Abstract We study perturbativeness of chiral nuclear forces in the 3P0 channel. In previous works, the focus has been on the one-pion exchange, and the applicable window of perturbative pion exchanges has been shown to span from the threshold to center-of-mass momentum k≃180 MeV. We will examine, instead, whether the cancellation of short- and long-range parts can sufficiently soften the 3P0 chiral force to make it more amenable to perturbation theory. The result is encouraging, as the combined 3P0 force is shown to be perturbative up to k≃280 MeV, covering many nuclear-structure calculations. Keywords：chiral nuclear force;perturbation theory;3P0 channel
PDF (421KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Rui Peng, Songlin Lyu, Bingwei Long. Perturbative chiral nucleon–nucleon potential for the 3P0 partial wave. Communications in Theoretical Physics, 2020, 72(9): 095301- doi:10.1088/1572-9494/aba251
1. Introduction
In the framework of chiral effective field theory (EFT), one-pion exchange (OPE) is the leading-order (LO) long-range part of two-body nuclear forces. Not only does it naturally give rise to tensor structure of nuclear forces, which is well known to play an important role in accounting for nuclear phenomena, but renormalization of OPE tells a lot about short-range nuclear forces [1, 2]. However, whether OPE is perturbative must first be decided and renormalization of OPE can be qualitatively different for perturbative and nonperturbative scenarios [3–13]. It may be desirable for some of the few-body or many-body methods to have perturbation theory as widely applicable as possible, so that computational model space is smaller at least for LO.
Suppression by centrifugal barriers has been the main mechanism to render OPE perturbative; therefore, power counting of chiral nuclear forces has been discussed by partial wave. It has been established that OPE is perturbative in 3P0 up to k≃180 MeV [14–16], where k is the center-of-mass momentum. (For such low momenta, one might as well resort to pionless theory.) On the other hand, the ${}^{3}{P}_{0}$ phase shifts are much smaller than those of ${}^{1}{S}_{0}$ and ${}^{3}{S}_{1}-{}^{3}{D}_{1}$, rising to the maximum of 10.8° around k≃160 MeV and vanishing around k≃295 MeV. One may be wondering why no perturbation theory would work for momenta higher than k≃180 MeV. In the present paper, we consider a novel perturbative scenario for 3P0: singular attractive OPE and the 3P0 short-range force, repulsive for soft cutoff values, cancel each other partially to bring about a smaller total force.
The singular attraction of OPE in some of the triplet channels, e.g. ${}^{3}{S}_{1}-{}^{3}{D}_{1}$, 3P0, and ${}^{3}{P}_{2}-{}^{3}{F}_{2}$, etc, does not stabilize NN system by itself, so a counterterm is needed at LO to make the problem well-defined. Systematic investigation of how long-range forces affect the role of short-range counterterms is often recast in the language of renormalization-group (RG) analysis. RG analysis studies the dependence of observables on the ultraviolet cutoff in the dynamic equation. High sensitivity to the cutoff value, if any, indicates the lack of our understanding of short-range physics; therefore, more low-energy constants in the effective Lagrangian must be called upon to parametrize short-range forces. We will not focus on the topic of renormalization in the paper, but the lessons learned so far will be applied to study perturbativeness of the 3P0 nuclear force.
By naive dimensional analysis (NDA), the 3P0 counterterm is at most next-to-next-to-leading order (N2LO), suppressed by ${ \mathcal O }({Q}^{2}/{M}_{\mathrm{hi}}^{2})$ compared to LO. On the ground of renormalization, however, it is promoted to LO [4] (for a more comprehensive review on renormalization of chiral nuclear forces, see [17]). For relatively small cutoff values, the 3P0 counterterm is repulsive [4, 14], so we are motivated to look into whether the attraction of OPE and the repulsion of the 3P0 counterterm can combine to form so small a sum that perturbation theory for 3P0 is viable for nuclear physics.
The paper is structured as follows. In section 2 we explain the related power counting schemes, followed by numerical results and comparisons to the empirical phase shifts in section 3. Finally, we summarize and discuss in section 4.
2. Power counting
This is probably a good place to formulate in power-counting language what we mean by ‘perturbation theory’. LO is always nonperturbative, i.e. the LO potential being iterated to all orders using the Lippmann–Schwinger equation:$ \begin{eqnarray}\begin{array}{rcl}{T}^{(0)} & = & {V}_{\mathrm{LO}}+{V}_{\mathrm{LO}}{G}_{0}{T}^{(0)}={V}_{\mathrm{LO}}\\ & & +\,{V}_{\mathrm{LO}}{G}_{0}{V}_{\mathrm{LO}}+{V}_{\mathrm{LO}}{\left({G}_{0}{V}_{\mathrm{LO}}\right)}^{2}+...,\end{array}\end{eqnarray}$where G0 is the free-particle propagator. More specifically, the integral equation for uncoupled partial waves to implement such iterations is given by$ \begin{eqnarray}\begin{array}{rcl}{T}^{(0)}(p^{\prime} ,p;k) & = & {V}_{\mathrm{LO}}^{{\rm{\Lambda }}}(p^{\prime} ,p)\\ & & +\,{\displaystyle \int }_{0}^{\infty }{\rm{d}}l\,{l}^{2}\,{V}_{\mathrm{LO}}^{{\rm{\Lambda }}}(p^{\prime} ,l;k)\displaystyle \frac{{T}^{(0)}(l,p;k)}{{k}^{2}-{l}^{2}+{\rm{i}}\epsilon },\end{array}\end{eqnarray}$where full off-shell form of partial-wave amplitude T is shown, p (p′) is incoming (outgoing) momentum, and k is related to the center-of-mass energy E by the usual nonrelativistic kinematics:$ \begin{eqnarray}{k}^{2}=2{m}_{N}E,\end{eqnarray}$where the nucleon mass mN=938.27 MeV. The amplitude is physical when momenta are on-shell $p^{\prime} =p=k$: $T(k,k;k)$. The superscript Λ is there to remind us that ${V}_{\mathrm{LO}}^{{\rm{\Lambda }}}$ has ultraviolet regularization as follows:$ \begin{eqnarray}{V}_{\mathrm{LO}}^{{\rm{\Lambda }}}(p^{\prime} ,p)=\exp \left(-\displaystyle \frac{p{{\prime} }^{4}}{{{\rm{\Lambda }}}^{4}}\right){V}_{\mathrm{LO}}(p^{\prime} ,p)\exp \left(-\displaystyle \frac{{p}^{4}}{{{\rm{\Lambda }}}^{4}}\right).\end{eqnarray}$Normalization of T and V can be inferred from the integral equation and it is consistent with the following relation between $T(k,k;k)$ and the ${}^{3}{P}_{0}$ phase shifts:$ \begin{eqnarray}\langle {}^{3}{P}_{0}| T(k)| {}^{3}{P}_{0}\rangle =-\displaystyle \frac{2}{\pi }\displaystyle \frac{{{\rm{e}}}^{{\rm{i}}\delta }\sin \delta }{k},\end{eqnarray}$where δ is the phase shift at k.
NLO and higher orders are perturbations on top of LO. Defining operator function,$ \begin{eqnarray}F(V)\equiv (1+{T}^{(0)}{G}_{0})V(1+{G}_{0}{T}^{(0)}),\end{eqnarray}$we can write subleading corrections to the amplitude as$ \begin{eqnarray}\,\begin{array}{c}\begin{array}{c}\begin{array}{l}{T}^{\left(1\right)}\,=\,F({V}_{{\rm{NLO}}}),\\ {T}^{\left(2\right)}\,=\,F({V}_{{{\rm{N}}}^{2}{\rm{LO}}})+F({V}_{{\rm{NLO}}}{G}_{0}{V}_{{\rm{NLO}}})\\ \,+\,F({V}_{{\rm{NLO}}}{G}_{0}{T}^{\left(0\right)}{G}_{0}{V}_{{\rm{NLO}}}),\\ {T}^{\left(3\right)}\,=\,F({V}_{{{\rm{N}}}^{3}{\rm{LO}}})\\ \,+\,F({V}_{{{\rm{N}}}^{2}{\rm{LO}}}{G}_{0}{V}_{{\rm{NLO}}})\\ \,+\,F({V}_{{\rm{NLO}}}{G}_{0}{V}_{{{\rm{N}}}^{2}{\rm{LO}}})+F({V}_{{\rm{NLO}}}{G}_{0}{V}_{{\rm{NLO}}}{G}_{0}{V}_{{\rm{NLO}}})\\ \,+\,F({V}_{{\rm{NLO}}}{G}_{0}{T}^{\left(0\right)}{G}_{0}{V}_{{\rm{NLO}}}{G}_{0}{V}_{{\rm{NLO}}})\\ \,+\,F({V}_{{\rm{NLO}}}{G}_{0}{V}_{{\rm{NLO}}}{G}_{0}{T}^{\left(0\right)}{G}_{0}{V}_{{\rm{NLO}}})\\ \,+\,F({V}_{{\rm{NLO}}}{G}_{0}{T}^{\left(0\right)}{G}_{0}{V}_{{\rm{NLO}}}{G}_{0}{T}^{\left(0\right)}{G}_{0}{V}_{{\rm{NLO}}})\\ \,+\,F({V}_{{{\rm{N}}}^{2}{\rm{LO}}}{G}_{0}{T}^{\left(0\right)}{G}_{0}{V}_{{\rm{NLO}}})\\ \,+\,F({V}_{{\rm{NLO}}}{G}_{0}{T}^{\left(0\right)}{G}_{0}{V}_{{{\rm{N}}}^{2}{\rm{LO}}})\\ \,\cdots \end{array}\end{array}.\end{array}\end{eqnarray}$What we mean by ‘perturbation theory’ is when ${V}_{\mathrm{LO}}$ vanishes, therefore, the amplitude consists of only a finite orders of the Born expansion terms.
The OPE potential [18–20] is given by$ \begin{eqnarray}{V}_{1\pi }(\vec{q})=-\displaystyle \frac{{g}_{A}^{2}}{4{f}_{\pi }^{2}}{{\boldsymbol{\tau }}}_{1}\,\cdot \,{{\boldsymbol{\tau }}}_{2}\displaystyle \frac{{\vec{\sigma }}_{1}\cdot \vec{q}\,{\vec{\sigma }}_{2}\cdot \vec{q}}{{\vec{q}}^{2}+{m}_{\pi }^{2}},\end{eqnarray}$where the pion mass mπ=139.57 MeV, pion decay constant fπ=92.2 MeV and the axial coupling gA=1.289. This potential is singular and attractive in 3P0. When treated nonperturbatively, a counterterm ${V}_{\mathrm{ct}}^{(0)}$, playing the role of short-range force, is needed at LO to renormalize OPE [4]:$ \begin{eqnarray}{V}_{\mathrm{LO}}={V}_{1\pi }+{V}_{\mathrm{ct}}^{(0)}.\end{eqnarray}$The partial-wave decomposition of OPE onto 3P0 is numerically performed with the following integral [21]:$ \begin{eqnarray}\begin{array}{l}{\langle }^{3}{P}_{0};p| {V}_{1\pi }{| }^{3}{P}_{0};k\rangle =-\displaystyle \frac{{m}_{N}}{4{\pi }^{2}}\displaystyle \frac{{g}_{A}^{2}}{4{f}_{\pi }^{2}}\\ \quad \times \,{\displaystyle \int }_{-1}^{1}{\rm{d}}z\left[2{pk}-z({p}^{2}+{k}^{2})\right]\displaystyle \frac{1}{\vec{q}{}^{2}+{m}_{\pi }^{2}},\end{array}\end{eqnarray}$where $\vec{q}{}^{2}\equiv {p}^{2}+{k}^{2}-2{pkz}$.
Together with consideration for multiple-pion exchanges [5, 9], nonperturbative OPE induces more ${}^{3}{P}_{0}$ LECs at given order than NDA:$ \begin{eqnarray}\begin{array}{rcl}\mathrm{LO}:\ {V}_{\mathrm{LO}} & = & {V}_{1\pi }+{C}_{0}{pp}^{\prime} ,\\ \mathrm{NLO}:\ {V}_{\mathrm{NLO}} & = & 0,\\ {{\rm{N}}}^{2}\mathrm{LO}:\ {V}_{{{\rm{N}}}^{2}\mathrm{LO}} & = & {V}_{2\pi }^{(0)}+{C}_{1}{pp}^{\prime} +{D}_{0}({p}^{2}+p{{\prime} }^{2}){pp}^{\prime} ,\\ {{\rm{N}}}^{3}\mathrm{LO}:\ {V}_{{{\rm{N}}}^{3}\mathrm{LO}} & = & {V}_{2\pi }^{(1)}+{C}_{2}{pp}^{\prime} +{D}_{1}({p}^{2}+p{{\prime} }^{2}){pp}^{\prime} ,\end{array}\end{eqnarray}$where ${V}_{2\pi }^{(0,1)}$ are leading and subleading two-pion exchanges (TPEs). Matrix elements of ${V}_{\mathrm{ct}}$ in ${}^{3}{P}_{0}$ are defined as$ \begin{eqnarray}\begin{array}{l}\langle p^{\prime} | {V}_{\mathrm{ct}}| p\rangle ={Cpp}^{\prime} +D({p}^{2}+p{{\prime} }^{2}){pp}^{\prime} \\ \,+\,{{Ep}}^{2}p{{\prime} }^{2}{pp}^{\prime} \,+\cdots ,\end{array}\end{eqnarray}$with ${Cpp}^{\prime} $ (and similarly other counterterms) being split into formally different pieces at each order [9]$ \begin{eqnarray}{Cpp}^{\prime} =({C}_{0}+{C}_{1}+{C}_{2}+\,...){pp}^{\prime} ,\end{eqnarray}$where C1 and C2 are corrections to C0 [3, 22].
We are interested here in scenarios within chiral EFT that supports perturbation theory in ${}^{3}{P}_{0}$. The first scenario is to treat OPE in perturbation theory in ${}^{3}{P}_{0}$, arguing that the centrifugal barrier suppresses OPE so much to the point where ${V}_{1\pi }$ must be demoted to NLO while LO vanishes [15]:$ \begin{eqnarray}\begin{array}{rcl}\mathrm{LO}:\,{V}_{\mathrm{LO}} & = & 0,\\ \mathrm{NLO}:\,{V}_{\mathrm{NLO}} & = & {V}_{1\pi },\\ {{\rm{N}}}^{2}\mathrm{LO}:\,{V}_{{{\rm{N}}}^{2}\mathrm{LO}} & = & {C}_{0}{pp}^{\prime} ,\\ {{\rm{N}}}^{3}\mathrm{LO}:\,{V}_{{{\rm{N}}}^{3}\mathrm{LO}} & = & {V}_{2\pi }^{(0)}+{C}_{1}{pp}^{\prime} ,\\ {{\rm{N}}}^{4}\mathrm{LO}:\,{V}_{{{\rm{N}}}^{4}\mathrm{LO}} & = & {V}_{2\pi }^{(1)}+{C}_{2}{pp}^{\prime} +{D}_{0}({p}^{2}+p{{\prime} }^{2}){pp}^{\prime} .\end{array}\end{eqnarray}$In [15], the above scheme was found to describe the phase shifts in 3P0 well only below k≃180 MeV, which is not significantly better than pionless EFT.
The second proposition, also the main message of the paper, is to examine the cancellation of the long and short-range forces in ${}^{3}{P}_{0}$, and explore the possibility that it supports perturbation theory of some kind. This is motivated by the observation that, at least for cutoff values Λ not too high, the counterterm is repulsive. Shown in figure 1 are the on-shell matrix elements of potentials ${V}_{1\pi }(p,p^{\prime} )$, ${C}_{0}{pp}^{\prime} $ and ${V}_{1\pi }(p,p^{\prime} )$ + ${C}_{0}{pp}^{\prime} $ as functions of k, where $p=p^{\prime} $ for ${\rm{\Lambda }}=400\,\,\mathrm{MeV}$, and the value of C0 is obtained with the nonperturbative OPE scheme(11). The cancellation between ${V}_{1\pi }(p,p^{\prime} )$ and ${C}_{0}{pp}^{\prime} $ is quite obvious: the maximum absolute value of ${V}_{1\pi }(p,p^{\prime} )$ + ${C}_{0}{pp}^{\prime} $ is half of that of ${V}_{1\pi }(p,p^{\prime} )$. Although this is an encouraging sign, we need to go beyond the tree level and demonstrate the convergence of a perturbation theory in powers of ${V}_{1\pi }(p,p^{\prime} )$ + ${C}_{0}{pp}^{\prime} $. To that end, we need a specific power counting scheme to account for the accidental smallness of the ${}^{3}{P}_{0}$ force before we can carry out actual calculations.
Figure 1.
New window|Download| PPT slide Figure 1.On-shell matrix elements of ${}^{3}{P}_{0}$ potentials as functions of the center-of-mass momentum. The red dashed, blue dashed–dotted and black solid correspond respectively to ${V}_{1\pi }(k,k)$, ${C}_{0}{k}^{2}$ and ${V}_{1\pi }(k,k)+{C}_{0}{k}^{2}$. C0 is fitted in the nonperturbative OPE scheme(11) for ${\rm{\Lambda }}=400\,\,\mathrm{MeV}$.
Unlike perturbative OPE scheme (14), OPE is not considered to be suppressed by the centrifugal barrier for power-counting purpose, nor is the ${}^{3}{P}_{0}$ counterterm. In other words, ${V}_{1\pi }(p,p^{\prime} )$ and ${C}_{0}{pp}^{\prime} $ are still formally considered LO when separated, only the sum of the terms is considered to be NLO. The consequence of this thinking is that TPEs are not to be counted smaller than in NDA. As for counterterms with higher powers of momenta, we will let renormalization of multiple iteration of ${V}_{1\pi }+{C}_{0}{pp}^{\prime} $ decide. That is, we add whatever counterterms necessary to remove divergence in the Born-expansion series of ${V}_{1\pi }+{C}_{0}{pp}^{\prime} $. The divergences of the terms in the expansion can be straightforwardly estimated by powers of large loop momenta appearing in the integration. Up to N3LO, the aforementioned guideline leads to the following power counting:$ \begin{eqnarray}\begin{array}{rcl}\mathrm{LO}:\,{V}_{\mathrm{LO}} & = & 0,\\ \,\mathrm{NLO}:\,{V}_{\mathrm{NLO}} & = & {V}_{1\pi }+{C}_{0}{pp}^{\prime} ,\\ {{\rm{N}}}^{2}\mathrm{LO}:\,{V}_{{{\rm{N}}}^{2}\mathrm{LO}} & = & {V}_{2\pi }^{(0)}+{C}_{1}{pp}^{\prime} +{D}_{0}({p}^{2}+p{{\prime} }^{2}){pp}^{\prime} ,\\ {{\rm{N}}}^{3}\mathrm{LO}:\,{V}_{{{\rm{N}}}^{3}\mathrm{LO}} & = & {V}_{2\pi }^{(1)}+{C}_{2}{pp}^{\prime} +\,{D}_{1}({p}^{2}+p{{\prime} }^{2}){pp}^{\prime} \\ & & +\,{E}_{0}{p}^{2}p{{\prime} }^{2}{pp}^{\prime} ,\end{array}\end{eqnarray}$here, renormalizing multiple iterations of ${V}_{1\pi }+{C}_{0}{pp}^{\prime} $, in fact, brings about even more ${}^{3}{P}_{0}$ counterterms than the nonperturbative OPE (11), and scaling of contact operators do not agree with NDA. This counting is somewhat a middle ground between the perturbative OPE (14) and the nonperturbative OPE schemes.
3. Results
We show and discuss in this section the numerical results according to power counting (15). The expressions of TPEs are taken from [21]. The couplings of $\nu =1$$\pi \pi {NN}$ seagull vertexes called ci's determine the size of subleading TPEs ${V}_{2\pi }^{(1)}$, where ν is the chiral index [23, 24]. The value of ci are: ${c}_{1}=-0.74$, ${c}_{3}=-3.61$, c4=2.44, all in GeV−1, taken from [25]. They were extracted from an analysis based on the Roy–Steiner equation of $\pi N$ scattering. We use the NLO HB-NN Q2 values of ci from table 1 in [25].
We fit the expanded amplitudes to the empirical phase shifts provided by the SAID program at the George Washington University [26, 27]. The results are shown in figure 2. For NLO, phase shifts near $k={m}_{\pi }$ are taken as inputs in the fitting. For N2LOand N3LO, phase shifts from threshold up to $k\simeq 300$ MeV are used. For comparison, the nonperturbative scheme (11) is also applied and shown, except for ${\rm{\Lambda }}=2400\,\,\mathrm{MeV}$ where the algorithm we use fails to carry out higher-order calculations for the nonperturbative OPE scheme(11). The values of the fitted counterterms are tabulated in tables 1 and 2, where they are multiplied by powers of Λ to be dimensionless.
Figure 2.
New window|Download| PPT slide Figure 2.The solid circles are the empirical phase shifts from the SAID program [26, 27]. The black and red lines correspond to the nonperturbative and perturbative cases, respectively. The dashed, dashed–dotted and solid correspond respectively to LO, N2LOand N3LOfor the nonperturbative case, NLO, N2LOand N3LOfor the perturbative case. The values of cutoff Λ (MeV) are marked at top left of each figure.
Table 1. Table 1.The values taken by the counterterms in the nonperturbative OPE scheme(11).
An important feature independent of the cutoff value is that the mixed perturbative scheme (15) converges near $k\simeq 280\,\,\mathrm{MeV}$, whereas the pure OPE perturbative scheme (8) can describe the ${}^{3}{P}_{0}$ phase shifts only up to k≃180 MeV [15]. The improved convergence confirms our speculation: the cancellation between OPE and the ${}^{3}{P}_{0}$ survives in quantum fluctuations manifested by higher-order terms in the Born expansion(7). Therefore, the perturbation theory based on$ \begin{eqnarray}{V}_{\mathrm{NLO}}={V}_{1\pi }+{C}_{0}{pp}^{\prime} \end{eqnarray}$furnishes a more convergent EFT expansion than that based on$ \begin{eqnarray}{V}_{\mathrm{NLO}}={V}_{1\pi }.\end{eqnarray}$
As successful as it is, the perturbation theory for ${}^{3}{P}_{0}$ still has a smaller convergence radius than the nonperturbative scheme(11) does, which has even fewer short-range parameters. Since OPE in ${}^{3}{P}_{0}$ will eventually become nonperturbative for sufficiently high momenta, the nonperturbative scenario is conceptually the underlying theory for the perturbative one. From this perspective, it would be less surprising that the nonperturbative theory ‘knows’ more physics than the perturbative one, thus needs fewer parameters.
One may ask whether it is TPEs or iterations of ${V}_{1\pi }+{C}_{0}{pp}^{\prime} $ that drive the expansion scheme (15) to break down near $k\simeq 280\,\,\mathrm{MeV}$. To answer this question, we ‘turn off’ TPEs and investigate what happens:$ \begin{eqnarray}\begin{array}{rcl}\mathrm{LO}:\,{V}_{\mathrm{LO}} & = & 0,\\ \,\mathrm{NLO}:\,{V}_{\mathrm{NLO}} & = & {V}_{1\pi }+{C}_{0}{pp}^{\prime} ,\\ {{\rm{N}}}^{2}\mathrm{LO}:\,{V}_{{{\rm{N}}}^{2}\mathrm{LO}} & = & {C}_{1}{pp}^{\prime} +{D}_{0}({p}^{2}+p{{\prime} }^{2}){pp}^{\prime} ,\\ {{\rm{N}}}^{3}\mathrm{LO}:\,{V}_{{{\rm{N}}}^{3}\mathrm{LO}} & = & {C}_{2}{pp}^{\prime} +{D}_{1}({p}^{2}+p{{\prime} }^{2}){pp}^{\prime} \\ & & +\,{E}_{0}{p}^{2}p{{\prime} }^{2}{pp}^{\prime} .\end{array}\end{eqnarray}$The results are shown in figure 3. The fact that the phase shifts with TPEs and without TPEs sit almost on top of each other tells us that the Born expansion of ${V}_{1\pi }+{C}_{0}{pp}^{\prime} $, although softer than ${V}_{1\pi }$ alone, still becomes too strong for momenta where TPEs are expected to make impacts.
Figure 3.
New window|Download| PPT slide Figure 3.The solid circles are the empirical phase shifts from the SAID program [26, 27]. The black and red lines correspond to the results with and without TPEs respectively. The dashed and solid correspond respectively to N2LOand N3LO. The values of cutoff Λ (MeV) was marked at top left of each figure.
4. Summary and discussion
Given the relatively small values of the ${}^{3}{P}_{0}$ phase shifts, we were motivated to study perturbativeness of chiral potentials in ${}^{3}{P}_{0}$ by examining the cancellation between OPE and the short-range forces represented by the ${}^{3}{P}_{0}$ counterterms within the framework of chiral EFT. The numerical calculation showed that the perturbative scheme (15) proposed in the paper can describe the empirical phase shifts up to $k\simeq 280\,\,\mathrm{MeV}$, where k is the center-of-mass momentum.
As shown in [15, 16], OPE alone does not support a perturbation theory that would be as successful as the proposed perturbative scheme (15). So our exploration explains satisfactorily from the viewpoint of chiral EFT why ${}^{3}{P}_{0}$ phase shifts appear so small while OPE by itself is quite strong in ${}^{3}{P}_{0}$.
Since there is a shallow bound state in ${}^{3}{S}_{1}-{}^{3}{D}_{1}$, so the nuclear force must not be perturbative in ${}^{3}{S}_{1}-{}^{3}{D}_{1}$. But what prevents the same rationale employed in the paper to argue for perturbative ${}^{3}{P}_{0}$ being applied to ${}^{3}{S}_{1}-{}^{3}{D}_{1}$? After all, OPE is singularly attractive too in ${}^{3}{S}_{1}-{}^{3}{D}_{1}$ and the ${}^{3}{S}_{1}$ counterterm [4]. There is a fundamental difference between partial-wave projections of OPE in ${}^{3}{P}_{0}$ and ${}^{3}{S}_{1}-{}^{3}{D}_{1}$ that is often overlooked. The singular attraction of OPE acts in the orbital mixing, ${}^{3}{S}_{1}\to {}^{3}{D}_{1}$, but the counterterm is part of ${}^{3}{S}_{1}\to {}^{3}{S}_{1};$ therefore, they can not cancel each other at the tree level.
Acknowledgments
The work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11775148 and 11735003.
,College of Physics, 
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