Scalar, Axial, and Tensor Interactions of Light Nuclei from Lattice QCD

Complete flavor decompositions of the matrix elements of the scalar, axial, and tensor currents in the proton, deuteron, diproton, and ${}^3\mathrm{He}$ at SU(3)-symmetric values of the quark masses corresponding to a pion mass $m_{\pi }\sim 806\mathrm{MeV}$ are determined using lattice quantum chromodynamics. At the physical quark masses, the scalar interactions constrain mean-field models of nuclei and the low-energy interactions of nuclei with potential dark matter candidates. The axial and tensor interactions of nuclei constrain their spin content, integrated transversity, and the quark contributions to their electric dipole moments. External fields are used to directly access the quark-line connected matrix elements of quark bilinear operators, and a combination of stochastic estimation techniques is used to determine the disconnected sea-quark contributions. The calculated matrix elements differ from, and are typically smaller than, naive single-nucleon estimates. Given the particularly large, $\mathcal{O}(10\%)$, size of nuclear effects in the scalar matrix elements, contributions from correlated multinucleon effects should be quantified in the analysis of dark matter direct-detection experiments using nuclear targets.

Figure 1

The bare effective matrix element of the connected isoscalar ${}^3\mathrm{He}$ scalar charge, $g_{{}^3\text{He},S}^{(8)}=⟨{}^3\text{He}|\overline{q}{\mathrm{\Lambda }}^{(8)}q|{}^3\text{He}⟩$, where the blue circles and orange diamonds denote SP and SS results, respectively. The blue band illustrates a correlated two-state fit of the form $\sim A+B{e}^{-\mathrm{\Delta }t}$, to the SP correlation functions, while the green band denotes the final fit result with combined statistical and systematic uncertainties.

Figure 2

The strange-quark (disconnected) scalar matrix element in ${}^3\mathrm{He}$, $g_{{}^3\text{He},S}^{(s)}=⟨{}^3\text{He}|\overline{s}s|{}^3\text{He}⟩$. The left (right) panel shows results obtained using the SS (SP) correlation functions for a range of current insertion times $\tau$ and sink insertion times. The green band corresponds to the extracted matrix element determined as described in the text, and the blue (orange) curves and bands illustrate a correlated two-state fit to the data shown as discussed in the Supplemental Material [50].

Figure 3

The calculated values of $\mathrm{\Delta }{R}_X^{(f)}$ for the deuteron (circles), diproton (diamonds), and ${}^3\mathrm{He}$ (squares) to those in the proton. The panels display the results obtained for the scalar (top), axial (middle), and tensor (bottom) interactions, and the columns within the panels display results for the different flavor structures of the currents, as indicated at the top of the figure. In each case, the statistical and systematic uncertainties have been combined in quadrature. The points exactly at zero are constrained to vanish by spin and/or isospin symmetry, while ratios are not given for the strange quark axial and tensor charges, as both the numerators and denominators are consistent with zero.