We evaluate the impact of recent developments in hadron phenomenology on extracting possible fundamental tensor interactions beyond the standard model. We show that a novel class of observables, including the chiral-odd generalized parton distributions, and the transversity parton distribution function can contribute to the constraints on this quantity. Experimental extractions of the tensor hadronic matrix elements, if sufficiently precise, will provide a, so far, absent testing ground for lattice QCD calculations. DOI: 10.1103/PhysRevLett.115.162001 PACS numbers: 13.60.Hb, 13.40.Gp, 24.85.+p High precision measurements of beta decay observables play an important role in beyond the standard model (BSM) physics searches, as they allow us to probe couplings other than of the V − A type, which could appear at the low energy scale. Experiments using cold and ultracold neutrons [1][2][3][4], nuclei [5][6][7][8], and meson rare decays [9] are being performed, or have been planned, that can reach the per-mil level or even higher precision. Effective field theory (EFT) allows one to connect these measurements and BSM effects generated at TeV scales. In this approach that complements collider searches, the new interactions are introduced in an effective Lagrangian describing semileptonic transitions at the GeV scale including four-fermion terms, or operators up to dimension six for the scalar, tensor, pseudoscalar, and V þ A interactions (for a review of the various EFT approaches, see Ref.[10]). Because the strength of the new interactions is defined with respect to the strength of the known SM interaction, the coefficients of the various terms, ϵ i , (i ¼ S; T; P; L; R) depend on the ratio m 2 W =Λ 2 i , where Λ i is the new physics scale relevant for these nonstandard interactions, and mW Þ. Therefore, the precision with which ϵ i ∝ m 2 W =Λ 2 i , is known determines a lower limit for Λ i . The scalar (S) and tensor (T) operators, in particular, contribute linearly to the beta decay parameters through their interference with the SM amplitude, and they are, therefore, more easily detectable. The matrix elements or transition amplitudes between neutron and proton states of all quark bilinear Lorentz structures in the effective Lagrangian which are relevant for beta decay observables, involve products of the BSM couplings, ϵ i , and the corresponding hadronic charges, g i , i.e., considering only terms with left-handed neutrinos,whereSðTÞ , characterize nucleon structure; however, at variance with the electroweak currents, there exists no fundamental coupling to these charges in the standard model. Therefore, they cannot be measured directly in elastic scattering processes. This Letter is concerned with an alternative approach aimed at extracting the hadronic charges from experimental data obtained in electron scattering. In previous work, various approaches have been developed to calculate these quantities including lattice QCD [11][12][13][14][15], and most recently, Dyson-Schwinger equations [16,17]. Lattice QCD ...