A perspective on the contemporary use of Dyson-Schwinger equations, focusing
on some recent phenomenological applications: a description and unification of
light-meson observables using a one-parameter model of the effective
quark-quark interaction, and studies of leptonic and nonleptonic nucleon form
factors.Comment: 7 pages, sprocl.sty, epsfig.sty. Contribution to the Proceedings of
  the Workshop on Light-Cone QCD and Nonperturbative Hadron Physics, Adelaide,
  Australia, 13-22 Dec 199

Hecht, M. B.

Roberts, C. D.

Schmidt, S. M.

English

arXiv

A perspective on the contemporary application of Dyson-Schwinger equations, focusing on some recent phenomenological applications: a description and unification of light-meson observable using a one-parameter model of the effective quark-quark interaction, and studies of leptonic and nonleptonic nucleon form factors. The theory and phenomenological application of Dyson-Schwinger equations (DSEs) have seen something of a renaissance. For example, they have been applied simultaneously to phenomena as apparently unconnected as low-energy {pi}{pi} scattering, B {r_arrow} D{sup *} decays and the equation of state for a quark gluon plasma, and there are renewed attempts to understand the origin of the infrared enhancement necessary in the kernel of the quark DSE (QCD gap equation) to generate dynamical chiral symmetry breaking (DCSB). Also significant is the appreciation that in this approach current algebra's anomalies remain a feature of the global aspects of DCSB

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.,-DSE HADRON PHENOMENOLOGYM. B. HECHT, C. D. ROBERTS and S. M. SCHMIDTPhysics Division, Bldg. 203, Argonne National LaboratoryArgonne IL 60439-~843, USAA perspective on the contemporary application of Dyson-Schwinger equations, fo-cusing on some recent phenomenological applications: a description and unificationof light-meson observable using a one-parameter model of the effective quark-quark interaction, and studies of Ieptonic and nonleptonic nucleon form factors.1 Introduction. The theory and phenomenologica,l application of Dyson-Schwinger equations (DSES) have seen something of a renaissance. For ex-ample, they have been applied simultaneously to phenomena as apparentlyunconnected as low-energy mr scattering B + D* decays2 and the equationof state for a quark gluon plasma: and there are renewed attempts4 to un-derstand the origin of the infrared enhancement necessary5 in the kernel ofthe quark DSE (QCD gap equation) to generate dynamical chiral symmetrybreaking (DCSB). Also significant is the appreciation that in this approachcurrent algebra’s anomalies remain a feature of the global aspects of DCSB.2 Meson Observable. The renormalised gap equation in Refs. [7-10] iswhere J’L := J’A d’q/(2m)4 represents mnemonically a translationally-invariantregularisation of the integral, with A the regularisation mass-scale. 22 is thequark wave function renormalisation constant, which depends on the A andthe renormalisation point, ~, and the renormalised current-quark mass ism({) := r7Zb.r,(A)/&(<2, A2); Z~({2, A2) = 24(<2, A2)/Z2((2, A2) , (2)where 24 is the renormaJisation constant for the mass-term in the QCD action.The model is specified by a choice for the effective interactionf3(fk2) 47r2~Dk2e–k2/”2 y ~~‘ym?r— =~2[(z F(F),)](3)~ln t-+ 1 +k2/A&~with $(k2) = [1 – exp(-k2/[4m~])] /k2, ~m = 12/25, AQCD = 0.234 GeV, andfixed values of w = 0.3 GeV(= 1/[.66 fro]) and m, = 0.5 GeV(= 1/[.39 fro]).The sole parameter is the mass-scale: D.1The submitted manuscript has been created’:by the University of Chicago as OperatorofArgonne National Laboratory (“Argonne”)under Contract No. W-31 -109-ENG-38 withthe U.S. Department of Energy. The U.S.Government retains for itself, and others act-ing on its behalf, a paid-up, nonexclusive,irrevocable worldwide license in said afiicleto reproduce, prepare derivative works, dis-tribute copies to the public, and perform pub-licly and display publicly, by or on behalf ofthe Government.DISCLAIMERThis report was prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.DISCLAIMERPortions of this document may be illegiblein electronic imageproduced from thedocument.products. Images arebest available original. 1,- ,-‘((@?)!GeV)*’3 ‘. ~K ‘. rnK- ~.$ ~z \K fp fK” f~Ohs. 0.236’2 0.1390.4960.7700.892 1.0200.1300.1600.216 0.2250.238Calc. 0.242 0.1390.4960.7470.956 1.088 Q 0.1540.1970.2460.255Table 1: Masses and decay constants fin GeV] of Iight vector and flavour nonsingletpseudoscalar mesons calculated using a renormalisation-group-improved rainbow-laddertruncation. The underlined quantities were fitted. The ‘Y3bs.” values of the masses aretaken from Ref. [13], s-s are ~m, ~K. The analogous vector meson decay constants arediscussed in Refs. [2,8] (Adapted from Ref. [8].)The qualitative features of Eq. (3) are clear: the first term provides forstrength in the infrared that is known to be necessary to support DCSB; thesecond term is proportional to the one-loop QCD running-coupling at largespacelike-k2, has no sir+ydarity on the real-k2 axis, and ensures that calculatedquantities exhibit the one-loop renormaJisation group flow of QCD. The lattercharacteristic follows necessarily from the fact that a weak-coupling expansionof the DSES yields all the diagrams of perturbation theory. This limits model-dependence to the infrared. Once a truncation of the quark DSE is specified,the form of the meson Bethe-Salpeter equation (BSE) follows immediatelyusing the systematic Ward-Takahashi identity preserving procedure of Ref. [11].The single model parameter, and the u = d and s current-quark masseswere varied7’8 so as to obtain a good description of low-energy z- and K-mesonproperties, using a renormalisation point ~ = 19 GeV that is large enough tobe in the perturbative domain. The fitting requires the repeated solving ofthe quark DSES and meson BSES. It could self-consistently yieId D s O,which would indicate that agreement with observable phenomena precludes aninfrared enhancement in the effective interaction. However, that was not thecase and a good fit requiredD = (1.12 GeV)> and m~~ev = 5.5 MeV, m~Gev = 124 NIeV, (4)and yields the results in Table 1, which are characterised by a root-mean-square error over predicted quantities of just 3.670. The quaMative featuresof the dressed-quark propagator obtained in these studies have recently beenconfirmed in lattice simulations!4The same model can be employed without modification in calculating theimpulse approximation to the light pseudoscalar meson form factors? ’l” In thkapplication one needs additionally to solve the inhomogeneous vector vertexequation, which describes the dressed-quark-photon coupling and exhibits apole at P2 + m? = O; i.e., at the particular flavour channel’s vector meson2,4.03.52.01.51.00.5— BSE, F1-F8— BSE, F1-F5 “F“”%0.0 I I-0.4 -0.2 0.0 0.2 0.4 0.6 0.8Q’ [GeV2]Figure 1: Pion form. factor calculated directly from the interaction in Eq. (3): solid line.The data are from Refs. [15]. No parameters were varied to obtain the result. The othercurves depict results from simplified calculations; in particular, the dash-dot line usesr~(k; P) = -yP, which violates the Ward-Takahashi identity. (Adapted from Ref. [9].)mass. It is only by solving the inhomogeneous vertex equation that one canunambiguously evolve the form factor from the spacelike into the timelikeregion and vice versa.The result for the pion form factor is illustrated in Fig. 1, with r: =(0.67fm)2 cf. expt?5 (0.663+ 0.007 fm)2. Good results are also obtainedfor the charged and neutral kaon form factors, with ~~+ = (0.62 fm)2 and@O = –(0.29 fro)’ cf. expt!5’16 rj+ = (0.583+0.043 fro)’ and r~o = –(0.23+0.06 fm)z. These results are too good because the calculations neglect meson-rescattering contributions that, in all channels, are additive in magnitude, with17Nevertheless they are a significant step, providing acorrections of up to 1570.manifestly Poincar6 invariant calculation and unification of light-meson observ-able, with the particular feature that all calculated quantities are independentof the definition of the relative momentum, which is arbitrary in a covariantformulation. A three-dimensional reduction of the Bethe-Salpeter equation isunnecessary and is not employed. Furthermore, current conservation is au-tomatic and the neutral mesons are neutral without fine-tuning. The sameframework predicts~s Q2Fme.on(Q2) = constant, as Q2 + co, in accordancewith the pQCD expectation, which is only possible because the pseudoscalarmeson Bethe-Salpeter amplitude necessarily has pseudovector components!g3#- ,3 Nucleon Observable. Reference [20] is an extensive study of the octetand decuplet baryon spectrum based on a quark-dlquark Fadde’ev equation. Itrepresents the nucleon as a composite of a quark and pointlike diquark, whichare bound together by a repeated exchange of roles between the dormant anddiquark-participant quarks, and demonstrates conclusively that an accuratedescription of the spectrum is possible using these degrees of freedom. Thismotivates and supports the product Ansatz for the nucleon’s amplitude usedin Refs. [21,22] to calculate a wide range of Ieptonic and nonleptonic nucleonform factors:V@~;Ct’~,Ti;Q,7_)= j’”’?/&, (pl + P2,P3)A(P1 + p2) r~1%2(PI #2), (5)with the nucleon’s momentum: P = PI +P2 -t-p3, P2 = –M2. In Eq. (5),(~il ~i) are quark spinor and isospin labels, and (a, I-) are those of the nucleon,with f = (1/3)/1 – (2/3)?2 and ~2 = -1, is a Bethe-Salpeter-like amplitudecharacterizing the correlation between the quark and the dlquark. In thk form,*I describes the upper-component of the positive-energy nucleon spinor and23 In addition, A(K) describes the pseud~particle+2 the lower-component.propagation characteristics of the diquarkj andrepresents the momentum-dependence, and spin and isospin character of thediquark correlation; i.e.,, it corresponds to a Bethe$alpeter-like amplitude forwhat here is a nonpointlike diquark.Equations (5)-(7) describe the nucleon as a composite of a quark anda scalar-diquark correlation, and in Refs. [21,22] the scalar functions w-ereparametrised:$(y) = (1 – e-y)/y, 42 = O, with Afq and A$ being the calculated nucleon and(ud)-diquark normalisation constants, which ensure composite electric chargesof 1 for the proton and 1/3 for the diquark. The parameters w+ = 0.2 GeV,Wi = 1.4 GeV and mA = 0.63 GeV were fixed22 in a least-squares fit to the pro-ton’s charge form factor on Q2 c [0,3] GeV2. A good description is obtainedand the parameter values demonstrate the internal consistency of the model.dr := l/tir is a measure of the mean separation between the quarks constitut-ing the scalar &quark and d+ := l/w@ is the analogue for the quark-dlquark4,separation. dr < d+ is necessary if the quark-quark clustering interpretation isto be valid. t(Ud)O+= l/mA is a measure of the range over which the diquarkmay propagate and that must be significantly less than the nucleon’s diameter.One particular highlight of the calculations is the result that the nonpoint-like nature of the diquark allows a better description of the nucleons’ magneticform factors than is possible in pointlike diquark models and, importantly,IPJvPI= 055 cf. expt. 0.68, whereas pointlike scal~-diquark models alwaysyield lp~/pP[ <0.5. Another is a prediction for the ratio pPG~(q2)/G~ (q2)that is in semi-quantitative agreement with recent results from TJNAF?4 Adefect is that IT; I is 60% too large.There are two obvious improvements – include: the lower component ofthe nucleon spinor, +2 # O; and the axial-vector diquark. The first is underwayand we report preliminary results for +2 = @l. A fit to G~ (p2) now requiresbut this represents only a small change in the fitting parameters that preservesthe model’s internal consistency [cf. after Eq. (8)]. These values yieldr2 (fm2) & (fm2) PP(KN) &n (PN) Pn/PPEmp. &87)2 -(0.34)’ 2.79 -1.91 -0.68Calc. (0.78)2 -(0.33)2 2.81 –1.61 –0.57(lo)Old Calc?l (0.79)’ -(0.43)2 2.88 -1.58 -0.55and the recalculated neutron electric form factor in Fig. 2. As elucidatedin Refk. [21], of these calculated quantities only G& involves a cancellationbetween contributions from the five diagrams that constitute the impulse ap-proximation to the nucleon’s electromagnetic current in this model, and onlyobservable tied to this form factor are dramatically fiected by the improve-ment of the product Ansatz, Eq. (5). We anticipate that this is a generalfeature; i.e., only observable receiving interfering contributions; e.g., gA andthe isoscabr tensor coupling fuATpI,are significantly modified by improvements.If thk were not the case, simple models could not be generally efficacious.Epilogue. This overviewis necessarily brief. It does no more than point torecent successes and says nothing of contemporary challenges. One such is tocomprehend the origin of the infrared enhancement in the kernel of the QCDgap equation that is necessary to ensure DCSB. Another is to develop a BSEbased understanding of the light scalar mesons and the ~-q’ mass-splitting.These aspects and more are cahvassed in Ref. [27].5, .,.0.21/ . ./ ‘.. \\1’ \\\/’ \ \0.1 ;\\.i*Ai-\,~ \‘Illc1 / ‘\\‘..I’0.10.0 1.0 2.0 3.0q’ (Gel/)Figure 2: SoIid line: neutron electric form fader calculated with 42 = +1; i.e., includ~ngthe lower component of the neutron spinor. Dashed line: the result with @2 = O?lData from Ref. [25], extracted using the Argonne V18 potentialz6 Improving the productAnsatz, Eq. (5), by including 42 significantly improves the estimate of G%. However, th:spreliminary result is too good as we still ignore the axial vector diquark correlation.Acknowledgments. CDR is grateful to the staff of the Special ResearchCentre for the Subatomic Structure of Matter at the University of Adelaidefor their hospitaMy and support during this workshop, and in the two pre-ceding weeks; and we acknowledge helpful communications with J.C.R. Bloch.Thk work was supported by the US Department of Energy, Nuclear Physics Di-vision, under contract no. W-31 -109-ENG-38, and benefited from the resourcesof the National Energy Research Scientific Computing Center.References1.2.3.4.5.C.D. Roberts, R.T. Cahill, M.E. Sevior and N. Iannella, Phys. Rev. D49 (1994) 125.M.A. Ivanov, Yu.L. Kalinovsky and C.D. Roberts, Phys. Rev. D 60(1999) 034018.D. Blaschke, C.D. Roberts and S.M. Schmidt, Phys. Lett. B 425 (1998)232.L. v. Smekal, A. IIauck and R. Alkofer, Annals Phys. 267 (1998) 1;D. Atkinson and J. C. Bloch, Mod. Phys. Lett. A 13 (1998) 1055.F.T. Hawes, P. Maris arid C.D. Roberts, Phys. Lett. B 440 (1998) 353.6● ✜✎✎,.6. M. Bando, M. Harada and T. Kugo, Prog. Theor. Phys. 91 (1994) 927;C.D. Roberts, Nucl. Phys. A 605 (1996) 475; P. Maris andC.D. Roberts, Phys. Rev. C 58 (1998) 3659.7. P. Maris and C.D. Roberts, Phys. Rev. C 56 (1997) 3369.8. P. Maris and P.C. Tandy, Phys. Rev. C 60 (1999) 055214.9. P. Maris and P.C. Tandy, Phys. Rev. C 61 (2000) 45202.10. P. Maris and P.C. Tandy, “The r, K+ and K“ electromagnetic formfactors,” nucl-th/0005015.11. A. Bender, C.D. Roberts and L. v. Smekal, Phys. Lett. B 380 (1996) 7.12. D. B. Leinweber, Annals Phys. 254 (1997) 328.13. C. Caso et al., Eur. Phys. J. C 3 (1998) 1.14. J.I. Skullerud and A.G. Williams, “The quark propagator inmomentum space,” hep-lat/9909142; and these proceedings.15. Open circles, C-J. Bebek et a2., Phys. Rev. D 13 (1976) 25; Squares,L.M. Barkov et al., NUCLPhys. B 256 (1985) 365; Filled circles,S.R. Amendolia et al. [NA7 Collaboration], Nucl. Phys. B 277 (1986)168.16. W. R. Molzon et al., Phys. Rev. Lett. 41 (1978) 1213.17. R. Alkofer, A. Bender and C.D. Roberts, Int. J. Mod. Phys. A 10(1995) 3319.18. P. Maris and C.D. Roberts, Phys. Rev. C 58 (1998) 3659.19. P. Maris, C.D. Roberts and P.C. Tandy, Phys. Lett. B 420 (1998) 267.20. G. Hellstern, R. Alkofer, M. Oettel and H. Reinhardt, Nucl. Phys. A627 (1997) 679.21. J.C.R. Bloch, C.D. Roberts, S.M. Schmidt, A. Bender and M.R. Frank,Phys. Rev. C 60 (1999) 062201.22. J.C.R. Bloch, C.D. Roberts and S.M. Schmidt, “Selected nucleon formfactors and a composite scalar diquark,” nucl-th/991 1068, to appear inPhys. Rev. C.23. K. Kusaka, G. Piller, A.W. Thomas and A.G. Williams, Phys. Rev. D55 (1997) 5299.24. M.K. Jones et al. [Jefferson Lab Hall A Collaboration], Phys. Rev.Lett., 84, 1398 (2000).25. S. Platchkov et aL, Nucl. Phys. A51O (1990) 740.26. R.B. Wiringa, private communication; R.B. Wiringa, V.G. Stoks andR. Schiavilla, Phys. Rev. C51 (1995) 38.27. C.D. Roberts and S.M. Schmidt, “Density, Temperature andContinuum Strong QCD~ preprint no. ANL-PHY-9530-TH-2000.7

DSE hadron phenomenology.

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DSE Hadron Phenomenology

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