙ Several commutators are summarized below: [yµ,yν]=[yµ,y¯ν]=[yµ,θα]=[yµ,θ¯α˙]=0, [y¯µ,y¯ν]=4θ¯θ¯Cµν, [xµ,θα]=iCαβσµ θ¯α˙, [xµ,xν]=θ¯θ¯Cµν, Cµν Cαβǫ (σµν)γ = Cνµ. (6) βα˙ ≡ βγ α − These commutators make the structure of the deformed superspace more clearly. Due to the commutation relations [yµ,yν] = 0, the supercovariant derivatives can be chosen so as to satisfy the Leibniz rule, then it is possible for us to define a chiral superfield on the deformed superspace [7]. As a consequence, the deformed superspace will be parametrized by a set (y,θ,θ¯) as coordinates in the chiral basis. We have to keep this fact in mind to perform any computations of quantities given by (anti)chiral superfields on the deformed superspace. The supercovariant derivatives and the supercharges are defined by ∂ ∂ ∂ D +2iσµ θ¯α˙ , D , (7) α ≡ ∂θα αα˙ ∂yµ α˙ ≡−∂θ¯α˙ and ∂ ∂ ∂ Q , Q +2iθασµ , (8) α ≡ ∂θα α˙ ≡−∂θ¯α˙ αα˙∂yµ respectively. Clearly, they satisfy 0 = D ,D = D ,D = D ,Q = D ,Q = D ,Q = D ,Q = Q ,Q , { α β} { α˙ β˙} { α β} { α˙ β} { α β˙} { α˙ β˙} { α β} ∂ ∂ ∂2 D ,D = 2iσµ , Q ,Q =2iσµ , Q ,Q = 4Cαβσµ σν . (9) { α˙ α} − αα˙ ∂yµ { α˙ α} αα˙∂yµ { α˙ β˙} − αα˙ ββ˙∂yµ∂yν AnyproductsofWeyl-orderedfunctionsofθ havetobeWeylre-ordered,anditisimplementedbythestarproduct: Cαβ ←∂− −→∂ f(y,θ)⋆g(y,θ) f(y,θ)exp g(y,θ) ≡ − 2 ∂θα∂θβ (cid:16) Cα(cid:17)β ∂ ∂ ∂ ∂ = f(y,θ)g(y,θ)+( 1)degf f(y,θ) g(y,θ) detC f(y,θ) g(y,θ), (10) − 2 ∂θα ∂θβ − ∂θθ ∂θθ 3 where, ∂ ǫαβ ∂ ∂ 1 =Q2, detC = CµνC . (11) ∂θθ ≡ 4 ∂θα∂θβ 4 µν The derivatives ∂ should apply at fixed yµ. From the definitions given above, one finds ∂θα ǫαβ Cαβ θα⋆θβ = θθ+ , θα⋆θθ =Cαβθ , θθ⋆θα = Cαβθ , β β − 2 2 − 1 1 θθ⋆θθ = detC, θσµθ¯⋆θσνθ¯= ηµνθθθ¯θ¯ Cµνθ¯θ¯. (12) − −2 − 2 Because of the commutator [y¯µ,y¯ν]=4θ¯θ¯Cµν =0, the definition of the star product (10) is applicable when f and g are given as functions of only y, θ and θ¯. For a6 star product of f¯(y¯,θ¯) and g¯(y¯,θ¯), the following re-expression, ←−∂ −→∂ f¯(y¯,θ¯)⋆g¯(y¯,θ¯)=f¯(y¯,θ¯)exp 2θ¯θ¯Cµν g¯(y¯,θ¯) (13) ∂y¯µ∂y¯ν (cid:16) (cid:17) is also useful. Because Q have θα, they do not commute with the star product: Q are broken generators. In other words, α˙ α˙ the ”translation” symmetry of ξ¯α˙-directions are explicitly broken, and we must choose the point of the direction for constructingafieldtheory. Ontheotherhand,Q arenotbrokengenerators,hencethedeformedsuperspaceiscalled α as =1/2 superspace. The definition of a chiral superfield Φ is N Φ(y,θ) φ(y)+√2θψ(y)+θθF(y), D Φ=0. (14) α˙ ≡ Hence Φ(y,θ) has the Weyl-ordered form. An antichiral superfield Φ¯ is defined as follows: Φ¯(y¯,θ¯) φ¯(y¯)+√2θ¯ψ¯(y¯)+θ¯θ¯F¯(y¯), D Φ¯ =0. (15) α ≡ Because Φ¯(y¯,θ¯) includes y¯, it has to be Weyl ordered. It will be written in the chiral coordinates yµ under the Weyl-ordered form as a function of θ: Φ¯(y¯,θ¯)=Φ¯(y 2iθσθ¯,θ¯)=φ¯(y)+√2θ¯ψ¯(y)+θ¯θ¯F¯(y)+√2θ iσµ∂ ψ¯(y)θ¯θ¯ i√2σµθ¯∂ φ¯(y) +θθθ¯θ¯(cid:3)φ¯(y). (16) µ µ − { − } Onthedeformedsuperspace,ΦandΦ¯ areindependentwitheachother: (Φ)† =Φ¯. Fromthepreservationofchiralities 6 in both a chiral and an antichiral fields on non(anti)commutative superspace, we should utilize the chiral projectors for a variation of (anti)chiral superfields, in perturbative and nonperturbative calculations, so forth, as same as the ordinary =1 case. N A star product of (anti)chiral superfields is again a (anti)chiral superfield. For example, Φ (y,θ)⋆Φ (y,θ) = Φ (y,θ)Φ (y,θ) Cαβ(ψ ) (y)(ψ ) (y) detCF (y)F (y) 1 2 1 2 1 α 2 β 1 2 − − +√2Cαβθ [(ψ ) (y)F (y) (ψ ) (y)F (y)], (17) β 1 α 2 2 α 1 − ∂ ∂ Φ¯ (y¯,θ¯)⋆Φ¯ (y¯,θ¯) = Φ¯ (y¯,θ¯)Φ¯ (y¯,θ¯)+2θ¯θ¯Cµν Φ¯ (y¯,θ¯) Φ¯ (y¯,θ¯), (18) 1 2 1 2 ∂y¯µ 1 ∂y¯ν 2 Cαβ ←∂− −→∂ Φ¯ (y¯,θ¯)⋆Φ (y,θ) = Φ¯ (y 2iθσθ¯,θ¯)⋆Φ (y,θ)=Φ¯ (y 2iθσθ¯,θ¯)exp Φ (y,θ). (19) 1 2 1 − 2 1 − − 2 ∂θα∂θβ 2 (cid:16) (cid:17) Any products of (anti)chiral superfields do not commute by the non(anti)commutativity, Φ ⋆Φ = Φ ⋆Φ , etc., 1 2 2 1 6 there are ambiguities to construct a Lagrangian in terms of products of chiral superfields. In this paper, we will use the symmetrized star products between (anti)chiral superfields themselves defined as follows [15,16]: 1 Φ ⋆Φ ⋆ ⋆Φ (Φ ⋆Φ ⋆ ⋆Φ +permutations), (20) 1 2 n sym 1 2 n ··· | ≡ n! ··· 1 Φ¯ ⋆Φ¯ ⋆ ⋆Φ¯ (Φ¯ ⋆Φ¯ ⋆ ⋆Φ¯ +permutations), (21) 1 2 n sym 1 2 n ··· | ≡ n! ··· and then 1 Φ ⋆ ⋆Φ ⋆Φ¯ ⋆ ⋆Φ¯ (Φ ⋆ ⋆Φ +permutations) 1 i sym i+1 n sym 1 i ··· | ··· | ≡ i!(n i)! ··· − ⋆(Φ¯ ⋆ ⋆Φ¯ +permutations). (22) i+1 n × ··· 4 These symmetrized products could be interpreted as a kind of ordering in terms of (anti)chiral superfields. There are a few choices of symmetrization, namely, taking the symmetrization between a chiraland an antichiral parts of a productornot,andingeneralthedifferenceofchoicesgivesdifferentresultsoftheircomponentfieldexpressions[16], thoughatleastinthemodelLagrangianwewillconsiderinthispaper,thereisonlyadifferenceofanumericalfactor. For example, 1 ∂ ∂ Φ ⋆Φ (Φ ⋆Φ +Φ ⋆Φ )=Φ Φ detC Φ Φ =Φ Φ detC(Q2Φ )(Q2Φ ), (23) 1 2 sym 1 2 2 1 1 2 1 2 1 2 1 2 | ≡ 2 − ∂θθ ∂θθ − 1 Φ¯ ⋆Φ¯ (Φ¯ ⋆Φ¯ +Φ¯ ⋆Φ¯ )=Φ¯ Φ¯ . (24) 1 2 sym 1 2 2 1 1 2 | ≡ 2 Hence, the deformation will modify an action functional of a theory through a product of chiral multiplets. There is no contribution coming from the deformation in a symmetrized product of antichiral superfields, because Cµν = Cνµ. Fortunately, such Lorentz-symmetry breaking parameters are explicitly removed from a theory under the − symmetrization in the superfield level, and a theory includes the deformation parameters only in the Lorentz-scalar form detC, in spite of the Lorentz-symmetry breaking relation defined in Eq. (2). III. THE GENERALIZED SUPERSYMMETRIC NAMBU−JONA-LASINIO MODEL A. Symmetries and Structure We employ the following U(N) U(N) -invariant Lagrangian of an = 1 generalized SNJL model on the L R × N four-dimensional deformed superspace: (C) K(Φ¯ ,Φ ) = + (1)(C)+ (2)(C), (25) L ≡ ± ± θθθ¯θ¯ L0 LI LI h i Φ¯ ⋆Φ +Φ¯ ⋆Φ = Φ¯ Φ +Φ¯ Φ , (26) 0 + + − − + + − − L ≡ θθθ¯θ¯ θθθ¯θ¯ hG i h i (1)(C) 1 Φ¯ ⋆Φ¯ ⋆Φ ⋆Φ , LI ≡ N + −|sym + −|sym θθθ¯θ¯ G h i = 1 Φ¯ Φ¯ Φ Φ detC(Q2Φ )(Q2Φ ) , (27) + − + − + − N − θθθ¯θ¯ G h (cid:16) (cid:17)i (2)(C) 2 (Φ¯ ⋆Φ¯ +Φ¯ ⋆Φ¯ )⋆(Φ ⋆Φ +Φ ⋆Φ ) , LI ≡ 4N + + − − + + − − θθθ¯θ¯ G h i = 2 (Φ¯ Φ¯ +Φ¯ Φ¯ ) Φ Φ +Φ Φ detC(Q2Φ )2 detC(Q2Φ )2 . (28) + + − − + + − − + − 4N − − θθθ¯θ¯ h n oi Here, K(Φ¯ ,Φ ) is the K¨ahler potential of our model, N is a number of flavor. G and G are coupling constants ± ± 1 2 ( G ,G > 0 ) they have mass dimension [Mass]−2, thus this model is nonrenormalizable under a power-counting 1 2 analysisinthe =1case. IntheobservationoftheWZmodel,theoperatorscomingfromthedeformationcarrymass N dimensionslargerthanfour,thoughtheyobtainradiativecorrectionswithdivergencesatmostlogarithmically[11,12]. ThisfactisoneofthebasesoftheproofofrenormalizabilityofthedeformedWZmodel. Weconsiderthatthecharacter of the divergent nature and non-renormalizability of the SNJL model is unchanged by the deformation. This model has no superpotential at this stage, and by applying the method of SUSY auxiliary fields of composites, the model will get (anti)holomorphic superpotentials. The parity symmetry of spatial inversion is maintained in the form of the Lagrange function in the ordinary = 1 case [30,31], and the deformation explicitly breaks the symmetry. N The global gauge symmetry of the theory is chosen as U(1), and Φ and Φ are oppositely charged. According to + − Eqs. (20)and(21),thesymmetrizationofproductsof(anti)chiralsuperfieldsthemselveshasbeentakenintheK¨ahler potentialgivenabove. Intheordinary =1SUSYtheory, (1) waspreparedforthedynamicalgenerationofaDirac N LI mass, while (2) is suitable for obtaining left-handed and right-handed Majorana mass terms [30,31]. Historically, LI the SNJL modelwas firstintroducedto investigatethe phenomenonofdynamicalchiralsymmetry breakinginSUSY field theory [26,27], and applied to the top quark condensation of electroweak symmetry breaking of the minimal SUSY Standard Model [28]. An SU(N ) SNJL model was used to describe phenomena of phase transitions in the c early universe [29]. Investigations on dynamical chiral symmetry breaking and (color)superconductivity of SUSY condensed matter systems were done by (25) at the case of the ordinary = 1 superspace with a SUSY-breaking N mass [30,31]. 5 Our model has following (pseudo)symmetries. The global flavor symmetry SU(N) SU(N) will be introduced L R × in the theory by the following definitions: N2−1 Φ e−iΓ+Φ , Φ¯ Φ¯ eiΓ+, Φ e−iΓ−Φ , Φ¯ Φ¯ eiΓ−, Γ ΓI T , (29) + → + + → + − → + − → − ± ≡ ± I I=1 X where,Φ (Φ )belongstothe(anti)fundamentalrepresentationN (N¯ )ofSU(N),andallofΓI arerealc-numbers. + − ± Thus, the normalization factor T(f) and the second-order Casimir invariant C(f) of the Hermitian generators T of I SU(N) ( I =1, ,N2 1 ) satisfy ··· − 1 N2−1 N2 1 N trT T =T(f)δ = δ , C(f)δ = (T T ) = − δ , T(f)= C(f). (30) I J IJ 2 IJ ij I I ij 2N ij N2 1 I=1 − X It should be mention that, (e−iΓ±Φ )† = Φ¯ eiΓ± in general. If the symmetry of the unitary group are gauged, ± ± 6 the discussion becomes more involved because of the deformation. Since the BCS-NJL mechanism breaks a global symmetry,thedefinitionsofsymmetriesweconcernhereareenoughforourpurpose. TheLagrangianhasthefollowing global Abelian symmetries: U(1) : Φ eiαvΦ , Φ e−iαvΦ , Φ¯ e−iαvΦ¯ , Φ¯ eiαvΦ¯ , (31) V + + − − + + − − → → → → U(1) : Φ eiαaΦ , Φ eiαaΦ , Φ¯ e−iαaΦ¯ , Φ¯ e−iαaΦ¯ , (32) A + + − − + + − − → → → → where, α and α are real. On the other hand, the R-symmetry, v a U(1)R :θ eiαrθ, θ¯ e−iαrθ¯, φ± e2inαrφ±, ψ± e2i(n−12)αrψ±, F± e2i(n−1)αrF±, (33) → → → → → was broken by the deformation. The deformation parameters Cαβ carry a non-vanishing U(1) -charge: R Cαβ e2iαrCαβ. (34) → In fact, Cαβ act as R-symmetry-breakingparameters. Next, we examine the component field expressionof the Lagrangian(25). In Refs. [15,16],component field expres- sions for =1/2 generic chiral sigma models on the deformed superspace were obtained: It is given by a sum of an N undeformedpartandadeformedpart, andthe latter is givenas aninfinite-order powerseriesofdetC with including infinite-order partial derivatives of a K¨ahler potential taken with respect to scalar fields. We follow the results of Ref. [16], and add the following extra terms of the first-order in detC to the undeformed model: 1 ∂3K(φ¯ ,φ ) 1 ∂4K(φ¯ ,φ ) (C) detC F F ± ± (cid:3)φ¯ + ± ± ∂ φ¯ ∂µφ¯ . (35) Ldef ≡ − i j 2! ∂φ ∂φ ∂φ¯ k 2!∂φ ∂φ ∂φ¯ ∂φ¯ µ k l i j k i j k l i,Xj=± nkX=± kX,l=± o In our case, the deformed part of the Lagrangianis given as a function of metrics, connections and curvature tensors ofthe K¨ahlermanifold. With the well-knownprocedureofthe theoryofK¨ahlermanifold,the undeformedpartofthe Lagrangian(25), (C =0), will be obtained as follows: L (C) = (C =0)+ (C), (36) def L L L 1 1 (C =0) = gij∗ F¯jFi ∂µφ¯j∂µφi iψ¯jσ¯µ∂µψi + gij∗,k∗( Fiψ¯jψ¯k)+gij∗,k( F¯jψiψk) L − − −2 −2 i,Xj=± (cid:0) (cid:1) i,jX,k=±h i 1 + gij∗,k( iψ¯jσ¯µψk∂µφi)+ gij∗,kl∗ (ψiψkψ¯jψ¯l). (37) − 4 i,j,k=± i,j,k,l=± X X The K¨ahler metric of our model is given in the following matrix form: gˆ g++∗ g−+∗ = 1+ GN1φ¯−φ−+ GN2φ¯+φ+, GN1φ¯−φ++ GN2φ¯+φ− , (38) ≡ (cid:18)g+−∗ g−−∗ (cid:19) (cid:18) GN1φ¯+φ−+ GN2φ¯−φ+, 1+ GN1φ¯+φ++ GN2φ¯−φ−,(cid:19) and the notations of the definitions gij∗,k =∂gij∗/∂φk, etc., have been used in Eq. (37). In the K¨ahler metric given above, the Hermiticity may be lost, (gˆ)† = gˆ because (φ )† = φ¯ in general. Obviously, the deformed Lagrangian ± ± 6 6 (25) still has the invariance under the K¨ahler transformation: K(Φ¯ ,Φ ) K(Φ¯ ,Φ )+ (Φ )+ ¯(Φ ), (39) ± ± ± ± ± ± → F F 6 where, ( ¯ ) is an arbitrarilyfunction of Φ ( Φ¯ ). Now, the Euler equations for the auxiliary fields are found to ± ± F F be ∂ (C) 1 1 1 0= L = gij∗F¯j gij∗,k∗ψ¯jψ¯k 2detCFj gjk∗,i(cid:3)φ¯k+ gjk∗,il∗∂µφ¯k∂µφ¯l , (40) ∂Fi " − 2 − 2! 2! # jX=± kX=± (cid:16)kX=± kX,l=± (cid:17) ∂ (C) 1 0= L∂F¯ = gji∗Fj − 2 gji∗,k∗ψjψk , (41) i jX=±h kX=± i ( i= ). From these results, one finds ± F 1 ∂gˆ ψ ψ + = gˆ−1 + k , (42) (cid:18)F− (cid:19) 2 k=±∂φ¯k (cid:18)ψ−ψk (cid:19) X F¯+ = 1(gˆT)−1 ∂gˆT ψ¯+ψ¯k +(gˆT)−1detC g+k∗,+ g−k∗,+ (cid:3)φ¯ (cid:18)F¯− (cid:19) 2 k=± ∂φ¯k (cid:18)ψ¯−ψ¯k (cid:19) "k=±(cid:18)g+k∗,− g−k∗,− (cid:19) k X X + g+k∗,+l∗ g−k∗,+l∗ ∂ φ¯ ∂µφ¯ 1gˆ−1 ∂gˆ ψ+ψk . (43) k,l=±(cid:18)g+k∗,−l∗ g−k∗,−l∗ (cid:19) µ k l#2 k=±∂φ¯k (cid:18)ψ−ψk (cid:19) X X We find that the auxiliary fields F ,F¯ will be completely eliminated through these Euler equations in our model ± ± without any ambiguities. It is found that detC enters into four-fermion interaction terms under quite unusual forms throughthe eliminationofallthe auxiliaryfieldsF andF¯ . Thus,the quantumdynamics ofthe =1/2deformed ± ± N SNJL model might not resemble to the ordinary case, though it is difficult for us to estimate the strength of the interaction coming from the deformation in the form obtained above. B. SUSY Auxiliary Fields of Composites In this subsection, we examine the method of SUSY auxiliary fields of composites on the deformed superspace. In the Lagrangiangiveninthe previoussubsection,wehavechosenthe formofthe K¨ahlerpotentialsuitable togenerate holomorphicand antiholomorphicsuperpotentials of SUSY composites. Our model Lagrangian(25)will be rewritten in the following form through the method of SUSY auxiliary fields [27-31]: L(C) = L0+LH +L1(C)+L¯1+L2(C)+L¯2, (44) where, N 4N Φ¯ ⋆Φ +Φ¯ ⋆Φ , H¯ ⋆H + H¯ ⋆H , 0 + + − − H 1 1 2 2 L ≡ θθθ¯θ¯ L ≡ G1 G2 θθθ¯θ¯ h i h i N N L1(C) ≡ S1⋆ G1H1−Φ+⋆Φ− |sym θθ, L¯1 ≡ S¯1⋆ G1H¯1−Φ¯+⋆Φ¯− |sym θ¯θ¯, h (cid:0)4N (cid:1) i h (cid:0) 4N (cid:1) i L2(C) ≡ S2⋆ G2H2−Φ+⋆Φ+−Φ−⋆Φ− |sym θθ, L¯2 ≡ S¯2⋆ G2H¯2−Φ¯+⋆Φ¯+−Φ¯−⋆Φ¯− |sym θ¯θ¯.(45) h (cid:0) (cid:1) i h (cid:0) (cid:1) i Here,the methodofSUSYauxiliaryfieldshasbeenappliedundertakingthe symmetrizedstarproducts. Only (C) 1 and (C) havecontributionscoming fromthe deformation. Ourmodelhas superpotentialswithconstraints: SL, S¯ , 2 1 1 S anLd S¯ were introduced as Lagrange multiplier multiplets for the purpose to enforce 2 2 G G H = 1Φ ⋆Φ , H¯ = 1Φ¯ ⋆Φ¯ , 1 + − sym 1 + − sym N | N | G G H = 2(Φ ⋆Φ +Φ ⋆Φ ), H¯ = 2(Φ¯ ⋆Φ¯ +Φ¯ ⋆Φ¯ ). (46) 2 + + − − 2 + + − − 4N 4N H and H¯ ( i= 1,2 ) are regarded as SUSY composites. Because the Hermiticity was lost in our theory, (S )† = S¯ i i i i and(H )† =H¯ ingeneral. LetusconfirmtheequivalenceofLagrangefunctions(25)and(44)underthesymmetr6 ized i i 6 star products. To convert the expression of the Lagrangian (44) in the point-product form, we take into account Q2(Q2Φ ) = 0. Because detC(Q2S )(Q2H ) coming from S ⋆H do not contribute to the component form of ± i i i i sym | 7 the holomorphic superpotential of the Lagrangian through θθ-integration, we drop these terms. After performing integrations of θ by parts, L1(C), L¯1, L2(C) and L¯2 will be rewritten in the forms of ordinary point-products: N (C) = S H S Φ Φ +detCS (Q2Φ )(Q2Φ ) , 1 1 1 1 + − 1 + − L "G1 − # θθ 4N (C) = S H S (Φ Φ +Φ Φ )+detCS (Q2Φ )2+(Q2Φ )2 , 2 2 2 2 + + − − 2 + − L "G2 − { }# θθ N 4N L¯1 = S¯1 G1H¯1−Φ¯+Φ¯− θ¯θ¯, L¯2 = S¯2 G2H¯2−Φ¯+Φ¯+−Φ¯−Φ¯− θ¯θ¯. (47) h (cid:16) (cid:17)i h (cid:16) (cid:17)i In this expression, it is clear for us that the constraints given in (46) are satisfied under taking the symmetrized star products. AfterS andS¯ wereeliminated, (44)recovers(25). The componentfieldexpressionofthe Lagrangian(44) i i is found to be: = F¯ F ∂ φ¯ ∂µφ iψ¯ σ¯µ∂ ψ +F¯ F ∂ φ¯ ∂µφ iψ¯ σ¯µ∂ ψ , (48) 0 + + µ + + + µ + − − µ − − − µ − L − − − − N 4N = F¯ F ∂ φ¯ ∂µφ iψ¯ σ¯µ∂ ψ + F¯ F ∂ φ¯ ∂µφ iψ¯ σ¯µ∂ ψ , (49) LH G H1 H1 − µ H1 H1 − H1 µ H1 G H2 H2 − µ H2 H2 − H2 µ H2 1 2 (cid:16) (cid:17) (cid:16) (cid:17) N (C) = φ F +F φ ψ ψ L1 G S1 H1 S1 H1 − S1 H1 1 (cid:16) (cid:17) F φ φ +φ F φ +φ φ F φ ψ ψ φ ψ ψ φ ψ ψ +detCF F F , (50) − S1 + − S1 + − S1 + −− S1 + −− + S1 −− − S1 + S1 + − L¯1 = (L(cid:16)1(C =0))†, (cid:17) (51) 4N (C) = φ F +F φ ψ ψ F (φ2 +φ2) L2 G S2 H2 S2 H2 − S2 H2 − S2 + − 2 (cid:16) (cid:17) (cid:16) +φ (2F φ +2F φ ψ ψ ψ ψ ) 2ψ (φ ψ +φ ψ ) +detCF (F2 +F2), (52) S2 + + − −− + +− − − − S2 + + − − S2 + − L¯2 = (L2(C =0))†. (cid:17) (53) We observe (C)= (C =0)+detC F F F +F (F2+F2) ,asumofordinaryLagrangianwithdeformedpart, the result ofLRef. [16L]holds in the cas{e oSf1th+e m−odelSw2ith+SUSY− a}uxiliary fields of composites. If φ andφ¯ ( Dirac S1 S1 mass ) or φ and φ¯ ( Majorana mass ) obtain finite VEVs, superfields Φ and Φ¯ become massive. The Euler S2 S2 ± ± equations for the auxiliary fields of chiral multiplets are found to be ∂ (C) ∂ (C) 0 = L =F¯ φ φ 2φ φ +detC(F F +2F F ), 0= L =F φ¯ φ¯ 2φ¯ φ¯ , ∂F +− S1 −− S2 + S1 − S2 + ∂F¯ +− S1 −− S2 + + + ∂ (C) ∂ (C) 0 = L =F¯ φ φ 2φ φ +detC(F F +2F F ), 0= L =F φ¯ φ¯ 2φ¯ φ¯ , ∂F −− S1 +− S2 − S1 + S2 − ∂F¯ −− S1 +− S2 − − − ∂ (C) N ∂ (C) N 0 = L = φ φ φ +detCF F , 0= L = φ¯ φ¯ φ¯ , ∂F G H1 − + − + − ∂F¯ G H1 − + − S1 1 S1 1 ∂ (C) 4N ∂ (C) 4N 0 = L = φ φ2 φ2 +detC(F2 +F2), 0= L = φ¯ φ¯2 φ¯2, ∂F G H2 − +− − + − ∂F¯ G H2 − +− − S2 2 S2 2 ∂ (C) N ∂ (C) N 0 = L = (F¯ +φ ), 0= L = (F +φ¯ ), ∂F G H1 S1 ∂F¯ G H1 S1 H1 1 H1 1 ∂ (C) 4N ∂ (C) 4N 0 = L = (F¯ +φ ), 0= L = (F +φ¯ ). (54) ∂F G H2 S2 ∂F¯ G H2 S2 H2 2 H2 2 Hence, F , F¯ , F , and F¯ ( i = 1,2 ) can be eliminated from (C) in the component field form. After the ± ± Hi Hi L elimination of auxiliary fields through the Euler equations, one finds G G φ = 1 φ φ detC(φ¯ φ¯ +2φ¯ φ¯ )(φ¯ φ¯ +2φ¯ φ¯ ) , φ¯ = 1φ¯ φ¯ , H1 N + −− S1 − S2 + S1 + S2 − H1 N + − G (cid:16) (cid:17) G φ = 2 φ2 +φ2 detC (φ¯ φ¯ +2φ¯ φ¯ )2+(φ¯ φ¯ +2φ¯ φ¯ )2 , φ¯ = 2(φ¯2 +φ¯2). (55) H2 4N + −− S1 − S2 + S1 + S2 − H2 4N + − (cid:16) (cid:8) (cid:9)(cid:17) From these expressions, again we have found that the expressions of the constraints of SUSY composites given in (46) are satisfied in terms of the scalar fields. The Lagrange multiplier multiplets S and S¯ ( i=1,2 ) work well to i i 8 keep the constraints of SUSY composites, and guarantee the equivalence of the Lagrange functions (25) and (44). It should be remarked that, (φ )† = φ¯ ( i = 1,2 ): The Hermiticity was explicitly lost in the component field level Hi 6 Hi of the SUSY composites at detC =0. Now, we find the tree-level potential V of this model (44): tree 6 N V F¯ F +φ F +φ¯ F¯ +φ F +φ¯ F¯ tree ≡ −G H1 H1 S1 H1 S1 H1 H1 S1 H1 S1 1 (cid:16) (cid:17) 4N F¯ F +φ F +φ¯ F¯ +φ F +φ¯ F¯ . (56) −G H2 H2 S2 H2 S2 H2 H2 S2 H2 S2 2 (cid:16) (cid:17) Thus, from ∂V N ∂V N ∂V 4N ∂V 4N 0= tree = F , 0= tree = F¯ , 0= tree = F , 0= tree = F¯ , ∂φ −G S1 ∂φ¯ −G S1 ∂φ −G S2 ∂φ¯ −G S2 H1 1 H1 1 H2 2 H2 2 ∂V N ∂V N ∂V 4N ∂V 4N 0= tree = φ , 0= tree = φ¯ , 0= tree = φ , 0= tree = φ¯ , (57) ∂F −G H1 ∂F¯ −G H1 ∂F −G H2 ∂F¯ −G H2 S1 1 S1 1 S2 2 S2 2 one finds that the global minimum of V locates at F = F¯ = F = F¯ = 0. Through the Euler equations of tree S1 S1 S2 S2 F , F¯ , F and F¯ given in (54), we find H1 H1 H2 H2 N 4N N 4N V = (φ F +φ¯ F¯ φ¯ φ ) (φ F +φ¯ F¯ φ¯ φ )= φ¯ φ + φ¯ φ .(58) tree −G H1 S1 H1 S1 − S1 S1 − G H2 S2 H2 S2 − S2 S2 G S1 S1 G S2 S2 1 2 1 2 Here, we have used the conditions (57). Thus, from the Euler equations of φ , φ¯ , φ and φ¯ , one finds V has S1 S1 S2 S2 tree a global minimum at φ =φ¯ =φ =φ¯ =0. (59) S1 S1 S2 S2 Obviously,thevacuumenergyvanishesattheglobalminimum. ItshouldbenoticedthatV attheglobalminimum tree is Hermite in spite ofthe non-Hermitiancharacterof (C). Atthe classicaltree-level,the symmetries are notbroken spontaneously in our model. From the discussion givLen above, one recognizes that F and F¯ may have a special S1 S1 roleto keepthe symmetriesofthe theoryunbroken. We willgobeyondthe classicaltree-levelanalysis,examinewhat kind of modifications will happen under the nonperturbatibe quantum dynamics in the SNJL model, and whether thereisacrucialchangeintheeffectiveactionfortheinvestigationofthedynamicalmassgenerationintheBCS-NJL mechanism. C. The One-loop Effective Action In this subsection, we evaluate the effective action at one-loop level [33,34], derive and solve the gap equation for the examination whether the dynamical mass generation takes place or not in the = 1/2 SNJL model. There N are several methods for this purpose. For example, we can employ the method of SUSY auxiliary fields with the steepest descent technique for a self-consistent evaluation of composite condensates with superfield or component field formalisms of the Lagrangian. Another choice for us is the method of Schwinger-Dyson equation based on a diagrammatic consideration under keeping star products or after converting them into ordinary point-products. For introducing the method ofCornwall-Jackiw-Tombouliseffective actionof composites [25],we haveto extend it to the = 1/2 SUSY case, and it may have external bilocal sources in superspace as conjugates of composites with the N operations of star products, thus it will demand us to do a complicated work. We choose the method of effective actionofthe LagrangianwithSUSY auxiliaryfields insuperspace,andperformthe large-N expansion(akindofthe steepestdescenttechnique)inthissubsection. ItisclearfromtheLagrangian(44),thecontributioncomingfromthe loop-expansion ( except the tree level ) does not includes φ , φ¯ , F and F¯ . After Φ and Φ¯ were integrated Hi Hi Hi Hi ± ± out, the contribution of loop-expansion to the effective action gives quantum corrections expressed as a function of φ , φ¯ , F and F¯ . Si Si Si Si Hereafter, we consider the case G =0. The effective action Γ of the theory is given by 2 Γ iln [dΦ ][dΦ¯ ][dΦ ][dΦ¯ ][dH ][dH¯ ][dS ][dS¯ ]exp i d4y , (60) + + − − 1 1 1 1 T ≡ − L (C)Z+ mΦ ⋆Φ + m¯Φ¯ ⋆Φ¯ = (Ch)+Z mΦ Φi + m¯Φ¯ Φ¯ . (61) LT ≡ L + −|sym θθ + −|sym θ¯θ¯ L + − θθ + − θ¯θ¯ Here, (C) takes the form(cid:2)of Eq. (44). W(cid:3)e hav(cid:2)e introduced a(cid:3)holomorphic (cid:2)and an an(cid:3)ti-ho(cid:2)lomorphic(cid:3)mass terms ( L correspondtoaDiracmasstermforfermionsinthe ordinary =1SUSYcase)tomaketheeffectiveactioninfrared N 9 well-defined [11,35]. In general, both m and m¯ are matrices in the flavor space. When the mass terms take diagonal and flavor-independentforms in the flavorspace, they breakU(N) U(N) into the diagonalsubgroup U(N) . L R L+R Inthecaseofdeformedsuperspace,we cantakem† =m¯ ingeneral. T×heresultofcomponentfieldexpressionof (C) 6 L given in (48)-(53) should be kept in our mind, and calculation will be performed in the similar way to the ordinary = 1 case [32,33,34]. All of symmetrized star products are reducible to ordinary point-products. First, we convert N all of the symmetrized star products in our Lagrangian to ordinary point products, and then put the model to the formalism of effective action in superspace. When Φ are chiral superfields, Q2Φ are also chiral superfields. Thus, ± ± the Feynman rules of the deformed superspace are obtained exactly the same way with the ordinary case, if we treat Q2Φ as independent chiral superfields [11,12]. We can utilize this fact for the diagrammatic consideration for our ± calculation. The evaluation of the effective action should be performed under the superfield formalism because we considera nonperturbativeproblem, andwe haveto sumup a subset ofFeynman diagramsofthe theory. In the case we concern in this paper, it seems difficult ( or, hard ) to achieve this purpose under the component field formalism. Now, we take a steepest descent approximation in the path integrations of the auxiliary fields H , H¯ , S and S¯ 1 1 1 1 in (60), regard all of the SUSY auxiliary fields as classical constant c-numbers. We expand in terms of quantum T fluctuating fields Φ , Φ¯ around the origin Φ =Φ¯ =0: L ± ± ± ± = [Φ =Φ¯ =0]+ + , (62) T T ± ± (2) L L L ··· Φ + 1 Φ L(2) ≡ 2 d2θd2θ¯(Φ¯+,Φ¯−,Φ+,Φ−)M Φ¯−+ . (63) Z Φ¯  −    The rules for functional derivatives of the variation of chiral and antichiral superfields δΦ±(z′) = 1D2δ8(z z′), δΦ¯±(z′) = 1D2δ8(z z′) (64) δΦ (z) −4 − δΦ¯ (z) −4 − ± ± ( z,z′ =(y,θ,θ¯) ) are still satisfied also in our case. Therefore, →− ←− →− ←− δ→−Φ¯δ±LT[Φ±,Φ¯±]Φ←−δ±|Φ±=Φ¯±=0 δ→−Φ¯δ±LT[Φ±,Φ¯±]Φ¯←−δ±|Φ±=Φ¯±=0 M ≡  δΦδ±LT[Φ±,Φ¯±]Φδ±|Φ±=Φ¯±=0 δΦδ±LT[Φ±,Φ¯±]Φ¯δ±|Φ±=Φ¯±=0   D2D2 σ ( D2)(m¯ +S¯ ) σ = (−D42) m+S−1−1d6et⊗C(Q02S1)Q2 ⊗σ1 − 4−D126D2 ⊗1σ0⊗ 1 !δ8(z−z′), (65) where, we have used the equival(cid:2)ent relations δ2(θ¯) = D(cid:3)2/4(cid:3) and δ2(θ) = D2/4(cid:3) inside the integration of the − − action functional. The relations D2D2D2 = 16(cid:3)D2, D2D2D2 = 16(cid:3)D2 were also used. If we consider the case G =0 with G =0, the off-diagonalpart of will be replaced as D2[m¯ +S¯ ] σ /4 D2[m¯ σ +S¯ 1]/4 2 1 1 1 1 2 6 M − ⊗ →− ⊗ ⊗ and D2[m+S detC(Q2S )Q2] σ /4 D2[m σ +(S detC(Q2S )Q2) 1]/4. Exceptthesereplacements, 1 1 1 1 2 2 − − ⊗ →− ⊗ − ⊗ the formulae we will obtain hereafter are essentially the same for the G = 0 case. The component field expressions 2 6 of the SUSY auxiliary fields are given by S =φ +θ2F , S¯ =φ¯ +θ¯2F¯ . (66) 1 S1 S1 1 S1 S1 Itshouldbenotedthatθ2(θ¯2)actsasa(anti)chiralsuperfield. Afterputtingtheseexpressionsto(60)andperforming the Berezinian integration, one gets Γ = d4y [Φ =Φ¯ =0]+Γ˜, (67) T ± ± L Z i Γ˜ iln [dΦ ][dΦ¯ ][dΦ ][dΦ¯ ]exp i d4y + = lnDet + . (68) ≡ − + + − − L(2) ··· 2 M ··· Z h Z i Hence, the effective action at the one-loop level is found to be 1 1 1 i Γ = N d4y H¯ H + S H δ2(θ¯)+ S¯ H¯ δ2(θ) + lim N d8ztr(ln )δ8(z z′). (69) 1 1 1 1 1 1 Z "G1 G1 G1 #θθθ¯θ¯ z′→z 2 Z M − 10 At S = S¯ = 0, Γ˜ identically vanishes because it corresponds to a tadpole graph in the = 1 case. The effect of 1 1 deformationcontainedin the operator [detC(Q2S )]Q2 has to act in the inside of loop inteNgrals. The evaluationof Γ˜ 1 is just to extract the D-term contribution after some manipulations of Q , D and D . We use several identities in α α α˙ the integrals of the loop expansion: 116D2D2δ2(θ−θ′)δ2(θ¯−θ¯′)|θ=θ′,θ¯=θ¯′ = 1, (70) 116Q2D2δ2(θ−θ′)δ2(θ¯−θ¯′)|θ=θ′,θ¯=θ¯′ = 1, (71) Q2116D2D2δ2(θ−θ′)δ2(θ¯−θ¯′)|θ=θ′,θ¯=θ¯′ = −4θ¯θ¯(cid:3). (72) will be divided into the following form: M Σ, 0 M ≡ M − D2D2 0 1 D2D2 0 16 , −1 = 16 , M0 ≡ 0 D2D2 ! M0 (cid:3)2 0 D2D2 ! 16 16 0 ( D2)(m¯ +S¯ ) σ Σ ≡ −(−D42) m+S1−detC(Q2S1)Q2 ⊗σ1 − − 4 0 1 ⊗ 1 !. (73) (cid:2) (cid:3) Thus, i i i i Γ˜ = Tr ln Σ = Trln + Trln 1 −1Σ = Trln 1 −1Σ . (74) 2 M0− 2 M0 2 −M0 2 −M0 h (cid:0) (cid:1)i (cid:16) (cid:17) (cid:16) (cid:17) We have dropped iTrln because it does not contribute to the integral of Γ˜. Because the inverse matrix of chiral 2 M0 projectorsobeystherelations −1 −1 = 1 −1,[ −1,Σ]=0,andalloftheoperatorsin −1 andΣarebosonic, M0 M0 (cid:3)M0 M0 M0 the expression given above will be converted as 1 1 Σ Trln(1 −1Σ)= Tr −1Σ+ −1Σ −1Σ+ −1Σ −1Σ −1Σ+ =Tr ln 1 (cid:3) −1 .(75) −M0 − M0 2M0 M0 3M0 M0 M0 ··· " − (cid:3) M0 # n o (cid:16) (cid:2) (cid:3)(cid:17) Therefore, one gets Γ˜ in the following form: i 1 D2D2 Γ˜ = lim Ntr d4yd2θd2θ¯ln 1 (m¯ +S¯ ) m+S detC(Q2S )Q2 δ8(z z′) z′→z 2 − (cid:3) 1 1− 1 16(cid:3) − i Z dh4p 1 1(cid:8) (cid:9)i D2D2 = Ntr d4yd2θd2θ¯ ln 1+ (m¯ +S¯ ) m+S detC(Q2S )Q2 δ2(θ θ′)δ2(θ¯ θ¯′). −2 (2π)4p2 p2 1 1− 1 16 − − Z Z (cid:16) (cid:8) (cid:9)(cid:17) (76) The effective potential is defined by Γ Γ˜ V , V˜ . (77) ≡− d4y ≡− d4y It is a quite difficult ( hard ) problem to evaluRate Γ˜ or V˜ exactlyR. Let us first consider the small deformation limit detC 0, and expand V˜ as a power series of detC: | |→ i d4p 1 V˜ = Ntr d2θd2θ¯ ln p2+(m¯ +S¯ )(m+S ) 2 (2π)4p2" { 1 1 } Z Z ∞ (detC)n (m¯ +S¯ )(Q2S ) n D2D2 1 1 Q2 δ2(θ θ′)δ2(θ¯ θ¯′) −nX=1 n (cid:16)p2+(m¯ +S¯1)(m+S1) (cid:17) # 16 − − = V˜(detC =0)+(detC)V˜(1)+(detC)2V˜(2)+ . (78) ···