The nuclear matrix element determining the $pp\to d{e}^{+}\nu$ fusion cross section and the Gamow-Teller matrix element contributing to tritium $\beta$ decay are calculated with lattice quantum chromodynamics for the first time. Using a new implementation of the background field method, these quantities are calculated at the SU(3) flavor–symmetric value of the quark masses, corresponding to a pion mass of $m_{\pi }\sim 806\mathrm{MeV}$. The Gamow-Teller matrix element in tritium is found to be 0.979(03)(10) at these quark masses, which is within $2\sigma$ of the experimental value. Assuming that the short-distance correlated two-nucleon contributions to the matrix element (meson-exchange currents) depend only mildly on the quark masses, as seen for the analogous magnetic interactions, the calculated $pp\to d{e}^{+}\nu$ transition matrix element leads to a fusion cross section at the physical quark masses that is consistent with its currently accepted value. Moreover, the leading two-nucleon axial counterterm of pionless effective field theory is determined to be $L_{1,A}=3.9(0.2)(1.0)(0.4)(0.9){\mathrm{fm}}^3$ at a renormalization scale set by the physical pion mass, also agreeing within the accepted phenomenological range. This work concretely demonstrates that weak transition amplitudes in few-nucleon systems can be studied directly from the fundamental quark and gluon degrees of freedom and opens the way for subsequent investigations of many important quantities in nuclear physics.

• Received 21 November 2016

DOI:https://doi.org/10.1103/PhysRevLett.119.062002