CALCULATION OF THE FOUR-QUARK CONDENSATES IN THE QCD MOTIVATED EXTENDED NAMBU-JONA-LASINIO MODEL 1
The four-quark condensates appearing in QCD sum rule are calculated nonperturbatively in a realistic QCD motivated extended Nambu-Jona-Lasinio model. The calculation is in the framework of the effective potential for local composite operators up to next-to-the-leading order in the 1/N expansion in which the non-factorized parts of the condensates are included. We show in this paper the possibility of explaining the phenomenological enhancement factor κ = 2.4 , needed for fitting the data in the factorization approximation, as the contribution of the non-factorized parts of the fourquark condensates. PACS numbers: 11.30.Qe, 11.15.Pg, 12.40.Aa 1 Work supported by the National Natural Science Foundation of China and the Fundamental Research Foundation of Tsinghua University Mailing address 1 One of the interesting but difficult problems in nonperturbative QCD is the calculation of the quark and gluon condensates. The two-quark condensate breaks the chiral symmetry which explains the lightness of the low lying 0 mesons, while all condensates are relevant to the well-known QCD sum rule . QCD sum rule incorporates the quark and gluon condensates with perturbative QCD calculations. For example, in the study of the sum rule for mesons, the vacuum expectation value (VEV) of two current operators < Tj(x)j(0) > is considered, and the operator product expansion (OPE) gives i ∫ dx e < Tj(x)j(0) >= ∑ n C n (q) < On > , (1) where C n (q) ’s are perturbatively calculable Wilson coefficients and < On > ’s are various nonperturbative vacuum condensates. In practical applications, on the RHS of (1), only C n (q) ’s are calculated from perturbative QCD while the condensates < On > ’s are left as free parameters determined by experimental inputs. To reduce the number of free parameters, people often make the simple factorization approximation to express the four-quark condensate in terms of the two-quark condensate, i.e. < ψ̄Γψψ̄Γψ >≈ FΓ < ψ̄ψ > , (2) where the Lorentz structure of Γ is 1 for scalar (S), iγ5 for pseudoscalar (P), γμ for vector (V), γ5γμ for axial-vector (A), σμν for tensor (T) (for color and flavor non-singlet, the color and flavor group generators λα/2 , ti should also be included ), and FΓ is a constant depending on the structure of Γ and can be easily calculated . However, applications of QCD sum rule to various hadronic processes with the approximation (2), e.g. the sum rules for baryons  ρ-meson  and pseudoscalar meson , show that the theoretical results are not quite good and phenomenologically the FΓ < ψ̄ψ > 2 term should be enlarged by a factor κ ≈ 3.6 for fitting the data [2][3][4]. Further study of the perturbative QCD corrections to the Wilson coefficient of the operator (ψ̄Γψ)2 shows that the O(αs) correction increases the Wilson coefficient by a factor of 1.5  which means that the enhancement factor should actually be κ ≈ 2.4 . So far there is no theoretical explanation of the enhancement factor κ ≈ 2.4 . A possible and natural conjecture is that the need for an enhancement factor might be due to the neglect of the nonfactorized part of the four-quark condensate < ψ̄Γψψ̄Γψ >NF in (2), but this is difficult to prove from the first principles of QCD. Several attempts have been made to discuss this problem. A careful calculation of the anomalous dimensions of the four-quark operators shows that the deviation from (2) 2 varies significantly with the renormalization scale, i.e. even if (2) is good at low energy it may fail at some high energy scale . In Ref.[7], a general formalism of the effective potential for local composite operators in the framework of 1/N expansion was developed and was applied to calculate the fourfermion condensate < ψ̄ψψ̄ψ > nonperturbatively in the Gross-Neveu model. It is shown that (2) holds only in the N → ∞ limit, i.e. < ψ̄ψψ̄ψ >NF is nonvanishing to order-1/N . Up to order1/N , for N = 3 , the numerical result shows that the ratio R ≡< ψ̄ψψ̄ψ >NF /[FS < ψ̄ψ >2] runs rapidly with the renormalization scale, and the minimal value of R is about 1/3 . This means that (2) is not a good approximation even at low energy. However, the Gross-Neveu model is a toy model which can not predict the value of κ to compare with the experiment. In recent years, there have been several papers studying QCD motivated Nambu-Jona-Lasinio (NJL) models as effective theories for low lying hadrons[8]. These models lead to quite successful phenomenological predictions, especially the extended Nambu-Jona-Lasinio (ENJL) model in Ref. , so that they may reflect some main features of nonperturbative QCD. Since the NJL model is a four-fermion interaction theory similar to the GrossNeveu model, the method developed in Ref.[7] can be directly applied to it. In this paper, we take the EJNL model of Ref.[8] and calculate the four-quark condensates appearing in the QCD sum rule nonperturbatively by using the method of Ref.[7]. The four-quark condensates are generally related to the sum rule in the form ∑ Γ C 6 < ψ̄Γψψ̄Γψ > (cf. (1)). Once the condensates < ψ̄ψ > and < ψ̄Γψψ̄Γψ > are calculated, the enhancement factor κ can be obtained from

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