Abstract
We present the first calculation of the hadronic tensor on the lattice for the nucleon. The hadronic tensor can be used to extract the structure functions in deep inelastic scatterings and also provide information for the neutrino-nucleon scattering which is crucial to the neutrino-nucleus scattering experiments at low energies. The most challenging part in the calculation is to solve an inverse problem. We have implemented and tested three algorithms using mock data, showing that the Bayesian Reconstruction method has the best resolution in extracting peak structures while the Backus-Gilbert and Maximum Entropy methods are somewhat more stable for the flat spectral function. Numerical results are presented for both the elastic case (clover fermions on domain wall configuration with $m_\pi\sim$ 370 MeV and $a\sim$ 0.06 fm) and a case (anisotropic clover lattice with $m_\pi\sim$ 380 MeV and $a_t\sim$ 0.035 fm) with large momentum transfer. For the former case, the reconstructed Minkowski hadronic tensor gives precisely the vector charge which proves the feasibility of the approach. While for the latter case, the nucleon resonances and possibly shallow inelastic scattering contributions around $\nu=1$ GeV are clearly observed but no information is obtained for higher excited states with $\nu>2$ GeV. A check of the effective masses of $\rho$ meson with different lattice setups indicates that, in order to reach higher energy transfers, using lattices with smaller lattice spacings is essential.
Highlights
In scattering processes involving nucleons such as deep inelastic scattering (DIS) and neutrino-nucleon scattering at low energies, the hadronic tensor Wμν is used to characterize the nonperturbative nature of the nucleon structure
To explore the reason why there is no contribution for ν ≳ 2 GeV in Figs. 6 and 7, we calculate the effective mass of the four-point functions which is a quick way to check the highest energy of intermediate states that our Euclidean hadronic tensor contains
This should be due to the lattice artifacts, since the lattice we are using has finite volume, finite lattice spacing, and unphysical pion mass
Summary
In scattering processes involving nucleons such as deep inelastic scattering (DIS) and neutrino-nucleon scattering at low energies, the hadronic tensor Wμν is used to characterize the nonperturbative nature of the nucleon structure. There is no direct requirement of large nucleon momentum p⃗ since the structure functions in the hadronic tensor are frame independent (N.B. many DIS experiments are done with the nucleon at rest), we find that a slightly boosted nucleon makes it easier in practice to reach large ν on the lattice Another advantage of our method is that the hadronic tensor is scale independent, such that no renormalization is needed except for the finite lattice normalization of the vector current if local operators are used. In the RES, SIS, and DIS regions, inelastic neutrino-nucleon scatterings emerge and one will need to have the hadronic tensor to cover all the inclusive contributions In this sense, calculating the hadronic tensor is so far the only way we know that lattice QCD can serve the neutrino experiments in the whole energy range [17].1.
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