Calculation of tensor susceptibility beyond rainbow-ladder approximation of Dyson-Schwinger equations approach
aa r X i v : . [ h e p - ph ] M a r Calculation of tensor susceptibility beyond rainbow-ladderapproximation of Dyson-Schwinger equations approach
Yuan-mei Shi , Hui-xia Zhu , , Wei-min Sun , , Hong-shi Zong , Department of Physics, Nanjing Xiaozhuang College, Nanjing 211171, China Department of Physics, Nanjing University, Nanjing 210093, China Department of Physics, Anhui Normal University, Wuhu, 241000, China and Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China
Abstract
In this paper, we extend the calculation of tensor vacuum susceptibility in the rainbow-ladderapproximation of the Dyson-Schwinger (DS) approach in [Y.M.Shi, K.P.Wu, W.M.Sun, H.S.Zong,J.L.Ping, Phys. Lett. B , 248 (2006)] to that of employing the Ball-Chiu (BC) vertex. Thedressing effect of the quark-gluon vertex on the tensor vacuum susceptibility is investigated. Ourresults show that compared with its rainbow-ladder approximation value, the tensor vacuum sus-ceptibility obtained in the BC vertex approximation is reduced by about 10%. This shows thatthe dressing effect of the quark-gluon vertex is not large in the calculation of the tensor vacuumsusceptibility in the DS approach.
PACS numbers: 12.38.Aw, 11.30.Rd, 12.38.Lg, 24.85.+p J Γ ( y ) V Γ ( y ) ≡ ¯ q ( y )Γ q ( y ) V Γ ( y ) ( q ( y ) is the quarkfield, Γ stands for the appropriate combination of Dirac, flavor, color matrices and V Γ ( y ) isthe variable external field of interest) [1–3]. Here following Ref. [11], we adopt the followingdefinition for the tensor vacuum susceptibility χ Z χ Z = { T r [ σ ηζ S Γ · ZS ] − T r [ σ ηζ S Γ · S ] } Z ηζ h ˜0 | : ¯ q (0) q (0) : | ˜0 i , (1)where Z ηζ denotes the variable external tensor field, Γ and Γ ((Γ ) µν = σ µν ) denote the fulland free tensor vertex, S and S are the full and free quark propagators. h ˜0 | : ¯ q (0) q (0) : | ˜0 i denotes the chiral quark condensate. Here it should be noted that Eq. (1) is essentially thetensor vacuum polarization, regularized by subtraction of the free vacuum polarization, andscaled by the scalar vacuum condensate. It can be obtained by external field differentiationof the propagator contracted with a bare vertex. Such a differentiation of a trace of apropagator, essentially a condensate, to produce a susceptibility as proportional to theassociated vacuum polarization has recently been used by some of the same authors in Ref.[26] for the vector and axial-vector vacuum susceptibility, and also in Refs. [27, 28] for thescalar and pseudoscalar vacuum susceptibility.From Eq. (1) we can see that the tensor vacuum susceptibility is closely related to thedressed quark propagator and the dressed tensor vertex at zero total momentum. Now weturn to the calculation of the dressed quark propagator and the dressed tensor vertex atzero total momentum in the DS approach. In the DS approach, the gap equation for the3ressed quark propagator S in the chiral limit can be written as S ( p ) − = Z iγ · p + Σ( p ) (2)with Σ( p ) = Z Z Λ q g D µν ( p − q ) λ a γ µ S ( q ) λ a gν ( q, p ) , (3)where D µν ( k ) is the dressed gluon propagator and Γ gν ( q, p ) is the dressed quark-gluon vertex.The quark-gluon vertex and quark wave-function renormalization constants, Z , ( ζ , Λ ),also depend on the gauge parameter.The gap equation’s solution has the form S ( p ) − = iγ · p A ( p , ζ ) + B ( p , ζ ) (4)and the mass function M ( p ) = B ( p , ζ ) /A ( p , ζ ) is renormalisation point independent.The quark propagator can be obtained from Eq. (2) with the following renormalisationcondition (since QCD is asymptotically free, one can choose this renormalization condition): S ( p ) − (cid:12)(cid:12) p = ζ = iγ · p. (5)The renormalized fully-dressed tensor vertex Γ µν satisfies an inhomogeneous Bethe-Salpeterequation: Γ µν ( k, P ; ζ ) = Z T σ µν + Z Λ q [ S ( q + )Γ µν ( q, P ) S ( q − )] sr K rstu ( q, k ; P ) . (6)Here k is the relative and P the total momentum of the quark-antiquark pair; q ± = q ± P/ r, s, t, u represent colour and Dirac indices; and K is referred to as the fully-amputatedquark-antiquark scattering kernel. Z T is the renormalisation constant for the tensor vertex.For the specific calculation of χ Z , one only requires the tensor vertex at P = 0. Fromgeneral Lorentz structure analysis and the asymmetry of the tensor vertex Γ µν with respectto the indices µ and ν , we can write down the general form of the tensor vertexΓ µν ( p,
0) = σ µ,ν E ( p ) + ( γ µ p ν − γ ν p µ ) F ( p ) + iγ · p ( γ µ p ν − γ ν p µ ) G ( p ) . (7)Substituting Eqs. (4) and (7) into Eq. (1), we can obtain the final expression for calcu-lating the tensor vacuum susceptibility χ Z = 316 π a Z ∞ dss ( E ( s ) (cid:20) B ( s ) sA ( s ) + B ( s ) (cid:21) − sG ( s ) sA ( s ) + B ( s ) ) , (8)4here a = h ˜0 | : ¯ q (0) q (0) : | ˜0 i is the two-quark condensate. Here we note that in obtainingthe above equation, we have made use of the fact that the subtraction term vanishes.In phenomenological applications, one may proceed by considering the truncation schemefor the DSEs and BSEs, especially for the dressed gluon propagator, the dressed quark-gluon vertex and the four-point dressed quark-antiquark scattering kernel. The importantinformation about the kernel of QCD’s gap equation can be phenomenologically drawn by adialogue between DSE studies and results from numerical simulations of lattice-regularizedQCD [29–32]. The ansatz that is typically implemented in the quark propagator’s gapequation can be written as Z g D ρσ ( p − q )Γ aσ ( q, p ) → G (( p − q ) ) D free ρσ ( p − q ) λ a σ ( q, p ) , (9)wherein D free ρσ ( ℓ ) is the Landau-gauge free gauge-boson propagator, G ( ℓ ) is a model effective-interaction and Γ σ ( q, p ) is a vertex ansatz.Over the past few years, the most usually used approximation is the rainbow-ladderapproximation [12–17], where the dressed quark-gluon vertex Γ µ ( q, p ) is replaced by the barevertex γ µ , and in the BS equation the ladder kernel is used. Rainbow-ladder approximationis the lowest order truncation scheme for the DSE. It is the nonperturbative symmetry-conserving truncation scheme because it satisfies the axial-vector WTI. Models formulatedusing the rainbow-ladder DSE to describe the quark dynamics within hadrons were foundto provide good and compact descriptions of the light pseudoscalar and vector mesons.However, the rainbow-ladder DSE cannot describe well the properties of scalar mesons. Sophysicists are trying to go beyond the rainbow-ladder approximation for years. The keypoints to go beyond the rainbow-ladder approximation are the dressed quark-gluon vertexand the four-point quark-antiquark scattering kernel.For the dressed quark-gluon vertex, we can employ the BC vertex [18, 19] i Γ σ ( k, ℓ ) = i Σ A ( k , ℓ ) γ σ + ( k + ℓ ) σ × (cid:20) i γ · ( k + ℓ )∆ A ( k , ℓ ) + ∆ B ( k , ℓ ) (cid:21) , (10)where Σ F ( k , ℓ ) = 12 [ F ( k ) + F ( ℓ )] , ∆ F ( k , ℓ ) = F ( k ) − F ( ℓ ) k − ℓ , (11)with F = A, B , viz., the scalar functions in Eq. (4). Here it should be noted that the BCvertex satisfies the vector WTI. 5ow one should find a kernel consistent with the BC vertex ansatz. This is a difficulttask that many scientists try to do. Recently, great progress has been done on this aspect.The authors in Ref. [25] have found a way to constrain the kernel for the general vertex.Following their method, an exact form of the inhomogeneous BSE for the tensor vertexΓ µν ( k,
0) can also be written asΓ µν ( k,
0) = Z T σ µν − Z q g D αβ ( k − q ) λ a γ α S ( q )Γ µν ( q, S ( q ) λ a β ( q, k )+ Z q g D αβ ( k − q ) λ a γ α S ( q ) λ a µνβ ( k, q ; 0) , (12)where Λ µνβ ( k, q ; 0) is a four-point Schwinger function that is completely defined via thequark self-energy [33, 34]. It satisfies the similar identity as those in Ref. [25]( k − q ) β i Λ µνβ ( k, q ; 0) = Γ µν ( k, − Γ µν ( q, . (13)Then we can obtain i Λ µνβ ( k, q ; 0) = 2 l β [∆ E ( q, k ; 0) + ( γ µ l ν − γ ν l µ )∆ F ( q, k ; 0)]+( γ µ δ νβ − γ ν δ µβ )Σ F ( q, k ; 0) + 2 l β γ · l ( γ µ l ν − γ ν l µ )∆ F ( q, k ; 0)+ γ · l ( γ µ δ νβ − γ ν δ µβ )Σ G ( q, k ; 0) + γ β ( γ µ l ν − γ ν l µ )Σ G ( q, k ; 0)+ 14 γ β ( k − q )[ γ µ ( q − k ) ν − γ ν ( q − k ) µ ]∆ G ( q, k ; 0) . (14)Herein we employ a simplified form of the renormalisation-group-improved effective in-teraction proposed in Refs. [12–17]; viz., we retain only that piece which expresses thelong-range behavior ( s = k ): G ( s ) s = 4 π ω D s e − s/ω . (15)This is a finite width representation of the form introduced in Ref. [36], which has beenrendered as an integrable regularisation of 1 /k [37]. Equation (15) delivers an ultravioletfinite model gap equation. Hence, the regularisation mass-scale can be removed to infinityand the renormalisation constants set equal to one.The active parameters in Eq. (15) are D and ω but they are not independent. Inreconsidering a renormalisation-group-improved rainbow-ladder fit to a selection of groundstate observables [13], Ref. [15] noted that a change in D is compensated by an alterationof ω . This feature has further been elucidated and exploited in Refs. [16, 17, 35]. For6 -4 -3 -2 -1 A ( p ) p (GeV ) -4 -3 -2 -1 B ( p ) p (GeV ) FIG. 1: Dressed quark propagator.
Left panel – A ( p ), right panel – B ( p ). In both panels,Dashed curve: calculated in rainbow-ladder truncation; solid curve: calculated with BC vertexansatz. ω ∈ [0 . , .
5] GeV, with the interaction specified by Eqs. (9), (10) and (15), fitted in-vacuumlow-energy observables are approximately constant along the trajectory ωD = (0 . =: m g . (16)Herein, we employ ω = 0 . D = m g /ω = 1 . .So now with the BC vertex ansatz and the model effective interaction, the equations ofthe DSE for the dressed quark propagator and the BSE for the dressed tensor vertex arereduced to a closed system of equations. We can numerically calculate them with iterationmethod. In Fig. 1 we plot the functions obtained through solving the gap equation and inFig. 2 those which describe the dressed tensor vertex.It is apparent in Fig. 1 that the vertex Ansatz has a quantitative impact on the magnitudeand point-wise evolution of the gap equation’s solution. That this should be anticipated isplain from Ref. [38]. Moreover, the pattern of behavior can be understood from Ref.[39]: the feedback arising through the ∆ B term in the BC vertex, Eq. (10), absent in therainbow approximation, always acts to alter the domain upon which A ( p ) and M ( p ) differsignificantly in magnitude from their respective free-particle values. Since E ( p ), F ( p )and G ( p ) are derived quantities, their behavior does not require explanation. We plot theintegrand in Eq. (8) in Fig. 3 for each vertex ansatz. From the figure we can see that thereis no far-ultraviolet tail in the integrand so that the we do not need regularization here. The7 -4 -3 -2 -1 E ( p ) p (GeV ) -4 -3 -2 -1 -0.12-0.10-0.08-0.06-0.04-0.020.00 F ( p ) p (GeV ) -4 -3 -2 -1 G ( p ) p (GeV ) FIG. 2: P = 0 scalar vertex, Eq. (7): upper left panel – E ( p ), upper right panel – F ( p ), lowerpanel – G ( p ). In all panels, Dashed curve: calculated in rainbow-ladder truncation; solid curve:calculated with BC vertex ansatz. resulting tensor vacuum susceptibilities are χ ZBC = 0 . − , χ ZRL = 0 . − . (17)The above result shows that the numerical value of the tensor vacuum susceptibility ob-tained in the BC vertex approximation is much smaller than that in the rainbow-ladderapproximation. Here it should be noted that in the above calculations of tensor vacuumsusceptibility using the effective interaction (15) in the rainbow-ladder truncation and theBC vertex, we have chosen the same model parameters for the effective interaction. As isshown in Ref. [25], the amount of chiral symmetry breaking (as measured by the chiral con-densate) and related quantities such as the pion decay constant are very different betweenthese two truncation schemes. Therefore, when calculating the tensor vacuum susceptibil-ity employing the BC vertex, a reasonable approach is to use refitted model parameters inthe effective interaction (15) in the calculation. Because the active parameters D and ω inEq. (15) are not independent, one can refit the model parameters from one physical quan-8 -4 -3 -2 -1 p (GeV ) FIG. 3: Integrand in Eq. (8) – Dashed curve: calculated in rainbow-ladder truncation; solid curve:calculated with BC vertex ansatz. tity, for example, the chiral condensate. Under the BC vertex, the value of the parameter D fitted from the chiral condensate is D = GeV (see Ref. [27]). The results for thedressed quark propagator, the scalar functions E ( p ) , F ( p ) , G ( p ), and the integrand in Eq.(14), calculated from both the rainbow-ladder truncation and the BC vertex with refittedmodel parameters are shown in Figs. 4 to 6. With refitted parameters in the BC vertexapproximation, the resulting tensor vacuum susceptibility are χ ZBC = 0 . − , χ ZRL = 0 . − . (18)So, compared with the rainbow-ladder truncation result, the value of χ Z in the BC vertexapproximation is reduced by about 10%. Therefore, one can draw the conclusion that inthe calculation of the tensor vacuum susceptibility in the framework of the DS approach thedressing effect of the quark-gluon vertex is not large.In Fig. 7 we depict the evolution of the tensor vacuum susceptibility with increasinginteraction strength, I = D/ω . The behavior may readily be understood. For I = 0 onehas a noninteracting theory and the “vacuum” is unperturbed by the external tensor field.Hence, the susceptibility is zero. The tensor vacuum susceptibility remains zero until theinteraction strength I reaches a critical value, I = I c . When I > I c , the tensor vacuumsusceptibility becomes larger quickly and then goes down slowly for both the rainbow-ladderapproximation and the BC vertex approximation. Those critical values for the interactionstrength are: I RLc = 1 . , I BCc = 1 .
41. It can be seen that the critical point in the rainbow-ladder approximation is larger than that in the BC vertex approximation. This is easy tounderstand, because the effect of the BC vertex itself amounts to enhancing the interaction9 -4 -3 -2 -1 A ( p ) p (GeV ) -4 -3 -2 -1 B ( p ) p (GeV ) FIG. 4: Dressed quark propagator.
Left panel – A ( p ), right panel – B ( p ). In both panels,Dashed curve: calculated in rainbow-ladder truncation; solid curve: calculated with BC vertexansatz with refitted model parameters. -4 -3 -2 -1 E ( p ) p (GeV ) -4 -3 -2 -1 -0.035-0.030-0.025-0.020-0.015-0.010-0.0050.0000.005 F ( p ) p (GeV ) -4 -3 -2 -1 G ( p ) p (GeV ) FIG. 5: P = 0 scalar vertex, Eq. (7): upper left panel – E ( p ), upper right panel – F ( p ), lowerpanel – G ( p ). In all panels, Dashed curve: calculated in rainbow-ladder truncation; solid curve:calculated with BC vertex ansatz with refitted model parameters. -4 -3 -2 -1 -0.00020.00000.00020.00040.00060.00080.0010 p (GeV ) FIG. 6: Integrand in Eq. (8) – Dashed curve: calculated in rainbow-ladder truncation; solid curve:calculated with BC vertex ansatz with refitted model parameters. Z FIG. 7: Dependence of the chiral susceptibility on the interaction strength in Eq. (15); viz., I := D/ω : dashed curve , RL vertex; solid curve , BC vertex. strength. The authors in Ref. [27] has explained the nature of the critical interactionstrength which denotes a second-order phase transition.For I < I c , the interaction strength is not sufficient to generate a non-zero scalar termin the dressed quark self-energy in the chiral limit. That means below the critical value,dynamical chiral symmetry breaking is impossible. The situation changes at I c , for I > I c a B = 0 solution is always possible. Moreover, when I < I c , the interaction strength is alsonot sufficient to generate the non-zero F and G functions in the dressed tensor vertex. Thatis the reason why the tensor vacuum susceptibility remains zero when I < I c .To summarize, using the expression obtained in the QCD sum rule external field approachin Ref. [11], we extend the calculation of tensor vacuum susceptibility in the rainbow-ladder approximation of the DS approach in Ref. [11] to that of employing the BC vertex11pproximation. Here a key problem is how to construct a consistent Bethe-Salpeter kernelfor a dressed quark-gluon vertex ansatz whose diagrammatic content is unknown. Recently,significant progress in this problem was achieved in Ref. [25]. In this paper, following thework of Ref. [25], we construct the kernel for the dressed tensor vertex at P = 0 whichis needed in the calculation of tensor vacuum susceptibility. Then we perform a consistentcalculation of the tensor vacuum susceptibility beyond the rainbow-ladder aproximation.Our results show that compared with its rainbow-ladder approximation value, the tensorvacuum susceptibility in the BC vertex approximation is reduced by about 10%. This showsthat the dressing effect of the quark-gluon vertex is not large in the calculation of thetensor vacuum susceptibility in the framework of the DS approach. In this paper we alsodemonstrate that the tensor vacuum susceptibility can be used to demarcate the domainof coupling strength within a theory upon which chiral symmetry is dynamically broken.For couplings below the associated critical value and in the absence of confinement, thetensor vacuum susceptibility remains zero. This situation changes until the interactionstrength is larger than a critical point. It is found that the critical point in the rainbow-ladder approximation is larger than that in the BC vertex approximation. This is easy tounderstand, because the effect of the BC vertex itself amounts to enhancing the interactionstrength.This work is supported in part by the National Natural Science Foundation of China(under Grant Nos. 10775069 and 10935001) and the Research Fund for the Doctoral Programof Higher Education (under Grant Nos. 20060284020 and 200802840009). [1] B. L. Ioffe, A. V. Smilga, Nucl. Phys. B , 109 (1984).[2] I. I. Balitsky, A. V. Yung, Phys. Lett. B , 328 (1983).[3] S. V. Mikhailov, A. V. Radyushkin, JETP Lett. , 712 (1986); Phys. Rev. D , 1754 (1992);A. P. Bakulev, A. V. Radyushkin, Phys. Lett. B , 223 (1991).[4] R. L. Jaffe, X. D. Ji, Phys. Rev. Lett. , 552 (1991); R. L. Jaffe, X. D. Ji, Nucl. Phys. B , 527 (1992).[5] H. X. He, X. D. Ji, Phys. Rev. D , 6897 (1996).[6] V. M. Belyaev, A. Oganesian, Phys. Lett. B , 307 (1997).
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