Nucleon Tomography and Generalized Parton Distribution at Physical Pion Mass from Lattice QCD
MMSUHEP-20-014
Nucleon Tomography and Generalized Parton Distribution at Physical Pion Massfrom Lattice QCD
Huey-Wen Lin
1, 2, ∗ Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824 Department of Computational Mathematics, Science and Engineering,Michigan State University, East Lansing, MI 48824
We present the first lattice calculation of the nucleon unpolarized generalized parton distribution(GPD) at the physical pion mass using a lattice ensemble with 2+1+1 flavors of highly improvedstaggered quarks (HISQ) generated by MILC Collaboration, with lattice spacing a ≈ .
09 fm andvolume 64 ×
96. We use momentum-smeared sources to improve the signal at nucleon boostmomentum P z ≈ . . , .
9] GeV .Nonperturbative renormalization in RI/MOM scheme is used to obtain the quasi-distribution beforematching to the lightcone GPDs. The three-dimensional distributions H ( x, Q ) and E ( x, Q ) at ξ = 0 are presented, along with the three-dimensional nucleon tomography and impact-parameter–dependent distribution for selected Bjorken x at µ = 3 GeV in MS scheme. PACS numbers: 12.38.-t, 11.15.Ha, 12.38.Gc
Nucleons (that is, protons and neutrons) are the building blocks of all ordinary matter, and the study of nucleonstructure is a central goal of many worldwide experimental efforts. Gluons and quarks are the underlying degreesof freedom that explain the properties of nucleons, and fully understanding how they contribute to the properties ofnucleons (such as their mass or spin structure) helps to decode the last part of the Standard Model that rules ourphysical world. In the theory of quantum chromodynamics (QCD), a branch of Standard Model, gluons stronglyinteract with themselves and with quarks, binding both nucleons and nuclei. However, due to their confinementwithin these bound states, we cannot single out individual constituents, quarks and gluons, to study them. More thanhalf a century since the discovery of nucleon structure, our understanding of it has improved greatly; however, thereis still long way to go in unveiling the nucleon’s detailed structure, which is characterized by functions such as thegeneralized parton distributions (GPDs) [1–3]. The GPDs can be viewed as a hybrid of parton distributions (PDFs),form factors and distribution amplitudes. They play an important role in providing a three-dimensional spatial pictureof the nucleon [4] and in revealing the spin structure of the nucleon [2]. Experimentally, GPDs can be accessed inexclusive processes such as deeply virtual Compton scattering [5] or meson production [6]. Determining GPDs drawsglobal scientific interest: Experimental collaborations and facilities worldwide have been devoted to searching forthese last unknowns of the nucleon, including HERMES at DESY, COMPASS at CERN, GSI in Europe, BELLE andJPAC in Japan, Halls A, B and C at Jefferson Laboratory, and PHENIX and STAR at RHIC (Brookhaven NationalLaboratory) in the US. There has also been planning for future facilities to continue this work in the decades to come:an upcoming electron-ion collider (EIC) [7] at Brookhaven National Laboratory in the US. In Europe, CERN plansto add an electron accelerator to the existing LHC hadron accelerator, creating the Large Hadron-Electron Collider(LHeC) [8, 9]. In Asia, there are also plans for an EIC in China (EIcC) [10, 11].Although interest in GPDs has grown enormously, we are still in need of fresh theoretical and phenomenologicalideas. The development of reliable and model-independent techniques is required. Most QCD models have issuesassociated with three-dimensional structure that are not yet fully understood, so a reliable framework for extractingthree-dimensional parton distributions and fragmentation functions from experimental observables does not yet exist.Theoretically, there are factorization issues in hadron production from hadronic reactions, and theoretical efforts atcollaboration centers and workshops are striving to answer key questions that lie along the path to a precise mappingof the three-dimensional nucleon structure from experimental data. It has become common understanding that weneed to develop a program in both theory and experiment that will allow an accurate flavor decomposition of thenucleon GPDs, including flavor differences in the quark structure, the gluon structure and the nucleon sea-quarkGPDs. Most current theoretical issues are associated with the nonperturbative features of QCD, that is, where thestrong coupling is too large for analytic perturbative methods to be valid. Using a nonperturbative theoretical methodthat starts from the quark and gluon degrees of freedom, lattice QCD (LQCD), allows us to compute these propertieson supercomputers.Probing hadron structure with lattice QCD was for many years limited to only the first few moments, due to com-plications arising from the breaking of rotational symmetry by the discretized Euclidean spacetime. The breakthroughfor LQCD came in 2013, when a technique was proposed to connect quantities calculable on the lattice to those on thelightcone: This large-momentum effective theory (LaMET), also known as the “quasi-PDF method” [12–14], allows a r X i v : . [ h e p - ph ] A ug us to calculate the full Bjorken- x dependence of distributions for the first time. There has been much progress madesince the first LaMET paper [14–104]. A majority of the work has been done using only one lattice ensemble, butthere has been some progress made toward determining the size of lattice systematic uncertainties. For example,finite-volume systematics were studied in Refs. [31, 86]. Machine-learning algorithms have been applied to the inverseproblem [84, 105] and to making predictions for larger boost momentum (and larger Wilson-displacement link) matrixelements [106]. On the lattice discretization errors, a N f = 2+1+1-flavor superfine ( a ≈ .
042 fm) lattice at 310-MeVpion mass was used to study nucleon PDFs in Ref. [91], and results using multiple lattice spacings were reported inRefs. [89, 92, 105]. The first attempt to obtain strange and charm distributions of the nucleon have also only beenrecently reported [90]. However, beyond one-dimensional hadron structure, there is little work available: Last spring,the first lattice study of generalized parton distributions was made for the pion case [32]. During the completionof this work, ETM Collaboration reported preliminary findings on both unpolarized and polarized nucleon GPDs atboost momentum 1.67 GeV at pion mass M π ≈
260 MeV in a conference proceeding [107]. Both of these works usedunphysically heavy pion mass.In this work, we present the first lattice-QCD calculation of the generalized parton distribution of the nucleon atphysical pion mass, using LaMET method. Our calculation is done using clover valence fermions on an ensembleof gauge configurations with N f = 2 + 1 + 1 (degenerate up/down, strange and charm) flavors of highly improvedstaggered quarks (HISQ) [108], generated by the MILC Collaboration [109] with a lattice spacing a = 0 .
09 fm withvolume of 64 ×
96 (with spatial lattice extension of L ≈ . H ( x, ξ, t ) and E ( x, ξ, t ), and the polarized ˜ H ( x, ξ, t ) and ˜ E ( x, ξ, t ), are defined in terms ofthe matrix elements F q ( x, ξ, t ) = (cid:90) dz − π e ixp + z − (cid:68) p (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) ¯ ψ (cid:16) − z (cid:17) γ + L (cid:16) − z , z (cid:17) ψ (cid:16) z (cid:17)(cid:12)(cid:12)(cid:12) p (cid:48) (cid:69) z + =0 ,(cid:126)z ⊥ =0 = 12 p + (cid:20) H ( x, ξ, t )¯ u ( p (cid:48)(cid:48) ) γ + u ( p (cid:48) ) + E ( x, ξ, t )¯ u ( p (cid:48)(cid:48) ) iσ + ν ∆ ν m u ( p (cid:48) ) (cid:21) , ˜ F q ( x, ξ, t ) = (cid:90) dz − π e ixp + z − (cid:68) p (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) ¯ ψ (cid:16) − z (cid:17) γ + γ L (cid:16) − z , z (cid:17) ψ (cid:16) z (cid:17)(cid:12)(cid:12)(cid:12) p (cid:48) (cid:69) z + =0 ,(cid:126)z ⊥ =0 = 12 p + (cid:20) ˜ H ( x, ξ, t )¯ u ( p (cid:48)(cid:48) ) γ + γ u ( p (cid:48) ) + ˜ E ( x, ξ, t )¯ u ( p (cid:48)(cid:48) ) γ ∆ + m u ( p (cid:48) ) (cid:21) , (1)where L ( − z/ , z/
2) is the gauge link along the lightcone and p µ = p (cid:48)(cid:48) µ + p (cid:48) µ , ∆ µ = p (cid:48)(cid:48) µ − p (cid:48) µ , t = ∆ , ξ = p (cid:48)(cid:48) + − p (cid:48) + p (cid:48)(cid:48) + + p (cid:48) + . (2)In the limit ξ, t → H and ˜ H reduce to the usual unpolarized and polarized parton distributions, while the informationencoded in E and ˜ E cannot be accessed, since they are multiplied by the momentum transfer ∆. Only in exclusiveprocesses with a nonzero momentum transfer can E and ˜ E be probed. The one-loop matching [35] for the GPD H ( ˜ H )turns out to be similar to that for the parton distribution, whereas the matching for E ( ˜ E ) is trivial, since the quasiand lightcone definitions yield the same result for E ( ˜ E ) at one-loop and leading-power accuracy. This implies thatthe lightcone GPD E ( ˜ E ) can be smoothly approached in the large-momentum limit of its quasi-GPD counterpart.In this work, we focused on the unpolarized GPDs and their quasi-GPD counterparts defined in terms of spacelikecorrelations. We use clover valence fermions on an ensemble with lattice spacing a ≈ .
09 fm, box size L ≈ . M π ≈
135 MeV and N f = 2 + 1 + 1 (degenerate up/down, strange and charm) flavors ofhighly improved staggered dynamical quarks (HISQ) [108] generated by MILC Collaboration [109]. The gauge linksare one-step hypercubic (HYP)-smeared [110] to suppress discretization effects. The clover parameters are tuned torecover the lowest sea pion mass of the HISQ quarks. There have been a number of works using such “mixed-action”approaches in the past, and there has been promising agreement between the calculated lattice nucleon charges,moments and form factors and the experimental data when applicable [111–123]. Gaussian momentum smearing [124]is used on the quark field to improve the overlap with ground-state nucleons of the desired boost momentum, allowingus to reach higher boost momentum for the nucleon states. We use high-statistics measurements, 125,440 total, similarto our previous PDF works [24, 27, 29], to drive down the increased statistical noise at high boost momenta.To better extract the boosted-momentum ground-state nucleon, we apply the variational method [125] to extractthe principal correlators corresponding to pure energy eigenstates from a matrix of correlators. We use a 3 × Q ( GeV ) z R e M E Q ( GeV ) - - - z I m M E FIG. 1: H ( p f , q , z ) matrix elements at selected Q ∈ { . , . , . } GeV . smeared-smeared correlation matrix, which can be decomposed as C ij = N (cid:88) n =1 v n ∗ i v nj e − tE n (3)with eigenvalues λ n ( t, t r ) = e − ( t − t r ) E n by solving the generalized eigensystem problem C ( t ) V = λ ( t, t r ) C ( t r ) V ,where V is the matrix of eigenvectors ( v i,j ) and t r is a reference time slice. The resulting 3 eigenvalues (principalcorrelators) λ n ( t, t r ) are then further analyzed to extract the energy levels E n . Since they have been projected ontopure eigenstates of the Hamiltonian, the principal correlator should be fit well by a single exponential and doublechecked for the consistency of the obtained energies. The leading contamination due to higher-lying states is anotherexponential having higher energy; we use a two-exponential fit to help remove this contamination. The overlapfactors ( A n ) between the interpolating operators and the n th state are derived from the eigenvectors obtained in thevariational method.To calculate the GPD matrix elements at nonzero momentum transfer, we first calculate the matrix element (cid:104) χ N ( (cid:126)p f ) | O µ | χ N ( (cid:126)p i ) (cid:105) , where χ N is the nucleon spin-1/2 interpolating field, O µ = qγ µ L ( z ) q is the LaMET Wilson-linedisplacement operator with q being either an up or down quark field, and (cid:126)p { i,f } are the initial and final nucleonmomenta. We integrate out the spatial dependence and project the baryonic spin, leaving a time-dependent three-point correlator of the formΓ (3) ,Tµ,AB ( t i , t, t f , (cid:126)p i , (cid:126)p f ) = Z O (cid:88) n (cid:88) n (cid:48) f n,n (cid:48) ( p f , p i , E (cid:48) n , E n , t, t i , t f ) × (cid:88) s,s (cid:48) T αβ u n (cid:48) ( (cid:126)p f , s (cid:48) ) β (cid:104) N n (cid:48) ( (cid:126)p f , s (cid:48) ) | O µ | N n ( (cid:126)p i , s ) (cid:105) u n ( (cid:126)p i , s ) α , (4)where f n,n (cid:48) ( p f , p i , E (cid:48) n , E n , t, t i , t f ) contains kinematic factors involving the energies E n and overlap factors A n obtainedin the two-point variational method, n and n (cid:48) are the indices of different energy states and Z O is the operatorrenormalization constant (which is determined nonperturbatively). We use one final momentum (cid:126)p f = πL { , , } a − ,and vary the initial momentum (cid:126)p i to generate momentum transfer (cid:126)q = (cid:126)p f − (cid:126)p i = πL { n x , n y , } a − with integer n x,y and q = n x + n y ∈ { , , , , , , } . By fitting the time dependence of the three-point correlators to the form ofEq. 4 with n and n (cid:48) restricted to 0 and 1, we extract the ground state and matrix elements involving the first excitedstate. In this work, we concentrate on the ground-state matrix element with n = n (cid:48) = 0. The overdetermined systemof linear equations (using multiple (cid:126)q , O µ ) allows for solution of H ( p f , q , z ) and E ( p f , q , z ) as shown in Figs. 1 and2. The real matrix elements decrease quickly to zero due to the large boost momentum used in this calculation. Thishelps us to use smaller-displacement data to avoid large contributions from higher-twist effects in the larger- z region.To obtain quasi-GPDs, we first apply nonperturbative renormalization (NPR) in RI/MOM scheme to the barematrix elements, using the NPR done in previous work using the same lattice ensembles [24, 27, 29], itself followingthe same strategy described in Refs. [22, 41]. In this work, we focus on ξ = 0 GPDs, where the matching formula isthe same as that in PDFs, as discussed in Ref. [126]. We normalize all matrix elements by H ( p f , Q = 0 , z = 0), asin our previous PDF work [24, 27, 29]; using matrix-element ratios help to reduce the lattice systematic error, as in Q ( GeV ) z R e M E Q ( GeV ) - - - - - z I m M E FIG. 2: E ( p f , q , z ) matrix elements at selected Q ∈ { . , . , . } GeV . Q quasi - GPDmatched GPD - - x H Q quasi - GPDmatched GPD - - x E FIG. 3: H and E quasi-GPDs and matched GPDs at momentum transfer Q = 0 .
48 GeV . the continuum limit H ( p f , Q = 0 , z = 0), the vector charge, goes to 1. The nonperturbatively renormalized matrixelements are then Fourier transformed into quasi-GPDs and matched-to the physical GPDs. Examples of the GPDsat momentum transfer Q ≈ . are shown in Fig. 3. Figure 3 compares the H and E GPDs at Q ≈ . with the quasi-distribution and matched distribution using P z ≈ . x tolarge- x distribution, as expected; as one approaches lightcone limit, the probability of a parton to carrying a largerfraction of its parent nucleon’s momentum should become smaller. However, due to the limited zP z reach of thiscalculation, we found that the small- x region is unreliable, due to lack of precision lattice data to constrain it. Asa result, the antiquark (negative- x ) distribution can also be sensitive to the usage of P z to conserve charge. It hasbeen found in past works [18, 24, 27, 29] that higher boosted momenta are needed to improve the antiquark region.Therefore, for the rest of the work, we will focus on the x > .
05 region. The full three-dimensional shape of H and E as functions of x and Q can be found in Fig. 4. Our GPDs at zero transfer momentum, H ( Q = 0 , x ), areconsistently within errors of the earlier study in 2018 using the same ensemble. In the ξ = 0 limit, the H and E GPDdecrease near monotonically as x ( Q ) increases.The Fourier transform of the non–spin-flip GPD H ( x, ξ = 0 , Q ) gives the impact-parameter–dependent distribution q ( x, b ) [127] q ( x, b ) = (cid:90) d q (2 π ) H ( x, ξ = 0 , t = − q ) e i q · b , (5)where b is the transverse distance from the center of momentum. Figure 5 shows the first results of impact-parameter–dependent distribution from lattice QCD: a three-dimensional distribution as function of x and b , and two-dimensionaldistributions at x = 0 .
3, 0.5 and 0.7. The impact-parameter–dependent distribution describes the probability density
FIG. 4: H and E GPDs at ξ = 0 as functions of x and momentum transfer Q . FIG. 5: (Left) Nucleon tomography: three-dimensional impact parameter–dependent parton distribution as a function of x and b using lattice H at physical pion mass. (Right) The two-dimensional impact-parameter–dependent distribution for x = 0 . for a parton with momentum fraction of x located in the transverse plane at distance b . As seen in Fig. 5, theprobability decreases quickly as x increases. Using Eq. 5 and H ( x, ξ = 0 , t = − q ) obtained from direct latticecalculation at the physical pion mass allows us to take a snapshot of the nucleon in the transverse plane to performnucleon tomography.In this work, we compute the isovector nucleon unpolarized GPDs at physical pion mass using boost momentum2 . ξ = 0. There are residual lattice systematics are not yet included in thecurrent calculation: In our past studies, we found the finite-volume effects to be negligible for isovector nucleonquasi-distributions calculated within the range M val π L ∈ { . , . } . We anticipate such systematics should be smallcompared to the statistical errors. The lattice discretization has been studied in Ref. [105] with three-lattice spacingsof 0.06, 0.09, 0.12 fm in the LaMET study of pion and kaon distribution amplitudes; there was mild lattice-spacingdependence for a majority of the Wilson-link displacements studied with similar largest boost momenta. Futuredirections will be investigating ensembles with smaller lattice spacing to reach even higher boost momentum (eitherthrough direct calculation or with the aid of machine learning as previously done in Ref. [106]) so that we can pushtoward reliable determination of the smaller- x and antiquark regions. Acknowledgments
We thank the MILC Collaboration for sharing the lattices used to perform this study. The LQCD calculationswere performed using the Chroma software suite [128]. This research used resources of the National Energy ResearchScientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S.Department of Energy under Contract No. DE-AC02-05CH11231 through ERCAP; facilities of the USQCD Collabo-ration, which are funded by the Office of Science of the U.S. Department of Energy, and supported in part by MichiganState University through computational resources provided by the Institute for Cyber-Enabled Research (iCER). Thework of HL is partially supported by the US National Science Foundation under grant PHY 1653405 “CAREER:Constraining Parton Distribution Functions for New-Physics Searches” and by the Research Corporation for ScienceAdvancement through the Cottrell Scholar Award “Unveiling the Three-Dimensional Structure of Nucleons”. ∗ Electronic address: [email protected][1] D. Mller, D. Robaschik, B. Geyer, F.-M. Dittes, and J. Hoˇrejˇsi, Fortsch. Phys. , 101 (1994), arXiv:hep-ph/9812448 .[2] X.-D. Ji, Phys. Rev. Lett. , 610 (1997), arXiv:hep-ph/9603249 [hep-ph] .[3] A. Radyushkin, Phys. Lett. B , 417 (1996), arXiv:hep-ph/9604317 .[4] M. Burkardt, Phys. Rev. D62 , 071503 (2000), [Erratum: Phys. Rev.D66,119903(2002)], arXiv:hep-ph/0005108 [hep-ph] .[5] X.-D. Ji, Phys. Rev.
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