title: PDFs and Neutrino-Nucleon Scattering from Hadronic Tensor
authors: Jian Liang Keh-Fei Liu
text: 
  
  
  
  § INTRODUCTION
  
  
  Lepton-nucleon scattering cross-section is a product of the leptonic tensor and the hadronic
  tensor. Being an inclusive reaction, the hadronic tensor includes all intermediate states
  
  
      W_μν(q^2, ν) = 1/2∑_n ∫∏_i =1^n [d^3 p⃗_i/(2π)^3 2E_i]  
      ⟨ N|J_μ(0)|n⟩⟨ n|J_ν(0) | N⟩_ spin   ave.(2π)^3 δ^4 (p_n - p - q),
  
  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^2, ν) measures the absorptive part of the  
  Compton scattering, 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^2, ν) = 1/π Im  T_μν(q^2, ν)
      = ∫d^4x/4π  e^ i q · x⟨ N| J_μ(x)  J_ν(0) | N⟩_ spin   ave..
   
  
  The hadronic tensor can be further decomposed, according to its Lorentz
  structure, into structure functions, e.g.,
  
      W_μν=(-g_μν+q_μq_ν/q^2)F_1(x,Q^2)+p̂_μp̂_ν/p· qF_2(x,Q^2)
  
  for the unpolarized case where p̂_μ=p_μ-p· q/q^2q_μ.
  
  The hadronic tensor is a function of Q^2 (=-q^2) and energy transfer ν. 
  The expected spectral density of the neutrino-nucleon scattering cross-section or structure
  functions is illustrated in Fig. <ref>,
  
  
  
  
  which shows that there are several kinematic regions in the spectral density in the energy 
  transfer  elastic scattering, 
  the inelastic reactions (π N, ππ N, η N etc.) and 
  resonances (Δ, Roper, S_11, etc.), 
  the shallow inelastic scattering (SIS), and the deep inelastic scattering (DIS) regions. 
  To determine how large a ν is needed for DIS, we look at W, the total invariant mass of the hadronic state 
  
  
      W^2 = (q+p)^2 = m_N^2 - Q^2 + 2 m_N ν,
  
  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^2 > 10  GeV^2. 
  To be qualified in the DIS region, the energy transfer  ν needs to be 
  
  
      ν  > 4.86  GeV + 0.533 ( GeV^-1)  Q^2.
  
  
  If we take  Q^2 = 4  GeV^2, this implies ν > 7 GeV.
  In the DIS region where the leading twists dominate, 
  the cross-section or the structure functions can be used to extract 
  the PDFs through the QCD factorization theorem
  F_i=∑_ac_i^a⊗ f_a, where the convolution kernel
  c_i^a is perturbatively calculable.
  
  On the other hand, for scatterings at lower energies, 
  the nucleon hadronic tenor is needed, 
  together with many-body calculation of the nucleus, to delineate 
  the experiments of neutrino-nucleus scattering, e.g. LBNF/DUNE <cit.>
  at Fermilab, which aims to study the neutrino properties. The beam
  energy of DUNE is in the range ∼1 to ∼7 GeV and at different
  beam energies, different contributions (quasi-elastic (QE), resonance
  (RES), SIS and DIS) will dominate the total cross
  section <cit.>. The hadronic tensor is useful
  in the whole energy region. For example in the QE region, the hadronic
  tensor is actually the square of the elastic form factors of nucleon and the
  cross-section of neutrino-nucleus scattering can be calculated by
  combining the nucleon form factors and nuclear models about the nucleon
  distribution inside a nucleus. In the RES, SIS and DIS region, inelastic
  neutrino-nucleon scatterings emerge and one will need to have inclusive
  hadronic tensor to cover the contributions of all final-state particles.
  In this sense, calculating the hadronic tensor is so far the only
  way we know that Lattice QCD can serve the neutrino experiments.
  
  
  
  § EUCLIDEAN PATH-INTEGRAL FORMULATION OF THE HADRONIC TENSOR
    
  
  The Euclidean hadronic tensor was formulated in the path-integral formalism 
  to identify the origin of the Gottfried sum rule violation <cit.>. 
  It is a current-current correlator in the nucleon and can
  be obtained by the following four-point-to-two-point correlator ratio
  
  
      W_μν(q⃗,p⃗,τ)    =   E_p /m_N Tr (Γ_e G_pWp(t_0,t_1,t_2,t_f))/ Tr (Γ_e G_pp(t_0,t_f))[                                          ;                                          ; t_f -t_2 ≫ 1/Δ E_p,   t_1 - t_0 ≫ 1/Δ E_p ]
         =   ⟨ N(p⃗)| ∫ d^3x⃗1/4π e^-iq⃗·x⃗
      J_μ(x⃗,τ) J_ν(0,0)|N(p⃗)⟩,
  
  
  where τ = t_2 - t_1 is the Euclidean time separation between the current J_ν(t_2) and 
  J_μ(t_1). 
  The source/sink of the nucleon interpolation fields at t_0/t_f are much larger than the 
  inverse of the energy difference between the nucleon and its first excited state (Δ E_p), 
  so that the nucleon excited states are filtered out. 
  Formally, the inverse Laplace transform converts W_μν(q⃗,p⃗,τ) to the Minkowski hadronic tensor
  
  
      W_μν(q⃗,p⃗,ν) = 1/2m_Ni∫_c-i ∞^c+i ∞ dτ 
      e^ντW_μν(q⃗,p⃗, τ),
   
  
  with c > 0. This is basically doing the anti-Wick rotation to go back to the Minkowski space. 
  In practice with the lattice calculation, 
  it is not possible to perform the inverse Laplace transform in Eq. (<ref>), 
  as there is no data at imaginary τ. 
  Instead, one can turn this into an inverse problem and find a solution from the Laplace transform <cit.>
  
  
      W_μν (q⃗,p⃗,τ) = ∫ d ν  e^-ντ
      W_μν(q⃗,p⃗,ν).
  
    
  This has been studied <cit.> with the inverse algorithms such as 
  the Backus-Gilbert, Maximum Entropy, and Bayesian Reconstruction methods. 
  
  
  
  It is shown <cit.> that, when the time ordering 
  t_f > t_2 > t_1 > t_0 is fixed, the four-point function Tr (Γ_e G_pWp(t_0,t_1,t_2,t_f))
  can be grouped in terms of 6 topologically distinct and gauge invariant path-integral insertions 
  as illustrated in Figs. <ref> and <ref>,
  according to different Wick contractions among the Grassmann numbers in the two currents and the source/sink interpolation fields.  
  They can be denoted as connected insertions (CI) (Fig. <ref> 
  Fig. <ref>, and Fig. <ref>) 
  where the quark lines are all connected and disconnected insertions (DI) 
  (Fig. <ref>, Fig. <ref>, and Fig. <ref>) 
  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 loops from the fermion determinant. 
  Only the quark lines associated with the interpolation fields and currents are drawn in these quark skeleton diagrams. 
  
  At low q⃗ and ν appropriate for a ρ - N intermediate state, 
  all CIs in Figs. <ref>, <ref>, and <ref> 
  contribute to the ρ-N scattering. 
  It is worth pointing out that Fig. <ref> includes the exchange contribution to prevent the u/d quark in the loop in Fig. <ref> 
  from occupying the same Dirac eigenstate in the nucleon propagator, 
  enforcing the Pauli principle. In fact, Figs. <ref> and 
  Fig. <ref> are analogous to the direct and exchange diagrams in time-ordered Bethe-Goldstone diagrams in many-body theory.
  
  
  
   §.§ Parton Degrees of Freedom
  
  
  In the DIS region (e.g. Q^2 ≥ 4  GeV^2 and ν > 7 GeV in Eq. (<ref>)) as illustrated in Fig. <ref>, 
  insertions in Fig. <ref> contain leading and higher twists (hand-bag diagrams), 
  while those in Fig. <ref> contain only higher twists (cat's ears diagrams). 
  As far as the leading-twist DIS structure functions F_1, F_2, and F_3 are concerned,
  the three diagrams in Fig. <ref> are additive with contributions classified 
  as the valence and sea quarks q^v+cs in Fig. <ref>, the connected sea (CS) antiquarks q̅^cs in Fig. <ref>, 
  and disconnected sea (DS) quark q^ds and antiquarks q̅^ds 
  in Fig. <ref> <cit.>. 
  It was pointed out that the Gottfried sum rule violation comes entirely from
  the CS difference u̅^cs - d̅^cs in the F_2 structure functions 
  at the isospin symmetric limit <cit.>. 
  
  Owing to the factorization theorem <cit.> 
  which separates out the long-distance and short-distance behaviors, 
  the structure function F_1 of the hadronic tensor can be factorized as
  
  
      F_1 (x, Q^2) = ∑_i ∫_x^1 dy/y  C_i (x/y, Q^2/μ^2, μ_f^2/μ^2, α_s(μ^2))  f_i (y, μ_f, μ^2),
  
  
  where i is summed over quark q_i, anti-quark q̅_i, and glue g. 
  C_i are the Wilson coefficients and f_i are the PDFs. μ_f is the factorization scale, 
  and μ is the renormalization scale. In practice, 
  the global fitting programs adopt the parton degrees of freedom as u, d, u̅, d̅, s, s̅ and g. 
  We see that from the path-integral formalism, each of the
  u and d has two sources, one from the CI (Fig. <ref>) and one
  from the DI (Fig. <ref>), so are u̅ and d̅ from Fig. <ref>
  and Fig. <ref>. On the other hand, s and s̅ only come from the DI (Fig. <ref>).
  In other words,
  
  
      u   =    u^v+cs + u^ds,  d =  d^v+cs + d^ds
      u̅   =   u̅^cs + u̅^ds, d̅  =  d̅^cs + d̅^ds, 
      
      s   =    s^ds,   s̅  =  s̅^ds.
  
  
  This classification of the parton degrees of freedom is richer than those in terms of q and
  q̅ in the global analysis,
  in which there are two sources for the partons – q^v+cs and q^ds – and two sources for the antipartons – q̅^cs and q̅^ds. They have different small
  x behaviors. For the CI part, q^v+cs, q̅^cs∼ x^-1/2 where 
  q = u,d; whereas, for the DI part, 
  q^ds, q̅^ds∼ x^-1 
  where q = u,d,s,c <cit.>. 
  It is discerning to follow these degrees of freedom in moments which further 
  reveals the roles of CI and DI in nucleon matrix elements. 
  They have been intensively studied in lattice calculations which are beginning to take into account all the systematic corrections. 
  
  
  
  § LARGE MOMENTUM TRANSFER
  
  
  Numerically, a substantial challenge of this approach is to convert the hadronic
  tensor from Euclidean space back to Minkowski space, which involves
  solving an inverse problem of a Laplace transform <cit.>. In order to tackle
  this problem, three methods (i.e., the Backus-Gilbert method <cit.>, the
  maximum entropy method <cit.> and the Bayesian reconstruction method <cit.>)
  have been implemented and tested <cit.>. It is believed that the Bayesian
  reconstruction method with further improvement <cit.> is the best
  method so far to solve the problem. On the other hand, the inverse problem
  is a common problem in PDF calculations, and can be tackled by
  using model-inspired fitting functions.
  In order to study parton physics, another challenge of this approach is
  to access high 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 ≤ 0.04 fm) 
  is needed to reach such high energy excitation as ν∼ 7 GeV in order to be in the DIS region as described
  in Eq. (<ref>) <cit.>. The need of fine lattices appear to be also a
  common challenge for the lattice PDF community. In order to alleviate excited-state contaminations on the source/sink side, 
  one can consider variational calculations with multiple interpolation fields <cit.>. 
  
  We show the results in Fig. <ref> of a case with Q^2 ∼ 4  GeV^2 so that the elastic scattering
  contribution (at ν=0) is suppressed. With Bayesian reconstruction, we see that there are substantial
  contribution from the resonance and inelastic scattering at ν < 2 GeV. After ν= 2 GeV, 
  the default models
  dominate the results, which is due to the fact that the lattice spacing of the ensemble used 
  (clover lattice <cit.> with a_s=0.12 fm, ξ=3.5) 
  is not fine enough to
  reveal higher intermediate states.
  
  
  
  
  
  § NEUTRINO-NUCLEON SCATTERING
  
  
  For the hadronic tensor
  calculation, one important point is that it works in all the energy ranges (from
  elastic scattering to inelastic scattering and on to deep inelastic scattering). 
  At low energy lepton-nucleon scattering, all the 6 diagrams in Figs. <ref> and 
  <ref> contribute and they are not separable.
  For the elastic scattering case, the hadronic tensor W_μν 
  as a function of Q^2 is basically the product of the relevant nucleon form factors. 
  For example, it is verified numerically (shown in Fig. <ref>)
  that it is the sum of Fig. <ref> and Fig. <ref> 
  that gives rise to the square of the total charges of the u quarks in the proton 
  in the forward limit when J_μ and J_ν are the charge currents, i.e.W_44 (p⃗ = q⃗ = 0, ν = 0) (u  quark) = (2 e_u)^2, 
  while the other diagrams are zero due to charge conservation.
  
  
  
  For the off-forward cases which are shown in Fig. <ref>, 
  the contribution from Fig. <ref> (labelled as C1 in the figure) is not zero,
  as there is no charge symmetry in this case. A very interesting point 
  is that the contribution from C2 (Fig. <ref>, the cat's ear diagram) becomes much smaller 
  when the momentum transfer increases from |q⃗|^2 = (2π/L)^2 to 3 (2π/L)^2 in the three panels in Fig. <ref>,
  while the contributions from C0 and C1 do not change much. This observation is consistent
  with the fact that in DIS where the momentum transfer is large, the contribution from C2 (Fig. <ref>)
  is from the higher twist and is suppressed. Physically, this is because the two currents are acted on different quark lines
  and extra gluon exchanges are needed for the off-forward matrix element of the hadronic tensor. 
  Thus, the calculation of the hadronic tensor on the lattice
  can be used to study the higher-twist effects explicitly.
  
  
  
  
  
  All the numerical check of this approach in terms of nucleon electric form factors
  are done
  on the RBC/UKQCD 32Ifine lattice <cit.>.
  The structure function of the elastic scattering from
  the hadronic tensor, a four-point function, is the product
  of the elastic nucleon form factors for the currents
  involved. Fig. <ref> shows that the electric form factors 
  (connected insertions only) calculated by means of the
  three-point functions for both u and d quarks are found to
  be consistent within errors with those deduced from the
  hadronic tensor for the elastic scattering.
  This lays a solid foundation for further calculations of the neutrino scattering cross-sections.
  
  
  
  § SUMMARY AND OUTLOOK
  
  
  We formulate the hadronic tensor on the lattice and point out the
  fact that this approach works for scattering processes at all
  energy regions. We show a result with a relatively large momentum transfer 
  which reveals the resonance and inelastic-scattering contributions in neutrino-nucleon scattering.
  For elastic scattering,
  we checked numerically for the case of two charge vector currents (V_4) with the electric form factor 
  calculated from the three-point function and found they agree within errors. 
  
  Currently, lattices with lattice spacing ∼0.04 fm are
  suitable, for instance, to study the neutrino-nucleus
  scattering for DUNE experiments, where the neutrino
  energy is between ∼1 to ∼7 GeV. At the same
  time, these lattices can also be used to explore the
  pion PDFs. In the future, working on lattices with lattice
  spacing of ∼0.03 fm or even smaller would be desirable for
  studying the nucleon substructure in the DIS region.
  
  
  
  
  § 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 research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This work used Stampede time under the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1053575.
  We also thank the National Energy Research Scientific Computing Center (NERSC) for providing HPC resources that have contributed to the research results reported within this paper.
          
  
  
  
  
  
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