adnan bashir michoacán university morelia,...

AdnanAdnan BashirBashir Michoacán Michoacán University University

Morelia, MexicoMorelia, Mexico

KhépaniKhépani Raya, Raya, LeiLei Chang, Javier Cobos,Chang, Javier Cobos, XiomaraXiomara Guerrero, Craig Roberts, Peter Guerrero, Craig Roberts, Peter TandyTandy

Illuminating Illuminating pionspions with photonswith photons

IntroductionIntroduction

•• DynamicalDynamical massmass generationgeneration forfor masslessmassless quarksquarks;; (DCSB)(DCSB).. NoNo degeneracydegeneracy betweenbetween JJP=+P=+ andand JJP=P=--.. •• DynamicalDynamical massmass generationgeneration forfor masslessmassless quarksquarks;; (DCSB)(DCSB).. NoNo degeneracydegeneracy betweenbetween JJP=+P=+ andand JJP=P=--..

•• Color degrees of freedom (quarks and gluons) are notColor degrees of freedom (quarks and gluons) are not observable observable (confinement).(confinement). •• Color degrees of freedom (quarks and gluons) are notColor degrees of freedom (quarks and gluons) are not observable observable (confinement).(confinement).

•• PartonsPartons of of perturbativeperturbative QCD produced at high energyQCD produced at high energy convert into color singlet convert into color singlet hadronshadrons of of non non perturbativeperturbative QCD by the time they reach the detector.QCD by the time they reach the detector.

•• PartonsPartons of of perturbativeperturbative QCD produced at high energyQCD produced at high energy convert into color singlet convert into color singlet hadronshadrons of of non non perturbativeperturbative QCD by the time they reach the detector.QCD by the time they reach the detector.

•• QCDQCD hashas twotwo facetsfacets:: PerturbativePerturbative && nonnon perturbativeperturbative.. PerturbativePerturbative -->> AsymptoticAsymptotic freedomfreedom..

•• QCDQCD hashas twotwo facetsfacets:: PerturbativePerturbative && nonnon perturbativeperturbative.. PerturbativePerturbative -->> AsymptoticAsymptotic freedomfreedom.. •• Non Non perturbativeperturbative QCD has QCD has emergent phenomenaemergent phenomena notnot treatable in perturbation theory.treatable in perturbation theory. •• Non Non perturbativeperturbative QCD has QCD has emergent phenomenaemergent phenomena notnot treatable in perturbation theory.treatable in perturbation theory.

•• Challenge is to understand Challenge is to understand hadronshadrons starting from thestarting from the fundamental degrees of freedom of fundamental degrees of freedom of quarks and gluonsquarks and gluons.. •• Challenge is to understand Challenge is to understand hadronshadrons starting from thestarting from the fundamental degrees of freedom of fundamental degrees of freedom of quarks and gluonsquarks and gluons..

ObservingObserving thethe transitiontransition ofof thethe pionpion (form(form factors,factors, suchsuch asas )) fromfrom quarksquarks andand gluonsgluons toto oneone withwith valencevalence quarksquarks alonealone cancan bebe studiedstudied naturallynaturally throughthrough SDESDE..

ObservingObserving thethe transitiontransition ofof thethe pionpion (form(form factors,factors, suchsuch asas )) fromfrom quarksquarks andand gluonsgluons toto oneone withwith valencevalence quarksquarks alonealone cancan bebe studiedstudied naturallynaturally throughthrough SDESDE..

SchwingerSchwinger--DysonDyson equationsequations areare thethe fundamentalfundamental equationsequations ofof QCDQCD andand combinecombine itsits UVUV andand IRIR behaviourbehaviour.. SchwingerSchwinger--DysonDyson equationsequations areare thethe fundamentalfundamental equationsequations ofof QCDQCD andand combinecombine itsits UVUV andand IRIR behaviourbehaviour..

IntroductionIntroduction

TheThe transitiontransition isis studiedstudied throughthrough processprocess:: TheThe transitiontransition isis studiedstudied throughthrough processprocess::

IntroductionIntroduction IntroductionIntroduction

TheThe transitiontransition formform factorfactor:: TheThe transitiontransition formform factorfactor::

CELLOCELLO H.J. Behrend et.al., Z. Phys C49 401 (1991). 0.7 – 2.2 GeV2

CLEOCLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998). 1.7 – 8.0 GeV2

BaBarBaBar R. Aubert et. al., Phys. Rev. D80 052002 (2009). 4.0 – 40.0 GeV2

TheThe leadingleading twisttwist asympasymp QDCQDC calculationcalculation:: TheThe leadingleading twisttwist asympasymp QDCQDC calculationcalculation::

G.P. G.P. LepageLepage, and S.J. Brodsky,, and S.J. Brodsky, PhysPhys. Rev. D22, 2157 (1980).. Rev. D22, 2157 (1980).



CELLOCELLO H.J. Behrend et.al., Z. Phys C49 401 (1991). 0.7 – 2.2 GeV2

CLEOCLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998). 1.7 – 8.0 GeV2

BaBarBaBar R. Aubert et. al., Phys. Rev. D80 052002 (2009). 4.0 – 40.0 GeV2

BelleBelle S. Uehara et. al., Phys. Rev. D86 092007 (2012). 4.0 – 40.0 GeV2

H.L.L. H.L.L. RobertesRobertes, C.D. Roberts, AB, L.X. , C.D. Roberts, AB, L.X. GutiérrezGutiérrez and P.C. Tandy, and P.C. Tandy, PhysPhys. Rev. C82, . Rev. C82, (065202:1(065202:1--11) 2010.11) 2010.



LowestLowest orderorder inin perturbationperturbation theorytheory andand thethe leadingleading twisttwist asymptoticasymptotic QCDQCD calculationcalculation:: LowestLowest orderorder inin perturbationperturbation theorytheory andand thethe leadingleading twisttwist asymptoticasymptotic QCDQCD calculationcalculation::

G.P. G.P. LepageLepage, and S.J. Brodsky,, and S.J. Brodsky, PhysPhys. Rev. D22, 2157 (1980).. Rev. D22, 2157 (1980).



Collinear factorization:Collinear factorization: Collinear factorization:Collinear factorization:

Transition form factor is the Transition form factor is the correlatorcorrelator of two of two currents:currents: Transition form factor is the Transition form factor is the correlatorcorrelator of two of two currents:currents:

T: hard scattering amplitude with quark gluon subT: hard scattering amplitude with quark gluon sub--processes.processes. T: hard scattering amplitude with quark gluon subT: hard scattering amplitude with quark gluon sub--processes.processes.

is the is the pionpion distribution amplitude:distribution amplitude: is the is the pionpion distribution amplitude:distribution amplitude:


is the is the pionpion distribution amplitude:distribution amplitude: is the is the pionpion distribution amplitude:distribution amplitude:

In asymptotic QCD:In asymptotic QCD: In asymptotic QCD:In asymptotic QCD:


P. Maris and P.C. Tandy., Phys. Rev. C65 045211 (2002)P. Maris and P.C. Tandy., Phys. Rev. C65 045211 (2002)..

BSE:BSE: Integration domain hits singularities at large QIntegration domain hits singularities at large Q22.. BSE:BSE: Integration domain hits singularities at large QIntegration domain hits singularities at large Q22..

•• TheThe triangletriangle diagramdiagram •• TheThe triangletriangle diagramdiagram


•• SchwingerSchwinger--Dyson EquationsDyson Equations •• SchwingerSchwinger--Dyson EquationsDyson Equations

Quark Propagator: Quark Mass Function Quark Propagator: Quark Mass Function Quark Propagator: Quark Mass Function Quark Propagator: Quark Mass Function

QuarkQuark--Photon VertexPhoton Vertex QuarkQuark--Photon VertexPhoton Vertex

PionPion BetheBethe--SalpeterSalpeter Amplitude Amplitude PionPion BetheBethe--SalpeterSalpeter Amplitude Amplitude

Quark propagator:Quark propagator: Quark propagator:Quark propagator:

The Quark PropagatorThe Quark Propagator The Quark PropagatorThe Quark Propagator


QuarkQuark propagatorpropagator QuarkQuark propagatorpropagator

Quark Quark propagator:propagator:

•• TheThe gluongluon propagatorpropagator andand thethe quarkquark--gluongluon vertexvertex areare directlydirectly responsibleresponsible forfor thethe quarkquark gapgap equationequation toto havehave infraredinfrared enhancedenhanced kernelkernel..

•• TheThe gluongluon propagatorpropagator andand thethe quarkquark--gluongluon vertexvertex areare directlydirectly responsibleresponsible forfor thethe quarkquark gapgap equationequation toto havehave infraredinfrared enhancedenhanced kernelkernel..

Several SDE and lattice Several SDE and lattice results support decoupling results support decoupling solution for the gluon solution for the gluon propagator.propagator.

Several SDE and lattice Several SDE and lattice results support decoupling results support decoupling solution for the gluon solution for the gluon propagator.propagator.

Momentum dependent gluon mass is reminiscent of the Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass function.momentum dependent quark mass function. Momentum dependent gluon mass is reminiscent of the Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass function.momentum dependent quark mass function. It is also in accord with the revised GZIt is also in accord with the revised GZ--picture.picture. It is also in accord with the revised GZIt is also in accord with the revised GZ--picture.picture.

The Gluon Propagator:The Gluon Propagator: The Gluon Propagator:The Gluon Propagator:



Phenomenology

Gauge

Covariance Lattice

Multiplicative

Renormalization

Perturbation

Theory

Quark-photon/

quark-gluon

vertex

The QuarkThe Quark--Gluon Vertex:Gluon Vertex: The QuarkThe Quark--Gluon Vertex:Gluon Vertex:

•• ThusThus DCSBDCSB manifestsmanifests itselfitself notnot onlyonly inin thethe quarkquark propagatorpropagator butbut alsoalso thethe quarkquark--gluongluon vertexvertex.. •• ThusThus DCSBDCSB manifestsmanifests itselfitself notnot onlyonly inin thethe quarkquark propagatorpropagator butbut alsoalso thethe quarkquark--gluongluon vertexvertex..

•• QuarkQuark gluongluon vertexvertex consistsconsists ofof 1212 linearlylinearly independentindependent DiracDirac structuresstructures.. •• QuarkQuark gluongluon vertexvertex consistsconsists ofof 1212 linearlylinearly independentindependent DiracDirac structuresstructures..

•• 55 ofof thesethese 1212 structuresstructures areare generatedgenerated dynamicallydynamically inin thethe chiralchiral limitlimit.. •• 55 ofof thesethese 1212 structuresstructures areare generatedgenerated dynamicallydynamically inin thethe chiralchiral limitlimit..

•• TheThe quarkquark--gluongluon vertexvertex cancan bebe expandedexpanded inin termsterms ofof thethe thethe followingfollowing quantitiesquantities.. •• TheThe quarkquark--gluongluon vertexvertex cancan bebe expandedexpanded inin termsterms ofof thethe thethe followingfollowing quantitiesquantities..

The Quark PropagatorThe Quark Propagator The Quark PropagatorThe Quark Propagator The QuarkThe Quark--Gluon Vertex:Gluon Vertex: The QuarkThe Quark--Gluon Vertex:Gluon Vertex:

•• TheThe rainbowrainbow ladderladder truncationtruncation andand thethe BallBall--ChiuChiu verticesvertices describedescribe ππ andand thethe ρρ mesonsmesons wellwell butbut failfail toto dodo soso forfor theirtheir parityparity partnerspartners σσ andand aa11..

•• TheThe rainbowrainbow ladderladder truncationtruncation andand thethe BallBall--ChiuChiu verticesvertices describedescribe ππ andand thethe ρρ mesonsmesons wellwell butbut failfail toto dodo soso forfor theirtheir parityparity partnerspartners σσ andand aa11..

The QuarkThe Quark--Gluon VGluon Vertexertex

•• TheThe nonnon perturbativeperturbative detailsdetails ofof thethe vertexvertex explainexplain thethe massmass splittingsplitting ofof thethe parityparity partnerpartner mesonsmesons throughthrough thethe generationgeneration ofof quarkquark anomalousanomalous chromochromo--magneticmagnetic momentmoment..

•• TheThe nonnon perturbativeperturbative detailsdetails ofof thethe vertexvertex explainexplain thethe massmass splittingsplitting ofof thethe parityparity partnerpartner mesonsmesons throughthrough thethe generationgeneration ofof quarkquark anomalousanomalous chromochromo--magneticmagnetic momentmoment..

•• TheThe correctionscorrections cancelcancel inin thethe pseudoscalarpseudoscalar andand vectorvector channelschannels butbut addadd inin thethe scalarscalar andand axialaxial vectorvector channelschannels.. •• TheThe correctionscorrections cancelcancel inin thethe pseudoscalarpseudoscalar andand vectorvector channelschannels butbut addadd inin thethe scalarscalar andand axialaxial vectorvector channelschannels..

L. Chang, C.D. Roberts, Phys. Rev. L. Chang, C.D. Roberts, Phys. Rev. LettLett. 103 081601 (2009) . 103 081601 (2009)

The QuarkThe Quark--Gluon Vertex:Gluon Vertex: The QuarkThe Quark--Gluon Vertex:Gluon Vertex:

JJ.. SkullerudSkullerud,, PP.. Bowman,Bowman, AA.. KizilersuKizilersu,, DD.. LeinweberLeinweber,, AA.. Williams,Williams, JJ.. HighHigh EnergyEnergy PhysPhys.. 0404 047047 ((20032003))

MM.. BhagwatBhagwat,, MM.. PichowskyPichowsky,, CC.. Roberts,Roberts, PP.. Tandy,Tandy, PhysPhys.. RevRev.. CC6868 015203015203 ((20032003))..

AB,AB, LL.. GutiérrezGutiérrez,, MM.. TejedaTejeda,, AIPAIP ConfConf.. ProcProc.. 10261026 262262 ((20082008))..

The QuarkThe Quark--GluonGluon Vertex:Vertex: One of the 12 One of the 12 form factorsform factors

The QuarkThe Quark--GluonGluon Vertex:Vertex: One of the 12 One of the 12 form factorsform factors



Complex conjugate Complex conjugate pole representation.pole representation. Complex conjugate Complex conjugate pole representation.pole representation.

BetheBethe--SalpeterSalpeter amplitudeamplitude forfor thethe pionpion:: BetheBethe--SalpeterSalpeter amplitudeamplitude forfor thethe pionpion::

GoldbergerGoldberger--TriemannTriemann relations:relations: GoldbergerGoldberger--TriemannTriemann relations:relations:

The BetheThe Bethe--SalpeterSalpeter AmplitudesAmplitudes The BetheThe Bethe--SalpeterSalpeter AmplitudesAmplitudes

The BetheThe Bethe--SalpeterSalpeter AmplitudesAmplitudes The BetheThe Bethe--SalpeterSalpeter AmplitudesAmplitudes

ParametricParametric formform ofof thethe BSAsBSAs.. ParametricParametric formform ofof thethe BSAsBSAs..

Nakanishi, Phys. Rev. 130 Nakanishi, Phys. Rev. 130 1230 (1963).1230 (1963).

L. Chang et. al., Phys. Rev. L. Chang et. al., Phys. Rev. LettLett. 110 132001 (2013). 110 132001 (2013)..

AB, M.R. Pennington Phys. Rev. D50 7679 (1994) AB, M.R. Pennington Phys. Rev. D50 7679 (1994)

D.C. Curtis and M.R. Pennington Phys. Rev. D42 4165 (1990)D.C. Curtis and M.R. Pennington Phys. Rev. D42 4165 (1990)

A. A. KizilersuKizilersu and M.R. Pennington Phys. Rev. D79 125020 (2009)and M.R. Pennington Phys. Rev. D79 125020 (2009)

L. Chang, C.D. Roberts, Phys. Rev. L. Chang, C.D. Roberts, Phys. Rev. LettLett. 103 081601 (2009) . 103 081601 (2009)

AB, R. Bermudez, L. Chang, C.D. Roberts, AB, R. Bermudez, L. Chang, C.D. Roberts, PhysPhys. Rev. C85 045205 (2012). Rev. C85 045205 (2012)..

The QuarkThe Quark--Photon VertexPhoton Vertex The QuarkThe Quark--Photon VertexPhoton Vertex

To conserve current conservation and Ward identities, aTo conserve current conservation and Ward identities, a proper proper quarkquark--photon vertexphoton vertex is essential.is essential.

The quarkThe quark--photon vertex must also take into account the photon vertex must also take into account the separation of scales involved. (Gauge Technique).separation of scales involved. (Gauge Technique).

R. R. DelbourgoDelbourgo, A. Salam, Phys. Rev. 135 1398 (1964)., A. Salam, Phys. Rev. 135 1398 (1964).

It must ensure It must ensure chiralchiral anomaly at zero momentum transfer.anomaly at zero momentum transfer.

The QuarkThe Quark--Photon VertexPhoton Vertex The QuarkThe Quark--Photon VertexPhoton Vertex

The vertex used satisfies the longitudinal WardThe vertex used satisfies the longitudinal Ward--GreenGreen--Takahashi (WGT) identity.Takahashi (WGT) identity.

It is free of kinematic singularities.It is free of kinematic singularities.

It reduces to the bare vertex in the freeIt reduces to the bare vertex in the free--field limit.field limit.

It has the same Poincare transformation properties as the It has the same Poincare transformation properties as the bare vertex.bare vertex.

This This AnsatzAnsatz has the additional merit that it reproduces the has the additional merit that it reproduces the correct ultraviolet limit of the photoncorrect ultraviolet limit of the photon--quark interaction quark interaction when one or both photons is hard.when one or both photons is hard.

PionPion Distribution AmplitudeDistribution Amplitude

PionPion Distribution AmplitudeDistribution Amplitude

This can be projected onto the This can be projected onto the GegenbauerGegenbauer 3/2 basis.3/2 basis.

One now evolves it to different scales and builds thisOne now evolves it to different scales and builds this evolution into the BSA amplitudes.evolution into the BSA amplitudes.

PionPion Form FactorsForm Factors TheThe quarkquark propagatorpropagator (NCCP),(NCCP), electronelectron--photonphoton vertexvertex (WTI(WTI--PT),PT), thethe BetheBethe SalpeterSalpeter AmplitudeAmplitude (Nakanishi)(Nakanishi) andand PDAPDA evolutionevolution provideprovide thethe ingredientsingredients forfor thethe pionpion formform factorfactor calculationscalculations..

TheThe quarkquark propagatorpropagator (NCCP),(NCCP), electronelectron--photonphoton vertexvertex (WTI(WTI--PT),PT), thethe BetheBethe SalpeterSalpeter AmplitudeAmplitude (Nakanishi)(Nakanishi) andand PDAPDA evolutionevolution provideprovide thethe ingredientsingredients forfor thethe pionpion formform factorfactor calculationscalculations..

PionPion Electromagnetic Form FactorElectromagnetic Form Factor

γγ**ππ00γ Transition Form Factorγ Transition Form Factor


•• BelleBelle IIII willwill havehave 4040 timestimes moremore luminosityluminosity.. •• BelleBelle IIII willwill havehave 4040 timestimes moremore luminosityluminosity..

PrecisePrecise measurementsmeasurements atat largelarge QQ22 willwill provideprovide aa stringentstringent constraintconstraint onon thethe patternpattern ofof chiralchiral symmetrysymmetry breakingbreaking.. PrecisePrecise measurementsmeasurements atat largelarge QQ22 willwill provideprovide aa stringentstringent constraintconstraint onon thethe patternpattern ofof chiralchiral symmetrysymmetry breakingbreaking..

Vladimir Vladimir SavinovSavinov:: 55thth Workshop of the APSWorkshop of the APS Topical Group on Topical Group on HadronicHadronic Physics, 2013.Physics, 2013.

Vladimir Vladimir SavinovSavinov:: 55thth Workshop of the APSWorkshop of the APS Topical Group on Topical Group on HadronicHadronic Physics, 2013.Physics, 2013.

γγ**ππ00γ Transition Form Factorγ Transition Form Factor

ConclusionsConclusions Dynamical Dynamical chiralchiral symmetry breaking and the symmetry breaking and the momentum momentum dependence of the quark mass function in QCD have dependence of the quark mass function in QCD have experimental signalsexperimental signals which enable us to differentiate SDE which enable us to differentiate SDE predictions from others.predictions from others.

Dynamical Dynamical chiralchiral symmetry breaking and the symmetry breaking and the momentum momentum dependence of the quark mass function in QCD have dependence of the quark mass function in QCD have experimental signalsexperimental signals which enable us to differentiate SDE which enable us to differentiate SDE predictions from others.predictions from others.

In a fully consistent treatment of In a fully consistent treatment of pionpion in SDE QCD, the in SDE QCD, the asymptotic QCD limit of the product Qasymptotic QCD limit of the product Q22G(QG(Q22) for ) for γγ**ππ00γγ, , seems to approach the asymptotic QCD results from below.seems to approach the asymptotic QCD results from below.

In a fully consistent treatment of In a fully consistent treatment of pionpion in SDE QCD, the in SDE QCD, the asymptotic QCD limit of the product Qasymptotic QCD limit of the product Q22G(QG(Q22) for ) for γγ**ππ00γγ, , seems to approach the asymptotic QCD results from below.seems to approach the asymptotic QCD results from below.

The large QThe large Q22 evolution of the form factors through SDEs evolution of the form factors through SDEs provides us with provides us with a unified description of the physics from a unified description of the physics from purely non purely non perturbativeperturbative to asymptotic regionto asymptotic region and shouldand should be systematically extended to all other form factors of be systematically extended to all other form factors of interest such as interest such as NN--> N*(1535)> N*(1535)..

The large QThe large Q22 evolution of the form factors through SDEs evolution of the form factors through SDEs provides us with provides us with a unified description of the physics from a unified description of the physics from purely non purely non perturbativeperturbative to asymptotic regionto asymptotic region and shouldand should be systematically extended to all other form factors of be systematically extended to all other form factors of interest such as interest such as NN--> N*(1535)> N*(1535)..

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