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The hadron mass can be obtained through the calculation of the trace of the energy-momentum tensor in the hadron which includes the trace anomaly and sigma terms. The anomaly due to conformal symmetry breaking is believed to be an important ingredient for hadron mass generation and confinement. In this work, we will present the calculation of the glue part of the trace anomaly form factors of the pion up to $Q^2\sim 4.3{\mathrm{GeV}}^2$ and the nucleon up to $Q^2\sim 1{\mathrm{GeV}}^{more »}$ 2 . The calculations are performed on a domain wall fermion ensemble with overlap valence quarks at seven valence pion masses varying from $\sim 250$ to $\sim 540\mathrm{MeV}$ , including the unitary point $\sim 340\mathrm{MeV}$ . We calculate the radius of the glue trace anomaly for the pion and the nucleon from the $z$ expansion. By performing a two-dimensional Fourier transform on the glue trace anomaly form factors in the infinite momentum frame with no energy transfer, we also obtain their spatial distributions for several valence quark masses. The results are qualitatively extrapolated to the physical valence pion mass with systematic errors from the unphysical sea quark mass, discretization effects in the renormalization sum rule, and finite-volume effects to be addressed in the future. We find the pion’s form factor changes sign, as does its spatial distribution, for light quark masses. This explains how the trace anomaly contribution to the pion mass approaches zero toward the chiral limit. Published by the American Physical Society 2024« less

A lattice QCD approach to the calculation of the long-distance contributions to ${\epsilon }_K$ is presented. This parameter describes indirect $CP$ violation in $K\to \pi \pi$ decay. While the short-distance contribution to ${\epsilon }_K$ can be accurately calculated in terms of standard model parameters and a single hadronic matrix element, $B_K$ , there is a long-distance part which is estimated to be approximately 5% of the total and is more difficult to determine. A method for determining this smallmore » but phenomenologically important contribution to ${\epsilon }_K$ using lattice QCD is proposed and a complete exploratory calculation of the contribution is presented. This exploratory calculation uses an unphysical light quark mass corresponding to a 339 MeV pion mass and an unphysical charm quark mass of 968 MeV, expressed in the $\overline{\mathrm{MS}}$ scheme at 2 GeV. This calculation demonstrates that future work should be able to determine this long-distance contribution from first principles with a controlled error of 10% or less. Published by the American Physical Society 2024« less