PDFs and Neutrino-Nucleon Scattering from Hadronic Tensor
PPDFs and Neutrino-Nucleon Scattering fromHadronic Tensor
Jian Liang ∗ Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA
Keh-Fei Liu
Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA ( χ QCD Collaboration)
We review the Euclidean path-integral formulation of the nucleon hadronic tensor and classify thegauge invariant and topologically distinct insertions in terms of connected and disconnected in-sertions and also in terms of leading and higher-twist contributions in the DIS region. Convertingthe Euclidean hadronic tensor back to the Minkowski space requires solving an inverse problemof the Laplace transform. We have investigated several inverse algorithms and studied the prosand cons of each. We show a result with a relatively large momentum transfer ( Q ∼ ) tosuppress the elastic scattering and reveal the contributions from the resonance and inelastic regionof the neutrino-nucleon scattering. For elastic scattering, the hadronic tensor is the the product ofthe elastic form factors for the two corresponding currents. We checked numerically for the caseof two charge vector currents ( V ) with the electric form factor calculated from the three-pointfunction and found they agree within errors. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] A ug adronic Tensor Jian Liang
1. Introduction
Lepton-nucleon scattering cross-section is a product of the leptonic tensor and the hadronictensor. Being an inclusive reaction, the hadronic tensor includes all intermediate states W µν ( q , ν ) = ∑ n (cid:90) n ∏ i = (cid:20) d (cid:126) p i ( π ) E i (cid:21) (cid:104) N | J µ ( ) | n (cid:105)(cid:104) n | J ν ( ) | N (cid:105) spin ave . ( π ) δ ( p n − p − q ) , (1.1)where p is the 4-momentum of the nucleon, p n is the 4-momentum of the nth intermediate state,and q is the momentum transfer. Since W µν ( q , ν ) measures the absorptive part of the Comptonscattering, it is the imaginary part of the forward amplitude, and can be expressed as the current-current correlation function in the nucleon, i.e. W µν ( q , ν ) = π Im T µν ( q , ν ) = (cid:90) d x π e iq · x (cid:104) N | J µ ( x ) J ν ( ) | N (cid:105) spin ave . . (1.2)The hadronic tensor can be further decomposed, according to its Lorentz structure, into structurefunctions, e.g., W µν = (cid:18) − g µν + q µ q ν q (cid:19) F ( x , Q ) + ˆ p µ ˆ p ν p · q F ( x , Q ) (1.3)for the unpolarized case where ˆ p µ = p µ − p · qq q µ .The hadronic tensor is a function of Q ( = − q ) and energy transfer ν . The expected spectraldensity of the neutrino-nucleon scattering cross-section or structure functions is illustrated in Fig. 1,which shows that there are several kinematic regions in the spectral density in the energy transfer Figure 1:
Illustrated spectral density of the ν -N scattering to show the elastic, the resonance, the SIS, andthe DIS regions at different energy transfer ν . ν – the elastic scattering, the inelastic reactions ( π N , ππ N , η N etc.) and resonances ( ∆ , Roper, S , etc.), the shallow inelastic scattering (SIS), and the deep inelastic scattering (DIS) regions. Todetermine how large a ν is needed for DIS, we look at W , the total invariant mass of the hadronicstate W = ( q + p ) = m N − Q + m N ν , (1.4)where m N is the nucleon mass. The global fittings of the high energy lepton-nucleon and Drell-Yan experiments to extract the parton distribution functions (PDFs) usually make a cut with W > adronic Tensor Jian Liang
10 GeV . To be qualified in the DIS region, the energy transfer ν needs to be ν > .
86 GeV + . ( GeV − ) Q . (1.5)If we take Q = , this implies ν > F i = ∑ a c ai ⊗ f a , where the convolution kernel c ai is perturbatively calculable.On the other hand, for scatterings at lower energies, the nucleon hadronic tenor is needed, to-gether with many-body calculation of the nucleus, to delineate the experiments of neutrino-nucleusscattering, e.g. LBNF/DUNE [1] at Fermilab, which aims to study the neutrino properties. Thebeam energy of DUNE is in the range ∼ ∼
2. Euclidean Path-integral Formulation of the Hadronic Tensor
The Euclidean hadronic tensor was formulated in the path-integral formalism to identify theorigin of the Gottfried sum rule violation [3]. It is a current-current correlator in the nucleon andcan be obtained by the following four-point-to-two-point correlator ratio (cid:101) W µν ( (cid:126) q ,(cid:126) p , τ ) = E p m N Tr ( Γ e G pW p ( t , t , t , t f )) Tr ( Γ e G pp ( t , t f )) t f − t (cid:29) / ∆ E p , t − t (cid:29) / ∆ E p = (cid:28) N ( (cid:126) p ) | (cid:90) d (cid:126) x π e − i (cid:126) q · (cid:126) x J µ ( (cid:126) x , τ ) J ν ( , ) | N ( (cid:126) p ) (cid:29) , (2.1)where τ = t − t is the Euclidean time separation between the current J ν ( t ) and J µ ( t ) . Thesource/sink of the nucleon interpolation fields at t / t f are much larger than the inverse of the energydifference between the nucleon and its first excited state ( ∆ E p ), so that the nucleon excited statesare filtered out. Formally, the inverse Laplace transform converts (cid:101) W µν ( (cid:126) q ,(cid:126) p , τ ) to the Minkowskihadronic tensor W µν ( (cid:126) q ,(cid:126) p , ν ) = m N i (cid:90) c + i ∞ c − i ∞ d τ e ντ (cid:101) W µν ( (cid:126) q ,(cid:126) p , τ ) , (2.2)with c >
0. This is basically doing the anti-Wick rotation to go back to the Minkowski space. Inpractice with the lattice calculation, it is not possible to perform the inverse Laplace transform inEq. (2.2), as there is no data at imaginary τ . Instead, one can turn this into an inverse problem andfind a solution from the Laplace transform [6] (cid:101) W µν ( (cid:126) q ,(cid:126) p , τ ) = (cid:90) d ν e − ντ W µν ( (cid:126) q ,(cid:126) p , ν ) . (2.3)2 adronic Tensor Jian Liang
This has been studied [7, 8] with the inverse algorithms such as the Backus-Gilbert, MaximumEntropy, and Bayesian Reconstruction methods. It is shown [3, 5, 4] that, when the time ordering (a) (b) (c)
Figure 2:
Three gauge invariant and topologically distinct insertions in the Euclidean path-integral formu-lation of the nucleon hadronic tensor, where the currents couple to the same quark propagator. In the DISregion, the parton degrees of freedom are (a) the valence and connected sea (CS) partons q v + cs , (b) the CSanti-partons ¯ q cs , and (c) the disconnected sea (DS) partons q ds and anti-partons ¯ q ds with q = u , d , s , and c .Only u and d are present in (a) and (b) for the nucleon hadronic tensor. (a) (b) (c) Figure 3:
Three other gauge invariant and topologically distinct insertions where the currents are insertedon different quark propagators. In the DIS region, they contribute to higher twists. t f > t > t > t is fixed, the four-point function Tr ( Γ e G pW p ( t , t , t , t f )) can be grouped in termsof 6 topologically distinct and gauge invariant path-integral insertions as illustrated in Figs. 2 and3, according to different Wick contractions among the Grassmann numbers in the two currents andthe source/sink interpolation fields. They can be denoted as connected insertions (CI) (Fig. 2(a)Fig. 2(b), and Fig. 3(a)) where the quark lines are all connected and disconnected insertions (DI)(Fig. 2(c), Fig. 3(b), and Fig. 3(c)) where there is vacuum polarization associated with the current(s)in disconnected quark loop(s). Note, these diagrams depict the quarks propagating in the non-perturbative gauge background which includes the fluctuating gauge fields and virtual quark loopsfrom the fermion determinant. Only the quark lines associated with the interpolation fields andcurrents are drawn in these quark skeleton diagrams.At low (cid:126) q and ν appropriate for a ρ − N intermediate state, all CIs in Figs. 2(a), 2(b), and 3(a)contribute to the ρ − N scattering. It is worth pointing out that Fig. 2(b) includes the exchangecontribution to prevent the u / d quark in the loop in Fig. 2(c) from occupying the same Dirac3 adronic Tensor Jian Liang eigenstate in the nucleon propagator, enforcing the Pauli principle. In fact, Figs. 2(c) and Fig. 2(b)are analogous to the direct and exchange diagrams in time-ordered Bethe-Goldstone diagrams inmany-body theory.
In the DIS region (e.g. Q ≥ and ν > F , F , and F are concerned, the three diagrams in Fig. 2 are additive with contributions classifiedas the valence and sea quarks q v + cs in Fig. 2(a), the connected sea (CS) antiquarks ¯ q cs in Fig. 2(b),and disconnected sea (DS) quark q ds and antiquarks ¯ q ds in Fig. 2(c) [3, 4, 5, 6]. It was pointedout that the Gottfried sum rule violation comes entirely from the CS difference ¯ u cs − ¯ d cs in the F structure functions at the isospin symmetric limit [3].Owing to the factorization theorem [9] which separates out the long-distance and short-distancebehaviors, the structure function F of the hadronic tensor can be factorized as F ( x , Q ) = ∑ i (cid:90) x dyy C i (cid:16) xy , Q µ , µ f µ , α s ( µ ) (cid:17) f i ( y , µ f , µ ) , (2.4)where i is summed over quark q i , anti-quark ¯ q i , and glue g . C i are the Wilson coefficients and f i arethe PDFs. µ f is the factorization scale, and µ is the renormalization scale. In practice, the globalfitting programs adopt the parton degrees of freedom as u , d , ¯ u , ¯ d , s , ¯ s and g . We see that from thepath-integral formalism, each of the u and d has two sources, one from the CI (Fig. 2(a)) and onefrom the DI (Fig. 2(c)), so are ¯ u and ¯ d from Fig. 2(b) and Fig. 2(c). On the other hand, s and ¯ s onlycome from the DI (Fig. 2(c)). In other words, u = u v + cs + u ds , d = d v + cs + d ds ¯ u = ¯ u cs + ¯ u ds , ¯ d = ¯ d cs + ¯ d ds , s = s ds , ¯ s = ¯ s ds . (2.5)This classification of the parton degrees of freedom is richer than those in terms of q and ¯ q inthe global analysis, in which there are two sources for the partons – q v + cs and q ds – and two sourcesfor the antipartons – ¯ q cs and ¯ q ds . They have different small x behaviors. For the CI part, q v + cs , ¯ q cs ∼ x − / where q = u , d ; whereas, for the DI part, q ds , ¯ q ds ∼ x − where q = u , d , s , c [4, 5, 10, 11]. It isdiscerning to follow these degrees of freedom in moments which further reveals the roles of CI andDI in nucleon matrix elements. They have been intensively studied in lattice calculations which arebeginning to take into account all the systematic corrections.
3. Large Momentum Transfer
Numerically, a substantial challenge of this approach is to convert the hadronic tensor fromEuclidean space back to Minkowski space, which involves solving an inverse problem of a Laplacetransform [6, 7, 8]. In order to tackle this problem, three methods (i.e., the Backus-Gilbert method [12,13], the maximum entropy method [14, 15] and the Bayesian reconstruction method [16]) have been4 adronic Tensor
Jian Liang implemented and tested [7, 8]. It is believed that the Bayesian reconstruction method with furtherimprovement [17, 18] is the best method so far to solve the problem. On the other hand, the inverseproblem is a common problem in PDF calculations, and can be tackled by using model-inspiredfitting functions. In order to study parton physics, another challenge of this approach is to accesshigh momentum and energy transfers such that the calculation can cover the DIS region.It is shown recently in a lattice calculation that small lattice spacing (e.g. a ≤ .
04 fm) isneeded to reach such high energy excitation as ν ∼ Q ∼ so that the elastic scattering con-tribution (at ν =
0) is suppressed. With Bayesian reconstruction, we see that there are substantialcontribution from the resonance and inelastic scattering at ν < ν = a s = .
12 fm, ξ = .
5) is not fine enough to reveal higher intermediatestates.
Figure 4:
The Minkowski hadronic tensor W M as a function of energy transfer ν reconstructed using theBR method for both u and d quarks. In the legend, “te” denotes the end point of t we use in BR and “c” isthe value of the constant default model. After ∼
4. Neutrino-Nucleon Scattering
For the hadronic tensor calculation, one important point is that it works in all the energyranges (from elastic scattering to inelastic scattering and on to deep inelastic scattering). At lowenergy lepton-nucleon scattering, all the 6 diagrams in Figs. 2 and 3 contribute and they are notseparable. For the elastic scattering case, the hadronic tensor W µν as a function of Q is basicallythe product of the relevant nucleon form factors. For example, it is verified numerically (shownin Fig. 5) that it is the sum of Fig. 2(a) and Fig. 3(a) that gives rise to the square of the totalcharges of the u quarks in the proton in the forward limit when J µ and J ν are the charge currents,i.e. W ( (cid:126) p = (cid:126) q = , ν = )( u quark ) = ( e u ) , while the other diagrams are zero due to chargeconservation. 5 adronic Tensor Jian Liang t ( t )0.000.250.500.751.001.251.501.752.00 W ( p = q = , = ) C0C1 C2
Figure 5:
The contributions (CI only) of different insertions to the square of the total charges of the u quarksin the proton in the forward limit. Labels C0, C1, and C2 in the figure denote the contributions from Fig. 2(a),Fig. 2(b), and Fig. 3(a), respectively. The horizontal axis is the current position t (for C0 and C2) or t (forC1). For the off-forward cases which are shown in Fig. 6, the contribution from Fig. 2(b) (labelledas C1 in the figure) is not zero, as there is no charge symmetry in this case. A very interesting pointis that the contribution from C2 (Fig. 3(a), the cat’s ear diagram) becomes much smaller whenthe momentum transfer increases from | (cid:126) q | = ( π / L ) to 3 ( π / L ) in the three panels in Fig. 6,while the contributions from C0 and C1 do not change much. This observation is consistent withthe fact that in DIS where the momentum transfer is large, the contribution from C2 (Fig. 3(a)) isfrom the higher twist and is suppressed. Physically, this is because the two currents are acted ondifferent quark lines and extra gluon exchanges are needed for the off-forward matrix element ofthe hadronic tensor. Thus, the calculation of the hadronic tensor on the lattice can be used to studythe higher-twist effects explicitly. t ( t )0.500.250.000.250.500.751.001.25 W ( p = , | q | = ( L ) ) C0C1 C2 4 6 8 10 12 14 t ( t )0.500.250.000.250.500.751.001.25 W ( p = , | q | = ( L ) ) C0C1 C2 4 6 8 10 12 14 t ( t )0.500.250.000.250.500.751.001.25 W ( p = , | q | = ( L ) ) C0C1 C2
Figure 6:
Similar to Fig. 5 but for off-forward cases. The results of the first 3 momentum transfers arecollected. A black dashed line of ˜ w = All the numerical check of this approach in terms of nucleon electric form factors are doneon the RBC / UKQCD 32Ifine lattice [21]. The structure function of the elastic scattering from thehadronic tensor, a four-point function, is the product of the elastic nucleon form factors for thecurrents involved. Fig. 7 shows that the electric form factors (connected insertions only) calculatedby means of the three-point functions for both u and d quarks are found to be consistent withinerrors with those deduced from the hadronic tensor for the elastic scattering. This lays a solid6 adronic Tensor Jian Liang q | /( L ) G E ( C I ) d u d u Figure 7:
The comparison of the electric form factors (CI contributions only) calculated by using three-pointfunctions and four-point functions for the first four momentum transfers (including zero) and for both u and d quarks. foundation for further calculations of the neutrino scattering cross-sections.
5. Summary and Outlook
We formulate the hadronic tensor on the lattice and point out the fact that this approach worksfor scattering processes at all energy regions. We show a result with a relatively large momentumtransfer which reveals the resonance and inelastic-scattering contributions in neutrino-nucleon scat-tering. For elastic scattering, we checked numerically for the case of two charge vector currents( V ) with the electric form factor calculated from the three-point function and found they agreewithin errors.Currently, lattices with lattice spacing ∼ ∼ ∼ ∼
6. Acknowledgement
This work is partially support by the U.S. DOE grant DE-SC0013065 and DOE Grant No.DE-AC05-06OR23177 which is within the framework of the TMD Topical Collaboration. This re-search used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge NationalLaboratory, which is supported by the Office of Science of the U.S. Department of Energy underContract No. DE-AC05-00OR22725. This work used Stampede time under the Extreme Scienceand Engineering Discovery Environment (XSEDE), which is supported by National Science Foun-dation Grant No. ACI-1053575. We also thank the National Energy Research Scientific ComputingCenter (NERSC) for providing HPC resources that have contributed to the research results reportedwithin this paper. 7 adronic Tensor
Jian Liang
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