A systematic approach to sketch Bethe-Salpeter equation

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by  Si-xue Qin

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0 A systematic approach to sketch Bethe-Salpeter equation
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n
a Si-xueQin1,a
J
1PhysicsDivision,ArgonneNationalLaboratory,Argonne,Illinois60439,USA
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Abstract. To study meson properties, one needs to solve the gap equation for the
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h quarkpropagatorandtheBethe-Salpeter(BS)equationforthemesonwavefunction,self-
t consistently. The gluon propagator, the quark-gluon vertex, and the quark–anti-quark
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l scatteringkernelarekeypiecestosolvethoseequations. Predictedbylattice-QCDand
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Dyson-SchwingeranalysesofQCD’sgaugesector, gluonsarenon-perturbativelymas-
u
n sive. Inthemattersector,themodeledgluonpropagatorwhichcanproduceaveracious
[ descriptionofmesonpropertiesneedstopossessamassscale,accordingly. Solvingthe

well-knownlongitudinalWard-Green-Takahashiidentities(WGTIs)andtheless-known
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transversecounterpartstogether, oneobtainsanontrivialsolutionwhichcanshedlight
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onthestructureofthequark-gluonvertex. Itishighlightedthatthephenomenologically
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proposedanomalouschromomagneticmoment(ACM)vertexoriginatesfromtheQCD
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Lagrangian symmetries and its strength is proportional to the magnitude of dynamical
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3 chiralsymmetrybreaking(DCSB).Thecolor-singletvectorandaxial-vectorWGTIscan
0 relatetheBSkernelandthedressedquark-gluonvertextoeachother.Usingtherelation,
. onecantruncatethegapequationandtheBSequation,systematically,withoutviolating
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crucialsymmetries,e.g.,gaugesymmetryandchiralsymmetry.
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1 Introduction
i
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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
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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]
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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
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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)
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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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