Pion Parton Distribution Function in Light-Front Holographic QCD
PPion Parton Distribution Function in Light-Front Holographic QCD
Lei Chang, Kh´epani Raya, and Xiaobin Wang School of Physics, Nankai University, Tianjin 300071, China (Dated: January 22, 2020)The pion parton distribution function, u π ( x ), is reexamined by a universal reparametrizationfunction, w τ ( x ), in the light-front holographic QCD (LFHQCD) approach. We show that, owing tothe flexibility of w τ ( x ), the large- x behavior u π ( x ) ∼ (1 − x ) can be contained within the LFHQCDformalism. From this fact, augmented by perturbative QCD and recent lattice QCD results, westate that such behavior cannot be excluded. Motivation — During the rise of parton models, aroundthe 1970s, a connection between the proton electromag-netic form factors (obtained via exclusive process) andits structure functions (inferred from deep inelastic scat-tering) was realized by Drell-Yann [1] and West [2].Their findings yielded the so-called Drell-Yan-West re-lation (DYW), which entails that, when the momentumtransfer ( − t = Q ) becomes asymptotically large, theproton electromagnetic form factor (EFF) falls as F p ( t ) ∼ − t ) τ − , (1)while the corresponding parton distribution function(PDF) behaves, at large- x ( i.e. , x → u p ( x ) ∼ (1 − x ) τ − . (2)Here, x is the longitudinal momentum fraction carried bythe parton - or Bjorken- x [3] - and τ , called twist , denotesthe number of τ -components of the hadron state. In asubsequent work by Ezawa [4], it was shown that the pionviolates the DYW relation. This can be attributed tothe different number of constituents and spin. It is seenthat, while the EFF exhibits the same asymptotic profilefor both mesons, Eq. (1), the pion parton distributionfunction adopts the large- x form u π ( x ) ∼ (1 − x ) τ − . (3)The leading-twist ( τ = 3 for proton, τ = 2 for pion)entails the well-known 1 / ( − t ) and 1 / ( − t ) falls of theproton and pion EFFs [5], respectively, and the x → u p ( x ) ∼ (1 − x ) , (4) u π ( x ) ∼ (1 − x ) . (5)Those patterns are further supported by perturbativeQuantum Chromodynamics (pQCD) [5–7]. In fact, as-suming a theory in which the quarks interact via theexchange of a vector-boson, asymptotically damped as(1 /k ) β , Eq. (5) generalizes as [8]: u π ( x ) ∼ (1 − x ) β . (6)Hence, the large- x behavior of the valence-quark PDFis a direct measure of the momentum-dependence of theunderlying interaction [6–9]. In the novel approach of light-front holographic QCD(LFHQCD) [10, 11], it is suggested that the DYW re-lation is preserved for both the proton and pion [12].Thereby, it predicts a valence pion PDF that, from theleading-twist-2 term, falls as u π ( x ) ∼ (1 − x ) , (7)feeding the controversy provoked by the E615-Experiment leading order (LO) analysis [13], which fa-vors a large- x exponent of “1”, in apparent contradictionwith the parton models and pQCD. Many theoretical andphenomenological approaches have been participants inthis debate, e.g. [8, 9, 12, 14–25]. Playing a key role inthis controversy, the analysis of Aicher et al. [15] showsthat, if a next-to-leading order (NLO) treatment of thedata is performed and soft-gluon resummation is consid-ered, it is possible to recover the pQCD prediction. Ondifferent grounds, the x → masslessness of the pion and the much larger size of theproton mass [26–28]. Similarly, it is vital to obtain a clearpicture of the proton and pion parton distributions in thesame approach. QCD predicts the profiles of Eqs. (4)-(5), thus we need to explain how those behaviors can (orcannot) take place.In this letter, we revisit Ref. [12]. There, the authorspresent an appealing way to parametrize the PDFs andgeneralized parton distributions (GPDs), from an inte-gral representation of the EFFs, but they claim that thefalloff of the pion PDF at x → x behavior ofEq. (3) can be perfectly accommodated within the sameLFHQCD formalism, while also maintaining the correctcounting rules for the proton. a r X i v : . [ h e p - ph ] J a n Counting rules in LFHQCD — Following Ref. [12], theform factor is expressed in an integral representation as F τ ( t ) = 1 N τ (cid:90) dy (1 − y ) τ − y − t/ λ − (8)= 1 N τ B ( τ − , − t λ ) , (9)where N τ = √ π Γ( τ − / Γ( τ −
2) and √ λ = 0 .
548 GeV; B ( u, v ) corresponds to the Euler Beta Function. Theuniversal scale, λ , is fixed by the ρ meson mass [10, 29].Under the change of variable y = w τ ( x ) one can write,more generally: F τ ( t ) = 1 N τ (cid:90) dx (1 − w τ ( x )) τ − w τ ( x ) − t/ λ − ∂w τ ( x ) ∂x , (10)where the reparametrization function, w τ ( x ), is con-strained by the conditions: w τ (0) = 0; w τ (1) = 1; ∂w τ ( x ) ∂x ≥ . (11)Notice that we have introduced a τ -dependence in w τ ( x ).This is a key difference, with respect to [12], that we willexploit later. At zero skewness, the valence-quark GPDis conveniently expressed as H ( x, t ) = q τ ( x )e tf τ ( x ) (12)where we identify the PDF and profile function, q τ ( x )and f τ ( x ) respectively, as q τ ( x ) = 1 N τ (1 − w τ ( x )) τ − w τ ( x ) − ∂w τ ( x ) ∂x , (13) f τ ( x ) = 14 λ log (cid:18) w τ ( x ) (cid:19) . (14)Then, a simple form for w τ ( x ) is suggested: w τ ( x ) = x (1 − x ) g ( τ ) e − a τ (1 − x ) g ( τ ) , (15)with g ( τ ) , a τ >
0. The adopted profile of w τ ( x ) preservesthe desired Regge behavior at small- x [11, 12], while alsosatisfying the constraints of Eqs. (11). Thus, owing to thereparametrization invariance of the Euler Beta Function, F τ ( t ) exhibits the large- t falloff: F τ ( Q ) ∼ (cid:18) − t (cid:19) τ − , (16)which implies that the correct asymptotic behavior of theform factor [4–6] is faithfully reproduced. On the otherhand, the x → q τ ( x ) will exhibit the τ -dependence as follows: q τ ( x ) ∼ (1 − x ) h ( τ ) , (17)with h ( τ ) = ( τ − g ( τ ) −
1. Due to the arbitrariness onthe choice of g ( τ ), LFHQCD cannot predict its precise form, and so the exact counting rules. However, it is thisflexibility what allows us to recover the correspondingcounting rules for both pion and proton, Eqs. (4)-(5).Given the simplicity of Eq. (17), we propose the followingfor the PDFs:Rule-I : (1 − x ) τ − , with g ( τ ) = 2 . (18)Rule-II : (1 − x ) τ − , with g ( τ ) = 2 + 1 τ − . (19)Thus, it follows from (17) that the spin − relation (2)can be satisfied if Rule-I is chosen, while the spin − Pion valence-quark PDF — Consider the twist-4 pionvalence-quark PDF as u π ( x ; ζ ) = (1 − γ ) q τ =2 ( x ; ζ ) + γq τ =4 ( x ; ζ ) , (20)with normalization (cid:82) dx u π ( x ; ζ ) = 1 and γ = 0 . ζ = ζ , which is set as ζ =1 . ± .
24, obtained at ζ := 2 GeV after NLOevolution, as compared to the lQCD estimates fromRefs. [21, 23–25]. To account for the impact of the twist-4 term, we also vary the ratio a /a from 0 . x are observed.Figure 1 displays the valence-quark PDFs, evolved to ζ := 5 . t -dependence of the valence-quarkGPD, for Rule-II, is presented in Figure 2.It is clear that Rule-I produces a PDF that is closerto the original experimental data [13], while the analo-gous for Rule-II matches the rescaled data from Ref. [15].Either rule will give the correct large- t fall of the EFFin Eq. (16), but only in the second case one obtainsthe x → x behavior of the valence-quarkPDF [8]. Moreover, state-of-the-art lQCD results [20]also establish that the asymptotic form of Eq. (5) is pre-ferred. It is noteworthy that, even though the pion PDF x x u π ( x ; ζ ) Lattice CSDSE E615 - OriginalE615 - RescaledRule - IRule - II FIG. 1.
Valence-quark pion PDF.
Obtained NLO results at ζ = 5 . ζ = 1 . ± . a /a =0 . Data points: (triangles) LO extraction“E615-Original” [13] and (circles) the NLO analysis “E615-Rescaled” of Ref. [15].FIG. 2.
Valence-quark pion GPD. t -dependence of thevalence-quark pion GPD at zero skewness. The plot abovecorresponds to Rule-II in Eq. (18), at the initial scale ζ . obtained from Rule-I differs from that of [12], the evo-luted results are compatible. This is unsurprising sincethe corresponding reparametrization function is not dra-matically different from its counterpart in [12]. Thus,although it is not be included in the present letter, weexpect Rule-I to produce a congruent picture for thevalence-quark PDF of the proton. These observationsencourage us to select Rule-I for the case of the protonand Rule-II when studying pions, for an internally con-sistent description based on the LFHQCD formalism. Summary and conclusions — We have reanalized theLFHQCD approach of Ref. [12] to study the valence-quark PDF of the pion. It has been proven that, giventhe flexibility of the universal reparametrization function, w τ ( x ), it is in fact possible to accomodate a large- x be-havior of u π ( x ) ∼ (1 − x ) τ − within this framework.Besides the agreement with the rescaled experimentaldata [15], this makes it compatible with the Ezawa find-ings [4] and the predictions from pQCD [5–7]. Recentcontinuum [18, 19] and sophisticated lQCD studies [20]also favor this endpoint form. Due to this confluence ofvastly different approaches, and given our observations,we state that the u π ( x ) ∼ (1 − x ) profile can not onlybe contained within the LFHQCD formalism, but alsocannot be excluded. Besides, we sketched how a simul-taneous description of the proton and pion distributionfunctions, that agrees with pQCD, can be achieved if thecounting rules are chosen accordingly: we encourage theuse of Rule-I for proton and Rule-II for pion.We acknowledge helpful conversations with Yuan Sun.This work is supported by: the Chinese GovernmentThousand Talents Plan for Young Professionals. [1] S. D. Drell and Tung-Mow Yan. Connection of Elas-tic Electromagnetic Nucleon Form-Factors at Large Q**2and Deep Inelastic Structure Functions Near Threshold. Phys. Rev. Lett. , 24:181–185, 1970.[2] Geoffrey B. West. Phenomenological model for the elec-tromagnetic structure of the proton.
Phys. Rev. Lett. ,24:1206–1209, 1970.[3] J. D. Bjorken. Asymptotic Sum Rules at Infinite Mo-mentum.
Phys. Rev. , 179:1547–1553, 1969.[4] Z. F. Ezawa. Wide-Angle Scattering in Softened FieldTheory.
Nuovo Cim. , A23:271–290, 1974.[5] G. Peter Lepage and Stanley J. Brodsky. Exclusive Pro-cesses in Perturbative Quantum Chromodynamics.
Phys.Rev. , D22:2157, 1980.[6] Glennys R. Farrar and Darrell R. Jackson. Pion andNucleon Structure Functions Near x=1.
Phys. Rev. Lett. ,35:1416, 1975.[7] Edmond L. Berger and Stanley J. Brodsky. Quark Struc-ture Functions of Mesons and the Drell-Yan Process.
Phys. Rev. Lett. , 42:940–944, 1979.[8] Roy J. Holt and Craig D. Roberts. Distribution Functionsof the Nucleon and Pion in the Valence Region.
Rev. Mod.Phys. , 82:2991–3044, 2010.[9] M. B. Hecht, Craig D. Roberts, and S. M. Schmidt.Valence quark distributions in the pion.
Phys. Rev. ,C63:025213, 2001.[10] Stanley J. Brodsky, Guy F. de Teramond, Hans GunterDosch, and Joshua Erlich. Light-Front Holographic QCDand Emerging Confinement.
Phys. Rept. , 584:1–105,2015.[11] Liping Zou and H. G. Dosch. A very Practical Guide toLight Front Holographic QCD. 2018.[12] Guy F. de Teramond, Tianbo Liu, Raza Sabbir Sufian,Hans G¨unter Dosch, Stanley J. Brodsky, and Alexan-dre Deur. Universality of Generalized Parton Distribu-tions in Light-Front Holographic QCD.
Phys. Rev. Lett. ,120(18):182001, 2018.[13] J. S. Conway et al. Experimental Study of Muon Pairs
Produced by 252-GeV Pions on Tungsten.
Phys. Rev. ,D39:92–122, 1989.[14] K. Wijesooriya, P. E. Reimer, and R. J. Holt. The pionparton distribution function in the valence region.
Phys.Rev. , C72:065203, 2005.[15] Matthias Aicher, Andreas Schafer, and Werner Vogel-sang. Soft-gluon resummation and the valence par-ton distribution function of the pion.
Phys. Rev. Lett. ,105:252003, 2010.[16] Lei Chang, C´edric Mezrag, Herv´e Moutarde, Craig D.Roberts, Jose Rodr´ıguez-Quintero, and Peter C. Tandy.Basic features of the pion valence-quark distributionfunction.
Phys. Lett. , B737:23–29, 2014.[17] Chen Chen, Lei Chang, Craig D. Roberts, Shaolong Wan,and Hong-Shi Zong. Valence-quark distribution functionsin the kaon and pion.
Phys. Rev. , D93(7):074021, 2016.[18] Minghui Ding, Kh´epani Raya, Daniele Binosi, Lei Chang,Craig D Roberts, and Sebastian M. Schmidt. Symmetry,symmetry breaking, and pion parton distributions. 2019.[19] Minghui Ding, Kh´epani Raya, Daniele Binosi, Lei Chang,Craig D Roberts, and Sebastian M Schmidt. Drawinginsights from pion parton distributions. 2019.[20] Raza Sabbir Sufian, Colin Egerer, Joseph Karpie,Robert G. Edwards, B´alint Jo´o, Yan-Qing Ma, KostasOrginos, Jian-Wei Qiu, and David G. Richards. Pion Va-lence Quark Distribution at Large x from Lattice QCD.2020.[21] B´alint Jo´o, Joseph Karpie, Kostas Orginos, Anatoly V.Radyushkin, David G. Richards, Raza Sabbir Sufian, andSavvas Zafeiropoulos. Pion valence structure from Ioffe-time parton pseudodistribution functions. Phys. Rev. , D100(11):114512, 2019.[22] Raza Sabbir Sufian, Joseph Karpie, Colin Egerer, KostasOrginos, Jian-Wei Qiu, and David G. Richards. PionValence Quark Distribution from Matrix Element Calcu-lated in Lattice QCD.
Phys. Rev. , D99(7):074507, 2019.[23] M. Oehm, C. Alexandrou, M. Constantinou, K. Jansen,G. Koutsou, B. Kostrzewa, F. Steffens, C. Urbach, andS. Zafeiropoulos. (cid:104) x (cid:105) and (cid:104) x (cid:105) of the pion PDF fromlattice QCD with N f = 2+1+1 dynamical quark flavors. Phys. Rev. , D99(1):014508, 2019.[24] Dirk Brommel et al. Quark distributions in the pion.
PoS , LATTICE2007:140, 2007.[25] William Detmold, W. Melnitchouk, and An-thony William Thomas. Parton distribution functionsin the pion from lattice QCD.
Phys. Rev. , D68:034025,2003.[26] Tanja Horn and Craig D. Roberts. The pion: an enigmawithin the Standard Model.
J. Phys. , G43(7):073001,2016.[27] Craig D. Roberts. Perspective on the origin of hadronmasses.
Few Body Syst. , 58(1):5, 2017.[28] Craig D Roberts. Insights into the Origin of Mass. In , 2019.[29] Stanley J. Brodsky, Guy F. de T´eramond, Hans GunterDosch, and C´edric Lorc´e. Universal Effective HadronDynamics from Superconformal Algebra.