Constraints on charm-anticharm asymmetry in the nucleon from lattice QCD
Raza Sabbir Sufian, Tianbo Liu, Andrei Alexandru, Stanley J. Brodsky, Guy F. de Téramond, Hans Günter Dosch, Terrence Draper, Keh-Fei Liu, Yi-Bo Yang
CConstraints on charm-anticharm asymmetry in the nucleon from lattice QCD
Raza Sabbir Sufian a , Tianbo Liu a , Andrei Alexandru b,c , Stanley J. Brodsky d , Guy F. de T´eramond e ,Hans G¨unter Dosch f , Terrence Draper g , Keh-Fei Liu g , Yi-Bo Yang h,i,j a Thomas Je ff erson National Accelerator Facility, Newport News, VA 23606, USA b Department of Physics, The George Washington University, Washington, DC 20052, USA c Department of Physics, University of Maryland, College Park, MD 20742, USA d SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA e Laboratorio de F´ısica Te´orica y Computacional, Universidad de Costa Rica, 11501 San Jos´e, Costa Rica f Institut f¨ur Theoretische Physik der Universit¨at, D-69120 Heidelberg, Germany g Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA h CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China i School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China j International Centre for Theoretical Physics Asia-Pacific, Beijing / Hangzhou, China
Abstract
We present the first lattice QCD calculation of the charm quark contribution to the nucleon electromagnetic form fac-tors G cE , M ( Q ) in the momentum transfer range 0 ≤ Q ≤ . . The quark mass dependence, finite lattice spacingand volume corrections are taken into account simultaneously based on the calculation on three gauge ensembles in-cluding one at the physical pion mass. The nonzero value of the charm magnetic moment µ cM = − . stat (5) sys ,as well as the Pauli form factor, reflects a nontrivial role of the charm sea in the nucleon spin structure. The nonzero G cE ( Q ) indicates the existence of a nonvanishing asymmetric charm-anticharm sea in the nucleon. Performing a non-perturbative analysis based on holographic QCD and the generalized Veneziano model, we study the constraints on the[ c ( x ) − ¯ c ( x )] distribution from the lattice QCD results presented here. Our results provide complementary informationand motivation for more detailed studies of physical observables that are sensitive to intrinsic charm and for futureglobal analyses of parton distributions including asymmetric charm-anticharm distribution. Keywords:
Intrinsic charm, Form factor, Parton distributions, Lattice QCD, Light-front holographic QCD,JLAB-THY-20-3155, SLAC-PUB-17515
1. Introduction
The charm-anticharm sea in the nucleon has receivedgreat interest in nuclear and particle physics for its par-ticular significance in understanding high energy re-actions associated with charm production. QuantumChromodynamics (QCD), the underlying theory of thestrong interaction, allows heavy quarks in the nucleon-sea to have both perturbative “extrinsic” and nonpertur-bative “intrinsic” origins. The extrinsic sea arises fromgluon splitting triggered by a probe in the reaction. Itcan be calculated order-by-order in perturbation theoryif the probe is hard. The intrinsic sea is encoded in thenucleon wave functions.The existence of nonperturbative intrinsic charm (IC)was originally proposed in the BHPS model [1] and inthe subsequent calculations [2, 3, 4] following the orig-inal proposal [1]. Proper knowledge of the existence of IC and an estimate of its magnitude will elucidate somefundamental aspects of nonperturbative QCD. There-fore, the main goal of this article is to investigate theexistence of nonzero “intrinsic” charm of nonperturba-tive origin in the nucleon. In the case of light-front (LF)Hamiltonian theory, the intrinsic heavy quarks of theproton are associated with higher Fock states such as | uudQ ¯ Q (cid:105) in the hadronic eigenstate of the LF Hamil-tonian; this implies that the heavy quarks are multi-connected to the valence quarks. The probability for theheavy-quark Fock states scales as 1 / m Q in non-AbelianQCD. Since the LF wavefunction is maximal at mini-mum o ff -shell invariant mass; i.e., at equal rapidity, theintrinsic heavy quarks carry large momentum fraction x Q . A key characteristic is di ff erent momentum and spindistributions for the intrinsic Q and ¯ Q in the nucleon, asmanifested, for example, in charm-anticharm asymme-try [5, 6], since the comoving quarks can react di ff er- Preprint submitted to Elsevier September 22, 2020 a r X i v : . [ h e p - l a t ] S e p ntly to the global quantum numbers of the nucleon [7].IC was also proposed in meson-baryon fluctuationmodels [8, 9]. The possible direct and indirect relevanceof IC in several physical processes has led to many phe-nomenological calculations involving the existence ofa non-zero IC to explain anomalies in the experimen-tal data and possible signatures of IC in upcoming ex-periments [7]. Unfortunately, the normalization of the | uudc ¯ c (cid:105) intrinsic charm Fock component in the light-front wavefunctions (LFWF) is unknown. Also, theprobability to find a two-body state ¯ D ( u ¯ c ) Λ + c ( udc ) inthe proton within the meson-baryon fluctuation modelscannot be determined without additional assumptions:precise constraints from future experiments and / or first-principles calculations are required.The e ff ect of whether the IC parton distribution iseither included or excluded in the determinations ofcharm parton distribution functions (PDFs) can inducechanges in other parton distributions through the mo-mentum sum rule, which can indirectly a ff ect the analy-ses of various physical processes that depend on the in-put of various PDFs. An estimate of intrinsic charm ( c )and anticharm (¯ c ) distributions can provide importantinformation to the understanding of charm quark pro-duction in the EMC experiment [10]. The enhancementof charm distribution in the measurement of the charmquark structure function F c compared to the expecta-tion from the gluon splitting mechanism in the EMCexperimental data has been interpreted as evidence fornonzero IC in several calculations [2, 3, 11, 12]. A pre-cise determination of c and ¯ c PDFs by considering boththe perturbative and nonperturbative contributions isimportant in understanding charmonia and open charmproductions, such as the J /ψ production at large mo-mentum from pA collisions at CERN [13], from π A col-lisions at FNAL [14], from pp collisions at LHC [15],and charmed hadron or jet production from pp colli-sions at ISR, FNAL, and LHC [15, 16, 17, 18]. LHCmeasurements associated with cross section of inclusiveproduction of Higgs, Z , W bosons via gluon-gluon fu-sion, and productions of charm jet and Z [19, 20, 21,22], J /ψ and D mesons at LHCb experiment [15] canalso be sensitive to the IC distribution. The J /ψ photo-or electro-productions near the charm threshold is be-lieved to be sensitive to the trace anomaly componentof the proton mass, and some experiments have beenproposed at JLab [23] as well as for the future EIC tomeasure the production cross section near the thresh-old. The existence of IC in the proton will provide ad-ditional production channels and thus enhance the crosssection, especially near the threshold. Similarly, opencharm production will also be enhanced by IC. If c and ¯ c quarks have di ff erent distributions in the proton, theenhancements on D and ¯ D productions will appear atslightly di ff erent kinematics. IC has also been proposedto have an impact on estimating the astrophysical neu-trino flux observed at the IceCube experiment [24].In global analyses of PDFs there are di ff erent ap-proaches to deal with heavy quarks in which a transi-tion of the number of active quark flavors is made atsome scale around the charm quark mass µ c ∼ m c [25,26, 27, 28]. The transition scale defines where the ex-trinsic charm-anticharm sea enters. However, the intrin-sic charm-anticharm sea can exist even at a lower scale.In many global fits [29, 30, 31, 32, 33, 34], the charmquark PDF is set to zero at µ c , but this is an assumptionof no IC. In recent years, several PDF analyses startedto investigate the possibility of nonzero charm and an-ticharm distributions at the scale µ c [35, 36, 37, 38, 39],but none of them can provide conclusive evidence orexclusion for the intrinsic charm due to the absence ofprecise data. A nonzero charm quark PDF at µ c is notnecessarily evidence of intrinsic charm, because suchestimation depends on the heavy-quark scheme and the µ c value used in the fit. This explains the specula-tion in [35] that the estimation of intrinsic charm maystrongly depend on the choice of the transition scale µ c .Fortunately, there is an ideal quantity, the asymmetriccharm-anticharm distribution [ c ( x ) − ¯ c ( x )], which wouldbe a clear signal for IC. Such asymmetry is allowed inQCD because the nucleon has nonzero quark number,the number of quarks minus the number of antiquarks,and thus the c quark and the ¯ c quark in a nucleon would“feel” di ff erent interactions, leading to an asymmetriccharm-anticharm distribution. Although the absence ofsuch asymmetry does not exclude intrinsic charm, anonzero [ c ( x ) − ¯ c ( x )] can serve as strong evidence, be-cause the extrinsic part of such asymmetry arising at thenext-to-next-to-leading order level is negligible [40].Although the global fits [35, 36, 37, 38, 39] considerthe possibility of IC, all these fits assume [ c ( x ) − ¯ c ( x )] =
0; constraints on [ c ( x ) − ¯ c ( x )] have been warranted inthe global fit [38]. It was found in [38] that a preciseand accurate parametrization of the charm PDFs willbe useful for more reliable phenomenology using thedata from LHC experiments and will eliminate possi-ble sources of bias arising from the assumptions of onlyperturbatively-generated charm PDFs. It is thereforeimportant to determine if [ c ( x ) − ¯ c ( x )] (cid:44)
0, and how orwhether the IC will have significant e ff ect in the physi-cal processes [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Aprecise and accurate knowledge of the IC will also havea direct impact on determining the unknown normaliza-tion constants of di ff erent model calculations associated2ith the nonperturbative c ( x ) and ¯ c ( x ) distributions.An important question to ask is whether the first-principles lattice QCD (LQCD) calculation can pro-vide some constraints or complementary information re-garding the existence of IC. Recently, there have beenLQCD calculations of the strange ( s ) quark electromag-netic form factors [41, 42, 43, 44, 45, 46] with higherprecision and accuracy than previously attained by ex-periments. These LQCD calculations provided indirectevidence for a nonzero strange quark-antiquark asym-metry in the nucleon by pinning down the nonzero valueof the strange electric form factor G sE ( Q ) at Q > G sE , M ( Q ) from the LQCD cal-culation in [42] has led to precise determination ofneutral current weak axial and electromagnetic formfactors [44, 47]. Using LQCD results from [42, 43]as constraints, it has been shown recently in [48],within the light-front holographic QCD (LFHQCD) ap-proach [49, 50, 51, 52] and the generalized Venezianomodel [53, 54, 55], that the [ s ( x ) − ¯ s ( x )] distribution isnegative at small- x and positive at large- x . This showsthe possibility of applying LQCD results for the phe-nomenological study of the [ s ( x ) − ¯ s ( x )] asymmetry inthe absence of precise experimental data and global fitsof the strange quark PDFs.The main goal of this article is to calculate thecharm electric and magnetic form factors, G cE ( Q ) and G cM ( Q ), from LQCD at nonzero momentum transferand discuss their connection to the existence of IC anda nonzero [ c ( x ) − ¯ c ( x )] asymmetry distribution in thenucleon. Using the LQCD calculation of G cE , M ( Q ) asconstraint, we determine the [ c ( x ) − ¯ c ( x )] distributionusing the nonperturbative framework described in [56].We note that the electromagnetic current is odd undercharge conjugation and the Dirac form factor F c ( Q >
0) provides a measure of the c -quark minus the ¯ c -quarkcontribution due to the opposite charges of the quarkand antiquark. While F c ( Q = =
0, required bythe quantum numbers of the nucleon, a positive F c ( Q )at Q > c -quark distribution is morecentralized than the ¯ c quark distribution in coordinatespace. This, in turn, results in a [ c ( x ) − ¯ c ( x )] asymme-try in momentum space, thereby providing possible evi-dence for the nonperturbative IC in the nucleon. On theother hand, a nonzero charm Pauli form factor F c ( Q )and a nonzero charm magnetic moment µ cM (cid:44) L z = L z = G cE , M ( Q ) anddetermine these in the physical limit. In Section 3, weuse LQCD results for G cE , M ( Q ) as input for quantitativeanalysis of the [ c ( x ) − ¯ c ( x )] asymmetry distribution inthe nucleon within the specific framework of LFHQCDand the generalized Veneziano model. We also presenta brief qualitative discussion of our results in Section 4.
2. Lattice QCD calculation of G cE , M ( Q ) We present in this section the first lattice QCD cal-culation of the charm quark electromagnetic form fac-tors in the nucleon. This first-principles analysis re-quires a disconnected insertion calculation. By “discon-nected insertion,” one refers to the nucleon matrix el-ements involving self-contracted quark graphs (loops),which are correlated with the valence quarks in the nu-cleon propagator by the fluctuating background gaugefields. (
Notice the distinction with the term “discon-nected diagram” used in the continuum Quantum FieldTheory literature. ) Numerical expense and complexityof the disconnected insertion calculations in LQCD, thedeficit of good signal-to-noise ratio in the matrix ele-ments, and the possibility for a very small magnitude ofthe c quark matrix elements make it di ffi cult to obtain aprecise determination of G cE , M ( Q ). We, therefore, needto accept several limitations while performing this cal-culation. For example, the data is almost twice as noisycompared to the matrix elements of the strange electro-magnetic form factors G sE , M ( Q ) [42, 43, 44] and wedo not see any signal for one of the gauge ensembles(32ID with a lattice spacing of a = .
143 fm [58]) usedin the previous calculations [42, 44]. Moreover, we areonly able to perform the widely used two-states summedratio fit of the nucleon three-point (3 pt ) to two-point(2 pt ) correlation functions instead of a simultaneous fitto the summed ratio and conventional 3 pt / pt -ratio aswas done in [44]. The reason is that the 3 pt / pt -ratio fitfor extracting G cE , M ( Q ) is not stable for the ensembleat the physical pion mass m π =
139 MeV. We also keepin mind that the O ( m c a ) errors associated with the lat-tice spacing can be larger than the case for the G sE , M ( Q )matrix elements.Our calculation comprises numerical compu-tation with valence overlap fermion on threeRBC / UKQCD domain-wall fermion gauge config-urations [58, 59]: (ensemble ID, L × T , β , a (fm), m π (MeV), N config ) = { (48I, 48 ×
96, 2.13, 0.1141(2),139, 81), (32I, 32 ×
64, 2.35, 0.0828(3), 300, 309),324I, 24 ×
64, 2.13, 0.1105(3), 330, 203) } . Here L is spatial and T is temporal size, a is lattice spacing, m π is the pion mass corresponding to the degeneratelight-sea quark mass and N config is the number ofconfigurations. We use 17 valence quark masses acrossthese ensembles to explore the quark-mass dependenceof the charm electromagnetic form factors. The detailsof the numerical setup of this calculation can be foundin [60, 61, 62, 44]. m c was determined in a global fiton the lattice ensembles with β = .
13 fm and 2 . M D ∗ s , M D ∗ s − M D s , and M J /ψ in [63]. Our statistics arefrom approximately 100k to 500k measurements acrossthe 24I to 48I ensembles. The quark loop is calculatedwith the exact low eigenmodes (low-mode average)while the high modes are estimated with 8 sets of Z noise [64] on the same (4 , , ,
2) grid with odd-evendilution and additional dilution in time. We refer thereaders to previous work [44] for a detailed discussionof the similar numerical techniques which have beenused for this calculation. G cE , M ( Q ) can be obtained by Figure 1: Two-state fits of the 32I and 48I ensembles 3pt / G cE , M ( Q ) matrix elements at the unitary points. The col-ored bands show the fit results. The upper panel shows the fit to matrixelements for G cE ( Q ) and the lower panel shows that for G cM ( Q ). the ratio of a combination of 3pt and 2pt correlations as, R µ ( (cid:126) q , t , t ) ≡ Tr[ Γ m Π ptV µ ( (cid:126) q , t , t )]Tr[ Γ e Π pt ( (cid:126) , t )] × e ( E q − m ) · ( t − t ) E q E q + m N . (1)Here, Π pt is the nucleon 2pt function, Π ptV µ is the nu-cleon 3pt function with the bilinear operator V µ ( x ) = c ( x ) γ µ ¯ c ( x ), E q = (cid:113) m N + (cid:126) q and m N is the nucleonmass, (cid:126) q = (cid:126) p (cid:48) − (cid:126) p is the three-momentum transfer withsink momentum (cid:126) p (cid:48) and the source momentum (cid:126) p = G cE is Γ m = Γ e = (1 + γ ) / G cM is Γ m = Γ k = − i (1 + γ ) γ k γ / k = , , R µ contains a ratio Z P ( q ) / Z P (0) (cid:44)
1, where Z P ( q ) is the wavefunction overlap for the point sink withmomentum | (cid:126) q | . As estimated in [44], the error intro-duced by neglecting this factor is about ∼
5% comparedto the statistical error ≥
30% in the matrix elements andthus it is ignored in this work. We extract G cE , M ( Q )matrix elements using the two-states fit of the nucleon3 pt / pt summed ratio SR ( t ) for a given Q and fixedindex in Eq. (1): SR ( t ) ≡ t ≤ ( t − t (cid:48)(cid:48) ) (cid:88) t ≥ t (cid:48) R ( t , t ) = ( t − t (cid:48) − t (cid:48)(cid:48) + C + C e − ∆ mt (cid:48)(cid:48) − e − ∆ m ( t − t (cid:48) + − e − ∆ m + C e − ∆ mt (cid:48) − e − ∆ m ( t − t (cid:48)(cid:48) + − e − ∆ m . (2)Here, R ( t , t ) is the 3 pt / pt -ratio, t and t are thesource and sink temporal positions, respectively, and t is the time at which the bilinear operator ¯ c ( x ) γ µ c ( x )is inserted, t (cid:48) and t (cid:48)(cid:48) are the number of time slices wedrop at the source and sink sides, respectively, and wechoose t (cid:48) = t (cid:48)(cid:48) = C i are the spectral weights involv-ing the excited-state contamination. Ideally, ∆ m is theenergy di ff erence between the first excited state and theground state but in practice, this is an average of themass di ff erence between the proton and the lowest fewexcited states. As shown in [44], the 3 pt / pt -ratio datapoints are almost symmetric between the source andsink within uncertainty and introducing two ∆ m doesnot change the fit results of C . The excited states in the SR ( t ) fit fall o ff faster as e − ∆ mt compared to the two-states fit case of the 3 pt / pt -ratio where the excited-state falls o ff at a slower rate as e − ∆ m ( t − t ) . Thesefaster-decreasing excited-state e ff ects allow for fittingthe matrix elements starting from shorter time extents,as was demonstrated in [65]. In Fig. 1, we present asample extraction of G cE , M matrix elements on 48I at m π =
139 MeV pion mass at Q = .
25 GeV which4nables us to demonstrate the extraction of ∆ m withsignal-to-noise ratio better than at other Q data pointson the 48I ensemble. We also present a similar ex-ample on the 32I ensemble at m π =
300 MeV andat the largest Q where the excited-state contributionis expected to be the largest. For example, we obtain ∆ m = . C = . ∆ m = . C = . G cE ( Q ) fits. m . m . m . m . m . m .
48I ( a = 0 . m . m . m . m . m .
24I ( a = 0 . a = 0 . m . m . m . m . m . Figure 2: G cE , M ( Q ) matrix elements obtained from the 48I, 32I, and24I ensembles. Corresponding legends for di ff erent pion masses areincluded in the lower panel of the figure. The numbers in the legends,such as m m
251 represent the data points corresponding to pionmass 139 MeV and 251 MeV, respectively at di ff erent Q -values. Thecyan band indicates G cE , M ( Q ) | physical . The outer (lighter tinted) cyanmargins represent an estimate of systematic uncertainty. Matrix el-ements at the same Q -value but at di ff erent pion masses are shownwith small o ff sets for better visibility. We present the matrix elements of G cE , M ( Q ) ob-tained from the fit Eq. (2) in the upper and lower pan-els of Fig. 2. With the extracted 102 matrix elementsfrom three gauge ensembles (for each of G cE ( Q ) and G cM ( Q )) at di ff erent pion masses and Q , we performa simultaneous correlated and model-independent z -expansion fit [66, 67] to G cE , M ( Q ) in the momentumtransfer range of 0 ≤ Q ≤ . and perform chi-ral, continuum (lattice spacing a → L → ∞ ) extrapolationsto obtain the form factors in the physical limit. For sucha fit to G cE ( Q ), we adopt the following fit form G cE ( Q , m π , m π, vs , m J /ψ , a , L ) = k max (cid:88) k = λ k z k × (cid:18) + A m π + A m π, vs + A m J /ψ + A a + A √ L e − m π L (cid:19) , (3)where z = (cid:112) t cut + Q − √ t cut (cid:112) t cut + Q + √ t cut . (4)In fit Eq. (3), m π, vs is the partially quenched pion mass m π, vs = / m π + m π, ss ) with m π, ss the pion mass corre-sponding to the sea quark mass. The m J /ψ masses forthe lattice ensembles are obtained in [63] and extrapo-lated to the physical value m J /ψ = .
097 GeV [68]. A includes the mixed-action parameter ∆ mix [69]. The vol-ume correction in fit (3) has been adopted from [70] tobest describe the data. We use t cut = m J /ψ , the pole of c ¯ c pair production. We note that this choice is di ff erentfrom the fit to the strange quark form factor where the t cut is chosen at 4 m K , because the mass of two kaons isless than the mass of φ , while the mass of two D mesonsis greater than the mass of J /ψ . One may also consider η c , which is a bit lighter, but J /ψ is more likely to beproduced from a vector current.The inclusion of higher-order terms beyond k max = z -expansion fit (3). We obtain χ / d . o . f . = . λ = λ = . λ = − . λ = . . λ = − . . A = − . A = − . A = . A = − . A = − . A m π by A m π results innegligible change in the final result. A faster decreasingvolume correction exp( − m D L ) correction gives A = . − m π L ) as expected and they are in statistical agree-ment. The significant increase of the uncertainty in thephysical value of G cE ( Q ) at larger Q is due to the factthat the data points on the 24I and 32I ensembles are atmuch heavier pion mass compared to the matrix elementat the physical m π =
139 MeV on the 48I ensemble andthere exist no LQCD data points at Q ≥ .
31 GeV on the 48I ensemble. We also see a similar feature for5 cM ( Q ) shown in the lower panel of Fig. 2. The cyanband in Fig. 2 represents G cE ( Q ) | physical in the physicallimit after the quark mass, finite lattice spacing and vol-ume corrections have been implemented using the fit pa-rameters listed above. Since most of the A i correctionsdo not have statistical significance, we explore the abovefit with separate combinations of A i , for example, with A & A , A & A , and A , A , & A correction terms. Forthese fits, we obtain { A , A } = {− . , − . } , { A , A } = {− . , − . } , and { A , A , A } = {− . , − . , − . } , while the physical G cE ( Q ) remains essentially unchanged with slightlysmaller final uncertainties compared to when all A i cor-rections are included. A similar investigation for the G cM ( Q ) fit results in a similar conclusion.The systematic uncertainty is estimated by calculat-ing the di ff erences between G cE ( Q ) | physical and G cE ( Q )obtained from the fit Eq. (3) by considering the correc-tions of the A , A , A terms from the m J /ψ -value on the24I ensemble obtained in [63], the smallest lattice spac-ing from the 32I ensemble, and the 48I ensemble withthe largest volume, respectively. The systematic uncer-tainty has been added as lighter-tinted margins to thestatistical uncertainty band in Fig. 2.To obtain G cM ( Q ) in the physical limit and in the0 ≤ Q ≤ . momentum transfer region, weadopt the following empirical fit form with the volumecorrection term adopted from [71]: G cM ( Q , m π , m π, vs , m J /ψ , a , L ) = k max (cid:88) k = λ k z k × (cid:18) + A m π + A m π, vs + A m J /ψ + A a + A m π (cid:20) − m π L (cid:21) e − m π L (cid:19) . (5)We limit the k max = z -expansion have no statistically significant e ff ecton G cM ( Q ) | physical . With the χ / d . o . f . = .
14 in thefit (5), we obtain the fit parameters λ = − . λ = . λ = − . λ = . . A = . A = − . A = . A = . A = − . G cM ( Q ) | physical is obtained in a simi-lar way as for the case of G cE ( Q ) | physical .The electric and magnetic radii can be extracted fromthe slope of the G cE , M ( Q ) form factors as Q → (cid:104) r E , M (cid:105) c = − dG cE , M ( Q ) dQ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q = = − (cid:18) λ t cut (cid:19) . (6) Using the λ -values from above, we obtain (cid:104) r E (cid:105) c = − . , (cid:104) r M (cid:105) c = − . . (7)While the existence of a nonzero G cE ( Q ) (and thusthe Dirac F c ( Q ) form factor) is related to a nonzeroasymmetry of the [ c ( x ) − ¯ c ( x )] distribution, a nonzero G cM ( Q ) (and thus Pauli F c ( Q ) form factor) imme-diately implies that there is a nonzero orbital angu-lar momentum contribution to the nucleon from thecharm quarks. The Pauli form factor F c ( Q ) has theLFWF representation as the overlap of the states dif-fering by one unit of orbital angular momentum. Itis thus closely related to the spin sector of the charmquark sea in the proton. F c ( Q ) is also given by thefirst x moment of the generalized parton distribution E c ( x , ξ, t ), t = − Q , which contributes to the secondterm of Ji’s sum rule [72]. Therefore, our result µ cM = − . stat (5) sys indicates a nontrivial role of thecharm quark sea in understanding the spin content ofthe proton.
3. A nonperturbative model for computing the in-trinsic [ c ( x ) − ¯ c ( x )] asymmetry in the nucleon Although a model-independent determination of ICdistributions from the charm quark form factors is notpossible at the moment, the underlying physics, gov-erned by QCD, does imply some important connectionsand constraints, such as the QCD inclusive-exclusiveconnection [73, 74, 75]: It relates hadron form factorsat large Q to the hadron structure function at x → c and ¯ c carry opposite charges, the charm quark formfactor measures the di ff erence of the transverse chargedensity [76] between the c and ¯ c quarks. For a posi-tive charge form factor, like those shown in Fig. 2, thecharm quark distribution is more spread out than the an-ticharm distribution in q T -space, where q T is the Fourierconjugate variable of the transverse coordinate b T and q T = Q . As a feature of Fourier transform, the charmquark density ρ ( b T ) is more centralized in b T -space thanthe anticharm quark. Representing the density as thesquare of the wave function, ρ ( b T ) = | ˜ ψ ( b T ) | , one caneasily find the k T -space wave function ψ ( k T ) is morespread out for the charm quark, where k T is the intrin-sic transverse momentum. Thus the actual distributionfor a positive charge form factor favors the c quark car-rying higher momentum than the ¯ c quark. As a result,the [ c ( x ) − ¯ c ( x )] distribution will favor negative values6n the low- x region and positive values in the high- x re-gion. We note that a strict definition of the transversecharge density is given by the Fourier transform of theDirac form factor F c ( Q ), which dominates the chargeform factor G cE ( Q ) in the low- Q regime.For a quantitative estimation of [ c ( x ) − ¯ c ( x )], onecurrently has to rely on additional assumptions, al-though the form factor result does indicate some qual-itative features of the distribution function based onthe discussions above. Here we take the nonpertur-bative phenomenological model in [48], which relatesthe form factor and sea-quark distribution functionswith minimal parameters. The formalism is basedon the gauge / gravity correspondence [77], light-frontholographic mapping [51, 52, 56], and the generalizedVeneziano model [53, 54, 55]. In the following, we re-fer to this model as LFHQCD. The charm quark Diracand Pauli form factors are given by [78] F c ( Q ) = (cid:88) τ c τ [ F τ ( Q ) − F τ + ( Q )] , (8) F c ( Q ) = (cid:88) τ χ τ F τ + ( Q ) , (9)where τ is the number of constituents of the Fock statecomponent. The leading Fock state with nontrivial con-tribution to the charm form factor is | uudc ¯ c (cid:105) , which is a τ = e.g. | uudu ¯ uc ¯ c (cid:105) , | uudd ¯ dc ¯ c (cid:105) , etc., will con-tribute to τ = | uudc ¯ cg (cid:105) , τ =
6, and / or higher Fock states.Form factor F τ can be expressed in a reparametriza-tion invariant form [56] F ( t ) τ = N τ (cid:90) dx w (cid:48) ( x ) w ( x ) − α ( t ) [1 − w ( x )] τ − , (10)where α ( t ) is the Regge trajectory, and N τ is a normal-ization factor; w ( x ) is a flavor independent function with w (0) = , w (1) = w (cid:48) ( x ) ≥
0. We use the same uni-versal form of the function w ( x ) from [56] w ( x ) = x − x e − a (1 − x ) , (11)with a = .
480 [83]: It incorporates Regge behaviorat small x with the J /ψ intercept (14), w ( x ) → x as x →
0, and the inclusive-exclusive counting rule at large x , q τ ( x ) → (1 − x ) τ − , as x → ffi cients c τ and χ τ are parameters to be determined from the LQCD computation of G cE ( Q ) | physical and G cM ( Q ) | physical to ob-tain F c ( Q ). The constraint that the numbers of charmand anticharm quarks are identical for each Fock statecomponent has been incorporated in Eq. (8).Then the asymmetric charm-anticharm distributionfunction is c ( x ) − ¯ c ( x ) = (cid:88) τ c τ [ q τ ( x ) − q τ + ( x )] , (12)where τ ≥ q τ ( x ) = N τ w ( x ) − α (0) [1 − w ( x )] τ − w (cid:48) ( x ) . (13)We should note here that the coe ffi cients c τ in Eq. (12)are the same as those in Eq. (8). Therefore, once theyare determined by the form factor, one can make predic-tions for the distribution functions.The form factors and distribution functions aboveare derived at the massless quark limit and one mayhave di ff erent approaches to incorporate quark masscorrections. For small quark masses (up, down andstrange) the latter can be treated perturbatively, leav-ing the Regge slope unchanged and leading to a mod-erate change of the intercept. The resulting spectra arein very good agreement with experiment [51, 79]. Thesituation is more intricate for the case of heavy quarks,like c quarks, since now conformal symmetry is stronglybroken and the occurrence of linear trajectories is farfrom obvious. It has been shown, however, that the for-malism can indeed be extended to heavy quark boundstates [80, 81], leading to a fair agreement with the data.In this case, the Regge trajectories are still linear, but theslope depends on the heavy quark mass. The interceptchanges quite drastically with the quark mass.The J /ψ Regge trajectory obtained in [81] is α ( t ) J /ψ = t κ c − . , (14)where κ c = .
874 GeV. This result agrees with the oneobtained in a phenomenological potential model [82].The large change of the intercept as compared to lightquarks removes the small- x singularity of quark dis-tribution functions while keeping the counting rules atlarge Q and at large x unchanged. The change of theslope a ff ects only the generalized parton distributionfunction. The quark distribution di ff erence [ c ( x ) − ¯ c ( x )]is not sensitive to the choice of the mass correction pro-cedure, since the quark mass a ff ects equally charm andanticharm distributions.In practice, one needs to truncate the expansion inEq. (8) to have numerical results. For simplicity, weonly keep the lowest Fock state containing the charm7uark components, i.e. , τ =
5. The coe ffi cient c τ is de-termined, through Eqs. (8) and (9) by the lattice resultsof G cE ( Q ) and G cM ( Q ) at the physical limit. We per-form a fit to the extracted results of G cE ( Q ) | physical and G cM ( Q ) | physical , i.e., the bands in Figs. 2. Since the lat-tice data from di ff erent ensembles are evaluated at dif-ferent Q values, and have been utilized to determine thequark mass, lattice spacing, and finite volume e ff ects,the e ff ective number of data points in the physical limitis 6 for G cE ( Q ) | physical and 6 for G cM ( Q ) | physical1 . To re-ally capture the uncertainty, we create 200 replicas fromthe extracted bands. Each replica is firstly generated byrandomly sampling 6 data points of G cE ( Q ) | physical and6 data points of G cM ( Q ) | physical from the extracted bandswithin 0 < Q < . , which are covered by thelattice data. Then for each data point, the central valueis resampled with a Gaussian distribution according toits uncertainty. In addition, we also randomly shift thevalue of κ c within ±
5% in each single fit of one replica toincorporate the theoretical uncertainty. The coe ffi cientdetermined from the fit is c τ = = . ffi cient c τ = from thelattice computation, we use Eq. (12), to obtain the asym-metric charm-anticharm distribution function x [ c ( x ) − ¯ c ( x )] shown in Fig. 3. The result from the fit is in agree-ment with the qualitative analysis at the beginning ofthis section, namely, that the charm quark tends to carrylarger momentum than the anticharm quark based on thelattice results for the charm quark form factors. Fromthe x [ c ( x ) − ¯ c ( x )] distribution obtained by combiningLQCD results from G cE , M ( Q ) and the LFHQCD formal-ism, we can calculate the first moment of the di ff erenceof c ( x ) and ¯ c ( x ) PDFs to be (cid:104) x (cid:105) c − ¯ c = (cid:90) dx x [ c ( x ) − ¯ c ( x )] = . , (15)where the total uncertainty is obtained from the fittingerror in c τ = and 5% variation in κ c . The [ c ( x ) − ¯ c ( x )]distribution result is about 3 times smaller in magnitudethan the s ( x ) − ¯ s ( x ) distribution obtained with the sameformalism [48]. Although a small asymmetry couldbe a result of the cancellation of two relatively large c ( x ) and ¯ c ( x ) distributions, it is possible that the intrin-sic charm and anticharm distributions are both small.Furthermore, the charm and anticharm distributions athigh energy scales are dominated by the extrinsic sea For each ensemble we have data points at 6 di ff erent Q . A si-multaneous fit of the data from three ensembles (48I, 32I, 24I) withdi ff erent quark masses, lattice spacings, and volumes leads to the re-sults in the physical limit. -2 -1 x x [ c ( x ) − ¯ c ( x ) ] LFHQCD
Figure 3: The distribution function x [ c ( x ) − ¯ c ( x )] obtained from theLFHQCD formalism using the lattice QCD input of charm electro-magnetic form factors G cE , M ( Q ). The outer (lighter tinted) cyan mar-gins represent an estimate of systematic uncertainty in the x [ c ( x ) − ¯ c ( x )] distribution obtained from a variation of the hadron scale κ c by5%. from perturbative radiation. The experimental observa-tion and isolation of the intrinsic charm e ff ect are ex-tremely challenging in such cases. Thus it is not sur-prising that the recent measurement of J /ψ and D pro-ductions by the LHCb collaboration [15] found no in-trinsic charm e ff ect. An ideal place to investigate intrin-sic charm would be the J /ψ or open charm productionsat relatively low energies, e.g. , at JLab, although it isalso possible to see intrinsic charm e ff ects in very ac-curate measurements of high energy reactions. In addi-tion, lepton-nucleon scattering may provide a cleanerprobe than nucleon-nucleon scattering to help reducebackgrounds and increase the chance to observe the in-trinsic charm e ff ect, and therefore the future EIC willprovide such opportunities.The nonzero value of G cE ( Q ) can also originatefrom the interference of the q → gq → c ¯ cq and q → ggq → c ¯ cq sub-processes, without the exis-tence of IC. However, as mentioned earlier, this extrin-sic [ c ( x ) − ¯ c ( x )] asymmetry which arises at the next-to-next-to-leading order level is negligible [40]. Moreover,according to [40], this extrinsic asymmetry would re-sult in a much smaller and negative value of the firstmoment of [ c ( x ) − ¯ c ( x )] distribution (cid:104) x (cid:105) c − ¯ c compared to (cid:104) x (cid:105) c − ¯ c = . (cid:104) x (cid:105) c − ¯ c would also result in a positive[ c ( x ) − ¯ c ( x )] distribution at small x and a negative dis-tribution at large x , in contrast to the [ c ( x ) − ¯ c ( x )] distri-bution we have obtained here. But the evidence based8n the [ s ( x ) − ¯ s ( x )] distribution in [48], the EMC mea-surement [10], and perturbative QCD computation [40]seem to indicate extremely small values of extrinsiccharm for x > .
1. The present determination of the[ c ( x ) − ¯ c ( x )] distribution from LQCD supports the ex-istence of nonperturbative intrinsic heavy quarks in thenucleon wavefunction at large x ∼ . − . x F > . pp collisions at the LHC from the direct couplingof the Higgs to the intrinsic heavy quark pair [84].
4. Conclusion and outlook
In this article, we have presented the first latticeQCD calculation of the charm quark electromagneticform factors in the physical limit. This first latticeQCD calculation indicates that a nonzero charm elec-tric form factor corresponds to the intrinsic charm-anticharm asymmetry in the nucleon sea, thereby pro-viding an indication of the existence of nonzero intrinsiccharm based on a first-principles calculation. In addi-tion, the nonzero value of the charm magnetic form fac-tor indicates a nonzero orbital angular momentum con-tribution to the nucleon coming from the charm quarks.We have discussed that the existence of IC is supportedby QCD and how an accurate knowledge of the intrinsiccharm can help to remove bias in the global fits of PDFsand related phenomenological studies.Motivated by the new lattice results, we have used thenonperturbative light-front holographic framework in-corporating the QCD inclusive-exclusive connection atlarge x to determine the [ c ( x ) − ¯ c ( x )] asymmetry up to anormalization factor, which is constrained by the latticeQCD calculation. Since the LFHQCD calculation startsfrom a nucleon Fock state with hidden charm, the partondistributions determined in this model refer exclusivelyto intrinsic charm where the small- x behavior is deter-mined by the J /ψ intercept. On the other hand, con-tributions from gluon splitting are supposed to be de-termined by the pomeron trajectory with a much higherintercept. These features will be discussed in a separatepublication.The new determination of the [ c ( x ) − ¯ c ( x )] asymme-try presented here gives additional elements and furtherinsights into the existence of intrinsic charm. It also canprovide complementary information to the global fits ofPDFs which look for the possibility of IC in the absenceof ample experimental data. Acknowledgements
RSS thanks Jeremy R. Green, Luka Leskovec, Jian-Wei Qiu, Anatoly V. Radyushkin, and David G.Richards for useful discussions. The authors thankthe RBC / UKQCD collaborations for providing theirDWF gauge configurations. This work is supportedby the U.S. Department of Energy, O ffi ce of Science,O ffi ce of Nuclear Physics under contract DE-AC05-06OR23177. A. Alexandru is supported in part by U.S.DOE Award Number DE-FG02-95ER40907. T. Draperand K.F. Liu are supported in part by DOE Award Num-ber DE-SC0013065. Y. Yang is supported by Strate-gic Priority Research Program of Chinese Academy ofSciences, Grant No. XDC01040100. This researchused resources of the Oak Ridge Leadership ComputingFacility at the Oak Ridge National Laboratory, whichis supported by the O ffi ce of Science of the U.S. De-partment of Energy under Contract No. DE-AC05-00OR22725. This work used Stampede time under theExtreme Science and Engineering Discovery Environ-ment (XSEDE), which is supported by National Sci-ence Foundation grant number ACI-1053575. We alsothank the National Energy Research Scientific Comput-ing Center (NERSC) for providing HPC resources thathave contributed to the research results reported withinthis paper. We acknowledge the facilities of the USQCDCollaboration used for this research in part, which arefunded by the O ffi ce of Science of the U.S. Departmentof Energy. References [1] S. J. Brodsky, P. Hoyer, C. Peterson, and N. Sakai, “The intrinsiccharm of the proton,” Phys. Lett. , 451 (1980).[2] S. J. Brodsky, J. C. Collins, S. D. Ellis, J. F. Gunion, andA. H. Mueller, “Intrinsic Chevrolets at the SSC,” in
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