EPJWebofConferenceswillbesetbythepublisher DOI:willbesetbythepublisher (cid:13)c Ownedbytheauthors,publishedbyEDPSciences,2016 6 1 0 A systematic approach to sketch Bethe-Salpeter equation 2 n a Si-xueQin1,a J 1PhysicsDivision,ArgonneNationalLaboratory,Argonne,Illinois60439,USA 3 1 Abstract. To study meson properties, one needs to solve the gap equation for the ] h quarkpropagatorandtheBethe-Salpeter(BS)equationforthemesonwavefunction,self- t consistently. The gluon propagator, the quark-gluon vertex, and the quark–anti-quark - l scatteringkernelarekeypiecestosolvethoseequations. Predictedbylattice-QCDand c Dyson-SchwingeranalysesofQCD’sgaugesector, gluonsarenon-perturbativelymas- u n sive. Inthemattersector,themodeledgluonpropagatorwhichcanproduceaveracious [ descriptionofmesonpropertiesneedstopossessamassscale,accordingly. Solvingthe well-knownlongitudinalWard-Green-Takahashiidentities(WGTIs)andtheless-known 1 transversecounterpartstogether, oneobtainsanontrivialsolutionwhichcanshedlight v onthestructureofthequark-gluonvertex. Itishighlightedthatthephenomenologically 4 proposedanomalouschromomagneticmoment(ACM)vertexoriginatesfromtheQCD 3 Lagrangian symmetries and its strength is proportional to the magnitude of dynamical 1 3 chiralsymmetrybreaking(DCSB).Thecolor-singletvectorandaxial-vectorWGTIscan 0 relatetheBSkernelandthedressedquark-gluonvertextoeachother.Usingtherelation, . onecantruncatethegapequationandtheBSequation,systematically,withoutviolating 1 crucialsymmetries,e.g.,gaugesymmetryandchiralsymmetry. 0 6 1 : v 1 Introduction i X r The visible mass of the universe is mainly contributed by hadrons – bound states of fundamental a blocks,i.e.,quarksandgluons. Theirdynamicsisdescribedbyquantumchromodynamics(QCD)– the strong interaction sector of the Standard Model. QCD has two fascinating features: dynamical chiral symmetry breaking (DCSB) and confinement, which are not apparent in its Lagrangian. It is believedthatDCSBisresponsibleforthegenerationofmassfromnothing(themassofconstituent lightquarksisseveralhundredsMeV),andthustheHiggsmechanismhasalmostnothingtodowith theoriginofthemassofthevisiblematter(themassofcurrentlightquarksisonlyseveralMeV).On theotherhand,thefundamentaldegreeoffreedoms,i.e.,quarksandgluons,areconfinedandcannot directlybedetected. Tounderstandconfinementisoneofthegreatestchallengesinmodernphysics. Neither DCSBnor confinement is understood perturbatively. The keyto the whole storyis to solve QCDnon-perturbatively. In order to solve QCD non-perturbatively, we try to study its equations of motion, i.e., Dyson- Schwingerequations(DSEs)[1]. Specifically,weneedtodealwiththreecategoriesofequations,i.e., theone-bodygapequationforquarks,thetwo-bodyBethe-Salpeter(BS)equationformesons,andthe ae-mail:[email protected] EPJWebofConferences three-bodyFaddeevequationforbaryons. Intherestofthispaper,wewillfocusonthegapequation andtheBSequation,i.e., −1 −1 and = + K(2) , (1) = + wheregraycircularblobsdenotedressedpropagatorsandvertices, K(2) denotesthefullquark–anti- quarkscatteringkernel,andblackdotsdenotebarepropagatorsorvertices.Oncethegluonpropagator, thequark-gluonvertex,andthescatteringkernelarespecified,thetwoequationscanbesolved,and thusmesonpropertiescanbestudiedaccordingly. 2 Gluon propagator InpureSU(3)Yang-Millstheory,theLagrangianisgaugeinvariantandthusdoesnotincludeamass scale. Certainly, for the standard perturbative solution of QCD, the gluon is massless. However, as discussedinRef. [2],thegluoncanbenon-perturbativelymassive. InEuclideanspaceandLandau-gauge,thegluonpropagatorisdecomposedas (cid:32) (cid:33) k k Z(k2) D (k)= δ − µ ν D(k2), D(k2)= , (2) µν µν k2 k2+m2(k2) g whereD(k2)isthedressingfunctionofgluonandhasamassive-typeformaccordingtolatticeQCD [3] or DSE [4] calculations; Z(k2) is the wavefunction renormalization function, and m (k2) is the g runninggluonmass. Inspiredbytheresultinone-loopperturbativeQCD,onecanparameterizeZ(k2) andm (k2)as g Z(k2)=z0lnk2+Λr2Mg2−γ, m2g(k2)= k2M+g4M2 , (3) g whereγisthegluonanomalousdimension;z ,r,Λ,and M arethefittingparameters. Usingsucha 0 g form,latticedatacanbewellfittedwithatypicalgluonmassscaleM ∼700MeV[3]. g Puttingthenon-perturbativelymassivegluonintoconsideration, weproposedarealisticinterac- tionmodel[5]as g2D (k)=k2G(k2)Dfree(k)=[k2G (k2)+α˜ (k2)]Dfree(k), (4) µν µν IR pQCD µν wherein Dfree(k) is the free-gauge-boson propagator, G (k2) is an ansatz which dominates the in- µν IR fraredinteraction,andα˜ (k2)isaregularcontinuationoftheperturbative-QCDrunningcoupling pQCD constant. Explicitly, 8π2 8π2γ F(s) 4πα(s) G(s)= ω4 De−s/ω2 + ln(cid:104)τ+(1+ms/Λ2 )2(cid:105) ≈ s+m2g(s), (5) QCD whereγ =12/25,Λ =0.234GeV;τ=e2−1;F(s)=[1−exp(−s)]/s;andα(s)istheeffective m QCD runningcouplingconstant. Insuchaform, theone-looprenormalization-groupbehaviorofQCDis preserved. The two parameters, i.e., the interaction strength D and the interaction width ω, shape theinfraredinteractionandmayimplicitlyrepresenthigher-orderultravioletcontributions. Withthe product Dω fixed, one can obtain a uniformly good description of pseudoscalar and vector mesons [5]. Inthefavorableparameterspace,thevaluesofM aretypical. Theinfraredvalueofthecoupling g is, typically, α (0)/π ∼ 10 and α (0)/π ∼ 2, where RL stands for the simplest rainbow-ladder RL DB approximationandDBfortheDCSB-improvedapproximation. 21stInternationalConferenceonFew-BodyProblemsinPhysics 3 Quark-gluon vertex In perturbation theory, the quark-gluon vertex can be calculated order by order in loop expan- sion. However, since QCD is non-perturbative, the dynamics dressing effect fundamentally alters the appearance of the vertex. In order to expose the structure of the quark-gluon vertex (or the fermion–gauge-boson vertex in general), we solve the familiar longitudinal WGTI and its less well knowntransversecounterpartstogether,whereweworkintheAbelianapproximationtoavoiddealing withghostfields. Asaconsequenceofgaugeinvariance,thedivergenceofthedressed-fermion–gauge-bosonvertex canberelatedtothedressed-fermionpropagators(q=k−pandS(p)=1/[iγ·pA(p2)+B(p2)]) iq Γ (k,p)=S−1(k)−S−1(p). (6) µ µ CombiningtheLorentztransformationwiththegaugeandchiraltransformations,onecanderive thevectorandaxial-vectortransverseWGTIs(t=k+p),respectively: q Γ (k,p)−q Γ (k,p)=S−1(p)σ +σ S−1(k)+2imΓ (k,p)+t ε ΓA(k,p)+AV (k,p), (7) µ ν ν µ µν µν µν λ λµνρ ρ µν q ΓA(k,p)−q ΓA(k,p)=S−1(p)σ5 −σ5 S−1(k)+t ε Γ (k,p)+VA(k,p), (8) µ ν ν µ µν µν λ λµνρ ρ µν wheremisthefermionmass;tr[γ γ γ γ γ ]=−4ε ;σ =(i/2)[γ ,γ ],σ5 =γ σ ;Γ (k,p)is 5 µ ν ρ σ µνρσ µν µ ν µν 5 µν µν aninhomogeneoustensorvertex;andAV (k,p)andVA(k,p)standsforcontributionsfromhigh-order µν µν Greenfunctions. ThetransverseWGTIsexpresscurlsofthevertices. AfirstobservationforthosetransverseWGTIsisthattheidentitiesfordifferentverticesarecou- pledtogether. However,withthewell-definedprojectiontensors[6] 1 1 T1 = ε t q I , T2 = ε γ q , (9) µν 2 αµνβ α β D µν 2 αµνβ α β onecandecoupletheidentitiesandisolatethetransverseWGTIsforthevectorvertexas (cid:104) (cid:105) q·tt·Γ(k,p) = T1 S−1(p)σ5 −σ5 S−1(k) + t2q·Γ(k,p)+T1 VA(k,p), (10) µν µν µν µν µν (cid:104) (cid:105) q·tγ·Γ(k,p) = T2 S−1(p)σ5 −σ5 S−1(k) + γ·tq·Γ(k,p)+T2 VA(k,p). (11) µν µν µν µν µν Choosing a proper basis for the matrix-valued equations, one can obtain a unique solution for the vector vertex. The solution confirms the longitudinal Ball-Chiu vertex [7] as a precise piece of the fullfermion–gauge-bosonvertexandexposesthetransverseACMterms,e.g., ΓACM5(k,p)=−∆ (k2,p2)σ q , ∆ (k2,p2)=[B(k2)−B(p2)]/[k2−p2], (12) µ B µν ν B whichiscomparablewiththatinRef.[8].Apparently,theACMstrengthisproportionaltothemagni- tudeofDCSB.Thus,itisexpectedthattheACMtermshavesignificantconsequencesinobservables. 4 Scattering kernel It has been shown that, in the bare vertex approximation, the one-gluon exchange form for the BS scatteringkernelfailstoproducemesonspectrumabove1GeV,e.g.,a ,b meson,radialexcitation 1 1 statesofpionandrhomeson[9,10]. Anexactkernelmustbevalidwhenthequark-gluonvertexis fullydressed. ThesolutionsoftheinhomogeneousBSequationsmustsatisfythecolor-singletvectorandaxial- vectorWGTIs(k =k±P),respectively, ± iPµΓµ(k,P) = S−1(k+)−S−1(k−), (13) PµΓ5µ(k,P)+2imΓ5(k,P) = S−1(k+)iγ5+iγ5S−1(k−). (14) EPJWebofConferences The vector WGTI guarantees current conservation, e.g., the pion form factor F (Q2 = 0) = 1; and π theaxial-vectorcounterpartguaranteespion’stwofoldrole: pseudo-scalarbound-stateandGoldstone boson,i.e.,pionmustbemasslessinthechirallimit[11,12]. Ascatteringkernelwhichisusefulfor studyingmesonpropertiesmustpreservesuchidentities. InsertingtheinhomogeneousBSequationsandthequarkgapequationintotheWGTIs,onecan relatethescatteringkerneltothequark-gluonvertexas ∫ K(2)αα(cid:48),β(cid:48)β[S(q−)−S(q+)]α(cid:48)β(cid:48) =∫ Dµν(k−q)γµ[S(q+)Γν(q+,k+)−S(q−)Γν(q−,k−)], (15) q q ∫ K(2)αα(cid:48),β(cid:48)β[S(q+)γ5+γ5S(q−)]α(cid:48)β(cid:48) =∫ Dµν(k−q)γµ[S(q+)Γν(q+,k+)γ5−γ5S(q−)Γν(q−,k−)]. (16) q q Usingtherelationandaproperansatz,onecanconstructthescatteringkernelforaspecifiedvertex, andviceversa. 5 Epilogue This paper explained a systematic approach to sketch BS equation: modeling the gluon propagator, investigating the structure of the quark-gluon vertex, and constructing the quark–anti-quark scatter- ing kernel. Using this approach, the spectrum of light-flavor mesons including ground and radial excitationstatescanbewelldescribed. Moreover,theapproachcanpotentiallybeextendedtomore applications,e.g.,mesonformfactorsandbaryonpropertiesinthediquarkpicture. Acknowledgements Theauthor wouldlike tothank C.D. Robertsfor helpfuldiscussions. The workwas supportedby theOffice oftheDirectoratArgonneNationalLaboratorythroughtheNamedPostdoctoralFellowshipProgram–Maria GoeppertMayerFellowship. 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