Renormalization of the off-shell chiral two-pion exchangeNNinteractions

Abstract

The renormalization, finiteness and off-shellness of short distance power like singular interactions is discussed. We show analytically that the renormalizability of the off-shell scattering amplitude relies completely on the corresponding on-shell amplitude without proliferation of new counterterms. We illustrate the result by complementary calculations both in coordinate as well as in momentum space in the simplest 1 S0 channel for chiral np interactions including Two Pion Exchange. A traditional source of theoretical uncertainty in the study of nuclear physics and nuclear reactions has been the relevance and significance of off-shellness in the NN force (For a review up to the mid seventies see e.g. [1] and references therein). As is well known, any off-shell ambiguity should cancel in the final results and off-shellness itself cannot be measured as a matter of principle. This does not necessarily mean, however, that off-shellness can generally be completely disposed of and most few-and many body calculations do involve off-shell quantities as intermediate stages. This feature relies on the fundamental fact that quantum mechanics is naturally formulated in terms of wave functions while they are not directly measurable quantities except at asymptotically large distances. A paralell statement for Quantum Field Theory applies for the fields themselves as well as the associated Green functions.Potential approaches to the NN interaction need the half off-shell extrapolated potential and the half off-shell T-matrix is used to determine the on-shell S−matrix. Moreover, a knowledge of the off-shell T-matrix is needed e.g. for nucleon-nucleon bremsstrahlung, the three nucleon problem as well as nuclear matter calculations and thus a phenomenological determination of the off-shell T-matrix has been the subject of intense research in the past [1]. A relevant issue in this regard is that the definition of off-shellness is largely conventional. Actually, the quantum mechanical trading between two body off- . Moreover, unitarity for the three body problem rests on off-shell unitarity for the two body problem, imposing constraints on the acceptable off-shellness [5,6].Within the Effective Field Theory (EFT) approach to nuclear physics based on chiral symmetry [7,8] (for comprehensive reviews see e.g. Ref. [9,10,11]) the ambiguities related to off-shellness can be rephrased in the freedom to undertake field dependent transformations and using the equations of motion. Actually, in purely contact EFT's, where the interaction is represented by a polynomial in momenta and/or energy, off-shellness can be completely ignored from the start by using local field redefinitions [4,12]. This fact does not generally hold for finite range interactions stemming from particle exchange if the exchanged momentum becomes comparable to the exchanged particle mass and where non-local and singular field redefinitions would be needed. Similarly to phenomenological potentials, Chiral potentials are not free from these ambiguities since by construct...

Renormalization of the leading-order chiral nucleon-nucleon interaction and bulk properties of nuclear matter

“…(22) [48]. The proof rests on the finiteness of the off-shell K-matrix [49]. This gives some confidence on the stability of the numerics for increasing cut-off values.…”

“…Turning to non-perturbative schemes, it was noted in Ref. [49] that off-shell properties of renormalized chiral potentials do not look much different from more conventional ones as the singularities are effectively removed. Furthermore, we know that the inclusion of ∆ degrees of freedom in chiral potentials [10] lead to a pattern of better convergence, where e.g.…”

“…In Ref. [49] the finiteness of the K-matrix off-shell for short distance singular interactions was established solely on the basis of on-shell renormalization conditions. This means that if K is finite on shell then the off-shell extension remains finite as well.…”

“…In this equation, the regulator may be effectively removed from the K-matrix at the operator level even if the energy w = w f does not correspond to the scattering energy [49]. If the interaction is attractive the single particle potential U (n) < 0 and w < w f and the (finite cutoff) K(w) involves states below threshold.…”

Nucleon-nucleon scattering within a multiple subtractive renormalization approach

“…Renormalization of the NN interaction in chiral effective theory in LO, up to next-to-leading order (NLO), and to next-to-next-to-leading order (NNLO), in S, P , and D waves, has been done with success in coupled and uncoupled waves [5,6,10,[30][31][32][33][34]. References [30,31] apply the renormalization approach in coordinate space, while the works [5,6,10,33,34] use subtracted-resolvent two-body techniques in momentum space.…”