Contact potentials in presence of a regular finite-range interaction using dimensional regularization and the $N/D$ method

TL;DR

This work shows that solving the Lippmann-Schwinger equation with a regular finite-range potential plus derivative contact terms using dimensional regularization reproduces the exact dispersive (N/D) amplitude for the ${}^1S_0$ NN channel, preserving unitarity and the OPE left-hand cut. By systematically increasing the short-range operator basis to ${\cal O}(Q^{6})$ and exploring two DR renormalization schemes (finite and infinite threshold subtraction), the authors establish a precise mapping between DR-LSE and the $N/D_{nd}$ framework, with the low-energy constants (via the ERE) playing the role of the renormalization inputs. They show that the two approaches are equivalent when the correct number and type of subtractions are chosen, and they analyze how many ERE parameters must be fixed at each order to fully specify the theory. The results provide a controlled, regulator-free route to generate $T$-matrix amplitudes with well-defined left- and right-hand cuts and offer insights into consistent power counting and potential extensions to longer-range or more singular interactions. Practically, the work connects DR-based LSE renormalization with dispersive methods, enabling quasi-analytic expressions for the $T$-matrix and clarifying how to incorporate higher-order short-range terms and future long-range corrections such as TPE.

Abstract

We solve the Lippman-Schwinger equation (LSE) with a kernel that includes a regular finite-range potential and additional contact terms with derivatives. We employ distorted wave theory and dimensional regularization, as proposed in Physics Letters B 568 (2003) 109. We analyze the spin singlet nucleon-nucleon $S-$wave as case of study, with the regular one-pion exchange (OPE) potential in this partial wave and up to ${\cal O}(Q^6)$ (six derivatives) contact interactions. We discuss in detail the renormalization of the LSE, and show that the scattering amplitude solution of the LSE fulfills exact elastic unitarity and inherits the left-hand cut of the long-distance OPE amplitude. Furthermore, we proof that the LSE amplitude coincides with that obtained from the exact $N/D$ calculation, with the appropriate number and typology of subtractions to reproduce the effective range parameters taken as input to renormalize the LSE amplitude. The generalization to higher number of derivatives is straightforward.

Figures (10)

• Figure 1: Comparison of the $^1S_0$ phase-shifts obtained at order ${\cal O}(Q^0)-$DR (blue line with dots) and the non-perturbative $N/D_{11}$ (red line) methods, fixing in both cases the scattering length to the same common value, as a function of the $NN$ center of mass momentum $k$. One can identify this solution with those obtained in Refs. PavonValderrama:2005wv and Nogga2005 using renormalization with boundary conditions and with one counter-term, respectively.
• Figure 2: Comparison of the phase-shifts obtained in the ${\cal O}(Q^2)-$DR (blue line with dots) and the non-perturbative $N/D_{22}$ (red line) methods fixing $a$, $r$ and $v_2$. The green line stands for the result of the $N/D_{12}$ solution fixing $a$ and $r$ and the magenta curve shows the prediction from the effective range approximation including terms up to $v_2$. The black dots with error bars are the experimental phase-shifts from Stoks:1993tb used to fit the order 2 ($n=1$) solution. The right panel is the same but using a log-scale for $k$. Furthermore, the red dashed curves represent the phase-shifts obtained from the long-range OPE potential $V_\pi$.
• Figure 3: Phase-shifts for the ${\cal O}(Q^2)$ DR$_{\infty}$ scheme (blue line with dots) ($n=1$) compared with the results from the $N/D_{12}$ method (red line).
• Figure 4: Red line: Dependence of $\delta_3$ on $J_0^R$ as deduced from the ${\cal O}(Q^4)$ Eq. \ref{['eq:delta3Orden2']}, and using the numerical values of $\delta_0$, $\delta_1$ and $\delta_2$ given in the ${\cal O}(Q^2)$ Eq. \ref{['eq:delta13org']}. The green line stands for the minimum value $\delta_3|_{\rm min}=\delta_2^2/\delta_1=-39.0717\,m^{-5}$, which is reached for $J_0^R=(\delta_1^2-\delta_0\delta_2)/\delta_2=-3.2070\, m$ [Eq. \ref{['eq:num-valPLB']}].
• Figure 5: Phase-shifts obtained from the $N/D_{33}$ (red lines) and ${\cal O}(Q^4)$ ($n=2$) DR (blue lines with dots) methods, fixing the ERE parameters up to $v_4$. Scenarios 1 [Eq. \ref{['eq:sol1']}] and 2 [Eq. \ref{['eq:sol2']}] are shown in the left and right panels of the figure, respectively. The green curve in both panels stands for the original DR calculation of Ref. Nieves:2003uu calculated at order ${\cal O}(Q^2)$ ($n=1$), and as detailed in the text, it has a different value for $v_4$ than the two ${\cal O}(Q^4)$ scenarios depicted in the plots.