The exclusive limit of the pion-induced Drell–Yan process

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Physics Letters B

www.elsevier.com/locate/physletb

The exclusive limit of the pion-induced Drell–Yan process

S.V. Goloskokov

^a

, P. Kroll

^b,c,∗

aBogoliubovLaboratoryofTheoreticalPhysics,JointInstituteforNuclearResearch,Dubna141980,Moscowregion,Russia bFachbereichPhysik,UniversitätWuppertal,D-42097Wuppertal,Germany

cInstitutfürTheoretischePhysik,UniversitätRegensburg,D-93040Regensburg,Germany

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received23June2015

Receivedinrevisedform10July2015 Accepted10July2015

Availableonline14July2015 Editor:A.Ringwald

Basedonpreviousstudiesofhardexclusiveleptoproductionofpionsinwhichtheessentialroleofthe pionpoleandthetransversitygeneralizedpartondistributions(GPDs)hasbeenpointedout,wepresent predictionsforthefourpartialcrosssectionsoftheexclusiveDrell–Yanprocess,π⁻p→^l−l+n.

©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Inrecent years hard exclusive leptoproductionof mesons and photonshas beenstudiedintensivelybybothexperimentalistsand theoreticians.Itbecameevidentinthecourseoftimethat within thehandbag approachwhichisbasedonQCDfactorizationinthe generalized Bjorken regime of large photon virtuality and large photon–proton center-of-mass energy but fixed x-Bjorken, it is possibleto interprettheseprocesses intermsof generalizedpar- ton distributions and hard perturbatively calculable subprocesses with, however, occasionally strong power corrections for meson production(forarecentreview see[1]).Exploiting theuniversal- itypropertyofthe GPDs,one mayusetheset ofGPDsextracted frommesonleptoproduction, inthe calculationof other hardex- clusiveprocesses. Of particular interest are processes with time- like virtual photons. Thus in [2] predictions for time-like DVCS (

γ

p→^{l−l+p) havebeengiven, their}experimentalexaminationis stillpending.Thehigh-energypionbeamatJ-PARCputintoopera- tioninthenearfuture,offersthepossibilityofmeasuringanother exclusiveprocesswithtime-likevirtualphotons,namelytheexclu- sivelimitoftheDrell–Yanprocess,

π

⁻p→^{l−l+n.The purposeof} thisletter is to presentpredictions forthe cross sections ofthis process taking into account what has been learned in the anal- yses ofpion leptoproduction[3,4]. The dataon the cross section for

π

⁺ leptoproduction [5,6] demonstrate the prominent role of thecontributionfromthepionpoleatsmallinvariantmomentum transfer,t, andit becameevident that it isto be calculated asa

*

Correspondingauthorat:FachbereichPhysik,Universität Wuppertal,D-42097 Wuppertal,Germany.

E-mailaddresses:goloskkv@theor.jinr.ru(S.V. Goloskokov), kroll@physik.uni-wuppertal.de(P. Kroll).

one-particle-exchange(OPE)termratherthanfromtheGPDE [7].

Inthelattercasethepion-polecontributiontothe

π

⁺crosssec- tion is underestimated by order of magnitude. A second impor- tantobservationhasbeenmadein[3,4]:Theinterpretationofthe transverse target spin asymmetries in

π

⁺ leptoproduction mea- suredby theHERMES Collaboration[8]necessitatescontributions fromtransversely polarizedphotonswhichare to be modeled by transversity GPDs within thehandbag approach.This observation is supported by a recent CLAS measurement of

π

⁰ leptoproduc- tion[9].

Sinceforthe process

π

⁻p→^{l−l+nthe same GPDscontribute} as for pion leptoproduction and the corresponding subprocesses arejustˆ^s↔ ˆ^{ucrossedones1}

H^{π−→γ∗}

(ˆ

s

,

u

ˆ ) = −

H^γ∗→π+

(

u

ˆ , ˆ

s

)

(1) where ˆ^{s andu}ˆ ^{denote thesubprocess Mandelstam variables,one} can exploit the knowledge acquired there. One thus gains pre- dictivepower, thereis nofree parameteror softhadronicmatrix elementleft fortheDrell–Yanprocess.Ouranalysismarkedlydif- fers froma previous studyperformedby Bergeretal.[11] where onlypredictionsforthelongitudinalcrosssection atleading-twist accuracyhave beengiven.Itshouldbestressedthattheirandour predictions for that cross section differ by about a factor of 40 dueto thedifferenttreatment ofthepion polecontribution.Our findings maybe ofhelp in thepreparationof a Drell–Yanexper- iment [12]. Future data on the exclusive pion-induced Drell–Yan process mayreveal whetheror notour presentunderstanding of hardexclusiveprocessesintermsofconvolutionsofGPDsandhard subprocessesalsoholdsfortime-likephotons.Thisisanon-trivial

1 A detaileddiscussionofthespace- andtime-likeconnectionoftheleading-twist amplitudescanbefoundin[10].

http://dx.doi.org/10.1016/j.physletb.2015.07.016

0370-2693/©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

Fig. 1.TheexclusiveDrell–Yanprocess.Thesymbols inbracketsdenotethemo- mentaoftherespectiveparticles.

issue because the physics in the time-like region is complicated and often not understood. Thus, for instance, there is no expla- nation of the time-like electromagnetic form factors of hadrons [13].Eventhesemi-inclusiveDrell–Yanprocesswasdifficulttoun- derstand.It took alongtime beforethe discrepancybetweenthe theoreticalpredictionsandexperiment,knownastheK-factor,has beenexplainedasthresholdlogarithms[14,15]representinggluon radiationresumed tonext-leading-log(NLL)accuracy.

2. Thehandbagapproach

Here,inthissection,werecapitulatethehandbagapproach.For moredetailsofitwerefertoourpreviouswork[3,4].Theprocess

π

⁻p→^{l−l+n is depictedinFig. 1.We work ina} center-of-mass frameinwhichp+^{ppointsalongthepositive3-axisandwecon-} siderthekinematicalrangeoflargeMandelstams(=(p+^q)²)and largephotonvirtuality,² Q²,butsmall

τ =

^Q2

s

−

m²

,

(2)

thetime-likeanalogueofBjorken-x(mbeingthemassofthenu- cleon).Hence,skewness,definedas

ξ =

^p+

−

^p+ p⁺

+

^p+

≈ τ

2

− τ ^,

⁽³⁾

isalsosmall.

Assumingfactorizationwe canexpressthehelicityamplitudes for

π

⁻p→

γ

^∗nin termsof convolutionsofGPDs andhard sub- processamplitudes

M0+,0+

=

1

− ξ

^{2 e0} Q

×

H⁽³⁾

− ξ

²

1

− ξ

²

E⁽_n³_.p⁾_.

+

²

ξ

m 1

− ξ

²

_π

t

−

^m2_π

,

M0−,0+

=

√ −

t 2m

e0

Q

ξ

E⁽_n³_.p⁾_.

−

^2m

_π

t

−

^m2_π

,

M−−,0+

=

1

− ξ

^{2 e0}

Q²

μ

π

^H(_T³⁾

,

M_±+,0+

=

√ −

^t 4m

e₀ Q²

μ

π

¯

^E(_T³⁾

∓

⁸

√

2m²

ξ

π t

−

^m2_π

,

M_+−,0+

≈

⁰

.

(4) Explicithelicitiesarelabeledbytheirsignsorby zero,e0 denotes thepositronchargeandt=^t−^t0 wheret0= −^4m2ξ²/(1−ξ²)is theminimalvaluet correspondingtoforwardscattering.Termsof ordert/Q² areneglectedthroughout.Theamplitudesfornegative helicityoftheinitialstateprotonareobtainedfromthesetofam- plitudes(4)byparityconservation.Theresidueofthepionpoleis givenby

π

= √

2g_{πN N}F_{πN N}

(

t

)

Q²F_π

(

Q²

)

(5)

2 TheQ²-regionsofquarkoniastateshavetobeexcluded.

where gπN N (= ¹³.1±⁰.3) is the familiar pion–nucleon cou- pling constant and FπN N is a form factor that describes the t-dependence ofthe couplingof the virtual pionto the nucleon.

Thepionmass,mπ,isneglectedexceptinthepionpropagator.As we mentioned in the introduction we treat the pion pole asan OPEterm.Thereforethefulltime-likeelectromagneticformfactor occursin(5).CalculatingthepionpolecontributionfromtheGPD E asitisdone in[11],oneobtains thesameexpressionforitbut withthe leading-order (LO)perturbative resultfor thepionform factor. In(4)it isalsoallowed fora possiblenon-pole (n.p.)part ofE.

Forincident

π

⁻mesonsthep→^{ntransitionGPDsarerequired} which, as a consequence of isospin invariance, are given by the isovectorcombinationofprotonGPDs[7]

K⁽³⁾

=

^Ku

−

^Kd

.

(6)

The convolutionsoftheGPDsandtheamplitudesH ^{forthe sub-} process

π

⁻q→

γ

^∗q read[3,4]

^K(3)

=

dxHμλ,0+

(

x

, ξ,

Q²

,

t

⁰

)

K⁽³⁾

(

x

, ξ,

t

) .

(7)

The helicity of the final state quark is λ=

μ

+¹/2 with the photon helicity,

μ

,beingeither zeroor−^{1.Thus, theasymptoti-} callyleadinglongitudinalamplitudeisrelatedtoahelicity-non-flip subprocessamplitude while,fortransversephotons, ahelicity-flip amplitude is convoluted with the transversity GPDs H_T and the combination E¯T =²HT +^ET. As made explicit in (4) the trans- verse amplitudes are suppressed by

μ

_π/Q as compared to the longitudinalones. Themassparameter

μ

π isrelatedtothechiral condensate

μ

π

=

^m2π m_u

+

^md

(8)

(mu,m_d arecurrentquarkmasses).Thesubprocessamplitudesare calculatedtoLOofperturbation theoryretainingquark transverse momenta,k_⊥,andtakingintoaccountSudakovsuppressionswhile the emission and reabsorption of partons by the nucleon hap- pens collinearlyto thenucleonmomenta.Thisso-calledmodified perturbative approach turns intothe leading-twist result[11] for Q²→ ∞^.

SincetheSudakovfactor,exp[−^S]^{,comprisesgluonicradiation,} resumed toallordersofperturbationtheoryinNLLapproximation [16]whichcanonlybeefficientlyperformedintheimpactparam- eter space, canonically conjugatedto the k_⊥-space, one isforced toworkintheb-space.Hence,

Hμλ,0+

=

dzd²b

ˆ

₋λ+

(

z

,−

^b

)

F

ˆ

_μλ,0+

(

x

, ξ,

z

,

Q²

,

b

)

× α

s

( μ

R

)

exp

[−

^S

(

z

,

b

,

Q²

)] .

(9) TheFouriertransformsofthehardscatteringkernelandthelight- cone wavefunction ofthepionaredenoted by Fˆ andˆ, respec- tively.Themomentumfractionofthehelicity+¹/2 quarkentering the pionisdenotedby z;the helicityoftheantiquarkis−λ^.For the renormalization scale we choose

μ

R =^max(z Q, (1−^z)Q, 1/b) andthefactorizationscale is1/b.Following LiandSterman [16]weonlyretainthemostimportantquarktransversemomenta which appear in the denominators of the parton propagators in the hard scatteringkernels. Therefore,we canuse the light-cone projector ofaqq¯ paironan ingoing pionincollinearapproxima- tion[17]

Pπ

=

^fπ 2

√

2N_c

γ

5

√

2

q/

(

z

)

− μ

π

P

(

z

) −

ⁱ

σ

μν 2N_c

_q^μ_n^ν

q

·

ⁿ

σ

(

z

) −

^qμd

σ

(

z

)

dz

∂

∂

k_⊥ν (10) andreplacethe distributionamplitudesby light-cone wave func- tions.In(10) fπ (=^{132 MeV) isthepiondecayconstant, N}c the number of colors and n is a light-like vector which in a frame where the massless pion moves along the z-direction is n= [⁰,1,0_⊥]^. Three-particle configurations,qqg,¯ are neglected. Dirac, flavorandcolorlabelsareomittedforconvenience. Thefirstterm in(10)is thewell-known twist-2part whichis employed inthe calculationofH0±,⁰+.Fortheaccompanyinglight-conewavefunc- tionwetake,asin[3,4],

₋₊

=

√

2N_c

f_π exp

[−

^a2_π^k2_⊥

/(

z

(

1

−

^z

))]

⁽¹¹⁾

withthe transverse size parameter aπ=√ 8

π

fπ

₋1

fixed from

π

⁰→

γ γ

decay[18].Thiswavefunctionhasfrequentlybeenused, see for instance [18,19]. The twist-3 part of (10) is utilized in thecalculationofH−−,⁰+.Sinceforthispartquarkandantiquark formingthe pionhave thesame helicityorbital angular momen- tum,representedby factorsof k_⊥,is required.Asthe calculation revealsH−−,⁰+ isdominatedby thecontributionfromP while thetensortermprovidesacorrectionofordert/Q²whichisne- glected for consistency. For the wave function associated to P (≡^{1 asfollowsfromtheequationofmotion[17,20]),weuseasin} ourpreviouswork³

₊₊

=

¹⁶

π

^3/2

√

2Nc

f_πa³_P

|

^k⊥

|

^exp

[−

^a2Pk²_⊥

] .

(12)

The simple z-independent exponential is forced by the require- mentsofaconstantdistributionamplitudeandthenormalizability of the wave function. For the transverse size parameter, aP, we take1.8 GeV⁻¹.

3. Predictionsforthepartialcrosssections

Before we presentour predictions for the exclusive Drell–Yan processwespecifythevariousparametersandsofthadronicfunc- tionsweuseintheevaluation.Theformfactor FπN N appearingin (5),isparametrizedas

F_πN N

=

²_N

−

m²_π

²_N

−

^{t (13)}

withN=⁰.44±⁰.04 GeV.Forthetime-likepionelectromagnetic formfactoralsooccurring in(5),wetake theaverageofthedata fromCLEO[22]andBaBar[23]aswellasavaluederivedfromthe

J/→

π

⁺

π

⁻decay[24]

Q²

|

^Fπ

(

Q²

)| =

⁰

.

88

±

⁰

.

04 GeV²

.

(14) Foritsphase,exp[ⁱδ(Q²)]^{,werelyonarecentdispersionanalysis} [25]which,for2 GeV^{2Q25 GeV2,provides}

δ =

1

.

014

π +

0

.

195

(

Q²

/

GeV²

−

2

) −

0

.

029

(

Q²

/

GeV²

−

2

)

²

.

(15)

3 Itmayseemappropriatetouseanlz= ±1 wavefunction(foraparticlemoving alongthez-direction).Suchawavefunctionhasbeenproposedin[21].Itispro- portionaltok^±_⊥=k^x_⊥±ik_⊥^y.Itscollinearreductionleadstothetensorpiecein(10) whichgoesalongwithσ anditsderivative;theimportantterm∼P islacking inthisansatz.

Intheabsenceofanyotherinformationonthisphaseweusethis parametrizationup to≈⁸.9 GeV² whereδ=

π

. Forlarger values of Q² wetake δ=

π

,theasymptoticphaseofthetime-likepion formfactor obtainedby analytic continuation ofthe perturbative resultforthespace-likeformfactor[13].

The GPDs are constructed with the help of the familiar dou- ble distribution ansatz from the zero-skewness GPDs which are parametrizedas⁴

K

(

x

, ξ =

⁰

,

t

) =

^k

(

x

)

exp

[

^t

(

b

+ α

lnx

) ]

⁽¹⁶⁾

where the forward limit, k(x), is an appropriate parton distribu- tion(PDF)orisparametrizedlikeaPDFwithparametersfittedto experiment. The GPD H(x,ξ=⁰,t) (including an error estimate) istakenfromtherecentanalysisofthenucleonformfactors[26]

which, for this GPD, is based on the DSSV polarized PDFs [27].

A non-polecontributiontoE isneglected,thereisnoclearsignal foritin thedataonpionleptoproduction. Forthezero-skewness transversity GPDs, H_T and E¯_T, the actual values of the parame- tersarespecifiedin[28].TheyareorientedonlatticeQCDresults [29,30]andleadtofairfitsofthepionleptoproductiondata[5,8,9]

aswell asofthespindensitymatrixelements andtransversetar- getspinasymmetriesforvectormesons[31,28].Theerrorsofthe transversity GPDsareestimatedfromthe p-polefits presentedin [29,30]. Forthemass parameter (8)that controlsthe strength of thetwist-3amplitudes,weadoptthevalue⁵

μ

_π=2 GeV validat the scale 2GeV. Forits errorwe choose +⁰.55 and−⁰.15 [24].

The QCD coupling constant,

α

s, is evaluated from the one-loop expression forfourflavorsandQCD=^{182MeV.Thetime-likeSu-} dakovfactoris unknown,thecontinuation fromthespace-liketo thetime-likeregionisnotwellunderstood(see[32]).Thereplace- ment of Q² by −Q² (see [32,33]) leads to an oscillating phase but it is unclear whether these oscillations are physical or not.

WethereforefollowGoussetandPire[32] andusethespace-like Sudakovfactor,asutilizedinourpreviouswork,alsointhetime- likeregion(with Q²→^{Q2).Asshownin[19],forQ2lessthan} 10GeV² the Sudakov factor isalways closeto unity except near b=¹/QCD where it drops to zero sharply. With the exception of this region the wave function provides the main suppression.

Therefore,thedetailedbehavioroftheSudakov factorisnotvery important.

Thefour-folddifferentialcrosssectionfor

π

⁻p→^l−l+nreads d

σ

dtd Q²dcos

θ

d

φ

=

³ 8

π

sin²

θ

^d

σ

L

dtd Q²

+

¹

+

cos²

θ

2

d

σ

T

dtd Q²

+

^sin

(

2

θ )

cos

φ

√

2

d

σ

LT

dtd Q²

+

^sin2

θ

cos

(

2

φ)

^d

σ

T T

dtd Q²

(17)

wheretheanglesφandθ,specifyingthedirectionsoftheleptons, are definedinFig. 2.The partialcrosssections arerelatedto the

π

⁻p→

γ

^∗nhelicityamplitudes(4)by d

σ

L

dtd Q²

= κ

ν

|

M0ν,0+

|

²

,

d

σ

T

dtd Q²

= κ

μ=±¹,ν

|

Mμν,0+

|

²

,

4 A morecomplicatedprofilefunctionisadoptedforH,see[26].

5 Accordingtotherecentparticledatatables[24]μπ israther2.6 GeV.Using thisvaluethenormalizationsofHT and E¯T havetobealteredaccordinglysince thefittothepionleptoproductiondatafixestheproductofμπandthetransversity GPDs.

Fig. 2.Definition of the anglesφandθ. The latter angle is defined in the rest frame of the virtual photon.

Fig. 3.ThelongitudinalcrosssectionsdσL/dtd Q²(left)atQ²=^{4 GeV2 versust} anddσL/d Q²(right)versusQ².Thethinsolidlineswitherrorbandsrepresent ourfullresultsats=^{20 GeV2,thethickdashedonesthoseat30 GeV2.Thethick} solid(dotted,thindashed)lineistheinterferenceterm(contributionfrom|H⁽³⁾|², leadingtwist).Thelattertworesultsaremultipliedby10fortheeaseoflegibility.

d

σ

LT

dtd Q²

= κ

Re

ν

M^∗_0ν,0+

(

M₊ν,0+

−

M₋ν,0+

) ,

d

σ

T T

dtd Q²

= κ

Re

ν

M^∗_+ν,0+M₋ν,0+

.

(18)

Thenormalizationfactorreads(leptonmassesareneglected)

κ = α

em

48

π

²

1

(

s

−

m²

)

²Q²

.

(19)

In Fig. 3 we show our predictions for d

σ

L/dtd Q² at Q²= 4 GeV² ands=^{20 GeV2 andd}

σ

L/d Q² integratedovert from0 to−⁰.5 GeV².Thelongitudinalcrosssectionisheavilydominated bythe contributionfromthepionpole,that onefrom H,includ- ingitsinterferencewiththepionpole,amountsonlytoabout10%

in the kinematical range of interest. The full result is markedly largerthanourleading-twistresultwhichisofthesameorderas that one quoted in [11]. This amplification is due to the use of theexperimental valueofthepionformfactor(14)insteadofits leading-twistresult(≈⁰.15 GeV²).Westress thattheOPEcontri- butionfromthepionpole doesneither relyonQCD factorization noronahardscattering.Itisthereforenotsubjecttoevolutionand higher-order perturbative QCD corrections. Because of the domi- nantcontribution fromthepion-pole andsincewe onlyconsider asmallrangeof Q² around 4 GeV² theevolutionoftheGPDs is insignificantandisthereforeneglected.Asopposedto[11]ourin- terferencetermispositive.Itis generatedbytheimaginaryparts

Fig. 4.ThetransversecrosssectionsdσT/dtd Q²(left)at Q²=^{4 GeV2versust} and dσT/d Q² (right)versus Q². Thethinsoliddashedlineswith errorbands representthefullresultats=^{20 GeV2,thethickdashedones thoseat30 GeV2} whilethedottedlineisthecontributionfromHT.Thethicksolidlinerepresents thelongitudinal–transverseinterferencecrosssection.

of H ^{and the pion-pole} contribution while, in an LO leading- twistcalculation,itisevidentlyundercontrolofthecorresponding realparts.Constructing H fromthepolarizedPDFsderivedin[34]

insteadfromtheDSSVones[27]altersthepredictionsforthelon- gitudinalcrosssectionbylessthantheestimatederrorsdisplayed inFig. 3.

The transverse cross section is shown in Fig. 4. It is sub- stantially smaller than the longitudinal cross section but much largerthantheleading-twistresult.Theuncertaintyofourpredic- tions isratherlargeandasymmetricduetotheasymmetricerror of

μ

π.Thetransversecrosssectioncanbedecomposedas(cf.(18) and(4))⁶

d

σ

T

dtd Q²

= κ |

M−−,0+

|

²

+

²

|

M++,0+

( π )|

²

+

²

|

M++,0+

(

E

¯

_T

)|

²

.

(20) Thefirsttermin(20),beingrelatedtotheGPDHT,isdisplayed in Fig. 4 separately; it dominates this cross section. The second term, thepion-pole contribution,is rathersmall;itgenerates the littledifferencebetweenthecontribution from H_T andthefullre- sultford

σ

T.ThecontributionfromE¯_T istiny.

6 TheE¯T(π)termbehavesas anatural(unnatural)parityexchangewhiletheHT hasnospecificparitybehavior[3,4].

The longitudinal–transverse interference cross section is also shown in Fig. 4. The width of its error band is about a half of thatofthetransversecrosssection.d

σ

LT canbewrittenas

d

σ

LT

dtd Q²

= κ

Re

2M^∗_0+,0+M₊₊0+

( π ) −

M^∗_0−,0+M_−−,0+

.

(21) Both the terms significantly contribute to d

σ

LT. The transverse–

transverseinterferencecrosssectionisgivenby d

σ

T T

dtd Q²

= κ |

M₊₊,0+

(

_E

¯

_T

)|

²

− |

M₊₊,0+

( π )|

²

.

(22)

Thiscrosssectionisverysmall.Forinstance,atQ²=^{4GeV2 and} s=^{20 GeV2itislessthan}≈⁰.3 pb/GeV⁴.

Thecrosssectionsdecreasewithgrowing s.Asanexamplewe showresultsats=^{30 GeV2 intheplots.At,say,s}≈^{360 GeV2 as} is available from the pion beamat CERN, the longitudinal cross section is about 30 fb/GeV² at Q²=^{4GeV2. This is likely too} smalltobemeasured.

4. Conclusions

WecalculatedthepartialcrosssectionsfortheexclusiveDrell–

Yanprocess,

π

⁻p→^{l−l+n,withinthehandbag approach.Incon-} trast to a previous studyof this process [11] we treat the pion poleasan OPEtermandtakeintoaccounttransversityGPDs.The parametrizationsoftheGPDsH, H_T andE¯_T aswellasthevalues ofotherparametersappearinginthepresentcalculationaretaken frompreviouswork[3,4,26].Thegeneralizationofourapproachto K⁻p→^l−l+isstraightforward.

Futuredata on

π

⁻p→^{l−l+n measured at J-PARC may allow} fora test offactorizationofthe process amplitudesin hard sub- processesandsoftGPDs.Incontrasttopionleptoproductionwhere thereisarigorousproofforfactorizationoftheamplitudesforlon- gitudinallypolarizedphotons,factorizationoftheexclusiveDrell–

Yanprocessisanassumptionalthoughitseemsplausiblethatthe factorizationarguments alsoholdfortime-like photons. However, Qiu[35] conjecturedthatfactorizationmaybebrokenforthe ex- clusiveDrell–Yanprocess. Ifhoweverfactorizationholdstoa suf- ficientdegree of accuracy futuredata on the exclusive Drell–Yan processmayimproveourknowledgeoftheGPDs.

Theexclusive Drell–Yanprocess also offersthe opportunity to check the dependence of the

π π γ

vertex on the pion virtual- ityby comparing data on the time-like form factor measured in l⁺l⁻→

π

⁺

π

⁻ withparametrizationsof

π

⁻

π

^+∗→^{l−l+ aspartof} theDrell–Yananalysis.Theextractionofthespace-likeformfactor fromlp→l

π

⁺ndatamaybenefitfromthatcheck.

Acknowledgements

Oneofus(P.K.)likes tothankMarkus DiehlandOleg Teryaev forusefuldiscussionsandremarks.Theworkissupported inpart

by the Heisenberg–Landau program and by the BMBF, contract number05P12WRFTE.

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefoundon- lineathttp://dx.doi.org/10.1016/j.physletb.2015.07.016.

References

[1]P. Kroll,EPJWebConf.85(2015)01005.

[2]H. Moutarde,B. Pire,F. Sabatie,L. Szymanowski,J. Wagner,Phys.Rev. D87 (5) (2013)054029.

[3]S.V. Goloskokov,P. Kroll,Eur.Phys.J. C65(2010)137.

[4]S.V. Goloskokov,P. Kroll,Eur.Phys.J. A47(2011)112.

[5]A. Airapetian,etal.,HERMESCollaboration,Phys.Lett. B659(2008)486.

[6]H.P. Blok,etal.,JeffersonLabCollaboration,Phys.Rev. C78(2008)045202.

[7]L. Mankiewicz,G. Piller,A. Radyushkin,Eur.Phys.J. C10(1999)307.

[8]A. Airapetian,etal.,HERMESCollaboration,Phys.Lett. B682(2010)345.

[9]I. Bedlinskiy,etal.,CLASCollaboration,Phys.Rev.Lett.109(2012)112001.

[10]D. Mueller,B. Pire,L. Szymanowski,J. Wagner,Phys.Rev. D86(2012)031502.

[11]E.R. Berger,M. Diehl,B. Pire,Phys.Lett. B523(2001)265.

[12] W.C. Chang, Presentation at the J-PARC Workshop in2015, http://research.

kek.jp/group/hadron10/j-parc-hm-2015/.

[13]A.P. Bakulev,A.V. Radyushkin,N.G. Stefanis,Phys.Rev. D62(2000)113001.

[14]G.F. Sterman,Nucl.Phys. B281(1987)310.

[15]S. Catani,L. Trentadue,Nucl.Phys. B327(1989)323.

[16]H.n. Li,G.F. Sterman,Nucl.Phys. B381(1992)129.

[17]M. Beneke,T. Feldmann,Nucl.Phys. B592(2001)3.

[18]S.J. Brodsky,T. Huang,G.P. Lepage,Conf.Proc. C810816(1981)143.

[19]R. Jakob,P. Kroll,Phys.Lett. B315(1993)463;

R. Jakob,P. Kroll,Phys.Lett. B319(1993)545(Erratum).

[20]V.M. Braun,I.E. Filyanov,Z. Phys. C48(1990)239;

V.M. Braun,I.E. Filyanov,Sov.J.Nucl.Phys.52(1990)126;

V.M. Braun,I.E. Filyanov,Yad.Fiz.52(1990)199.

[21]X.d. Ji,J.P. Ma,F. Yuan,Eur.Phys.J. C33(2004)75.

[22]K.K. Seth,S. Dobbs,Z. Metreveli,A. Tomaradze,T. Xiao,G. Bonvicini,Phys.Rev.

Lett.110 (2)(2013)022002.

[23]B. Aubert,etal.,BaBarCollaboration,Phys.Rev.Lett.103(2009)231801.

[24]K.A. Olive,et al.,ParticleDataGroupCollaboration,Chin. Phys. C38(2014) 090001.

[25]M. Belicka, S. Dubnicka, A.Z. Dubnickova, A. Liptaj,Phys. Rev. C 83 (2011) 028201,arXiv:1102.3122[hep-ph].

[26]M. Diehl,P. Kroll,Eur.Phys.J. C73 (4)(2013)2397.

[27]D. deFlorian,R. Sassot,M. Stratmann,W. Vogelsang,Phys.Rev. D80(2009) 034030.

[28]S.V. Goloskokov,P. Kroll,Eur.Phys.J. C74(2014)2725.

[29]M. Gockeler,etal.,QCDSFandUKQCDCollaborations,Phys.Lett. B627(2005) 113.

[30]M. Gockeler,etal.,QCDSFandUKQCDCollaborations,Phys.Rev.Lett.98(2007) 222001.

[31]S.V. Goloskokov,P. Kroll,Eur.Phys.J. A50 (9)(2014)146.

[32]T. Gousset,B. Pire,Phys.Rev. D51(1995)15.

[33]L. Magnea,G.F. Sterman,Phys.Rev. D42(1990)4222.

[34]J. Blümlein,H. Böttcher,Nucl.Phys. B636(2002)225.

[35] J.Qiu,Talkpresentedat the KEKworkshopon HadronPhysicswith High- MomentumHadronBeamsatJPARC,March2015.