Regulator constraints for the perturbative renormalizability of attractive triplets

TL;DR

The paper investigates whether perturbative corrections in the attractive $^3P_0$ channel of nuclear EFT are regulator-independent by examining exceptional cutoffs that appear in distorted-wave perturbation theory. Using concrete calculations with various regulator schemes, it shows that exceptional cutoffs are regulator-dependent artifacts, not universal features, and that a well-defined $\Lambda\to\infty$ limit exists for suitable regulators; however, certain nonlocal or separable contact representations can introduce genuine exceptional zeros. The work demonstrates that finite-cutoff EFTs can be practically reliable within EFT truncation errors, and discusses strategies to design regulators that avoid exceptional cutoffs or to compensate for regulator effects via reshaped EFT expansions. It highlights the role of locality, compact support, and regulator choice in ensuring perturbative renormalizability, while aligning with Wilsonian RG intuition that the observable content can be regulator-insensitive when renormalization is performed properly. Overall, the findings clarify the regulator constraints in perturbative renormalizability of attractive triplets and offer guidance for practical regulator design in nuclear EFT.

Abstract

Nuclear effective field theory organizes the calculation of observables as a power series in terms of the ratio of soft and hard momentum scales. The rigorous implementation of this idea requires a mixture of perturbative and non-perturbative methods: on the one hand, nuclei are bound states that require the iteration of part of the nuclear potential, while on the other corrections that are small in the aforementioned power series should be perturbative in principle. Recently, it has been noted that these corrections are not cutoff independent as there are a set of exceptional cutoffs for which the couplings cannot be determined, as exemplified with the subleading order $^3P_0$ phase shifts in two-nucleon scattering. Yet, here it is shown by means of concrete calculations that exceptional cutoffs are a regulator-dependent feature. There exists a well-defined limit when the cutoff is removed, which implies that not every regulator choice (understood not only as the regulator itself, but in tandem with renormalization conditions) is acceptable within the effective field theory framework. The practical implications are minor, though: except if one is trying to explicitly probe the cutoff independence of the theory, most sensible regulator and cutoff choices are compatible with the renormalized limit within truncation errors.

Figures (12)

• Figure 1: Subleading $^3P_0$ phase shift at ${\rm NLO}$ (Weinberg notation: refers to $Q^2$ or the order at which leading TPE is included) for $k = 300\,{\rm MeV}$. The renormalization conditions for the phase shift are the reproduction of the Nijmegen II phase shifts at $k = 100\,{\rm MeV}$ and $200\,{\rm MeV}$ (i.e. two data points). Calculations have been done with the regularization procedures and the numerical codes of Ref. Valderrama:2011mv, which are referred to as "regularization 1" or "$R_1$" in the figure.
• Figure 2: $^3P_0$ phase shifts at ${\rm NLO}$ as a function of the center-of-mass momentum. They are calculated with regularization 1 ($R_1$, i.e. as in Fig. \ref{['fig:3P0-R1']}) and compared with the Nijmegen II phase shifts. The two circles indicate the $k = 100$ and $200\,{\rm MeV}$ Nijmegen II phase shifts, whose reproduction serves as the renormalization conditions for the ${\rm NLO}$ ones. The cutoff radius is $R_c = 0.05\,{\rm fm}$, for which calculations have effectively converged to their $R_c \to 0$ limit.
• Figure 3: Subleading $^3P_0$ phase shift at ${\rm NLO}$ where the ${\rm LO}$ contact is regularized with a delta-shell, Eq. (\ref{['eq:bc-contact']}). The renormalization conditions for the subleading phases are as in Fig. (\ref{['fig:3P0-R1']}). This setup is called regularization 2, or "$R_2$" in the figure.
• Figure 4: Subleading $^3P_0$ phase shift at ${\rm NLO}$ where the ${\rm LO}$ contact is regularized with a delta-shell, Eq. (\ref{['eq:bc-contact']}), and with derivatives instead of energy dependence in the subleading contacts. The renormalization conditions for the subleading phases are as in Fig. (\ref{['fig:3P0-R1']}). This is a variation of regularization 2 that is denoted as "$R_{2d}$" in the figure.
• Figure 5: Perturbative determinant for the $^3P_0$ phase shifts for a subleading contact with derivatives (with regularization 2 for the ${\rm LO}$ results, i.e. using what is called "$R_{2d}$" in Fig. \ref{['fig:3P0-R2d']}). The units are chosen as for the largest local maximum of the determinant to be $1$ within the cutoff range in which it is plotted.