# Radiative proton capture on $\nuclide[15]{N}$ within effective field theory

Sangyeong Son<EMAIL_ADDRESS>Department of Physics, Kyungpook National
University, Daegu 41566, Korea  Shung-Ichi Ando<EMAIL_ADDRESS>Department of Display and Semiconductor Engineering, Sunmoon University, Asan,
Chungnam 31460, Korea  Yongseok Oh<EMAIL_ADDRESS>Department of Physics,
Kyungpook National University, Daegu 41566, Korea  Asia Pacific Center for
Theoretical Physics, Pohang, Gyeongbuk 37673, Korea

###### Abstract

The astrophysical $S$ factor for the radiative proton capture process on the
[15]N nucleus, i.e., $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$, at stellar
energies are studied within the framework of the cluster effective field
theory. The thermonuclear $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ reaction
links the type-I to type-II cycles of the carbon-nitrogen-oxygen cycle and
affects the abundances of elements in the univere. For investigating this
reaction in the effective field theory formalism, we first construct an
effective Lagrangian that is appropriate for this reaction at low-energies.
Since the intermediate excited states of the [16]O nucleus have a crucial role
in this reaction, we include these resonances in the formalism. The
corresponding radiative capture amplitudes and cross section are calculated,
which lead to the astrophysical $S$ factor. The low energy constants
introduced in the effective Lagrangian are determined by fitting the
theoretical results to the observed $S$ factors in the range of
$130~{}\mathrm{keV}<E_{p}<2500~{}\mathrm{keV}$ using three different
experimental data sets. Considering the recent data sets, we obtain
$S(0)=29.8\mbox{--}34.1~{}\mathrm{keV\ b}$, which is in a good agreement with
the estimates from $R$-matrix approaches in the literature. The values of $S$
at the Gamow energy are found to be larger than $S(0)$ values by about 10%.

##  I Introduction

Thermonuclear fusion is the energy source of stars. Among thermonuclear fusion
processes, the proton-proton chain and the carbon-nitrogen-oxygen (CNO) cycle
are two major processes [1, 2]. The neutrinos produced in the Sun by these
processes have been observed in underground experiments [3, 4, 5, 6], which
brings information on the solar neutrinos and confirmed the thermonuclear
reactions of the proton-proton chain and the CNO cycle inside the Sun. In the
CNO cycle, the proton capture reaction on [15]N allows two possible channels,
namely, $\nuclide[15]{N}(p,\alpha)\nuclide[12]{C}$ and
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$. The former is responsible for the
type-I cycle (CN cycle) and the latter for the type-II cycle (NO cycle). At
low energies the CN cycle is dominant but at higher temperatures the NO cycle
becomes active. Since the CN and NO cycles intersect at the nucleus [15]N, the
relative strength of the two reactions,
$\nuclide[15]{N}(p,\alpha)\nuclide[12]{C}$ and
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$, determines the probability of the
path in the CNO cycle. It turns out that the rate for the former reaction is
much larger than for the latter, which means that the CN cycle is more
probable than the NO cycle. Although its contribution to the energy production
in stars would be small, the $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$
reaction is crucial to understand nucleosynthesis and the observed oxygen
abundances. Therefore, the evaluation of its cross sections at stellar energy
scales is strongly required for resolving such issues [7, 8].

Because of the Coulomb barrier, the nuclear reaction cross section $\sigma(E)$
at the center-of-mass (c.m.) energy $E$ is parameterized by the astrophysical
$S$ factor defined as

$S(E)=E\sigma(E)\exp(2\pi\eta),$ (1)

where $\eta$ is the dimensionless Sommerfeld parameter,

$\eta=Z_{A}Z_{B}\mu\alpha_{\rm em}/p$ (2)

with the fine structure constant $\alpha_{\rm em}=e^{2}/(4\pi)$. Here, $\mu$
is the reduced mass of the system, $\mu\equiv m_{A}m_{B}/(m_{A}+m_{B})$, where
$m_{A}$ and $m_{B}$ are the masses of the initial state nuclei whose charges
are $Z_{A}$ and $Z_{B}$, respectively. The magnitude of the relative spatial
momentum between the two nuclei in the c.m. frame is represented by $p$.

Thus the astrophysical $S$ factor is the main characteristic of any
thermonuclear reaction at low energies. Determination of its value by
experiments is, however, highly nontrivial as the most experiments are carried
out at the energies above 100 keV, while the realistic energy scale for
astrophysical environments is about 0.1–100 keV. Thus, we need the value of
$S(0)$ in practical calculations as $S(E)$ is a very slowly varying function
of energy at low energy region.111The cold and hot CNO burning cycles have the
Gamow peak energy regions at $E_{G}\simeq 26$ keV and $150-200~{}\mathrm{keV}$
for the corresponding core temperatures at $T_{c}\simeq 1.5\times 10^{7}$ K
and $\mbox{(2-3)}\times 10^{8}$ K, respectively [7, 9].  Therefore, most
realistic method for obtaining the astrophysical $S$ factor at zero energy
would be the extrapolation of experimentally determined $S(E)$ to the lower
energy range. However, because of the difficulties of experiments near
threshold, larger error bars at low energy region are inevitable and this
causes the uncertainties of $S(0)$. The extrapolation would, of course, depend
on the theoretical models adopted for the description of the reaction.

Despite the importance of the $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$
reaction, experimental measurements of its cross sections at low energies are
rare and only two measurements were reported until 2009 [10, 11].222The
experimental data reported in Ref. [12] are for $E_{p}\geq 860$ keV and do not
constrain the value of $S(0)$, where $E_{p}$ is the kinetic energy of the
incident proton in the laboratory frame. We also note that the data of Ref.
[13] covered the region of $E_{p}>250$ keV. The first measurement was made
about 60 years ago [10], which was then followed by the work of Ref. [11] in
mid 1970s. The former experiment reported $S(0)\simeq 32$ keV b, while the
latter estimated $S(0)=64\pm 6$ keV b. This shows that the value of the latter
experiment is about twice that of the former one. However, these experiments
verified that the reaction is dominated by the first two interfering
$J^{\pi}=1^{-}$ resonances at $E_{R}=312$ and $964$ keV, respectively, where
$E_{R}$ is the resonance energy in the center-of-mass frame [14].333The
corresponding excitation energies are 12.45 MeV and 13.09 MeV, respectively.
In the laboratory frame, these values correspond to $E_{p}=338$ keV and $1028$
keV, respectively. These two data sets are used for the analyses in Refs. [15,
14, 16]. In particular, it was pointed out in Ref. [14] that the contribution
from the direct capture process was overestimated in Ref. [11].

In the last decade, more measurements were performed at the Laboratory for
Underground Nuclear Astrophysics (LUNA) at Gran Sasso underground laboratory
(LNGS) and at the Notre Dame Nuclear Science Laboratory [17, 18, 19]. These
efforts are summarized in Refs. [20, 21] and the data are collected, for
example, in Refs. [22, 23] which are used for updating the estimation of the
$S$ factors in Refs. [24, 25, 26, 27].444The experimental data of Ref. [18]
are compiled in Ref. [26]. In the present work, we use the experimental data
compiled in Experimental Nuclear Reaction Data (EXFOR) [23]. The newly
estimated values of $S(0)$ based on these data sets, called the post-NACRE
data, are as low as $33.1$ keV b [24] and as high as $45^{+9}_{-7}$ keV b
[22].

Figure 1: Collected experimental data for (a) the cross sections and (b) the
astrophysical $S$ factors $S(E)$ of the
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ reaction. The data from Refs. [10,
11, 17, 18, 19, 25] are compiled in Ref. [23].

Shown in Fig. 1 are currently available experimental data of
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ compiled in Ref. [23]. Figure 1(a)
shows the total cross section data, while Fig. 1(b) presents the derived
astrophysical $S$ factors. They are given as functions of $E_{\rm cm}$, the
kinetic energy of the system in the center-of-mass frame. These figures
explicitly show that the reaction is dominated by the two broad resonances
mentioned previously. Except the earlier data of Refs. [10, 11], the reported
data show a good agreement. The cross section data of Ref. [10] indicate a
small structure at vey low energies and it is further exaggerated in the
astrophysical $S$ factor as shown by black dots in Fig. 1(b). It is also
evident that the data of Ref. [11] have larger error bars compared with the
data of the next generation experiments. Furthermore, the data from Ref. [19,
17] do not cover the second resonance region. Therefore, in the present work,
we focus on the data covering the both resonance regions, which include the
data of Ref. [11] and the post-NACRE data of Refs. [18, 25], and apply our
calculations to these three experimental data sets.

In the present work, we apply the effective field theory (EFT) formalism to
investigate the [15]N($p$,$\gamma$)[16]O reaction. EFTs allow for a systematic
calculation by introducing a scale $\Lambda_{H}$ which separates relevant
degrees of freedom at low energies from irrelevant degrees of freedom at high
energies. Then the effective Lagrangian is constructed by the expansion with
the number of derivatives, and the reaction amplitudes are calculated in
powers of $Q/\Lambda_{H}$ where $Q$ is a typical momentum scale of the
reaction in question. The irrelevant high energy degrees of freedom are
integrated out and the coefficients in the effective Lagrangian, called low-
energy constants (LECs), are determined by fitting the experimental data [28].
EFT methods have been applied to the studies of various reactions including
thermonuclear reactions such as $\alpha$ capture on [12]C [29], radiative
proton capture on [12]C [30], and proton and neutron capture on light nuclei
[31, 32, 33, 34, 35]. It was also applied to the reactions of [15]C [36],
$\alpha$-[12]C elastic scattering [37, 38, 39], and $\beta$ delayed $\alpha$
emission from [16]N [40], and so on. Reviews on these topics can be found, for
example, in Refs. [41, 42, 43, 44].

The construction of EFT for the $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$
reaction involves three open channels, namely, $p$-[15]N, $\alpha$-[12]C, and
$\alpha$-$\nuclide[12]{C}^{*}$ where $\nuclide[12]{C}^{*}$ denotes the first
excited $2^{+}$ state of [12]C. In the present study, by focusing on the
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ reaction data, we perform the
single channel calculations leaving the coupled-channel calculations to a
future work. The energy range of the data of our interests covers up to
$E\simeq 2$ MeV, which includes the aforementioned two $s$-wave resonant
states of [16]O. The inclusion of a resonant state in EFT was investigated,
for example, by Gelman [45] and by Habashi, Fleming, and van Kolck [46], and,
following their method, we here employ the effective Lagrangian for the
$p$-[15]N system with the two resonant states of [16]O.

We choose the breakup energy of [16]O into the $n$-[15]O channel, $\Delta
E=3.54$ MeV, as the large energy scale, which gives
$\Lambda_{H}=\sqrt{2\mu\Delta E}=80$ MeV, where $\mu$ is the reduced mass of
the system. On the other hand, the resonant energies, $E_{1}=0.312$ MeV and
$E_{2}=0.964$ MeV, are chosen as the typical energy scales of the theory
leading to $Q_{1}=\sqrt{2\mu E_{1}}=24$ MeV and $Q_{2}=\sqrt{2\mu E_{2}}=41$
MeV. Therefore, our expansion parameters are $Q_{1}/\Lambda_{H}\simeq 0.3$ and
$Q_{2}/\Lambda_{H}\simeq 0.5$, and the terms in the amplitudes are expanded in
powers of $(Q_{1,2}/\Lambda)^{2n}$ as the effective range expansion (ERE). We
include the terms up to the next-to-leading order for the first resonant state
and up to the fourth order terms for the second so that the theoretical
uncertainties of the present calculations are estimated as
$(Q_{1}/\Lambda_{H})^{2}\simeq 0.09$ and $(Q_{2}/\Lambda_{H})^{6}\simeq
0.016$.

Our preliminary results in this approach were reported in Ref. [47] where only
the first resonance was considered with the experimental data of Ref. [19].
The estimated astrophysical $S$ factor is $S(0)=30.4$ keV b that is close to
the lower limit of $S(0)$ obtained in the recent analyses of Ref. [24]. In the
present work, we perform more complete analysis working with the data covering
the energy regions of both resonances which were reported in Refs. [11, 18,
25].

This paper is organized as follows. In the next section, we develop the
cluster EFT formalism for the reaction of
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ by introducing the effective
Lagrangian for this reaction. In Sec. III, the propagator of the di-field,
which is introduced for the description of resonances, is discussed. Then the
capture amplitudes are derived in Sec. IV, and Sec. V presents our numerical
results which are compared with the estimates in the literature. Section VI
summarizes this work and the derivation of the loop integral formulas is given
in Appendix.

##  II Effective Lagrangian

The cluster EFT, which is applied to the investigation of nuclear reactions at
stellar environments, is similar to the pionless EFT [48, 49] by adopting
contact couplings among the participating particles. In the present approach,
we treat the nuclei involved in the the reaction of
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ as point-like particles, and
describe resonance states $\nuclide[16]{O}^{*}$ as bound systems of the proton
and the $\nuclide[15]{N}$ nucleus. The high energy scale is determined by the
breakup energy of [16]O and the resonant energies are used to estimate the low
energy scales.555The momentum scale corresponding to the radius of
$\nuclide[16]{O}$ is about 70 MeV which is larger than our typical scales and
is comparable to the large scale. The finite-range effects would be
investigated by the expansion in powers of spatial derivatives.

Since we have two resonant $\nuclide[16]{O}^{*}$, we construct the effective
Lagrangian for this reaction as

$\displaystyle\mathscr{L}$ $\displaystyle=$
$\displaystyle\psi_{p}^{\dagger}\left\\{iv\cdot
D+\frac{1}{2M_{p}}\left[(v\cdot
D)^{2}-D^{2}\right]\right\\}\psi_{p}+\psi_{N}^{\dagger}\left\\{iv\cdot
D+\frac{1}{2M_{N}}\left[(v\cdot D)^{2}-D^{2}\right]\right\\}\psi_{N}$ (3)
$\displaystyle\mbox{}+\sum_{n=0}^{n_{\rm
max}}C_{n}^{(I=0)}d_{i}^{(I=0)\dagger}\left\\{iv\cdot
D+\frac{1}{2(M_{p}+M_{N})}\left[(v\cdot
D)^{2}-D^{2}\right]\right\\}^{n}d_{i}^{(I=0)}$
$\displaystyle\mbox{}+\sum_{m=0}^{m_{\rm
max}}C_{m}^{(I=1)}d_{ia}^{(I=1)\dagger}\left\\{iv\cdot
D+\frac{1}{2(M_{p}+M_{N})}\left[(v\cdot
D)^{2}-D^{2}\right]\right\\}^{m}d_{ia}^{(I=1)}$
$\displaystyle\mbox{}-y_{t}^{(I=0)}\biggl{\\{}d_{i}^{(I=0)\dagger}\biggl{(}\psi_{p}^{T}P^{(^{3}S_{1})}_{i}\psi_{N}\biggr{)}+\mbox{H.c.}\biggr{\\}}-y_{t}^{(I=1)}\biggl{\\{}d_{ia}^{(I=1)\dagger}\biggl{(}\psi_{p}^{T}P^{(^{3}S_{1})}_{ia}\psi_{N}\biggr{)}+\mbox{H.c.}\biggr{\\}}$
$\displaystyle\mbox{}-y_{s}\biggl{\\{}\phi_{O}^{\dagger}\biggl{(}\psi_{p}^{T}P^{(^{3}P_{0})}\psi_{N}\biggr{)}+\mbox{H.c.}\biggr{\\}}-y_{t}^{(I=0)}h^{(I=0)}\biggl{\\{}\phi_{O}^{\dagger}(-iD_{i})d_{i}^{(I=0)}+\mbox{H.c.}\biggr{\\}}$
$\displaystyle\mbox{}-y_{t}^{(I=1)}h^{(I=1)}\biggl{\\{}\phi_{O}^{\dagger}(-iD_{i})d_{ia}^{(I=1)}\delta_{a3}+\mbox{H.c.}\biggr{\\}},$

where $\psi_{p}$, $\psi_{N}$, and $\phi_{O}$ are the fields of the proton,
$\nuclide[15]{N}$, and the ground state of $\nuclide[16]{O}$, respectively,
and the masses of the proton and [15]N are denoted by $M_{p}$ and $M_{N}$,
respectively. The interactions with higher derivatives are suppressed in the
effective Lagrangian of Eq. (3). The di-fields $d^{(I=0)}$ and $d^{(I=1)}$ are
introduced as auxiliary fields to describe the first and second resonance
states of $\nuclide[16]{O}$, respectively, where $I$ in the superscript
indicates its isospin quantum number. The coupling constants $y_{s}$,
$y_{t}^{(I)}$, and $h^{(I)}$ are the LECs which determine the strength of the
contact interactions among the proton, $\nuclide[15]{N}$, and
$\nuclide[16]{O}$ nucleus.

Figure 2: Fully-dressed propagator of the di-field. In the right hand side,
the double line stands for the bare propagator and the shaded region
represents the Coulomb interaction between two nuclei.

The four-velocity vector $v^{\mu}$ is chosen to be $v^{\mu}=(1,\bm{0})$, and
$D_{\mu}=\partial_{\mu}+ie\hat{Q}A_{\mu}$ is the covariant derivative where
$e$, $\hat{Q}$, and $A_{\mu}$ are the elementary charge, charge operator, and
the photon field, respectively. The projection operators in the interaction
terms are defined as

$\displaystyle P^{(^{3}S_{1})}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sigma_{2}\sigma_{i}\tau_{2},\qquad
P^{(^{3}S_{1})}_{ia}=\frac{1}{2}\sigma_{2}\sigma_{i}\tau_{2}\tau_{a},$ (4)
$\displaystyle P^{(^{3}P_{0})}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\tau_{2}\sigma_{2}\vec{\sigma}\cdot\frac{(-i\overrightarrow{D})}{M_{N}}-\frac{(-i\overleftarrow{D})}{M_{p}}\cdot\tau_{2}\sigma_{2}\vec{\sigma}\right),$
(5)

where the superscript in the projection operator stands for the orbital
angular momentum of the interaction between the proton and the
$\nuclide[15]{N}$ nucleus. Here, $\sigma_{i}$ are the Pauli spin matrices and
$\tau_{i}$ are the Pauli isospin matrices. The integers $n_{\rm max}$ and
$m_{\rm max}$ are the numbers of the considered effective range parameters of
the effective range expansion (ERE) which determine the coefficients
$C^{(I=0)}_{n}$ and $C^{(I=1)}_{m}$ for the iso-singlet and iso-triplet di-
fields, respectively. For the two resonances these integers are chosen by
$n_{\rm max}=1$ and $m_{\rm max}=3$ corresponding to the terms up to the
second and the fourth order.666As we mentioned, the orders associated with the
two resonances differ. In order to further exploit the uncertainty control, we
have increased the order with the lower resonance to $(Q_{1}/\Lambda_{H})^{4}$
so that the theoretical uncertainties from the two resonances are similar,
namely, 0.008 and 0.016, respectively. However, we could not find any
improvement with this choice and we have a feature of over-fitting. Through
these trials, we found that ($n_{\rm max}=1$, $m_{\rm max}=3$) is the optimal
choice.

##  III Di-field propagator

In the present work, we depict the intermediate resonance states by
considering the fully-dressed propagator of di-field $d^{(I)}$. As shown in
Fig. 2, the fully-dressed propagator is expressed as an infinite series of the
bare propagator $D_{0}$ and the self-energy $\Sigma$ as

$D=\left(D_{0}^{-1}-\Sigma\right)^{-1}.$ (6)

In this figure, the self-energy is represented by the bubble diagram with a
shaded region between the two bare propagators. From the Lagrangian (3), the
bare propagator for $d^{(I)}$ is written as

$D_{0}^{(I)}(p)=\frac{1}{C_{0}^{(I)}+C_{1}^{(I)}p^{2}+C_{2}^{(I)}p^{4}+\cdots},$
(7)

where $p$ is the magnitude of the relative spatial momentum between the two
nuclei in the c.m. frame. Following the approach of Refs. [50, 51], the
Coulomb interactions between the two nuclei at low-energies are described by
using the Coulomb Green’s function $G_{C}(E)$, which leads to

$\displaystyle\Sigma_{ij}(E)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(y_{t}^{(I)}\right)^{2}\Braket{0}{G_{C}(E)}{0}\delta_{ij}$
(8) $\displaystyle=$
$\displaystyle\frac{1}{2}\left(y_{t}^{(I)}\right)^{2}\left(-\frac{\mu}{\pi}\kappa
H(\eta)+J_{0}^{\rm div}\right)\delta_{ij},$

where $E=p^{2}/2\mu$ is the total kinetic energy and

$H(\eta)=\psi(i\eta)-\ln(i\eta)-\frac{i}{2\eta},$ (9)

where $\psi(z)$ is the digamma function defined as
$\psi(z)=d[\ln\Gamma(z)]/dz$ with $\Gamma(z)$ being the gamma function. The
Sommerfeld parameter $\eta$ is defined as $\eta=\kappa/p$ with
$\kappa=Z_{p}Z_{N}\mu\alpha_{\rm em}$, where $Z_{A}$ is the charge number of
nucleus $A$. In our case, $\kappa=44.9$ MeV.

The divergent part $J_{0}^{\rm div}$ arising from the Coulomb Green’s function
reads

$J_{0}^{\rm
div}=\frac{\mu}{\pi}\kappa\left(\frac{1}{\epsilon}+\ln\frac{\Lambda\sqrt{\pi}}{2\kappa}+1-\frac{3}{2}\gamma_{E}-\frac{\Lambda}{2\kappa}\right),$
(10)

where $\gamma_{E}$ is the Euler’s constant and $\Lambda$ is the regularization
scale introduced by performing the dimensional regularization in space-time
dimension of $d=4-\epsilon$. The term linear in $\Lambda$ comes from the power
divergence subtraction scheme [52]. One may write the fully-dressed propagator
by making use of the ERE so that the coefficients $C^{(I)}_{i}$ can be fixed
by the effective range parameters [53], which gives

$D^{(I)}(p)=-\left(y_{t}^{(I)}\right)^{-2}\frac{4\pi}{\mu}\frac{1}{K^{(I)}(p)-2\kappa
H(\eta)},$ (11)

and the ERE allows to write

$K^{(I)}(p)=-\frac{1}{a_{R}^{(I)}}+\frac{1}{2}r^{(I)}p^{2}-\frac{1}{4}P^{(I)}p^{4}+\cdots,$
(12)

where $a_{R}^{(I)}$, $r^{(I)}$, and $P^{(I)}$ are the scattering length,
effective range, and the shape parameter, respectively, and the divergent part
$J_{0}^{\rm div}$ of Eq. (10) is absorbed by the scattering length.

Following Ref. [54], the effective range parameters appearing in the
intermediate propagator are related to the resonance energy and its decay
width. It is achieved by comparing the obtained amplitude of the elastic
scattering process with the one obtained with the Breit-Wigner form. In Refs.
[55, 39], it is modified by including additional terms in the denominator of
the propagator as a result of expanding the denominator near the resonance
energy. The real part of the denominator of the propagator is then expanded as

$\displaystyle\tilde{K}^{(I)}$ $\displaystyle=$ $\displaystyle
K^{(I)}-2\kappa\,\mathrm{Re}\,H(\eta)$ (13) $\displaystyle=$
$\displaystyle-\frac{1}{a_{R}^{(I)}}+\frac{1}{2}\tilde{r}^{(I)}p^{2}-\frac{1}{4}\tilde{P}^{(I)}p^{4}+\tilde{Q}^{(I)}p^{6}$

up to $O(p^{6})$, where777This is an expansion by $p/\kappa$ and, since $p$ of
the second resonance region is close to the value of $\kappa$, the expansion
would be questionable. In our case, however, the expansion of the function
$\mbox{Re}\,H(\eta)$ converges fast and is valid within 5% even in the second
resonance region ($E_{\rm cm}\lesssim 1.5$ MeV).

$\displaystyle\tilde{r}^{(I)}$ $\displaystyle=$ $\displaystyle
r^{(I)}-1/3\kappa,$ $\displaystyle\tilde{P}^{(I)}$ $\displaystyle=$
$\displaystyle P^{(I)}+1/15\kappa^{3},$ $\displaystyle\tilde{Q}^{(I)}$
$\displaystyle=$ $\displaystyle Q^{(I)}-1/126\kappa^{5}.$ (14)

Here, the term containing $\kappa$ comes from the Coulomb self-energy,
$-2\kappa H(\eta)$. The imaginary part of the propagator is
$-2\kappa\,\mbox{Im}[H(\eta)]=-pC_{\eta}^{2}$, where

$C_{\eta}=\sqrt{\frac{2\pi\eta}{e^{2\pi\eta}-1}}$ (15)

is the Gamow factor.

Figure 3: Feynman diagrams for the radiative capture process of
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$. The thin and thick solid lines
denote the proton and the [15]N nucleus, respectively, and the wavy line
stands for the out-going photon. The dotted line represents the ground state
of the [16]O nucleus in the final state. The shaded blobs represent the
Coulomb repulsion between the proton and [15]N. The circle with a cross in (g)
is introduced as a counterterm of the loop diagrams.

Considering up to the order of $p^{6}$ in the ERE, the real part of the
denominator of propagators can be expanded around the resonance energy as

$\tilde{K}^{(I)}(E)=\left.\sum_{n=0}^{3}\frac{1}{n!}\frac{\partial^{n}\tilde{K}^{(I)}}{\partial
E^{n}}\right|_{E=E_{R}}(E-E_{R})^{n},$ (16)

with $\tilde{K}^{(I)}(E_{R})=0$ where $E_{R}$ is the resonance energy with
$p_{R}=\sqrt{2\mu E_{R}}$. The di-field propagator with the effective range
parameters can also be represented in terms of the resonance energy and width
as the Breit-Wigner formula, which reads

$D(p)=\left(y_{t}^{(I)}\right)^{-2}\frac{4\pi}{\mu
p}\frac{\frac{1}{2}\Gamma_{R}(E)C_{\eta}^{-2}}{E-E_{R}+R(E)+i\frac{1}{2}\Gamma_{R}(E)},$
(17)

with $\eta_{R}=\kappa/p_{R}$, and the energy dependence of the resonance width
is written as

$\displaystyle\Gamma_{R}(E)$ $\displaystyle=$
$\displaystyle-\frac{4\pi\kappa}{\mu\left(\tilde{r}-\tilde{P}p_{R}^{2}+6\tilde{Q}p_{R}^{4}\right)}\frac{1}{e^{2\pi\eta}-1}$
(18) $\displaystyle=$
$\displaystyle\Gamma_{R}(E_{R})\frac{e^{2\pi\eta_{R}}-1}{e^{2\pi\eta}-1}.$

In Eq. (17), $R(E)$ contains the higher order corrections in the expansion
around the resonance energy, which reads

$R(E)=a(E-E_{R})^{2}+b(E-E_{R})^{3},$ (19)

where

$\displaystyle
a=\frac{-\mu\tilde{P}+12\mu\tilde{Q}p_{R}^{2}}{\tilde{r}-\tilde{P}p_{R}^{2}+6\tilde{Q}p_{R}^{4}},\quad
b=\frac{8\mu^{2}\tilde{Q}}{\tilde{r}-\tilde{P}p_{R}^{2}+6\tilde{Q}p_{R}^{4}}.$
(20)

Through Eqs. (17)–(20) we omit isospin index $I$ for simplicity. We take the
resonance energy, width, $a$, and $b$ as free parameters to be fixed by
fitting the experimental data of the astrophysical $S$ factors, which will
then give the effective range parameters through Eqs. (13), (18), (20), and
the condition that $\tilde{K}(E_{R})=0$.

##  IV Capture amplitudes

The diagrams of the radiative capture amplitudes for the
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ reaction are depicted in Fig. 3.
The capture amplitude $\mathscr{M}$ can be decomposed into the iso-singlet and
iso-triplet parts as

$\mathscr{M}=\frac{e}{2}\chi_{p}^{T}\sigma_{2}(\bm{\epsilon}_{\gamma}^{*}\cdot\bm{\sigma})\tau_{2}\left(X^{(I=0)}-\tau_{3}X^{(I=1)}\right)\chi_{N},$
(21)

where the spinors for the proton and the $\nuclide[15]{N}$ nucleus in the
initial state are represented by $\chi_{p}$ and $\chi_{N}$, respectively, and
$\bm{\epsilon}_{\gamma}$ is the photon polarization vector. In Fig. 3, the
shaded blobs represent the Coulomb repulsion between the proton and [15]N. The
circle with a cross ($\otimes$) in Fig. 3(g) is the photon coupling between
[16]O and the di-fields which is introduced as a counterterm of the loop
diagrams. The isospin-dependent terms can be written as

$\displaystyle X^{(I=0)}$ $\displaystyle=$ $\displaystyle
X_{(a+b)}+X_{(c)}+X^{(I=0)}_{(d+e)}+X^{(I=0)}_{(f)}+X^{(I=0)}_{(g)}\,,$ (22)
$\displaystyle X^{(I=1)}$ $\displaystyle=$ $\displaystyle
X^{(I=1)}_{(d+e)}+X^{(I=1)}_{(f)}+X^{(I=1)}_{(g)}\,,$ (23)

where the amplitudes, $X_{(a,b,c)}$, $X^{(I=0)}_{(d,e,f)}$, and
$X^{(I=1)}_{(d,e,f)}$ are calculated from the corresponding Feynman diagrams
in Fig. 3.

In order to obtain the radiative capture amplitudes, we follow the approach
advocated by the authors of Refs. [56, 57]. Using the effective Lagrangian of
Eq. (3), the amplitude for non-resonant process depicted in Fig. 3(a) in the
c.m. frame is obtained as

$\displaystyle\mathscr{M}_{(a)}$ $\displaystyle=$
$\displaystyle-\frac{y_{s}}{\mu}\frac{Z_{p}}{M_{p}}I_{(a)}^{ij}\frac{e}{2}\chi_{p}^{T}\epsilon_{\gamma}^{*i}\sigma_{2}\sigma_{j}\tau_{2}\chi_{N},$
(24)

where $I_{(a)}^{ij}$ comes from the loop integration defined as

$\displaystyle I_{(a)}^{ij}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}\ell\,d^{3}\ell^{\prime}}{(2\pi)^{6}}d^{3}r\,d^{3}r^{\prime}\,d^{3}r^{\prime\prime}\,\ell_{i}\,\ell^{\prime}_{j}\Braket{\bm{\ell^{\prime}}}{{\bf
r}^{\prime}}$ $\displaystyle\mbox{}\times\Braket{{\bf
r}^{\prime}}{G_{C}(-B)}{{\bf r}}\Braket{{\bf
r}}{\bm{\ell}-\frac{\mu}{M_{p}}{\bf k}}\Braket{\bm{\ell}}{{\bf
r}^{\prime\prime}}\Braket{{\bf r}^{\prime\prime}}{\psi_{\bf p}}.$

Here, $B$ is the binding energy of $\nuclide[16]{O}$ relative to the
$p$-$\nuclide[15]{N}$ breakup threshold, ${\bf k}$ is the outgoing photon
momentum, and $\Braket{{\bf r}}{\psi_{\bf p}}=\psi_{\bf p}({\bf r})$ is the
Coulomb wave function. One can expand the Coulomb wave function and Green’s
function by partial waves as

$\displaystyle\psi_{\bf p}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}(2l+1)i^{l}e^{i\sigma_{l}}P_{l}(\hat{\bf
r}\cdot\hat{\bf p})\frac{F_{l}(\eta,\rho)}{\rho},$ (26)
$\displaystyle\Braket{{\bf r}^{\prime}}{G_{C}(E)}{{\bf r}}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}(2l+1)G_{C}^{(l)}(E;r^{\prime},r)P_{l}(\hat{\bf
r}^{\prime}\cdot\hat{\bf r}),$ (27)

where $\rho=pr$, $\sigma_{l}=\mbox{arg}\left[\Gamma(l+1+i\eta)\right]$ is the
Coulomb phase shift, $P_{l}(x)$ is the Legendre polynomial, and
$F_{l}(\eta,\rho)$ is the regular Coulomb function. It can be easily verified
that the $s$-wave ($l=0$) contribution in Eq. (27) vanishes by symmetry
consideration and only the $p$-wave ($l=1$) part contributes. In Ref. [57],
the partial-wave-expanded Coulomb Green’s function $G_{C}^{(l)}$ for the bound
state is shown to be simplified when it is written in terms of the regular and
irregular Coulomb functions, $F_{l}(\eta,\rho)$ and $G_{l}(\eta,\rho)$, which
leads to

$G_{C}^{(l)}(-B,\rho^{\prime},\rho)=-\frac{\mu
p}{2\pi}\frac{F_{l}(\eta,\rho^{\prime})}{\rho^{\prime}}\frac{iF_{l}(\eta,\rho)+G_{l}(\eta,\rho)}{\rho}.$
(28)

Using the identities of the Coulomb functions [58] the loop integration
$I_{(a)}^{ij}$ is rewritten as

$\displaystyle I_{(a)}^{ij}$ $\displaystyle=$
$\displaystyle\delta_{ij}\frac{2\mu\gamma}{3}\Gamma(2+\kappa/\gamma)e^{i\sigma_{0}}$
$\displaystyle\mbox{}\times\int_{0}^{\infty}drrW_{-\kappa/\gamma,3/2}(2\gamma
r)j_{0}\left(\frac{\mu}{M_{p}}kr\right)\frac{\partial}{\partial
r}\left(\frac{F_{0}(\eta,pr)}{pr}\right),$

where $j_{l}(z)$ and $W_{i\eta,l+1/2}(2i\rho)$ are the spherical Bessel
function and the Whittaker $W$-function, respectively. Here,
$\gamma=\sqrt{2\mu B}$ is the binding momentum determined by the binding
energy $B$ of [16]O relative to the $p$-[15]N threshold. Hence, the amplitude
$\mathscr{M}_{(a)}$ is obtained as

$\displaystyle\mathscr{M}_{(a)}$ $\displaystyle=$ $\displaystyle-
y_{s}\frac{2\gamma Z_{p}}{3M_{p}}\Gamma(2+\kappa/\gamma)e^{i\sigma_{0}}$ (30)
$\displaystyle\mbox{}\times\int_{0}^{\infty}drrW_{-\kappa/\gamma,3/2}(2\gamma
r)j_{0}\left(\frac{\mu}{M_{p}}kr\right)\frac{\partial}{\partial
r}\left(\frac{F_{0}(\eta,pr)}{pr}\right)$
$\displaystyle\mbox{}\qquad\times\chi_{p}^{T}\frac{e}{2}\sigma_{2}\left(\bm{\epsilon}_{\gamma}^{*}\cdot\bm{\sigma}\right)\tau_{2}\chi_{N}.$

The rest of the amplitudes can be calculated in a similar way, and the
isospin-dependent terms in Eq. (21) are obtained as

$\displaystyle X_{(a+b)}$ $\displaystyle=$ $\displaystyle
y_{s}\frac{2\gamma}{3}\Gamma(2+\kappa/\gamma)e^{i\sigma_{0}}$ (31)
$\displaystyle\mbox{}\times\int_{0}^{\infty}drrW_{-\kappa/\gamma,3/2}(2\gamma
r)\left[\frac{Z_{N}}{M_{N}}j_{0}\left(\frac{\mu}{M_{N}}kr\right)-\frac{Z_{p}}{M_{p}}j_{0}\left(\frac{\mu}{M_{p}}kr\right)\right]\frac{\partial}{\partial
r}\left(\frac{F_{0}(\eta,pr)}{pr}\right),$ $\displaystyle X_{(c)}$
$\displaystyle=$ $\displaystyle
y_{s}\left(\frac{Z_{N}}{M_{N}}-\frac{Z_{p}}{M_{p}}\right)C_{\eta}e^{i\sigma_{0}},$
(32) $\displaystyle X^{(I)}_{(d+e)}$ $\displaystyle=$ $\displaystyle
y_{s}\frac{2\gamma}{3}\Gamma(1+i\eta)\Gamma(2+\kappa/\gamma)\frac{C_{\eta}e^{i\sigma_{0}}}{K^{(I)}(p)-2\kappa
H(\eta)}$ (33)
$\displaystyle\mbox{}\times\int_{0}^{\infty}drrW_{-\kappa/\gamma,3/2}(2\gamma
r)\left[\frac{Z_{N}}{M_{N}}j_{0}\left(\frac{\mu}{M_{N}}kr\right)-\frac{Z_{p}}{M_{p}}j_{0}\left(\frac{\mu}{M_{p}}kr\right)\right]\frac{\partial}{\partial
r}\left(\frac{W_{-i\eta,1/2}(-2ipr)}{r}\right),$ $\displaystyle X^{(I)}_{(f)}$
$\displaystyle=$ $\displaystyle
y_{s}\frac{C_{\eta}e^{i\sigma_{0}}}{K^{(I)}(p)-2\kappa
H(\eta)}\left(\frac{Z_{N}}{M_{N}}-\frac{Z_{p}}{M_{p}}\right)\left(2\kappa
H(\eta)-\frac{2\pi}{\mu}J_{0}^{\mathrm{div}}\right),$ (34) $\displaystyle
X^{(I)}_{(g)}$ $\displaystyle=$ $\displaystyle-h^{(I)}\frac{4\pi
Z_{O}}{\mu}\frac{C_{\eta}e^{i\sigma_{0}}}{K^{(I)}(p)-2\kappa H(\eta)}.$ (35)

The integrals of Eq. (31) and Eq. (33) can be rewritten in terms of the
confluent hypergeometric functions instead of the Coulomb and Whittaker
functions [58]. In order to perform numerical integration, we simplify these
integrals as

$\displaystyle L_{(a+b)}(p)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dr\mbox{
}e^{(-\gamma+ip)r}r^{3}U(2+\kappa/\gamma,4,2\gamma
r)\left[\frac{Z_{N}}{M_{N}}j_{0}\left(\frac{\mu}{M_{N}}kr\right)-\frac{Z_{p}}{M_{p}}j_{0}\left(\frac{\mu}{M_{p}}kr\right)\right]$
(36)
$\displaystyle\mbox{}\times\left[M(1+i\eta,2,-2ipr)-(1+i\eta)M(2+i\eta,3,-2ipr)\right],$
$\displaystyle L_{(d+e)}(p)$ $\displaystyle=$
$\displaystyle\int_{r_{c}}^{\infty}dr\mbox{
}e^{(-\gamma+ip)r}r^{3}U(2+\kappa/\gamma,4,2\gamma
r)\left[\frac{Z_{N}}{M_{N}}j_{0}\left(\frac{\mu}{M_{N}}kr\right)-\frac{Z_{p}}{M_{p}}j_{0}\left(\frac{\mu}{M_{p}}kr\right)\right]$
(37)
$\displaystyle\mbox{}\times\left[U(1+i\eta,2,-2ipr)+2(1+i\eta)U(2+i\eta,3,-2ipr)\right],$

so that $X_{(a+b)}$ and $X_{(d+e)}^{(I)}$ can be reexpressed as

$\displaystyle X_{(a+b)}$ $\displaystyle=$ $\displaystyle
iy_{s}\frac{8\gamma^{3}p}{3}\Gamma(2+\kappa/\gamma)C_{\eta}e^{i\sigma_{0}}L_{(a+b)},$
(38) $\displaystyle X_{(d+e)}^{(I)}$ $\displaystyle=$ $\displaystyle
y_{s}\frac{16\gamma^{3}p^{2}}{3}\Gamma(1+i\eta)\Gamma(2+\kappa/\gamma)$ (39)
$\displaystyle\mbox{}\times\frac{C_{\eta}e^{i\sigma_{0}}}{K^{(I)}(p)-2\kappa
H(\eta)}L_{(d+e)},$

where $M(a,c,z)$ and $U(a,c,z)$ are the confluent hypergeometric functions of
the first and second kinds, respectively. The detailed derivation is given in
the Appendix.

The cut-off $r_{c}$ is introduced in the integration of Eq. (37) to avoid
divergence. The divergence from its rest part and the bubble diagram in Fig.
3(f) are absorbed by the counter term containing the LEC $h^{(I)}_{R}$, which
is defined by

$h^{(I)}_{R}=h^{(I)}+\frac{y_{s}}{Z_{O}}\left(\frac{Z_{N}}{M_{N}}-\frac{Z_{p}}{M_{p}}\right)\left(\frac{1}{2}J_{0}^{\mathrm{div}}+L_{(d+e)}^{\mathrm{div}}\right),$
(40)

where $J_{0}^{\mathrm{div}}$ is given in Eq. (10) and
$L_{(d+e)}^{\mathrm{div}}$ is the divergence part coming from the loop in the
diagram (d) and (e) in Fig. 3. Explicitly, it reads

$L_{(d+e)}^{\mathrm{div}}=\frac{\mu}{6\pi}\int_{0}^{r_{c}}\frac{dr}{r^{2}}-\frac{\mu\kappa}{2\pi}\int_{0}^{r_{c}}\frac{dr}{r},$
(41)

which has both logarithmic and linear divergences. The dependence of the LECs
and the astrophysical $S$ factor on the cut-off value of $r_{c}$ will be
examined in the next section.

##  V Numerical results

The purpose of the present work is to extrapolate the astrophysical $S$
factors, or equivalently the total cross sections, of the
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ reaction to the extremely low
energy regions corresponding to stellar environments which are hard to reach
in laboratory. The astrophysical $S$ factor is defined by the total cross
section $\sigma(E)$ that is given by

$\sigma(E)=\frac{1}{4\pi}\frac{\gamma^{2}+p^{2}}{4p}\sum_{\mathrm{spins}}|\mathscr{M}|^{2},$
(42)

in the non-relativistic limit up to the leading terms in $1/M$ expansion [53],
where $\mathscr{M}$ is the capture amplitude defined in Eq. (21), and $\gamma$
and $p$ are introduced in the previous section. Using the measured masses of
the nuclei we obtain $\gamma\approx 146$ MeV.888In the present work, we use
$M_{p}=938.272$ MeV for the proton mass, $M_{N}=13968.936$ MeV for the [15]N
mass, and $M_{O}=14895.081$ MeV for the [16]O mass.

The effective Lagrangian (3) contains the following LECs: $y_{s}$,
$y_{t}^{I=0,1}$, and $h^{I=0,1}$. In addition, since we include two $s$-wave
resonance states of [16]O, one with $I=0$ and the other with $I=1$, by
employing ERE, additional parameters for each resonance state are introduced
as shown in Eq. (13). However, as Eq. (11) shows, the inverse of the squared
$y_{t}^{I=0,1}$ is multiplied to the corresponding di-field propagator and
this factor is canceled out when multiplied by the vertex functions.
Therefore, these two LECs are redundant and cannot be determined through the
analysis of the present work. The coupling constant $y_{s}$ determines the
strength of the coupling between the ${}^{3}P_{0}$ state of the $p$-[15]N
system and the ground state of [16]O. The $h_{R}^{(I=0,1)}$ terms are the
counter terms which absorb the divergences from the loop integrals and the
values of $h_{R}^{(I=0,1)}$ depend on the cut-off $r_{c}$ in Eq. (41).
Furthermore, throughout numerical computations, we found that the ERE for the
first resonance of iso-singlet gives stable results with the terms up to
$O(p^{2})$, while $O(p^{6})$ expansion is needed for the second resonance of
iso-triplet. These ERE parameters are rephrased in terms of the resonance
energies ($E_{R0}$, $E_{R1}$), widths ($\Gamma_{R0}$, $\Gamma_{R1}$), and
higher order corrections $R(E)$ for the second resonance, which introduces two
parameters ($a$, $b$). As a result, we have totally 9 parameters, namely,
$y_{s}$, $h^{I=0,1}$, $E_{R0}$, $E_{R1}$, $\Gamma_{R0}$, $\Gamma_{R1}$, $a$,
and $b$ on top of the cut-off $r_{c}$ value.

In this study, we work with three different data sets reported by Refs. [11,
18, 25]. The old data set of Ref. [11], referred to as RR, contains 53 data
points which cover the energy range of
$150~{}\mbox{keV}<E_{p}<2500~{}\mbox{keV}$. The 80 data points reported by
Ref. [18], referred to as LeB, cover the energy range of
$130~{}\mbox{keV}<E_{p}<1800~{}\mbox{keV}$. The most recent data reported by
Ref. [25], referred to as Imb, cover the similar energy region,
$140~{}\mbox{keV}<E_{p}<1800~{}\mbox{keV}$ with 78 data points.999The data set
of Ref. [18] was revised in Ref. [26]. We use the revised data set in the
present work. We use these data sets separately to determine our model
parameters, which enables us to predict the astrophysical $S$ factor at zero
energy.

Figure 4:  The astrophysical $S$ factor. The parameters are fitted by the
three experimental data sets from Ref. [11] (RR), [18] (LeB), and [25] (Imb).
The bands indicate the error range obtained in the MCMC calculations.

For the fitting process, we adopt the Markov chain Monte Carlo (MCMC) method
[59] to minimize $\chi^{2}$ defined as $\chi^{2}=\sum_{i}(S^{\rm
Theor.}_{i}-S^{\rm Expt.}_{i})^{2}/(S^{\rm Err.}_{i})^{2}$. Figure 4 presents
our results for the astrophysical $S$ factor of the [15]N($p$,$\gamma$)[16]O
reaction as a function of the c.m. energy $E$. In Fig. 5, the same results are
shown in logarithmic scales in both axes to highlight the extrapolated $S$
factor in the very low energy region, which explicitly shows that the $S$
factor becomes nearly flat as $E\to 0$. The bands in these figures indicate
the errors in the $S$ factor estimation of the MCMC calculations, which are
estimated from the elements of the covariance matrix.

Figure 5:  Same as Fig. 4 but with logarithmic scales in both axes.

The fitted parameter values and the predicted $S(0)$ for the three data sets
are given in Table 1. This shows that the resonance parameters are consistent
to each other, which is expected as the three data sets share the same
structure arising from the two resonances. The differences in the magnitudes,
in particular, between the old [11] and new data [18, 25] are reflected mostly
in the differences in the couplings $y_{s}$ and $h_{R}^{I=0}$. This behavior
is also understood as the gaps among data sets lie mostly in the very low
energy region where the direct coupling ($y_{s}$) and the coupling to the iso-
singlet resonance of [16]O ($h_{R}^{I=0}$) dominate.

Data Set | $y_{s}$ | $h^{(I=0)}_{R}$ | $h^{(I=1)}_{R}$ | $E_{R0}$ | $\Gamma_{R0}$ | $E_{R1}$ | $\Gamma_{R1}$ | $a$ | $b$ | $\chi^{2}/\mbox{datum}$ | $S(0)$ | $S(E_{G})$
---|---|---|---|---|---|---|---|---|---|---|---|---
RR [11] | $2.759$ | $3.410$ | $-29.98$ | $364.6$ | $259.6$ | $967.2$ | $155.9$ | $-0.389$ | $1.158$ | 2.43 | $75.3\pm 12.1$ | $80.9\pm 12.5$
LeB [18] | $0.603$ | $0.585$ | $-31.46$ | $369.3$ | $347.4$ | 962.9 | 160.0 | $-0.278$ | $1.417$ | 5.04 | $34.1\pm 0.9$ | $38.2\pm 1.0$
Imb [25] | $0.897$ | $0.978$ | $-31.05$ | $359.8$ | $252.1$ | $962.6$ | $154.6$ | $-0.326$ | $1.636$ | $2.58$ | $29.8\pm 1.1$ | $33.2\pm 1.1$
Table 1:  The fitted values of the parameters for each data set and the
derived astrophysical $S$ factor at $E=0$ and at the Gamow energy $E_{G}=26$
keV. The units of $y_{s}$ are $10^{-2}\mbox{ MeV}^{-1/2}$ and those of
$h_{R}^{I}$ are $10^{-3}\mbox{ MeV}^{1/2}$. The energies and widths of the
resonances are given in the units of keV. The units of $a$ and $b$ are
$\mbox{MeV}^{-1}$ and $\mbox{MeV}^{-2}$, respectively, and the astrophysical
$S$ factors are in the units of keV b. For the cut-off value, we use
$r_{c}=1.0$ fm.

The fitted resonance parameters are $E_{R0}\approx\mbox{360 -- 370}$ keV,
$\Gamma_{R0}\approx\mbox{250 -- 350}$ keV, $E_{R1}\approx\mbox{960 -- 970}$
keV, and $\Gamma_{R1}\approx\mbox{155 -- 160}$ keV. These should be compared
with the experimentally measured values of Ref. [60], namely, $E_{R0}^{\rm
Expt.}=312$ keV, $\Gamma_{R0}^{\rm Expt.}=91$ keV, $E_{R1}^{\rm Expt.}=926$
keV, and $\Gamma_{R1}^{\rm Expt.}=130$ keV. We can see that the differences
between resonance parameters in the present work and the empirical values are
within 5–20% except $\Gamma_{R0}$. Our fitted $\Gamma_{R0}$ is more than three
times the empirical value $\Gamma_{R0}^{\rm Expt.}$. Such a difference is also
seen in the $\alpha\alpha$ scattering study within EFT in Ref. [54].
Furthermore, the structures in the astrophysical $S$ factors shown in Fig. 4
are reasonably reproduced by the present resonance parameters. In fact, our
results indicate that the resonance structure in the astrophysical $S$ factor
is not sensitive to the value of $\Gamma_{R0}$, and, therefore, constraining
the value of $\Gamma_{R0}$ by this reaction is not easy.101010The fitting was
tried with forcing $\Gamma_{R0}^{\rm Expt.}=91$ keV. In this case, although
the broadness of the curve of $S(E)$ in the first resonance region does not
change, it overestimates the height of the resonance peak. Thus the resolution
of the difference in resonance parameters deserve more detailed and
comprehensive studies [46].

As in $R$-matrix analyses [10, 11, 26], we find that there is a destructive
interference between the two resonance contributions at $E\approx 0.5$ MeV,
where the contributions from the two resonances overlap. We also find that the
resonance contributions dominate the cross sections and the direct emission of
the photon without a resonance shown in Figs. 3(a,b,c) is suppressed.

Data Set | RR [11] | LeB [18] | Imb [25]
---|---|---|---
$a_{R}^{(I=0)}$ ($10^{4}$ fm) | $-1.722$ | $-1.761$ | $-1.824$
$\tilde{r}^{(I=0)}$ ($10^{-3}$ fm) | $-7.053$ | $-6.993$ | $-6.746$
$a_{R}^{(I=1)}$ (fm) | $-21.44$ | $-19.04$ | $-18.87$
$\tilde{r}^{(I=1)}$ (fm) | $-4.342$ | $-5.250$ | $-5.319$
$\tilde{P}^{(I=1)}$ ($\mathrm{fm}^{3}$) | $-144.07$ | $-189.5$ | $-192.6$
$\tilde{Q}^{(I=1)}$ ($\mathrm{fm}^{5}$) | $-246.26$ | $-339.5$ | $-345.2$
$r^{(I=0)}$ (fm) | $1.457$ | $1.457$ | $1.458$
$r^{(I=1)}$ (fm) | $-2.878$ | $-3.786$ | $-3.855$
$P^{(I=1)}$ ($\mathrm{fm}^{3}$) | $-149.72$ | $-195.1$ | $-198.2$
$Q^{(I=1)}$ ($\mathrm{fm}^{5}$) | $-233.28$ | $-326.5$ | $-332.2$
Table 2:  Reduced effective range parameters and effective range parameters
for each data set.

Shown in Table 2 are the effective range parameters and the reduced effective
range parameters derived from the resonance parameters given in Table 1. As we
have seen above, all three data sets share the similar structures in the
astrophysical $S$ factor coming from the resonances, and it is expected that
they have similar ERE parameters. For the first resonance we take ERE up to
$O(p^{2})$ and for the second resonance we take up to $O(p^{6})$. Our results
show that the effects from the $\kappa$ terms in the reduced effective range
parameters are non-trivial, in particular, for the effective range
$\tilde{r}$. We find that there is a big cancellation in $\tilde{r}^{(I=0)}$
between $r^{(I=0)}$ and the $\kappa$ term, which leads to a fine tuning of the
value of $\tilde{r}^{(I=0)}$.

$S(0)$ ($\mathrm{keV\ b}$)
---
Data of Ref. [10]
Ref. [10] | 32111111This value is obtained for the proton energy of 25 keV in the laboratory frame, which gives the equivalent value of $S(0)\approx 26$ $\mathrm{keV\ b}$ [11].
Ref. [15] | 35–40
Data of Ref. [11]
Ref. [11] | $64\pm 6$
Ref. [15] | 50
This work (RR) | $75.3\pm 12.1$
Data of Refs. [10, 11]
Ref. [14] | $36.0\pm 6$
Ref. [16] | $21.1$
Data of Ref. [18]
Ref. [18] | $39.6\pm 2.6$
Ref. [24] |  $33.1$–$40.1$
Ref. [26] | $40\pm 3$
This work (LeB) | $34.1\pm 0.9$
Compiled data
Ref. [22] | $45^{+9}_{-7}$
Ref. [27] |  $39.5$–$43.35$ 121212The authors of Ref. [27] obtained these values for $E_{p}=50$–$60$ keV and accepted it as $S(0)$.
Data of Ref. [19]
Ref. [47] | $30.4$
Data of Ref. [25]
This work (Imb) | $29.8\pm 1.1$
Table 3:  Estimated values of $S(0)$ for the
$\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ reaction in the units of
$\mathrm{keV\ b}$. The estimated values are categorized by the used data
set(s).

Since we have introduced the cut-off $r_{c}$ in the loop integral, the results
presented in the present work are obtained by taking $r_{c}=1$ fm. In order to
see the $r_{c}$-dependence of our results, we varied the values of $r_{c}$
from 0.1 fm to 1.0 fm to confirm that the $r_{c}$-dependence of our results on
the astrophysical $S$ factor are very weak although the coupling strengths
would change.

In Table 3, we list the previous estimates of the astrophysical $S$ factor at
zero energy $S(0)$ from the $R$-matrix analysis [11, 10, 18, 15, 24, 14, 26]
or potential model calculations [16, 22, 27]. Those values of $S(0)$ are
scattered from 22 $\mathrm{keV\ b}$ to 64 $\mathrm{keV\ b}$, depending on the
data used for the fit. We first note that our estimate on the central value of
the astrophysical $S(0)$ using the data of Ref. [11] is somehow larger than
the values of the $R$ matrix analyses of Refs. [11, 15] but the large error
ranges overlap each other. However, these estimates differ from the $S(0)$
value extracted from the post-NACRE data of Refs. [18, 25], which give new
values around 30 $\mathrm{keV\ b}$. Our results are also close to the values
extracted from the data sets up to the energy region of the first resonance
reported in Refs. [22, 27, 47]. The comparison of our results for $S(0)$ with
the values in the literature are visualized as well in Fig. 6. The results
presented in Table 3 also shows that the $S$ factor at the Gamow energy is
about 10% larger than $S(0)$.

Figure 6:  The obtained $S(0)$ values of the present work compared with the
values in the literature. The abbreviated references are as follows: Heb [10],
Ba [15], RR [11], Muk [14], HBG [16], LeB [18], MLK [24], deB [26], Xu [22],
DD [27], Caci [19], SAO [47], and Imb [25]. The references in the upper part
indicate the sources of the experimental data and the references in the lower
part indicate the sources of the calculated results.

##  VI Summary and discussion

In the present work, we studied the radiative proton capture process,
[15]N($p$,$\gamma$)[16]O, that has an important role in the CNO cycle by
connecting the CN cycle and the NO cycle. Our study was performed by employing
the EFT formalism, and, to our knowledge, there was no application of EFT
approach for the investigation of this reaction. Three open channels of
$\alpha$-[12]C, $\alpha$-[12]C∗, and $p$-[15]N can be involved to describe the
resonant states of [16]O, and the experimental data of the
[15]N($p$,$\gamma$)[16]O reaction clearly show the structure of the two
resonant $1^{-}$ states of $\nuclide[16]{O}^{*}$. One of them is the isospin
singlet state and the other is the isospin triplet state, which have the
excitation energies of 12.45 MeV and 13.09 MeV, respectively.

In order to describe this reaction, we employ a minimum set of effective
Lagrangian with the terms for the two $1^{-}$ states with $I=0$ and $I=1$ in
the $p$-[15]N channel. The resonances are described by the di-fields and the
structure of their propagators are rephrased in terms of ERE and by taking the
Breit-Wigner form. The parameters introduced in the low-energy effective
Lagrangian are fitted to the avilable experimental data of the astrophysical
$S$ factor of the [15]N($p$,$\gamma$)[16]O reaction in the range of
$E_{p}=130$ keV to 2500 keV. In the present analysis, we employ the
experimental data reported in Refs. [11, 18, 25] and the fitting process was
performed for each data set using the MCMC method. Then the astrophysical $S$
factors at $E=0$ and $E=E_{G}$ are extrapolated and compared with the
estimates in the literature. Our results show that the value of $S(0)$ is in
the range of 30–35 $\mathrm{keV\ b}$ based on the data of Refs. [18, 25],
which agrees with other estimates based on the post-NACRE data. We also found
that the values of $S(E_{G})$ are about 10% larger than those of $S(0)$.

The present work shows that the EFT approach can be applied to the study of
[15]N($p$,$\gamma$)[16]O. However, the extracted resonance parameters are
puzzling as they show some discrepancies with the empirical values, in
particular for the first resonance state although the resonance structures in
the astrophysical $S$ factor are well reproduced. The overestimate of a
resonance width was also found in $\alpha\alpha$ scattering study within EFT,
but the experimental value of a width of [16]O could be well reproduced in the
study of the $S$ factor of [12]C($\alpha$,$\gamma$)[16]O within EFT [29]. This
deserves more detailed and rigorous investigations including the effects of
other open channels.

###### Acknowledgements.

We are grateful to T.-S. H. Lee for fruitful discussions. This work was
supported by the National Research Foundation of Korea (NRF) under Grants No.
NRF-2019R1F1A1040362, No. NRF-2022R1F1A1070060, No. NRF-2020R1A2C1007597, and
No. NRF-2018R1A6A1A06024970 (Basic Science Research Program). *

##  Appendix A Loop integrals

In this Appendix, we derive the integrals of Eqs. (36) and (37) explicitly by
using identities of the Coulomb functions, Whittaker functions, and the
confluent hypergeometric functions [58]. The Coulomb function
$F_{l}(\eta,\rho)$ is written as

$F_{l}(\eta,\rho)=C_{l}(\eta)\rho^{l+1}e^{i\rho}M(l+1+i\eta,2l+2,-2i\rho),$
(43)

where $M(a,c,z)$ is the confluent hypergeometric function of the first kind or
the Kummer’s function. The factor of $C_{l}(\eta)$ is defined as

$C_{l}(\eta)=\frac{2^{l}}{(2l+1)!}|\Gamma(l+1+i\eta)|e^{-\frac{\pi}{2}\eta},$
(44)

which becomes the Gamow factor for $l=0$, namely, $C_{0}(\eta)$ becomes
$C_{\eta}$ of Eq. (15). On the other hand, the Whittaker function
$W_{\kappa,\mu}(z)$ is written as

$W_{\kappa,\mu}(z)=\textstyle
e^{-\frac{z}{2}}z^{\mu+\frac{1}{2}}U(\mu-\kappa+\frac{1}{2},1+2\mu,z),$ (45)

where $U(a,c,z)$ is the confluent hypergeometric function of the second kind
or the Tricomi’s function.

Using Eqs. (43) and (45), it is straightforward to get

$\displaystyle W_{-\kappa/\gamma,3/2}(2\gamma r)$ $\displaystyle=$
$\displaystyle 4\gamma^{2}r^{2}e^{-\gamma r}U(2+\kappa/\gamma,4,2\gamma r),$
(46) $\displaystyle\frac{W_{-i\eta,1/2}(-2ipr)}{r}$ $\displaystyle=$
$\displaystyle-2ipe^{ipr}U(1+i\eta,2,-2ipr),$ (47)
$\displaystyle\frac{F_{0}(\eta,pr)}{pr}$ $\displaystyle=$ $\displaystyle
C_{\eta}e^{ipr}M(1+i\eta,2,-2ipr).$ (48)

As the loop integrals of Eqs. (31) and (33) contain derivatives of the above
functions, we use the identities of the derivative of the confluent
hypergeometric functions as

$\displaystyle\frac{d}{dz}M(a,c,z)$ $\displaystyle=$
$\displaystyle\frac{a}{c}M(a+1,c+1,z),$ (49)
$\displaystyle\frac{d}{dz}U(a,c,z)$ $\displaystyle=$ $\displaystyle-
aU(a+1,c+1,z)$ (50)

to obtain the formulas of Eqs. (38) and (39). This finally leads to the
integrals of Eqs. (31) and (33) in the simplified form of

$\displaystyle\int_{0}^{\infty}drrW_{-\kappa/\gamma,3/2}(2\gamma
r)\left[\frac{Z_{N}}{M_{N}}j_{0}\left(\frac{\mu}{M_{N}}kr\right)-\frac{Z_{p}}{M_{p}}j_{0}\left(\frac{\mu}{M_{p}}kr\right)\right]\frac{\partial}{\partial
r}\left(\frac{F_{0}(\eta,pr)}{pr}\right)$ (51) $\displaystyle=$ $\displaystyle
4\gamma^{2}ipC_{\eta}\int_{0}^{\infty}dr\mbox{
}e^{(-\gamma+ip)r}r^{3}U(2+\kappa/\gamma,4,2\gamma
r)\left[\frac{Z_{N}}{M_{N}}j_{0}\left(\frac{\mu}{M_{N}}kr\right)-\frac{Z_{p}}{M_{p}}j_{0}\left(\frac{\mu}{M_{p}}kr\right)\right]$
$\displaystyle\qquad\mbox{}\times\left[M(1+i\eta,2,-2ipr)-(1+i\eta)M(2+i\eta,3,-2ipr)\right],$

and

$\displaystyle\int_{0}^{\infty}drrW_{-\kappa/\gamma,3/2}(2\gamma
r)\left[\frac{Z_{N}}{M_{N}}j_{0}\left(\frac{\mu}{M_{N}}kr\right)-\frac{Z_{p}}{M_{p}}j_{0}\left(\frac{\mu}{M_{p}}kr\right)\right]\frac{\partial}{\partial
r}\left(\frac{W_{-i\eta,1/2}(-2ipr)}{r}\right)$ (52) $\displaystyle=$
$\displaystyle 8\gamma^{2}p^{2}\int_{0}^{\infty}dr\mbox{
}e^{(-\gamma+ip)r}r^{3}U(2+\kappa/\gamma,4,2\gamma
r)\left[\frac{Z_{N}}{M_{N}}j_{0}\left(\frac{\mu}{M_{N}}kr\right)-\frac{Z_{p}}{M_{p}}j_{0}\left(\frac{\mu}{M_{p}}kr\right)\right]$
$\displaystyle\qquad\mbox{}\times\left[U(1+i\eta,2,-2ipr)+2(1+i\eta)U(2+i\eta,3,-2ipr)\right].$

Embedding the cut-off $r_{c}$ on the integration range of Eq. (52) gives Eqs.
(38) and (39).

## References

  * [1] E. M. Burbidge, G. R. Burbidge, W. A. Fowler, and F. Hoyle,  Synthesis of the elements in stars, Rev. Mod. Phys. 29, 547 (1957).
  * [2] E. G. Adelberger et al.,  Solar fusion cross sections. II. The $pp$ chain and CNO cycles, Rev. Mod. Phys. 83, 195 (2011).
  * [3] G. Bellini et al. (Borexino Collaboration),  First Evidence of $pep$ Solar Neutrinos by Direct Detection in Borexino,  Phys. Rev. Lett. 108, 051302 (2012).
  * [4] M. Agostini et al. (Borexino Collaboration),  Comprehensive measurement of $pp$-chain solar neutrinos, Nature 562, 505 (2018).
  * [5] M. Agostini et al. (Borexino Collaboration),  Experimental evidence of neutrinos produced in the CNO fusion cycle in the Sun,  Nature 587, 577 (2020).
  * [6] G. D. O. Gann, K. Zuber, D. Bemmerer, and A. Serenelli,  The future of solar neutrinos,  Annu. Rev. Nucl. Part. Sci. 71, 491 (2021).
  * [7] M. Wiescher, J. Görres, E. Uberseder, G. Imbriani, and M. Pignatari,  The cold and hot CNO cycles,  Annu. Rev. Nucl. Part. Sci. 60, 381 (2010).
  * [8] A. Boeltzig et al.,  Shell and explosive hydrogen burning: Nuclear reaction rates for hydrogen burning in RGB, AGB and novae,  Eur. Phys. J. A 52, 75 (2016).
  * [9] J. José and M. Hernanz,  Nucleosynthesis in classical novae: CO versus ONe white dwarfs,  Astrophys. J. 494, 680 (1998).
  * [10] D. F. Hebbard,  Proton capture by $\nuclide[15]{N}$,  Nucl. Phys. 15, 289 (1960).
  * [11] C. Rolfs and W. S. Rodney,  Proton capture by $\nuclide[15]{N}$ at stellar energies,  Nucl. Phys. A235, 450 (1974).
  * [12] A. Schardt, W. A. Fowler, and C. C. Lauritsen,  The disintegration of N15 by protons,  Phys. Rev. 86, 527 (1952).
  * [13] F. Brochard, P. Chevallier, D. Disdier, V. Rauch, and F. Scheibling,  Étude des désexcitations électromagnétiques des niveaux 1- situés à 12,44 et 13,09 MeV dans le noyau [16]O, J. Phys. France 34, 363 (1973).
  * [14] A. M. Mukhamedzhanov et al.,  New astrophysical $S$ factor for the $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ reaction via the asymptotic normalization coefficient (ANC) method,  Phys. Rev. C 78, 015804 (2008).
  * [15] F. C. Barker,  $\nuclide[15]{N}(p,\gamma_{0})\nuclide[16]{O}$ $S$ factor,  Phys. Rev. C 78, 044612 (2008).
  * [16] J. T. Huang, C. A. Bertulani, and V. Guimarães,  Radiative capture of nucleons at astrophysical energies with single-particle states,  Atom. Data Nucl. Data Tabl. 96, 824 (2010).
  * [17] D. Bemmerer et al. (LUNA Collaboration),  Direct measurement of the $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ total cross section at novae energies,  J. Phys. G 36, 045202 (2009).
  * [18] P. J. LeBlanc et al.,  Constraining the $S$ factor of $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ at astrophysical energies,  Phys. Rev. C 82, 055804 (2010),  Phys. Rev. C 84, 019902(E) (2011).
  * [19] A. Caciolli et al.,  Revision of the $\nuclide[15]{N}(p.\gamma)\nuclide[16]{O}$ reaction rate and oxygen abundance in H-burning zones,  Astron. Astrophys. 533, A66 (2011).
  * [20] A. Guglielmetti,  Recent results on $(p,\gamma)$ and $(\alpha,\gamma)$ fusion reactions at LUNA,  EPJ Web Conf. 17, 06001 (2011).
  * [21] C. Broggini, D. Bemmerer, A. Caciolli, and D. Trezzi,  LUNA: Status and prospects,  Prog. Part. Nucl. Phys. 98, 55 (2018).
  * [22] Y. Xu, K. Takahashi, S. Goriely, M. Arnould, M. Ohta, and H. Utsunomiya,  NACRE II: An update of the NACRE compilation of charged-particle-induced thermonuclear reaction rates for nuclei with mass number $A<16$,  Nucl. Phys. A918, 61 (2013).
  * [23] Experimental Nuclear Reaction Data (EXFOR),  https://www-nds.iaea.org/nrdc/about/about-exfor.html.
  * [24] A. M. Mukhamedzhanov, M. La Cognata, and V. Kroha,  Astrophysical $S$ factor for the $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ reaction,  Phys. Rev. C 83, 044604 (2011).
  * [25] G. Imbriani et al.,  Measurement of $\gamma$ rays from $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ cascade and $\nuclide[15]{N}(p,\alpha_{1}\gamma)\nuclide[12]{C}$ reactions,  Phys. Rev. C 85, 065810 (2012),  86, 039902(E) (2012).
  * [26] R. J. deBoer, J. Görres, G. Imbriani, P. J. LeBlanc, E. Uberseder, and M. Wiescher,  $R$-matrix analysis of [16]O compound nucleus reactions,  Phys. Rev. C 87, 015802 (2013).
  * [27] S. Dubovichenko and A. Dzhazairov-Kakhramanov,  Study of the neutron and proton capture reactions $\nuclide[10,11]{B}(n,\gamma)$, $\nuclide[11]{B}(p,\gamma)$, $\nuclide[14]{C}(p,\gamma)$, and $\nuclide[15]{N}(p,\gamma)$ at thermal and astrophysical energies,  Int. J. Mod. Phys. E 23, 1430012 (2014).
  * [28] S. Weinberg,  Phenomenological Lagrangians,  Physica 96A, 327 (1979).
  * [29] S.-I. Ando,  $S_{E1}$ factor of radiative $\alpha$ capture on [12]C in cluster effective field theory,  Phys. Rev. C 100, 015807 (2019).
  * [30] M. M. Khansari, H. Khalili, and H. Sadeghi,  Cross section and astrophysical $S$-factor for $\nuclide[12]{C}(p,\gamma)\nuclide[13]{N}^{*}$ reaction with halo effective field theory at low-energies,  New Astron. 57, 76 (2017).
  * [31] X. Zhang, K. M. Nollett, and D. R. Phillips,  Combining ab initio calculations and low-energy effective field theory for halo nuclear systems: The case of $\nuclide[7]{Li}+n\to\nuclide[8]{Li}+\gamma$,  Phys. Rev. C 89, 024613 (2014).
  * [32] X. Zhang, K. M. Nollett, and D. R. Phillips,  Halo effective field theory constrains the solar $\nuclide[7]{Be}+p\to\nuclide[8]{B}+\gamma$ rate,  Phys. Lett. B 751, 535 (2015).
  * [33] X. Zhang, K. M. Nollett, and D. R. Phillips,  Models, measurements, and effective field theory: Proton capture on [7]Be at next-to-leading order,  Phys. Rev. C 98, 034616 (2018).
  * [34] G. Rupak and R. Higa,  Model-Independent Calculation of Radiative Neutron Capture on Lithium-7,  Phys. Rev. Lett. 106, 222501 (2011).
  * [35] R. Higa, G. Rupak, and A. Vaghani,  Radiative $\nuclide[3]{He}(\alpha,\gamma)\nuclide[7]{Be}$ reaction in halo effective field theory,  Eur. Phys. J. A 54, 89 (2018).
  * [36] L. Moschini, J. Yang, and P. Capel,  [15]C: From halo effective field theory structure to the study of transfer, breakup, and radiative-capture reactions,  Phys. Rev. C 100, 044615 (2019).
  * [37] S.-I. Ando,  Elastic $\alpha$-[12]C scattering at low energies in cluster effective field theory,  Eur. Phys. J. A 52, 130 (2016).
  * [38] S.-I. Ando,  Elastic $\alpha$-[12]C scattering at low energies with the bound states of [16]O in effective field theory,  Phys. Rev. C 97, 014604 (2018).
  * [39] S.-I. Ando,  Elastic $\alpha$-[12]C scattering at low energies with the resonant $2_{2}^{+}$ and $2_{3}^{+}$ states of [16]O, arXiv:2108.02386.
  * [40] S.-I. Ando,  Cluster effective field theory and nuclear reactions,  Eur. Phys. J. A 57, 17 (2021).
  * [41] P. F. Bedaque and U. van Kolck,  Effective field theory for few nucleon systems,  Annu. Rev. Nucl. Part. Sci. 52, 339 (2002).
  * [42] G. Rupak,  Radiative reactions in halo effective field theory,  Int. J. Mod. Phys. E 25, 1641004 (2016).
  * [43] H.-W. Hammer, C. Ji, and D. R. Phillips,  Effective field theory description of halo nuclei,  J. Phys. G 44, 103002 (2017).
  * [44] P. Capel,  Combining halo-EFT descriptions of nuclei and precise models of nuclear reactions,  Few-Body Syst. 63, 14 (2022).
  * [45] B. A. Gelman,  Narrow resonances and short-range interactions,  Phys. Rev. C 80, 034005 (2009).
  * [46] J. B. Habashi, S. Fleming, and U. van Kolck,  Nonrelativistic effective field theory with a resonance field,  Eur. Phys. J. A 57, 169 (2021).
  * [47] S. Son, S.-I. Ando, and Y. Oh,  Determination of astrophysical ${S}$ factor for $\nuclide[15]{N}(p,\gamma)\nuclide[16]{O}$ at low-energies within effective field theory, New Phys.: Sae Mulli 72, 291 (2022).
  * [48] C. A. Bertulani, H.-W. Hammer, and U. van Kolck,  Effective field theory for halo nuclei: shallow $p$-wave states,  Nucl. Phys. A712, 37 (2002).
  * [49] P. F. Bedaque, H.-W. Hammer, and U. van Kolck,  Narrow resonances in effective field theory,  Phys. Lett. B 569, 159 (2003).
  * [50] X. Kong and F. Ravndal,  Proton–proton scattering lengths from effective field theory,  Phys. Lett. B 450, 320 (1999),  458, 565(E) (1999).
  * [51] X. Kong and F. Ravndal,  Coulomb effects in low energy proton–proton scattering,  Nucl. Phys. A665, 137 (2000).
  * [52] D. B. Kaplan, M. J. Savage, and M. B. Wise,  A new expansion for nucleon-nucleon interactions,  Phys. Lett. B 424, 390 (1998).
  * [53] S.-I. Ando and C. H. Hyun,  Effective field theory on the deuteron with dibaryon field,  Phys. Rev. C 72, 014008 (2005).
  * [54] R. Higa, H.-W. Hammer, and U. van Kolck,  $\alpha\alpha$ scattering in halo effective field theory, Nucl. Phys. A809, 171 (2008).
  * [55] S.-I. Ando,  Elastic $\alpha$-[12]C scattering at low energies with the sharp resonant $0_{3}^{+}$ state of [16]O, Phys. Rev. C 102, 034611 (2020).
  * [56] E. Ryberg, C. Forssén, H.-W. Hammer, and L. Platter,  Effective field theory for proton halo nuclei,  Phys. Rev. C 89, 014325 (2014).
  * [57] E. Ryberg, C. Forssén, H.-W. Hammer, and L. Platter,  Constraining low-energy proton capture on beryllium-7 through charge radius measurements,  Eur. Phys. J. A 50, 170 (2014).
  * [58] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors,  NIST Handbook of Mathematical Functions (Cambridge University Press, 2010).
  * [59] D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman,  emcee: The MCMC hammer, PASP 125, 306 (2013).
  * [60] D. R. Tilley, H. R. Weller, and C. M. Cheves,  Energy levels of light nuclei $A=16$–$17$,  Nucl. Phys. A564, 1 (1993).