Towards the nucleon hadronic tensor from lattice QCD

Abstract

We present the first calculation of the hadronic tensor on the lattice for the nucleon. The hadronic tensor can be used to extract structure functions in deep inelastic scatterings and to provide information for neutrino-nucleon scattering which is crucial to 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 smooth functions. Numerical results are presented for both the elastic case (clover fermions on domain wall configurations with ${m}_{\ensuremath{\pi}}\ensuremath{\sim}370\text{ }\text{ }\mathrm{MeV}$ and $a\ensuremath{\sim}0.06\text{ }\text{ }\mathrm{fm}$) and a case (anisotropic clover lattice with ${m}_{\ensuremath{\pi}}\ensuremath{\sim}380\text{ }\text{ }\mathrm{MeV}$ and ${a}_{t}\ensuremath{\sim}0.035\text{ }\text{ }\mathrm{fm}$) 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. For the latter case, the resonance and possibly shallow inelastic scattering contributions around energy transfer $\ensuremath{\nu}=1\text{ }\text{ }\mathrm{GeV}$ are clearly observed but no information is obtained for higher excited states with $\ensuremath{\nu}>2\text{ }\text{ }\mathrm{GeV}$. A check of the effective masses of the $\ensuremath{\rho}$ meson with different lattice setups indicates that, in order to reach higher energy transfers, using lattices with smaller lattice spacings is essential.