Scattering amplitudes versus potentials in nuclear effective field theory: search for a potential compromise

TL;DR

The paper probes whether nuclear EFT amplitudes, when expanded perturbatively, follow the same power counting as the expanded nuclear potential in the Weinberg framework. It shows that amplitude-based expansions can violate the nominal counting, with breakdown scales that depend on the regulator and potential generation, notably in the ${}^1S_0$ and ${}^3P_0$ channels. To address this, the author constructs EFT-inspired potentials by minimally modifying the Weinberg counting, adding targeted counterterms to preserve power counting upon iteration. The resulting modified potentials extend the range over which amplitude and potential expansions align, providing a practical approach that preserves EFT error estimates while bridging ideas from RG analysis and traditional non-perturbative nuclear physics.

Abstract

In effective field theory physical quantities, in particular observables, are expressed as a power series in terms of a small expansion parameter. For non-perturbative systems, for instance nuclear physics, this requires the non-perturbative treatment of at least part of the interaction (or the potential, if one is dealing with a non-relativistic system), while the rest of the interaction is included as perturbations. This is not entirely trivial and as a consequence different interpretations on how to treat these systems have appeared. A practical approach is to expand the effective potential, where this potential is later fully iterated in the Schroedinger equation for obtaining amplitudes and observables. The expectation is that this will lead to observables that will have an implicit power counting expansion. Here I explicitly check whether the amplitudes (when expanded according to the counting) are actually following the same power counting as the potential. It happens that reality does not necessarily conform to expectations and the amplitudes will sometimes violate the power counting with which the potential has been expanded. A more formal approach is to formulate the expansion directly in terms of amplitudes and observables, which is the original aim of the effective field theory idea. Yet this second approach is technically complicated. I explore here the possibility of constructing potentials that when fully iterated will make sure that amplitudes are indeed expansible in terms of a small expansion parameter.

Figures (10)

• Figure 1: Phase shifts in the Weinberg counting as a function of the center-of-mass momentum $k_{\rm c.m.}$, depending on whether one applies the power counting to the potential (W, for Weinberg in the way it is usually applied) or to the scattering amplitude (EFT, for indicating that the phase shifts are expanded according to the literal EFT expansion for the Weinberg counting, Eq. (\ref{['eq:T-exp']})). The calculation is done at ${\rm N^2LO}$ (i.e. including terms up to $\nu=3$ or $Q^3$) with a momentum space cutoff $\Lambda = 400\,{\rm MeV}$ (see Appendix \ref{['app:p-space']}). At ${\rm LO}$ the potential- and amplitude-based expansions coincide. At ${\rm N^2LO}$ the two expansions are approximately compatible within the proposed error bands, which assume a breakdown scale of $M = 210\,{\rm MeV}$ (where the details are explained around Eqs. (\ref{['eq:err-sum-abridged']}) and (\ref{['eq:err-breakdown']}) and in Appendix \ref{['app:err']}).
• Figure 2: Phase shifts in an alternative power counting (read below), depending on whether one applies the power counting to the potential (W) or to the scattering amplitude (EFT). In this alternative power counting I have promoted the leading and subleading two pion exchange diagrams to ${\rm LO}$ and demoted one-pion exchange to ${\rm N^2LO}$. I use the terms ${\rm LO}'$ and ${\rm N^2LO'}$ for referring to the orders in this counting. This alternative power counting is indistinguishable from the Weinberg one when expanding in terms of the potential. The error bands of the ${\rm N^2LO' (T)}$ phase shifts assume a breakdown scale of $M = 400\,{\rm MeV}$ (see Appendix \ref{['app:err']}).
• Figure 3: Phase shifts for secondEpelbaum:2003grEpelbaum:2003xx and third generationGezerlis:2014zia$\rm N^2LO$ potentials in the Weinberg counting, depending on whether one expands the potential (W) or the scattering amplitude (EFT). For the second generation potential of Refs. Epelbaum:2003grEpelbaum:2003xx the Lippmann-Schwinger and SFR cutoffs fall within the $\Lambda = (450-650)$ and $\tilde{\Lambda} = (500-700)\,{\rm GeV}$ ranges, where here I use $(\Lambda, \tilde{\Lambda}) = (550,600)\,{\rm MeV}$ for obtaining the phase shifts. The third generation potential of Ref. Gezerlis:2014zia is formulated in coordinate space and uses a Gaussian type cutoff in the range $R_c = (1.0-1.2)\,{\rm fm}$ and the SFR cutoff $\tilde{\Lambda} = (1.0-1.4)\,{\rm GeV}$. In the figures for the potential of Ref. Gezerlis:2014zia I have generated the phase shifts for $R_c = 1.1\, {\rm fm}$ and $\tilde{\Lambda} = 1.0\,{\rm GeV}$. I compute the phase shifts for the $^1S_0$ and $^3P_0$ partial waves, where the roman numeral in parentheses indicates whether the calculation corresponds to the second (II) or third generation (III) potential. The errors bands in the amplitude expansion are generated from Eqs. (\ref{['eq:err-sum-abridged']}) and (\ref{['eq:err-breakdown']}) with a breakdown scale of $M = 310\,{\rm MeV}$ and $270\,{\rm MeV}$ ($M = 380\,{\rm MeV}$ and $190\,{\rm MeV}$) for the $^1S_0$ and $^3P_0$ partial waves for the second (third) generation potential.
• Figure 4: Same as Fig. \ref{['fig:W23']} but for the ${}^3S_1$, $E_1$ and ${}^3D_1$ partial waves (nuclear bar phase shifts).
• Figure 5: Same as Fig. \ref{['fig:W23']} but for the ${}^1P_1$, ${}^3P_1$ partial waves.