Valence-quark distribution functions in the kaon and pion

Valence-quark distribution
functions in the kaon and pion
ChenChen
IFT-UNESP
May 1-7 , 2016
Valence-quark distributionfunctionsinthekaon andpion
ChenChen,Lei Chang,CraigD.Roberts,Shao-longWanandHong-ShiZong
Phys.Rev.D93(2016)074021/1-11
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Partondistributionfunctions(PDFs)
Ø The hadronic tensor relevant to inclusive deep inelastic lepton–pseudoscalar meson
scattering may be expressed in terms of two invariant structure functions. In the
deep-inelastic Bjorken limit, that tensor can be written
Ø F1(x): the structure function
Ø The structure function may be computed from the imaginary part of the virtualphoton–meson forward Compton scattering amplitude:
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Handbag
Ø Thetextbookhandbag contributiontovirtualComptonscattering
Ø InRLtruncation,H(P,k)isaninfinitesumofladder-likerungs
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Ø Overlooked diagrams…amistake!
Ø ForwardlimitofBC.Itexpressesaphotonbeingabsorbedbya
dressed-quark,whichthenproceedstobecomepartofthepion
bound-statebeforere-emittingthephoton.
Ø Thiscontributionisofpreciselythesameorderasthehandbagcontribution.
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Ø Therearesomeoverlooked diagrams…amistake!
Ø Thatcontributionisofpreciselythesameorderasthehandbagcontribution.
Ø Itexpressesaphotonbeingabsorbedbyadressed-quark,whichthen
proceedstobecomepartofthepion bound-statebeforere-emitting
thephoton.
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Ø TheimaginarypartintheBjorken limit.Itproducestheleading
contribution:thevertexinsertioncanappearbetweenanypairof
interactionlines.
Ø ThecompoundvertexiscorrespondtodifferentiationoftheBethe–
Salpeter amplitudeitself.
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Ø Finalexpression
Ø Thestartpoint!
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AlgebraicModel
Ø Propagators:
u Parameters:
Ø Pion andkaon Bethe-Salpeter amplitudes:
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AlgebraicModel
Ø Propagators:
ü Decayconstants
Ø Pion andkaon Bethe-Salpeter amplitudes:
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Pion:chiral limit
Ø Chiral limit:mπ =0
Ø Analyticexpression:
Lei Chang, Cédric Mezrag, Hervé Moutarde, Craig D. Roberts, Jose RodríguezQuintero and Peter C. Tandy, Phys. Lett. B 737 (2014)
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Pion:chiral limit
Ø Chiral limit:mπ =0
Ø Analyticexpression:
Ø QCD-likescalingbehavior:
Ø Thepower-lawpredictedbytheQCDparton model.
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Moments
Ø Ifmπ/K≠0,thePDFscannotbeobtainedanalytically.
Ø FirstwecancomputethePDFs’moments:
Ø ThenuseGegenbauer polynomialstoreconstructthePDFs.
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Moments
Ø Reconstructionprocedureyields:
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PDFs
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PDFs:features
Ø Theratiosareζ-independent:
Ø TheshiftinthepeakofuK(x)awayfromx=0.5,itmeasuresthe
scaleofSU(3)- flavour symmetrybreakingandindicateitsorigin.
Ø Onefindsashiftof17%,thisresultisnearlyidenticaltothe16%
shiftinthepeakoftheleading-twists-quarkparton distribution
amplitude(PDA)inthekaon.Additionally,italmostmatchesthe
relativeshiftindressedquarkmasses,i.e. MswithrespecttoMu.
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Includingsea-quarks&glue
Ø TheRLtruncationincludesnomechanismthatcanshiftmomentum
fromthedressed-quarksintosea-quarksandgluons:aRLmesonis
constitutedsolelyfromadressed-quarkanddressed-antiquark.
Ø ThecorrectionstotheRLtruncationcanbeseparatedintotwo
classes:
[C1]redistributesbaryon-number andmomentum intothedressed-quark sea;
and
[C2]shiftsmomentum intothedressed-gluon distribution.
Ø Inasymmetrypreservingtreatment,correctionsin[C2]haveno
impactonnetbaryonnumberbuttheydorobmomentumfromthe
baryon-number-carryingdressed-partons
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Sea-quarks&gluons
Ø Inasymmetrypreservingtreatment,correctionsin[C2]haveno
impactonnetbaryonnumberbuttheydorobmomentumfromthe
baryon-number-carryingdressed-partons
Ø Thecompletedressed-quarkdistributionfunction:
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Sea-quarks&gluons
Ø Notefirstthatwithrealisticmasses,meson-loopcorrections
totheRLresultforthepion electromagneticformfactorat
Q2=0areanO(5%)effect.
Ø Zπ =0.05.
M.Gluck,E.Reya,I.Schienbein,
Ø πNDrell–Yan:
Eur.Phys.J.C10(1999)313–317.
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Sea-quarks&gluons
Ø Notefirstthatwithrealisticmasses,meson-loopcorrectionstothe
RLresultforthepion electromagneticformfactoratQ2=0arean
O(5%)effect.
Ø Zπ =0.05.
Ø Sea:WesetZK=0,becauseoursubsequentanalysiscannot
reasonablybeexpectedtoexhibitareliablesensitivityto1%
effects.
Ø Glue:Usecg^K (ζH)asaparametertobedeterminedbyfitting
extantDrell-YandataontheratiouK(x)/uπ(x),requiringthatgluons
removethesamefractionofmomentumfromu- ands-quarksin
thekaon,
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Sea-quarks&gluons
Ø BeginwithRLresults,obtainedataparticularscaleζH;thenproceed
systematicallytoaddthecorrectionsfromsea-quarksandgluons;
and,finally,useDGLAPevolutiontoobtaintheresultatanyother
scaleζ>ζH.
Ø GDLAPequationsareonlyvalidontheperturbative domain.
Ø OneshoulduseζH ≥2ΛQCD≈0.5GeV.
R.J. Holt, C.D. Roberts,
Rev. Mod. Phys. 82 (2010) 2991–3044
Ø Itisimpossibletobeginatasmallerscale.
L. Chang, C.D. Roberts, D.J. Wilson,
in: PoS QCD-TNT-II, 2012, p.039.
Ø ζH=0.51GeV →ζ5=5.2GeV
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Sea-quarks&gluons
Ø Theevolveddistributionsmaysatisfactorilybeinterpolatedbythe
followingexpression:
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Ratio:uK(x)/uπ(x)
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Ratio:uK(x)/uπ(x)
Ø TheRatioisindependentofthekaon’s gluon(andsea)contentat
ζH.Thisfeatureoftheratioatx=1isacorollaryofitsζindependence.
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PDFs
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ζ =2GeV
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ζ =2GeV
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Conclusionsandprospects
Ø Thehandbagdiagramusedtodefinethedressed-quarkdistribution
functionisincorrectowingtoomissionofcontributionsfromthe
gluons.
Ø Thepurevalence-quarkdistributionbehavesas(1−x)2onx≈1.
Ø ThecorrectionstotheRLpredictionforthePDFsmaybedivided
intotwoclasses:[C1]:fromdressed-quarksea;and[C2]:from
dressed-gluondistribution.
Ø Webuiltasimplealgebraicmodeltoexpresstheprincipalimpactof
bothclassesofcorrections,which,coupledwiththeRLprediction,
permittedarealisticcomparisonwithexistingexperiment.
Ø Prediction:thegluecontentofkaon ismuchsmallerthanpion at
thesamescale.
Ø Extension:usingarealisticmodeltocalculatePDFs.&Computation
ofafirstrealisticapproximationtotheGPDs.
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Thankyou!
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