Nuclear Drell-Yan effect in a covariant model C. L. Korpa Department of Theoretical Physics, University of P´ecs, Ifju´sa´g u´tja 6, 7624 P´ecs, Hungary A. E. L. Dieperink Kernfysisch Versneller Instituut, Zernikelaan 25, NL-9747AA Groningen, The Netherlands (Dated: January 14, 2013) We investigate effects of nuclear medium on antiquark distribution in nuclei applying theresults of a recently developed relativistically-covariant self-consistent model for the pion and the isobar. We take into account Fermi motion including Pauli blocking and binding effects on the nucleons and medium effects on the isobar and pion leading to modest enhancement of the pion light-cone- momentum distribution in large nuclei. As a consequence the Drell-Yan cross-section ratio with 3 respect to thedeuteron exceeds one only for small values of thelight-cone momentum. 1 0 2 I. INTRODUCTION which use a computed spectral function that accounts n forlargeremovalenergy(50MeVinnuclearmatter)and a Fermi motion due to correlations. This approachcan re- Improving our understanding of the quark and gluon J produce the observed slope of the reduction for x < 0.5 degrees of freedom of nucleons bound in nuclei necessi- 1 in the EMC ratio 2FA(x)/AFd(x) but not the behavior tates further efforts in spite of numerous investigations 2 2 1 around the minimum in the ratio at x=0.8. The latter and successes [1]. The planned new nuclear Drell-Yan seems to require ad-hoc off-shell effects [9]. In the sec- ] (DY) scattering experiment SeaQuest at Fermilab (E- h ondcategory[10,11])oneusuallystartsfromtheWalecka 906) [2–4] strongly motivates recent advancements in t modelinthemean-fieldapproximationwhichhasasmall - analysis of nuclear parton distributions [5] as well as l netbindingeffect(8MeVpernucleon)andhenceyieldsa c attempts directed at more reliable and accurate calcu- verysmallEMCeffect,andthenoneaddstheeffectofex- u lation of experimentally observable quantities. In line n ternalscalarandvectorfields. Sinceinthepresentstudy with the second objective is our aim to update and ex- [ weareinterestedintheantiquarkdistributionforx<0.4 tend to larger x values previous calculations of nuclear we rely on the conventional convolution approach using 1 DY effect presented in Refs. [6, 7]. Furthermore, the a parameterizednucleon distribution, which has the two v physics interest is twofold: (i) to investigate the role of 3 non-perturbativephysics(pioncloud)inthe u¯ d¯asym- above mentioned parameters related to the removal en- 6 − ergy η and the Fermi momentum. metry in the nucleon, and (ii) to study possible anti- 4 In Sec. II we study the antiquark distributions in the quark enhancement due to presence of virtual mesons in 2 free nucleon and determine the off-shell πNN and πN∆ . the nuclearmedium. FromsymmetrypropertiesofQCD 1 form-factorswhichleadtogooddescriptionoftheisovec- we may infer that pion is one possible source of non- 0 torpartofthe protonantiquarkdistribution by the pion perturbativequark-antiquarkphysics,inparticularu d 3 − cloud. In Sec. III we turn to general discussion of the asymmetry. 1 nuclear effects consisting of binding and Fermi motion : In conventional models based upon meson exchange v of nucleons and modification of the pion cloud. Detailed nuclear binding of nuclear matter comes for about 50% i considerationofthemediumeffectsonthenucleon’spion X from (virtual) pions present in the nucleus. Can we ob- cloudfollowsinSec.IVwhereexpressionsarederivedfor r servethese? SomeindicationforpionsispresentinEMC a thepionlight-conedistributionsoriginatingfromthe πN effect enhancement around x = 0.1, which can be as- and π∆ states. We emphasize the careful treatment of cribed to the fact that pions (and heavier mesons) carry the in-medium delta baryon based on a complete rela- afractionofthe momentumsumrule. Canoneseethese tivisticallycovariantbasisforitsdressedpropagator. Nu- pionsmoreexplicitly, e.g.inthe formofanenhancement merical results for the in-medium pion distribution and ofanti-quarksinthenucleus? Anti-quarkscanbeprobed DYcross-sectionratiosfornucleartargetsrelativetothe directlyinDrell-Yanscatteringbutpreviousexperiments deuteron are presented and discussed in Sec. V. Finally, [8]withintheexperimentaluncertaintyofabout10%did Sec. VI contains a summary of our results. not show a nuclear enhancement; results of calculations varied strongly. In practice one can distinguish two main types of the- II. ANTIQUARKS IN FREE NUCLEONS oretical interpretations of the classical EMC effect: (i) in terms of nucleon as constituents which are bound but notmodifiedinthe medium, and(ii) intermsofoff-shell Before turning to the nuclear case we want to inves- nucleons with medium modified structure functions, e.g. tigate whether the pion cloud approach we will use in through scalar and vector fields acting on the quarks. the medium can reproduce the observed flavor asym- In the first category one has the non-relativistic models metry in the free nucleon. The distribution of anti- 2 quarks in the nucleon can be decomposed into a flavor- 10 Lorentz scalar functions [14, 15] which contains both symmetricisoscalarpart(originatingfromgluonsplitting spin-3/2 and spin-1/2 sectors [16]. However, using the andpossiblymesoncloud)andanon-perturbativeisovec- convenient basis from Ref. [14] it turns out that a single tor meson-cloud contribution.The latter can be consid- term, namely the (on mass shell) positive energy spin- ered to be the source of the u¯ d¯asymmetry. In addi- 3/2contributiongivesthedominantcontributionandall − tion to the pion cloud of the nucleon we include also the others(sometermsinthepropagatorareidenticallyzero) isobarwithitspioncloudsince itwasshowntogivesub- arecompletelynegligible. InthenotationofRef.[14]this stantialcontributions[12]. Inthis wayonecanconstrain isthecoefficientoftheprojectorsumQ′µν Qµν +Pµν [11] ≡ [11] [55] the πNN and πN∆ form factors using the empirical an- whichwe denote by G(Q′)(p). The pionlight-cone distri- tiquark u-d flavor asymmetry. The physical free nucleon [11] bution originating from the ∆π state then can be ex- state is expressed approximately as pressed as: Neglecting|Noffi-=sh√ellZe|ffNecibtsarteh+eαlig|Nhtπ-cio+neβm|∆oπmi.entum d(i1s)- fπ−∆/N(y)= yM6gππ23N∆ Z−−∞(My+mπ)dp′3 Z0∞p′⊥dp⊥′ tribution of a quark with flavor f in a proton can be (M +p pˆ′) t+(k pˆ′)2 written as (B =N,∆): ·Fπ(πN)∆(t)2Fπ(∆N)∆(p′)2 ·(t+(cid:0)m2)2 · (cid:1) π 1 dy ImG(Q′)(p′), (5) qf(x)=Zqf,bare(x)+ ci(cid:20)Z y fBi/N(y)qfB,ibare(x/y) · [11] XB,i x wherep andp′ arethe four-momentaofthe nucleonand 1 dy isobar,t (p p′)2 and F(π,∆) arethe form factorsof + fπi/N(y)qπi(x/y) , (2) ≡− − πN∆ Z y (cid:21) the πN∆ vertex. The form factor x where ci (i labels the charge states) are the appropriate F(∆) (p)=exp p2−(M +mπ)2 (6) isospinClebsch-Gordancoefficients, qBi (x) isthe par- πN∆ (cid:20)− Λ2 (cid:21) f,bare ton distribution in the bare B baryon and qπ(x) is the i with Λ = 0.97GeV (and g = 20.2GeV−1) was used pion parton distribution function. πN∆ Attributing the asymmetry in the u¯ and d¯antiquark in Ref. [15] and shown to give a good fit to the relevant pion-nucleon phase shift. For the πNN and πN∆ off- distributions to the nucleon meson cloud we are con- shell form factors which take into account the off-shell cerned with the pion light-cone distribution in the nu- pion we take a dipole form: cleon which gets contributions from final states with ei- ther nucleon or isobar: Λ2 m2 2 F(π) (t)= πX − π , (7) fπ/N(y)=fπN/N(y)+fπ∆/N(y). (3) πNX (cid:18) Λ2 +t (cid:19) πX The nucleon term was calculated by Sullivan [13]: with X standing for N or ∆. In orderto calculate the d¯ u¯ distributionfor the free fπ0N/N(y)= gπ2NNy ∞ dt|Fπ(πN)N(t)|2t, (4) prircolteoanviwnge oansslyumtheetchoanttrtihbeutp−ioionnosfevaaliesnicseosdpiisntrisbyumtimonets-. 16π2 ZM2y2(1−y) (t+m2π)2 Thefinalstatewithnucleoncontributesthroughthepres- with y =(k +k )/M being the pion light-cone momen- ence of π+ with distribution 2fπ0N/N, while the isobar 0 3 tumfraction,withM thephysicalmassofthenucleon(as final state can have also a π+ with isospin weight 1/3 − a convenient scale), F(π) (t) is the off-shell form-factor or a π with minus sign (because of the pion valence u¯ of the πNN vertex, wπhNilNe g is the π0NN coupling. distribution) with respect to fπ−∆/N giving in total: πNN The free-pion propagator, D0, appears in the above ex- qprdesesniootnesinthtehepifoonrmfo(utr+-mmom2ππ)e−n1tu,mw.heTrehet i≡sob−aqr2cwonhterrie- (d¯−u¯)p(x)=Zx1 dyy (cid:18)2fπ0N/N(y)− 23fπ−∆/N(y)(cid:19)qvπ(x/y), butionalsoplaysanimportantrole[12]despite the kine- (8) maticalsuppressioncomingfromtheisobar-nucleonmass with qvπ(x) denoting the valence parton distribution of difference. In Ref. [12] it was calculated using the free charged pion. In Fig. 1 we show the pion distributions isobar propagator, i.e. neglecting its width. A complete with nucleonfinalstate (solid line) andisobarfinalstate relativistically covariant treatment of the isobar in vac- (dashline). Fortheform-factorcutoffweusedthefollow- uumandnuclearmediumwasintroducedinRef.[14]and ing values: Λ(π) = 0.95GeV and Λ(π) = 0.75GeV. The πN π∆ we use that formalism to take into account the vacuum bare-nucleon probability then takes the value Z = 0.69 width consistent with the measured pion-nucleon scat- which suggests that higher-order terms with more than tering phase shift in the spin-3/2 isospin-3/2 channel. one pion do not contribute significantly. The calculated The full Lorentz structure of the vacuum propagator of valueforthed¯ u¯asymmetryinthefreeprotonisshown − the Rarita-Schwinger field can be expressed in terms of in Fig. 2 by solid line and compared to the result using 3 (dash line) and isobar final state (dash-dot line). These resultsarequitesimilartothed¯ u¯obtainedinRef.[18], − although there the infinite-momentum-frame formalism 0.25 was used with suitably adjusted values of the πN and π∆ form factors. 0.20 n o stributi0.15 Isobar f. s. Nucleon f. s. III. NUCLEAR EFFECTS di n Firstcalculationsofthe nuclearDrell-Yanprocess[19] o0.10 Pi suggested an enhancement coming from the medium modification of the pion cloud. However, the experi- 0.05 mental data [8] did not show that enhancement within a 10% uncertainty. Later onother groups reportedmore detailed calculations of the Drell-Yan ratio with a large 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 variationin resultsasshowninRefs.[2–4]. Herewe con- y sidertheratioofthecrosssectionsofproton-nucleusand proton-deuteron scattering, FIG. 1. (Color online) Pion distribution in the free proton: fπ0N/N(y)shown bysolid line and fπ−∆/N(y)bydash line. R = 2 dσpA/dx1dx2. (9) A/d A dσpd/dx dx 1 2 whereAdenotesthenucleusanditsnucleonnumber. We the u¯ and d¯fits CT10 [17] (dot line). Also shown are specialize for the case of isoscalar targets for which the separately the contributions from the nucleon final state cross-sectionratio becomes: e2 qp(x ) qp/A(x )+qn/A(x ) +qp(x ) qp/A(x )+qn/A(x ) f f f 1 f 2 f 2 f 1 f 2 f 2 R = n h i h io. (10) A/d P e2 qp(x ) qp(x )+qn(x ) +qp(x ) qp(x )+qn(x ) f f f 1 f 2 f 2 f 1 f 2 f 2 n h i h io P Inthecasewhenx1 islarge,sayx1 >0.3thesecondterm 2qn (x/y) + A dyfπ0N/A(y) qπ0(x/y)+2qπ+(x/y(1)3), in the numerator becomes negligible and only medium f,bare (cid:3) Zx y h f f i effect on the antiquarks plays a role. where isospin-symmetric nuclear medium was assumed. The (anti-)quark distribution in the medium can be Addingthedifferenceoftheleft-handsideandright-hand modifiedintwoways,(i)throughFermimotionandbind- side of (11) to the right-hand side of (13) and repeating ing of the nucleon, and (ii) modification of the nucleon’s the same procedure for the neutron we obtain pion cloud. To establish the connection to the (anti- 1 )quark distribution of the free nucleon we use Eq. (2) q˜p(x)+q˜n(x)=qp(x)+qp (x) fN/N(y)dy which for the free proton gives (with isobar terms not f f f f,bare Z − 0 written out for brevity): 1 dy A fN/N(y)qp (x/y) qp (x) fN/A(y)dy+ qp(x)=Zqp (x)+ 1 1 dyfN/N(y) qp (x/y)+ Zx y f,bare − f,bare Z0 f f,bare 3Zx y h f,bare A dyfN/A(y)qp (x/y)+(p n) 2qn (x/y) + 1 dyfπ0N/N(y) qπ0(x/y)+2qπ+(x/y(1)1), Zx y f,bare → f,bare (cid:3) Zx y h f f i +2 A dy fπ0N/A(y) fπ0N/N(y) with Zx y h − i 1 1 Z ≡1−Z dyfN/N(y)=1−3Z dyfπ0N/N(y), (12) ·hqπf0(x/y)+qπf+(x/y)+qπf−(x/y)i, (14) 0 0 where we used that where the last equality expresses flavor-chargeconserva- 1 tion. Similarly, the quark distribution for the nuclear Z 1 dyfN/A(y). proton can be written: A ≡ −Z 0 1 A dy Toproceedwiththeaboveexpressiononeneedsthebare q˜p(x)=Z qp (x)+ fN/A(y) qp (x/y)+ f A f,bare 3Zx y h f,bare antiquark distributions which could be determined from 4 the antiquarkdistributionofthe free “isoscalar”nucleon for twousedpion parametersets. For the distributionof the bare nucleon in the free nucleon we use the relation 1.75 fN/N(y)=3fπ0N/N(1 y)whichfollowsfromtheproba- 1.50 bilistic interpretation o−f these functions [18] and its ana- Total etry 1.25 Fit CT10 log for the in-medium case, fN/A(y) = 3fπ0N/A(1 y). mm 1.00 Nucleon f. s. We remark that the latter relationship can only be−ap- sy Isobar f. s. proximatesincethesupportofthepionin-mediumdistri- a 0.75 E866 data ark 0.50 butionis notstrictly limited byvalue one,butin viewof u thesimilarityofthepiondistributionsinthetwocasesit q nti 0.25 shouldbeareasonableapproximationfortheestimateof A 0.00 -0.25 0.025 -0.50 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 x ction 0.020 pion (1) e pion (2) FIG. 2. (Color online) The pion-cloud result for the d¯−u¯ orr c0.015 asymmetry in free proton (solid line) compared to the differ- n enceoftheprotond¯andu¯distributions(dotline)fromthefit cleo CT10[17]anddatapointsbyFermilabE866/NuSeaCollabo- nu0.010 ration[20]. Theisobarcontribution(dash-dotline)isnegative are and much smaller than the nucleon term (dash line). Used B 0.005 parameter values: Λ(π) =0.95GeV and Λ(π) =0.75GeV. πN π∆ 0.000 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 x Eq. (11) and its analog for the neutron. A much sim- pler, though approximate procedure is just to subtract FIG. 3. (Color online) Sum of the second, third, fourth and the meson-cloud contribution form the antiquark distri- fifth terms in Eq. (14) devided by the antiquark distribution bution of the physical nucleon. Indeed, using the fact of the free “isoscalar” nucleon. Solid line is for the pion pa- that antiquark distributions at small x behave as 1/x rameter set (1) and dash line for parameter set (2). one can confirm that the first and second term on the right-hand side of (11) combine to give the bare distri- bution if one adds to (11) its neutron analog. Using the asmalleffect. Weobservethatthiscontributionisindeed same argument about the small x behavior of antiquark quite smallas expected fromthe formofthe nucleonan- distributions one can establish an approximate cancella- tiquarkdistributionandpion(aswellasrelatednucleon) tion of the second and third as well as the fourth and light-cone-momentum distributions. The parameter set fifth terms on the right-hand side of Eq. (14) and of the (1)′is: M∗ =′0.89GeV,′ΣvN = 0,Σs∆ = −0.1GeV,Σv∆ = corresponding terms involving the neutron. This simpli- 0,g =0.9,g =0.3,g =0.3,whiletheset(2)isgiven 11 12 22 fication was used in our previous work [6], but in the by:′M∗ = 0.′89GeV,ΣvN′ = 0,Σs∆ = −0.05GeV,Σv∆ = present calculation we want to take into account these 0,g11 = 1.0,g12 = 0.4,g22 = 0.3; where M∗ is the mean- contributions with the bare antiquark distributions de- field shifted nucleon mass, Σv the energy shift of the N terminedbytheabovementionedsubtractionofthepion nucleon,Σs andΣv the delta’s mean-fieldshifts andg′ ∆ ∆ ij contribution fromthe physicaldistribution. In Fig. 3 we the Migdal four-fermion interaction parameters. show the proton-neutron average of the sum of the sec- Introducing a shorter notation for the bare nucleon ond, third, fourth and fifth terms in Eq. (14) divided by contribution to the in-medium antiquark distribution: 1 1 dy qp+n (x)=qp(x)+qp (x) fN/N(y)dy fN/N(y)qp (x/y) f,bare f f,bare Z −Z y f,bare 0 x A A dy qp (x) fN/A(y)dy+ fN/A(y)qp (x/y)+(p n), (15) − f,bare Z Z y f,bare → 0 x 5 we can finally write the sum of in-medium proton and neutron antiquark distribution as qp/A(x)+qn/A(x)= A dyfN(y)qp+n (x/y)+2 A dy fπ0/A(y) fπ0/N(y) qf (x/y)+qf (x/y)+qf (x/y) . f f Zx y Fb f,bare Zx y h − ih π0 π+ π− i (16) 1.08 1.06 Convolution Convolution + pion (1) 1.04 Convolution + pion (2) x) 1.02 d F(2 A 1.00 x) / 0.98 A 2 F(20.96 0.94 0.92 0.90 0.0 0.1 0.2 0.3 0.4 0.5 0.6 x FIG. 5. Types of diagrams contributing to the in-medium piondistribution with outgoing nucleon. Thesametypesap- FIG. 4. (Color online) The ratio of F (x) per nucleon for pear also with outgoing delta baryon. The dash line denotes 2 isospin symmetric nuclear medium with parameters: Fermi the dressed pion propagator, while the solid and the double momentum p = 250MeV, η = 0.97, and deuteron. Pion line correspond to nucleon and delta. F contribution with parameter sets (1) and (2) is included in theresults shown by solid and dash-dot lines. IV. ANTIQUARKS IN BOUND NUCLEONS: PION CONTRIBUTION The convolution with fN(z) takes into account Fermi Fb motion and binding effects on the in-medium nucleons. For the function fN(z) we take the result of Birse [21]: We nowturn to considerationofthe pioncontribution Fb toantiquarkdistributionsinnucleonsboundinlargenu- fN(z)= 3 ǫ2 (z η)2 Θ(ǫ z η ), (17) cleiwhichwe modelbyaninfinite systemwithappropri- Fb 4ǫ3 − − −| − | ate average nuclear density. (cid:2) (cid:3) For corresponding pion properties in the nuclear mat- where ǫ p /M, η is a parameter with value slightly F ≡ ter we use the results of a fully covariant self-consistent below one which takes into account the nuclear binding model developed in Ref. [23]. Compared to the case of andΘ(x)istheunitstepfunction. InFig.4weshowthe the free nucleon the nuclear environment changes the ratiooftheF (x)structurefunctionsfortheisospinsym- 2 pion propagator appearing in the Sullivan formula (4) metricnuclearmatterandthedeuteron. Weassumeneg- and renormalizes the πNN as well as the πN∆ ver- ligiblemediumeffectsinthedeuteronandforthenuclear tices through nucleon-nucleon correlations modeled by mediumusetheconvolutionmodelwithlight-conedistri- the Migdalfour-fermioninteractions [24]. Nucleon prop- bution(17)andparametersp =250MeVandη =0.97. F ertiesarealsoaffectedandwetakeintoaccountthebind- In this way one can reproduce the negative slope in the ingeffectsthroughmean-fieldmassandenergyshiftscon- classicalEMCeffectfor0.1<x<0.5asshownforexam- sistent with approach in Ref. [23]. pleinRef.[22]. Theexperimentalenhancementobserved around x = 0.1 can be attributed to the pion enhance- The inclusion of the dressed pion propagator is ment as shown in the figure for the two parameter sets straightforward but the dressing of the πNN and the (1) and (2) used also for the plots in Fig. 3 and given πN∆ vertices requires summation of nucleon-hole and above. The pion enhancement term was calculated by delta-hole bubbles. The types of relevant diagrams are the convolution of the in-medium pion light-cone distri- shown in Fig. 5. For resummation of these diagrams we bution enhancement relative to the free nucleon and the use the relativisticallycovariantformalismintroducedin pion F distribution. Note that expected shadowing ef- Ref. [25] and applied for pion and delta self energy cal- 2 fects would lead to decrease of the nuclear cross section culation in Ref. [23]. In the present case it concerns a for x 0.05. different type of contribution which for the nucleon in ≤ 6 the final state can be written: the nonzero contribution involving one particle-hole (nucleon-hole or delta-hole) loop comes from K =u¯(p′)γ γµu(p) Π (q)qν, (18) N 5 µν · whereq =p p′ isthe pion4-momentum,u(p)isthenu- Π(µ1ν) =g1′1 Π(1N1h)(q)L(µ1ν1)(q)+Π(2N1h)(q)L(µ2ν1)(q) − (cid:16) (cid:17) cleon in-medium spinor (with mean-field shifts of mass +g′ Π(∆h)(q)L(11)(q)+Π(∆h)(q)L(21)(q) (,20) andenergy)andΠµν(q) is the resummedcontributionof 12(cid:16) 11 µν 21 µν (cid:17) nucleon-hole and delta-hole loops. Using the decomposi- tion of nucleon-hole and delta-hole loops [23]: where g′ ,g′ ,g′ are the usual Migdal parameters [23] 11 12 22 where index 1 refers to the nucleon and index 2 to the 2 Π(Nh)(q)= Π(Nh)(q)L(ij)(q)+Π(Nh)(q)T (q), delta and we took into account that: µν ij µν T µν iX,j=1 L(12)(q) qν =L(22)(q) qν =T (q) qν =0. (21) 2 µν · µν · µν · Π(∆h)(q)= Π(∆h)(q)L(ij)(q)+Π(∆h)(q)T (q()1,9) µν ij µν T µν iX,j=1 In order to perform the summation involving arbitrary number of nucleon-hole or delta-hole loops it is conve- where L(ij)(q) and T (q) have projector properties, nient to introduce the following matrices [25]: µν µν g1′1 0 g1′2 0 Π(1N1h) Π(1N2h) 0 0 0 g′ 0 g′  Π(Nh) Π(Nh) 0 0  g(L) =g01′2 0g11′12 g02′2 0g12′22  , Π(L) = 0021 0022 ΠΠ(1(∆∆1hh)) ΠΠ(1(∆∆2hh))  . (22)  21 22  The lowest order contribution (20) can then be written as: Π(1) =[(Π(L)g(L)) +(Π(L)g(L)) ]L(11)(q) µν 11 31 µν +[(Π(L)g(L)) +(Π(L)g(L)) ]L(21)(q). (23) 21 41 µν Higher order terms are accounted for by taking appropriate matrix elements of products of (Π(L)g(L)) matrices and the summation of terms with arbitrary number of loops is simply achieved by replacing (Π(L)g(L)) in (23) by Π(L)g(L)(1 Π(L)g(L))−1 leading to: − Π = [Π(L)g(L)(1 Π(L)g(L))−1] +[Π(L)g(L)(1 Π(L)g(L))−1] L(11)(q) µν (cid:16) − 11 − 31(cid:17) µν + [Π(L)g(L)(1 Π(L)g(L))−1] +[Π(L)g(L)(1 Π(L)g(L))−1] L(21)(q). (24) (cid:16) − 21 − 41(cid:17) µν Adding g to Π in (18) we obtain the full contribution of the diagram with nucleon final state. Squaring its µν µν absolutevalue andperformingsummationoverthe spinprojectionsofthe nucleoninthe finalstate andaveragingfor the nucleon in the initial state we obtain: 1 2 |KN|2 =Aqq· 2(p−ΣvNu)·q(p′−ΣvNu)·q−q2[M∗2+(p′−ΣvNu)·(p−ΣvNu)] Xs,s′ (cid:0) (cid:1) +2A ((p Σv u) u(p′ Σv u) q+(p′ Σv u) u(p Σv u) q qu· − N · − N · − N · − N · q u[M2+(p′ Σv u) (p Σv u)] − · ∗ − N · − N +A 2(p Σv u) u(p′ Σv u) u(cid:1) M2 (p′ Σv u) (p Σv u) , (25) uu· − N · − N · − ∗ − − N · − N (cid:0) (cid:1) where u is the 4-velocityofthe medium (implicitly presentalsoin Eqs.(18)−(24)), M∗ =MN+ΣsN, ΣsN and ΣvN are the nucleon mean-field mass and energy shifts. The factors A ,A ,A are given by: qq qu uu 2 A =2 1+A+q uB/ q2 (q u)2 , qq (cid:12) · p − · (cid:12) (cid:12) (cid:12) A = (cid:12)2Re q2B/ q2 (q u)2(1+A¯(cid:12) q uB¯/ q2 (q u)2) qu − h p − · − · p − · i 2 A =2 q2B/ q2 (q u)2 , (26) uu (cid:12) p − · (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 7 where the bar denotes complex conjugation, A [Π(L)g(L)(1 Π(L)g(L))−1] +[Π(L)g(L)(1 Π(L)g(L))−1] , 11 31 ≡ − − B [Π(L)g(L)(1 Π(L)g(L))−1] +[Π(L)g(L)(1 Π(L)g(L))−1] , (27) 21 41 ≡ − − and we used that: where D (q) is the in-medium dressed pion propa- π L(11)(q)qν =q , gator, b ≡ My − p3 − M∗2+p23+p2⊥, p′⊥min = µν µ p L(µ2ν1)(q)qν = q2 q·(uq u)2 qµ− q2 q2(q u)2 uµ(2.8) qan2dbpp~′⊥M, a∗2n+dpt2Fhe−πMN∗N2−fobr2m, ϑfacistotrhFeπ(aπNn)Ng(le−qb2e)twweaesnap~ls⊥o − · − · included. p p To compute the pion light-cone distribution per nu- The contribution coming from the delta baryonin the cleon in the medium we integrate over incoming nucle- final state is made more involved by the complicated ons in the Fermi sea and outgoing ones above the Fermi structure of the in-medium delta propagator [14]. How- sea,restrictingthepionlight-conemomentumfractionto ever, considerable simplification can be achieved by in- the specified value by inserting a delta function and fi- cluding only the two dominant contributions in the con- nally divide by the nucleon density. The final expression venientrelativistically covariantdecompositionsince the obtained is: imaginary part of the other components is typically two orders of magnitude smaller at nuclear saturation den- fN(y)=3My(cid:18)mfNπ(cid:19)2 32π13p3F Z−ppFF dp3Z0pp2f−p23p⊥dp⊥ dsQietµ[l1ytν1a][2ac6na]sd.e,PTb[µ5hu5νe]ttaderoremmdsiniffwaenhrtiecnhctowninterritebhudeteimgoenensdeircuaomtmee[i1n4fr]t.ohmeSuftrmheee- ∞ 2π 1 p′ dp′ dϑ K 2 mationof particle-holeloops dressingthe πN∆ vertexis ⊥ ⊥ N ·Zp′⊥min Z0 2b Xs,s′ | | athnealroegleovuasnttomthaetrπixNeNlemceansets,wofitΠh(Lon)gly(Ld)i(ff1ereΠn(cLe)gb(eLin))g−i1n, F(π) ( q2)D (q) 2, (29) replacing the expression (24) with: − ·(cid:12) πNN − π (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Π∆ = [Π(L)g(L)(1 Π(L)g(L))−1] +[Π(L)g(L)(1 Π(L)g(L))−1] L(11)(q) µν (cid:16) − 13 − 33(cid:17) µν + [Π(L)g(L)(1 Π(L)g(L))−1] +[Π(L)g(L)(1 Π(L)g(L))−1] L(21)(q). (30) (cid:16) − 23 − 43(cid:17) µν The expression analogous to (25) in this case takes the form: 1 1 2 |K∆|2 = 2Tr (/p−ΣvNu/+M∗)ImGµν(p′) gµαgνβ +Π∆µαΠ¯∆νβ qαqβ, (31) Xs,s′ (cid:2) (cid:3)(cid:0) (cid:1) ′ where ImG (p) denotes the imaginarypartof the in-medium ∆ propagatorfor which we take the dominantcontri- µν bution given in the basis used in Ref. [14] by just two terms: Gµν(p′)=Qµν (p′)G(Q)(p′)+Pµν(p′)G(P)(p′). (32) [11] [11] [55] [55] In this way the expression (31) takes the form: 1 K 2 = A(∆)c(Q)+2A(∆)c(Q)+A(∆)c(Q) ImG(Q)(p′) 2 Xs,s′ | ∆| h qq qq qu qu uu uu i [11] + A(∆)c(P)+2A(∆)c(P)+A(∆)c(P) ImG(P)(p′), (33) qq qq qu qu uu uu [55] h i where the expressions for A(∆),A(∆),A(∆),c(Q),c(Q),c(Q),c(P),c(P),c(P) are given in the Appendix. The pion light- qq qu uu qq qu uu qq qu uu cone momentum distribution stemming from the process with the nucleon emitting a pion and a delta baryon is analogous to expression (29) and reads: f∆ 2 1 pF p2f−p23 ∞ ′ f∆(y)=3My(cid:18)mπ(cid:19) 32π4p3F Z−pF dp3Z0p p⊥dp⊥Z−∞dp3 ∞ 2π 1 2 p′ dp′ dϑ K 2 F∆ (p′)F(π) ( q2)D (q) . (34) ·Z0 ⊥ ⊥Z0 2 Xs,s′ | ∆| (cid:12)(cid:12) πN∆ πN∆ − π (cid:12)(cid:12) (cid:12) (cid:12) 8 We checked by explicit numerical calculation that both Eqs.(29)and(34)havethe correctlow-densitylimit, i.e. reproduce the free nucleon and delta results. 0.30 free nucleon M*=0.89GeV, g’11=0.8, g’12=0.3, g’22=0.3 V. NUMERICAL RESULTS AND DISCUSSION 0.25 M*=0.85GeV, g’11=1.0, g’12=0.4, g’22=0.3 n M*=0.89GeV, g’11=1.0, g’12=0.4, g’22=0.3 o uti0.20 proFpoerrttihees wcoemrpeluytaotniotnheofreicne-nmtleydiduemvelpoipoendarnedlatiisvoibstair- distrib0.15 callycovariantself-consistentmodelpresentedinRef.[23] n o and used for the nuclear photoabsorption calculation in Pi0.10 the isobar regionin Ref. [27]. For the medium computa- tionweusethesameπNNandπN∆formfactorsasinthe 0.05 vacuum one and include them in the model of Ref. [23]. ′ The values ofthe Migdal g parameterswhich model the 0.00 short-range nucleon and isobar correlations were taken 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 y in the range preferred by the results of Ref. [27]; in this work a good description of the nuclear photo-absorption cross section in the isobar region was obtained. Binding FIG. 6. (Color online) Nucleon contribution to the pion effects for the nucleon are taken into account by the ef- distribution(fπ0N/A(y))forthein-mediumnucleoncompared fective(mean-field)massM∗ andtheenergyshiftΣvN. A to thecase of free nucleon. consequence of the use of the mean-field approximation isareductionofthein-mediumpiondistributioncoming is the result. This is not completely surprising since the from the nucleon final state. Namely the dominant con- computationsofRef.[23]donotleadtoappreciablesoft- tributiontoitisthetermproportionaltoA inEq.(25) qq ening of the pion spectrum in the medium which would with: result in enhanced pion distribution. In this respect the 2(p−ΣvNu)·q(p′−ΣvNu)·q−q2[M∗2+(p′−ΣvNu) pfrioomn adnreoslsdinergcaolfcuRlaeft.io[n23[3]0i]swnhoicthsuigsnedifiacannotnlyreldaitffiveirsetnict (cid:0)·(p−ΣvNu)])=−2M∗2q2, (35) treatment of the isobar and a softer pion-nucleon-delta form factor. On the other hand a significant enhance- which is the same expression as for the free nucleon, ex- ment is observed for the contribution originating from cept that M∗ appears instead of M. the transition N π∆ which is not Pauli suppressed. Since M∗/M < 1 a further suppression in addition to → These results emphasize the importance of careful treat- that from the Pauli blocking is obtained, depending on ment of the in-medium isobar self energy and propaga- the actual value of M∗/M. The latter is difficult to con- tor, which is made possible by the convenient complete strainsinceobservablesgenerallyareonlysensitivetothe basis introduced in this context in Ref. [14]. As a conse- combination M∗ +ΣvN. Since our aim is to make com- quence in the nuclear medium the combinedeffects from parison with experiments on finite nuclei (rather than the pion and from the isobar can produce sizeable in- nuclearmatter)withanaveragedensitysmallerthanthe crease in the pion light-cone distribution. The latter is saturationdensity weassumesmallvaluesforthe energy constrainedto smaller light-cone-momentumratio y val- shift in the range zero to Σv =0.04GeV, corresponding N ues because of kinematical effect of the isobar-nucleon to effective mass values in the range of of 0.85GeV and mass difference but can still have significant effects on 0.89GeV.Thesevaluesareclosetotheonesusedinmore the DY cross-sectionratio. elaboratetreatmentsofnuclearmatter[28,29]whereval- Since we are considering isospin symmetric nuclear ues of 0.8 0.85GeV at saturation density give good −− medium and make a comparison with the deuteron it agreementwithobservables. The mean-fieldshifts ofthe is advantageous to consider the pion distribution in an isobar mass and energy are chosen in such a way so that “isoscalar”nucleon,i.e. to consider a proton-neutronav- they reproduce the isobar-nucleon mass difference used erage. Taking into account pions of all charges gives the inRef.[27]. This meansΣs = 0.05GeVand 0.1GeV ∆ − − complete pion distribution of an “isoscalar” nucleon: and zero for the energy shift. In Figs. 6 and 7 we show the pion distributions fπ/A(y)=3fπ0N/A(y)+2fπ−∆/A(y). (36) fπ0N/A(y)andfπ−∆/A(y)forin-mediumnucleonsfordif- ferentparametersets. For the nucleonic distributionone InFig.8weshowthe functionfπ/A(y)fordiferentinput observesareductioncomingpartlyfromthePauliblock- parameter values compared to the pion distribution of ingofthenucleonsinthemediumandpartlyfromtheM∗ the free “isoscalar” nucleon. The probability ZA of the effect(whichleadstoasuppressionroughlybythefactor bare nucleon in the medium takes the values from 0.6 to (M∗/M)2). The pion broadening in the nuclear medium 0.65, i.e. just slightly smaller than in the free nucleon onlypartlycompensatestheseeffectsandanetreduction case. 9 0.40 1.20 0.35 free nucleon pion (1) pion (2) g’11=0.8, g’12=0.3, g’22=0.3 1.15 n0.30 g’11=1.0, g’12=0.4, g’22=0.3 o distributi00..2205 uark ratio11..0150 Pion 0.15 Antiq1.00 0.10 0.95 0.05 0.00 0.90 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 y x FIG. 7. (Color online) Isobar contribution to the pion dis- FIG. 9. (Color online) The u¯(pmedium)/u¯(pfree) (solid lines) tribution(fπ−∆/A(y))inthein-mediumnucleoncomparedto andd¯(pmedium)/d¯(pfree) (dashlines) ratiosofantiquarkdistribu- thefree nucleon. For both in-medium curvesM∗ =0.89GeV tions of an in-medium proton relative to the free proton for and Σv =0. parameter sets (1) and (2). N due to the delta-baryon final state as compared to quite modest enhancement and even suppression for the down antiquarkasaconsequenceoflargerweightofnucleonfi- 1.2 free nucleon nalstateandsmallerweightofdelta-baryonfinalstateas M*=0.8GeV, g’11=1.0, g’12=0.4, g’22=0.3 n M*=0.85GeV, g’11=0.6, g’12=0.3, g’22=0.3 compared to the up antiquark. This difference points to utio1.0 M*=0.89GeV, g’11=1.0, g’12=0.4, g’22=0.3 the possibility of distinguishing between effects coming b from the medium modification of the nucleon and delta stri0.8 baryon by examining observables to which up and down di n antiquarks contribute with different weights. o0.6 pi We now turn to the DY cross-section ratio (10). In Full 0.4 Figs. 10 and 11 we show the cross-sectionratio (10) as a functionofx forfixedvaluesofx . Theinputparameter 2 1 0.2 values are given in the figure caption. We observe an enhancement only for small values of x , typically less 2 0.0 than 0.2, and for x2 >0.1 a decreasing trend as a result 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 of the convolution with nucleon distribution (17). y For comparison with the measurements of Ref. [8] we computed the ratio of the nuclear and deuteron cross FIG.8. (Coloronline)Fullpiondistribution(fπ/A(y))inthe sections for given x and integrating over x satisfying 2 1 in-medium nucleon compared to the case of free “isoscalar” the condition x > x +0.2 corresponding to the exper- 1 2 nucleon. imental cut-off. Fig. 12 shows the measured values with error bars and the calculated curves for different input parameters. WeconsiderthelowestcurveinFig.12with Before examining the DY cross-section we show the M∗ =0.8GeVandcorrespondingratherpronouncedsup- ratioofthe antiquarkdistributioninthe in-mediumpro- pression of the order (M∗/M)2 probably exaggerating ton and the same distribution in the free proton. The the effect of the nucleon mean-field approximation and upanddownantiquarkdistributionsexperiencedifferent regard the other two curves as representing better our in-medium modification due to different weights of nu- results based on the preferred parameter sets. cleon and delta contributions even in isospin-symmetric nuclearmedium. InFig.9weshowtheratiosofantiquark distributionsforanin-mediumprotonrelativetothefree VI. SUMMARY onefortwotypicalparametersetsdenotedby“pion(1)” and “pion (2)”, already used for plots in Figs. 3 and 4. We observe pronounced enhancement for the up an- Inthisworkwepresentedananalysisofnucleareffects tiquark coming from the substantial pion enhancement on the Drell-Yan process. The approach is based on the 10 1.06 1.10 x1=0.25 1.08 1.04 x1=0.35 1.06 n ratio1.02 x1=0.45 n ratio11..0024 o o Cross-secti1.00 Cross-secti001...990680 0.98 0.94 0.92 0.96 0.90 0.88 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 x2 x2 FIG.10. (Coloronline)Thecross-sectionratio(10)asafunc- FIG. 12. (Color online) Experimental results from Ref. [8] tion of x for fixed values of x . The used parameter values compared to our calculation for different parameter values. 2 1 are: M∗ =0.85GeV,ΣvN =0.04GeV,Σs∆ =−0.1GeV,Σv∆ = Short dash line: M∗ = 0.89GeV,ΣvN = 0; solid line: 0,g1′1 =0.8,g1′2 =0.3,g2′2 =0.3. M∗ = 0.85GeV,ΣvN = 0.05GeV; dash-dot line: M∗ = 0.8GeV,Σv = 0.09GeV. For all three curves: Σs = N ∆ −0.1GeV,Σv =0,g′ =1.0,g′ =0.4,g′ =0.3. ∆ 11 12 22 We took into accountthe change of the pion cloud origi- 1.06 x1=0.25 nating from both the pion-nucleon and pion-delta states and a small correction (neglected in previous work) at- x1=0.35 n ratio1.04 x1=0.45 triFbeurtmedimtootthioenbainnddibnigndeffinegctofofnubcalreearnunculceloeno.nswereac- o secti1.02 cdoisutnritbeudtfioonr b(y17t)hwehtwicoh-preapraromdeutceerslitghhet-nceognaet-imveomsleonpteumof s- s the classical EMC effect in the region 0.1 < x < 0.5 o Cr1.00 as shown in Fig. 4. Taking into account the pion en- hancement which comes from the pion-delta state of the nucleon (which is significant only for small light-cone- 0.98 momentum(y 0.2)values)leadsto someenhancement ≈ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 of the F2(x) ratio for x 0.2. x2 Pion and delta prope≤rties in the nuclear medium are calculated in a recently developed fully covariant self- consistent model [23] which consistently takes into ac- FIG. 11. (Color online) The same as Fig. 10 but with used parameter values: M∗ = 0.89GeV,ΣvN = 0,Σs∆ = count the πNN and πN∆ vertex corrections due to −0.1GeV,Σv =0,g′ =1.,g′ =0.4,g′ =0.3. Migdal short-range correlations. Pronounced softening ∆ 11 12 22 ofthe in-mediumpionspectrumpresentinsimplermod- els does not appear in this approach and consequently pion-cloudmodelofthe nucleonandarelativisticallyco- Pauli blocking causes some suppression of the pion dis- variant self-consistent in-medium calculation of the pion tribution coming from the pion-nucleon state for an in- and delta baryon propagators taking into account nu- medium nucleon. However, enhancement results from a clear effects in the mean-field approximation. Starting carefultreatmentofthepion-deltastateasaconsequence with the free nucleonwe showedthat the observedd¯ u¯ of pion broadening and delta shift and broadening. − antiquark distribution can be well reproduced by suit- Theneteffectforpreferredparametervaluesisamod- able choiceofthe πNN andπN∆formfactorswithdelta est enhancement of pion light-cone-momentum distribu- vacuum propagator taking into account its free width. tion mostly concentrated around y 0.2 value. As a ≈ Using the same values for the form factors we com- consequence the DY cross-section ratio exceeds one for putedthepionlight-cone-momentumdistributionfornu- small values, typically less than 0.2, of the x variable 2 cleonsinanisospinsymmetricmediumwithdensity cor- forfixedx valuesorintegrationoveritcorrespondingto 1 responding to average densities of medium mass nuclei. some experimental cuts. The convolution with distribu-