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The hadronic tensor from four-point functions on the lattice

Christian Zimmermann, Terrence Draper, Jian Liang, Keh-Fei Liu, Raza Sabbir Sufian, Bigeng Wang

Abstract

The hadronic tensor is the central non-perturbative object in the calculation of the cross section of lepton-hadron interactions like neutrino-nucleon scattering. It is usually parameterized in terms of structure functions, which encode all necessary information for all kinematic regions. Moreover, the structure functions can be factorized in terms of parton distribution functions (PDFs) and contains information on hadron resonances. On the lattice, we can calculate the corresponding matrix element of two quark-bilinear currents with a relative Euclidean time separation. The reconstruction of the hadronic tensor in Minkowski space requires appropriate dealing with the corresponding inverse problem. In our current work, we extend previous calculations on the nucleon by considering a much larger range of momentum transfers, which is inevitable in the context of structure functions. This can be achieved by using stochastic sources, which allows us to calculate the required four-point functions in a broad kinematic region. We employ a clover fermion ensemble at pion mass $m_π= 223~\mathrm{MeV}$ and lattice spacing $a=0.085~\mathrm{fm}$. In these proceedings, we will give an overview of our simulation and present some first preliminary results.

The hadronic tensor from four-point functions on the lattice

Abstract

The hadronic tensor is the central non-perturbative object in the calculation of the cross section of lepton-hadron interactions like neutrino-nucleon scattering. It is usually parameterized in terms of structure functions, which encode all necessary information for all kinematic regions. Moreover, the structure functions can be factorized in terms of parton distribution functions (PDFs) and contains information on hadron resonances. On the lattice, we can calculate the corresponding matrix element of two quark-bilinear currents with a relative Euclidean time separation. The reconstruction of the hadronic tensor in Minkowski space requires appropriate dealing with the corresponding inverse problem. In our current work, we extend previous calculations on the nucleon by considering a much larger range of momentum transfers, which is inevitable in the context of structure functions. This can be achieved by using stochastic sources, which allows us to calculate the required four-point functions in a broad kinematic region. We employ a clover fermion ensemble at pion mass and lattice spacing . In these proceedings, we will give an overview of our simulation and present some first preliminary results.
Paper Structure (5 sections, 12 equations, 3 figures, 1 table)

This paper contains 5 sections, 12 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The hadronic tensor in Euclidean space time
  3. Lattice calculation of four-point functions
  4. First results
  5. Conclusion

Figures (3)

  • Figure 1: Five types of Wick contractions contributing to the four-point function in \ref{['eq:4ptdef']} for the case of baryons. The explicit expressions of the diagrams $C_1$, $C_2$, and $S_1$ depends on the quark flavor of the currents.
  • Figure 2: Panel (a): the dependence of $W^{\mathrm{E},u}_{44}$ on $\bar{\tau}$ for $\vec{q}=(2,3,4)2\pi/L$, several source-sink separations $t$, and two different values of $\tau$. Panel (b): the $\tau$-dependence of $W^{\mathrm{E},d}_{44}$ for small momentum transfers $\vec{q}$. Panel (c): the $\tau$-dependence of $W^{\mathrm{E},d}_{44}$ for larger $\vec{q}$. Panel (d): the $\tau$-dependence of $W^{\mathrm{E},\mathrm{em}}_{44}$ for several $\vec{q}$. Panels (b), (c), and (d) show the results for different source-sink separations $t$: solid line: $t=8a$, dashed line: $t=10a$, dotted line: $t=12a$. For better visibility, we applied a small offset w.r.t. the horizontal axis for different data sets in panel (d).
  • Figure 3: $|\vec{q}|$-dependence of the Euclidean structure functions $A$ (a) and $B$ (b) for different values of $\tau$ and $t=10a$. The results are shown for the case of electromagnetic currents in an unpolarized proton.