Chiral dynamics and partonic structure at large transverse distances
aa r X i v : . [ h e p - ph ] J un JLAB-THY-09-1012
Chiral dynamics and partonic structure at large transverse distances
M. Strikman and C. Weiss Department of Physics, Pennsylvania State University, University Park, PA 16802, USA Theory Center, Jefferson Lab, Newport News, VA 23606, USA
We study large–distance contributions to the nucleon’s parton densities in the transverse coordi-nate (impact parameter) representation based on generalized parton distributions (GPDs). Chiraldynamics generates a distinct component of the partonic structure, located at momentum fractions x < ∼ M π /M N and transverse distances b ∼ /M π . We calculate this component using phenomenolog-ical pion exchange with a physical lower limit in b (the transverse “core” radius estimated from thenucleon’s axial form factor, R core = 0 .
55 fm) and demonstrate its universal character. This formula-tion preserves the basic picture of the “pion cloud” model of the nucleon’s sea quark distributions,while restricting its application to the region actually governed by chiral dynamics. It is found that(a) the large–distance component accounts for only ∼ / d − ¯ u at x ∼ .
1; (b) the strange sea quarks, s and ¯ s , are significantly more localized thanthe light antiquark sea; (c) the nucleon’s singlet quark size for x < . h b i q +¯ q > h b i g , as suggested by the t –slopes of deeply–virtual Compton scattering and exclusive J/ψ production measured at HERA and FNAL. We show that our approach reproduces the general N c –scaling of parton densities in QCD, thanks to the degeneracy of N and ∆ intermediate statesin the large– N c limit. We also comment on the role of pionic configurations at large longitudinaldistances and the limits of their applicability at small x . PACS numbers: 13.60.Hb, 12.39.Fe, 14.20.Dh, 11.15.Pg
Contents
I. Introduction II. Chiral dynamics and partonic structure b asymptotics 6D. Contribution to nucleon parton densities 7 III. Pion cloud model in impact parameterrepresentation b SU (3) flavor 11 IV. Large–distance component of the nucleonsea d − ¯ u u + ¯ d s, ¯ s u + ¯ d − s V. Transverse size of nucleon
VI. Pion cloud and large– N c QCD VII. Small x –regime and longitudinaldistances VIII. Summary and outlook Acknowledgments References A. Meson–baryon couplings from SU (3) symmetry B. Evaluation of coordinate–spacedistributions I. INTRODUCTION
Parton densities summarize the structure of the nu-cleon probed in high–momentum transfer processes suchas deep–inelastic lepton–nucleon scattering and produc-tion of high–mass systems (jets, heavy particles) innucleon–nucleon collisions. They are defined in the con-text of a factorization procedure, by which the cross sec-tion of these processes is separated into a short–distancequark/gluon subprocess, calculable in perturbative QCD,and the distribution of the partons in the initial state,and thus represent long–distance, low–energy character-istics of the nucleon. As such, they are governed bythe same low–energy dynamics which determines othernucleon observables like the vector and axial couplings(to which they are related by the partonic sum rules),form factors, meson–nucleon couplings etc.
Of partic-ular interest are the charge ( u − ¯ u, d − ¯ d ) and flavor( u − d, ¯ u − ¯ d ) non–singlet quark densities, which exhibitonly weak scale dependence and are of non–perturbativeorigin; they represent quasi–observables which directlyprobes the QCD quark structure of the nucleon at lowresolution scales.The long–distance behavior of strong interactions atlow energies is governed by the spontaneous breaking ofchiral symmetry in QCD. The Goldstone boson natureof the pion explains its small mass on the hadronic scaleand requires its coupling to other hadrons to vanish inthe long–wavelength limit. The resulting “chiral dynam-ics” gives rise to a number of distinctive phenomena atdistance scales ∼ /M π , such as the ππ, πN and N N interaction at large distances, the pion pole in the axialcurrent matrix element, etc.
An important question ishow chiral dynamics affects the nucleon’s parton densi-ties, and whether one can see any signs of chiral effectsin observables of high–momentum transfer processes.The prime candidate for an effect of chiral dynam-ics in parton densities has been the flavor asymmetryof the light antiquark densities in the nucleon. Mea-surements of the proton–neutron structure function dif-ference in inclusive deep–inelastic scattering [1], semi–inclusive meson production [2], and particularly Drell–Yan pair production [3, 4, 5] have unambiguously shownthat (cid:2) ¯ d − ¯ u (cid:3) ( x ) > x < .
3, and havepartly mapped the x –dependence of the asymmetry; seeRefs. [6] for a review of earlier experimental results. Thebasic picture is that the “bare” proton can make a transi-tion to a virtual state containing a pion, and fluctuations p → nπ + are more likely than p → ∆ ++ π − , resulting inan excess of π + over π − in the “dressed” proton. Fol-lowing the original prediction of Ref. [7], which includedonly the nucleon intermediate state, this idea was imple-mented in a variety of dynamical models, which incorpo-rate finite–size effects through various types of hadronicform factors associated with the πN N and πN ∆ ver-tices; see Refs. [6] for a review of the extensive litera-ture. It was noted long ago [8] that in order to reproducethe fast decrease of the observed asymmetry with x oneneeds πN N form factors much softer than those com-monly used in meson exchange parametrizations of the N N interaction [9]. However, even with such soft formfactors the pion transverse momenta in the nucleon gen-erally extend up to values ≫ M π [10]. This raises thequestion to what extent such models actually describelong–distance effects associated with soft pion exchange(momenta ∼ M π ), and what part of their predictionsis simply a parametrization of short–distance dynamicswhich should more naturally be described in terms ofnon-hadronic degrees of freedom. More generally, onefaces the question how to formulate the concept of the“pion cloud” in the nucleon’s partonic structure [11] in amanner consistent with chiral dynamics in QCD.A framework which allows one to address these ques-tions in a systematic fashion is the transverse coordi-nate (or impact parameter) representation, in which thedistribution of partons is studied as a function of the longitudinal momentum fraction, x , and the transversedistance, b , of the parton from the transverse center–of–momentum of the nucleon [12, 13]. In this representation,chiral dynamics can be associated with a distinct compo-nent of the partonic structure, located at x < ∼ M π /M N and b ∼ /M π . In a previous work [14] we have shownthat in the gluon density this large–distance componentis sizable and causes the nucleon’s average gluonic trans-verse size, h b i g to grow if x drops below M π /M N , inagreement with the t –slopes observed in exclusive J/ψ photo– and electroproduction at HERA [15, 16], FNAL[17], and experiments at lower energies. Essential in thisis the fact that in the gluon density (more generally, inany isoscalar parton density) the pion cloud contribu-tions from N and ∆ intermediate states have the samesign and add constructively. The special role of the ∆compared to other excited baryon states is supported bythe fact that in the large– N c limit of QCD the N and ∆are degenerate and enter on an equal footing.In this article we perform a comprehensive study of thechiral large–distance component of the nucleon’s partonicstructure, considering both its contribution to the totalquark/antiquark/gluon densities and to the nucleon’s av-erage partonic transverse size. The method we use tocalculate this component is phenomenological pion ex-change formulated in the impact parameter representa-tion, restricted to the region of large transverse distances.A physical lower limit in b for πB ( B = N, ∆) configu-rations in the nucleon wave function is set by the trans-verse “core” radius, estimated from the nucleon’s axialform factor, R core = 0 .
55 fm, and we explicitly demon-strate the universal character of the pionic contributionsin the region b > R core . This formulation preserves thebasic physical picture of the “pion cloud” model of thenucleon’s sea quark distributions, while restricting its ap-plication to the region actually governed by chiral dy-namics. In fact, our study serves both a conceptual anda practical purpose. First, we want to establish in whichregion of transverse distances the results of the tradi-tional pion cloud model are model–independent and canbe associated with large–distance chiral dynamics. Sec-ond, we want to employ this model to actually calcu-late the universal large–distance component and studyits properties. A preliminary account of our study of theflavor asymmetry ¯ d − ¯ u was presented in Ref. [18].The investigation reported here proceeds in severalsteps. In Sec. II, we develop the theory of large–distancecontributions to the partonic structure from a general,model–independent perspective. We outline the para-metric region of πB configurations in the nucleon wavefunction, the properties of the b –dependent momentumdistribution of pions in the nucleon, its large– b asymp-totics, and the convolution formulas for the nucleon par-ton densities. In Sec. III, we investigate the phenomeno-logical pion cloud model in the impact parameter repre-sentation, and demonstrate that at large b its predictionsbecome independent of the πN B form factors modelingthe short–distance dynamics. We also comment on theextension of this model to SU (3) flavor. In Sec. IV wethen apply this model to calculate the large–distance con-tributions to the sea quark distributions in the nucleon,including the isovector ( ¯ d − ¯ u ) and isoscalar (¯ u + ¯ d ) lightquark sea, the strange sea ( s, ¯ s ), and the SU (3)–flavorsymmetry breaking asymmetry (¯ u + ¯ d − s ). We com-pare the calculated large–distance contributions to em-pirical parametrizations of the parton densities and thusindirectly infer the contribution from the short–distanceregion (“core”), which cannot be calculated in a model–independent way. In the course of this we see how therestriction to large b solves several problems inherent inthe traditional pion cloud model which formally allowsfor pionic configurations also at small impact parameters.In Sec. V we consider the large–distance contributions tothe nucleon’s partonic transverse size h b i , which is acces-sible experimentally through the t –slope of hard exclusiveprocesses γ ∗ N → M + N ( M = meson , γ, etc. ). Becauseof the emphasis on large distances this quantity is cal-culable in a practically model–independent manner andrepresents a clean probe of chiral dynamics in the par-tonic structure. Specifically, we show that at x ∼ − the large–distance contribution to the nucleon’s singletquark transverse size, h b i q +¯ q , is larger than that to thegluonic size, h b i g , which is consistent with the observa-tion of a larger t –slope in deeply–virtual Compton scat-tering [19, 20] than in exclusive J/ψ production at HERA[15, 16]. In Sec. VI we discuss the correspondence of thephenomenological pion exchange contribution to the nu-cleon parton densities with the large– N c limit of QCD.In particular, we show that the large–distance contribu-tions obtained from pion exchange reproduce the general N c –scaling of parton densities in QCD, thanks to the de-generacy of N and ∆ intermediate states in the large– N c limit. This re-affirms the need to include intermediate∆ states on the same footing as the nucleon, and showsthat the phenomenological large–distance contributionsconsidered here are a legitimate part of the nucleons par-tonic structure in large– N c QCD. Finally, in Sec. VIIwe focus on the physical limitations to the picture of in-dividual πB configurations at small x , arising from thenon–chiral growth of the transverse sizes due to diffu-sion, and from chiral corrections to the structure of thepion. We also comment on the role of chiral dynamics atlarge longitudinal distances. Our summary and outlookare presented in Sec. VIII. The two appendices presenttechnical material related to the meson–nucleon couplingconstants for SU (3) flavor symmetry, and the numericalevaluation of the b –dependent pion momentum distribu-tions in the nucleon.In the context of our studies of the strange seaquark distributions, s ( x ) and ¯ s ( x ), and the SU (3) fla-vor symmetry–breaking asymmetry, (cid:2) ¯ u + ¯ d − s (cid:3) ( x ), weconsider also contributions from configurations contain-ing SU (3) octet mesons ( K Λ , K Σ , K Σ ∗ , ηN ) to the nu-cleon’s partonic structure at large distances. While suchhigh–mass configurations are not governed by chiral dy-namics and treated at a purely phenomenological level, it is interesting to compare their large–distance tails withthose of chiral contributions from pions. We note that theissue of the strange sea in the nucleon ( s, ¯ s ) and the ques-tion of possibly different x –distributions of s and ¯ s hasacquired new urgency following the results of the NuTeVexperiment in semi–inclusive charged–current neutrinoDIS, which can discriminate between s and ¯ s via the pro-cess W + + s → c [21, 22].Chiral contributions to the nucleon’s parton densitieshave been studied extensively within chiral perturbationtheory [23, 24], mostly with the aim of extrapolating lat-tice QCD results obtained at large pion masses towardlower values [25]. Chiral perturbation theory was alsoapplied to GPDs, including the impact parameter rep-resentation [26, 27, 28, 29]. Compared to these calcula-tions, which use methods of effective field theory basedon a power–counting scheme, we take here a more prag-matic approach. We study the pion distribution in thenucleon in a phenomenological approach which incorpo-rates the finite bare nucleon size through form factors,and investigate numerically in which region the resultsbecome insensitive to the form factors and can be at-tributed to universal chiral dynamics [62]. In this ap-proach we maintain exact relativistic kinematics (physi-cal pion and nucleon masses) and calculate distributionsof finite support, which are then analyzed in the differ-ent parametric regions and matched with the asymptotic“chiral” predictions. This also allows us to deal withthe strong cancellations between contributions from N and ∆ intermediate states in the isovector quark densi-ties, which are difficult to accommodate within a powercounting scheme. In fact, the cancellation becomes ex-act in the large– N c limit of QCD and ensures the proper1 /N c counting required of the isovector antiquark distri-bution in QCD [14].In this study we focus on chiral large–distance contri-butions to the nucleon’s partonic structure at moderatelysmall momentum fractions, x > ∼ − , which arise fromindividual πB ( B = N, ∆) configurations in the nucleonwave function. When extending the discussion towardsmaller x , several effects need to be taken into accountwhich potentially modify this picture. One is diffusion inthe partonic wave function, which causes the transversesize of the nucleon’s partonic configurations to grow atsmall x (however, this effect is suppressed at large Q ).Another effect are possible chiral corrections to the struc-ture of the pion itself, which were recently studied in anapproach based on resummation of chiral perturbationtheory in the leading logarithmic approximation [31]. Wediscuss the limitations to the applicability of the pictureof individual πB configurations in Secs. VII A and VII B.We also comment on the role of πB configurations atlarge longitudinal separations and arbitrary transversedistances, and point out that there may be a window fora chiral regime at x > ∼∼ − ; at smaller x coherenceeffects become dominant; see Sec. VII C. A detailed in-vestigation of this new regime will be the subject of aseparate study. II. CHIRAL DYNAMICS AND PARTONICSTRUCTUREA. Parametric region of chiral component
As the first step of our study we want to delineate theparametric region where parton densities are governedby chiral dynamics and establish its numerical limits, asimposed by other, non–chiral physical scales. The pri-mary object of our discussion is the pion longitudinalmomentum and transverse coordinate distribution in afast–moving nucleon, f π ( y, b ), where y is the pion mo-mentum fraction. Here we introduce this concept heuris-tically, appealing to its obvious physical meaning; its pre-cise definition in terms of GPDs and its region of appli-cability will be elaborated in the following.Chiral dynamics generally governs contributions to nu-cleon observables from large distances, of the order 1 /M π ,which is assumed here to be much larger than all otherhadronic length scales in question. These contributionsresult from exchange of “soft” pions in the nucleon restframe; in the time–ordered formulation of relativistic dy-namics these are pions with energies E π ∼ M π and mo-menta | k π | ∼ M π . Chiral symmetry provides that suchpions couple weakly to the nucleon and to each other, sothat their effects can be computed perturbatively. Boost-ing these weakly interacting pion–nucleon configurationsto a frame in which the nucleon is moving with large ve-locity, we find that they correspond to longitudinal pionmomentum fractions of the order [63] y ∼ M π /M N . (1)At the same time, the soft pions’ transverse momenta,which are not affected by the boost, correspond to trans-verse distances of the order b ∼ /M π . (2)Together, Eqs. (1) and (2) determine the parametric re-gion where the pion distribution in the fast–moving nu-cleon is governed by chiral dynamics, and the soft pioncan be regarded as a “parton” in the nucleon’s wave func-tion in the usual sense (see Fig. 1).The condition Eq. (1) implies that the pion momentumfraction in the nucleon is parametrically small, y ≪ i.e. , the soft pion is a “slow” parton. As a consequence,one can generally neglect the recoil of the spectator sys-tem and identify the distance b with the separation ofthe pion from the transverse center–of–momentum of thespectator system, r = b/ (1 − y ) [64] [12]. This circum-stance greatly simplifies the spatial interpretation of chi-ral contributions to the parton densities.Pionic configurations in the nucleon wave function arephysically meaningful only if the transverse separationof the pion and the spectator system is larger than thesum of the intrinsic “non–chiral” sizes of these objects.This basic fact imposes a limit on the applicability ofchiral dynamics, even though the dynamics itself may not ∼ b M π ∼ My π M N / / FIG. 1: Parametric region where the pion distribution in thenucleon is governed by chiral dynamics. The variables arethe pion longitudinal momentum fraction, y , and transverseposition, b . change dramatically at the limiting distance. In orderto make the picture of Fig. 1 quantitative, we have toestimate down to which values of b the concept of pionicconfigurations is applicable.The transverse size of the “core” in the nucleon’s par-tonic wave function in the valence region ( x > ∼ − ) canbe estimated from the transverse axial charge radius ofthe nucleon, which does not receive contributions fromthe pion cloud [14, 32]: h b i axial = h r i axial ≈ . , (3)where the factor 2 / R core = (cid:2) h b i axial (cid:3) / ≈ .
55 fm . (4)Equation (4) imposes a numerical lower limit for the pionimpact parameter, b , in pionic configurations. Note thatthis number represents a rough estimate, as the inter-pretation of RMS radius in terms of a “size” dependson the shape of the transverse distribution of partonsin the core. A more refined estimate, which takes intoaccount the intrinsic transverse size of the pion as wellas the effect of the recoil of the spectator system, is ob-tained by requiring that b/ (1 − y ) > ( R + R π ) / .Assuming that R π ranges between zero and R , andanticipating that the typical y –values in the pion distri-bution at b ∼ R core are y = (1 − × M π /M N ∼ .
2, weobtain b > . − .
62 fm, in good agreement with theestimate of Eq. (4). When considering the nucleon’s par-tonic structure at small x ( < − ) the above estimateof the nucleon core size needs to be modified to accountfor the non-chiral growth due to diffusion in the partonicwave function. Also, in this region the transverse size ofthe pion itself can grow due to chiral corrections. Theseeffects will be discussed separately in Secs. VII A andVII B.Chiral dynamics produces also configurations in thefast–moving nucleon characterized by large longitudinalseparations of the pion and the spectator system, l ∼ /M π , (5)with no restriction on b . The relevance of these config-urations for the nucleon’s partonic structure cannot beascertained without detailed consideration of the effec-tive longitudinal sizes of the subsystems and possible co-herence effects, and will be discussed in Sec. VII C. Inthe following we limit ourselves to chiral contributions atlarge transverse distances. B. Pion distribution in the nucleon
In its region of applicability defined by Eqs. (1) and (2),the b –dependent pion “parton” distribution can be cal-culated as the transverse Fourier transform of the “pionGPD” in the nucleon. The latter is defined as the transi-tion matrix element of the operator measuring the num-ber density of pions with longitudinal momentum frac-tion y in the fast–moving nucleon, integrated over thepion transverse momenta, and with a transverse momen-tum transfer ∆ ⊥ to the nucleon (see Fig. 2a): Z d k (2 π ) δ ( y − k k /P ) × h p | a † π,a ( k + ∆ / a π,a ( k − ∆ / | p i P →∞ = (2 π ) (2 P ) δ (3) ( p − p + ∆ ) H π ( y, t ) , (6)where p k = P → ∞ , ∆ k = 0, and t ≡ − ∆ ⊥ . (7)Here a † π,a and a π,a denote the pion creation and anni-hilation operators, and the sum over isospin projections(subscript a ) is implied. Eq. (6) refers to the helicity–conserving component of the nucleon transition matrixelement ( λ = λ ), and H π ( y, t ) is the correspondingGPD; the helicity–flip GPD is defined in analogously butwill not be needed in the present investigation. In termsof the pion GPD the transverse coordinate distributionis then obtained as ( b ≡ | b | ) f π ( y, b ) = Z d ∆ ⊥ (2 π ) e − i ( ∆ ⊥ b ) H π ( y, t ) . (8)We note that a manifestly covariant definition of the pionGPD, as the matrix element of a pionic light–ray operatorbetween nucleon states, was given in Ref. [14]; the equiv-alence of that definition to Eq. (6) is shown by going tothe frame where the nucleon is moving fast and expand-ing the pion fields in creation and annihilation operators.The pion GPD in the nucleon implies summation overall relevant baryonic intermediate states. Because the (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) k k p p PN π π N ∆ T Btt p t B (a)(b) s s yP FIG. 2: The pion GPD in the nucleon. (a) Transition ma-trix element of the density of pions with longitudinal momen-tum fraction y ∼ M π /M N and transverse momentum transfer | ∆ ⊥ | ∼ M π , Eq. (6). (b) Invariants used in modeling finite–size effects with form factors. t , are the pion virtualities inthe invariant formulation, Eq. (11); s , the invariant massesof the πB systems in the time–ordered formulation, Eq. (20). pion wavelength is assumed to be large compared to thetypical nucleon/baryon radius, only the lowest–mass ex-citations can effectively contribute to the GPD in theregion of Eqs. (1) and (2). We therefore retain only the N and ∆ intermediate states in the sum: H π = H πN + H π ∆ , (9) f π = f πN + f π ∆ . (10)The inclusion of the ∆, whose mass splitting with the nu-cleon introduces a non-chiral scale which is numericallycomparable to the pion mass, represents a slight depar-ture from strict chiral dynamics but is justified by thenumerical importance of this contribution; cf. the dis-cussion of the N c → ∞ limit in QCD in Sec. VI.To study the properties of the b –dependent pion dis-tribution at large distances we need a dynamical modelwhich allows us to calculate the pion GPD in the rele-vant region of momenta. Here we follow a heuristic ap-proach and start from the simplest possible system ofpointlike pions and nucleons interacting according to aphenomenological Lagrangian. We shall see later howthis definition can be amended to incorporate finite–sizeeffects. In which region the results should be regarded asphysical in the light of the discussion in Sec .II A will bethe matter of the following investigations.The pion GPD in the nucleon can be calculated usinginvariant perturbation theory, by evaluating the matrixelement in Eq. (6), or, equivalently, the matrix element ofthe pionic light–ray operator of Ref. [14], using the Feyn-man rules for pointlike πN interactions; see Ref. [14] fordetails. The resulting Feynman integral is computed byintroducing light–cone coordinates and performing theintegral over the “minus” (energy) component of the loopmomentum using Cauchy’s theorem. Closing the con-tour around the pole of the propagator of the spectatorbaryon, one arrives at a representation in which the spec-tator is on mass–shell, and the emitted and absorbed pionare off mass–shell, with virtualities [65] t , ≡ k , = − ( k ⊥ ∓ ¯ y ∆ ⊥ / / ¯ y + t min (11)(see Fig. 2b). Here k ⊥ is the transverse momentum ofthe spectator baryon, ¯ y ≡ − y, (12)and t min ≡ − (cid:2) y M N + y ( M B − M N ) (cid:3) / ¯ y (13)is the minimum virtuality required by kinematics for agiven pion momentum fraction, y . The πN and π ∆GPDs are then obtained as H πN ( y, t ) = 3 g πNN I ( y, t ; M π , M N ) , (14) H π ∆ ( y, t ) = 2 g πN ∆ I ( y, t ; M π , M ∆ ) . (15)Here g πNN and g πN ∆ are the coupling constants in theconventions of Ref. [14] and Appendix A, and the dis-tributions are the isoscalar pion GPDs, corresponding tothe sum of π + , π − and π distributions in the proton; cf. Eq. (6). The functions I and I denote the basictransverse momentum integrals arising in the calculationof the meson distribution with intermediate octet anddecuplet baryons, I , ( y, t ; M π , M B ) ≡ y π ¯ y Z d k ⊥ (2 π ) φ , ( t − M π )( t − M π ) , (16)where φ ≡ (cid:2) − t − t + ¯ y t + 2( M B − M N ) (cid:3) , (17) φ ≡ M N M (cid:2) M ( − t − t + t )+ ( M N − M − t )( M N − M − t ) (cid:3) × (cid:2) M ∆ + M N ) − t − t + ¯ y t (cid:3) . (18)Note that while the t , of Eq. (11) depend on the vector ∆ ⊥ , the integral Eq. (16) depends only on t ≡ − ∆ ⊥ because of rotational invariance in transverse space.As it stands, the transverse momentum integral inEqs. (16)–(18) is divergent. This divergence is related toshort–distance contributions in the pointlike particle ap-proximation and does not affect the chiral long–distancebehavior of the b –dependent distribution. Several waysof regularizing this divergence and extracting the chiralcontribution will be discussed in the following. The pion GPD can equivalently be evaluated in time–ordered perturbation theory, where Fig. 2a is interpretedas a process where the fast–moving nucleon (momentum P ≫ M N ) makes a transition to a πB intermediate state,in which we evaluate the operator measuring the densityof pions with longitudinal momentum yP , and then backto a nucleon state whose transverse momentum differsfrom the original one by ∆ ⊥ . In this formulation the in-termediate particles are on mass–shell, but the energies ofthe πB states before and after the operator are differentfrom that of the initial/final nucleon state. The invari-ant masses of the intermediate states, which are directlyproportional to the energies, are given by s , = ( k , + p B ) (19)= ( k ⊥ ∓ ∆ ⊥ / + M π y + k ⊥ + M B ¯ y − ∆ ⊥ y and k ⊥ ,∆ s , ≡ s , − M N = M π − t , y , (21)whence the denominators in Eq. (16) can also be inter-preted as “energy denominators.” The minimum value ofthe invariant mass difference, ∆ s min , for given momen-tum fraction y can be obtained by substituting t , by t min , Eq. (13). Both the invariant and the time–orderedformulation will be useful for discussing the propertiesof the chiral long–distance contribution following fromEqs. (16)–(18). C. Large– b asymptotics It is instructive to consider the asymptotic behavior ofthe distribution of pions for b → ∞ and fixed y . It isdetermined by the leading branch cut singularity of theGPD in the t –channel and can be calculated by applyingthe Cutkosky rules to the Feynman graphs of Fig. 2 withpointlike vertices [14]. The asymptotic behavior is of theform f πB ( y, b ) ∝ e − κ B bκ B b , (22)where B = N, ∆ , . . . denotes the intermediate baryon;the expression applies in principle also to higher–massstates, cf. the discussion below. The decay constant, κ B ,depends on the pion momentum fraction, y , and is di-rectly related to the minimum pion virtuality, Eq. (13), inthe invariant formulation, or the minimum invariant massdifference in the time–ordered formulation, cf . Eq. (21): κ B = 2 (cid:18) M π − t min ¯ y (cid:19) / . (23)To exhibit the y –dependence of the decay constant inthe parametric region of chiral dynamics, y ∼ M π /M N ,Eq. (1), we set y = η M π /M N , (24)where the scaling variable, η , is generally of order unity.Substituting Eq. (13) into Eq. (23) and dropping termssuppressed by powers of M π /M N , we obtain κ B = 2 (cid:20) (1 + η ) M π + η ( M B − M N ) M π M N (cid:21) / . (25)This result has several interesting implications:(a) For the nucleon intermediate state ( B = N ) thesecond term is zero, and one has κ N ∝ M π witha coefficient of order unity and depending on η .In this case the b –distribution exhibits a “Yukawatail” with a y –dependent range of the order 1 /M π ,as expected.(b) For a higher–mass intermediate state ( B = N )the decay constant is determined by competitionof the chiral scale, M π , and the non–chiral scale,( M B − M N ) M π /M N . The larger the N – B masssplitting, the smaller η has to be for the chiral scaleto dominate. This effect suppresses the contribu-tion of higher–mass baryons to f π ( y, b ) at large b and finite η . Note also that the pre-exponentialfactor, which is not shown in Eq. (22) for brevity,vanishes ∝ y for y → η → κ B → M π irrespective of the N – B mass splitting. In this limit the transverse“Yukawa tail” has the range one would naively ex-pect from the analogy with the 3–dimensional situ-ation. However, this limit is purely formal, as thisregion makes a vanishing contribution to the nu-cleon’s partonic structure at moderate x ; cf . thediscussion in Secs. II D and VII below.For pion momentum fractions parametrically of orderunity, y ∼
1, Eq. (23) gives a decay constant of the order κ B ∼ M N . An exponential decay with range ∼ /M N is not a chiral contribution to the pion distribution, asis expected, because the values of y lie outside the para-metric region of Eq. (1). In sum, the large– b asymptoticbehavior obtained from the naive pion distribution withpointlike πN couplings fully supports the general argu-ments of Sec. II A concerning the parametric region ofthe chiral component.One notes that the characteristic transverse rangeof the chiral contribution of the pion distribution,1 / (2 M π ) = 0 .
71 fm, is numerically not substantiallylarger than our estimate of the non–chiral “core” size,Eq. (4). This shows that an effective field theory ap-proach to chiral dynamics, which implicitly assumes thatthe core has zero size and builds up its structure bycounter terms, is not practical here, and underscoresthe rationale for our phenomenological approach, wherefinite–size effects are included explicitly.
D. Contribution to nucleon parton densities
The chiral contribution to the nucleon’s parton densi-ties is obtained as the convolution of the pion momen-tum distribution in the nucleon with the relevant par-ton distribution in the pion. For the gluon, the isoscalarquark/antiquark, and the isovector quark/antiquark den-sities it takes the form [66] g ( x, b ) chiral = Z x dyy [ f πN + f π ∆ ] ( y, b ) g π ( z ) , (26)[ u + d ] ( x, b ) chiral = (cid:2) ¯ u + ¯ d (cid:3) ( x, b ) chiral = Z x dyy [ f πN + f π ∆ ] ( y, b ) q tot π ( z ) , (27)[ u − d ] ( x, b ) chiral = (cid:2) ¯ d − ¯ u (cid:3) ( x, b ) chiral = Z x dyy (cid:2) f πN − f π ∆ (cid:3) ( y, b ) q val π ( z ) , (28)where z ≡ x/y (29)is the parton momentum fraction in the pion. Here f πN and f π ∆ are the isoscalar pion distributions (sumof π + , π − and π ) with N and ∆ intermediate states inthe conventions of Refs. [10, 14] and Appendix A; theisovector nature of the asymmetry, Eq. (28), is encodedin the numerical prefactors. The functions g π , q tot π , and q val π are the gluon, isoscalar (total), and isovector (va-lence) quark/antiquark densities in the pion, q tot π ( z ) = (cid:2) ¯ u + ¯ d (cid:3) π ± ,π ( z ) = [ u + d ] π ± ,π ( z )= (cid:2) u + ¯ u + d + ¯ d (cid:3) π ± ,π ( z ) , (30) q val π ( z ) = ± (cid:2) ¯ d − ¯ u (cid:3) π ± ( z ) = ± [ u − d ] π ± ( z )= ± (cid:2) u − ¯ u − d + ¯ d (cid:3) π ± ( z ); (31)the latter is normalized as Z dz q val π ( z ) = 1 . (32)The π does not have a valence distribution because ofcharge conjugation invariance, and we assume isospinsymmetry. Note that the parton densities in the pion, aswell as the result of the convolution integrals in Eqs. (26)–(28), depend on the resolution scale; we have suppressedthis dependence for brevity. The convolution formulasfor the strange antiquark density and the SU (3)–flavorsymmetry breaking asymmetry will be given in Sec. IV.The expressions in Eqs. (26)–(28) apply to parton mo-mentum fractions of the order x ∼ M π /M N but oth-erwise not exceptionally small, and transverse distances b ∼ /M π . In deriving them we have assumed thatthe “decay” of the pion into partons happens locally onthe transverse distance scale of the chiral b –distribution, b ∼ /M π (see Fig. 1). This is justified parametrically,as for the values of x under consideration the parton mo-mentum fraction in the pion does not reach small values( x < z < z .To see in which region of x the chiral contribution tothe isovector antiquark density is localized, it is conve-nient to write the convolution formula Eq. (28) in theform x (cid:2) ¯ d − ¯ u (cid:3) ( x, b ) chiral = Z x dy (cid:2) f πN − f π ∆ (cid:3) ( y, b ) × zq val π ( z ) , (33)where we have multiplied both sides of Eq. (28) by x andused Eq. (29) on the right–hand side. Now both functionsin the integrand vanish for small arguments: f πB ( y ) → y →
0, and zq val π ( z ) → z →
0. Noting that thevalence distribution zq val π ( z ) is localized around z ∼ / y ∼ M π /M N , we conclude that theconvolution produces a sea quark distribution in the nu-cleon centered around values x = yz ∼ (1 / × M π /M N ,in agreement with the general expectation. The sameargument applies to the bulk of the chiral isoscalar den-sity, Eq. (27), which arises mainly from the valence quarkcontent of the pion; only at very small x the non-valencequarks in the pion produce a distinct contribution. Notealso that the valence quark density in the pion at z ∼ / N and ∆ contributions add in the isoscalar sector, Eq. (27),while they partly cancel in the isovector sector, Eq. (28)[10]. This is contrary to the general expectation thatchiral effects manifest themselves mostly in the sea quarkflavor asymmetry ¯ d − ¯ u . The cancellation between N and∆ contributions in the isovector case becomes perfect inthe large– N c limit of QCD and restores the proper N c scaling of the isovector distributions; see Sec. VI.In principle one can use the asymptotic expressionsfor the pion distribution in the nucleon, Eqs. (22) and(23), to do a numerical estimate of the large–distancecontribution to the nucleon parton densities based onEqs. (26)–(28). This approach was taken in Ref. [14] toestimate the chiral contribution to the nucleon’s gluonictransverse size, h b i g , proportional to the b –weighted in-tegral of the impact–parameter dependent gluon density.Because of the weighting with b this quantity empha-sizes large transverse distances, and the estimates of the b –integrated chiral contribution are relatively insensitiveto the lower limit in b imposed in the integral (see alsoSec. V). In the present investigation we are interestedin the antiquark densities per se (not weighted with b ),where there is no such enhancement of large distances,and estimates of the chiral contribution are more sen-sitive to the lower limit in b . We therefore approachthis problem differently, by analyzing the phenomeno-logical pion cloud model (which incorporates finite–sizeeffects) and establishing down to which b the numericalpredictions are insensitive to the short–distance cutoff(Sec. III). The numerical evaluation of the long–distancecontribution based on Eqs. (26)–(28) will then be donebased on the results of this investigation (Secs. IV andV). III. PION CLOUD MODEL IN IMPACTPARAMETER REPRESENTATIONA. Modeling finite–size effects
For a quantitative study of the chiral large–distancecomponent in the nucleon’s partonic structure we needa dynamical model which allows us to compute the dis-tribution of pions beyond its leading asymptotic behav-ior. In addition, we must address the question down towhich values of b numerical study of this component ismeaningful, in the sense that it is not overwhelmed byshort–distance contributions unrelated to chiral dynam-ics. Ultimately, this question can only be answered in adynamical model which smoothly “interpolates” betweenthe chiral long–distance regime and the effective short–distance dynamics. Here we study this question in theframework of the phenomenological pion cloud model,where the short–distance dynamics is not treated explic-itly, but modeled by form factors implementing a finitehadronic size unrelated to chiral dynamics. This studyserves two purposes — it establishes what part of thepredictions of the traditional pion cloud model actuallyarises from the long–distance region governed by chiraldynamics, and it offers a practical way of computing thisuniversal long–distance contribution.In the phenomenological pion cloud model, the pionGPD in the nucleon is defined by the graph of Fig. 2, cf. Eqs. (16)–(18), in which now form factors are associatedwith the πN B vertices, rendering the transverse momen-tum integral explicitly finite. Two different schemes toimplement these form factors are commonly used andhave extensively been discussed in the literature. One,based on the invariant formulation in which the spectatorbaryon in on mass–shell, restricts the virtualities of theexchanged pions by inserting in Eq. (16) a form factor F (cid:18) M π − t , Λ (cid:19) (34)for each πN B vertex (see Fig. 2b). Here F ( a ) denotesa function of finite range which vanishes for a → ∞ ;for example, an exponential, exp( − a ), or the dipole formfactor, (1 + a ) − . These form factors can be comparedto those in the well–known meson exchange parametriza-tions of the N N interaction, where the exchanged pionis regarded as a virtual particle [9]. The other scheme,based on the time–ordered formulation, restricts the in-variant mass of the πB systems in the intermediate statesby form factors of the type [33] F (cid:18) s , − M N Λ (cid:19) (35)(see Fig. 2b). An advantage of this scheme is thatit preserves the momentum sum rule in the transition N → πB , i.e. , the longitudinal momentum distributionof the baryon B in the nucleon is given by f πB (1 − y ) for B = N, ∆ [33, 34]. The relation between the two differ-ent cutoff schemes can easily be derived from Eq. (21).Effectively, Λ = y Λ , (36) i.e. , a constant invariant mass cutoff amounts to a y –dependent virtuality cutoff which tends to zero as y → b region, the two schemeslead to rather different pion momentum distributions.The distributions at large b and y ∼ M π /M N , however,are dominated by vanishing pion virtualities viz. invariantmass differences, so that the results in the two schemesbecome effectively equivalent, up to small finite renor-malization effects. In the following numerical studies weshall employ the virtuality cutoff as used in Ref. [10]; theequivalence of the two schemes for our purposes will bedemonstrated explicitly in Sec. III C.We emphasize that we are interested in the pion cloudmodel with form factors only as a means to identify thechiral large–distance contribution and delineate the re-gion where it is universal and independent of the formfactors. We do not consider those aspects of the modelrelated to the fitting of data without restriction to largedistances (tuning of cutoff parameters, πN B couplings, etc. ); those have been discussed extensively in the litera-ture reviewed in Refs. [6]. B. Universality at large b We first consider the dependence of the pion distribu-tion in the nucleon on the impact parameter, b . Specif-ically, we want to demonstrate that it reproduces the“universal” chiral behavior Eq. (22) at large b , and in-vestigate for which values of b the distribution is sub-stantially modified by the form factors. To this end wecalculate the pion GPD by numerical evaluation of theloop integral, Eq. (16), with a virtuality cutoff of the typeof Eq. (34), and perform the transformation to the im-pact parameter representation according to Eq. (8); use-ful formulas for the numerical calculation are collected in π b f π N ( y , b ) [f m - ] b [fm] R core y = 0.07 y = 0.3formfactorpointlike FIG. 3: The transverse spatial distribution of pions in the nu-cleon, f πN ( y, b ), as a function of b , for values y = 0 .
07 and 0.3.Shown is the radial distribution 2 πb f πN ( y, b ), whose integralover b (area under the curve) gives the pion momentum dis-tribution. Solid lines:
Pion cloud model with virtuality cutoff(exponential form factor, Λ πN = 1 . Dashed line:
Distribution for pointlike particles, regulated by subtractionat ∆ ⊥ = 0; the integral over b does not exist in this case. Theestimated “core” radius, Eq. (4), is marked by an arrow. Appendix B. Figure 3 shows f πN ( y, b ) obtained with anexponential form factor (Λ virt = 1 . b for y = 0 .
07 and 0.3, which is 1/2 and 2times M π /M N , respectively. Also shown are the distribu-tions obtained with pointlike particles (no form factors),in which the loop integral was regularized by subtrac-tion at ∆ ⊥ = 0; this subtraction of a ∆ ⊥ –independentterm in the GPD corresponds to a modification of theimpact parameter distribution by a delta function term ∝ δ (2) ( b ), which is “invisible” at finite b [14]. One seesthat for b > ∼ . b both distributions in Fig. 3 exhibitthe universal asymptotic behavior derived earlier [14].It is interesting that the b –value where in Fig. 3 the“universal” behavior of f πN ( y, b ) sets in is numericallyclose to the transverse radius of the nucleon’s “core,”inferred earlier from independent considerations, R core ≈ .
55 fm, cf.
Eq. (4). This shows that the pion cloud modelcan safely be used to compute the large– b parton densitiesover the entire region defined by Eq. (4).Figure 4 illustrates the connection between the trans-verse distance and the pion virtualities in Eq. (16) from0 M e d i a n p i on v i r t u a lit y ( M π − t , ) [ G e V ] b [fm] R core y = 0.070.3 FIG. 4: The median pion virtuality in the unregularized in-tegral, Eqs. (16)—(18), as a function of b , for y = 0 .
07 (solidline) and 0 . , for which f πN ( y, b ) reaches half of itsvalue for Λ → ∞ , corresponding to the unregularized in-tegral. a different perspective. Shown there is the median pionvirtuality in the unregularized loop integral, defined asthe value of the virtuality cutoff, Λ , for which the reg-ularized f πN ( y, b ) reaches half of its value for Λ → ∞ ;the latter coincides with the value obtained by regular-ization through subtraction. The function f πN ( y, b ) isalways positive when evaluated with an exponential vir-tuality cutoff, and monotonously decreasing as a functionof Λ , so that the median value of Λ provides a sen-sible measure of the average virtualities in the integralEq. (16) for given y and b . One sees that the average pionvirtualities in the loop strongly decrease with increasing b , indicating the approach to the universal chiral region.We recall that the leading asymptotic behavior at b → ∞ is determined by quasi–on–shell pions, cf. the derivationin Sec. II C. C. Effective pion momentum distribution
We now want to investigate the distribution of pions atlarge transverse distances as a function of the momentumfraction, y . In keeping with our general line of approach,we do this by studying how the momentum distributionof the pion cloud model with form factors is modifiedwhen a restriction on the minimum b is imposed. Wedefine the effective momentum distribution of pions with ∫ d b Θ ( b > b ) f π N ( y , b ) y M π / M N b = 0 (full) b = 0.55 fm b = 1.1 fmvirtuality cutoff, Λ virt = 1.00 GeVinv. mass cutoff, Λ inv = 1.66 GeV FIG. 5: Effective momentum distribution of pions in πN configurations with impact parameters b > b , Eq. (37), inthe pion cloud model. Solid lines:
Distributions obtainedwith a virtuality cutoff, Eq. (34) (exponential form factor,Λ virt = 1 . b = 0 (full integral), b = 0 .
55 fm and b = 1 . Dashed lines:
Same for distributions obtainedwith an invariant mass cutoff, Eq. (34) (exponential form fac-tor, Λ virt = 1 .
66 GeV). The value of Λ virt was chosen suchthat it produces the same total number of pions ( y –integral)for the full distribution as the given virtuality cutoff. Thevalue y = M π /M N is indicated by an arrow. b > b as the integral Z d b Θ( b > b ) f πB ( y, b ) ( B = N, ∆); (37)for b = 0 we recover the momentum distribution ofpions in the traditional usage of the pion cloud model.Figure 5 (solid lines) shows the b –integrated distributionEq. (37), obtained with an exponential virtuality cutoff(Λ virt = 1 . b = 0 (full integral) as well as b = 0 .
55 fm and 1 . R core . One sees thatthe restriction to large b –values strongly suppresses largepion momentum fractions and shifts the strength of thedistribution toward values of the order y ∼ M π /M N ,in agreement with the general expectations formulatedin Sec. II. From the perspective of the traditional pioncloud model, the results of Fig. 5 show that less than halfof the pions in that model arise from the region b > R core ,where the pion cloud can be regarded as a distinct com-ponent of the nucleon wave functionAlso shown in Figure 5 (dashed lines) are the corre-sponding distributions obtained with an invariant masscutoff, Eq. (35). For the sake of comparison the cutoff pa-rameter Λ was chosen here such that it gives the sametotal number of pions ( y –integral) for the “full” distri-1butions in which no restriction on b is imposed ( b = 0);the value of Λ inv = 1 .
66 GeV thus obtained is withinthe range considered in phenomenological applications ofthe pion cloud model [35]. One sees that the full dis-tributions are quite different for the virtuality and theinvariant mass cutoff, as dictated by the relation (21).However, when restricted to large b the y –distributionsin the two regularization schemes become more and morealike, as their strength shifts toward values of the order y ∼ M π /M N . This explicitly demonstrates the equiva-lence of the virtuality and the invariant mass regulariza-tion in the context of our approach, as announced above. D. Extension to SU (3) flavor In our studies of the strange sea and the SU (3)–breaking flavor asymmetry below we shall consider alsothe contributions from K and η mesons to the sea quarkdistributions at large distances. Because the masses ofthese mesons are numerically comparable to the typicalhadronic mass scale (as given, say, by the vector mesonmass), their contributions to the partonic structure of thenucleon cannot be associated with chiral dynamics, evenat large transverse distances. Still, in the context of thepresent discussion of the pion cloud model, it is instruc-tive to study the distribution of K and η in the impactparameter representation, and contrast it with that ofthe π .The pseudoscalar octet meson couplings to the nu- ∫ d b Θ ( b > b ) f K Λ ( y , b ) yb = 0 (full) b = 0.55 fm b = 1.1 fm FIG. 6: Effective momentum distribution of kaons in K Λ con-figurations with impact parameters b > b , cf. Eq. (37), in themeson cloud model with virtuality cutoff (exponential formfactor, Λ virt = 1 . Solid line: b = 0 (full integral). Dashed lines: b = 0 .
55 fm and b = 1 . cleon, as determined by SU (3) flavor symmetry, andthe definition of their impact parameter–dependent mo-mentum distributions are summarized in Appendix A.Significant contributions come only from the K Λ and K Σ ∗ channels. The large– b behavior of these distribu-tions is formally governed by the asymptotic expression,Eqs. (22) and (23), with the π mass replaced by the K mass. Figure 6 shows the numerically computed effectivemomentum distribution of K in K Λ configurations, withand without restriction to large b , cf. Eq. (37), whichshould be compared to the corresponding distributionsfor the π in Fig. 5. One sees that the overall magnitudeof f K Λ is substantially smaller than that of f πN , becauseof the smaller coupling constant (no isospin degeneracy, cf . Appendix A) and the larger meson and intermediatebaryon mass. More importantly, one notes that the re-striction to large b suppresses the K distribution muchmore strongly than the π distribution; only about 1 / b > .
55 fm, and less than 1% are found at b > . K (and η ) contribution to the partonicstructure above the nucleon’s core radius, R core = 0 .
55 fmis extremely small.
IV. LARGE–DISTANCE COMPONENT OF THENUCLEON SEAA. Isovector sea ¯ d − ¯ u We now apply the formalism developed in Secs. II andIII to study the chiral large–distance contributions to thesea quark distributions in the nucleon. To this end, weevaluate the convolution formulas, Eqs. (27)–(28), withthe b –integrated pion distribution, Eq. (37), where thelower limit, b , is taken sufficiently large to exclude themodel–dependent small–distance region, cf. Fig. 3. Ourstandard value for this parameter is the phenomenolog-ical “core” radius, Eq. (4); variation of this value willallow us to estimate the sensitivity of the results to un-known short–distance dynamics. While not permitting acomplete description of the sea quark distributions, ourresults allow us to quantify how much comes from the“universal” large–distance region, providing guidance forfuture comprehensive models of the partonic structure.We first consider the isovector antiquark distributionin the proton, [ ¯ d − ¯ u ]( x ), which experiences only non–singlet QCD evolution and is largely independent of thenormalization scale. The convolution formula Eq. (28)involves the valence quark distribution of the pion; thenormalization of this distribution is fixed by Eq. (32), andits shape has been determined accurately by fits to the πN Drell–Yan data; see Ref. [36] and references therein.We use the leading–order parametrization of the valencedistribution provided in Ref. [36]; the differences to thenext–to–leading order parametrization are minor in thiscase. Figure 7 shows the chiral long–distance contribu-2 [ − d − − u ] ( x ) x Q = 54 GeV FNAL E866 data π N + π∆ , b > R core b > 0.8 (1.2) R core FIG. 7:
Solid line:
Large–distance contribution to the anti-quark flavor asymmetry, [ ¯ d − ¯ u ]( x ), obtained from πN and π ∆configurations restricted to impact parameters b > R core =0 .
55 fm, cf.
Eq. (37).
Dotted lines:
Same with b > . R core (upper line) and 1 . R core (lower line). Data:
Result of anal-ysis of final FNAL E866 Drell–Yan data [5]; statistical andsystematic errors were added in quadrature. All curves anddata points refer to the scale Q = 54 GeV . tion obtained when b is taken to be the phenomeno-logical “core” radius, R core = 0 .
55 fm (solid line), aswell as the band covered when b is changed from thisvalue by ±
20% (dotted lines). Also shown in the fig-ure are the results of an analysis of the final data fromthe FNAL E866 Drell–Yan experiment, presented at acommon scale Q = 54 GeV [5]. One sees that thelarge–distance contribution to the asymmetry is prac-tically zero for x > .
3, as expected from the generalconsiderations of Sec. II. At x ∼ . ∼
30% of the measured asym-metry, indicating that most of it results from the nu-cleon’s core at small transverse distances. This conclu-sion is robust and, as demonstrated in Sec. III, does notdepend on the form factors employed in the calculationwithin the pion cloud model (the specific results shownhere were obtained with an exponential virtuality cutoffwith Λ πN = 1 . π ∆ = 0 . x ( ∼ .
01) the large–distance contribution obtained inour approach comes closer to the data; however, the qual-ity of the present data is rather poor, and it is difficult toinfer the magnitude of the required “core” contributionby comparing the present estimate of the large–distancecontribution to the data in this region of x .One sees from Eq. (28) that the isovector antiquark dis-tribution involves strong cancellations between the con-tributions from πN and π ∆ intermediate states. This isnot accidental — the cancellation between the two be-comes exact in the large– N c limit of QCD, and is in fact necessary to restore the proper N c scaling of the isovectordistribution; see Sec. VI. B. Isoscalar sea ¯ u + ¯ d The isoscalar light antiquark distribution, [¯ u + ¯ d ]( x ),is subject to singlet QCD evolution and thus exhibitsstronger scale dependence than the isovector distribution.The convolution formula for this distribution, Eq. (27),involves the total (singlet) antiquark distribution in thepion, which we may write in the form q tot π ( z ) = q val π ( z ) + 2 q sea π ( z ) , (38)where q val π is the valence distribution, Eq. (31), and q sea π the “sea” distribution [67] q sea π = ¯ u π + = d π + = u π − = ¯ d π − . (39)The pion sea was determined within a radiative partonmodel analysis, supplemented by a constituent quark pic-ture which relates the pion to nucleon parton densities,and found to be relatively small [36]. Again, we use theleading–order parametrization for the parton densities inthe pion.Figure 8 shows our result for the chiral large–distancecontribution to the isoscalar antiquark distribution, sep-arately for the valence and sea distributions in the pionas well as the total, at the scale Q = 2 GeV . Onesees that the sea in the pion becomes important only at x ≪ M π /M N , where the antiquark momentum fraction x [ − u + − d ] ( x ) x Q = 2 GeV π N + π∆ , val, b > R core π N + π∆ , sea, b > R core π N + π∆ , tot, b > R core MSTW2008LO
FIG. 8:
Solid/dashed/dashed–dotted line:
Large–distancecontribution to the isoscalar antiquark density, x [¯ u + ¯ d ], result-ing from πN and π ∆ configurations restricted to b > R core =0 .
55 fm. The plot shows separately the contributions aris-ing from the valence, sea, and total antiquark density in thepion, cf.
Eq. (38).
Dotted line:
MSTW2008LO leading–orderparametrization [37]. z ≪
1. Altogether, thelarge–distance contribution accounts for only ∼ / u + ¯ d in the nucleon at x ∼ . πB ( B = N, ∆) configurations cannot be larger than the total an-tiquark distribution in the nucleon, which includes theradiatively generated sea. The large–distance contribu-tion calculated in our approach easily satisfies this the-oretical constraint, as can be seen from the comparisonwith the parametrization obtained in the recent leading–order global fit of Ref. [37] (MSTW20008LO), see Fig. 8.We note that the traditional pion cloud model without re-striction to large b , which generates pions with transversemomenta of the order ∼ ∼ ,produces an isoscalar sea which comes close to saturat-ing the empirical ¯ u + ¯ d at large x with the usual rangeof parameters parameters, and can even overshoot it forcertain choices [10, 35]. The restriction of πB configura-tions to large b in our approach solves this problem in amost natural way. C. Strange sea s, ¯ s The strange sea ( s, ¯ s ) in the nucleon at large distanceshas two distinct components. One is the chiral compo-nent, arising from s and ¯ s in the pion in πN and π ∆configurations. It is given by a similar convolution for-mula as the isoscalar sea, ¯ u + ¯ d , Eq. (27),¯ s ( x, b ) chiral = Z x dyy [ f πN + f π ∆ ] ( y, b ) ¯ s π ( z ) , (40)and similarly for s , where ¯ s π ( z ) and s π ( z ) are the strange(anti–) quark distributions in the pion. Assuming thatthe sea in the pion is mostly generated radiatively [36],we take them to be equal and proportional to the non–strange sea in the pion, Eq. (39),¯ s π ( z ) = s π ( z ) = q sea π ( z ) . (41)The other component comes from valence ¯ s quarks in KY ( Y = Λ , Σ , Σ ∗ ) and ηN configurations in the nu-cleon. Because the masses of these mesons are numer-ically comparable to the typical hadronic mass scale (asgiven, say, by the vector meson mass), their contributionto the partonic structure of the nucleon cannot strictly beassociated with chiral dynamics, even at large transversedistances. We include them in our numerical studies be-cause (a) it is instructive to contrast their contributionto those of πN and π ∆; (b) they contribute to ¯ s onlyand could in principle generate different x –distributionsfor s and ¯ s , as suggested by the model of Ref. [38] (weshall comment on this model below). The couplings ofthe octet mesons to the nucleon, as determined by SU (3)symmetry and the quark model value of the F/D ratio,as well as the definitions of the corresponding meson mo-mentum distributions are summarized in Appendix A. x s ( x ) , x − s ( x ) x Q = 2 GeV s = − s , π N + π∆ , b > R core − s , K Λ + K Σ * , b > R core s = − s , MSTW2008LO × FIG. 9:
Dashed line:
Large–distance contribution to thestrange sea in the nucleon, s = ¯ s , from πN and π ∆ con-figurations ( b > R core = 0 .
55 fm).
Solid line:
Large–distancecontribution to ¯ s from K Λ and K Σ ∗ configurations, involvingthe valence strange quark distribution in the kaon; K Σ and ηN are numerically negligible. Dotted line:
MSTW2008LOleading–order parametrization of the total s = ¯ s [37], multi-plied by 1 /
10 for easier comparison.
The contribution of K and η to ¯ s ( x, b ) in the proton isobtained as¯ s ( x, b ) = Z x dyy (cid:8) f ηN ( y, b ) ¯ s η ( z )+ [ f K Λ + f K Σ + f K Σ ∗ ] ( y, b ) ¯ s K ( z ) } , (42)where the factor 2 / η to be in a configuration with a valence ¯ s quark (weassume a pure octet state of the η and do not take intoaccount singlet–octet mixing, as the η contribution turnsout to be negligibly small anyway). The functions ¯ s η ( z )and ¯ s K ( z ) are the normalized momentum distributionsof ¯ s in η and K , Z dz ¯ s η,K ( z ) = 1 . (43)Assuming SU (3) symmetry, we will approximate thesedistributions by the valence quark distribution in thepion, ¯ s η,K ( z ) ≈ q val π ( z ) . (44)We again use the leading–order parametrization ofRef. [36] for the valence quark density in the pion. Nu-merical evaluation of the meson distributions shows thatthe contributions from ηN and K Σ in Eq. (42) are negli-gible because of their relatively small coupling; we retainonly the K Λ and K Σ ∗ terms in the following.Figure 9 shows the different large–distance contribu-tions to the strange sea, integrated over b > R core =40 .
55 fm. One sees that for x > . s coming from the valence ¯ s in K Λ and K Σ ∗ configurations; the precise magnitude ofthis contribution is sensitive to the lower limit in b , cf. Fig. 6. For x < . πB ( B = N, ∆) configurations, which contributesequally to s and ¯ s . The different mechanisms resultin s ( x ) = ¯ s ( x ) for the large–distance component of thestrange sea. However, the overall magnitude of the large–distance component represents only ∼ /
20 of the empir-ically determined average strange sea, [ s + ¯ s ] ( x ) [37],so that one cannot draw any conclusions about the x –distributions of the total s and s in the nucleon from thelarge–distance component. Note that the large–distancecomponent at b > R core represents a much smaller frac-tion of the total sea in the case of s and ¯ s than for ¯ u + ¯ d ,at least in the region x > ∼ . s + ¯ s ]( x ) obtained in the globalfits of Refs. [37] and [39]; up to a factor ∼ x = 0 . s and ¯ s are only a small fraction of thetotal. Also, some of the next–to–leading order fits by sev-eral groups [37, 40] have begun to extract information onthe shapes of s ( x ) and ¯ s ( x ) individually, by incorporatingneutrino scattering data which discriminate between thetwo. The difference [ s − ¯ s ]( x ) is very poorly determinedby the existing data, and the fits serve mostly to limitthe range of allowed values. We note that our approachto large–distance contributions and the convolution for-mulas of Eqs. (26)–(28) remain valid also for next–to–leading order parton densities, if the parton densities inthe pion are taken to be the next–to–leading order ones.In the present study we restrict ourselves to the lead-ing order, because at this order the parton densities arerenormalization–scheme–independent and possess a sim-ple probabilistic interpretation, and because the presentcomparison of our results with the data does not warranthigh accuracy.We would like to comment on the approach of Ref. [38],where the shapes of s ( x ) and ¯ s ( x ) were investigated ina light–front wave function model with K Λ components,whose amplitude was adjusted to fit the observed totalstrange sea, [ s +¯ s ]( x ). As just explained, our results showthat only a very small fraction of the total s and ¯ s seaarise from transverse distances b > R core ≈ .
55 fm wherethe notion of meson–baryon components in the nucleonwave function is physically sensible. Even in the tradi-tional meson cloud model without restriction to large b , K Λ configurations with standard form factors [10, 41]would account only for ∼ / s + ¯ s [37]. This shows that the assumption of saturationof the strange sea by K Λ configurations made in Ref. [38]would require a KN Λ coupling ∼ SU (3) value and is not realistic. While we see indica-tions for s ( x ) = ¯ s ( x ) in the large–distance contribution,and certainly nothing requires the shapes to be equal, the magnitude of the effect cannot be reliably predictedon the basis of the model of Ref. [38]. D. Flavor asymmetry ¯ u + ¯ d − s The antiquark SU (3) flavor asymmetry ¯ u + ¯ d − s isa non–singlet combination of the isoscalar non–strangeand strange sea, which exhibits only weak scale depen-dence. Since we assume SU (3) flavor symmetry of thesea quarks in the pion, Eq. (41), only the valence π and K components of ¯ u + ¯ d and ¯ s enter in this combination(we neglect the ηN and K Σ contributions): (cid:2) ¯ u + ¯ d − s (cid:3) ( x, b ) ≈ Z x dyy [ f πN + f π ∆ − f K Λ − f K Σ ∗ ] ( y, b ) ¯ q val π ( z ) . (45)The large–distance contribution to this asymmetry isshown in Fig. 10. One sees that the asymmetry over-whelmingly results from the valence ¯ u and ¯ d content ofthe pion in πN and π ∆ configuration; the ¯ s in the kaonof K Λ and K Σ ∗ contributes only at the level of < ∼ / SU (3) flavor asymmetryat x ∼ . x [ − u + − d − − s ] ( x ) x Q = 2 GeV π N + π∆ , , val, b > R core K Λ + K Σ * , val, b > R core MSTW2008LO
FIG. 10:
Solid line:
Large–distance contribution to the an-tiquark SU (3) flavor asymmetry asymmetry in the nucleon¯ u + ¯ d − s from valence ¯ u + ¯ d in the pion in πN and π ∆configurations, cf. Fig. 8 ( b > R core = 0 .
55 fm).
Dashedline:
Contribution from the valence ¯ s in the kaon in K Λ and K Σ ∗ configurations, cf. Fig. 9.
Dotted line:
Leading–orderparametrization of Ref. [37]. V. TRANSVERSE SIZE OF NUCLEONA. Transverse size and GPDs
An interesting characteristic of the nucleon’s partonicstructure is the average squared transverse radius of thepartons with given longitudinal momentum fraction x . Itis defined as h b i f ( x ) ≡ Z d b b f ( x, b ) f ( x ) ( f = q, ¯ q, g ) , (46)where f ( x, b ) is the impact parameter–dependent distri-bution of partons, related to the total parton density by Z d b f ( x, b ) = f ( x ) . (47)The average is meaningful thanks to the positivity of f ( x, b ) [12, 13]. Physically, Eq. (46) measures the aver-age transverse size of configurations in the nucleon wavefunction contributing to the parton density at given x .The transverse size implicitly depends also on the scale, Q ; this dependence arises from the DGLAP evolutionof the impact–parameter dependent parton distributionand was studied in Ref. [42].The average transverse quark/antiquark/gluon size ofthe nucleon is directly related to the t –slope of the cor-responding nucleon GPD at t = 0, h b i f ( x ) = 4 ∂∂t (cid:20) H f ( x, t ) H f ( x, (cid:21) t =0 . (48)Here H f ( x, t ) ≡ H f ( x, ξ = 0 , t ) denotes the “diagonal”GPD (zero skewness, ξ = 0), with H f ( x,
0) = f ( x ), whichis related to the impact parameter–dependent distribu-tion as ( b ≡ | b | ) H f ( x, t = − ∆ ⊥ ) = Z d b e − i ( ∆ ⊥ b ) f ( x, b ) . (49)For a general review of GPDs and their properties werefer to Refs.[43]. B. Transverse size from hard exclusive processes
By virtue of its connection with the GPDs, the trans-verse size of the nucleon is in principle accessible experi-mentally, through the t –slope of hard exclusive processes, γ ∗ ( Q ) + N → M + N ( M = meson , γ, . . . ) , at Q ≫ and | t | < ∼ , whose amplitudecan be calculated using QCD factorization and is propor-tional to the nucleon GPDs. In general, such processesrequire a longitudinal momentum transfer to the nucleonand probe the “non–diagonal” GPDs ( ξ = 0), so thatthe connection between the observable t –slope and the transverse size can be established only with the help of aGPD parametrization which relates the distributions at ξ = 0 to those at ξ = 0. The connection becomes simplein the limit of high–energy scattering, ξ ≈ x B / ≪ x ( φ, ρ ) and heavy vector mesonphoto/electroproduction ( J/ψ,
Υ) are proportional to thediagonal gluon GPD, and thus( dσ/dt ) γ ∗ N → V + N ∝ H g ( x = x B , t ) . (50)The gluonic average transverse size can be directly in-ferred from the relative t –dependence of the differentialcross section h b i g = 4 ∂∂t (cid:20) dσ/dt ( t ) dσ/dt (0) (cid:21) / t =0 . (51)The universal t –dependence of exclusive ρ and φ elec-troproduction at sufficiently large Q and exclusive J/ψ photo/electroproduction, implied by Eq. (50), is indeedobserved experimentally and represents an importanttest of the approach to the hard reaction mechanism;see Ref. [46] for a recent compilation of results.Experimental information on the nucleon’s gluonicsize and its dependence on x comes mainly from theextensive data on the t –dependence of exclusive J/ψ photo/electroproduction, measured in the HERA H1[15] and ZEUS [16] experiments, as well as the FNALE401/E458 [17] and other fixed–target experiments; seeRef. [47] for a recent summary. The t –dependence of thecross section measured in the HERA experiments is welldescribed by an exponential,( dσ/dt ) γN → J/ψ + N ∝ exp( B J/ψ t ) , (52)and assuming that this form is valid near t = 0, thenucleon’s average gluonic transverse size is obtained as h b i g = 2 B J/ψ . (53)For a more accurate estimate, the measured t –slopeis reduced by ∼ . − to account for the finitesize of the produced J/ψ . The exponential slope mea-sured by H1 at h W i = 90 GeV is B J/ψ = 4 . ± . +0 . − . GeV − [15], and ZEUS quotes a value of B J/ψ = 4 . ± . +0 . − . GeV − [16]. The central valuescorrespond to a transverse gluonic size at x ∼ − in therange h b i g = 0 . − .
35 fm , substantially smaller thanthe transverse size of the nucleon in soft hadronic interac-tions. It is also found that the gluonic size increases withlog(1 /x ) with a coefficient much smaller than the soft–interaction Regge slope, cf. the discussion in Sec. VII A.Comparatively little is known about the quark sizeof the nucleon at small x . As explained above, lightvector meson production at small x couples mainly to6the gluon GPD. Interesting new information comes fromthe t –dependence of deeply–virtual Compton scatter-ing (DVCS) recently measured at HERA. The H1 ex-periment [19] obtained an exponential slope of B γ =5 . ± . ± .
34 GeV − by measuring t through the pho-ton transverse momentum; larger by one unit than the J/ψ slope measured by the same experiment. ZEUS [20]extracted a DVCS slope of B γ = 4 . ± . ± . bymeasuring the transverse momentum of the recoiling pro-ton, again larger than the J/ψ slope measured by thatexperiment; however, the exponential fit to the ZEUSdata is rather poor and the extracted B γ has large errors.We note that in both experiments the B γ values were de-termined by an exponential fit over the entire measuredregion of t and thus reflect the average t –dependence, notdirectly the slope at t = 0. Still, the data provide someindication that the t –slope of DVCS at t = 0 is largerthan that of J/ψ production (the Q in the DVCS ex-periments here are comparable to the effective scale in J/ψ photoproduction, Q ≈ ). In leading–order(LO) QCD factorization, the DVCS amplitude is propor-tional to the singlet quark GPDs, and the t –slope of thisprocess is directly related to the nucleon’s singlet quarksize, h b i q +¯ q = 2 B γ , (54) cf. Eq. (53). One would thus conclude that h b i q +¯ q > h b i g . (55)At next–to–leading order (NLO) the DVCS amplitudealso involves the gluon GPD, and substantial cancella-tion is found between the gluon and singlet quark contri-butions to the amplitude. This cancellation amplifies theeffect of a difference in h b i q +¯ q and h b i g on the DVCS t –slope. Because the gluon contribution is negative andcancels ∼ / ∼ ξ = 0), and the analysis relies onGPD parametrizations. It is interesting that the t –slopeof ρ production measured in the recent CLAS experi-ment [50] seems to be compatible with the Regge–basedGPD parametrization of Ref. [51] (however, it is presentlyunclear how to describe the absolute cross section withinthis framework). A detailed phenomenological study ofthe transverse distribution of valence quarks, based onparton densities and form factor data, was performed inRef. [52]. C. Chiral contribution
We now want to study the contribution of the chirallarge–distance region, b ∼ /M π , to the nucleon’s averagetransverse size. Adopting a two–component description,we define h b i f = Z d b b [ f ( x, b ) core + Θ( b > b ) f ( x, b ) chiral ] f ( x ) ≡ h b i f, core + h b i f, chiral . (56)Here f ( x, b ) core denotes the parton density arising fromaverage configurations in the nucleon, distributed overtransverse distances b ∼ R core . The function f ( x, b ) chiral is the chiral component of the parton distribution, ex-tending over distances b ∼ /M π . Following the sameapproach as above, we integrate it over b with a lowercutoff, b , of the order of the core radius, Eq. (4); thesensitivity of the results to the precise value of b will beinvestigated below. Note that in Eq. (56) the b –weightedintegral in the numerator is computed in two separatepieces, while the denominator in both cases is the totalparton density (core plus chiral) at the given value of x ;the h b i f, chiral thus defined represents the contribution ofthe chiral component to the overall transverse size of thenucleon, not the “intrinsic” size of the chiral componentalone.The “core” contribution to h b i was estimated inSec. II A and Ref. [14], by relating it to the slope ofthe nucleon’s axial form factor, which does not receivecontributions from the pion cloud: h b i core ≈ h r i axial = 0 . . (57)We have already used this result to fix the short–distance cutoff in the integral over the chiral contribu-tion. A more quantitative determination of the “non–chiral” transverse sizes of the nucleon, including the dif-ferences between quarks, antiquarks and gluons and their x –dependence, requires a dynamical model of the nucleonwhich smoothly interpolates between small and large dis-tances and will be the subject of a separate study. Herewe focus on the chiral contribution, h b i f, chiral , whichcan be calculated in a model–independent manner; wecompare it to the “generic” core size given by Eq. (57),keeping in mind that the latter may have a richer struc-ture than reflected by this simple estimate.The chiral contribution to the transverse size, Eq. (56),is obtained by calculating the b –weighted integral of the b –dependent pion momentum distribution in the nucleonstudied in Sec. III, cf. Eq. (37), and substituting the re-sult in the convolution formula for the nucleon partondensity, Eq. (27) et seq.
Useful formulas for the numeri-cal evaluation of the b –weighted integrals are presentedin Appendix B. Because of the weighting factor b , thechiral contribution to the transverse size is much less sen-sitive to unknown short–distance dynamics ( i.e. , to thecutoff b ) than the contribution to the parton density7 -2 -1 〈 b 〉 f , c h i r a l [f m ] x f = gu + − u + d + − ds + − s FIG. 11: The chiral large–distance contribution to the nu-cleon’s average transverse size, h b i f , as defined by Eq. (56),as a function of x ( Q = 3 GeV ). Solid line:
Gluonicsize ( f = g ), cf. Ref. [14].
Dashed line:
Singlet quark size( f = u + ¯ u + d + ¯ d ) Dotted line:
Strange quark size ( f = s + ¯ s ).In all cases, the curves show the sum of contributions from πN and π ∆ configurations. itself, and thus represents a much more interesting quan-tity for studying effects of chiral dynamics in the par-tonic structure. Furthermore, the b –weighted integralcan reliably be computed using the asymptotic form ofthe distribution of pions at large b , Eq. (22), as was donein Ref. [14]. We can use this to estimate analytically thesensitivity of h b i f, chiral to the lower limit, b . Evaluatingthe integral I ≡ Z d b Θ( b > b ) b f πN ( y, b ) (58)with the asymptotic expression Eq. (22), and taking thelogarithmic derivative with respect to b , we obtain − b I ∂I∂b ≈ ≪ y = M π /M N ) , (59)where we have used Eq. (25) for κ N and b = R core =0 .
55 fm. This shows that the sensitivity of h b i chiral isindeed low — a 20% change in b causes only a 4% changein h b i f, chiral .Our results for the chiral contribution to the nucleon’saverage transverse size and its dependence on x are sum-marized in Fig. 11, for the scale Q = 3 GeV . The curvesshown are the sum of contributions from πN and π ∆ con-figurations; heavier mesons make negligible contributionsat large distances. For reasons of consistency the nucleonparton densities in the denominator of Eq. (56) were eval-uated using the older parametrization of Ref. [53], whichserved as input to the analysis of the pion parton dis-tributions of Ref. [36]. The following features are worthmentioning: (a) The chiral contribution to the transverse size ispractically zero above x ∼ M π /M N ∼ . x drops below this value, in agree-ment with the basic picture described in Sec. II.The rise of h b i f, chiral with decreasing x is morepronounced than that of the parton density itselfbecause the former quantity emphasizes the contri-butions from large distances.(b) The singlet u – and d –quark size grows more rapidlywith decreasing x than the gluonic radius. This hasa simple explanation: the quark/antiquark densityin the pion sits at relatively large momentum frac-tions z ∼ .
5, while the gluon density in the pionrequires z < . z = x/y in the convolution integral, and the pion momen-tum fractions are of the order y ∼ M π /M N , therelevant values of z are reached much earlier forthe quark than for the gluon as x decreases below M π /M N . Thus, the chiral large–distance contribu-tion suggests that the transverse quark size of thenucleon at x < ∼ .
01 is larger than the transversegluon size, cf.
Eq. (55). The difference between thechiral contribution to the average sizes at x = 0 . h b i q +¯ q, chiral − h b i g, chiral = 0 .
09 fm . (60)Assuming identical core sizes for the quark andgluon distribution, this would correspond to a dif-ference of the leading–order DVCS and J/ψ t –slopes, cf.
Eqs. (53) and (54), B γ − B J/ψ = 1 . , (61)well consistent with the HERA results summarizedin Sec. V B. It should be remembered that the chi-ral prediction, Eq. (61), is for the exact t –slope ofthe cross section at t = 0, while the HERA resultsrepresent effective slopes, obtained by fitting theempirical t –dependence over the measured range;the comparison may be affected by possible devia-tions of the true t –dependence from the exponen-tial shape. More quantitative conclusions wouldrequire detailed modeling of the core contributionsto the transverse size, which themselves can growwith decreasing x due to diffusion, see Sec. VII A.(c) The chiral contribution to the transverse strangequark size of the nucleon closely follows that to thegluonic size. This is natural, as s + ¯ s is mostlygenerated radiatively, by conversion of gluons into s ¯ s , in both the pion and the nucleon. VI. PION CLOUD AND LARGE– N c QCD
The relation of the chiral component of the large– b par-ton densities to the large– N c limit of QCD is a problemof both principal and practical significance. First, in the8large– N c limit QCD is expected to become equivalent toan effective theory of mesons, in which baryons appear assolitonic excitations, establishing a connection with thephenomenological notion of meson exchange. Second, ourcalculations show that contributions from ∆ intermediatestates are numerically large, and the large– N c limit pro-vides a conceptual framework which allows one to treat N and ∆ states on the same footing and relate theirmasses and coupling constants. We now want to verifythat the large–distance component of the nucleon’s par-tonic structure, calculated from phenomenological pionexchange, exhibits the correct N c –scaling required of par-ton densities in QCD ( cf. also the discussion in Ref. [14]).The general N c scaling of the unpolarized quark den-sities in the nucleon in QCD is of the form [54] g ( x ) ∼ N c × function( N c x ) , (62)[ u + d ]( x ) , [¯ u + ¯ d ]( x ) ∼ N c × function( N c x ) , (63)[ u − d ]( x ) , [¯ u − ¯ d ]( x ) ∼ N c × function( N c x ) , (64)where the scaling functions are stable in the large– N c limit but can be different between the various distribu-tions. Equations (63) and (64) were derived by assum-ing non–exceptional configurations ( x ∼ N − c ) and fixingthe normalization of the scaling function from the lowestmoments of the parton densities, i.e. , from the condi-tions that the total number of quarks scale as N c , andthe nucleon isospin as N c . The transverse coordinate–dependent parton distributions should generally scale inthe same manner as the total densities, Eqs. (63) and(64), as the nucleon radius is stable in the large– N c limit(this applies even to the nucleon’s chiral radii, because M π ∼ N c ).Turning now to the pion cloud contribution to the par-ton densities at large b , it follows from the expressionsof Eqs. (14)–(18) and their Fourier transform, Eq. (8),that the b –dependent distributions of pions in the nu-cleon scale as [14] f πN ( y, b ) , f π ∆ ( y, b ) ∼ N c × function( N c x ) . (65)This behavior applies to values y ∼ M π /M N ∼ N − c and values b ∼ N c , corresponding to | t | ∼ N c in thepion GPD. In arriving at Eq. (65) we have used that M N , M ∆ ∼ N c ; that g πNN ∼ N / c , as implied theGoldberger–Treiman relation; and that g πN ∆ scales inthe same way as g πNN . Equation (65) states that themomentum distribution of pions in the nucleon at large N c scales like that of isoscalar quarks or gluons. At thesame time, the parton densities in the pion scale as g π ( z ) , q π ( z ) ∼ function( z ) , (66)where z ∼ N c in typical configurations; that is, theyhave no explicit N c dependence at large N c . One thusconcludes that the N c –scaling of the convolution integralfor the pion cloud contribution to the nucleon’s antiquarkdensities, for both B = N and ∆ intermediate states, is Z x dyy f πB ( y, b ) q π ( z ) ∼ N c × function( N c x ) . (67) This correctly reproduces the general N c –scaling of theisoscalar quark and gluon distribution, Eq. (63), wherethe N and ∆ contributions are added, cf. Eq. (28). How-ever, it may seem that the pion cloud contribution atlarge b cannot reproduce the N c scaling of the isovectordistribution, Eq. (63), which is suppressed by one power.The paradox is resolved when one notes that in the large– N c limit the N and ∆ become degenerate, M ∆ − N N ∼ N − c , (68)and their couplings are related by [55] g πN ∆ = g πNN . (69)Using these relations one has f π ∆ ( y, b ) = 2 f πN ( y, b ) ( y ∼ N − c ) , (70)as can be seen from Eqs. (14)–(18) and Eq. (8), keepingin mind that t, t , t ∼ N c in the region of interest. Byvirtue of Eq. (70) the N and ∆ contributions at large N c cancel exactly in the isovector convolution integral,Eq. (64), ensuring that the result has the proper N c –scaling behavior as Eq. (64).In sum, our arguments show that the pion exchangecontribution at large b is a legitimate part of the nu-cleon’s partonic structure in large– N c QCD, exhibitingthe same scaling behavior as the corresponding “aver-age” distributions. The inclusion of π ∆ configurationsat the same level as πN is essential because they repro-duce the proper N c –scaling of the isovector distributions,and because they make numerically sizable contributions— twice larger than πN — to the isoscalar distributions.In Ref. [14] we have shown that the isoscalar large– b pion distribution in the nucleon [ f πN + f π ∆ ] ( y, b ) ob-tained from phenomenological soft–pion exchange, canequivalently be computed in the chiral soliton pictureof the nucleon at large N c , as a certain longitudinalFourier transform of the universal classical pion fieldof the soliton at large transverse distances. Extend-ing this connection to the isovector pion distribution, (cid:2) f πN − f π ∆ (cid:3) ( y, b ), which is suppressed in the large– N c limit, remains an interesting problem for furtherstudy. In particular, this requires establishing the con-nection between soft–pion exchange and the collectiverotations of the classical soliton. VII. SMALL x –REGIME AND LONGITUDINALDISTANCESA. Growth of core size through diffusion In our studies so far we have focused on chiral contri-butions to the nucleon’s partonic structure at moderatelysmall momentum fractions, x > ∼ − , which arise fromindividual πB ( B = N, ∆) configurations in the nucleon9wave function. When considering smaller values of x sev-eral effects must be taken into account which potentiallylimit the validity of the present approximations.One of them is the growth of the transverse size of“average” partonic configurations in the nucleon due todiffusion. Generally, the partons at small x are decayproducts of partons at larger x ; the decay process has thecharacter of a random walk in transverse space and leadsto a logarithmic growth of the transverse area occupiedby the partons: h b i parton = h b i parton ( x ) + 4 α ′ parton ln( x /x )( x < x ∼ − ) . (71)The rate of growth — the effective Regge slope, α ′ parton — depends on the type of parton and generally decreaseswith increasing scale Q , because higher Q increase theeffective transverse momenta in the decay process [42].Measurements of the energy dependence of the t –slope ofexclusive J/ψ production at HERA H1 and ZEUS [15, 16]indicate that the rate of growth for gluons at a scale Q ≈ is approximately α ′ g ≈ .
14 GeV − [68]significantly smaller than the rate of growth of the trans-verse nucleon size in soft hadronic interactions, α ′ soft ≈ .
25 GeV − . Using the former value as a general measureof the rate of growth of the nucleon’s transverse size dueto diffusion, we estimate that at Q ≈ the trans-verse size of the “core” increases from R = 0 . at x = 10 − to 0 .
35 (0 .
4) fm at x = 10 − (10 − ). Inprinciple this effect pushes the region of πB configura-tions governed by chiral dynamics out to larger b as x decreases. However, the rate of growth at this scale is stillrather small, leaving ample room for such configurationsin the region x > − . Note that at lower scales the rateof growth is larger; studies based on DGLAP evolutionshow that α ′ g approaches the soft value at Q ∼ . [42]. B. Chiral corrections to pion structure
Another effect which needs to be taken into accountat small x are modifications of the parton density inthe pion itself due to chiral dynamics. The same mech-anism as discussed above for the nucleon in principleworks also in the pion itself — the pion can fluctuateinto configurations containing a “slow” pion and a two–pion spectator system. When evaluated in chiral pertur-bation theory, the momentum fraction of the slow pionrelative to its parent in such configurations is of the or-der y ( π in π ) ∼ M π / (4 πF π ), where F π is the pion decayconstant, and 4 πF π represents the generic short–distancescale appearing in the context of the renormalization ofthe loop integrals. Such contributions to the parton den-sity and the GPD in the pion were recently computedin an all–order resummation of the leading logarithmicapproximation to chiral perturbation theory [31], whichdoes not require knowledge of the higher–order terms in the chiral Lagrangian. For the nucleon parton densitiesthis mechanism could become important for x ≪ − ,where the effective parton momentum fractions in thepion can reach small values z < ∼ .
1. In the presentstudy we restrict ourselves to nucleon parton densities at x > ∼ − , for which the convolution integrals are dom-inated by “non–chiral” values of z . The incorporationof such corrections to the partonic structure of the pionand extension of the present nucleon structure calcula-tion toward smaller x remains an interesting problem forfuture study. In particular, it should be investigated howthe expressions derived in the leading–log approximationof chiral perturbation theory compare to a “single–step”calculation of pion structure including finite mass andsize (form factors), along the lines done here for the nu-cleon. C. Chiral dynamics at large longitudinal distances
In our studies in Secs. II–V we considered chiral con-tributions to the nucleon’s partonic structure at largetransverse distances, which arise from πB configurationsat large transverse separations, b ∼ /M π . As already in-dicated in Sec. II A, there is in principle another class of πB configurations governed by chiral dynamics, namelythose corresponding to large longitudinal separations inthe nucleon rest frame, l ∼ /M π , (72)and arbitrary values of the transverse separation, downto b = 0. We now want to discuss in which region of x such configurations can produce distinct contributions tothe partonic structure.The main limitation in admitting πB configurations ofthe type Eq. (72) as part of the partonic structure arisesfrom the possible longitudinal overlap of the relevant par-tonic configurations in the pion and the “core.” To deter-mine the region where this effect plays a role, it is usefulto consider instead of the parton densities the structurefunction for γ ∗ N scattering and appeal to the notion ofthe coherence length of the virtual photon. Contributionsto the partonic structure of the type of the convolutionintegrals of Eqs. (26)–(28) correspond to the impulse ap-proximation of γ ∗ N scattering, which requires that thecoherence length of the process be smaller than the longi-tudinal distance between the constituents, so that inter-ference effects can be neglected; see e.g. Ref. [56]. Gen-erally, the coherence length for γ ∗ N scattering in thenucleon rest frame is given by l coh = (2 M N x ) − , (73)where M N is the nucleon mass and x ≈ Q /W ≪ W is the center–of–mass energy ofthe scattering process. Thus, one would naively thinkthat in scattering from a πN system with longitudinalseparation ∼ (2 M π ) − coherence effects set in if x < M π /M N ∼ .
1. However, this argument neglects the factthat in the fast–moving nucleon the pion carries only afraction of the order y ∼ M π /M N ∼ . γ ∗ π scattering is actually lower by this factor, and thecoherence length smaller by this factor, than in the γ ∗ N process. Interference effectively takes place only whenthe coherence length for both scattering on the pion andon the baryon in the πB configuration is ∼ (2 M π ) − ,which requires x < ∼ . . (74)For larger values of x coherence effects are small, andthere is in principle room for a chiral component of thepartonic structure at small b and longitudinal distances ∼ /M π . In order to calculate this component one wouldneed to model the finite–size effects limiting the longi-tudinal extension of the pion and the spectator system,which is related to the “small– x behavior” of the par-ton densities of the respective systems. We leave thisproblem to a future study. Interestingly, this could re-sult in partial “readmission” of the small– b component ofthe pion cloud model which was excluded in the presentstudy, potentially affecting e.g. the comparison with themeasured flavor asymmetry ¯ d − ¯ u in Sec. IV A.We note that the interference effects in scattering from πB configurations described here are large in the regionin which chiral corrections to the structure of the pionwould become important, cf. the discussion in Sec. VII B.An interesting question is whether in the chiral perturba-tion theory approach these effects come into play alreadyat the level of the leading logarithmic approximation [31],or only at the level of subleading or finite terms. VIII. SUMMARY AND OUTLOOK
The transverse coordinate representation based onGPDs represents a most useful framework for study-ing the role of chiral dynamics in the nucleon’s partonicstructure. It allows one to identify the parametric re-gion of the chiral component ( x < ∼ M π /M N , b ∼ /M π )and provides a practical scheme for calculating it in amodel–independent way. Let us briefly summarize themain results of our investigation.(a) The contributions from πB ( B = N, ∆) configu-rations to the parton distributions become inde-pendent of the πN B form factors at transversedistances b > ∼ . b approximately coincides with the nucleon’s coreradius, R core = 0 .
55 fm, inferred previously fromother phenomenological considerations.(b) Only ∼ / d − ¯ u ]( x ) at x > − comes from the large–distance region b > R core , showing that most of it resides in the nucleon’s core at small transversedistances. The traditional pion cloud model, whichattempts to explain the entire asymmetry from pio-nic contributions, gets most of the effect from small b where the concept of πB configurations is not ap-plicable.(c) The isoscalar antiquark distribution [¯ u + ¯ d ]( x ) ob-tained from pions at large b remains safely be-low the total antiquark distribution determined byQCD fits to deep–inelastic scattering data, leav-ing room for the (non–perturbatively and pertur-batively generated) antiquarks in the core. Thisnaturally solves a problem of the traditional pioncloud model, where the pionic contribution can sat-urate or even exceed the total antiquark density forcertain non-exceptional parameter values.(d) The strange sea quark distributions, s ( x ) and ¯ s ( x ),overwhelmingly sit at small transverse distances, b < R core . Neither chiral ( πN, π ∆) nor K Λ con-figurations at large b account for more than a fewpercent of the empirical s + ¯ s . The predictions ofRef. [38] for the x –dependence of s ( x ) and ¯ s ( x ) from K Λ fluctuations rely on the region where the con-cept of distinct meson–baryon configurations is notapplicable and require a probability of K Λ fluctu-ations several times larger than what is obtainedfrom the standard SU (3) couplings.(e) The pionic contributions to the nucleon’s transversesize, h b i , are much less sensitive to short–distancedynamics than those to the parton distributionsthemselves, and thus furnish a new set of clean chi-ral observables. The large–distance contributionsto the nucleon’s singlet quark size at x < . h b i q +¯ q > h b i g , in agreement with the pat-tern of t –slopes of deeply–virtual Compton scat-tering and exclusive J/ψ production measured atHERA and FNAL.In the present study we have limited ourselves to theuniversal large–distance contributions to the partonicstructure, which are governed by soft pion exchange andcan be calculated in a model–independent way. A com-plete description should include also a model of the short–distance part, which actually carries most of the partondensities. One way of combining the two would be atwo–component picture, in which the constituents in the“core” act as a source of the chiral pion fields which prop-agate out to distances ∼ /M π . Such an approach wouldbe very effective if the characteristic transverse sizes ofthe “cloud” and the “core” were numerically very differ-ent. However, this is not the case — the characteristicrange of two–pion exchange 1 / (2 M π ) = 0 .
71 fm is numer-ically not much larger than our estimate of the “core”size, R core = 0 .
55 fm. Another approach, which appearsmore promising, is based on the idea of a smooth “in-terpolation” between the chiral large–distance dynamics1and the short–distance regime. In particular, the effec-tive theory of Ref. [57], which is based on the large– N c limit of QCD, uses constituent quarks as interpolatingdegrees of freedom; it is valid in a wide region, from dis-tances of the order ∼ /M π down to distances of theorder ρ ≈ . N c description ofnucleon structure is equivalent to phenomenological soft–pion exchange at large transverse distances, thanks to theuniversality of chiral dynamics; it thus, in a sense, con-tains the result of the present work as a limiting case. Us-ing this large– N c picture as a script to model the impactparameter–dependent parton densities at all b , would cer-tainly be an interesting problem for further study.Direct experimental study of the chiral component ofthe nucleon’s partonic structure through hard exclusiveprocesses at x < . t –dependences of the differential cross sections forvarious channels ( J/ψ, φ, ρ, π ) at | t | ≪ . , andtheir change with x ; at sufficiently large Q such mea-surements can be related directly to the t –dependence ofthe gluon and quark GPDs at small t . In particular, suchmeasurements should be able to resolve variations of the t –slope with t and possible deviations from exponential t –dependence. Measurements of exclusive processes requirehigh luminosity and the capability to detect the recoilingbaryon at small angles, which is possible with appropriateforward detectors. Another interesting option are pionknockout processes, corresponding to exclusive scatter-ing from a pion at transverse distances b ∼ /M π , whereboth the recoiling pion and the nucleon are identified inthe final state; see Ref. [14] for a detailed discussion.The partonic content of the nucleon’s pion cloud can inprinciple also be probed in high–energy pp collisions with hard processes, such as dijet and Drell–Yan pair produc-tion. Such processes, including accompanying spectatorinteractions, are most naturally described in the trans-verse coordinate (impact parameter) representation em-ployed in our investigation here. Interesting new effectsappear in collisions at multi–TeV energies (LHC), wherethe cross sections for hard processes can approach the ge-ometric limit (black–disk regime) and the probability formultiple hard interactions becomes significant. In thissituation it is important to realize that the πB configu-rations participate in the high–energy scattering processwith a fixed transverse orientation, which is frozen dur-ing the collision; depending on this orientation one mayeither have a violent collision of the pion with the otherproton or no interaction at all. The averaging over theorientations of the πB configuration must be performedin the colliding pp system with given transverse geome-try, not in the partonic wave functions of the individualprotons. This circumstance affects e.g. the rate of mul-tijet events in peripheral collisions [59]. More generally,the pion cloud represents an example of transverse cor-relations in the nucleon’s partonic wave function, whichare neglected in the usual mean–field approximation forhigh–energy pp collisions. In particular, such correlationsplay a role in central inclusive diffraction, where they re-duce the rapidity gap survival probability relative to themean–field result [60]. Acknowledgments
The authors are indebted to A. Freund, J. Goity,V. Guzey, N. Kivel, P. Nadolsky, M. V. Polyakov, andA. W. Thomas for enlightening discussions and usefulhints.Notice: Authored by Jefferson Science Associates, LLCunder U.S. DOE Contract No. DE-AC05-06OR23177.The U.S. Government retains a non–exclusive, paid–up,irrevocable, world–wide license to publish or reproducethis manuscript for U.S. Government purposes. Sup-ported by other DOE contracts. [1] P. Amaudruz et al. [New Muon Collaboration], Phys.Rev. Lett. , 2712 (1991). M. Arneodo et al. [New MuonCollaboration], Phys. Rev. D , 1 (1994).[2] K. Ackerstaff et al. [HERMES Collaboration], Phys. Rev.Lett. , 5519 (1998).[3] A. Baldit et al. [NA51 Collaboration], Phys. Lett. B ,244 (1994).[4] E. A. Hawker et al. [FNAL E866/NuSea Collaboration],Phys. Rev. Lett. , 3715 (1998).[5] R. S. Towell et al. [FNAL E866/NuSea Collaboration],Phys. Rev. D , 052002 (2001).[6] S. Kumano, Phys. Rept. , 183 (1998). G. T. Garvey and J. C. Peng, Prog. Part. Nucl. Phys. , 203 (2001).[7] A. W. Thomas, Phys. Lett. B , 97 (1983).[8] L. L. Frankfurt, L. Mankiewicz, and M. I. Strikman, Z.Phys. A , 343 (1989);[9] R. Machleidt, K. Holinde and C. Elster, Phys. Rept. ,1 (1987).[10] W. Koepf, L. L. Frankfurt and M. Strikman, Phys. Rev.D , 2586 (1996).[11] J. D. Sullivan, Phys. Rev. D , 1732 (1972).[12] M. Burkardt, Int. J. Mod. Phys. A , 173 (2003).[13] P. V. Pobylitsa, Phys. Rev. D , 094002 (2002).[14] M. Strikman and C. Weiss, Phys. Rev. D , 054012 (2004).[15] A. Aktas et al. [H1 Collaboration], Eur. Phys. J. C ,585 (2006).[16] S. Chekanov et al. [ZEUS Collaboration], Nucl. Phys. B , 3 (2004).[17] M. Binkley et al. , Phys. Rev. Lett. , 73 (1982).[18] M. Strikman and C. Weiss, in: Proceedings of LIGHTCONE 2008 “Relativistic Nuclear and Particle Physics,”Mulhouse, France, July 7–11, 2008; arXiv:0811.3631[hep-ph].[19] F. D. Aaron et al. [H1 Collaboration], Phys. Lett. B ,796 (2008).[20] S. Chekanov et al. [ZEUS Collaboration],arXiv:0812.2517 [hep-ex].[21] M. Goncharov et al. [NuTeV Collaboration], Phys. Rev.D , 112006 (2001).[22] D. Mason et al. , Phys. Rev. Lett. , 192001 (2007).[23] J. W. Chen and X. D. Ji, Phys. Lett. B , 107 (2001);Phys. Rev. Lett. , 152002 (2001) [Erratum-ibid. ,249901 (2002)].[24] D. Arndt and M. J. Savage, Nucl. Phys. A , 429(2002).[25] W. Detmold, W. Melnitchouk, J. W. Negele, D. B. Ren-ner and A. W. Thomas, Phys. Rev. Lett. , 172001(2001).[26] A. V. Belitsky and X. Ji, Phys. Lett. B , 289 (2002).[27] S. I. Ando, J. W. Chen and C. W. Kao, Phys. Rev. D , 094013 (2006).[28] M. Diehl, A. Manashov and A. Schafer, Eur. Phys. J. A , 315 (2006). Eur. Phys. J. A , 335 (2007).[29] N. Kivel and M. V. Polyakov, arXiv:hep-ph/0203264.Phys. Lett. B , 64 (2008).[30] R. D. Young, D. B. Leinweber and A. W. Thomas, Nucl.Phys. Proc. Suppl. , 233 (2005).[31] N. Kivel, M. V. Polyakov and A. Vladimirov, Phys.Rev. D , 014028 (2009); Phys. Rev. Lett. , 262001(2008).[32] L. Frankfurt and M. Strikman, Phys. Rev. D , 031502(2002).[33] V. R. Zoller, Z. Phys. C , 443 (1992).[34] W. Melnitchouk and A. W. Thomas, Phys. Rev. D ,3794 (1993).[35] W. Melnitchouk, J. Speth and A. W. Thomas, Phys. Rev.D , 014033 (1999).[36] M. Gluck, E. Reya and I. Schienbein, Eur. Phys. J. C ,313 (1999).[37] A. D. Martin, W. J. Stirling, R. S. Thorne and G. Watt,arXiv:0901.0002 [hep-ph].[38] S. J. Brodsky and B. Q. Ma, Phys. Lett. B , 317(1996).[39] M. Gluck, P. Jimenez-Delgado and E. Reya, Eur. Phys.J. C , 355 (2008).[40] H. L. Lai, P. M. Nadolsky, J. Pumplin, D. Stump,W. K. Tung and C. P. Yuan, JHEP , 089 (2007).[41] H. Holtmann, A. Szczurek and J. Speth, Nucl. Phys. A , 631 (1996).[42] L. Frankfurt, M. Strikman and C. Weiss, Phys. Rev. D , 114010 (2004).[43] K. Goeke, M. V. Polyakov and M. Vanderhaeghen, Prog.Part. Nucl. Phys. , 401 (2001). M. Diehl, Phys. Rept. , 41 (2003). A. V. Belitsky and A. V. Radyushkin,Phys. Rept. , 1 (2005).[44] L. Frankfurt, A. Freund, V. Guzey and M. Strikman,Phys. Lett. B , 345 (1998) [Erratum-ibid. B , 414 (1998)].[45] A. G. Shuvaev, K. J. Golec-Biernat, A. D. Martin andM. G. Ryskin, Phys. Rev. D , 014015 (1999).[46] A. Levy, arXiv:0905.2034 [hep-ex].[47] M. Strikman and C. Weiss, in: Proceedings of the work-shop: HERA and the LHC workshop series on the impli-cations of HERA for LHC physics, Eds. H. Jung et al. ,arXiv:0903.3861 [hep-ph].[48] A. Freund and M. F. McDermott, Phys. Rev. D ,091901 (2002).[49] H. Lim, L. Schoeffel and M. Strikman, in: “Summaryof the ’Diffraction and vector mesons’ working group atDIS06,” Proceedings of 14th International Workshop onDeep Inelastic Scattering (DIS 2006), Tsukuba, Japan,20-24 Apr 2006, pp. 853-866; arXiv:hep-ph/0608107.[50] S. A. Morrow et al. [CLAS Collaboration], Eur. Phys. J.A , 5 (2009).[51] M. Guidal, M. V. Polyakov, A. V. Radyushkin andM. Vanderhaeghen, Phys. Rev. D , 054013 (2005).[52] M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Eur. Phys.J. C , 1 (2005).[53] M. Gluck, E. Reya and A. Vogt, Eur. Phys. J. C , 461(1998).[54] D. Diakonov, V. Petrov, P. Pobylitsa, M. V. Polyakovand C. Weiss, Nucl. Phys. B , 341 (1996); Phys. Rev.D , 4069 (1997).[55] G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B , 552 (1983).[56] L. L. Frankfurt and M. I. Strikman, Phys. Rept. , 215(1981).[57] D. Diakonov and M. I. Eides, JETP Lett. , 433 (1983)[Pisma Zh. Eksp. Teor. Fiz. , 358 (1983)]. D. Diakonovand V. Y. Petrov, Nucl. Phys. B , 457 (1986).[58] D. Diakonov, V. Y. Petrov and P. V. Pobylitsa, Nucl.Phys. B , 809 (1988).[59] T. C. Rogers, A. M. Stasto and M. I. Strikman, Phys.Rev. D , 114009 (2008).[60] L. Frankfurt, C. E. Hyde, M. Strikman and C. Weiss,Phys. Rev. D , 054009 (2007).[61] V. Guzey and M. V. Polyakov, arXiv:hep-ph/0512355.[62] The approach taken here bears some similarity to theuse of finite–size regulators in the chiral extrapolation oflattice QCD results [30].[63] In invariant perturbation theory soft pions have virtual-ities − k π ∼ M π , and Eq. (1) results from the conditionthat the minimum pion virtuality which is kinematicallyrequired for a given longitudinal momentum fraction, y ,be at most of the order M π ; cf . Eq. (13) below.[64] The relation between the distance of a constituent fromthe transverse center–of–momentum, b , and its distancefrom the center–of–momentum of the spectator system, r , can easily be derived for the case of a non–interactingsystem, by starting from the well–known expression forthe center–of–mass in the rest frame and performing aboost to large velocity, taking into account that for thenon-interacting system the longitudinal momentum frac-tions are given by the ratios of the constituent massesto the total mass of the system. A more formal deriva-tion, based on the light–cone components of the energy–momentum tensor, can be found in Ref. [12].[65] In Ref. [14] the pion virtualities were denoted by − s ± .Here we denote them by t , , reserving s , for the in-variant masses of the πB system, Eq. (20).[66] Equation (46) of Ref. [14] incorrectly writes the convo- lution formula for the antiquark flavor asymmetry withthe antiquark density in the pion rather than the va-lence quark density. The correct expression is with thevalence quark density, cf. Eq. (28). This does not affectthe conclusions about the large– N c behavior presentedin Ref. [14], which was the sole point of the discussionthere.[67] The relation of our conventions for the pion par-ton densities to those of Ref. [36] (GRS) is q val π = v π (GRS) , q sea π = ¯ q π (GRS).[68] The value quoted here corresponds to the arithmeticmean of the parametrizations of α ′ J/ψ quoted by theHERA H1 [15] and ZEUS [16] experiments; see Ref. [47]for details.
APPENDIX A: MESON–BARYON COUPLINGSFROM SU (3) SYMMETRY
In this appendix we summarize the meson–baryon cou-plings used in the calculation of SU (3) octet meson( π, K, η ) contributions to the sea quark distributions (seeSec. IV C) and derive the expressions for the correspond-ing meson momentum distributions in the nucleon.For the coupling constants governing the N ↔ M + B transitions we rely on SU (3) flavor symmetry, whichis known to describe the empirical couplings well; seeRef. [61] for a recent review. For ↔ × transitionsthere are two independent SU (3)–invariant structures,with coupling constants traditionally denoted by F and D . With the standard assignments of the meson andbaryon fields the Lagrangian takes the form (we showexplicitly only the terms describing transitions M + B → p ) L = g ¯ p h π + n + √ π p − √ (1 + 2 α ) K + Λ+ (1 − α ) (cid:16) √ K + Σ + K Σ + (cid:17) − √ (1 − α ) ηp i + h.c. (A1)Here g ≡ F + D is the overall octet coupling, which isrelated to our πN N coupling as g πNN ≡ g π pp = g / √
2; (A2)we use g πNN = 13 .
05 in our numerical calculations [10].The ratio α ≡ F/ ( F + D ) remains a free parameter inthe context of SU (3) flavor symmetry and can only bedetermined empirically or by invoking dynamical models.The SU (6) spin–flavor symmetry of the non-relativisticquark model implies F/D = 2 /
3, and thus α = 2 / .
4; (A3)we use this value in our numerical studies in Sec. IV C.For ↔ × transitions there is only a single SU (3)–invariant structure, and the Lagrangian is of the form L = g ¯ p (cid:16) √ π + ∆ + q π ∆ + − π − ∆ ++ + √ K + Σ ∗ − √ K Σ ∗ + (cid:17) + h.c. (A4)The decuplet coupling g coincides (up to the sign) withour πN ∆ coupling, g πN ∆ ≡ g π − p ∆ ++ = − g ; (A5)we use g πN ∆ = 20 .
22, which is close to the large– N c value of (3 / g πNN , cf. Eq. (69). Note that ourdefinition of the coupling constant g πN ∆ differs fromthe one of Ref. [10] by a factor, g πN ∆ (Ref. [10]) = √ g πN ∆ (this work).The GPDs of SU (3) octet meson in the nucleon, H MB ( y, t ), and the corresponding impact parameter–dependent distributions are obtained by straightforward4extension of the expressions for πN and π ∆ in Sec. II, cf. Eqs. (14) and (15), and Eqs. (16)–(18). We work with theisoscalar distributions, in which we sum over the isospincomponents of the intermediate meson–baryon system.Using the couplings provided by Eqs. (A1) and (A4), theyare obtained as (we omit the arguments for brevity) H πN ≡ H π + n + H π p = 3 g πNN I ,H ηN ≡ H ηp = g πNN (1 − α ) I ,H K Λ ≡ H K + Λ = g πNN (1 + 2 α ) I ,H K Σ ≡ H K + Σ + H K Σ + = 3 g πNN (1 − α ) I ,H π ∆ ≡ H π + ∆ + H π ∆ + + H π − ∆ ++ = 2 g πN ∆ I ,H K Σ ∗ ≡ H K + Σ ∗ + H K Σ ∗ + = g πN ∆ I . (A6)Here I and I are the basic momentum integrals ofEqs. (16)–(18), taken at the appropriate values of themeson and baryon masses. Note that because of isospinsymmetry the distributions for the individual isospincomponents are all proportional to the same function andcan be expressed in terms of the isoscalar distribution as H π + n = H πN , H π p = H πN , etc. (A7)where the proportions are determined by the squares ofthe coupling constants in the Lagrangian. The corre-sponding b –dependent distributions, f MB ( y, b ), are thenobtained by substituting these GPDs in Eq. (8).To determine the coefficients with which the differentmesons contribute to a given parton density in the pro-ton, one must account for the probability with which theparton occurs in the individual meson charge states. Forexample (in abbreviated notation)( ¯ d − ¯ u ) p = f π + n ( ¯ d − ¯ u ) π + + f π p ( ¯ d − ¯ u ) π = f πN q val π , (A8)where q val π denotes the valence quark distribution in thepion, Eq. (31), and we have used Eq. (A7) for the im-pact parameter distribution and the isospin and chargeconjugation relations for the parton densities in the pion,¯ u π + = d π + , ¯ u π = ¯ d π . APPENDIX B: EVALUATION OFCOORDINATE–SPACE DISTRIBUTIONS
In this appendix we present expressions suitablefor numerical evaluation of the transverse coordinate–dependent distribution of pions in the nucleon and theirpartial radial integrals. The b –dependent distribution,defined by Eq. (8), is evaluated as (we omit the argu-ment y and the subscript for brevity) f ( b ) = Z d ∆ ⊥ (2 π ) e − i ( ∆ ⊥ b ) H ( t ) (B1)= 12 π Z ∞ d ∆ ⊥ ∆ ⊥ J (∆ ⊥ b ) H ( t ) (B2) where b = | b | , ∆ ⊥ = | ∆ ⊥ | , t = − ∆ ⊥ , and J denotesthe Bessel function. Equation (B2) can be used to calcu-late the f ( b ) corresponding to a numerically given H ( t ),as obtained from evaluating the loop integral Eq. (16)with πN form factors. In practice, since H ( t ) shows onlya power–like fall–off at large − t >
0, we multiply theintegrand in Eq. (B2) by an exponential convergence fac-tor, exp( ǫt ), calculate the integral numerically for finite ǫ , and estimate the limiting value for ǫ → ǫ .From Eq. (B2) we can also derive expressions for thepartial radial integrals of f ( b ), including those with aweighting factor b . Using standard identities for inte-grals of the Bessel function multiplied by powers of itsargument, we obtain for the integrals over the region b < b Z d b Θ( b < b ) f ( b ) = b Z ∞ d ∆ ⊥ J ( z ) H ( t ) (B3) Z d b Θ( b < b ) f ( b ) b = b Z ∞ d ∆ ⊥ ∆ ⊥ [2 J ( z )+ ( z − /z ) J ( z )] H ( t ) , (B4)where z ≡ ∆ ⊥ b . (B5)The complementary integrals over the region b > b arecalculated by re-writing the original Fourier integral for f ( b ), Eq. (B1), in the form Z d b Θ( b > b ) = Z d b − Z d b Θ( b < b ) . (B6)The unrestricted integral on the R.H.S. then produces atwo–dimensional delta function at ∆ ⊥ = 0 and can beevaluated in terms of H ( t = 0) or its derivative. In thisway we obtain Z d b Θ( b > b ) f ( b )= H ( t = 0) − Z d b Θ( b < b ) f ( b ) , (B7) Z d b Θ( b > b ) b f ( b )= 4 ∂H∂t ( t = 0) − Z d b Θ( b < b ) b f ( b ) , (B8)where the right–hand side can be evaluated usingEqs. (B3) and (B4). Note that the ∆ ⊥ integrals rep-resenting the b –integrated distributions, Eqs. (B3) and(B4), converge more rapidly at large ∆ ⊥ than the in-tegral representing the original f ( bb