DYSON SCHWINGER EQUATIONS: CONNECTING SMALL AND LARGE LENGTH-SCALES CRAIG ROBERTS Physics Division, 203, Argonne National Laboratory, Argonne IL 60439, USA I illustrate the phenomenologicalapplication of Dyson-Schwinger equations to the 9 calculationof mesonproperties observable at TJNAF. Particularemphasis is given 9 to this framework’s ability to unify long-range effects constrained by chiral sym- 9 metry with short-range effects prescribed by perturbation theory, and interpolate 1 between them. n PACS numbers: 12.38.Lg, 13.40.Gp, 14.40.Aq, 24.85.+p a J Keywords: Dyson-SchwingerEquations,Nonperturbativemethods,ContinuumQCD, Confinement,Dynamicalchiralsymmetrybreaking,Pionelectromagneticandanoma- 7 2 lous transition form factors 1 1. Dressed Quarks v 1 The Dyson-Schwingerequations(DSEs)[1]provideanonperturbativeapproach 9 0 tostudyingthecontinuumformulationofQCD,makingaccessiblephenomenasuch 1 as confinement, dynamical chiral symmetry breaking (DCSB) and bound state 0 structure. However, they also provide a generating tool for perturbation theory 9 andhence their phenomenologicalapplicationis tightly constrainedathigh-energy. 9 This is the particular feature of the phenomenological application of DSEs: their / h abilityto furnishaunifieddescriptionofhigh-andlow-energyphenomenainQCD. t - It is elucidated in Refs. 2,3 and here I only illustrate this aspect, using as primary l ∗ c exemplars the electromagnetic pion form factor and the γ π → γ transition form u factor, which are particularly relevant to the TJNAF community. n A key element in the description of hadronic observables is the dressing of the : v quark propagator,which is described by the quark DSE: i X S (p)−1 := iγ·pA (p2)+B (p2)=A (p2) iγ·p+M (p2) (1) f f f f f r a Λ (cid:0) λa (cid:1) = Z (iγ·p+mbm)+ Z g2D (p−q) γ S (q)Γfa(q,p).(2) 2 f 1 Z µν 2 µ f ν q Here f is a flavour label, D (k) is the dressed-gluon propagator [4,5], Γfa(q,p) is µν ν the dressed-quark-gluon vertex, mbm is the Λ-dependent current-quark bare mass f and Λ := Λd4q/(2π)4 represents mnemonically a translationally-invariant regu- q larisaRtion oRf the integral, with Λ the regularisation mass-scale. The quark-gluon- vertex and quark wave function renormalisation constants, Z and Z , depend on 1 2 the renormalisationpoint, ζ, and the regularisationmass-scale. The qualitative features of the solution of Eq. (2) are known. In QCD the chiral limit is defined by mˆ =0, where mˆ is the renormalisation-point-independent current-quarkmass. Formˆ =0 there is no mass-likedivergenceinthe perturbative 1 101 100 V) Ge10−1 2p) ( b−quark M( c−quark 10−2 s−quark u,d−quark chiral limit M2(p2) = p2 10−3 10−2 10−1 100 101 102 p2 (GeV2) Figure 1: Quark mass function obtained as a solution of Eq. (2) using D (k) and µν Γfa(q,p) from Ref. 6, and mζ = 3.7MeV, mζ = 82MeV, mζ = 0.58GeV and ν u,d s c mζ = 3.8GeV (ζ = 19GeV). The indicated solutions of M2(p2) = p2 define the b Euclidean constituent-quark mass, ME, which takes the values: ME =0.56GeV, f u,d ME =0.70GeV, ME =1.3GeV, ME =4.6GeV. s c b evaluation of the quark self energy and hence for p2 > 20GeV2 the solution of Eq. (2) is [6] M (p2)larg=e−p2 2π2γm −hq¯qi0 , (3) 0 3 (cid:0) (cid:1) 1−γm p2 1ln p2/Λ2 2 QCD (cid:16) h i(cid:17) where γ = 12/(33−2N ) is the gauge-independent mass anomalous dimension m f andhq¯qi0 is the renormalisation-point-independentvacuum quarkcondensate. The momentum-dependence is a model-independent result. The existence of DCSB meansthathq¯qi0 6=0,however,itsactualvaluedependsonthelong-rangebehaviour ofD (k)andΓ0a(q,p),whichismodelledincontemporaryDSEstudies. Requiring µν ν a good description of light-meson observables necessitates hq¯qi0 ≈−(0.24GeV)3. In contrast, for mˆ 6=0, f M (p2)larg=e−p2 mˆf . (4) f γm 1ln p2/Λ2 2 QCD (cid:16) h i(cid:17) Anobviousqualitativedifferenceisthat,relativetoEq.(4),thechiral-limitsolution is 1/p2-suppressed in the ultraviolet. There is some quantitative model-dependence in the p2-evolution of M (p2) f into the infrared. However,for any forms of D and Γfa that provide an accurate µν ν description of f and m , one obtains profiles like those illustrated in Fig. 1. π,K π,K 2 Theevolutiontocoincidencebetweenthechiral-limitandu,d-quarkmassfunctions, apparent in this figure, makes clear the transition from the perturbative to the nonperturbative domain. The chiral limit mass-function is nonzero only because of the nonperturbative DCSB mechanism whereas the u,d-quark mass function is purely perturbative at p2 > 20GeV2, where Eq. (4) is accurate. The DCSB mechanismthus has asignificanteffect onthe propagationcharacteristicsofu,d,s- quarks,andthisisfundamentallyimportantinQCDwithobservableconsequences. 2. Bound States A meson is a bound state of a dressed-quark and -antiquark, and its internal structure is described by a Bethe-Salpeter amplitude obtained as the solution of Λ [Γ (k;Q)] = [χ (q;Q)] Krs(q,k;Q), (5) P tu Z P sr tu q where χP(q;Q) = S(q+)ΓP(q;Q)S(q−); S(q) = diag(Su(q),Sd(q),Ss(q),...); q+ = q+ηQQ, q− =q−(1−ηQ)Q, with Q the totalmomentum of the bound state and η ∈ [0,1] the relative-momentum partitioning parameter; and r,...,u represent Q colour-, Dirac- and flavour-matrix indices. For a pseudoscalar meson, such as the pion, the solution has the general form Γ (k;Q) = TPγ iE (k;Q)+γ·QF (k;Q) (6) P 5 P P (cid:20) +γ·kk·QG (k;Q)+σ k Q H (k;Q) , P µν µ ν P (cid:21) where TP is a flavour matrix identifying the meson; e.g., Tπ+ = 1 λ1+iλ2 , with 2 {λj,j =1...Nf2−1} the Gell-Mann matrices of SU(Nf). (cid:0) (cid:1) In Eq. (5), K is the renormalised, fully-amputated, quark-antiquark scattering kernel. Important in the successful application of DSEs is that K has a systematic skeletonexpansionintermsofthe elementarydressed-particleSchwingerfunctions; e.g.,the dressed-quarkand-gluonpropagators. TheexpansionintroducedinRef.7 providesameansofconstructingakernelthat,order-by-orderinthenumberofver- tices,ensuresthepreservationofvectorandaxial-vectorWard-Takahashiidentities; i.e., current conservation. Only with such a truncation is an accurate description of the light-quark mesons possible. In QCD the leptonic decay constant of a pseudoscalar meson is [6] Λ f Q =trZ (TP)Tγ γ χ (k;Q), (7) P µ 2 5 µ P Z k where the traceis overcolour,Dirac and flavourindices. Equation(7)is exact: the Λ-dependenceofZ ensuresthattheright-hand-side(r.h.s.) isfiniteasΛ→∞,and 2 its ζ- and gauge-dependence is just that necessaryto compensate that of χ (k;Q). P 3 In the chiral limit the axial-vector current is conserved, and employing any Ward-Takahashi identity preserving truncation of K one obtains f E (k;0)=B (k2), F (k;0)+2f F (k;0)=A (k2), GP (kP;0)+2f G0 (k;0)=2A′(k2), HR(k;0)+2fP HP (k;0)=00, (8) R P P 0 R P P where F , G and H are calculable functions in the dressed axial-vector ver- R R R tex, ΓH(k;Q). These identities are associated with Goldstone’s theorem and in 5µ fact one can show [6] that when chiral symmetry is dynamically broken: 1) the flavour-nonsinglet,pseudoscalarBSEhasamasslesssolution;2)theBethe-Salpeter amplitude for the massless bound state has a term proportional to γ alone, with 5 the momentum-dependence of E (k;0) completely determined by that of B (k2), P 0 in addition to terms proportional to other pseudoscalar Dirac structures that are nonzero; and 3) ΓP (k;Q) is dominated by the pseudoscalar bound state pole for 5µ Q2 ≃ 0. The converse is also true. Hence, in the chiral limit, the pion is a mass- less composite of a quark and an antiquark, each of which has an effective mass ME ∼0.5GeV. Fornonzerovaluesofthe current-quarkmass,whethersmallorlarge,insteadof Eqs. (8) one obtains [6] f2 m2 = −M hq¯qiP, (9) P P P ζ where M =tr M TP, TP T e.g., for the π: M =mζ +mζ, and P f (ζ) π+ u d h n (cid:0) (cid:1) oi Λ −hq¯qiP = f trZ TP Tγ χ (q;Q). (10) ζ P 4Z 5 P q (cid:0) (cid:1) Equation(9)isanexactmassformulaforflavournon-singletmesonsandInotethat ther.h.s. doesnotinvolveadifferenceofmassivequarkpropagators: aphenomeno- logicalassumptionoftenemployed. hq¯qiP inEq.(10)isan“in-hadron”condensate. ζ It is gauge-independent and its renormalisation point dependence is exactly that required to ensure that the r.h.s. of Eq. (9) is renormalisation point independent. Forsmallcurrent-quarkmasses,mˆ ∼0,Eq.(9)yieldswhatiscommonlycalled q the Gell-Mann–Oakes–Rennerrelation; i.e., m2 ∝mˆ , because P q limhq¯qiP =hq¯qi0. (11) mˆ→0 ζ ζ However, it also has an important corollary when the current-mass, mˆ , of one or Q both constituents becomes large, predicting [8] m ∝mˆ , (12) P Q whichfollowsbecausehq¯qiP ismˆ -independentforlarge-mˆ andf ∝m−1/2. The ζ Q Q P P transition from the quadratic to the linear mass-relation occurs at mˆ ≈ 2mˆ [9], q s at which point explicit chiral symmetry breaking overwhelms DCSB. 4 Twootherimportantmodel-independentresultscanbeobtained[6]fromEq.(5) and the systematic construction of K. The scalar functions in Eq. (6) depend on threeinvariants;e.g.,E (k;Q)=E (k2,k·Q,Q2). ThezerothChebyshevmoment P P of these functions; e.g., 2 π 0E (k2,Q2):= dx 1−x2E (k;Q), k·Q:=x k2Q2 (13) P P π Z 0 p p are dominant in the description of bound state properties, and 0E (k2,Q2)larg∝e−k2 M (k2), (14) P 0 with0F (k2,Q2),k20G (k2,Q2)andk20H (k2,Q2)behavinginpreciselythesame P P P way. Further k20G (k2,Q2)larg=e−k2 20F (k2,Q2). (15) P P These results determine the asymptotic form of the electromagnetic pion form fac- tor. 3. Electromagnetic Pion Form Factor The impulse approximation to the electromagnetic pion form factor is [10] (p +p ) F (q2):=Λ (p ,p ) (16) 1 2 µ π µ 1 2 Λ =2NctrDZ Γ¯π(k;−p2)S(k++)iΓγµ(k++,k+−)S(k+−)Γπ(k0−;p1)S(k−−), k where Γ¯ (q;−P)T := C−1Γ (−q;−P)C with C = γ γ , the charge conjugation π π 2 4 matrix, and k := k+αp /2+βq/2, p := p +q. No renormalisation constants αβ 1 2 1 appear explicitly in Eq. (16) because the renormalised dressed-quark-photon ver- tex, Γγ, satisfies the vector Ward-Takahashi identity. This also ensures current µ conservation: (p −p ) Λ (p ,p )=0. 1 2 µ µ 1 2 The calculation of F (q2) is simplified by using a model algebraic parametri- π sation of the dressed-quark propagator that efficiently characterises the essential elements of the solution of the quark-DSE and determines the pion Bethe-Salpeter amplitude via Eqs. (8) and (15), and an efficacious Ansatz [11] for Γγ in which the µ vertexis completely determined bythe dressed-quarkpropagator. The resultis de- picted in Fig. 2. The current uncertainty in the experimental data at intermediate q2 is apparentin the lowerpanel,as is the difference betweenthe results calculated with or without the pseudovector components: F, G, of the pion Bethe-Salpeter amplitude. ThesecomponentsprovidethedominantcontributiontoF (q2)atlarge π pion energy [10] because of the multiplicative factors: γ ·Q and γ ·kk·Q, which contribute an additional power of q2 in the numerator of those terms involving F2, FG and G2 relative to those proportionalto E. Including them one finds F (q2)larg∝e−q2 α(q2) (−hq¯qi0q2)2 ; (17) π q2 f4 π 5 1.0 0.8 0.6 2q) F(π 0.4 0.2 0.0 0.0 0.2 0.4 0.6 q2 (GeV2) 0.40 0.30 2V) e G 2q) ( 0.20 F(π 2q 0.10 0.00 0.0 5.0 10.0 15.0 q2 (GeV2) Figure2: Upperpanel: calculatedpionformfactorcomparedwithdataatsmall-q2. Lower panel: the large-q2 comparison, with the two solid lines showing the range of model-dependent uncertainty. In both panels the dashed line [13] assumes that F =0=G =H , and the data are taken from Refs. 12. π π π i.e., q2F (q2) ≈ const., up to calculable lnq2-corrections, in agreement with the π expectations raised by perturbative QCD. If the pseudovector components of Γ π are neglected, the additional numerator factor of q2 is missing and one obtains [13] q4F (q2)≈const.. π 4. γ∗π → γ Transition Form Factor The impulse approximation to this form factor is 1 T3 (k ,k ):= iε k k Tˆ(k2,k ·k ,k2) (18) µν 1 2 4π2 µνρσ 1ρ 2ρ 1 1 2 2 Λ =tr S(q )Γ (qˆ;−P)S(q )iQΓγ(q ,q )S(q )iQΓγ(q ,q ), Z 1 π 2 µ 2 12 12 ν 12 1 q 6 where Q = (1I +τ3)/2 = diag(2/3,−1/3), and k , k are the photon momenta 3 1 2 [on-shell: k2 = 0 = k2, 2k ·k = P2], q is the loop-momentum, and q := q−k , 1 2 1 2 1 1 q :=q+k , qˆ:= 1(q +q ), q :=q−k +k . The manner in whichthe “triangle 2 2 2 1 2 12 1 2 anomaly” is recovered with no dependence on model parameters is described in Ref. [10], and this same mechanism applies to all anomalous pion and photopion processes [14]. It is a unique feature of the DSE framework. Usingthisexpressiononecancalculate[15,16]Tˆ(k2,k ·k ,k2)whenoneorboth 1 1 2 2 ofthephotonsisoffshellandalsodeterminetheasymptoticbehaviouranalytically. The formal character of that derivation is presented in Ref. 16 but, in neglecting essential aspects of renormalisation, it is imprecise. I remedy that here for the illustrative case of one photon off-shell: k2 =Q2. 1 At large-Q2, k ·k ≈ −Q2/2, and in Eq. (18) the pion Bethe-Salpeter ampli- 1 2 tude focuses the integration support at qˆ = 0. As a consequence the asymptotic behaviour of the integral can be determined using 1 1 S(q )≈ iγ·(k −k ), Γγ ≈Z γ , (19) 12 Z Q2 1 2 µ 1 µ 2 which follow from Eq. (2) and the DSE for the quark-photonvertex, so that 1 Λ T3 (k ,k )≈i (k −k ) trZ 1τ3χ (qˆ;−P)γ γ γ , (20) µν 1 2 Q2 1 2 σ 2Z 6 π µ σ ν q where I have used the Ward identity: Z =Z . Hence, the transition form factor 1 2 1 iε k k T(Q2):=T3 (k ,k )+T3 (k ,k ) 4π2 µνρσ 1ρ 2σ µν 1 2 νµ 2 1 1 Λ ≈ − iε (k −k ) 2trZ 1τ3χ (qˆ;−P)γ γ (21) Q2 µνρσ 1 2 ρ 3 2Z 2 π 5 σ q 4 f π = iε k k , (22) µνρσ 1ρ 2σ 3 Q2 where the last line follows from Eq. (7); i.e., T(Q2)larg≈e−Q2 4 4π2fπ , (23) 3 Q2 in agreement with the expectations raised by perturbative QCD. Thus, as with Eq. (16), one equation unifies the small- and large-Q2 results and predicts the evolution between them [15-17]. 5. Epilogue I have been necessarily brief. There are many other applications of interest to this community, among them the diffractive electroproduction of neutral vector 7 mesons [18], the electromagnetic form factors of their charged states [19] and the unificationoflight-andheavy-mesonobservables[8]. Themostpressingcontempo- rary challenge relevant to this community is the extension of the framework to the calculation of baryon observables, which is underway. Acknowledgments I would like to thank Dubravko Klabucar and the organisers for their assistance, kindness and hospitality. This work was supported by the US Department of En- ergy,Nuclear Physics Division, under contractnumber W-31-109-ENG-38,and the National Science Foundation, under grant no. INT-9603385. References 1. C.D. Roberts and A.G. Williams, Prog. Part. Nucl. Phys. 33 (1994) 477. 2. P.C. Tandy, Prog. Part. Nucl. Phys. 39 (1997) 117. 3. C.D. Roberts, “Nonperturbative QCD with modern tools,” nucl-th/9807026. 4. M.R. Pennington, “Calculating hadronic properties in strong QCD,” hep- ph/9611242. 5. R. Alkofer, S. Ahlig and L.v. Smekal, “The infrared behavior of gluon, ghost, and quark propagators in Landau gauge QCD,” hep-ph/9901322, these pro- ceedings. 6. P. Maris and C.D. Roberts, Phys. Rev. C56 (1997) 3369. 7. A. 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M.A. Pichowsky and T.S. Lee, Phys. Rev. D56 (1997) 1644. 19. F.T.HawesandM.A.Pichowsky,“Electromagneticform-factorsoflightvector mesons,” nucl-th/9806025. 8