1. Introduction
The delta-isobar resonance Δ(1232) is the closest baryon to the nucleon, marked by the nucleon-delta mass splitting Δ ≃ 293 MeV, thus it has been long thought to contribute significantly to two-nucleon [1–5] and three-nucleon interactions [6–10], in addition to other scenarios of improving chiral nuclear forces [11–22]. Besides the derivation of chiral nuclear forces, there are efforts to investigate the role of the delta isobar in nuclear structures by accounting for these so-called deltaful chiral forces in ab initio calculations [23–25]. We are motivated in part by a recent analysis of delta-less, renormalization-group (RG) invariant chiral forces [26], where scattering data is described by such a power counting only up to the on-shell centre of mass (CM) momentum k ≃ 200 MeV. Although disappointing, the conclusion of the perturbative analysis is consistent with the expectation of deltaless power counting: it converges in a kinematic window limited by k ≲ Δ. This is evident from the factor 1/Δ in deltaful two-pion-exchange (TPE) equation (7 ).
In this paper we report the findings in our first investigation on the role of the delta isobar in nucleon–nucleon scattering. In particular, we are interested to know why the deltaful TPE potentials differ so significantly between two regularization schemes: dimensional regularization (DR) and spectral function regularization (SFR). If the DR is used [2], the TPE diagrams shown in figure 1 do not yield any explicit regularization-scale dependence, once the divergences are subtracted by constant or momentum-squared counterterms. It has become known in the literature that the DR version of the deltaful TPE forces appears to be overly attractive in comparison with the phase shifts from the partial-wave analyses (PWA) [27, 28]. The SFR scheme [3, 29], however, does have an explicit momentum cut-off $\tilde{{\rm{\Lambda }}}$. If $\tilde{{\rm{\Lambda }}}$ is taken to the limit $\tilde{{\rm{\Lambda }}}\to \infty $, the DR expressions will be reproduced, at least for these deltaful TPE nucleon–nucleon (NN) diagrams. $\tilde{{\rm{\Lambda }}}=700\,\mathrm{MeV}$ were advocated because it leads to better agreement with the PWA. This setup was popular in the various applications of deltaful chiral forces, including NN scattering [30–33], finite nuclei [23] and infinite nuclear matter [34]. Among the results, the Goteborg-Oak Ridge (GO) group reported promising results on finite nuclei and nuclear matter with deltaful chiral potential [23]. However, the NN scattering phase shift was found to be poorly reproduced, as highlighted in [32].
Figure 1. The leading TPE potentials with Δ intermediate states. The dashed, solid, and double-dashed lines correspond to the pion, the nucleon, and the Δ resonance, respectively. |
Adopting a particular choice of the cut-off value is equivalent to correlating the series of NN contact forces, or counterterms. Since ${ \mathcal O }({Q}^{2})$ counterterms with adjustable low-energy constants (LECs) are already included at the same level with TPEs in power counting, the said choice of $\tilde{{\rm{\Lambda }}}$ implies that a combination of ${ \mathcal O }({Q}^{4})$ counterterms, thought to be higher-order in naive dimensional analysis (NDA), are included implicitly and their values are related to $\tilde{{\rm{\Lambda }}}=700\,\mathrm{MeV}$. If they are indeed important for the deltaful TPE to agree with NN scattering data, there must be a rationale to power count them differently than in NDA. Therefore, understanding Q4 counterterms subsumed in the SFR at finite $\tilde{{\rm{\Lambda }}}$ is the lead at the beginning of our investigation.
On the other hand, the TPE diagrams are computed in complete perturbation theory, and the cut-off dependence of Feynman diagrams is known to follow NDA. As we will see, it is actually the numerical factor of certain diagrams that are less suppressed than estimated by $1/{\left(4\pi \right)}^{\nu }$ where ν is the chiral index defined by Weinberg [35, 36]. In addition, the appearance of Δ in the denominator also helps because Q ∼ Δ is the kinematic window we are interested in, where Q generically denotes the size of momenta involved in the processes. We will incorporate this enhancement into power counting and re-examine the role of the delta isobar by studying the peripheral partial-wave phase shifts.
The paper is organized as follows. In section 2 we identify the most dominant deltaful TPE force by studying the residual cut-off dependence. After adjusting the power counting, we compare this with the empirical NN phase shifts in section 3 . Finally, discussions and a summary are offered in section 4 .
2. Enhancement of deltaful TPE potentials
Following the conventions of [2], we break down the NN forces to a combination of spin and isospin operators:
$\begin{eqnarray}\begin{array}{rcl}V(\vec{p}^{\prime} ,\vec{p}) & = & {V}_{C}+{{\boldsymbol{\tau }}}_{1}\,{\boldsymbol{\cdot }}\,{{\boldsymbol{\tau }}}_{2}{W}_{C}+\left({V}_{S}+{{\boldsymbol{\tau }}}_{1}\cdot {{\boldsymbol{\tau }}}_{2}{W}_{S}\right){\vec{\sigma }}_{1}\cdot {\vec{\sigma }}_{2}\\ & & +\left({V}_{T}+{{\boldsymbol{\tau }}}_{1}\cdot {{\boldsymbol{\tau }}}_{2}{W}_{T}\right){\vec{\sigma }}_{1}\cdot \vec{q}\,{\vec{\sigma }}_{2}\cdot \vec{q},\end{array}\end{eqnarray}$
where $\vec{p}$ ($\vec{p}^{\prime} $) is the initial (final) relative momentum and $\vec{q}\equiv \vec{p}^{\prime} -\vec{p}$ is the momentum transfer. Vi and Wi are scalar functions of $\vec{p}$ and $\vec{p}^{\prime} $. For instance, the one-pion-exchange (OPE) potential is given by $\begin{eqnarray}{V}_{\mathrm{OPE}}(\vec{p}^{\prime} ,\vec{p})={W}_{T}^{(\pi )}(q){{\boldsymbol{\tau }}}_{1}\cdot {{\boldsymbol{\tau }}}_{2}{\vec{\sigma }}_{1}\cdot \vec{q}\,{\vec{\sigma }}_{2}\cdot \vec{q},\end{eqnarray}$
and $\begin{eqnarray}{W}_{T}^{(\pi )}(q)\equiv -\displaystyle \frac{{g}_{A}^{2}}{4{f}_{\pi }^{2}}\displaystyle \frac{1}{{q}^{2}+{M}_{\pi }^{2}},\end{eqnarray}$
where the pion decay constant fπ = 92.4 MeV, nucleon axial-vector coupling gA = 1.29, pion mass Mπ = 138 MeV. In both S waves and at least part of P waves [37–39], it has been established that OPE is nonperturbative for k ≃ Δ. Therefore, OPE is counted as leading order (LO) because it must be iterated nonperturbatively. In this paper, however, we will study peripheral partial waves where OPE is perturbative, therefore, it is considered NLO in those channels: $\begin{eqnarray}{V}_{\mathrm{NLO}}={V}_{\mathrm{OPE}}.\end{eqnarray}$
The leading deltaful TPE diagrams are those with chiral index ν = 0, as shown in figure 1. To avoid double counting contributions that are already accounted for at LO through the resummation of OPE, one must subtract contributions from purely two-nucleon intermediate states. Some of the physical constants needed are the NΔ axial-coupling hA = 1.40, the nucleon mass MN = 939 MeV, and the delta-nucleon mass splitting Δ = MΔ − MN = 293.7 MeV. The analytic expressions have been worked out in [2] with DR and in [3] with SFR. The SFR scheme is a dispersion integral, with the regularized potentials being obtained through an integral over spectral functions
$\begin{eqnarray}V(q)=\displaystyle \frac{2}{\pi }{\int }_{2{m}_{\pi }}^{\infty }{\rm{d}}\mu \,\mu \displaystyle \frac{\rho (\mu ;\tilde{{\rm{\Lambda }}})}{{\mu }^{2}+{q}^{2}}\end{eqnarray}$
where the spectral function $\rho (\mu ,\tilde{{\rm{\Lambda }}})$ is derived by completing the TPE Feynman diagrams with a sharp momentum cut-off, see more detail in [29].We categorize these potentials in three groups, as done in both papers. In the first group, the potential is composed of the triangle diagram figure 1(a) and its variants by the permutation of the vertexes:
$\begin{eqnarray}\begin{array}{rcl}{W}_{C}^{(a)} & = & -\displaystyle \frac{{h}_{A}^{2}}{216{\pi }^{2}{f}_{\pi }^{4}}\\ & & \times \left[(5{q}^{2}+8{M}_{\pi }^{2}-12{{\rm{\Delta }}}^{2}){L}^{\tilde{{\rm{\Lambda }}}}(q)\right.\\ & & \left.+12{{\rm{\Delta }}}^{2}({q}^{2}+2{M}_{\pi }^{2}-2{{\rm{\Delta }}}^{2}){D}^{\tilde{{\rm{\Lambda }}}}(q)\right].\end{array}\end{eqnarray}$
The second group are the box diagrams with a single-Δ excitation figures 1(b) and (c): $\begin{eqnarray}{V}_{C}^{(b+c)}=-\displaystyle \frac{{g}_{A}^{2}{h}_{A}^{2}}{12\pi {\rm{\Delta }}{f}_{\pi }^{4}}{\left({q}^{2}+2{M}_{\pi }^{2}\right)}^{2}{A}^{\tilde{{\rm{\Lambda }}}}(q),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{W}_{C}^{(b+c)} & = & -\displaystyle \frac{{g}_{A}^{2}{h}_{A}^{2}}{216{\pi }^{2}{f}_{\pi }^{4}}\left[(12{{\rm{\Delta }}}^{2}-20{M}_{\pi }^{2}-11{q}^{2}){L}^{\tilde{{\rm{\Lambda }}}}(q)\right.\\ & & \left.+6{\left({q}^{2}+2{M}_{\pi }^{2}-2{{\rm{\Delta }}}^{2}\right)}^{2}{D}^{\tilde{{\rm{\Lambda }}}}(q)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{V}_{T}^{(b+c)} & = & -\displaystyle \frac{1}{{q}^{2}}{V}_{S}^{(b+c)}=-\displaystyle \frac{{g}_{A}^{2}{h}_{A}^{2}}{48{\pi }^{2}{f}_{\pi }^{4}}\\ & & \times \left[-2{L}^{\tilde{{\rm{\Lambda }}}}(q)+({q}^{2}+4{M}_{\pi }^{2}-4{{\rm{\Delta }}}^{2}){D}^{\tilde{{\rm{\Lambda }}}}(q)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{W}_{T}^{(b+c)} & = & -\displaystyle \frac{1}{{q}^{2}}{W}_{S}^{(b+c)}\\ & = & -\displaystyle \frac{{g}_{A}^{2}{h}_{A}^{2}}{144\pi {\rm{\Delta }}{f}_{\pi }^{4}}\left({q}^{2}+4{M}_{\pi }^{2}\right){A}^{\tilde{{\rm{\Lambda }}}}(q).\end{array}\end{eqnarray}$
The double-Δ excitation diagrams figures 1(d) and (e) make up the third group: $\begin{eqnarray}\begin{array}{l}{V}_{C}^{(d+e)}=-\displaystyle \frac{{h}_{A}^{4}}{27{\pi }^{2}{f}_{\pi }^{4}}\left\{-4{{\rm{\Delta }}}^{2}{L}^{\tilde{{\rm{\Lambda }}}}(q)+({q}^{2}+2{M}_{\pi }^{2}-2{{\rm{\Delta }}}^{2})\right.\\ \quad \left.\times \left[{H}^{\tilde{{\rm{\Lambda }}}}(q)+({q}^{2}+2{M}_{\pi }^{2}+6{{\rm{\Delta }}}^{2}){D}^{\tilde{{\rm{\Lambda }}}}(q)\right]\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{W}_{C}^{(d+e)}=-\displaystyle \frac{{h}_{A}^{4}}{486{\pi }^{2}{f}_{\pi }^{4}}\left\{(11{q}^{2}+20{M}_{\pi }^{2}-24{{\rm{\Delta }}}^{2}){L}^{\tilde{{\rm{\Lambda }}}}(q)\right.\\ +3({q}^{2}+2{M}_{\pi }^{2}-2{{\rm{\Delta }}}^{2})\\ \left.\times \left[{H}^{\tilde{{\rm{\Lambda }}}}(q)+(-{q}^{2}-2{M}_{\pi }^{2}+10{{\rm{\Delta }}}^{2}){D}^{\tilde{{\rm{\Lambda }}}}(q)\right]\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{V}_{T}^{(d+e)}=-\displaystyle \frac{1}{{q}^{2}}{V}_{S}^{(d+e)}=-\displaystyle \frac{{h}_{A}^{4}}{216{\pi }^{2}{f}_{\pi }^{4}}\\ \times \left[6{L}^{\tilde{{\rm{\Lambda }}}}(q)+(-{q}^{2}-4{M}_{\pi }^{2}+12{{\rm{\Delta }}}^{2}){D}^{\tilde{{\rm{\Lambda }}}}(q)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{W}_{T}^{(d+e)}=-\displaystyle \frac{1}{{q}^{2}}{W}_{S}^{(d+e)}=-\displaystyle \frac{{h}_{A}^{4}}{1296{\pi }^{2}{f}_{\pi }^{4}}\\ \times \left[2{L}^{\tilde{{\rm{\Lambda }}}}(q)+({q}^{2}+4{M}_{\pi }^{2}+4{{\rm{\Delta }}}^{2}){D}^{\tilde{{\rm{\Lambda }}}}(q)\right].\end{array}\end{eqnarray}$
Here the momentum cut-off $\tilde{{\rm{\Lambda }}}$ is related to the SFR scheme. Various functions of q and $\tilde{{\rm{\Lambda }}}$ used above are defined as follows: $\begin{eqnarray}\begin{array}{l}{L}^{\tilde{{\rm{\Lambda }}}}(q)=\displaystyle \frac{\sqrt{{q}^{2}\,+\,4{M}_{\pi }^{2}}}{2q}\\ \times \mathrm{ln}\displaystyle \frac{{\tilde{{\rm{\Lambda }}}}^{2}\left({q}^{2}+4{M}_{\pi }^{2}\right)+{q}^{2}({\tilde{{\rm{\Lambda }}}}^{2}-4{M}_{\pi }^{2})+2\tilde{{\rm{\Lambda }}}q\sqrt{\left({q}^{2}+4{M}_{\pi }^{2}\right)({\tilde{{\rm{\Lambda }}}}^{2}-4{M}_{\pi }^{2})}}{4{M}_{\pi }^{2}({\tilde{{\rm{\Lambda }}}}^{2}+{q}^{2})},\end{array}\end{eqnarray}$
$\begin{eqnarray}{A}^{\tilde{{\rm{\Lambda }}}}(q)=\displaystyle \frac{1}{2q}\arctan \displaystyle \frac{q(\tilde{{\rm{\Lambda }}}-2{M}_{\pi })}{{q}^{2}\,+\,2\tilde{{\rm{\Lambda }}}{M}_{\pi }},\end{eqnarray}$
$\begin{eqnarray}{D}^{\tilde{{\rm{\Lambda }}}}(q)=\displaystyle \frac{1}{{\rm{\Delta }}}{\int }_{2{M}_{\pi }}^{\tilde{{\rm{\Lambda }}}}\displaystyle \frac{{\rm{d}}u}{{u}^{2}\,+\,{q}^{2}}\arctan \displaystyle \frac{\sqrt{{u}^{2}-4{M}_{\pi }^{2}}}{2{\rm{\Delta }}},\end{eqnarray}$
$\begin{eqnarray}{H}^{\tilde{{\rm{\Lambda }}}}(q)=\displaystyle \frac{2{q}^{2}\,+\,4{M}_{\pi }^{2}-4{{\rm{\Delta }}}^{2}}{{q}^{2}\,+\,4{M}_{\pi }^{2}-4{{\rm{\Delta }}}^{2}}\left[L\left(q\right)-L\left(2\sqrt{{{\rm{\Delta }}}^{2}-{M}_{\pi }^{2}}\right)\right].\end{eqnarray}$
The choice of $\tilde{{\rm{\Lambda }}}$ is arbitrary as long as it is higher than the breakdown scale of ChEFT Mhi. The dependence of the NN scattering amplitude on $\tilde{{\rm{\Lambda }}}$ should be no more important than higher-order effects, a requirement equivalent to renormalization-group (RG) invariance. The $\tilde{{\rm{\Lambda }}}$ dependence of the deltaful TPE potential listed in equations (6 )–(14 ) is ‘residual’ in the sense that it vanishes at the limit $\tilde{{\rm{\Lambda }}}\to \infty $. In fact, one arrives at the DR expressions given in [2] for $\tilde{{\rm{\Lambda }}}\to \infty $.
Using the analytical forms of equations (6 )–(14 ) and ${m}_{\pi }\sim q\sim {\rm{\Delta }}\ll \tilde{{\rm{\Lambda }}}$, one can expand the residual cut-off dependence in powers of $1/\tilde{{\rm{\Lambda }}}$. These ${\tilde{{\rm{\Lambda }}}}^{-n}$ terms all appear as polynomials in q2, that is, as NN contact forces. We focus on the q4 terms, because, as mentioned in section 1 , polynomials up to q2 are already included at the same order as TPEs. Therefore, the q4 counterterms are the first in the line to potentially upset NDA. We find that the most dominant piece of the residual cut-off dependence is contributed by ${V}_{C}^{(b+c)}$ and ${V}_{C}^{(d+e)}$:19 ) is proportional to π, as opposed to π2 estimated by the standard ChEFT counting.
$\begin{eqnarray}{V}_{C}^{({q}^{4})}=\displaystyle \frac{{g}_{A}^{2}{h}_{A}^{2}}{24\pi {f}_{\pi }^{4}{\rm{\Delta }}\tilde{{\rm{\Lambda }}}}\left(1+\displaystyle \frac{4}{9}\displaystyle \frac{{h}_{A}^{2}}{{g}_{A}^{2}}\right){q}^{4}.\end{eqnarray}$
By comparison, the second-largest term, ${W}_{C}^{({q}^{4})}$, is smaller by a factor of approximately 15%. With the usual scaling for ChEFT with the explicit delta degrees of freedom [40–43], $\begin{eqnarray}\begin{array}{l}4\pi {f}_{\pi }\sim \tilde{{\rm{\Lambda }}}\sim {M}_{\mathrm{hi}},q\sim {f}_{\pi }\sim {\rm{\Delta }}\sim Q,\end{array}\end{eqnarray}$
${V}_{C}^{({q}^{4})}$ is counted as $\begin{eqnarray}{V}_{C}^{({q}^{4})}\sim \displaystyle \frac{1}{{M}_{\mathrm{hi}}^{2}}.\end{eqnarray}$
We notice that ${V}_{C}^{({q}^{4})}$ is smaller than OPE by ${ \mathcal O }({Q}^{2}/{M}_{\mathrm{hi}}^{2})$. Its size becomes more considerable if one accounts for the numerical fact q ∼ 3fπ because we are interested in the kinematic region q ∼ Δ. At any rate, the unusual significance of this residual term suggests that the long-range part, i.e., the terms that survives at $\tilde{{\rm{\Lambda }}}\to \infty $, of the isoscalar central force $\begin{eqnarray}{V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}\equiv {V}_{C}^{(b+c)}+{V}_{C}^{(d+e)}\end{eqnarray}$
is even larger, and it is smaller than OPE by only one order. However, ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ is counted in NDA as two orders smaller than OPE. It follows that for peripheral waves interested in the paper, the central part of the deltaful TPE must be considered N2LO, compared with its NDA counting of N3LO. Development of ChEFT has seen prior enhancement of certain diagrams due to unexpectedly large numerical factors. For instance, in [44, 45], the seagull diagrams of deltaless TPEs with ν = 1 vertexes appear to be enhanced too. Similar to the numerical enhancement of the seagull diagrams, the denominator in equation (It is not totally clear to us where the origin of this enhancement lies. Instead, we proceed to examine its consequence in peripheral partial waves of NN scattering. It is instructive to review the perturbative counting for peripheral waves without complication due to Δ, i.e., power counting with the deltaless TPEs. As mentioned earlier, since none of the forces are treated nonperturbatively, the LO potential is considered vanishing and OPE appears at NLO. As argued in [38], the N2LO potential includes the ${ \mathcal O }({Q}^{2})$ counterterms and all the deltaless TPEs appear at N3LO. The deltaless counting scheme is tabulated as follows:
$\begin{eqnarray}\begin{array}{rcl}\mathrm{LO}\,:\ {V}_{\mathrm{LO}} & = & 0,\\ \mathrm{NLO}\,:\ {V}_{\mathrm{NLO}} & = & {V}_{\mathrm{OPE}},\\ {{\rm{N}}}^{2}\mathrm{LO}\,:\ {V}_{{{\rm{N}}}^{2}\mathrm{LO}} & = & {V}_{\mathrm{ct}}^{(2)},\\ {{\rm{N}}}^{3}\mathrm{LO}\,:\ {V}_{{{\rm{N}}}^{3}\mathrm{LO}} & = & {V}^{\mathrm{TPE}{/}\!\!\!{{\rm{\Delta }}}},\end{array}\end{eqnarray}$
where ${V}^{\mathrm{TPE}{/}\!\!\!{{\rm{\Delta }}}}$ refers to the TPE potentials with ν = 0 vertexes without the Δ field and ${V}_{\mathrm{ct}}^{(2)}$ represent the ${ \mathcal O }({Q}^{2})$ counterterms acting in the P waves $\begin{eqnarray}\left\langle \mathrm{chn},p^{\prime} \right|{V}_{\mathrm{ct}}^{(2)}\left|\mathrm{chn},p\right\rangle ={C}_{\mathrm{chn}}{pp}^{\prime} ,\end{eqnarray}$
with chn = 1P1, 3P1, 3P2. But we refrain from applying to the P wave because it is not certain whether OPE is still perturbative at the relative momentum k ≃ Δ.Now with the enhancement discovered above, the power counting of the deltaful TPEs needs to be adjusted. The central force of the deltaful TPEs ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ must be added to N2LO. We then have the rest of the TPEs and ${V}_{C}^{({q}^{4})}$ at N3LO:
$\begin{eqnarray}\begin{array}{rcl}\mathrm{LO}\,:\ {V}_{\mathrm{LO}} & = & 0,\\ \mathrm{NLO}\,:\ {V}_{\mathrm{NLO}} & = & {V}_{\mathrm{OPE}},\\ {{\rm{N}}}^{2}\mathrm{LO}\,:\ {V}_{{{\rm{N}}}^{2}\mathrm{LO}} & = & {V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}},\\ {{\rm{N}}}^{3}\mathrm{LO}\,:\ {V}_{{{\rm{N}}}^{3}\mathrm{LO}} & = & {V}_{C}^{({q}^{4})}+\,\mathrm{Rest}\ \mathrm{of}\ \mathrm{TPEs}.\end{array}\end{eqnarray}$
For definiteness, the TPE potentials quoted above all use the DR version, or $\tilde{{\rm{\Lambda }}}\to \infty $ in its SFR formulation. When the central counterterm ${V}_{C}^{({q}^{4})}$ is projected onto the D waves, the numerical prefactor is independent of the spin s and total angular momentum j: $\begin{eqnarray}\begin{array}{l}\langle D\,s\,j,p^{\prime} | {V}_{C}^{({q}^{4})}| D\,s\,j,p\rangle =\displaystyle \frac{8}{15}{p}^{2}{{p}^{{\prime} }}^{2}{D}_{C-{\rm{\Delta }}},\end{array}\end{eqnarray}$
where the value of DC−Δ is parametrized by $\tilde{{\rm{\Lambda }}}$: $\begin{eqnarray}{D}_{C-{\rm{\Delta }}}=\displaystyle \frac{{g}_{A}^{2}{h}_{A}^{2}}{24\pi {f}_{\pi }^{4}{\rm{\Delta }}\tilde{{\rm{\Lambda }}}}\left(1+\displaystyle \frac{4}{9}\displaystyle \frac{{h}_{A}^{2}}{{g}_{A}^{2}}\right).\end{eqnarray}$
Computation of the NN T matrix according to a perturbative counting is explained in detail in [38, 39]. In a nutshell, n insertions of order-ν potentials contributes to the T matrix at order-nν. Therefore,
$\begin{eqnarray*}\begin{array}{rcl}{T}_{\mathrm{LO}} & = & 0,\\ {T}_{\mathrm{NLO}} & = & {V}_{\mathrm{NLO}},\\ {T}_{{{\rm{N}}}^{2}\mathrm{LO}} & = & {V}_{{{\rm{N}}}^{2}\mathrm{LO}}+{V}_{\mathrm{NLO}}{G}_{0}{V}_{\mathrm{NLO}},\\ {T}_{{{\rm{N}}}^{3}\mathrm{LO}} & = & {V}_{{{\rm{N}}}^{3}\mathrm{LO}}+{V}_{\mathrm{NLO}}{G}_{0}{V}_{\mathrm{NLO}}{G}_{0}{V}_{\mathrm{NLO}}\\ & & \quad +{V}_{{{\rm{N}}}^{2}\mathrm{LO}}{G}_{0}{V}_{\mathrm{NLO}}+{V}_{\mathrm{NLO}}{G}_{0}{V}_{{{\rm{N}}}^{2}\mathrm{LO}},\end{array}\end{eqnarray*}$
where G0 is the free two-nucleon propagator. Iterations of the sort VG0V implies summing over NN intermediate states, which is done in the partial-wave projected form: $\begin{eqnarray}\int \displaystyle \frac{{\rm{d}}{{ll}}^{2}}{2{\pi }^{2}}V(p^{\prime} ,l)\displaystyle \frac{{m}_{N}}{{k}^{2}-{l}^{2}+{\rm{i}}\epsilon }V(l,p).\end{eqnarray}$
They are regularized by the separable Gaussian function: $\begin{eqnarray}V(p^{\prime} ,p;{\rm{\Lambda }})\equiv {{\rm{e}}}^{-{\left(\tfrac{p^{\prime} }{{\rm{\Lambda }}}\right)}^{4}}V(p^{\prime} ,p\,){{\rm{e}}}^{-{\left(\tfrac{p}{{\rm{\Lambda }}}\right)}^{4}}.\end{eqnarray}$
The cut-off Λ, which regularizes NN intermediate states in the distorted-wave expansion, is to be distinguished from $\tilde{{\rm{\Lambda }}}$ used in the SFR, which regularized the diagrams of figure 1. The dependence on Λ turns out to be much less sensitive than that on $\tilde{{\rm{\Lambda }}}$, therefore we fix its value Λ = 800 MeV throughout the paper.3. Results
We first look at the D waves. The perturbative counting is not applied to 3D1 because 3D1 is coupled to 3S1 and OPE contributes strongly to this coupled channel. The phase shifts produced by the deltaful TPEs are shown in figure 2. We use $\tilde{{\rm{\Lambda }}}=700$ and 800 MeV in equation (27 ) to represent the different choice of DC−Δ. The phase shifts generated by the deltaless TPE potentials according to power counting (23 ) are plotted in figure 3 for comparison. As ${V}_{\mathrm{ct}}^{(2)}$ vanishes in D and F waves, there are no free parameters. We observe that ${V}_{C-{\rm{\Delta }}}^{{TPE}}$ indeed takes up much of the strength of the deltaful TPEs. This is evident in 3D1 and 3D2 where N2LO appears to be close to N3LO. However, this is not quite the case in 3D3, where N3LO is seen to deviate from N2LO. We note that the 3D3 phase shifts are dominated by iterations of OPE, as demonstrated by the deltaless 3D3 in figure 3 and figure 9 in [39]. In addition, the difference is exaggerated somewhat by the relatively small scale in the 3D3 plot.
Figure 2. NN phase shifts as functions of center-of-mass (CM) momentum in various D waves with the deltaful TPEs. ‘PWA’ refers to the empirical phase shifts from [27]. The solid and dotted curves are plotted with different values of DC−Δ. See the text for more explanation. |
Figure 3. NN phase shifts of the D waves up to N3LO with the deltaless TPEs. |
Compared with the deltaless TPEs, the delta isobar provides much needed central forces, as exhibited in spin-singlet channels, e.g., 3D1. At N3LO, the newly promoted isoscalar central counterterm ${V}_{C}^{({q}^{4})}$ gives us a handle to reduce the attraction of ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ towards higher momenta. A more optimum value of DC−Δ could be obtained by a combined fit to various D-wave channels.
Because the counterterm ${V}_{C}^{({q}^{4})}$ does not act in the F waves, there is no need to tune its value there. For the same reason to have excluded 3D1, we do not include 3F3, as OPE could become strong enough to be nonperturbative in 3P2−3F2. In figures 4 and 5, one finds that the significant attraction provided by the deltaful central TPE remains the most prominent feature. In addition, we notice that the deltaless TPEs have little effect in the F waves by agreeing with the OPE phase shifts, while the deltaful TPEs turn away from the OPE curve (NLO) considerably.
Figure 4. NN phase shifts as functions of CM momentum in various F waves with the deltaful TPEs. |
Figure 5. NN phase shifts as a function of CM momentum in the F waves with the deltaless TPEs. |
4. Summary and discussions
Although the delta isobar has been long speculated to play a role in chiral nuclear forces due to its proximity in mass to the nucleon, incorporating Δ into the hierarchy of chiral forces is plagued by a few issues. The dimensionally regularized deltaful TPE potentials, first derived in [2], have an isoscalar central component that appears too strong for a force considered two orders smaller than OPE. Later a different regularization scheme—the spectral function regularization— was advocated in [3], and a particular choice of the cut-off value $\tilde{{\rm{\Lambda }}}=700\,\mathrm{MeV}$ was made in order to agree with the NN phase shifts. At the beginning, we investigated the reasons behind the significant disparity between $\tilde{{\rm{\Lambda }}}=700$ MeV and ∞ in SFR and why DR potentials are so strongly attractive compared with SFR.
Assisted by the analytic expressions of the SFR potentials given in [3], we were able to analyze the counterterms subsumed in the deltaful TPEs and found that the q4 counterterm associated with the isoscalar central force ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ has a numerical factor that is surprisingly large, as shown in equation (19 ). As a counterterm, ${V}_{C}^{({q}^{4})}$ is smaller than the long-range part of ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ by ${\tilde{{\rm{\Lambda }}}}^{-1};$ therefore, we realized that ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ is more important than in Weinberg’s scheme. That is, even though it is a one-loop diagram, ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ is suppressed by ${Q}^{2}/(4\pi {f}_{\pi }^{2})$ relative to OPE, instead of ${Q}^{2}/{\left(4\pi {f}_{\pi }\right)}^{2}$.
Examining the counterterms helped numerically identify the dominant long-range forces, but the origin of this enhancement remains unclear. While we did not seek to answer that question in the current paper, we surmise that this is related to the enhancement of the triangle diagram noticed by [44, 45], shown in figure 6. If the ππNN vertex is momentum-dependent, a similar enhancement arises: in the denominator, there is a factor of π instead of π2. (We note that the ν = 0 ππNN vertex, i.e., the famed Weinberg–Tomozawa term, is energy-dependent.) The connection between the triangle diagram and figure 1(b) is that when the delta pole is picked up in figure 1(b), the resultant three-momentum integral resembles that of the nucleon-pole contribution of the triangle diagram.
Figure 6. The seagull diagram for the TPE potential with a ππNN vertex. For the explanation of other symbols, see figure 1. |
The promotion of ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ to N2LO in peripheral waves also means that one has to adjust the counting of ${V}_{C}^{({q}^{4})}$ by assigning it to N3LO, one order down from NDA. This modification of power counting affords us a handle to moderate the attraction of ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ before going up to N4LO.
The phase shifts up to N3LO showed in general a stronger attraction than in the absence of the delta isobar. While this is welcome in some channels (3D1, 3F4), the attraction seems excessive in others (3D2, 3F3). Going forward, there are several improvements we plan to implement. The value of the NΔ coupling hA and the nucleon-delta mass splitting Δ may be adjusted in conjunction with the analysis of πN scattering. To be consistent with the promotion of ${V}_{C-{\rm{\Delta }}}^{\mathrm{TPE}}$ on grounds of factors involving π, the triangle diagram with ν = 1 vertexes shown in figure 6 is to be promoted by one order as well.