Commun. Theor. Phys. (Beijing, China) 40 (003) pp. 675 680 c International Academic Publishers Vol. 40, No. 6, December 5, 003 Vector Susceptibility of QCD Vacuum from an Effective Quark-Quark Interaction ZONG Hong-Shi,,, QI Shi, CHEN Wei, and WU Xiao-Hua 3 Department of Physics, Nanjing University, Nanjing 0093, China CCAST (World Laboratory), P.O. Box 8730, Beijing 00080, China 3 Department of Physics, Sichuan University, Chengdu 60064, China (Received May 6, 003) Abstract A new approach for calculating vacuum susceptibilities from an effective quark-quark interaction model is derived. As a special case, the vector vacuum susceptibility is calculated. A comparison with the results of the previous approaches is given. PACS numbers:.5.tk,.38.aw,.38.lg,.39.-x Key words: QCD vacuum, vector vacuum susceptibility, global color symmetry model The vacuum condensate is known to play an essential role for characterizing the non-perturbative aspect of the QCD vacuum. In fact, the gauge-invariant vacuum condensates such as the chiral quark condensate, the two-gluon condensate, and the mixed quark-gluon condensate are well-known examples studied extensively so far. Meanwhile susceptibilities of vacuum are also important quantities of strong interaction physics. They directly enter the determination of hadron properties in the QCD sum rule external field approach. 4 It is the aim of this paper to consider vacuum susceptibilities in general and in S GCM q, q; V = particular the vector vacuum susceptibility 5 in the framework of the global color symmetry model (GCM). Though GCM 6 violates local color SU(3) C gauge invariance and renormalizability, they provide a successful description of various nonperturbative aspects of strong interaction physics and hadronic phenomena at low energies. 7 0 Now let us consider the Euclidean action of the GCM in an external field (without loss of generality, we will focus in the variable external vector field V µ (x) in this paper), d 4 x{ q(x)γ x iγ ν V ν (x)q(x)} + g d 4 xd 4 y s ja µ(x)dµν(x ab y)jν(y) b, () where jµ(x) a denotes the color octet vector current jµ(x) a = q(x)γ µ λ a Cq(x)/, and Dµν(x y) ab denotes the gluon two-point Green function. For convenience we will use the Feynman-like gauge Dµν(x ab y) = δ µν δ ab D(x y) from now on. Introducing an auxiliary bilocal field B θ (x, y) and applying the standard bosonization procedure, 6 the generating functional of GCM ZV = D qdq exp ( S GCM q, q; V) () can be rewritten in terms of the bilocal fields B θ (x, y) ZV = DB θ exp ( S eff B θ ; V) (3) with the effective bosonic action S eff B θ, V = Tr ln G B θ ; V + and the quark operator d 4 xd 4 y Bθ (x, y)b θ (y, x) g sd(x y), (4) G B θ ; V = γ x iγ ν V ν (x)δ(x y) + Λ θ B θ (x, y). (5) The matrices Λ θ = D a F b C c are determined by Fierz transformation in Dirac, flavor, and color spaces of the current-current interaction in Eq. (5), and are given by Λ θ = { i i } { } { 4 D, iγ 5, γ ν, γ ν γ 5 F, λ a F 3 3 i } C, λ a C. (6) 3 The project supported in part by National Natural Science Foundation of China under Grant Nos. 075033 and 035030 E-mail: zonghs@chenwang.nju.edu.cn.
676 ZONG Hong-Shi, QI Shi, CHEN Wei, and WU Xiao-Hua Vol. 40 We will calculate the vector vacuum susceptibility from the saddle-point expansion, that is, we will work at the mean-field level. This is consistent with the large N c limit in the quark fields for a given model gluon two-point function. In the mean-field approximation, the fields B θ (x, y) are substituted simply by their vacuum value B0(x, θ y), which is defined as δs eff /δ B B0 = 0 and is given by B θ 0V(x, y) = g sd(x y) tr DC Λ θ G 0 V(x, y), (7) where tr DC denotes the trace taken in Dirac and color space and G0 V(x, y) denotes the inverse propagator with the self-energy ΣV(x, y) = Λ θ B0V(x, θ y) in the external field V µ (x). It should be noted that both B0V(x, θ y) and G0 V(x, y) have an implicit dependence on the external field V µ(x). If the external field V µ (x) is switched off, G 0 V goes into the dressed quark propagator G G 0 V µ = 0, which has the decomposition G (p) = iγ p + Σ(p) = iγ pa(p ) + B(p ) (8) with Σ(p) = d 4 x e ip x Λ θ B θ 0(x) = 4 3 (π) 4 g sd(p q)γ ν G(q)γ ν, (9) where the self-energy function A(p ) and B(p ) are determined by the rainbow Dyson Schwinger equation (DSE), A(p ) p = 8 A(q )p q 3 (π) 4 g sd(p q) q A (q ) + B (q ), (0) B(p ) = 6 B(q ) 3 (π) 4 g sd(p q) q A (q ) + B (q ). () This dressing comprises the notion of constituent quarks by providing a dynamical mass M(p ) = B(p )/A(p ), reflecting a vacuum configuration with dynamically broken chiral symmetry. In order to get the numerical solution of A(p ) and B(p ), one often uses model forms for gluon two-point function as input in Eqs. (0) and (). As a typical example, we choose g D(q ) = 3π (λ / ) e q / with three sets of different parameters for λ and, which are adjusted to reproduce the pion decay constant in the chiral limit f π = 87 MeV and the chiral low energy coefficients., Here we want to stress that the B(p ) in Eqs. (0) and () has two qualitatively distinct solutions. The Nambu Goldstone solution, for which B(p ) 0 () describes a phase in which the chiral symmetry is dynamically broken because one has a nonzero quark mass function and the dressed quarks are confined, because the propagator described by these functions does not have a Lehmann representation. 9 In Nambu Goldstone phase, the vacuum configuration B0(x θ y) in GCM (in the mean-field approximation) can be regarded as a good approximation to the exact vacuum in QCD. The alternative Wigner solution, for which B(p ) 0 (3) describes a phase in which the chiral symmetry is not broken and the dressed-quarks are not confined. In Wigner phase, the vacuum configuration B0(x θ y) in GCM (in the mean-field approximation) models can be considered as the perturbative vacuum in QCD. In Wigner phase (the perturbative quark scalar self-energy function B (p ) 0), the Dyson Schwinger equations (0) and () reduce to A (p ) p = 8 p q 3 (π) 4 g sd(p q) q A (q ), (4) where A (p ) denotes the perturbative quark vector self-energy function. Therefore, the perturbative quark propagator in GCM can be written as G P (q ) = iγ q/a (q )q = iγ qc(q ). Numerical studies show that for q Λ QCD, one has G(q ) G P (q ) = iγ qa(q ) + B(q ) A (q )q + B (q + iγ qc(q ) = 0. (5) )
No. 6 Vector Susceptibility of QCD Vacuum from an Effective Quark-Quark Interaction 677 This is just what one expected in advance. The above results can be seen from Figs. and. Fig. Comparison between σ A(s) and C(s) for gluon propagator g D(q ) = 3πχ e q / /. Fig. σ B(s) for gluon propagator g D(q ) = 3π χ e q / /. Let us now study linear response of dressed quark propagator in the presence of the external vector field. It is obvious that G0 V can be expanded in powers of V as which leads to the formal expansion with 0 V G0 V = G 0 V V µ=0 + δg δv Vµ=0 V + = G + V µ Γ µ +, (6) G 0 V = G GV µ Γ µ G + (7) δg 0 Γ µ (y, y ; z) = V(y, y ) δv µ (z). (8) V µ=0 In coordinate space the dressed vector vertex Γ µ (x, y; z) is given as the functional derivative of the inverse quark propagator G0 V with respect to the variable external field V µ(x). Taking the functional derivative in Eq. (5) and put it into Eq. (8), we have δσv(y, y ) Γ µ (y, y ; z) = iγ µ δ(y y )δ(y z) + δv µ (z). (9) V µ=0 The second term on the right-hand side of Eq. (9) can be determined by employing the stationary condition Eq. (7), which, after reversing Fierz recording, can be cast into ΣV(y, y ) = 4 3 g sd(y, y )γ ν G 0 V(y, y )γ ν. (0) Substituting Eqs. (7) and (0) into Eq. (9), we have the inhomogeneous ladder Bethe Salpeter (BS) equation of vector vertex, which reads Γ µ (y, y ; z) = iγ µ δ(y y )δ(y z) 4 3 g sd(y y ) du du γ ν G(y, u )Γ µ (u, u ; z)g(u, y )γ ν. ()
678 ZONG Hong-Shi, QI Shi, CHEN Wei, and WU Xiao-Hua Vol. 40 Fourier transformation of Eq. () leads then to the momentum space form of the inhomogeneous BS equation,3 Γ µ (P, q) = iγ µ 4 d 4 K ( 3 (π) 4 g sd(p K)γ ν G K + q ) ( Γ µ (K, q)g K q ) γ ν. () As was shown above, both the rainbow DS equations (0) and () and the ladder BS equation () can be consistently derived from the action of the GCM in a variable external vector field V µ (x). Here, it should be noted that it is important to take into account the effect of external field V to the vacuum configuration B0V θ in the process of obtaining the dressed vector vertex. If one neglect the implicit dependence of vacuum configuration B0V θ on the external field V, one only gets an inadequate bare vertex γ µ from Eq. (9) (see Eq. (37)). So far, we have derived the dressed vector vertex Γ µ (P, q) for a variable external vector field V µ (x) at the mean field level in the framework of GCM. This means that we have known how the quark propagates in a variable external vector field V µ (x) at the mean-field level. Therefore, in principle, once the A(p ) and B(p ) are determined by means of Eqs. (0) and (), we can use GCM generating functional (in Nambu Goldstone or Wigner phase) in the presence of variable external field to calculate the vacuum expectation of T-product of any quark operator. However, according to Eq. (), it is not easy to obtain the solution of the dressed vector vertex Γ µ (P, q) for a variable external vector field V µ (x).,3 Fortunately, in traditional external field QCD sum rule approach, one only needs to know the two-point correlation function in the presence of a constant external field (V µ (x) = V µ ). In the case of constant external field, there is zero momentum transfer (q = 0) for the dressed vector vertex Γ µ (P, q). Therefore, we can use Ward identity to calculate the dressed vector vertex Γ µ (P, q = 0) (see below). At the level of the dressed vector vertex Γ µ (P, q), we have the following Ward Takahashi identity (WTI), q µ Γ µ (P, q) = G ( P q ) G ( P + q ), (3) which can be verified by substituting it into Eq. (). Expanding the right hand of Eq. (3) in q µ and taking the limit q µ 0 lead to the Ward identity, Using Eq. (8) and substituting it into Eq. (4), one obtains Γ µ (P, 0) = G (P ) P µ. (4) Γ µ (p, 0) = iγ µ A(p ) iγ p A(p ) p p µ B(p ) p p µ. (5) In the case of constant external vector field V µ, by means of Eqs. (7) and (5), we have G 0 V (x) = G(x) d 4 u d 4 u G(x, u )Γ µ (u, u )G(u, 0) V µ + = G(x) Expanding G 0 V (x) to the first order in the external field, we have d 4 u d 4 d 4 p u G(x, u ) (π) 4 e ip (u u) Γ µ (p, 0) G(u, 0) V µ + (6) G 0 V (x) = G(x) + S V (x) V + 0 T q(x) q(0) 0 V =0 + 0 T q(x) q(0) 0 V V + (7) with G(x) 0 T q(x) q(0) 0 V =0 and S V (x) 0 T q(x) q(0) 0 V S V,P (x) + S V,NP (x) 0 T q(x) q(0) 0 P V + 0 : T q(x) q(0) : 0 NP V, (8) where G(x) denotes quark propagator in the absence of external field and S V (x) is the linear response term of the quark propagator in the presence of external field. 0 denotes the exact QCD vacuum (Nambu Goldstone phase) and 0 is perturbative QCD vacuum (Wigner phase), S V,P (x) 0 T q(x) q(0) 0 P V and SV,NP (x) 0 : T q(x) q(0) : 0 NP V are the quark propagator coupled perturbatively and nonperturbatively to the external current, respectively. Here, : : is normal product, which means that we subtract the contribution of the perturbative term S V,P (x) from S V (x). We will show that it is the term S V,NP (x) that is tightly related to the vacuum susceptibility in QCD two-point external field formula. Comparing Eq. (6) with Eq. (7), we can obtain the linear response term of the exact quark correlation function (Nambu Goldstone phase) in the presence of constant external vector field S V (x) = d 4 u d 4 d 4 p u G(x, u ) (π) 4 e ip (u u) Γ µ (p, 0) G(u, 0). (9)
No. 6 Vector Susceptibility of QCD Vacuum from an Effective Quark-Quark Interaction 679 with Similarly, we have the perturbative quark propagator (Wigner phase) in the presence of the external current V, S V,P (x) = d 4 u d 4 u G P d 4 p (x, u ) (π) 4 e ip (u u) Γ µ(p, 0) G P (u, 0) (30) Γ µ(p, 0) = iγ µ A (p ) iγ p A (p ) p p µ. (3) Now we turn to the calculation of vector vacuum susceptibilities. The vector vacuum susceptibility χ V in the QCD sum rule two point external field treatment can be defined as 0 : q a α(0) q b β(0) : 0 NP V V ( i) (γ V ) αβδ ab χ V, (3) where S abv,np αβ (x) 0 : qα(0) q a β b NP (0) : 0 V denotes the local quark condensate (for more details about the calculation of quark vacuum condensate, one can see Ref. 4). (ab) and (αβ) are color and spinor indexes of quark respectively. Multiplying Eq. (3) by δ ab γν βα and recalling Eq. (8), we have 0 : q(0)γ ν q(0) : 0 NP V V = ( i)χ V V ν = 0 T q(0)γ ν q(0) 0 V 0 T q(0)γ ν q(0) 0 P V V. (33) Based on GCM external field approach, it is not difficult to get the following expression by means of GCM generating functional in the presence of the constant external field, 0 T q(0)γ ν q(0) 0 V V = tr DC γ ν GΓ V G (Nambu Goldstone phase), (34) 0 T q(0)γ ν q(0) 0 P V V = tr DC γ ν G P Γ V G P (Wigner phase). (35) Putting Eqs. (34) and (35) into Eq. (33), the vector vacuum susceptibility χ V from QCD sum rule two-point external formula can be written as χ V = itr DCγ ν GΓ ν G tr DC γ ν G P Γ νg P = 6 { (π) 4 A (q )q + B (q ) q A 3 (q ) A (q ) A(q ) q q 4 q A(q ) B(q ) q B(q ) + A(q )B (q ) + q A(q ) A (q )q q A 3 (q ) A (q ) A (q ) q B (q ) (q ) q4}, (36) which is UV convergent (recall Eq. (5)). However, if we follow the previous approach proposed by Refs. 5 7, we have χ V = itr DCγ ν Gγ ν G tr DC γ ν G P γ ν G P = 6 { (π) 4 A (q )q + B (q ) q A 3 (q ) + A(q )B (q ) A (q )q q A 3 (q ) }. (37) Comparing Eq. (36) with Eq. (37), it is easy to see that the main difference between χ V and χ V comes from the fact that in the previous calculation, instead of using the complete vertex Γ ν, one uses the bare vertex γ ν to calculate the vector vacuum susceptibility, which neglects the implicit dependence of the vacuum configuration B0V θ on the external field V. Multiplying Eqs. (36) and (37) by a = (π), we obtain and χ V a = 3 χ V a = 3 { sds 0 A (s)s + B sa 3 (s) A (s) A(s) (s) + A(s)B (s) + s A(s) B (s) A (s)s 0 { sds A (s)s + B (s) s sa(s) B(s) B(s) sa 3 (s) A (s) A (s) sa 3 (s) + A(s)B (s) s } (38) sa (s)} 3 A. (39) (s)s
680 ZONG Hong-Shi, QI Shi, CHEN Wei, and WU Xiao-Hua Vol. 40 In order to get a quantitative analysis of the difference between χ V a and χ V a, we will calculate the vector vacuum susceptibility from Eqs. (38) and (39) respectively. In Table, we display the result for the vector vacuum susceptibility χ V a and χ V a for the model gluon two-point function. Table The vector vacuum susceptibility χ V a and χ V a for g D(q ) = 3π (λ / ) e q /. GeV λ GeV χ V a (GeV ) χ V a (GeV ) 0.00.3. 0 4 0.06 Table shows the result for the vacuum susceptibility χ V a is much smaller than χ V a obtained within the previous approach. This shows that it is important to take into account the effect of external field V to the vacuum configuration B θ 0V in the calculation of vacuum susceptibility. To summarize, in the framework of GCM, a new approach which is consistent with the definition of vacuum susceptibilities in QCD sum rule two-point external field method, is adopted to calculate vacuum susceptibility. Within this approach the vector vacuum susceptibility is free of UV divergence. This is reasonable and what is to be expected. The results of vector susceptibility we obtained is much smaller than the previous estimation, which can be used to provide a reference for further study. Finally, we want to stress that GCM is not renormalizable. Therefore, the scale at which a vacuum susceptibility is defined in our approach is a typical hadronic scale, which is implicitly determined by the model gluon propagator g D(q ) and the solution of the rainbow DS equations (9) and (0). This situation is very similar to the determination of vacuum condensates in the instanton liquid model where the scale is set by the inverse instanton size. 8,9 References M. Shifman, A. Vainshtein, and V. Zakharov, Nucl. Phys. B47 (979) 385; L. Reinders, H. Rubinstein, and S. Yazaki, Phys. Rep. 7 (985) ; S. Narison, QCD Spectral Sum Rules, World Scientific, Singapore (989) and the references therein. B.L. Ioffe and A.V. Smilga, Nucl. Phys. B3 (984) 09. 3 I.I. Balitsky and A.V. Yung, Phys. Lett. B9 (983) 38. 4 S.V. Mikhailov and A.V. Radyushkin, JETP Lett. 43 (986) 7; Phys. Rev. D45 (99) 754; A.P. Bakulev and A.V. Radyushkin, Phys. Lett. B7 (99) 3. 5 Mannque Rho, hep-ph/030336. 6 R.T. Cahill and C.D. Roberts, Phys. Rev. D3 (985) 49; P.C. Tandy, Prog. Part. Nucl. Phys. 39 (997) 7; R.T. Cahill and S.M. Gunner, Fiz. B7 (998) 7 and the references therein. 7 C.D. Roberts and A.G. Williams, Prog. Part. Nucl. Phys. 33 (994) 477, and the references therein. 8 M.R. Frank and T. Meissner, Phys. Rev C57 (998) 345. 9 C.D. Roberts, R.T. Cahill, and J. Praschiflca, Ann. Phys. (N.Y.) 88 (988) 0. 0 P. Maris, C.D. Roberts, and P.C. Tandy, Phys. Lett B40 (998) 67. C.D. Roberts, R.T. Cahill, M.E. Sevior, and N. Ianella, Phys. Rev. D49 (994) 5. M.R. Frank and T. Meissner, Phys. Rev. C53 (996) 40. 3 R.T. Cahill and S. Gunner, Phys. Lett. B359 (995) 8; Mod. Phys. Lett. A0 (995) 305. 4 P. Maris and C.D. Roberts, Phys. Rev. C56 (997) 3369. 5 Lü Xiao-Fu, Liu Yu-Xin, Zong Hong-Shi, and Zhao En- Guang, Phys. Rev. C58 (998) 95; Zong Hong-Shi, Lü Xiao-Fu, Gu Jian-Zhong, Chang Chao-Hsi, and Zhao En-Guang, Phys. Rev. C60 (999) 05508; Zong Hong- Shi, Lü Xiao-Fu, Wang Fan, Chang Chao-Hsi, and Zhao En-Guang, Commun. Theor. Phys. (Beijing, China) 34 (000) 563; Zong Hong-Shi, Liu Yu-Xin, Lü Xiao-Fu, Wang Fan, and Zhao En-Guang, Commun. Theor. Phys. (Beijing, China) 36 (00) 87; Zong Hong-Shi, Chen Xiang-Song, Wang Fan, Chang Chao-Hsi, and Zhao En- Guang, Phys. Rev. C66 (00) 050. 6 C. Burden, C.D. Roberts, and M. Thomson, Phys. Lett. B37 (996) 63. 7 C. Burden and D. Liu, Phys. Rev. D55 (997) 367; M.A. Ivanov, Yu. L. Kalinovskii, P. Maris, and C.D. Roberts, Phys. Lett. B46 (998) 9; Phys. Rev. C57 (998) 99. 8 A. Bender, D. Blaschke, Y. Kalinovskii, and C.D. Roberts, Phys. Rev. Lett. 77 (996) 374. 9 M.R. Frank, P.C. Tandy, and G. Fai, Phys. Rev. C43 (99) 808; M.R. Frank and P.C. Tandy, Phys. Rev. C46 (99) 338; C.W. Johnson, G. Fai, and M.R. Frank, Phys. Lett. 386 (996) 75. 0 R.T. Cahill, Nucl. Phys. A543 (99) 63. T. Meissner, Phys. Lett. B405 (997) 8. M.R. Frank, Phys. Rev. C5 (995) 987; M.R. Frank and P.C. Tandy, Phys. Rev. C49 (994) 478. 3 T. Meissner and L.S. Kisslinger, Phys. Rev. C59 (999) 986. 4 Zong Hong-Shi, Ping Jia-Lun, Yang Hong-Ting, Lü Xiao- Fu, and Wang Fan, Phys. Rev. D67 (003) 074004. 5 M.B. Johnson and L.S. Kisslinger, Phys. Rev. D57 (998) 847. 6 Yang Hong-Ting, Zong Hong-Shi, Ping Jia-Lun, and Wang Fan, Phys. Lett. B557 (003) 33. 7 L.S. Kisslinger, Phys. Rev. C59 (999) 3377. 8 M.V. Polyakov and C. Weiss, Phys. Lett. B387 (996) 84. 9 A.E. Dorokhov, S.V. Esaibegian, and S.V. Mikhailov, Phys. Rev. D56 (997) 406.