We present results for the isovector axial, scalar and tensor charges g^u-d_A, g^u-d_S and g^u-d_T of the nucleon needed to probe the Standard Model and novel physics. The axial charge is a fundamental parameter describing the weak interactions of nucleons. The scalar and tensor charges probe novel interactions at the TeV scale in neutron and nuclear β-decays, and the flavor-diagonal tensor charges g^u_T, g^d_T and g^s_T are needed to quantify the contribution of the quark electric dipole

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... (EDM) to the neutron EDM. The 9 ensembles, generated by the MILC Collaboration using the HISQ action with 2+1+1 dynamical flavors, span three lattice spacings a ≈ 0.06, 0.09 and 0.12 fm and light-quark masses corresponding to the pion masses M_π≈ 135, 225 and 315 MeV. High-statistics estimates on five ensembles using the all-mode-averaging method allow us to quantify all systematic uncertainties and perform a simultaneous extrapolation in the lattice spacing, lattice volume and light-quark masses for the connected contributions. Our final estimates, in the MS scheme at 2 GeV, of the isovector charges are g_A^u-d = 1.195(33)(20), g_S^u-d = 0.97(12)(6) and g_T^u-d = 0.987(51)(20). The first error includes statistical and all systematic uncertainties except that due to the extrapolation Ansatz, which is given by the second error estimate. Combining our estimate for g_S^u-d with the difference of light quarks masses (m_d-m_u)^ QCD=2.67(35) MeV given by FLAG, we obtain (M_N-M_P)^ QCD = 2.59(49) MeV. Estimates of the connected part of the flavor-diagonal tensor charges of the proton are g^u_T=0.792(42) and g^d_T=-0.194(14). Combining our new estimates with precision low-energy experiments, we update constraints on novel scalar and tensor interactions, ϵ_S,T, at the TeV scale.