The separation of the connected and disconnected sea partons, which were uncovered in the Euclidean path-integral formulation of the hadronic tensor, is accommodated with the CT18 parametrization of the global analysis of the parton distribution func
tions (PDFs). This is achieved with the help of the distinct small $x$ behaviors of these two sea parton components and the constraint from the lattice calculation of the ratio of the strange momentum fraction to that of the ${bar u}$ or ${bar d}$ in the disconnected insertion. This allows lattice calculations of separate flavors in both the connected and disconnected insertions to be directly compared with the global analysis results term by term.
Lattice results on sigma terms and global analysis of parton momentum fractions are used to give the quark and glue fractions of the proton mass and rest energy. The mass decomposition in terms of the trace of the energy-momentum tensor is renormaliz
ation group invariant. The decomposition of the rest energy from the Hamiltonian and the gravitational form factors are scheme and scale dependent. The separation of the energy-momentum tensor into the traceless part which is composed of the quark and glue parton momentum fractions and the trace part has the minimum scheme dependence. We identify the glue part of the trace anomaly $langle H_{beta}rangle $ as the vacuum energy from the glue condensate in the vacuum. From the metric term of the gravitational form factors, which is the stress part of the stress-energy-momentum tensor, we find that the trace part of the rest energy, dominated by $langle H_{beta}rangle$, gives a {it constant} restoring pressure which balances that from the traceless part of the Hamiltonian to confine the hadron, much like the cosmological constant Einstein introduced for a static universe. From a lattice calculation of $langle H_{beta}rangle$ in the charmonium, we deduce the associated string tension which turns out to be in good agreement with that from a Cornell potential which fits the charmonium spectrum.
We point out a problem of the phenomenological definition of the valence partons as the difference between the partons and antipartons in the context of the NNLO evolution equations. After demonstrating that the classification of the parton degrees o
f freedom (PDF) of the parton distribution functions (PDFs) are the same in the QCD path-intergral formulations of the hadronic tensor and the quasi-PDF with LaMET, we resolve the problem by showing that the proper definition of the valence should be in terms of those in the connected insertions only. We also prove that the strange partons appear as the disconnected sea in the nucleon.
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 present a determination of the neutral current weak axial charge $G^Z_A(0)=-0.654(3)_{rm stat}(5)_{rm sys}$ using the strange quark axial charge $G^s_A(0)$ calculated with lattice QCD. We then perform a phenomenological analysis, where we combine
the strange quark electromagnetic form factor from lattice QCD with (anti)neutrino-nucleon scattering differential cross section from MiniBooNE experiments in a momentum transfer region $0.24lesssim Q^2 lesssim 0.71$ GeV$^2$ to determine the neutral current weak axial form factor $G^Z_A(Q^2)$ in the range of $0lesssim Q^2leq 1$ GeV$^2$. This yields a phenomenological value of $G^Z_A(0)=-0.687(89)_{rm stat}(40)_{rm sys}$. The value of $G^Z_A(0)$ constrained by the lattice QCD calculation of $G^s_A(0)$, when compared to its phenomenological determination, provides a significant improvement in precision and accuracy and can be used to provide a constraint on the fit to $G^Z_A(Q^2)$ for $Q^2>0$. This constrained fit leads to an unambiguous determination of (anti)neutrino-nucleon neutral current elastic scattering differential cross section near $Q^2=0$ and can play an important role in numerically isolating nuclear effects in this region. We show a consistent description of $G^Z_A(Q^2)$ obtained from the (anti)neutrino-nucleon scattering cross section data requires a nonzero contribution of the strange quark electromagnetic form factor. We demonstrate the robustness of our analysis by providing a post-diction of the BNL E734 experimental data.
It is suggested in the paper by A.J. Chambers {it et al.} (Phys. Rev. Lett. 118, 242001 (2017), arXiv:1703.01153) that the time-ordered current-curent correlator in the nucleon calculated on the lattice is to be identified as the forward Compton ampl
itude so that it is related to the sum of the even moments of the structure function as in the Minkowski space in the continuum. We point out two problems with this identification. First of all, the current-current correlator defined in the Euclidean space is not analytic everywhere on the rest of the complex $ u$ or $omega$ plane, besides the cuts on the real axis. As such, there is no dispersion relation to relate it to its imaginary part and hence the moments of the structure function. On the lattice, there is an additional difficulty in that the higher dimensional local operators from the operator production expansion (OPE) of the current-current product can mix with lower dimensional higher-twist operators which leads to divergences in the powers of inverse lattice spacing. This mixing needs to be removed before their matrix elements can be identified as the moments of the structure function.
A feasibility study of fusion reactors based on accelerators is carried out. We consider a novel scheme where a beam from the accelerator hits the target plasma on the resonance of the fusion reaction and establish characteristic criteria for a worka
ble reactor. We consider the reactions $ d + t rightarrow n + alpha, d + {}^3H_e rightarrow p + alpha$, and $p + {}^{11}B rightarrow 3 alpha$ in this study. The critical temperature of the plasma is determined from overcoming the stopping power of the beam with the fusion energy gain. The needed plasma lifetime is determined from the width of the resonance, the beam velocity and the plasma density. We estimate the critical beam flux by balancing the energy of fusion production against the plasma thermo-energy and the loss due to stopping power for the case of an inert plasma. The product of critical flux and plasma lifetime is independent of plasma density and has a weak dependence on temperature. Even though the critical temperatures for these reactions are lower than those for the thermonuclear reactors, the critical flux is in the range of $10^{22} - 10^{24}/rm{cm^2/s}$ for the plasma density $rho_t = 10^{15}/{rm cm^3}$ in the case of an inert plasma. Several approaches to control the growth of the two-stream instability are discussed. We have also considered several scenarios for practical implementation which will require further studies. Finally, we consider the case where the injected beam at the resonance energy maintains the plasma temperature and prolongs its lifetime to reach a steady state. The equations for power balance and particle number conservation are given for this case.
It is a common problem in lattice QCD calculation of the mass of the hadron with an annihilation channel that the signal falls off in time while the noise remains constant. In addition, the disconnected insertion calculation of the three-point functi
on and the calculation of the neutron electric dipole moment with the $theta$ term suffer from a noise problem due to the $sqrt{V}$ fluctuation. We identify these problems to have the same origin and the $sqrt{V}$ problem can be overcome by utilizing the cluster decomposition principle. We demonstrate this by considering the calculations of the glueball mass, the strangeness content in the nucleon, and the CP violation angle in the nucleon due to the $theta$ term. It is found that for lattices with physical sizes of 4.5 - 5.5 fm, the statistical errors of these quantities can be reduced by a factor of 3 to 4. The systematic errors can be estimated from the Akaike information criterion. For the strangeness content, we find that the systematic error is of the same size as that of the statistical one when the cluster decomposition principle is utilized. This results in a 2 to 3 times reduction in the overall error.
It has been revealed from the path-integral formulation of the hadronic tensor that there are connected sea and disconnected sea partons. The former is responsible for the Gottfried sum rule violation primarily and evolves the same way as the valence
. Therefore, the DGLAP evolution equations can be extended to accommodate them separately. We discuss its consequences and implications vis-a-vis lattice calculations.
We report a lattice calculation of nucleon forward matrix elements on a $48^3 times 96$ lattice at the physical pion mass and a spatial size of 5.5 fm. The $2+1$ flavor dynamical fermion configurations are generated with domain-wall fermions (DWF) an
d the overlap fermions are adopted for the valence quarks. The isovector $g_A^3$ and $g_S^3$, and the connected insertion part of $g_S^0$ are reported for three source-sink separations. With local current, we obtain $g_A^3 = 1.18(4)$ from a two-state fit. For the quark momentum fraction $langle x rangle_{u-d}$, we have included smaller lattices (i.e. $24^3 times 64$ and $32^3 times 64$ lattice with pion mass at 330 and 290 MeV respectively) for a fit which includes partially quenched cases as well as finite volume and continuum corrections. A global fit with perturbative renormalization gives $langle x rangle_{u-d} (overline{MS},, mu = 2, {rm GeV}) = 0.170(14)$. We made a cost comparison of calculating the nucleon matrix elements with those from the twisted mass fermion on similar sized lattice at the physical pion point and the domain-wall fermion calculation on the same DWF lattice. We also compare cost with the clover fermion calculation on similar sized lattice at about the same quark mass. The comparison shows that with several improvements, such as many-to-all correlator with grid source and low-mode substitution in the connected insertion and low-mode average in the quark loop can make the overlap as efficient as the twisted-mass and clover fermions in calculating the three-point functions. It is more efficient than the DWF. When the multi-mass feature is invoked, the overlap can be more efficient in reaching the same precision than the single mass comparison made so far.
