Towards consistent nuclear interactions from chiral Lagrangians I: The path-integral approach

Low-energy nuclear interactions have been extensively studied in the framework of chiral effective field theory. The corresponding potentials have been worked out using dimensional regularization to evaluate ultraviolet divergent loop integrals. An additional cutoff is then introduced in the nuclear Schr\"odinger equation to calculate observables. Recently, we have shown that such a mixture of two regularization schemes violates chiral symmetry when applied beyond the two-nucleon system and/or to processes involving external probes. To solve this issue, three- and four-nucleon forces as well as exchange current operators need to be re-derived using symmetry-preserving cutoff regularization. While it is possible to introduce a symmetry-preserving cutoff already in the effective chiral Lagrangian, the appearance of high-order time derivatives of the pion field, caused by the regulator, makes the standard Hamiltonian-based methods not well suited for the calculation of nuclear potentials. Here, we propose a new approach to derive nuclear interactions using the path integral method with no reliance on the canonical quantization. To this aim, the interaction part of the action is brought to an instantaneous form via suitably chosen nonlocal field redefinitions. Loop contributions to the nuclear potentials are then generated through the functional determinant, induced by the field redefinitions. We discuss in detail the application of these ideas to the case of a regularized Yukawa-type model of pion-nucleon interactions. Our new method allows to perform a systematic quantum mechanical reduction within the quantum field theory framework and opens the way for deriving consistently regularized nuclear forces and current operators from the effective chiral Lagrangian.