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arxiv:0704.0829v2 [hep-th] 23 jun 2007quark-antiquark and diquark condensates in vacuum in a 3d two-flavorgross-neveu model ∗zhou bang-rongcollege of physical sciences, graduate school of the chinese academy of sciences, beijing 100049, china andccast (world laboratory), p.o.box 8730, beijing 100080, china(dated:)the ef f ective potential analysis indicates that, in a 3d two-f l avor gross-neveu model in vacuum,depending on less or bigger than the critical value 2/3 of g s /h p , where g s and h p are respectivelythe coupling constants of scalar quark-antiquark channel and pseudoscalar diquark channel, thesystem will have the ground state with pure diquark condensates or with pure quark-antiquarkcondensates, but no the one with coexistence of the two forms of condensates. the similarities anddif f erences in the interplay between the quark-antiquark and the diquark condensates in vacuum inthe 2d, 3d and 4d two-f l avor four-fermion interaction models are summarized.pacs numbers: 12.38aw; 12.38.lg; 12.10.dm; 11.15.pgkeywords: 3d gross-neveu model, quark-antiquark and diquark condensates, ef f ective potentiali. introductionit has been shown by ef f ective potential approach thatin a two-f l avor 4d nambu-jona-lasinio (njl) model [1],even when temperature t = 0 and quark chemical poten-tial µ = 0, i.e. in vacuum, there could exist mutual com-petition between the quark-antiquark condensates andthe diquark condensates [2]. similar situation has alsoemerged from a 2d two-f l avor gross-neveu (gn) model[3] except some dif f erence in the details of the results [4].an interesting question is that if such mutual competi-tion between the two forms of condensates is a generalcharacteristic of this kind of two-f l avor four-fermion in-teraction models? for answer to this question, on thebasis of research on the 4d njl model and the 2d gnmodel, we will continue to examine a 3d two-f l avor gnmodel in similar way. the results will certainly deepenour understanding of the feature of the four-fermion in-teraction models.we will use the ef f ective potential in the mean f i eldapproximation which is equivalent to the leading orderof 1/n expansion. it is indicated that a 3d gn model isrenormalizable in 1/n expansion [5].ii. model and its symmetriesthe lagrangian of the model will be expressed byl = ¯ qiγ µ ∂ µ q g s [(¯ qq) 2 (¯ q~ τq) 2 ] h pxa=2,5,7(¯ qτ 2 λ a q c )(¯ q c τ 2 λ a q). (1)all the denotations used in eq.(1) are the same as theones in the 2d gn model given in ref.[4], except that∗ the project supported by the national natural science founda-tion of china under grant no.10475113.the dimension of space-time is changed from 2 to 3 andthe coupling constant h s of scalar diquark interactionchannel is replaced by the coupling constant h p of pseu-doscalar diquark interaction channel. now the matricesγ µ (µ = 0,1,2) and the charge conjugate matrix c aretaken to be 2 × 2 ones and have the explicit formsγ 0 =?1 00 −1?, γ 1 =?0 ii 0?, γ 2 =?0 1−1 0?= c.(2)it is emphasized that, in 3d case, no ”γ 5 ” matrix canbe def i ned, hence the third term in the right-handed sideof eq.(1) will be the only possible color-anti-triplet di-quark interaction channel which could lead to lorentz-invariant diquark condensates, where we note that thematrix cτ 2 λ a is antisymmetric. without ”γ 5 ”, the la-grangian (1) will have no chiral symmetry. except this, itis not dif f i cult to verify that the symmetries of l include:1. continuous f l avor and color symmetries su f (2) ⊗su c (3) ⊗ u f (1);2. discrete symmetry r: q → −q;3. parity p: q(t,~ x) → γ 0 q(t,−~ x) and q c (t,~ x) →−γ 0 q c (t,−~ x);4. time reversal t : q(t,~ x) → γ 2 q(−t,~ x) andq c (t,~ x) → −γ 2 q c (−t,~ x);5. charge conjugate c: q ↔ q c ;6. special parity p 1 : q(t,x 1 ,x 2 ) → γ 1 q(t,−x 1 ,x 2 ) andq c (t,x 1 ,x 2 ) → −γ 1 q(t,−x 1 ,x 2 );7. special parity p 2 : q(t,x 1 ,x 2 ) → γ 2 q(t,x 1 ,−x 2 ) andq c (t,x 1 ,x 2 ) → −γ 2 q c (t,x 1 ,−x 2 ).if the quark-antiquark condensates h¯ qqi could be formed,then the time reversal t , the special parities p 1 and p 2will be spontaneously broken [6]. if the diquark conden-sates h¯ q c τ 2 λ 2 qi could be formed, then the color symme-try su c (3) will be spontaneously broken down to su c (2)