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Reexamining the perturbative renormalizability of the coupled triplets

Manuel Pavon Valderrama

TL;DR

This work reexamines the perturbative renormalizability of chiral two-pion exchange (TPE) in the two-nucleon sector when the one-pion exchange (OPE) tensor force is fully iterated at leading order. By analyzing uncoupled and coupled triplet channels via distorted-wave perturbation theory and exploiting an attractive/repulsive eigenchannel structure of the OPE tensor force, the author shows that only two counterterms are required to achieve cutoff independence, a reduction from previous counts. The analysis includes explicit calculations of the $^3S_1$-$^3D_1$ and $^3P_2$-$^3F_2$ phase shifts, using boundary conditions and delta-shell derivative contacts to demonstrate smooth Rc→0 behavior and the absence of genuine exceptional cutoffs for practical regulators. The results have significant implications for power counting in nuclear EFT, suggesting two counterterms may be the bare minimum in this framework, though broader RG and truncation-error considerations leave open whether more terms are preferable in some schemes.

Abstract

I reexamine the perturbative renormalizability of chiral two-pion exchange in two-nucleon scattering for coupled triplets when one-pion exchange has been fully iterated at leading order. Improving over previous works, it is shown that only two counterterms are required to obtain cutoff independent results, which is one less than in naive dimensional analysis. The explanation for this reduction is the existence of an attractive and repulsive eigenchannel in the one-pion exchange potential for the coupled triplets: the attractive eigenchannel can be renormalized like a regular attractive uncoupled triplet, while the repulsive eigenchannel is always finite regardless of whether there are counterterms or not. I discuss the implications of this finding for the power counting of the $^3S_1$-$^3D_1$ and $^3P_2$-$^3F_2$ partial waves.

Reexamining the perturbative renormalizability of the coupled triplets

TL;DR

This work reexamines the perturbative renormalizability of chiral two-pion exchange (TPE) in the two-nucleon sector when the one-pion exchange (OPE) tensor force is fully iterated at leading order. By analyzing uncoupled and coupled triplet channels via distorted-wave perturbation theory and exploiting an attractive/repulsive eigenchannel structure of the OPE tensor force, the author shows that only two counterterms are required to achieve cutoff independence, a reduction from previous counts. The analysis includes explicit calculations of the - and - phase shifts, using boundary conditions and delta-shell derivative contacts to demonstrate smooth Rc→0 behavior and the absence of genuine exceptional cutoffs for practical regulators. The results have significant implications for power counting in nuclear EFT, suggesting two counterterms may be the bare minimum in this framework, though broader RG and truncation-error considerations leave open whether more terms are preferable in some schemes.

Abstract

I reexamine the perturbative renormalizability of chiral two-pion exchange in two-nucleon scattering for coupled triplets when one-pion exchange has been fully iterated at leading order. Improving over previous works, it is shown that only two counterterms are required to obtain cutoff independent results, which is one less than in naive dimensional analysis. The explanation for this reduction is the existence of an attractive and repulsive eigenchannel in the one-pion exchange potential for the coupled triplets: the attractive eigenchannel can be renormalized like a regular attractive uncoupled triplet, while the repulsive eigenchannel is always finite regardless of whether there are counterterms or not. I discuss the implications of this finding for the power counting of the - and - partial waves.
Paper Structure (10 sections, 109 equations, 3 figures)

This paper contains 10 sections, 109 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Perturbative renormalizability
  3. Uncoupled triplets
  4. Coupled triplets
  5. Attractive and repulsive eigenchannels
  6. Regularization and renormalization
  7. Calculation of the phase shifts
  8. Cutoff dependence
  9. Derivative contact-range interactions
  10. Discussion and conclusions

Figures (3)

  • Figure 1: Phase shifts (nuclear bar) for the $^3S_1$-$^3D_1$ and $^3P_2$-$^3F_2$ channels at ${\rm N^3LO}$ and ${\rm N^4LO}$ (i.e. up to the inclusion of leading and subleading TPE, respectively). The ${\rm LO}$ contact-range interaction is regularized with the boundary conditions of Eqs. (\ref{['eq:uw-reg']}) and (\ref{['eq:orthogonality-mod']}) at the cutoff radius $R_c$, while the subleading orders use the delta-shell regularization and couplings of Eqs. (\ref{['eq:delta-shell-coupled']}) and (\ref{['eq:contacts-coupled']}). The finite-range interaction is always regularized with a sharp cutoff in coordinate space (that is, the potential is zero for $r < R_c$). The ${\rm LO}$ renormalization condition is the reproduction of the $^3S_1$ ($^3P_2$) scattering length (volume), $a_1({}^3S_1) = 5.419\,{\rm fm}$ ($a_1({}^3P_2) = -0.040\,{\rm fm}^3$), while at subleading orders the calculation reproduces the $^3S_1$ ($^3P_2$) nuclear bar phase shifts of the Nijmegen II potential at $k_{\rm cm} = 100\,{\rm MeV}$ and $200\,{\rm MeV}$ (these two data points are represented as circles in the $^3S_1$ and $^3P_2$ plots). The $^3S_1$-$^3D_1$ ($^3P_2$-$^3F_2$) phase shifts are calculated at $R_c = 0.15$, $0.25$, $0.35$ and $0.50\,{\rm fm}$ ($R_c = 0.25$, $0.35$, $0.50$ and $0.70\,{\rm fm}$) and are compared with the Nijmegen II ones.
  • Figure 2: Perturbative determinant for the $^3S_1$-$^3D_1$ and $^3P_2$-$^3F_2$ partial waves and the renormalization conditions used in Fig. \ref{['fig:3C1-3C2']}. The units have been normalized as to generate local maxima of one in the region between the first and second zero of the perturbative determinant (where for the $^3P_2$-$^3F_2$ case the second zero is not actually reached, but just implied). These zeros are quartic and correspond to the appearance of deeply bound states. The point calculated with the lowest numerically stable cutoff is indicated with a circle (provided it falls within the plot range).
  • Figure 3: Cutoff dependence of the $^3S_1$-$^3D_1$ and $^3P_2$-$^3F_2$ (nuclear bar) phase shifts at ${\rm N^3LO}$ and ${\rm N^4LO}$, calculated for a center-of-mass momentum of $k_{\rm cm} = 300\,{\rm MeV}$. The regularization and renormalization choices are identical to the ones in Fig. \ref{['fig:3C1-3C2']}. It can be appreciated that there is convergence with respect to the cutoff when $R_c \to 0$, though owing to the appearance of numerical instabilities at hard cutoffs (which are a consequence of the existence of a repulsive eigenchannel for the OPE potential) the calculations are limited to $R_c \geq 0.14\,{\rm fm}$ and $R_c \geq 0.18\,{\rm fm}$ in the $^3S_1$-$^3D_1$ and $^3P_2$-$^3F_2$ channels, respectively. The lowest point at which calculations are numerically stable is marked with a small circle.