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Register a new userParton Distribution Function from the Hadronic Tensor on the Lattice
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Keh-Fei Liu
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The path-integral formulation of the hadronic tensor W_{mu u} of deep inelastic scattering is reviewed. It is shown that there are 3 gauge invariant and topologically distinct contributions. The separation of the connected sea partons from those of the disconnected sea can be achieved with a combination of the global fit of the parton distribution function (PDF), the semi-inclusive DIS data on the strange PDF and the lattice calculation of the ratio of the strange to $u/d$ momentum fraction in the disconnected insertion. We shall discuss numerical issues associated with lattice calculation of the hadronic tensor involving a four-point function, such as large hadron momenta and improved maximum entropy method to obtain the spectral density from the hadronic tensor in Euclidean time. We also draw a comparison between the large momentum approach to the parton distribution function (PDF) and the hadronic tensor approach.
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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 $m_pisim$ 370 MeV and $asim$ 0.06 fm) and a case (anisotropic clover lattice with $m_pisim$ 380 MeV and $a_tsim$ 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 $ u=1$ GeV are clearly observed but no information is obtained for higher excited states with $ u>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.
We extend the study of lowest moments, $<x>$ and $<x^2>$, of the parton distribution function of the nucleon to include those of the sea quarks; this entails a disconnected insertion calculation in lattice QCD. This is carried out on a $16^3 times 24$ quenched lattice with Wilson fermion. The quark loops are calculated with $Z_2$ noise vectors and unbiased subtractions, and multiple nucleon sources are employed to reduce the statistical errors. We obtain 5$sigma$ signals for $<x>$ for the $u,d,$ and $s$ quarks, but $<x^2>$ is consistent with zero within errors. We provide results for both the connected and disconnected insertions. The perturbatively renormalized $<x>$ for the strange quark at $mu = 2$ GeV is $<x>_{s+bar{s}} = 0.027 pm 0.006$ which is consistent with the experimental result. The ratio of $<x>$ for $s$ vs. $u/d$ in the disconnected insertion with quark loops is calculated to be $0.88 pm 0.07$. This is about twice as large as the phenomenologically fitted $displaystylefrac{< x>_{s+bar{s}}}{< x>_{bar{u}}+< x>_{bar{d}}}$ from experiments where $bar{u}$ and $bar{d}$ include both the connected and disconnected insertion parts. We discuss the source and implication of this difference.
We present lattice results for the isovector unpolarized parton distribution with nonperturbative RI/MOM-scheme renormalization on the lattice. In the framework of large-momentum effective field theory (LaMET), the full Bjorken-$x$ dependence of a momentum-dependent quasi-distribution is calculated on the lattice and matched to the ordinary lightcone parton distribution at one-loop order, with power corrections included. The important step of RI/MOM renormalization that connects the lattice and continuum matrix elements is detailed in this paper. A few consequences of the results are also addressed here.
We present the first lattice calculation of the nucleon isovector unpolarized generalized parton distribution (GPD) at the physical pion mass using a lattice ensemble with 2+1+1 flavors of highly improved staggered quarks (HISQ) generated by MILC Collaboration, with lattice spacing $aapprox 0.09$~fm and volume $64^3times 96$. We use momentum-smeared sources to improve the signal at nucleon boost momentum $P_z approx 2.2$ GeV, and report results at nonzero momentum transfers in $[0.2,1.0]text{ GeV}^2$. Nonperturbative renormalization in RI/MOM scheme is used to obtain the quasi-distribution before matching to the lightcone GPDs. The three-dimensional distributions $H(x,Q^2)$ and $E(x,Q^2)$ at $xi=0$ are presented, along with the three-dimensional nucleon tomography and impact-parameter--dependent distribution for selected Bjorken $x$ at $mu=3$ GeV in $overline{text{MS}}$ scheme.
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