Commun. Theor. Phys. (Beijing, China) 45 (2006) pp. 675 680 c International Academic Publishers Vol. 45, No. 4, April 15, 2006 Applicability o Parametrized Form o Fully Dressed Quark Propagator ZHOU Li-Juan 1,2 and MA Wei-Xing 1,3 1 Collaboration Group o Hadron Physics and Non-perturbative QCD Study, Guangxi University o Technology, Liuzhou 545006, China 2 College o Physical Science and Engineering Technology, Guangxi University, Nanning 530004, China 3 Institute o High Energy Physics, the Chinese Academy o Sciences, Beijing 100049, China (Received August 15, 2005) Abstract According to extensive study o the Dyson Schwinger equations or a ully dressed quark propagator in the rainbow approximation with an eective gluon propagator, a parametrized ully dressed conining quark propagator is suggested in this paper. The parametrized quark propagator describes a conined quark propagation in hadron, and is analytic everywhere in complex p 2 -plane and has no Lehmann representation. The vector and scalar sel-energy unctions [1 A ( p 2 )] and [B ( p 2 ) m ], dynamically running eective mass o quark M ( p 2 ) and the structure o non-local quark vacuum condensates as well as local quark vacuum condensates are predicted by use o the parametrized quark propagator. The results are compatible with other theoretical calculations. PACS numbers: 14.65.Bt, 24.85.+p, 12.38.Lg Key words: quark propagator, Dyson Schwinger equations, non-perturbative QCD 1 Introduction The study o ully dressed quark propagator is one o the most important subjects in the investigation o QCD, since it is related to the determination o quark masses. Quark masses are undamental QCD input parameters o Standard Model, and accurate determination o these parameters is extremely important or both phenomenological and theoretical applications. In particular, the values o quark masses cannot be directly measured in experiments because quarks are conined inside hadrons, so that no ree quark survives in the ield o our vision, but can only be obtained indirectly. Thereore, the study o QCD ree parameters is o paramount importance. The quark propagator contains valuable inormation about non-perturbative QCD, and it is, in turn, completely determined by the non-perturbative vacuum o QCD, which is densely populated by long wave luctuations o gluon and quark ields. The order parameters o QCD vacuum state are characterized by the vacuum matrix elements o various singlet combinations o quark and gluon ields, called vacuum condensates. The nonzero quark vacuum condensate is responsible or the spontaneous break down o chiral symmetry and the nonzero gluon condensate deines the mass scale or hadron through trace anomaly. The non-local vacuum condensates describe the distribution o quarks and gluons in the non-perturbative vacuum. [1] Physically, it means that vacuum quarks and gluons have a non-zero meansquare momentum (virtuality). In this paper we propose a parameterized quark propagator and then we examine its validity o the parameterized orm. In Sec. 2, we briely introduce every aspect o Dyson Schwinger equations (DSEs), and build up a conined quark propagator according to the study o the solutions o DSEs in rainbow approximation with dierent eective gluon propagators. We use the parametrized quark propagator which is analytic and has no Lehmann representation to describe a ully dressed quark propagation inside hadron. In Sec. 3, we briely introduce the QCD vacuum condensates o quarks and quark gluon mixed vacuum condensate and then give our predictions o various QCD quantities. We also compare our theoretical predictions with the results o other model calculations. Finally, discussion and concluding remarks stemming rom this work are given in Sec. 4. 2 Quark Propagator We deine the quark mass by use o quark propagator. In principle, the quark propagator can be obtained rom the DSEs [2 4] i(s (p)) 1 = i(s 0 (p)) 1 + 4 d 4 k (2π) 4 γµ S (k)γ ν (k, p)g µν (p k).(1) The corresponding schematic representation o Eq. (1) is depicted in Fig. 1. In Eq. (1), S 0 (p) is the bare quark propagator, is 0(p) 1 = /p m. g s is the strong coupling constant o QCD, which relates to QCD running coupling constant α s (Q) by α s (Q) = gs/4π, 2 and γ µ are Dirac matrix operators. The G µν (p k) denotes an eective gluon propagator, which is know in the perturbative QCD region but has to be modeled in the non-perturbative region. In an arbitrary covariant gauge, speciied by ξ, the gluon propagator can be written as [5] G µν (q) = 1 [( q 2 δ µν q ) µq ν 1 q 2 1 Π + ξ q ] µq ν q 2, (2) The project supported in part by National Natural Science Foundation o China under Grant Nos. 10247004, 10565001, and by the Department o Science and Technology o Guangxi Province o China under Grant Nos. 0481030, 0575020, 0542042
676 ZHOU Li-Juan and MA Wei-Xing Vol. 45 where ξ = 0 is or Landau gauge and ξ = 1 or Feynman gauge. Π is gluon polarization unction and δ µν = diag (1, 1, 1, 1) is the Euclidean metric. [5] In this paper, we choose Feynman-like gauge with quench approximation Π = 0, which consequently corresponds to the ollowing model gluon propagator, [6,7] G µν (x) = δ ab δ µν G(x). (3) In Eq. (1), the Γ ν (k, p) is the Bethe Salpeter amplitude describing the ully dressed quark-gluon coupling vertex. Gauge covariance requires that the transverse part o the dressed quark-gluon coupling vertex is constrained by the Ward Takahashi identity, [8] (k p) µ iγ µ (k, p) = S 1 (k) S 1(p), (4) which is an extension to non-vanishing momentum transer o the Ward identity, [9] Γ µ (k, p) = Σ(p). (5) p µ As it is impossible to solve the complete set o DSEs, one has to truncate this ininite tower in a physically acceptable way to reduce them to something that is soluble. To this end, we make a rainbow or ladder approximation or DSEs. The rainbow approximation corresponds to the choice o Γ ν = γ ν. (6) Under the approximation o Eq. (6), the equation (1) then becomes i(s (p)) 1 = i(s 0 (p)) 1 + 4 d 4 k (2π) 4 γµ S (k)γ ν G µν (p k). (7) The schematic representation o Eq. (7) is shown in Fig. 2. The explanations are the same as those in Fig. 1. Fig. 1 Diagrammatic representation o DSEs or ully dressed quark propagator. S (p) and S (k) are quark propagators with momentum p and k, respectively. S 0 (p) is the bare quark propagator with momentum p, G µν(q) the ully dressed gluon propagator with momentum q = p k, and Γ ν (k, p) the coupling vertex o the gluon-quark interaction. Fig. 2 Schematic representation o DSEs or ully dressed conined quark propagators in the rainbow approximation, i.e., Γ ν = γ ν. All other explanations are the same as those in Fig. 1. by On the other hand, the general orm o the inverse ully dressed conining quark propagator S 1 (p) can be expressed S 1 (p) = i/pa (p 2 ) + B (p 2 ). (8) A and B satisy a set o coupled integral equations as shown in the ollowing: [A (p 2 ) 1]p 2 = 8 d 4 k (2π) 4 G(p k) A (k 2 ) k 2 A 2 (k2 ) + B 2(k2 ) p k, (9) B (p 2 ) = 16 d 4 k (2π) 4 G(p k) B (k 2 ) k 2 A 2 (k2 ) + B 2(k2 ). (10) Except or the current quark mass and perturbative corrections, the unctions [A (p 2 ) 1] and B (p 2 ) are nonperturbative quantities, which we reer to as the vectorand scalar-propagator unctions, respectively. Using the Feynman-like gauge, which corresponds to the gluon propagator, the unction G(x) [see Eq. (3)] can
No. 4 Applicability o Parametrized Form o Fully Dressed Quark Propagator 677 be taken as [6] g 2 sg(q 2 ) = 4πα i(q 2 ) q 2, (11) where α i (q 2 ) can be expressed by an eective orm with two-parameters. For example, three α i (i = 1, 2, 3) have been used, α 1 (q 2 ) = 3πq 2 χ2 2 4 2 e q / + [ q α 2 (q 2 2 χ 2 ) = πd q 2 + + 1 ln(q 2 /Λ 2 + ɛ) πd ln(q 2 /Λ 2 + ɛ), (12) ], (13) [ 1 + χe α 3 (q 2 q 2 / ] ) = πd ln(q 2 /Λ 2. (14) + ɛ) All o these determine the quark-quark interaction through the two parameters χ and, which are varied with the pion decay constant which is here ixed at π = 86 MeV. This value is appropriate at zero-momentum rather than the pion-mass-shell value o 93 MeV. However the result is not very sensitive to this small dierence. Where N = 3 is the number o quark lavor and Λ (we take Λ = 0.20 GeV) is the scale parameter o QCD. d = 12/(33 2N ) = 12/27. In Eqs. (12) (14) ɛ can be varied in the range o 1.0 2.50. In our present work we take ɛ to be 2.0. The other parameters o and χ are taken rom Res. [6] and [7]. In our early work, we have gotten some numerical results though Eqs. (9) (12). On the other hand, the quark propagator is deined by vacuum expectation value o time ordering operator product expansion S (x) = 0 T [q(x) q(0)] 0, (15) where q(x) is the quark ield and the T denotes the timeordering operator. The quark sel-energy Σ (p) is then deined by S 1 (x) = /p + m + Σ (p) (16) with m being the current quark mass in the QCD Lagrangian. At the same time, the quark sel-energy Σ (p) in Eq. (16) can be evidently expressed as Σ (p) = i/p[1 A (p 2 )] + [B (p 2 ) m )]. (17) The physical meaning o Eq. (17) will be given in later discussion. The loss o gauge covariance when one uses the rainbow approximation is direct consequence o the violation o the Ward Takahashi identity. Thereore, to go beyond rainbow approximation, one has to adopt a parametrized quark propagator. [11] We know that the quark eective mass undergoes an evolution rom the constituent quark mass in low-energy quark model to the current quark mass in the QCD Lagrangian as momentum increasing, i.e. as one goes rom the non-perturbative to perturbative regions o QCD with momentum transer increasing. This can be understood most directly by considering the quark propagator, which has the orm in the Euclidean space, [10,11] 1 S (p) = iγpa(p 2 ) + B(p 2 ) = iγp σ v (p 2 ) + σ s (p 2 ), (18) where the running quark mass is given by M( p 2 ) = B( p 2 )/A( p 2 ). The unctions A( p 2 ), B( p 2 ) are determined by a set o sel-consistent equations in the Dyson Schwinger ormalism. M( p 2 ) is expected to undergo the transition rom the constituent quark mass o about 330 MeV to the current quark mass o a ew MeV or u and d quark crossing the turn point at about 1 GeV. [10] The algebraic expressions o σv and σs have the ollowing orm, [11] σv = σ v Λ 2, σ s = σ s Λ, (19) where σ v and σ s are given, respectively, by σ v (x) = 2(x + m2 ) 1 + exp[ 2(x + m2 )] 2(x + m 2 )2, (20) σ s (x) = 1 exp( b 1 ) b 1 x 1 exp( b3 x) [ b 3 x b 0 + b 2 1 exp( Λ x) ] 1 exp[ 2(x + m )] Λ + m x x + m 2 (21) with m = m /Λ, x = p 2 /Λ 2, Λ = 10 4, and Λ = 0.566 GeV. The parameters b i (i = 0, 1, 2, 3) and m ( = u, d, s,...) are given in Table 1. [11] Table 1 The parameters o conined quark propagator in Eqs. (20) and (21). m is the current mass o quark with lavor in QCD Lagrangian. Flavor () b 0 b 1 b 2 b 3 m (MeV) u 0.131 2.90 0.603 0.185 5.1 d 0.131 2.90 0.603 0.185 5.1 s 0.105 2.90 0.740 0.185 127.5 At the same time, we can set up the relation between A ( p 2 ), B ( p 2 ) and σ v, σ s through Eq. (18), and in doing so we then arrive at [11] A ( p 2 ) = B ( p 2 ) = σ v (p 2 ) (σ s ) 2 [ p 2 (σ v /σ s ) 2 + 1], (22) 1 σ s [ p 2 (σ v /σ s ) 2 + 1]. (23) Thereore, A ( p 2 ) and B ( p 2 ) can be evaluated rom σ v and σ s. The sel-energy unctions [1 A (p 2 )] and [B ( p 2 ) m ] are shown in Figs. 3 and 4. The eective masses o quarks, M = B ( p 2 )/A ( p 2 ) predicted by
678 ZHOU Li-Juan and MA Wei-Xing Vol. 45 Eqs. (22) and (23) are given by Fig. 5. Fig. 3 p 2 -dependence o the sel-energy unctions 1 A( p 2 ) and B( p 2 ) m or u, d quarks. Fig. 4 p 2 -dependence o the sel-energy unctions 1 A( p 2 ) and B( p 2 ) m or s quarks. Fig. 5 p 2 -dependence o the eective mass o quarks, M ( p 2 ), or u, d and s quarks. Needless to say, the resulting A and B deined by Eqs. (22) and (23) have the ollowing eatures: (i) In the limit that the quark momentum p 2 becomes large and space-like, the unctions A and B approach an asymptotic limit lim ( p 2 ) = 1.0, p 2 (24) lim B ( p 2 ) = m. (25) p 2 Hence, the conined quark propagator S ( p) reduces to a ree quark propagator lim S 1 p 2 ( p) = i/p + m. (26) For the quark sel-energy Σ ( p), rom Eq. (17), we can see that when p 2, Σ ( p) 0 due to the renormalization requirements o A ( ) = 1 and B ( ) = m. That means sel-energy vanishes, and the ree quark propagator appears, we have an asymptotically ree quark. Thereore, quark propagator S ( p) has a correct behavior. On the other hand, when p 2 0, by deinition, S 1 (0) = B (0), Σ (0) = B (0) m, and M (0) = B (0)/A (0) relect the value o constituent quark mass. In act, when p 2 runs rom 0 to, M ( p 2 ) changes rom constituent quark mass to current quark mass, the dynamical transition o the eective quark mass are obtained. (ii) For small quark momentum, the quark propagator becomes quite dierent compared to the ree quark propagator S 0 ( p). In this momentum region, there is a strong non-perturbative enhancement o the mass unction B ( p 2 ). This enhancement is a maniestation o dynamical chiral symmetry breaking and quark coninement. We can see the change rom the transition o constituent quark mass to current quark mass, which is deined by M = B ( p 2 )/A ( p 2 ). 3 QCD Vacuum Condensates and Predictions o Various QCD Quantities As we discussed in Sec. 2, the quark propagator has been deined (see Eq. (15)) by S (x) = 0 T [q(x) q(0)] 0. (27) For the physical vacuum consisting o both perturbative and non-perturbative parts, the quark propagator S (x) can be divided into two parts ( perturbative and nonperturbative part ) as the ollowing, [12] S (x) = S PT (x) + S NPT (x). (28) In the non-perturbative vacuum state, the matrix elements o normal-ordered product o operators o S NP (x) does not vanish. For short distance, the operator product expansion or the scalar part o S NP (x) gives [13,14] S NP (x) 0 : q(x)q(0) : 0 = 0 : q(0)q(0) : 0 x2 4 0 : q(0)ig sσg(0)q(0) : 0 +, (29) in which the matrix elements o the local operator in the expression are the quark vacuum condensate, the quarkgluon mixed vacuum condensate, and so orth. In Re. [7] Kisslinger deined a unction g(x), dubbed susceptibility and relected the structure o non-local quark vacuum condensate by 0 : q(x)q(0) : 0 = g(x) 0 : q(0)q(0) : 0. (30)
No. 4 Applicability o Parametrized Form o Fully Dressed Quark Propagator 679 He parametrized g(x) in a phenomenological way. In this work, the non-local quark condensate 0 : q(x)q(0) : 0 is given by the scalar part o the Fourier transormed inverse quark propagator, [14] 0 : q(x)q(0) : 0 d 4 p B (p 2 ) = ( 4N c ) 0 2π 4 p 2 A 2 (p2 ) + B 2(p2 ) e ipx = ( ) 12 B (s) [ 16π 2 ds s 0 sa 2 (s) + B2 (s) 2 J 1( sx 2 ) ] (31) sx 2 with the notation that s = p 2. The expression o the local (x = 0) quark vacuum condensate 0 : q(0)q(0) : 0 is then naturally given by 0 : q(0)q(0) : 0 = 3 µ B (s) 4π 2 ds s 0 sa 2 (s) +. (32) B2 (s) According to Eq. (30), the non-locality unction g(x) can be easily obtained through dividing Eq. (31) by Eq. (32). Our analysis ignores eect rom hard gluonic radiative correction to the vacuum condensates, which are connected to a possible change o the renormalization scale µ, at which the condensates are deined. In our calculation, µ is taken to be 1 GeV 2. Finally, we obtain a structure o the non-local quark vacuum condensate, which is shown in Fig. 6, and the numerical predictions or values o the local quark vacuum condensates are also given in Table 2. Fig. 6 p 2 -dependence o the non-local quarks vacuum condensates, 0 : q(x)q(0) : 0, or u, d and s quarks. As the mixed quark-gluon vacuum condensate is considered, the expression can be written as 0 : q(0)ig s σg(0)q(0) : 0 = 9 µ { 4π 2 ds s s B (s)[2 A (s)] 0 sa 2 (s) + B2 (s) + 81 B (s){2sa (s)[a (s) 1] + B 2(s)} } 16[sA 2 (s) + B2 (s)], (33) which is derived in a 1/N c expansion o the gluon twopoint unction. [15] In the present calculations, we take µ to be 1 GeV 2. We also display our results o the mixed vacuum condensate 0 : q(0)ig s σg(0)q(0) : 0 in Table 2, and we compare our predictions with the corresponding values obtained by other theoretical approaches such as QCD sum rules, quench lattice QCD, the instanton liquid model, and so on. From Table 2, we can see that our predictions are compatible with other theoretical results. Table 2 The values o 0 : q(0)q(0) : 0 and 0 : q(0) ig sσg(0)q(0): 0 Reerence : qq : 1/3 (Mev) : qigσgq : 1/5 (MeV) Re. [16] 210 230 375 395 Re. [17] 225 402 429 Re. [18] 272 490 Re. [19] 217 429 Our work (u, d) 205 605 Our work (s) 260 647 4 Concluding Remarks Based on an extensively study o DSEs in the rainbow approximation with an eective gluon propagator, we suggest a parametrized quark propagator, which is analytic everywhere in the inite complex p 2 -plane and has no Lehmann representation, and hence there are no quark-production thresholds in the calculation o observable. The ully dressed conining quark propagator exhibits a dynamical symmetry breaking phenomenon and gives a constituent quark mass o about 330 MeV or u, d quarks, which is close to the value o commonly used constituent quark mass about 350 MeV in the chiral quark model. With the dressed quark propagator, we obtain the values o quark vacuum condensates 0 : qq : 0, and the quark gluon mixed condensates 0 : qgσgq : 0, which are compatible with other theoretical calculations. At the same time, the structures o non-local quark vacuum condensates are predicted by our theoretical calculations. Thereore, the Kislinger s susceptibility unction g(x) is naturally obtained. All the results are compatible with other theoretical calculations. Reerences [1] A.E. Dorokhov and S.V. Mikhailov, Physics o Particles and Nuclei (Suppl.) 32 (2001) 554; M. Bochicchio, et al., Nucl. Phys. B 268 (1983) 331; V. Lubicz, Nucl. Phys. (Proc. Suppl.) B 94 (2001) 116; C.D. Roberts and A.G. williams, Prog. Part. Nucl. Phys. 33 (1994) 477, and reerences therein. [2] F.J. Dyson, Phys. Rev. 75 (1949) 1736, [3] L.S. schwinger, Proc. Nat. Acad. Sci. 37 (1951) 452.
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