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Double Parton Distributions of the Pion 7 1 Christian Zimmermann∗† 0 UniversityofRegensburg 2 E-mail: [email protected] n a J The effects of double hard interactions are no longer negligible at energy scales reached at the 9 1 LHC.Doublepartonscattering(DPS)processesareoftendescribedbytakingtheproductoftwo singlepartonscatteringprocessesassumingthatinterferenceeffectsareverysmall. Wecalculate ] t fourpointfunctions(4pt-functions),whichappearinthetheDPScrosssection,employinglattice a l techniques. Weconsiderapionatrestandtestthevalidityoftheafore-mentionedfactorization - p assumption by convoluting two pion form factors and comparing the result to the 4pt data. For e h our calculations we use a N =2 gauge ensemble on a 403×64 lattice, with lattice spacing f [ a=0.071fmandpionmassmπ =288.8MeV. 1 v 9 7 4 5 0 . 1 0 7 1 : v i X r a 34thannualInternationalSymposiumonLatticeFieldTheory 24-30July2016 UniversityofSouthampton,UK ∗Speaker. †forRQCD (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/ DPDsofthePion ChristianZimmermann 1. Introduction Hadron-hadron collisions are an important subject when discovering new physics at high en- ergyscales. Thereforethecorrectdescriptionoftheunderlyingeventintheseinteractionsisessen- tial. At energy scales reached in LHC, multi parton interactions (MPIs) are no longer negligible. The most important MPI processes are double parton scattering (DPS) processes, which are often naivelydescribedbyneglectinganyinterferenceeffects: dσ dσ SPS SPS dσ = , (1.1) DPS Cσ eff whereσ isthesinglepartonscattering(SPS)crosssection,andσ issomeeffectivecrosssec- SPS eff tion,whichisrelatedwiththehadronsize.C=2or1denotessomesymmetryfactor[1]. A more fundamental description can be done by introducing so-called double parton distributions (DPDs), which involve two-operator matrix elements. Our intention is to obtain these matrix ele- mentsfromcalculationsof4pt-functionsonthelattice,startingwiththepionatrest. Wealsowant tochecktowhatextentthe"naivefactorization"ansatz(1.1)isvalidbycomparingtheconvolution oftwoformfactorswithourlatticeresultsforthe4pt-functions. 2. DoublePartonScatteringandDoublePartonDistributions Forthedescriptionofadoublepartonscattering(DPS)process,itisusuallyassumedthatthe processfactorizesintoasoftandahardpart. Inthatcase,onecanwritethecrosssectionas(inthe followinglightconecoordinatesareused): dσ σ σ (cid:90) = ∑ 1 2 d2y F(x,y )F(x¯,y ). dx dx¯ dx dx¯ C ⊥ i ⊥ i ⊥ (2.1) 1 1 2 2 polarization flavor A derivation from first principles is given in [2]. In the equation above the σ denote the parton i scattering cross sections. The variable y can be interpreted as the distance between the scattering partons. F(x,y ) are the so-called collinear double parton distributions (DPDs), which involve i ⊥ hadronicmatrixelementsoftwolightconeoperators. ThesecontainWilsonlines,onceweinclude higherordercontributions. This is different for Mellin moments of DPDs, where the two parton momentum fractions are integrated out, which has the consequence that the operators become local and therefore Wilson linesdonotappearanymore. Latticecalculationsbecomemorefeasibleinthiscase. Explicitlyone findsforthelowestmoment: (p+)−1(cid:90) Mp(y)= dy−Mp(y) (2.2) 2 (cid:12) Mp(y)= (cid:104)h(p)|Of1f1(cid:48)(0)Of2f2(cid:48)(y)|h(p)(cid:105)(cid:12) (2.3) 1 2 (cid:12) y+=0 Off(cid:48)(y)=q¯f(y)Γqf(cid:48)(y), (2.4) 1,2 where h(p) denotes a specific hadron with momentum p. Γ is a combination of Dirac matrices, whichdependsonthepolarizationofthequarkstakingpartontheinteraction. Wewillonlydiscuss 1 DPDsofthePion ChristianZimmermann thevectorandaxialvectorcaseandfurthermorethescalarandpseudoscalarchannels,althoughthe lattertwocorrespondtohighertwistcontributions. In the following we will use decompositions of the matrix elements w.r.t. their Lorentz structure intoinvariantfunctions,e.g.: Mp (y)=2m2A (py,y2) (2.5) SS/PP SS/PP (cid:104) (cid:105) T Mp,{µν}(y)=(cid:2)2pµpν−1gµνp2(cid:3)A (py,y2)+m2 2p{µyν}−1gµνpy B (py,y2) VV 2 VV 2 VV (2.6) +m4(cid:2)2yµyν−1gµνy2(cid:3)C (py,y2), 2 VV where T denotes trace subtraction and the {}-notation indicates symmetrization w.r.t. Lorentz indices. Asmentionedbeforeweintendtocheckthevalidityofthefactorizationassumption(1.1), i.e.the factorization of a two-operator matrix element into two one-operator matrix elements. It seems naturalthatasuitablewayistoinsertacompletesetofintermediatestates,assumingthatstatesof theobservedhadrondominate: (cid:90) d4p(cid:48) (cid:104)h(p)|O O |h(p)(cid:105)≈ (cid:104)h(p)|O |h(p(cid:48))(cid:105)(cid:104)h(p(cid:48))|O |h(p)(cid:105)δ(p(cid:48)2−m2). (2.7) 1 2 (2π)4 1 2 Thisansatzonecanapplytolightconematrixelementsaswellastomatrixelementsoflocaloper- ators. WewillcheckbothpossibilitiesinSection4. 3. DPDsfromLatticeStudies C1 A C2 π+(0) O1(τ,0) π+(t) π+(0) O1(τ,0) π+(t) π+(0) O1(τ,0) π+(t) O (τ,y) O (τ,y) 2 O (τ,y) 2 2 D S1 S2 O (τ,0) 1 π+(0) O (τ,0) π+(t) 1 π+(0) O (τ,0) π+(t) 1 π+(0) O (τ,y) π+(t) 2 O (τ,y) 2 O (τ,y) 2 Figure1: DepictionofthesixpossibleindependentWickcontractions. We are able to obtain some information about the matrix elements (2.3) by calculating 4pt- functions on the lattice. We start with the calculation for the pion with zero momentum taking only channels containing no derivative terms corresponding to the first Mellin moment. These 2 DPDsofthePion ChristianZimmermann Ensemble β a[fm] κ V m [GeV] N(N ) N π 4pt sm I 5.29 0.071 0.13632 403×64 0.2888(11) 2025(984) 400 Table1: Detailsoftheensembleusedforthesimulation. N labelsthenumberofconfigurations, which 4pt areusedforour4pt-calculations. N denotesthenumberofsmearingiterations. sm S P V A T Z 0.4577(18) 0.3538(92) 0.7365(48) 0.76487(64) 0.9141(26) Z 1.3543 1.3543 1 1 0.93313 conv Table2: RenormalizationfactorsZandfactorsZ fortheconversiontotheMS-scheme. S,P,V,AandT conv denotethedifferentoperatorinsertiontypes. matrixelementscanbeobtainedbycalculatingthefollowingratio,inalimit,whereexcitationsare suppressed(thisisanalogoustothecalculationofone-operatormatrixelements,seee.g.[3]): (cid:12) M(y)= 2mC4pt(t,τ,y)(cid:12)(cid:12) , (3.1) C (t) (cid:12) 2pt 0(cid:28)τ(cid:28)t whereC (t)istheusualpion2pt-functionandC (t,τ,y)denotesthe4pt-function: 2pt 4pt C (t,τ,y)=(cid:104)O (t)O (0,τ)O (y,τ)O† (0)(cid:105) (3.2) 4pt π+ 1 2 π+ C (t)=(cid:104)O (t)O† (0)(cid:105). (3.3) 2pt π+ π+ O† (t)denotesthepioninterpolator,whereweuse: π+ 1 O† (t)= ∑u¯(x,t)γ d(x,t). (3.4) π+ V 5 x For the calculation of the 4pt-function, we have to consider six independent Wick contractions, which are depicted in Figure 1. The two operator insertions are placed at the same time slice, whichhasatimeseparationτ tothepionsource. Forthetimeseparationbetweensourceandsink we chooset =15a. Hence we expect a plateau in the region 6(cid:46)τ (cid:46)9. For the evaluation of the 4pt-graphsweuseanensembleofN =2gaugeconfigurations(seeTable1, sameasensembleV f in [4]). The gauge fields are smoothed using APE smearing [5]. For our calculations we use the One-End-Trick incorporating stochastic Z ⊗Z -sources [6], which are improved by Wuppertal 2 2 smearing [7]. We renormalize our results and convert it to the MS-scheme at a scale µ =2GeV usingtherenormalizationandconversionfactorslistedinTable2(see[8]). ForthegraphsC1,C2,S1andAweobtainclearnon-zerosignals,whileforthegraphsS2and Dthesignalsareverynoisy. ForsomematrixelementstheseresultsareshowninFigure2. Weare able to project out the invariant functions appearing in (2.5) and (2.6), which are not shown here (theC1dataofthescalarandvectorchannelsisshowninFigure3comparedwiththeformfactor convolutions). Notice that disconnected graphs are less suppressed in the (pseudo)scalar channel thaninthe(axial)vectorchannel. 3 DPDsofthePion ChristianZimmermann Resultsfor𝑀SS(𝐿=40) ResultRsefsourlt𝑀sfSoSr(𝑀logPsPca(l𝐿e,=𝐿4=0)40) Resultsfor𝑀PP(logscale,𝐿=40) 20 15 S1 S1 4−2(|𝑦|)[10GeV]S −1150505 4𝑀(|𝑦|)|[GeV]SS4−2(|𝑦|)[10GeV]P111000−−−−13215005 CCA12 4𝑀(|𝑦|)|[GeV]PP 111000−−−321 CCA12 𝑀S−10 CS11 |𝑀P−10 CS11 | C2 C2 −15 A 10−−145 A 10−4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.02.2 0.04.4 0.06.6 0.08.8 11 1.12.2 1.14.4 1.16.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 |𝑦|[fm] |𝑦||𝑦[|f[mfm]] |𝑦|[fm] 2 Resultsfor𝑀V0V0(𝐿=40) 10−12 ResultRsefosrul𝑀tsVfo0Vr0𝑀(lAo0gAs0ca(l𝐿e,=𝐿4=0)40) 10−1 Resultsfor𝑀A0A0(logscale,𝐿=40) S1 S1 4eV] 1.15 4V]4eV]10−12.5 CCA21 4V] 10−2 CCA21 G eG 1 e −2(|𝑦|)[10V0−00..055 (|𝑦|)|[GVV−2(|𝑦|)[1000A01100−−043.05 (|𝑦|)|[GAA001100−−43 𝑀V0 −1 CS21 |𝑀𝑀A010−−05.5 CS21 |𝑀 10−5 C1 C1 −1.5 A 10−−61 A 10−6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.02.20.04.40.06.60.08.8 11 1.12.21.14.41.16.6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 |𝑦|[fm] |𝑦||𝑦[f|m[fm] ] |𝑦|[fm] Figure 2: Results for two-operator matrix elements for the channels SS, PP, V V and A A . Here the 0 0 0 0 resultsofthegraphsC1,C2,S1andAareshownseparately. 4. NaiveFactorization Finallywewanttocheckthe"naivefactorization"intoone-operatormatrixelements. Apply- ing(2.7)tomatrixelementsoflightconeoperatorsyieldsthefollowingrelation(test1): η (cid:90) 1 (cid:18) ζ(cid:19)2 (cid:90) d2r A (py=0,y2)≈ C dζ 1− (1−ζ)−1 ⊥ e−iy⊥·r⊥F (t(ζ,r )), (4.1) VV/SS π 2 (2π)2 V/S ⊥ 0 where t(ζ,r ) = −(ζ2m2+r2)/(1−ζ) and η = 1 for the scalar case and −1 for the vector ⊥ ⊥ C case. Ontheotherhand(2.7)maybeinsertedinamatrixelementoflocaloperators,whichcanbe broughtintothefollowingform(test2): (cid:90) ∞ d(r2) sin(|y||r|)(cid:0)m+E(r2)(cid:1)2 (cid:104)π+(p)|Ouu(0)Odd(y)|π+(p)(cid:105)≈− F2(t(r2)) V0 V0 4π2|y| 2E(r2) V 0 (4.2) (cid:90) ∞ d(r2) sin(|y||r|) (cid:104)π+(p)|Ouu(0)Odd(y)|π+(p)(cid:105)≈ F2(t(r2)), S S 4π2|y| 2E(r2) S 0 √ whereE(r)= r2+m2 andt(r)=2m2−2mE(r). Bothtestsinvolvetheelectromagnetic(F (t)) V or scalar form factor (F (t)) of the pion, respectively. These we obtain from 3pt-functions on the S lattice. Hereweconsiderforthemomentonlyconnectedgraphsandusecontributionsofmomenta, √ forwhich|p|≤ 2π 3. Noticethatbothteststriviallyfailforthepseudoscalaroraxialvectorcase, L sincethecorrespondingformfactorsareexactlyzeroforsymmetryreasons. Thisis,however,not thecasefortwo-operatormatrixelements(seeSection3). 4 DPDsofthePion ChristianZimmermann # quantity F M[GeV] p χ2/DOF 0 1 F 1(fixed) 0.777(12) 1(fixed) 6.010 em 2 1(fixed) 0.872(16) 1.173(69) 4.400 3 F 2.222(19)GeV 1.314(39) 1(fixed) 7.886 scal 4 2.212(19)GeV 2.023(50) 2(fixed) 9.877 Table3:Resultsforthefitsontheelectromagneticorscalarpionformfactor. Allfitstakeintoaccounterror correlations. Thelargestcontributingvalueof|t|inthefitis1.143GeV2. To perform the integrals in (4.1) and (4.2), we use the following parametrization of the pion form factor: (cid:16) t (cid:17)−p F(t)=F 1+ . (4.3) 0 M2 The parameters F and M may be obtained from a covariant fit on the 3pt data (in the el. m. case 0 F wasfixedto1accordingtochargeconservation). Togetanestimateofoursystematicerror,we 0 performseveralfitsvaryingthefixedvalueof p(fortheel.m.formfactorwealsotreat pasafree fitparameter,sincethedataqualityishighenough). ThefitresultsareshowninTable3. Comparisonfor𝐴uVdV:3pt∗3ptto4pt(𝐿=40) Comparisonfor𝐴uSdS:3pt∗3ptto4pt(𝐿=40) 0 3.0 4ptdata −1 3pt∗3pterror(𝑝=1) 2V] −2 2V] 2.5 33pptt∗∗33pp3ttpcctoo∗nn3vvpootlluuettriirooonnr(((𝑝𝑝𝑝===212))) Ge −3 Ge 2.0 ud−3𝐴(|𝑦|)[10VV −−−−−87654 3pt∗3p3t3pcpto∗tn∗3v3poptltueertriroornor4r(p((𝑝𝑝𝑝tf=d=rae11tea))) ud−2𝐴(|𝑦|)[10SS 011...505 3pt∗3ptconvolution(𝑝free) −9 0.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 𝑚−1 𝑚−1 𝜋 |𝑦|[fm] 𝜋 |𝑦|[fm] Figure3: Resultsforthefirstnaivefactorizationtestaccordingto(4.1)forourtwofitresults. Theconvolu- tionsarecomparedwiththe4ptdataforA andA . VV SS 0 Comparisonfor𝑀V4V4:3pt∗3ptto4pt(𝐿=40) ComparisCoonmfoprar𝑀isVon4Vf4or(l𝑀ogSsSc:al3ep)t:∗33pptt∗3topt4tpot4(𝐿pt=(𝐿40=)40) Comparisonfor𝑀SS(logscale):3pt∗3ptto4pt(𝐿=40) 10−124 4ptd4aptatdata 4ptdata 3pt∗33pptt∗er3rpotre(r𝑝ro=r1(𝑝)=1) 3pt∗3pterror(𝑝=1) 4−3𝑀(|𝑦|)[10GeV]VV44−−−11550 3pt∗3p3tpcto∗n3vpotluetriroonr4p((𝑝𝑝t=d=a11ta)) 4|𝑀(|𝑦|)|[GeV]VV444−2𝑀(|𝑦|)[10GeV]SS111000−−−11543022468 33pptt∗∗3333pppp3ttttp∗∗ccto33o∗nppn33vttvppootccltloouue∗nntrt3irivvpooooontnrlluue(((ttr𝑝𝑝𝑝iiroooff=nnrrree1(((ee𝑝𝑝𝑝)))===212))) 4𝑀(|𝑦|)[GeV]SS 1100−−21 33pptt∗∗33pp3ttpcctoo∗nn3vvpootlluuettriirooonnr(((𝑝𝑝𝑝===212))) 3pt∗3pterror(𝑝free) −20 3pt∗3ptconvolution(𝑝free) 0 10−3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 00 00..22 00..44 00.6.6 0.08.8 11 1.21.21.41.41.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 𝑚−𝜋1|𝑦|[fm] 𝑚𝑚−𝜋−𝜋11|𝑦|𝑦|[|f[mfm]] 𝑚−𝜋1 |𝑦|[fm] Figure4: Resultsfortheconvolution(4.2)comparedwiththe4ptdataofthetwo-operatormatrixelements fortheSSandVVcase(logarithmicscale). 5 DPDsofthePion ChristianZimmermann The integrals (4.1) and (4.2) are evaluated numerically and compared to the corresponding 4pt data, where again only connected contributions are used. The results for test 1 are presented in Figure 3. In the electromagnetic case one may observe a good agreement between the two curves for large distances, while for distances smaller than the pion wave length clear deviations arevisible. Forthescalarcaseasimilarbehaviorcanbefound,althoughtheagreementisslightly worse. This might be corrected by taking into account disconnected contributions, which are less suppressedinthescalarcase. The results of test 2 plotted in Figure 4 are very different. There is no region in y, where the 3pt data and the 4pt data show the same behavior. The naive factorization ansatz only predicts the correctorderofmagnitudeforboth,theelectromagneticandthescalarchannel. 5. Conclusion We have investigated 4pt-functions, which are needed for the description of DPS processes, by performing lattice calculations. The calculations have been done for all 4pt-graphs, where we obtainedgoodresultsforatleastfourgraphs. Fortheremainingtwographswestillhavetoincrease ourstatistics. Wehavecheckedthevalidityofthe"naivefactorization"assumptionbyperforming two tests, where we have found that this kind of factorization fails for small parton distances, but alsoforlargedistancesitmightnotbeavalidassumption. Theefforts,whichhavebeendonesofar,areplannedtobeextendedtofurtheraspects,likehigher moments and non-zero momenta. Finally we want to investigate other particles, especially the nucleon,sincethisistheparticlecollidedatLHC. References [1] G.CalucciandD.Treleani,Protonstructureintransversespaceandtheeffectivecross-section, Phys. Rev.D60(1999)054023[hep-ph/9902479]. [2] M.Diehl,D.Ostermeier,andA.Schäfer,ElementsofatheoryformultipartoninteractionsinQCD, JHEP1203(2012)089[arXiv:1111.0910]. [3] S.Capitaniet.al.,Nucleonform-factorsandO(a)improvement, Nucl.Phys.Proc.Suppl.73(1999) 294[hep-lat/9809172]. [4] G.S.Bali,S.Collins,B.Glässle,M.Göckeler,J.Najjar,R.H.Rödl,A.Schäfer,R.W.Schiel, W.Söldner,andA.Sternbeck,NucleonisovectorcouplingsfromN =2latticeQCD, Phys.Rev.D91 f (2015),no.~5054501[arXiv:1412.7336]. [5] M.Falcioni,M.Paciello,G.Parisi,andB.Taglienti,AgainonSU(3)glueballmass, NuclearPhysicsB 251(1985)624. [6] P.A.Boyle,A.Juttner,C.Kelly,andR.D.Kenway,Useofstochasticsourcesforthelattice determinationoflightquarkphysics, JHEP08(2008)086[arXiv:0804.1501]. [7] S.Gusken,AStudyofsmearingtechniquesforhadroncorrelationfunctions, Nucl.Phys.Proc.Suppl. 17(1990)361. [8] M.Gockeleret.al.,PerturbativeandNonperturbativeRenormalizationinLatticeQCD, Phys.Rev. D82(2010)114511[arXiv:1003.5756]. 6