Using the general connection between the upper limit on the neutrino mass and the upper limits on certain types of non-Standard Model interaction that can generate loop corrections to the neutrino mass, we derive constraints on some non-Standard Model d → ue −ν interactions. When cast into limits on n → pe −ν coupling constants, our results yield constraints on scalar and tensor weak interactions improved by more than an order of magnitude over the current experimental limits. When combined with the existing limits, our results yield |CS/CV | 5 × 10 −3 , |C ′ S /CV | 5 × 10 −3 , |CT /CA| 1.2 × 10 −2 and |C ′ T /CA| 1.2 × 10 −2 .PACS numbers: 23.40.Bw,14.60.PqHistorically, nuclear β decay has played an important role in establishing the V − A structure of the electroweak current of the Standard Model (SM). More recently, precision studies of nuclear and neutron β decay have been used to test the SM and to search for what may lie beyond it. f t-values and various angular correlations, for example, have been measured on various nuclear species for small deviations from what the V − A model of weak interactions predicts. These experiments have provided important constraints (for a recent review, see Ref.[1]). With an increased intensity of cold and ultracold neutrons becoming available, increasingly more precise β-decay measurements with free neutrons may probe physics beyond the SM. Neutron β-decay measurements have the special advantage of being free from uncertainties due nuclear structure corrections. The UCNA experiment [2] at the Los Alamos National Laboratory, and the future abBA experiment [3], planned for the Spallation Neutron Source at the Oak Ridge National Laboratory, both aim at precision neutron β-decay measurements that will provide stringent tests of the SM.On the other hand, various solar, atmospheric and reactor neutrino experiments have provided clear evidence of neutrino oscillation, hence establishing that not all the neutrinos are massless [4,5,6]. In addition, the recent remarkable progress in observational cosmology now allows us to study the "Particle Physics" of the early universe through precision measurements of the anisotropy of the Cosmic Microwave Background (CMB). In fact, the most stringent upper limit on the neutrino mass comes from combining WMAP [7] and SDSS [8] data.The fact that the neutrino masses are so much smaller than the other SM fermions -at least six orders of magnitude -together with the fact that the lepton mixing matrix is strikingly different from the quark mixing matrix, may be a window onto new physics. Accordingly, the neutrino mass matrix has become a subject of intensive experimental and theoretical research. At the same time, the search for new physics through lowenergy observables such as muon decay and β decay continues with increasing accuracy. In view of this situation, model-independent connections between the neutrino mass and other low-energy observables would provide valuable guidance in the search for physics beyond the SM.Recently, an important conne...