August 10, 2023

Radiative decay of the sub-threshold $1_{1}^{-}$ and $2_{1}^{+}$ states of 16O
in cluster effective field theory

Shung-Ichi<EMAIL_ADDRESS>
Department of Display and Semiconductor Technology, Sunmoon University, Asan,
Chungnam 31460, Republic of Korea

Radiative decay of the sub-threshold $1_{1}^{-}$ and $2_{1}^{+}$ states of 16O
is studied in cluster effective field theory. The wave function normalization
factors for initial and final states of the radiative decay amplitudes are
deduced by using the phase shift data of elastic $\alpha$-12C scattering for
$l=0,1,2$, which are related to the asymptotic normalization coefficients of
$0_{1}^{+}$, $1_{1}^{-}$, $2_{1}^{+}$ states of 16O for the two-body
$\alpha$-12C channel; then only one unfixed parameter remains in each of the
radiative decay amplitudes. We fit the parameters to the experimental
radiative decay rates and apply the fitted parameters to the study of
radiative $\alpha$ capture on 12C, 12C($\alpha$,$\gamma$)16O. The order of
magnitude of astrophysical $S$ factors of $E1$ and $E2$ transitions of
12C($\alpha$,$\gamma$)16O is reproduced compared to the experimental data, and
we discuss an improvement in the calculation of $S$ factors of
12C($\alpha$,$\gamma$)16O.

PACS(s): 11.10.Ef, 24.10.-i, 25.55.-e, 26.20.Fj

1\. Introduction

The radiative $\alpha$ capture on 12C, 12C($\alpha$,$\gamma$)16O, is a
fundamental reaction in nuclear astrophysics, which determines, together with
the triple $\alpha$ process, the C/O ratio in the core of a helium-burning
star [1]. It plays a crucial role in simulations of nucleosynthesis in star
evolutions [2, 3], e.g., the radiative $\alpha$ capture reaction is one of the
most uncertain nuclear processes in different models of type Ia Supernova
nucleosynthesis [4]. The $\alpha$ capture rate, or alternatively the
astrophysical $S$ factor of 12C($\alpha$,$\gamma$)16O at the energy of helium-
burning process, namely the Gamow-peak energy, $E_{G}=0.3$ MeV, has never been
measured in an experimental facility because of the Coulomb barrier. One needs
to employ a theoretical model, whose parameters are fitted to experimental
data measured at a few MeV energy, and extrapolate the reaction rate down to
$E_{G}$. During the last half-century, many experimental and theoretical
studies have been carried out; refer to, e.g., Refs. [5, 6, 7, 8, 9] for
review.

During the last decade, we have been studying an estimate of the $S$ factor of
12C($\alpha$,$\gamma$)16O by employing the methodology of effective field
theory (EFT) [10, 11, 12]. An EFT is constructed by introducing a scale that
separates relevant degrees of freedom at low energy from irrelevant degrees of
freedom at high energy. Then, one constructs the most general form of
effective Lagrangian so as to satisfy the symmetry requirement and expands it
in powers of the number of derivatives; the theory provides us with a
perturbative expansion scheme in terms of $Q$/$\Lambda_{H}$ where $Q$ is a
typical momentum scale of a reaction in question and $\Lambda_{H}$ is a
momentum scale of high energy for separation. The irrelevant degrees of
freedom are integrated out of the Lagrangian and their effect is embedded in
the coefficients of terms of Lagrangian. Those coefficients, in principle, can
be determined within its mother theory while they, in practice, are fixed by
using empirical values or fitted to experimental data. Furthermore, the
inclusion of a resonance state in theory was studied by, e.g., Galman [13] and
Habashi, Fleming, and van Kolck [14]. EFTs are applied to various studies for
nuclear reactions at low energies, such as radiative neutron capture on a
proton for big-bang nucleosynthesis [15, 16, 17], proton-proton capture in
hydrogen burning [18, 19, 20], and solar neutrino reactions on a deuteron [21,
22].

To construct an EFT for the calculation of the astrophysical $S$ factor of
12C($\alpha$,$\gamma$)16O at $E_{G}$, we choose the energy difference between
the open channels of $\alpha$-12C and $p$-15N states, $\Delta E=4$ MeV, for
the separation scale, and one has the large momentum scale $\Lambda_{H}$ as
$\Lambda_{H}=\sqrt{2\mu\Delta E}=160$ MeV where $\mu$ is the reduced mass of
$\alpha$ and 12C. The relevant degrees of freedom for the theory are $\alpha$
and 12C represented as non-relativistic point-like scalar fields. The
effective Lagrangian is constructed so as to be invariant under the Galilean
and gauge transformations, and expanded in terms of the number of covariant
derivatives. The typical momentum scale at $E_{G}$ is $Q=\sqrt{2\mu E_{G}}=40$
MeV, and the expansion parameter becomes $Q/\Lambda_{H}=1/4$. We introduce
auxiliary fields for bound and resonant states of 16O to carry out a momentum
expansion around the unitary limit [23, 24, 25], which reproduces the
expression of effective range expansion [26]. In addition, it is
straightforward to introduce the electromagnetic and weak interactions in the
theory. In the previous works, we constructed an EFT for an estimate of $S$
factor of 12C($\alpha$,$\gamma$)16O at $E_{G}$ studying elastic $\alpha$-12C
scattering for various cases of including the bound and resonant states of 16O
[27, 28, 29, 30], $E1$ transition of 12C($\alpha$,$\gamma$)16O [31], and
$\beta$ delayed $\alpha$ emission from 16N [9].

In this work, we study the radiative decay of excited $1_{1}^{-}$ and
$2_{1}^{+}$ states of 16O within the cluster EFT. The main aim of the present
work is to seek constraints on the radiative capture amplitudes of
12C($\alpha$,$\gamma$)16O. Feynman diagrams of the radiative decay process are
parts of those of 12C($\alpha$,$\gamma$)16O, and one can fix the coupling
constants of $O^{*}\gamma O$ vertex functions by using the experimental data
of decay rates of $1_{1}^{-}$ and $2_{1}^{+}$ states of 16O. We also employ
the wave function normalization factors, equivalently the asymptotic
normalization coefficients (ANCs) of $0_{1}^{+}$, $1_{1}^{-}$, $2_{1}^{+}$
states of 16O, to fix the coupling constants of $\alpha CO$ vertex function
for the final $0_{1}^{+}$ ground state of 16O and $O^{*}\alpha C$ vertex
functions for the initial $1_{1}^{-}$ and $2_{1}^{+}$ states of 16O. Moreover,
we improve the regularization method for logarithmic divergence appearing from
loop diagrams. In the previous work [31], we employed two regularization
methods, sharp cutoff regularization and dimensional regularization. In this
work, we apply the dimensional regularization method to the calculation of
logarithmic divergence, which previously was regularized by introducing a
sharp cutoff. Then, we fit the couplings of $O^{*}\gamma O$ vertex functions
to the experimental data of decay rates of $1_{1}^{-}$ and $2_{1}^{+}$ states
of 16O as functions of the sharp cutoff parameter $r_{C}$; a mild cutoff
dependence is found compared to the result in the previous study [31]. Now
there are no free parameters in the amplitudes of 12C($\alpha$,$\gamma$)16O.
We calculate the $S$ factors of $E1$ and $E2$ transitions of
12C($\alpha$,$\gamma$)16O by using the fitted parameters and plot the curves
of $S$ factors with the experimental data. We find that the order of magnitude
of $S$ factors is reproduced and discuss how the calculation of $S$ factors of
12C($\alpha$,$\gamma$)16O can be improved.

The present work is organized as the following. In Sec. 2 the expression of
effective Lagrangian is reviewed, and in Sec. 3, the radiative decay
amplitudes are derived considering the regularization methods of loop diagrams
and the wave function normalization factors, equivalently the ANCs of
$0_{1}^{+}$, $1_{1}^{-}$, $2_{1}^{+}$ states of 16O. In Sec. 4, the numerical
results of fitting the parameters and plotting the $S$ factors of
12C($\alpha$,$\gamma$)16O are discussed, and finally in Sec. 5, the results
and discussion of this work are presented. In Appendix A, masses and formulae
of the decay rates for the present study are summarized, and in Appendix B,
the derivation of ANC of ground $0_{1}^{+}$ state of 16O from the phase shift
of elastic $\alpha$-12C scattering is discussed.

2\. Effective Lagrangian

As discussed in the introduction, the theory has a perturbative scheme in
terms of the number of derivatives: the expansion parameter is obtained as
$Q/\Lambda_{H}=1/4$. While we discussed a modification of the counting rules
for the effective range expansion for elastic $\alpha$-12C scattering at low
energies [28]; one or two orders of magnitude larger contributions (than the
term estimated by using a phase shift datum at the lowest energy in the
experiment [35]222 The lowest energy of the experimental data for the elastic
$\alpha$-12C scattering is $E_{\alpha}=2.6$ MeV where $E_{\alpha}$ is the
$\alpha$ energy in the lab frame. ) appear from the Coulomb self-energy term.
To subtract those unnaturally large contributions, we include the effective
range terms up to $p^{6}$ order as counter terms. In the study of $E1$
transition of 12C($\alpha$,$\gamma$)16O, on the other hand, it is known that
the $E1$ transition is suppressed between initial and final isospin singlet
states. The suppression factor can be seen in the term,
$(Z_{\alpha}/m_{\alpha}-Z_{C}/m_{C})$, in the amplitudes, where $Z_{\alpha}$
and $m_{\alpha}$ ($Z_{C}$ and $m_{C}$) are the proton number and mass of
$\alpha$ (12C); if one ignores the binding energies of nuclei,
$m_{\alpha}\simeq 4m_{N}$ and $m_{C}\simeq 12m_{N}$ where $m_{N}$ is the
nucleon mass, and $Z_{\alpha}=2$ and $Z_{C}=8$, the term
$(Z_{\alpha}/m_{\alpha}-Z_{C}/m_{C})$ vanishes. The prime contribution to the
$E1$ transition amplitudes comes out of the coupling constant of $O^{*}\gamma
O$ vertex function. This constant is supposed to contain dynamics from high
energy: a significant contribution may come out of the isovector channel,
namely, $p$-15N state. The $p$-15N open channel can appear in loop diagrams
instead of $\alpha$-12C state. While the $p$-15N state is integrated out and
its effect is embedded in the coupling constant of $O^{*}\gamma O$ vertex
function; we fitted the coupling constant to experimental data of $S$ factor
for $E1$ transition of 12C($\alpha$,$\gamma$)16O in the previous study [31].

The effective Lagrangian for radiative decay of $1_{1}^{-}$ and $2_{1}^{+}$
states of 16O is the same as that for $E1$ and $E2$ transitions of
12C($\alpha$,$\gamma$)16O [31]. Thus, one has

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\phi_{\alpha}^{\dagger}\left(iD_{0}+\frac{\vec{D}^{2}}{2m_{\alpha}}+\cdots\right)\phi_{\alpha}+\phi_{C}^{\dagger}\left(iD_{0}+\frac{\vec{D}^{2}}{2m_{C}}+\cdots\right)\phi_{C}$
(1)
$\displaystyle+\sum_{n=0}^{3}C_{n}^{(0)}d^{\dagger}\left[iD_{0}+\frac{\vec{D}^{2}}{2(m_{\alpha}+m_{C})}\right]^{n}d-y^{(0)}\left[d^{\dagger}(\phi_{\alpha}\phi_{C})+(\phi_{\alpha}\phi_{C})^{\dagger}d\right]$
$\displaystyle+\sum_{n=0}^{3}C_{n}^{(1)}d_{i}^{\dagger}\left[iD_{0}+\frac{\vec{D}^{2}}{2(m_{\alpha}+m_{C})}\right]^{n}d_{i}-y^{(1)}\left[d_{i}^{\dagger}(\phi_{\alpha}O_{i}^{(1)}\phi_{C})+(\phi_{\alpha}O_{i}^{(1)}\phi_{C})^{\dagger}d_{i}\right]$
$\displaystyle+\sum_{n=0}^{3}C_{n}^{(2)}d_{ij}^{\dagger}\left[iD_{0}+\frac{\vec{D}^{2}}{2(m_{\alpha}+m_{C})}\right]^{n}d_{ij}-y^{(2)}\left[d_{ij}^{\dagger}(\phi_{\alpha}O_{ij}^{(2)}\phi_{C})+(\phi_{\alpha}O_{ij}^{(2)}\phi_{C})^{\dagger}d_{ij}\right]$
$\displaystyle-h^{(1)}\frac{y^{(0)}y^{(1)}}{\mu}\left[({\cal
O}_{l}^{(1)}d)^{\dagger}d_{l}+\mbox{\rm
H.c.}\right]-h^{(2)}\frac{y^{(0)}y^{(2)}}{\mu^{2}}\left[({\cal
O}_{ij}^{(2)}d)^{\dagger}d_{ij}+\mbox{\rm H.c.}\right]+\cdots\,,$

where $\phi_{\alpha}$ ($m_{\alpha}$) and $\phi_{C}$ ($m_{C}$) are fields
(masses) of $\alpha$ and 12C, respectively. $D_{\mu}$ is the covariant
derivative, $D_{\mu}\phi=(\partial_{\mu}+ieZA_{\mu})\phi$, where $e$ is the
electric charge, $Z$ is the number of protons in a nucleus, and $A_{\mu}$ is
the photon field. The combination of the terms, $iD_{0}+\vec{D}^{2}/(2m)$, is
required for the invariance under the Galilean transformation. The dots denote
higher-order terms. $d$, $d_{i}$, $d_{ij}$ are fields of composite states of
$\alpha$ and 12C for $l=0,1,2$, respectively, which represent the bound states
of 16O 333 We suppress the terms of Lagrangian for resonance states of 16O for
the sake of simplicity, which are involved when fitting the phase shift data
of elastic $\alpha$-12C scattering. One can find the expression of the terms
in Eq. (6) in Ref. [30]. , where the spin states are represented as Cartesian
tensors of lank $l$. The coefficients $C_{n}^{(l)}$ correspond with the
effective range parameters in the form of $C_{n}^{(l)}/y^{(l)2}$ with
$l=0,1,2$, where $y^{(l)}$ are the coupling constants of $O^{*}\alpha C$ and
$O\alpha C$ vertex functions. As mentioned before, we include the effective
range parameters up to $p^{6}$ order; namely, $n=0,1,2,3$. $h^{(1)}$ and
$h^{(2)}$ are bare coupling constants of contact $O^{*}\gamma O$ vertex
functions, with which the infinities from loop diagrams are renormalized. The
projection operators in $p$-wave and $d$-wave read

$\displaystyle
O_{l}^{(1)}=i\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{i}}{M}\equiv
i\left(\frac{\stackrel{{\scriptstyle\rightarrow}}{{D}}_{C}}{m_{C}}-\frac{\stackrel{{\scriptstyle\leftarrow}}{{D}}_{\alpha}}{m_{\alpha}}\right)_{i}\,,\
\ \
O_{ij}^{(2)}=-\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{i}}{M}\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{j}}{M}+\frac{1}{3}\delta_{ij}\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{D}}^{2}}{M^{2}}\,,$
(2) $\displaystyle{\cal O}_{i}^{(1)}=\frac{iD_{i}}{m_{O}}\,,\ \ \ {\cal
O}_{ij}^{(2)}=-\frac{1}{m_{O}^{2}}(D_{i}D_{j}-\frac{1}{3}\delta_{ij}D^{2})\,,$
(3)

where $m_{O}$ is the mass of ground $0_{1}^{+}$ state of 16O.

3\. Radiative decay amplitudes

In this section, we derive expressions of the decay amplitudes from the
effective Lagrangian. We discuss an improvement in regularization methods of
the log divergence from loop diagrams and the calculation of wave function
normalization factors by using the phase shift data of elastic $\alpha$-12C
scattering; final expressions of the decay amplitudes of $1_{1}^{-}$ and
$2_{1}^{+}$ states of 16O contain only one parameter for each of the
amplitudes, and the parameters are fitted to the experimental decay rates in
the next section.

3.1 Amplitudes

The radiative decay amplitudes $A^{(l)}$ with $l=1,2$ are calculated from
diagrams displayed in Fig. 1, which are parts of the diagrams for
12C($\alpha$,$\gamma$)16O (displayed in Fig. 2 in Ref. [31]).

Figure 1:  Diagrams of radiative decay amplitudes for 16O∗($1_{1}^{-}$)
$\to^{16}$O${}_{g.s.}+\gamma$ and 16O∗($2_{1}^{+}$)
$\to^{16}$O${}_{g.s.}+\gamma$. A thick (thin) dashed line represents a
propagator of 12C ($\alpha$), and a thick and thin double-dashed line with a
filled circle represents the initial and final states of 16O, and a wavy line
the outgoing photon. A shaded blob represents a set of diagrams consisting of
all possible one-potential-photon-exchange diagrams up to infinite order and
no potential-photon-exchange one, namely, the Coulomb greens function. A
filled box represents the contact interactions of $O^{*}\gamma O$ vertex
functions and renormalize the divergence terms from the loop diagrams.

The amplitudes $A^{(l)}$ with $l=1,2$ are represented as

$\displaystyle A^{(l=1)}$ $\displaystyle=$
$\displaystyle\vec{\epsilon}_{(\gamma)}^{*}\cdot\vec{\epsilon}_{(l=1)}Y^{(l=1)}\,,$
(4) $\displaystyle A^{(l=2)}$ $\displaystyle=$
$\displaystyle\epsilon_{(\gamma)}^{*i}\hat{k}_{2}^{j}\epsilon_{(l=2)}^{ij}Y^{(l=2)}\,,$
(5)

with

$\displaystyle Y^{(l)}$ $\displaystyle=$ $\displaystyle
Y_{(a+b)}^{(l)}+Y_{(c)}^{(l)}+Y_{(d)}^{(l)}\,,$ (6)

where $\vec{\epsilon}_{(\gamma)}^{*}$ is the polarization vector of out-going
photon, $\vec{\epsilon}_{(l=1)}$ is the polarization vector of initial
$p$-wave state of 16O∗($1_{1}^{-}$), and $\hat{k}_{2}$ is a unit vector of the
momentum $\vec{k}_{2}$ of out-going photon from the decay of $2_{1}^{+}$ state
of 16O, and $\epsilon_{(l=2)}^{ij}$ is the symmetric traceless tensor of
initial $d$-wave state of 16O∗($2_{1}^{+}$). The amplitudes,
$Y_{(a+b)}^{(l)}$, $Y_{(c)}^{(l)}$, $Y_{(d)}^{(l)}$, correspond to the
diagrams displayed in Fig. 1. Masses and decay rates for the present study are
summarized in Appendix A. Thus, the spin-averaged decay rates in Eq. (45)
become

$\displaystyle\Gamma^{(l=1)}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{E}k_{1}}{3\pi}|Y^{(l=1)}|^{2}\,,$ (7)
$\displaystyle\Gamma^{(l=2)}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{E}k_{2}}{5\pi}|Y^{(l=2)}|^{2}\,,$ (8)

where $\alpha_{E}$ is the fine structure constant, and $k_{1}$ and $k_{2}$ are
the magnitude of photon momenta in Eqs. (43) and (44) for radiative decay of
$1_{1}^{-}$ and $2_{1}^{+}$ states of 16O, respectively. The amplitudes,
$Y_{(a+b)}^{(l)}$, $Y_{(c)}^{(l)}$, and $Y_{(d)}^{(l)}$, are obtained as

$\displaystyle Y_{(a+b)}^{(l=1)}$ $\displaystyle=$
$\displaystyle+\frac{1}{9\pi}y^{(0)}y^{(1)}\gamma_{1}\Gamma(1+\kappa/\gamma_{0})\Gamma(2+\kappa/\gamma_{1})$
(9)
$\displaystyle\times\int^{\infty}_{0}drrW_{-\kappa/\gamma_{0},\frac{1}{2}}(2\gamma_{0}r)\left[\frac{\mu
Z_{\alpha}}{m_{\alpha}}j_{0}\left(\frac{\mu}{m_{\alpha}}k_{1}r\right)-\frac{\mu
Z_{C}}{m_{C}}j_{0}\left(\frac{\mu}{m_{C}}k_{1}r\right)\right]$
$\displaystyle\times\left\\{\frac{\partial}{\partial
r}\left[\frac{W_{-\kappa/\gamma_{1},\frac{3}{2}}(2\gamma_{1}r)}{r}\right]+2\frac{W_{-\kappa/\gamma_{1},\frac{3}{2}}(2\gamma_{1}r)}{r^{2}}\right\\}\,,$
$\displaystyle Y_{(c)}^{(l=1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}y^{(0)}y^{(1)}\mu\left(\frac{Z_{\alpha}}{m_{\alpha}}-\frac{Z_{C}}{m_{C}}\right)\left[\frac{2\pi}{\mu}J_{0}^{div}-2\kappa
H(-i\kappa/\gamma_{0})\right]\,,$ (10) $\displaystyle Y_{(d)}^{(l=1)}$
$\displaystyle=$ $\displaystyle-h^{(1)}y^{(0)}y^{(1)}\frac{Z_{O}}{\mu
m_{O}}\,,$ (11) $\displaystyle Y_{(a+b)}^{(l=2)}$ $\displaystyle=$
$\displaystyle-\frac{2}{75\pi}y^{(0)}y^{(2)}\frac{\gamma_{2}^{2}}{\mu}\Gamma(1+\kappa/\gamma_{0})\Gamma(3+\kappa/\gamma_{2})$
(12)
$\displaystyle\times\int_{0}^{\infty}drrW_{-\kappa/\gamma_{0},\frac{1}{2}}(2\gamma_{0}r)\left[\frac{\mu
Z_{\alpha}}{m_{\alpha}}j_{1}\left(\frac{\mu}{m_{\alpha}}k_{2}r\right)+\frac{\mu
Z_{C}}{m_{C}}j_{1}\left(\frac{\mu}{m_{C}}k_{2}r\right)\right]$
$\displaystyle\times\left\\{\frac{\partial}{\partial
r}\left[\frac{W_{-\kappa/\gamma_{2},\frac{5}{2}}(2\gamma_{2}r)}{r}+3\frac{W_{-\kappa/\gamma_{2},\frac{5}{2}}(2\gamma_{2}r)}{r^{2}}\right]\right\\}\,,$
$\displaystyle Y_{(c)}^{(l=2)}$ $\displaystyle=$ $\displaystyle 0\,,$ (13)
$\displaystyle Y_{(d)}^{(l=2)}$ $\displaystyle=$
$\displaystyle+h^{(2)}y^{(0)}y^{(2)}\frac{Z_{O}}{\mu^{2}m_{O}^{2}}k_{2}\,,$
(14)

where $\Gamma(x)$, $j_{l}(x)$ ($l=0,1$), and $W_{\alpha,\beta}(x)$ are the
gamma function, the spherical Bessel function, and the Whittaker function,
respectively. $\mu$ is the reduced mass of $\alpha$ and 12C,
$\mu=m_{\alpha}m_{C}/(m_{\alpha}+m_{C})$ = 2795 MeV. $\gamma_{0}$,
$\gamma_{1}$, and $\gamma_{2}$ are the binding momenta for $0_{1}^{+}$,
$1_{1}^{-}$, and $2_{1}^{+}$ states of 16O, respectively;
$\gamma_{l}=\sqrt{2\mu B_{l}}$ with $l=0,1,2$ where $B_{l}$ are the binding
energies. Thus, one has $\gamma_{0}=200.1$ MeV, $\gamma_{1}=15.88$ MeV, and
$\gamma_{2}=37.00$ MeV. $\kappa$ is the inverse of Bohr radius,
$\kappa=\alpha_{E}Z_{\alpha}Z_{C}\mu$, where $Z_{\alpha}$ and $Z_{C}$ (and
$Z_{O}$ in Eq. (14)) are the number of protons in $\alpha$ and 12C (and 16O);
$Z_{\alpha}=2$ and $Z_{C}=6$ (and $Z_{O}=8$), respectively; thus, $\kappa=245$
MeV. In addition, the function $H(\eta)$ in Eq. (10) reads

$\displaystyle H(\eta)$ $\displaystyle=$
$\displaystyle\psi(i\eta)+\frac{1}{2i\eta}-\ln(i\eta)\,,$ (15)

where $\psi(z)$ is the digamma function and $\eta$ is the Sommerfeld
parameter. (We will mention the function $J_{0}^{div}$ in Eq. (10) in the next
subsection.)

One may notice that there are five constants, {$y^{(0)}$, $y^{(1)}$,
$y^{(2)}$, $h^{(1)}$, $h^{(2)}$}, appearing in the amplitudes. Three of them,
$y^{(0)}$, $y^{(1)}$, $y^{(2)}$, are determined by using the wave function
normalization factors, and the remaining two parameters, $h^{(1)}$ and
$h^{(2)}$, are fitted to the radiative decay data. Before fixing the
parameters, we discuss the renormalization of infinities from one-loop
diagrams in the next subsection.

3.2 Renormalization of the loop diagrams

The one-loop diagrams (a), (b), and (c) in Fig. 1 diverge. The divergent terms
are subtracted by the counter terms, $h^{(1)}$ and $h^{(2)}$, in the diagram
(d) in Fig. 1. For the diagrams (a) and (b), the $r$-space integrals in Eqs.
(9) and (12) diverge: the log divergence appears in the limit where $r$ goes
to zero. In the previous work in Ref. [31], we introduced a sharp-cutoff
$r_{C}$ in the $r$-space integral for the radiative capture amplitude of $E1$
transition of 12C($\alpha$,$\gamma$)16O, and the divergent term was
renormalized by the counter term, $h^{(1)}$. We found that the cutoff
dependence was severe, and moreover, it was inconsistent with the
regularization method, namely the dimensional regularization, which was
applied to the calculation of diagram (c). In this work, we separate the
$r$-space integrals in Eqs. (9) and (12) into two parts by introducing a
cutoff parameter $r_{C}$, and the calculation of the log-diverging integral in
the short-range part is carried out by employing the dimensional
regularization in $d=4-2\epsilon$ space-time dimensions, ignoring the other
finite terms. Thus, a minor cutoff dependence remains in the numerical result.
The log divergence of the short-range part is replaced as

$\displaystyle\int_{0}^{r_{C}}\frac{dr}{r}$ $\displaystyle\to$
$\displaystyle\left(\frac{\mu_{DR}}{2}\right)^{2\epsilon}\int_{0}^{r_{C}}drr^{-1+2\epsilon}=\frac{1}{2\epsilon}+\ln\left(\frac{\mu_{DR}}{2}r_{C}\right)+O(\epsilon)\,,$
(16)

where $\mu_{DR}$ is a scale parameter from the dimensional regularization. For
the diagram (c) in Fig. 1, we employ dimensional regularization and calculate
the loop integral in $d=4-2\epsilon$ space-time dimension; one has the
divergence term $J_{0}^{div}$ as

$\displaystyle J_{0}^{div}$ $\displaystyle=$
$\displaystyle\frac{\kappa\mu}{2\pi}\left[\frac{1}{\epsilon}-3C_{E}+2+\ln\left(\frac{\pi\mu_{DR}^{2}}{4\kappa^{2}}\right)\right]\,,$
(17)

where $C_{E}$ is the Euler-Mascheroni constant, $C_{E}=0.5772\cdots$.

We subtract the divergent terms being proportional to $1/(2\epsilon)$ as well
as some constant terms by using the coupling constants, $h^{(1)}$ and
$h^{(2)}$, as

$\displaystyle
h^{(1)}+\frac{\kappa\mu^{2}}{9\pi}\frac{m_{O}}{Z_{O}}\left(\frac{Z_{\alpha}}{m_{\alpha}}-\frac{Z_{C}}{m_{C}}\right)\left[\left(\frac{\mu_{DR}}{2}\right)^{2\epsilon}\int_{0}^{r_{C}}\frac{dr}{r^{1-2\epsilon}}-\frac{9\pi}{\kappa\mu}J_{0}^{div}\right]$
(18)
$\displaystyle=h_{R}^{(1)}+\frac{\kappa\mu^{2}}{9\pi}\frac{m_{O}}{Z_{O}}\left(\frac{Z_{\alpha}}{m_{\alpha}}-\frac{Z_{C}}{m_{C}}\right)\left[\ln\left(\frac{\mu_{DR}}{2}r_{C}\right)-9\ln\left(\frac{\mu_{DR}}{2\kappa}\right)\right]\,,$
$\displaystyle
h^{(2)}+\frac{2\kappa\mu^{3}}{75\pi}\frac{m_{O}^{2}}{Z_{O}}\left(\frac{Z_{\alpha}}{m_{\alpha}^{2}}+\frac{Z_{C}}{m_{C}^{2}}\right)\left(\frac{\mu_{DR}}{2}\right)^{2\epsilon}\int_{0}^{r_{C}}\frac{dr}{r^{1-2\epsilon}}$
(19)
$\displaystyle=h_{R}^{(2)}+\frac{2\kappa\mu^{3}}{75\pi}\frac{m_{O}^{2}}{Z_{O}}\left(\frac{Z_{\alpha}}{m_{\alpha}^{2}}+\frac{Z_{C}}{m_{C}^{2}}\right)\ln\left(\frac{\mu_{DR}}{2}r_{C}\right)\,,$

where $h_{R}^{(1)}$ and $h_{R}^{(2)}$ are the renormalized coupling constants
and fitted to the experimental radiative decay rates in the next section.

3.3 Wave function normalization factors

The coefficients, {$y^{(0)}$, $y^{(1)}$, $y^{(2)}$}, appeared in the
amplitudes can be fixed by using the wave function normalization factors,
$\sqrt{{\cal Z}_{l}}$ with $l=0,1,2$, which are related to the ANCs of
$0_{1}^{+}$, $1_{1}^{-}$, $2_{1}^{+}$ states of 16O for the two-body
$\alpha$-12C system. The relations between $y^{(l)}$ and $\sqrt{{\cal Z}_{l}}$
may be given as [32]444 The wave function normalization factors, $\sqrt{{\cal
Z}_{l}}$ with $l=0,1,2$, appear only in the initial and final states of
reaction amplitudes. $y^{(l)}$ appearing in the intermediate states are
represented, excluding $\sqrt{{\cal Z}_{l}}$, as
$y^{(l)}=\sqrt{2(2l+1)\pi\mu^{2l-1}}$.

$\displaystyle y^{(l)}$ $\displaystyle=$
$\displaystyle\sqrt{2(2l+1)\pi\mu^{2l-1}}\sqrt{{\cal Z}_{l}}\,,$ (20)

with $l=0,1,2$. ${\cal Z}_{l}$ are defined in the the inverse of dressed 16O
propagators, $D_{l}(p)$, for the two-body $\alpha$-12C system, where $p$ is
the relative momentum between $\alpha$ and 12C. Diagrams of dressed 16O
propagators are depicted in Fig. 2,

Figure 2:  Diagrams for dressed 16O propagator. A thick and thin double dashed
line with and without a filled circle represents a dressed and bare 16O
propagator, respectively. See the caption of Fig. 1 as well.

and one has

$\displaystyle\frac{1}{D_{l}(p)}$ $\displaystyle=$
$\displaystyle\frac{1}{K_{l}(p)-2\kappa H_{l}(p)}=\frac{{\cal
Z}_{l}}{E+B_{l}}+\cdots\,,$ (21)

where $E$ is the kinetic energy of $\alpha$ and 12C, $E=p^{2}/(2\mu)$, and
$B_{l}$ are the binding energies of bound states of 16O as the $\alpha$-12C
two-body system. The dots denote finite terms at $E=-B_{l}$. The term,
$-2\kappa H_{l}(p)$, is the Coulomb self-energy term from the one-loop diagram
in Fig. 2, and one has

$\displaystyle H_{l}(p)$ $\displaystyle=$ $\displaystyle W_{l}(p)H(\eta)\,,$
(22) $\displaystyle W_{l}(p)$ $\displaystyle=$
$\displaystyle\left(\frac{\kappa^{2}}{l^{2}}+p^{2}\right)W_{l-1}(p)\,,\ \ \
W_{0}(p)=1\,.$ (23)

The term, $K_{l}(p)$, is represented in terms of the effective range
parameters as

$\displaystyle K_{l}(p)$ $\displaystyle=$
$\displaystyle-\frac{1}{a_{l}}+\frac{1}{2}r_{l}p^{2}-\frac{1}{4}P_{l}p^{4}+Q_{l}p^{6}\,,$
(24)

where one or two parameters of the effective range parameters are fixed by
using the condition that the inverse of the propagator $D_{l}(p)$ vanishes at
the binding momenta $\gamma_{l}=\sqrt{2\mu B_{l}}$, $D_{l}(i\gamma_{l})=0$.
Thus, two parameters are fixed by using the binding energies of $0_{1}^{+}$
and $0_{2}^{+}$ states of 16O for $l=0$, and one parameter is fixed by using
the binding energies of $1_{1}^{-}$ and $2_{1}^{+}$ states of 16O for $l=1,2$.
We fix the scattering length and effective range $a_{0}$ and $r_{0}$ for $l=0$
and the scattering lengths $a_{l}$ for $l=1,2$. Thus, one has the inverse of
propagators as [28, 29]

$\displaystyle D_{0}(p)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\left[\gamma_{01}^{2}\gamma_{02}^{2}+(\gamma_{01}^{2}+\gamma_{02}^{2})p^{2}+p^{4}\right]P_{0}$
(25)
$\displaystyle+\left[-\gamma_{01}^{4}\gamma_{02}^{2}-\gamma_{01}^{2}\gamma_{02}^{4}-(\gamma_{01}^{4}+\gamma_{01}^{2}\gamma_{02}^{2}+\gamma_{02}^{4})p^{2}+p^{6}\right]Q_{0}$
$\displaystyle-2\kappa\left[\frac{\gamma_{02}^{2}+p^{2}}{\gamma_{01}^{2}-\gamma_{02}^{2}}H_{0}(i\gamma_{01})-\frac{\gamma_{01}^{2}+p^{2}}{\gamma_{01}^{2}-\gamma_{02}^{2}}H_{0}(i\gamma_{02})+H_{0}(p)\right]\,,$
$\displaystyle D_{l}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}r_{l}(\gamma_{l}^{2}+p^{2})+\frac{1}{4}P_{l}(\gamma_{l}^{4}-p^{4})+Q_{l}(\gamma_{l}^{6}+p^{6})+2\kappa\left[H_{l}(i\gamma_{l})-H_{l}(p)\right]\,,$
(26)

with $l=1,2$: $\gamma_{01}(=\gamma_{0})$ and $\gamma_{02}$ are the bounding
momenta of $0_{1}^{+}$ and $0_{2}^{+}$ states of 16O, respectively;
$\gamma_{02}=\sqrt{2\mu B_{02}}$, where $B_{02}$ is the binding energy of the
$0_{2}^{+}$ state of 16O as the two-body $\alpha$-12C system, and one has
$\gamma_{02}=78.86$ MeV.

The other effective range parameters in Eqs. (25) and (26) are fitted to the
phase shift data of elastic $\alpha$-12C scattering.

Figure 3:  Diagram of the scattering amplitude. Shaded blobs in the initial
and final states represent non-perturbative Coulomb interaction, namely, the
Coulomb wave functions. See the captions of Figs. 1 and 2 as well.

The elastic scattering amplitudes are calculated from a diagram depicted in
Fig. 3. Expression of the $S$ matrices, amplitudes, and fitted values of some
parameters can be found in Ref. [30]. Thus, one may calculate the wave
function normalization factors $\sqrt{{\cal Z}_{l}}$ from the inverse of
propagators $D_{l}(p)$ in Eq. (21) as

$\displaystyle\sqrt{{\cal Z}_{l}}$ $\displaystyle=$
$\displaystyle\left(2\mu\left|\frac{\partial D_{l}(p)}{\partial
p^{2}}\right|_{p^{2}=-\gamma_{l}^{2}}\right)^{-1/2}\,,$ (27)

and through the relations in Eq. (20), those factors are multiplied to the
amplitudes, as the coupling constants, $y^{(l)}$, for the initial $1_{1}^{-}$
or $2_{1}^{+}$ state and final $0_{1}^{+}$ state of 16O. In addition, the wave
function normalization factors are related to ANCs through the formula derived
by Iwinski, Rosenberg, and Spruch [33]:

$\displaystyle|C_{b}|_{l}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{l}^{l}}{l!}\Gamma(l+1+\kappa/\gamma_{l})\sqrt{2\mu{\cal
Z}_{l}}\,.$ (28)

Now, we rewrite the amplitudes of radiative decay reactions in terms of the
wave function renormalization factors as

$\displaystyle Y_{(a+b)}^{(l=1)}=+\frac{16}{3\sqrt{3}}\sqrt{{\cal Z}_{01}{\cal
Z}_{1}}\,\Gamma(1+\kappa/\gamma_{0})\Gamma(2+\kappa/\gamma_{1})\gamma_{1}^{2}$
(29)
$\displaystyle\times\int^{\infty}_{r_{C}}dre^{-(\gamma_{0}+\gamma_{1})r}\gamma_{0}\gamma_{1}r^{2}U(1+\kappa/\gamma_{0},2,2\gamma_{0}r)\left[\frac{\mu
Z_{\alpha}}{m_{\alpha}}j_{0}\left(\frac{\mu}{m_{\alpha}}k_{1}r\right)-\frac{\mu
Z_{C}}{m_{C}}j_{0}\left(\frac{\mu}{m_{C}}k_{1}r\right)\right]$
$\displaystyle\times\left[(3-\gamma_{1}r)U(2+\kappa/\gamma_{1},4,2\gamma_{1}r)-2\gamma_{1}r(2+\kappa/\gamma_{1})U(3+\kappa/\gamma_{1},5,2\gamma_{1}r)\right]\,,$
$\displaystyle Y_{(c)}^{(l=1)}=+\sqrt{3}\sqrt{{\cal Z}_{01}{\cal
Z}_{1}}\,\mu\left(\frac{Z_{\alpha}}{m_{\alpha}}-\frac{Z_{C}}{m_{C}}\right)[-2\kappa
H(-i\kappa/\gamma_{0})]\,,$ (30) $\displaystyle
Y_{(d)}^{(l=1)}=-2\sqrt{3}\,\pi\sqrt{{\cal Z}_{01}{\cal
Z}_{1}}\frac{Z_{O}}{\mu m_{O}}$ (31)
$\displaystyle\times\left\\{h_{R}^{(1)}+\frac{\kappa\mu^{2}}{9\pi}\frac{m_{O}}{Z_{O}}\left(\frac{Z_{\alpha}}{m_{\alpha}}-\frac{Z_{C}}{m_{C}}\right)\left[\ln\left(\frac{\mu_{DR}}{2}r_{C}\right)-9\ln\left(\frac{\mu_{DR}}{2\kappa}\right)\right]\right\\}\,,$
$\displaystyle Y_{(a+b)}^{(l=2)}=-\frac{64\sqrt{5}}{75}\sqrt{{\cal
Z}_{01}{\cal
Z}_{2}}\,\gamma_{2}^{3}\,\Gamma(1+\kappa/\gamma)\Gamma(3+\kappa/\gamma_{2})$
(32)
$\displaystyle\times\int_{r_{C}}^{\infty}dre^{-(\gamma_{0}+\gamma_{2})r}\gamma_{0}\gamma_{2}^{2}r^{3}U(1+\kappa/\gamma_{0},2,2\gamma_{0}r)\left[\frac{\mu
Z_{\alpha}}{m_{\alpha}}j_{1}\left(\frac{\mu}{m_{\alpha}}k_{2}r\right)+\frac{\mu
Z_{C}}{m_{C}}j_{1}\left(\frac{\mu}{m_{C}}k_{2}r\right)\right]$
$\displaystyle\times\left[(5-\gamma_{2}r)U(3+\kappa/\gamma_{2},6,2\gamma_{2}r)-2\gamma_{2}r(3+\kappa/\gamma_{2})U(4+\kappa/\gamma_{2},7,2\gamma_{2}r)\right]\,,$
$\displaystyle Y_{(c)}^{(l=2)}=0\,,$ (33) $\displaystyle
Y_{(d)}^{l=2)}=+2\sqrt{5}\,\pi\sqrt{{\cal Z}_{01}{\cal
Z}_{2}}\frac{Z_{O}k_{2}}{\mu
m_{O}^{2}}\left[h_{R}^{(2)}+\frac{2\kappa\mu^{3}}{75\pi}\frac{m_{O}^{2}}{Z_{O}}\left(\frac{Z_{\alpha}}{m_{\alpha}^{2}}+\frac{Z_{C}}{m_{C}^{2}}\right)\ln\left(\frac{\mu_{DR}}{2}r_{C}\right)\right]\,,$
(34)

where we have also rewritten the Whittaker function, $W_{\kappa,\mu}(z)$, in
terms of the Kummer function, $U(a,b,c)$. One may notice that the combinations
for the ANCs appear in the amplitudes, $Y_{(a+b)}^{(l=1)}$ and
$Y_{(a+b)}^{(l=2)}$, in Eqs. (29) and (32).

4\. Numerical results

In this section, we first briefly review the calculation of the wave function
renormalization factors, $\sqrt{{\cal Z}_{01}}$, $\sqrt{{\cal Z}_{1}}$,
$\sqrt{{\cal Z}_{2}}$, by fitting the effective range parameters to the
experimental phase shift data of elastic $\alpha$-12C scattering at low
energies. In the following, we quote the values of the ANCs rather than those
of $\sqrt{{\cal Z}_{l}}$ because the ANCs are more reliable quantities for the
other theoretical studies. Then, we fit the remaining two parameters,
$h_{R}^{(1)}$ and $h_{R}^{(2)}$, to the radiative decay rates of sub-threshold
$1_{1}^{-}$ and $2_{1}^{+}$ states of 16O. After fitting the parameters, there
is no free parameter in $E1$ and $E2$ transitions of amplitudes of
12C($\alpha$,$\gamma$)16O, and we perform a parameter-free calculation of the
astrophysical $S$ factors for $E1$ and $E2$ transitions of
12C($\alpha$,$\gamma$)16O. We will see that the order of magnitude of the $S$
factors compared to the experimental data is reproduced.

As discussed in the previous section, the coupling constants, {$y^{(0)}$,
$y^{(1)}$, $y^{(2)}$}, are fixed by using the wave function normalization
factors, $\sqrt{{\cal Z}_{01}}$, $\sqrt{{\cal Z}_{1}}$, $\sqrt{{\cal Z}_{2}}$,
which are related to the ANCs through the relation in Eq. (28). In the
previous work, we obtained the ANCs for $1_{1}^{-}$ and $2_{1}^{+}$ states of
16O as [30]

$\displaystyle|C_{b}|_{1}$ $\displaystyle=$ $\displaystyle 1.727(3)\times
10^{14}\ \ \textrm{fm}^{-1/2}\,,$ (35) $\displaystyle|C_{b}|_{2}$
$\displaystyle=$ $\displaystyle 3.1(6)\times 10^{4}\ \ \textrm{fm}^{-1/2}\,,$
(36)

by fitting the effective range parameters to the phase shift data [35]. We
employ the fitted values of effective range parameters in TABLE II in Ref.
[30] to calculate $\sqrt{{\cal Z}_{1}}$ and $\sqrt{{\cal Z}_{2}}$. The ANC of
$1_{1}^{-}$ state agrees with the other results in literature while that of
$2_{1}^{+}$ state is about four times smaller than the other ones; the fit of
effective range parameters for $l=2$ is sensitive to the conditions of
effective range parameters imposed on the low-energy region where the
experimental phase shift data are not available [36].

To calculate $\sqrt{{\cal Z}_{01}}$, we need to fix the values of two
effective range parameters, $P_{0}$ and $Q_{0}$, in Eq. (25); we refit the
parameters to the phase shift data including the $0_{1}^{+}$, $0_{2}^{+}$,
$0_{3}^{+}$, $0_{4}^{+}$ states of 16O. Details of the fit of the parameters
$P_{0}$ and $Q_{0}$ are discussed in Appendix B. Thus, we have the ANC of
ground $0_{1}^{+}$ state of 16O as

$\displaystyle|C_{b}|_{01}=44.5(3)\ \ \textrm{fm}^{-1/2}\,,$ (37)

where the corresponding value of $y^{(0)}$ is $y^{(0)}=0.355(3)$ MeV-1/2.
Those quantities, $|C_{b}|_{01}$ and $y^{(0)}$, are related through Eqs. (20)
and (28);
$|C_{b}|_{01}=\frac{\mu}{\sqrt{\pi}}\Gamma(1+\kappa/\gamma_{01})y^{(0)}$. It
may be worth pointing out that this value of $|C_{b}|_{01}$ is close to that
used in the recent $R$-matrix analysis for 12C($\alpha$,$\gamma$)16O reported
by deBoer et al., $|C_{b}|_{01}=58$ fm-1/2 [8]. The reported values of the
ANC, $|C_{b}|_{01}$, in literature are still scattered, 13.9(24) to 4000
fm-1/2; see TABLE XVI in Ref. [8]. In addition, we fitted the values of
$y^{(0)}$, along with the constant $h_{R}^{(1)}$ to the experimental data of
$S_{E1}$ factor of 12C($\alpha$,$\gamma$)16O by using the sharp cutoff
regularization scheme in the previous work [31] and had $y_{0}=0.253(9)$ –
$2.249(84)$ MeV-1/2 with the cutoff values $r_{C}=0.01$ – $0.35$ fm.

$r_{C}$ (fm) |  $h_{R}^{(1)}\times 10^{-5}$ (MeV3) |  $h_{R}^{(1)}$ (MeV3)[31] |  $y^{(0)}$ (MeV-1/2)[31]
---|---|---|---
0.005 | 2.503(2), 2.635(2) | – | –
0.01 | 2.626(2), 2.757(2) | 5.268(1)$\times 10^{5}$ | 0.253(9)
0.035 | 2.775(2), 2.906(2) | 2.448(1)$\times 10^{5}$ | 0.310(11)
0.05 | 2.825(2), 2.957(2) | 1.529(1)$\times 10^{5}$ | 0.347(12)
0.10 | 2.957(2), 3.088(2) | -0.070(1)$\times 10^{5}$ | 0.495(18)
Table 1:  Two values of $h_{R}^{(1)}$ in the second column are obtained by
fitting to the radiative decay rate of $1_{1}^{-}$ state of 16O,
$\Gamma_{1}^{exp}$, as a function of the cutoff parameter $r_{C}$ employing
the dimensional regularization where the scale parameter $\mu_{DR}$ is fixed
as $\mu_{DR}=\Lambda_{H}$. Values of $h_{R}^{(1)}$ and $y^{(0)}$ in the third
and fourth columns are fitted values in the previous work [31], where
$h_{R}^{(1)}$ and $y^{(0)}$ are treated as free parameters and fitted to the
experimental data of $S_{E1}$ factor of radiative $\alpha$ capture on 12C
employing the sharp cutoff regularization method.

(The result of fitted values of $h_{R}^{(1)}$ and $y^{(0)}$ for the cutoff
values $r_{C}=0.01-0.10$ fm is included in Table 1.) The $y^{(0)}$ values are
converted to the ANC as $|C_{b}|_{01}=31.7(1)-282(10)~{}\textrm{fm}^{-1/2}$.
One may see that the value of ANC in Eq. (37) is reproduced at $r_{C}\simeq
0.05$ fm. In this work, we fix the value of $y^{(0)}$ by using the ANC of
ground $0_{1}^{+}$ state of 16O in Eq. (37) in the radiative decay amplitudes.

Now, the coupling constants $h_{R}^{(1)}$ and $h_{R}^{(2)}$ are fitted to the
experimental data of radiative decay rates of $1_{1}^{-}$ and $2_{1}^{+}$
states of 16O [34],

$\displaystyle\Gamma^{exp}_{1}$ $\displaystyle=$ $\displaystyle 0.055(3)\ \
\textrm{eV}\,,$ (38) $\displaystyle\Gamma^{exp}_{2}$ $\displaystyle=$
$\displaystyle 0.097(3)\ \ \textrm{eV}\,,$ (39)

as a function of the cutoff $r_{C}$ where the scale parameter $\mu_{DR}$ is
fixed as $\mu_{DR}=\Lambda_{H}$. The short-range part of integrals in the
amplitudes, $Y_{(a+b)}^{l=1)}$ and $Y_{(a+b)}^{(l=2)}$ in Eqs. (29) and (32),
respectively, are replaced by the $1/r$ integrals in Eq. (16); the finite
components are ignored and the cutoff dependence remains in the numerical
results. The fitted values of $h_{R}^{(1)}$ and $h_{R}^{(2)}$ are presented in
Tables 1 and 2, respectively. The fitted values of $h_{R}^{(1)}$ and $y^{(0)}$
in the previous work [31] are displayed in Table 1 as well.

$r_{C}$ (fm) |  $h_{R}^{(2)}\times 10^{-11}$ (MeV4)
---|---
0.005 | 8.414(2), 8.627(2)
0.01 | 7.759(2), 7.971(2)
0.035 | 7.013(2), 7.226(2)
0.05 | 6.774(2), 6.986(2)
0.10 | 6.146(2), 6.359(2)
Table 2:  Values of $h_{R}^{(2)}$ are obtained by fitting to the radiative
decay rate of $2_{1}^{+}$ state of 16O, $\Gamma_{2}^{exp}$, as a function of
the cutoff parameter $r_{C}$. See the caption of Table 1 as well.

We find that there are two values of $h_{R}^{(1)}$ and $h_{R}^{(2)}$ with
small error bars. This indicates a large cancelation between the amplitude of
the loop diagrams and that of the counter term resulting in two same-size
amplitudes with different signs. The cutoff dependence remains in the fitted
values of $h_{R}^{(1)}$ and $h_{R}^{(2)}$ while that of $h_{R}^{(1)}$ becomes
much milder than the previous fitted values obtained by means of the sharp
cutoff regularization method.

Figure 4:  Curves of $S_{E1}$ factor of radiative $\alpha$ capture on 12C are
plotted as a function of $\alpha$-12C energy $E$ in the center-of-mass frame
by using the large and small values of $h_{R}^{(1)}$ at $r_{C}=0.01$ fm in
Table 1. Experimental data are displayed in the figure as well.  Figure 5:
Curves of $S_{E2}$ factor of radiative $\alpha$ capture on 12C are plotted as
a function of $\alpha$-12C energy $E$ in the center-of-mass frame by using the
small and large values of $h_{R}^{(2)}$ at $r_{C}=0.01$ fm in Table 2.
Experimental data are displayed in the figure as well.

In Figs. 4 and 5, we plot the astronomical $S_{E1}$ and $S_{E2}$ factors for
$E1$ and $E2$ transitions of radiative $\alpha$ capture on 12C as a function
of the energy $E$ of $\alpha$-12C in the center-of-mass frame by using the
large and small values of $h_{R}^{(1)}$ and $h_{R}^{(2)}$ with $r_{C}=0.01$ fm
in Tables 1 and 2, respectively. Experimental data of the $S$ factors are
displayed in the figures as well. A minor cutoff dependence is observed in the
plots. (Figures for the other $r_{C}$ values are suppressed.) We note that the
curves plotted in the figures are not fitted to the data of $S$ factors; those
are parameter-free predictions. One can see that the curve of $S_{E1}$ factor
with the large value of $h_{R}^{(1)}$ in Fig. 4 and that of $S_{E2}$ factor
with the small value of $h_{R}^{(2)}$ in Fig. 5 relatively agree with the
data.

In Fig. 4, the data on the high-energy side are reproduced better, but those
on the low-energy side are not. Because this pattern cannot be fixed by
adjusting an overall factor, namely the value of $y^{(0)}$ coefficient, it may
be necessary to introduce higher order terms to fit the $S_{E1}$ data. The
$S_{E1}$ factor at $E_{G}=0.3$ MeV is deduced as
$S_{E1}=11.2~{}\textrm{keV}\,\textrm{b}$ with the large value of $h_{R}^{(1)}$
and $S_{E1}=10.1~{}\textrm{keV}\,\textrm{b}$ with the small vale of
$h_{R}^{(1)}$. Those values are significantly smaller than the value 86.3 keV
b recently estimated by means of an $R$ matrix analysis [8].

In Fig. 5, the sharp resonant peak at $E=2.68$ MeV is not reproduced by the
curves because the propagator of resonant $2_{2}^{+}$ state of 16O is not
included. Moreover, the shape of the curve with a small value of $h_{R}^{(2)}$
at small energies, $E<1.7$ MeV, is different from those one can find in
literature; the curves of $S_{E2}$ previously reported continuously increase
as the energy decreases; no maximum and minimum points appear in the low-
energy region. This is attributed to the fact that the effective range
parameters for $l=2$ are not well determined by using the phase shift data. It
may be necessary to fit the effective range parameters by using the $S_{E2}$
data. While, the $S_{E2}$ factor at $E_{G}=0.3$ MeV is deduced as
$S_{E2}=1.3~{}\textrm{keV}\,\textrm{b}$ for the small value $h_{R}^{(2)}$ and
$S_{E2}=0.71~{}\textrm{keV}\,\textrm{b}$ for the large value of $h_{R}^{(2)}$.
Once again, those values are significantly smaller than the value 45.3 keV b
recently estimated by an $R$ matrix analysis [8].

5\. Results and discussion

In the present work, we studied the radiative decay of sub-threshold
$1_{1}^{-}$ and $2_{1}^{+}$ states of 16O within the cluster EFT. We, first,
discussed an improvement of the regularization method: we employed the
dimensional regularization method to calculate the log-divergent term, which
was regularized by using the cutoff regularization method in the previous
work. Because the finite terms below the cutoff $r_{C}$ were ignored in the
$r$-space integral, a minor cutoff dependence remains in the numerical result.
In addition, we fixed the coupling constants, $y^{(0)}$, $y^{(1)}$, $y^{(2)}$,
of $O^{*}\alpha C$ and $\alpha CO$ vertex functions for the initial and final
states of 16O by using the wave function normalization factors $\sqrt{{\cal
Z}_{l}}$, equivalently the ANCs of $0_{1}^{+}$, $1_{1}^{-}$, $2_{1}^{+}$
states of 16O. The values of $\sqrt{{\cal Z}_{l}}$ (or those of the ANCs) were
deduced from the phase shift data of elastic $\alpha$-12C scattering at low
energies. After fixing the coupling constants, $y^{(0)}$, $y^{(1)}$,
$y^{(2)}$, only one parameter, $h_{R}^{(1)}$ or $h_{R}^{(2)}$, remained in
each of the radiative decay amplitudes, and we fitted them to the experimental
data of decay rates of $1_{1}^{-}$ and $2_{1}^{+}$ states of 16O. We, then,
perform a parameter-free calculation of $S$ factors for $E1$ and $E2$
transitions of 12C($\alpha$,$\gamma$)16O by using the fitted parameters in the
present study and found that the order of magnitude of $S$ factors, compared
to the experimental data, could be reproduced.

The ANCs (or the wave function normalization factors) can be a model-
independent quantity when a clear separation scale exists. For example, the
asymptotic normalization factor of $s$-wave deuteron wave function works quite
well [25, 37] where the deuteron binding energy is much smaller than the scale
of $NN$ interactions, namely the pion mass, $m_{\pi}$. Among the three ANCs
which we employed in the present work, the ANC of $1_{1}^{-}$ state of 16O
could be a model-independent one because the binding energy is so small,
$B_{1}=0.045$ MeV, compared to the separation scale, $\Delta E=4$ MeV. Our
estimate of the ANC for the $1_{1}^{-}$ state of 16O turns out to be
remarkably smaller, by about 20% (about one sigma difference with experimental
error bars), than those obtained from the $\alpha$-transfer reactions,
(6Li,$d$) [38, 39]. Recently, a new deduction of the ANCs from the
$\alpha$-transfer reactions is reported by considering the new estimate of ANC
of $1_{1}^{+}$ state of 6Li as a two-body $d$-$\alpha$ system [40], where the
ANC of $1_{1}^{+}$ state of 6Li is obtained by means of an ab initio
calculation of $\alpha$($d$,$\gamma$)6Li capture rate and elastic $d$-$\alpha$
scattering at energies below 3 MeV [41]; the new value of ANC of $1_{1}^{+}$
state of 6Li turns out to be significantly larger than the value previously
employed. The reported new value of the ANC of $1_{1}^{-}$ state of 16O is now
reduced and in better agreement with the result presented in Eq. (35).

Though the binding energy of $2_{1}^{+}$ state of 16O is also small,
$B_{2}=0.245$ MeV, there is a large uncertainty, at least a factor of 2, to
deduce the ANC of $2_{1}^{+}$ state of 16O, mainly because of the tiny phase
shifts at energies below the resonant $2_{2}^{+}$ state of 16O. (The
difficulty to fix the ANC of $2_{1}^{+}$ state of 16O from the phase shift of
elastic $\alpha$-12C scattering is discussed in Appendix in Ref. [30].) The
ambiguity in the value of ANC could be compensated by the fitted value of
$h_{R}^{(2)}$ in the radiative decay amplitudes. However, as mentioned before,
the energy dependence of $S$ factor for $E2$ transition of
12C($\alpha$,$\gamma$)16O is different from that of the experimental data of
$S$ factor in Fig. 5. It may be necessary to fix the effective range
parameters for $l=2$ by using the data of $S$ factor of $E2$ transition or
those of the other reactions.

The use of the ANC of ground $0_{1}^{+}$ state of 16O is questionable, though
it was encouraging to see that the value of ANC of $0_{1}^{+}$ state of 16O
obtained in Eq. (37) is close to that employed in the recent $R$-matrix
analysis [8] as well as those fitted to the $S$ factor of $E1$ transition of
12C($\alpha$,$\gamma$)16O, as discussed above. Because the length scale
between $\alpha$ and 12C in the ground state is quite short,
$\gamma_{0}^{-1}\simeq 1$ fm (and $\gamma_{0}$ is larger than the large scale
of the theory, $\Lambda_{H}$, $\gamma_{0}>\Lambda_{H}$), it seems that a model
dependence in the ANC of $0_{1}^{+}$ state of 16O is inevitable. Only the
order of magnitude of the ANC may be reliable, and thus, it would be practical
to fit the ANC (or the coupling constant $y^{(0)}$) to the data of $S$ factors
together with the coefficients of contact terms, $h_{R}^{(1)}$ and
$h_{R}^{(2)}$.

We found the double values of fitted parameters, $h_{R}^{(1)}$ and
$h_{R}^{(2)}$ in Tables 1 and 2. As mentioned, this implies a large
cancelation between the contributions from loop diagrams and counter terms.
The size of the radiative decay amplitudes may be estimated by using a
relation, $[h_{R}^{(l)}(L)-h_{R}^{(l)}(S)]/[h_{R}^{(l)}(L)+h_{R}^{(l)}(S)]$,
where $h_{R}^{(l)}(L/S)$ represent the large/small values of $h_{R}^{(l)}$,
and we have 0.025 for $l=1$ and 0.013 for $l=2$ by using the values at
$r_{C}=0.01$ fm in the tables. Two orders of magnitude large values of
$h_{R}^{(1)}$ and $h_{R}^{(2)}$ are obtained, compared to the size of
amplitudes for the radiative decay. This could be a similar situation to that
for the modification of counting rules introduced in the effective range
parameters of elastic $\alpha$-12C scattering at low energies. Thus, we might
need to introduce higher-order corrections to the $h_{R}^{(1)}$ and
$h_{R}^{(2)}$ contact vertex functions as counter terms. Such corrections
would play a role of the form factors, which describe a finite size effect of
16O.

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korean government (MSIT) (No. 2019R1F1A1040362 and
2022R1F1A1070060).

Appendix A

$m_{O^{*}(1_{1}^{-})}$ | 14902.196426(140) MeV
---|---
$m_{O^{*}(2_{1}^{+})}$ | 14901.9967(6) MeV
$m_{O}$ | 14895.079576 MeV
$m_{\alpha}$ | 3727.379 MeV
$m_{C}$ | 11174.862 MeV
Table 3:  Masses of nuclei in the unit of MeV, which are used in the present
work.

Masses of 16O*($1_{1}^{-}$), 16O*($2_{1}^{+}$), and 16O as well as $\alpha$
and 12C are presented in Table 3, and the mass differences (excited energies)
of $1_{1}^{-}$ and $2_{1}^{+}$ states with respect to the ground $0_{1}^{+}$
state of 16O are [34]

$\displaystyle\Delta_{1}$ $\displaystyle=$ $\displaystyle
m_{O^{*}(1_{1}^{-})}-m_{O}=7.11685(14)\ \ \textrm{MeV}\,,$ (40)
$\displaystyle\Delta_{2}$ $\displaystyle=$ $\displaystyle
m_{O^{*}(2_{1}^{+})}-m_{O}=6.9171(6)\ \ \textrm{MeV}\,.$ (41)

At the rest frame for 16O∗, the $1_{1}^{-}$ and $2_{1}^{+}$ states of 16O
decay to the ground state of 16O by emitting a photon. The energy-momentum
conservation including the terms up to $1/m_{O}$ corrections reads

$\displaystyle\Delta_{l}$ $\displaystyle=$ $\displaystyle
k_{l}+\frac{1}{2m_{O}}k_{l}^{2}\simeq k_{l}+\frac{1}{2m_{O}}\Delta_{l}^{2}\,,$
(42)

with $l=1,2$. The magnitudes of photon momenta $k_{1}$ and $k_{2}$ (or photon
energies) are obtained as

$\displaystyle k_{1}$ $\displaystyle=$
$\displaystyle\Delta_{1}-\frac{1}{2m_{O}}\Delta_{1}^{2}=7.1152(1)\ \
\textrm{MeV}\,,$ (43) $\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\Delta_{2}-\frac{1}{2m_{O}}\Delta_{2}^{2}=6.9155(6)\ \
\textrm{MeV}\,,$ (44)

where $\Delta_{1}^{2}/(2m_{O})\simeq 1.7\times 10^{-3}$ MeV and
$\Delta_{2}^{2}/(2m_{O})\simeq 1.6\times 10^{-3}$ MeV.

Thus, one has the spin-averaged decay rates as

$\displaystyle\Gamma_{l}$ $\displaystyle=$
$\displaystyle\frac{k_{l}}{2\pi}\frac{1}{2S_{O^{*}(l)}+1}\sum_{spins}|A^{(l)}|^{2}\,,$
(45)

where $S_{O^{*}(1_{1}^{-})}=1$ and $S_{O*(2_{1}^{+})}=2$, and $A^{(l)}$ are
the radiative decay amplitudes which we calculate from the effective
Lagrangian.

Appendix B

In the previous works, we calculated the ANCs of $0_{1}^{+}$ and $0_{2}^{+}$
states of 16O, not including the resonant states of 16O, by fitting the
parameters to the phase shift of elastic $\alpha$-12C scattering for $l=0$
below the energy of the first resonant $0_{3}^{+}$ state [29]. We also
calculated the ANC of $0_{2}^{+}$ state of 16O including the resonant
$0_{3}^{+}$ and $0_{4}^{+}$ states of 16O (but not including the $0_{1}^{+}$
state of 16O) by fitting the parameters to the phase shift data below the
$p$-15N breakup energy [30]. In this appendix, we discuss the derivation of
the ANC of the ground $0_{1}^{+}$ state of 16O including the excited
$0_{2}^{+}$ bound state and $0_{3}^{+}$ and $0_{4}^{+}$ resonant states of 16O
by fitting the parameters to the phase shift below the $p$-16N breakup energy.

We employ the expression of $S$ matrix for elastic $\alpha$-12C scattering at
low energies for $l=0$ in terms of the effective range parameters in Eq. (28)
in Ref. [30]:

$\displaystyle e^{2i\delta_{0}}$ $\displaystyle=$
$\displaystyle\frac{K_{0}(p)-2\kappa\textrm{Re}H_{0}(p)+ipC_{\eta}^{2}}{K_{0}(p)-2\kappa\textrm{Re}H_{0}(p)-ipC_{\eta}^{2}}\prod_{i=3}^{4}\frac{E-E_{R(0i)}+R_{(0i)}(E)-i\frac{1}{2}\Gamma_{(0i)}(E)}{E-E_{R(0i)}+R_{(0i)}(E)+i\frac{1}{2}\Gamma_{(0i)}(E)}\,,$
(46)

where $\delta_{0}$ is the phase shift of elastic $\alpha$-12C scattering for
$l=0$, and $K_{0}(p)=D_{0}(p)+2\kappa H_{0}(p)$; $D_{0}(p)$ is given in Eq.
(25) and has two zeros for the $0_{1}^{+}$ and $0_{2}^{+}$ bound states,
$D_{0}(i\gamma_{01})=0$ and $D_{0}(i\gamma_{02})=0$. Furthermore,

$\displaystyle R_{(0i)}(E)$ $\displaystyle=$ $\displaystyle
a_{(0i)}(E-E_{R(0i)})^{2}+b_{(0i)}(E-E_{R(0i)})^{3}\,,$ (47)
$\displaystyle\Gamma_{(0i)}(E)$ $\displaystyle=$
$\displaystyle\Gamma_{R(0i)}\frac{pC_{\eta}^{2}}{p_{r}C_{\eta_{r}}^{2}}\,,\ \
\ C_{\eta}^{2}=\frac{2\pi\eta}{e^{2\pi\eta}-1}\,,$ (48)

where $E_{R(0i)}$ and $\Gamma_{R(0i)}$ are resonant energies and widths, and
$a_{(0i)}$ and $b_{(0i)}$ are the coefficients of second order
$(E-E_{R(0i)})^{2}$ and third order $(E-E_{R(0i)})^{3}$ contributions in the
Breit-Wigner-like expression for the resonant states. $p_{r}$ are resonant
momenta, $p_{r}=\sqrt{2\mu E_{R(0i)}}$ and $\eta_{r}=\kappa/p_{r}$. (We
suppress the index $i$ for $p_{r}$ and $\eta_{r}$.)

After including the two zeros in $D_{0}(p)$ in Eq. (25), we have 10
parameters, $\\{P_{0}$, $Q_{0};$ $E_{R(03)}$, $\Gamma_{R(03)}$, $a_{(03)}$,
$b_{(03)};$ $E_{R(04)}$, $\Gamma_{R(04)}$, $a_{(04)}$, $b_{(04)}\\}$, in
general, however, we exclude $a_{(03)}$ and $b_{(03)}$ from the fit because
the resonant $0_{3}^{+}$ state is so sharp and the fit is insensitive to
$a_{(03)}$ and $b_{(03)}$. We fix $E_{R(04)}$ and $\Gamma_{R(04)}$ by using
the experimental data, $E_{R(04)}^{exp}=6.870(15)$ MeV and
$\Gamma_{R(04)}^{exp}=185(35)$ keV for the $0_{4}^{+}$ state [34] because its
peak does not appear in the data; it provides a background contribution from
high energy, where $a_{(04)}$ and $b_{(04)}$ describe the high energy tale
from the $0_{4}^{+}$ state of 16O. Thus, we have 6 parameters, $\\{P_{0}$,
$Q_{0}$; $E_{R(03)}$, $\Gamma_{R(03)}$; $a_{(04)}$, $b_{(04)}\\}$, to fit the
phase shift data.

The six parameters are fitted to the phase shift data of elastic $\alpha$-12C
scattering measured at $E_{\alpha}=2.6$ – $6.62$ MeV, reported by Tischhauser
et al. [35], where $E_{\alpha}$ is the $\alpha$ energy in the lab frame, and
we have

$\displaystyle P_{0}$ $\displaystyle=$ $\displaystyle-0.03452(2)\ \
\textrm{fm}^{3}\,,\ \ \ Q_{0}=0.001723(7)\ \ \textrm{fm}^{5}\,,$ (49)
$\displaystyle E_{R(03)}$ $\displaystyle=$ $\displaystyle 4.8883(1)\ \
\textrm{MeV}\,,\ \ \ \Gamma_{R(03)}=1.35(3)\ \ \textrm{keV}\,,$ (50)
$\displaystyle a_{(04)}$ $\displaystyle=$ $\displaystyle 0.756(7)\ \
\textrm{MeV}^{-1}\,,\ \ \ b_{(04)}=0.167(4)\ \ \textrm{MeV}^{-2}\,,$ (51)

where the chi-square per the number of data is $\chi^{2}/N=0.10$: $N$ is the
number of data, $N=252$. The fitted values of $E_{R(03)}$ and $\Gamma_{R(03)}$
agree well with the experimental values, $E_{R(03)}^{exp}=4.887(2)$ MeV and
$\Gamma_{R(03)}^{exp}=1.5(5)$ keV [34].

Figure 6:  Phase shift $\delta_{0}$ of elastic $\alpha$-12C scattering for
$l=0$ is plotted as a function of $E_{\alpha}$ by using the fitted parameters,
where $E_{\alpha}$ is the $\alpha$ energy in the lab frame. The experimental
data are displayed in the figure as well.

In Fig. 6, we plot the phase shift, $\delta_{0}$, using the fitted parameters
and include the experimental data in the figure. One can see that the plotted
curve reproduces the data well.

| w/o $0_{3}^{+}$, $0_{4}^{+}$[29] | w/o $0_{1}^{+}$[30] | This work
---|---|---|---
$|C_{b}|_{01}(\textrm{fm}^{-1/2})$ | 44(2) | – | 44.5(3)
$|C_{b}|_{02}(\textrm{fm}^{-1/2})$ | 605(44) | 370(25) | 621(9)
Table 4:  ANCs of $0_{1}^{+}$ and $0_{2}^{+}$ states of 16O, $|C_{b}|_{01}$
and $|C_{b}|_{02}$, respectively, for two-body $\alpha$-12C bound state.
Values in the second column are the result not including the resonant
$0_{3}^{+}$ and $0_{4}^{+}$ state in the parameter fit [29]. A value in the
third column is the result not including the ground $0_{1}^{+}$ state [30].
Values in the last column are the result of this work.

In Table 4, we display the values of ANCs of $0_{1}^{+}$ and $0_{2}^{+}$
states of 16O for the two-body $\alpha$-12C state, $|C_{b}|_{01}$ and
$|C_{b}|_{02}$, respectively, calculated by using the values of effective
range parameters, $P_{0}$ and $Q_{0}$, in Eq. (49). We also display the ANCs
by using the parameters fitted not including the resonant $0_{3}^{+}$ and
$0_{4}^{+}$ states and those not including the ground $0_{1}^{+}$ state in the
table. One can see that the results of this work and those not including the
resonant $0_{3}^{+}$ and $0_{4}^{+}$ states are in good agreement within the
error bars, while the result of ANC for $0_{2}^{+}$ state obtained not
including the ground $0_{1}^{+}$ state is small, about 60% of that of this
work.

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