March 26, 2023, a12Cwarsapw_ver6

The $S$ matrices of elastic $\alpha$-12C scattering at low energies in
effective field theory

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

The elastic $\alpha$-12C scattering at low energies for $l=0,1,2,3,4,5,6$ is
studied in effective field theory. We discuss the construction of the $S$
matrices of elastic $\alpha$-12C scattering in terms of the amplitudes of sub-
threshold bound and resonant states of 16O, which are calculated from the
effective Lagrangian. The parameters appearing in the $S$ matrices are fitted
to the phase shift data below the $p$-15N breakup threshold energy, and we
find that the phase shifts are well described within the theory.

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

1\. Introduction

Radiative $\alpha$ capture on carbon-12, 12C($\alpha$,$\gamma$)16O, is a
fundamental reaction in nuclear astrophysics, which determines the C/O ratio
along with the triple-$\alpha$ process in the helium burning process in stars
[1]. The radiative capture rate at the Gamow peak energy,
$E_{G}=0.3~{}\textrm{MeV}$, in the stars is, however, difficult to measure in
the experimental facilities because of the Coulomb barrier. One needs to
employ a theoretical model, fit the parameters of the model to the
experimental data measured at a few MeV energy or larger, and extrapolate the
reaction rate down to the Gamow peak energy, $E_{G}=0.3~{}\textrm{MeV}$. Over
the last half-century, many experimental and theoretical studies have been
carried out. See, e.g., Refs. [2, 3, 4, 5, 6, 7] for review.

The experimental data of elastic $\alpha$-12C scattering provide us important
information about the energies and widths of resonant states of 16O at low
energies, which are used to fix some parameters of theoretical models. The
first integrated phase shift analysis of the elastic scattering for
$l=0,1,2,3,4,5,6$ was reported with the data taken at the Ruhr-Universität
Bochum by Plaga et al in 1987 [8]. The upgraded precise phase shift analysis
for partial waves, $l=0,1,2,3,4,5,6$, was reported with the data taken at the
University of Notre Dame by Tischhauser et al. in 2009 [9] where the energy
range of the $\alpha$ particle is $2.6~{}\textrm{MeV}\leq E_{\alpha}\leq
6.62~{}\textrm{MeV}$; $E_{\alpha}$ is the $\alpha$ energy in the lab frame. In
this work, we study the elastic $\alpha$-12C scattering at low energies by
employing an effective field theory (EFT).

To construct an EFT, one first needs to introduce a large momentum scale,
$\Lambda_{H}$. An EFT is constructed by using the relevant degrees of freedom
at low energy, below the large momentum scale, $\Lambda_{H}$. Then, the theory
provides us a perturbative (derivative) expansion scheme in powers of
$Q/\Lambda_{H}$ where $Q$ is a typical momentum scale in a reaction in
question. The coefficients of effective Lagrangian are fixed by using
experimental data, though they, in principle, can be determined from its
mother theory [10]. An EFT is constructed for few-body nuclear systems, known
as pionless EFT, in which the pions are regarded as irrelevant degrees of
freedom, and one has $\Lambda_{H}=m_{\pi}$ where $m_{\pi}$ is the pion mass
[11]. In addition, the dibaryon fields which have the baryon number, $B=2$,
are introduced in the theory to expand the terms around the unitary limit [12,
13, 14]. This expansion scheme turns out to reproduce the effective range
expansion [15]. Furthermore, one can naturally extend the formalism to the
studies of reactions involving photo emission [16, 17, 18], $\beta$ emission
[19, 20, 21], and neutrino reactions [22, 23]. In the previous studies, we
constructed an EFT for the 12C($\alpha$,$\gamma$)16O reaction at the Gamow-
peak energy, $E_{G}=0.3$ MeV, and studied the elastic $\alpha$-12C scattering
at low energies for $l=0,1,2,3$ with and without the sub-threshold bound
states of 16O [24, 25]. We subsequently studied the $E1$ transition of the
12C($\alpha$,$\gamma$)16O reaction [26] and the $\beta$-delayed
$\alpha$-emission from 16N [7] in the EFT.

The inclusion of a resonant state in EFT has been studied by many authors,
e.g., by Gelman [27] and by Habashi, Fleming, and van Kolck [28]; one needs to
sum the leading order interactions up to the infinite order at the vicinity of
resonant energy, where in most cases ones consider a resonant state and a
background contribution. In real situations, on the other hand, a number of
resonant states are involved. In the previous work, we studied the elastic
$\alpha$-12C scattering for $l=2$ including the sub-threshold $2_{1}^{+}$
state and two resonant $2_{2}^{+}$ and $2_{3}^{+}$ states of 16O, in which the
$S$ matrix is constructed from the amplitudes of those sub-threshold and
resonant states; the amplitudes are derived from the effective Lagrangian and
represented in terms of the effective range parameters [29]. In the present
work, we apply the method to the study of the elastic $\alpha$-12C scattering
at low energies for $l=0,1,2,3,4,5,6$. The parameters in the $S$ matrices are
fitted to the phase shift data below the $p$-15N breakup threshold energy, and
we find that the phase shifts are well described within the theory. Then, we
discuss the implication of the result for the application of EFT to the study
of nuclear reactions in stellar evolution.

The present work is organized as the following. In section 2, an expression of
the $S$ matrices is introduced and the effective Lagrangian is presented, and,
in section 3, the elastic scattering amplitudes of the sub-threshold and
resonant states for the $l$-th partial wave states are derived from the
Lagrangian. In section 4, we discuss the numerical results of the parameter
fit to the phase shift data for $l=0,1,2,3,4,5,6$, and finally in section 5,
the results and discussion of this work are presented.

2\. $S$ matrices and effective Lagrangian

The construction of the $S$ matrix of the elastic $\alpha$-12C scattering for
$l=2$ was discussed in Ref. [29]. In this section, we review the method to
extend it to the cases for $l$-th partial wave states. The $S$ matrices of the
elastic $\alpha$-12C scattering for $l$-th partial wave channels are given as

$\displaystyle S_{l}=e^{2i\delta_{l}}\,,$ (1)

where $\delta_{l}$ is the phase shift of elastic scattering for $l$-th partial
wave channel whose experimental data for $l=0,1,2,3,4,5,6$ at
$2.6~{}\textrm{MeV}\leq E_{\alpha}\leq 6.62$ MeV are reported by Tischhauser
et al. [9]. 222 The $\alpha$ energy labeled by $E_{\alpha}$ is in the lab
frame, and the other energies are given in the center-of-mass frame.  The
scattering amplitude $\tilde{A}_{l}$ is related to the $S$ matrix as 333 There
is a common factor difference between the expression of the amplitude
$\tilde{A}_{l}$ and the standard form of the amplitude $A_{l}$;
$A_{l}=\frac{2\pi}{\mu}(2l+1)e^{2i\sigma_{l}}\tilde{A}_{l}$ where $\sigma_{l}$
is the Coulomb phase shift for $l$,
$e^{2i\sigma_{l}}=\Gamma(l+1+i\eta)/\Gamma(l+1-i\eta)$ with $\eta=\kappa/p$.

$\displaystyle S_{l}=1+2ip\tilde{A}_{l}\,.$ (2)

Various resonant states of 16O appear in the phase shift data, and the sub-
threshold bound states and resonant states at high energy (above the maximum
energy of the data) may give contributions to the $S$ matrices of elastic
$\alpha$-12C scattering.

$l$ | (Bound states) |  $2.6~{}\textrm{MeV}\leq E_{\alpha}\leq 6.62$ MeV, | 6.62 MeV $<E_{\alpha}$
---|---|---|---
0 | $0_{2}^{+}$ | $0_{3}^{+}$ | $0_{4}^{+}$
1 | $1_{1}^{-}$ | $1_{2}^{-}$ | $1_{3}^{-}$
2 | $2_{1}^{+}$ |  $2_{2}^{+}$, $2_{3}^{+}$ | $2_{4}^{+}$
3 | $3_{1}^{-}$ | $3_{2}^{-}$ | $3_{3}^{-}$
4 | – |  $4_{1}^{+}$, $4_{2}^{+}$ | $4_{3}^{+}$
5 | – | – | $5_{1}^{-}$
6 | (bg) | – | $6_{1}^{+}$
Table 1:  Bound and resonant ($l^{\pi}_{i-th}$) states of 16O, which are used
to construct the $S$ matrices for $l$-th partial wave states of elastic
$\alpha$-12C scattering; the resonant states in the second column appear in
the energy range, $2.6~{}\textrm{MeV}\leq E_{\alpha}\leq 6.62$ MeV, and those
in the third column do at $6.62~{}\textrm{MeV}<E_{\alpha}$.

In Table 1, we present a list of the sub-threshold bound states, the resonant
states appearing in the phase shift data, and the resonant states as
background contributions from high energy for the partial wave states of
$\alpha$-12C system. By employing those states in the table, we construct the
$S$ matrices of the elastic $\alpha$-12C scattering.

To construct an $S$ matrix, $S_{l}$, we may decompose a phase shift
$\delta_{l}$, for example, in the case of a sub-threshold bound state and two
resonant states, as [27]

$\displaystyle\delta_{l}=\delta_{l}^{(bs)}+\delta_{l}^{(rs1)}+\delta_{l}^{(rs2)}\,,$
(3)

where $\delta_{l}^{(bs)}$ is a phase shift from a sub-threshold bound state,
and $\delta_{2}^{(rsN)}$ with $N=1,2$ are those from resonant states. We now
assume that each of those phase shifts may have a relation to a corresponding
scattering amplitude as

$\displaystyle e^{2i\delta_{l}^{(ch)}}$ $\displaystyle=$ $\displaystyle
1+2ip\tilde{A}_{l}^{(ch)}\,,$ (4)

where $ch(annel)=bs,rs1,rs2$, and $\tilde{A}_{l}^{(bs)}$ and
$\tilde{A}_{l}^{(rsN)}$ with $N=1,2$ are the amplitudes for the binding part
and the first and second resonant parts of the amplitudes, which will be
constructed from the effective Lagrangian in the next section. Thus, the total
amplitude $\tilde{A}_{l}$ for the nuclear reaction part in terms of the three
amplitudes, $\tilde{A}_{l}^{(bs)}$ and $\tilde{A}_{l}^{(rsN)}$ with $N=1,2$,
is

$\displaystyle\tilde{A}_{l}$ $\displaystyle=$
$\displaystyle\tilde{A}_{l}^{(bs)}+e^{2i\delta_{l}^{(bs)}}\tilde{A}_{l}^{(rs1)}+e^{2i(\delta_{l}^{(bs)}+\delta_{l}^{(rs1)})}\tilde{A}_{l}^{(rs2)}\,.$
(5)

We note that the total amplitudes, $\tilde{A}_{l}$, are not obtained as the
summation of the amplitudes, $\tilde{A}_{l}^{(bs)}$ and
$\tilde{A}_{l}^{(rsN)}$ with $N=1,2$, but have the additional phase factors to
$\tilde{A}_{l}^{(rsN)}$ with $N=1,2$, and the order of the three amplitudes,
$\tilde{A}_{l}^{(bs)}$ and $\tilde{A}_{l}^{(rsN)}$ with $N=1,2$ are
exchangeable.

To study the elastic $\alpha$-12C scattering at low energies in EFT, we choose
the $p$-15N breakup threshold energy as the high energy scale. For the
relevant degrees of freedom of the theory, the ground $0^{+}$ states of
$\alpha$ and 12C are chosen as elementary-like scalar fields. We also
introduce the composite fields of $\alpha$ and 12C to describe the sub-
threshold and resonant states of 16O. An effective Lagrangian to derive the
scattering amplitude for the $l$-th wave elastic $\alpha$-12C scattering at
low energies including $l^{\pi}_{i-th}$ states of 16O may be written as [24,
25]

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\phi_{\alpha}^{\dagger}\left(iD_{0}+\frac{\vec{D}^{2}}{2m_{\alpha}}\right)\phi_{\alpha}+\phi_{C}^{\dagger}\left(iD_{0}+\frac{\vec{D}^{2}}{2m_{C}}\right)\phi_{C}$
(6)
$\displaystyle+\sum_{l=0}^{6}\sum_{i}\sum_{k=0}^{3}C_{(li)k}d_{(li)}^{\dagger}\left[iD_{0}+\frac{\vec{D}^{2}}{2(m_{\alpha}+m_{C})}\right]^{k}d_{(li)}$
$\displaystyle-\sum_{l=0}^{6}\sum_{i}y_{(li)}\left[d_{(nr)}^{\dagger}(\phi_{\alpha}O_{(l)}\phi_{C})+(\phi_{\alpha}O_{(l)}\phi_{C})^{\dagger}d_{(li)}\right]\,,$

where $\phi_{\alpha}$ ($m_{\alpha}$) and $\phi_{C}$ ($m_{C}$) are scalar
fields (masses) of $\alpha$ and 12C, respectively. $D^{\mu}$ is a covariant
derivative, $D^{\mu}=\partial^{\mu}+i{\cal Q}A^{\mu}$ where ${\cal Q}$ is a
charge operator and $A^{\mu}$ is the photon field. $d_{(li)}$ are the
composite fields for the $l^{\pi}_{i-th}$ states of 16O consisting of $\alpha$
and 12C fields in $l$-th partial wave states, which are introduced for
perturbative expansion around the unitary limit [12, 13, 14, 15]. The field
$d_{(li)}$ are tensors in general, which are represented as Cartesian tensors
of rank $l$ [30, 31, 32] (we suppressed the indices of the Cartesian tensors);
$O_{(l)}$ are also tensors to project the $\alpha$-12C system to $l$-th
partial wave states. The coupling constants, $C_{(li)k}$ with $k=0,1,2,3$,
correspond to the effective range parameters of elastic $\alpha$-12C
scattering; for those of the sub-threshold states of 16O, the first coupling
constants, $C_{(li)k}$ with $k=0$, are fixed by using the binding energies of
the sub-threshold states and the other parameters are fitted to the
experimental phase shift data with other parameters appearing in the $S$
matrices. For the coupling constants of the resonant parts, $C_{(li)k}$ with
$k=0,1,2,3$, the first two terms, $C_{(li)k}$ with $k=0,1$, are rewritten by
using the resonant energies and widths of the resonant states of 16O,
respectively. The third and fourth parameters for the resonant states,
$C_{(li)k}$ with $k=2,3$ are fitted to the phase shift data. The coupling
constants $y_{(li)}$ are convention-dependent [33], and we take the convenient
choice; $y_{(li)}=\sqrt{2\pi(2l+1)\mu^{2l-1}}$ where $\mu$ is the reduced mass
of $\alpha$ and 12C.

3\. Scattering amplitudes

Figure 1:  Diagrams for dressed 16O propagator. A thick (thin) dashed line
represents a propagator of 12C ($\alpha$), and a thick and thin double dashed
line with and without a filled circle represent a dressed and bare 16O
propagator, respectively. 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.

Figure 2:  Diagram of the scattering amplitude. See the caption of Fig. 1 as
well.

All the scattering amplitudes, $\tilde{A}_{l}^{(bs)}$ and
$\tilde{A}_{l}^{(rsN)}$ with $N=1,2,3$, are calculated from the diagrams
depicted in Figs. 1 and 2. The shaded blobs in the diagrams represent the
parts of the non-perturbative Coulomb interaction, the Coulomb propagator in
the one-loop diagrams and the Coulomb wavefunctions for the initial and final
state interactions. Here, the bubble diagrams are summed up to infinite order
in Fig. 1. For the bound parts of the amplitudes, we treat them non-
perturbatively though they can be expanded perturbatively at the energy region
of the experimental data. For the resonant parts of the amplitudes (for our
case, they are classified as narrow resonances because of $\Gamma_{r}\ll
E_{r}$ [28]), the counting rules of resonant states are carefully studied by
Gelman [27] and Habashi, Fleming, and van Kolck [28]. The energy range of
phase shift data covers the resonant states, and at the vicinities of the
resonant energies we should have the amplitudes for which the bubble diagrams
are summed up to the infinite order. While at the off-resonant energy regions,
one can expand the resonant amplitudes perturbatively and the $d_{(li)}$
fields may start mixing through the bubble diagram for corrections at higher
orders. We keep the summed amplitudes for the resonant states as leading
contributions and ignore the field mixing.

For the bound state amplitudes, $\tilde{A}_{l}^{(bs)}$ with $l=0,1,2,3$, we
have [24, 25]

$\displaystyle\tilde{A}^{(bs)}_{l}$ $\displaystyle=$
$\displaystyle\frac{C_{\eta}^{2}W_{l}(p)}{K_{l}(p)-2\kappa H_{l}(p)}\,,$ (7)

where the function $C_{\eta}^{2}W_{l}(p)$ in the numerator of the amplitude is
calculated from the initial and final state Coulomb interactions in Fig. 2;
$p$ is the magnitude of relative momentum of the $\alpha$-12C system in the
center of mass frame, $p=\sqrt{2\mu E}$, and one has

$\displaystyle C_{\eta}^{2}$ $\displaystyle=$
$\displaystyle\frac{2\pi\eta}{\exp(2\pi\eta)-1}\,,$ (8) $\displaystyle
W_{l}(p)$ $\displaystyle=$
$\displaystyle\left(\frac{\kappa^{2}}{l^{2}}+p^{2}\right)W_{l-1}(p)\,,\ \ \
W_{0}(p)=1\,,$ (9)

where $\eta=\kappa/p$: $\kappa$ is the inverse of the Bohr radius,
$\kappa=Z_{\alpha}Z_{12C}\alpha_{E}\mu$, where $Z_{A}$ is the number of
protons of the nuclei, $Z_{\alpha}=2$ and $Z_{12C}=6$, and $\alpha_{E}$ is the
fine structure constant. The function $-2\kappa H_{l}(p)$ in the denominator
of the amplitude is the Coulomb self-energy term which is calculated from the
loop diagram in Fig. 1, and one has

$\displaystyle H_{l}(p)$ $\displaystyle=$ $\displaystyle W_{l}(p)H(\eta)\,,\ \
\ H(\eta)=\psi(i\eta)+\frac{1}{2i\eta}-\log(i\eta)\,,$ (10)

where $\psi(z)$ is the digamma function. The nuclear interaction is
represented in terms of the effective range parameters in the function
$K_{l}(p)$ in the denominator of the amplitude in Eq. (7). As discussed in
Ref. [24], large and significant contributions to the series of effective
range expansion, compared to that evaluated from the phase shift data at the
lowest energy of the data, $E_{\alpha}=2.6$ MeV, appear from the Coulomb self-
energy term, $-2\kappa H_{l}(p)$. In order to subtract those contributions, we
include the effective range terms up to $p^{6}$ order for $l=0,1,2$ and those
up to $p^{8}$ order for $l=3$ as counter terms. Thus, we have

$\displaystyle K_{l}(p)$ $\displaystyle=$
$\displaystyle-\frac{1}{a_{l}}+\frac{1}{2}r_{l}p^{2}-\frac{1}{4}P_{l}p^{4}+Q_{l}p^{6}-R_{l}p^{8}\,,$
(11)

where $a_{l}$, $r_{l}$, $P_{l}$, $Q_{l}$, $R_{l}$ are effective range
parameters. (We note that $R_{l}=0$ for $l=0,1,2$.)

Now we fix a parameter among the five effective range parameters, $a_{l}$,
$r_{l}$, $P_{l}$, $Q_{l}$, and $R_{l}$ by using the condition that the inverse
of the scattering amplitude $\tilde{A}_{l}^{(bs)}$ vanishes at the binding
energy of the sub-threshold states of 16O. Thus, the denominator of the
scattering amplitude,

$\displaystyle D_{l}(p)=K_{l}(p)-2\kappa H_{l}(p)\,,$ (12)

vanishes at $p=i\gamma_{l}$ where $\gamma_{l}$ are the binding momenta of the
$0_{2}^{+}$, $1_{1}^{-}$, $2_{1}^{+}$, $3_{1}^{-}$ ($l^{\pi}_{i-th}$) states
of 16O; $\gamma_{l}=\sqrt{2\mu B_{l}}$ where $B_{l}$ are the binding energies
of the bound states of 16O from the $\alpha$-12C breakup threshold. At the
binding energies, we have the wavefunction normalization factors
$\sqrt{Z_{l}}$ for the bound states of 16O in the dressed 16O propagators as

$\displaystyle\frac{1}{D_{l}(p)}=\frac{Z_{l}}{E+B_{l}}+\cdots\,,$ (13)

where the dots denote the finite terms at $E=-B_{l}$. Thus, one has

$\displaystyle\sqrt{Z_{l}}=\left(\left|\frac{dD_{l}(p)}{dE}\right|_{E=-B_{l}}\right)^{-1/2}=\left(2\mu\left|\frac{dD_{l}(p)}{dp^{2}}\right|_{p^{2}=-\gamma_{l}^{2}}\right)^{-1/2}\,.$
(14)

The wavefunction normalization factor $\sqrt{Z_{l}}$ is multiplied to a
reaction amplitude when a bound state appears in the initial or final state of
the reaction.

Using the condition, $D_{l}(i\gamma_{l})=0$, we fix the effective range
parameter $a_{l}$ as

$\displaystyle-\frac{1}{a_{l}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\gamma_{l}^{2}r_{l}+\frac{1}{4}\gamma_{l}^{4}P_{l}+\gamma_{l}^{6}Q_{l}+\gamma_{l}^{8}R_{l}+2\kappa
H_{l}(i\gamma_{l})\,.$ (15)

Using the relation of Eq. (15), we rewrite the denominator of the amplitude
$D_{l}(p)$ as

$\displaystyle D_{l}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}r_{l}(\gamma_{l}^{2}+p^{2})+\frac{1}{4}P_{l}(\gamma_{l}^{4}-p^{4})+Q_{l}(\gamma_{l}^{6}+p^{6})+R_{l}(\gamma_{l}^{8}-p^{8})$
(16) $\displaystyle+2\kappa\left[H_{2}(i\gamma_{l})-H_{l}(p)\right]\,,$

where we have three constants, $r_{l}$, $P_{l}$, $Q_{l}$ for $l=0,1,2$ and
four constants, $r_{3}$, $P_{3}$, $Q_{3}$, $R_{3}$ for $l=3$ in the function
$D_{l}(p)$ for the non-resonant amplitude $\tilde{A}_{l}^{(bs)}$, which are
fitted to the phase shift data. For the study of the asymptotic normalization
coefficients (ANC), the exponential factors in Eq. (5) almost become ones at
the small energy region due to the Gamow factor in $C_{\eta}^{2}$, and the
amplitudes become

$\displaystyle\tilde{A}_{l}$ $\displaystyle=$
$\displaystyle\tilde{A}_{l}^{(bs)}+\tilde{A}_{l}^{(rs1)}+\tilde{A}_{l}^{(rs2)}+O(C_{\eta}^{4})\,,$
(17)

where the poles at the sub-threshold bound states of 16O exist in
$\tilde{A}_{l}^{(bs)}$, the ANCs $|C_{b}|$ for the sub-threshold bound states
of 16O are calculated by using the formula [34]

$\displaystyle|C_{b}|=\frac{\gamma_{l}^{l}}{l!}\Gamma(l+1+\kappa/\gamma_{l})\left(\left|\frac{dD_{l}(p)}{dp^{2}}\right|_{p^{2}=-\gamma_{l}^{2}}\right)^{-1/2}\
\ (\textrm{fm}^{-1/2})\,,$ (18)

where $\Gamma(x)$ is the gamma function, and one may notice that the ANCs are
proportional to the wavefunction normalization factor $\sqrt{Z_{l}}$. We note
that the ANCs themselves are not necessary for the calculations of EFT (while
they may become constraints when the experimental data are not available for
fitting the effective range parameters). In the present work, we display the
values of the ANCs (in the next section) as a demonstration of the comparison
with those obtained in our previous works as well as the other theoretical
models. In addition, we perform a test calculation to study the values of ANCs
from a potential model. Its results are presented in the appendix.

For the elastic scattering amplitudes for the resonant states of 16O, we may
first have those amplitudes as the same expression of the bound state
amplitudes in Eq. (7) in terms of the effective range parameters as

$\displaystyle\tilde{A}^{(rsN)}_{l}$ $\displaystyle=$
$\displaystyle\frac{C_{\eta}^{2}W_{l}(p)}{K^{(rsN)}_{l}(p)-2\kappa
H_{l}(p)}\,,$ (19)

with $N=1,2,3$, which correspond to the first, second, and third resonant
states of 16O, respectively, for $l$-th partial wave states and

$\displaystyle K^{(rsN)}_{l}(p)$ $\displaystyle=$
$\displaystyle-\frac{1}{a_{l}^{(rsN)}}+\frac{1}{2}r_{l}^{(rsN)}p^{2}-\frac{1}{4}P_{l}^{(rsN)}p^{4}+Q_{l}^{(rsN)}p^{6}\,,$
(20)

where we include the terms up to $p^{6}$ order for all the resonant
amplitudes.

We now introduce the Taylor expansion around the resonant energies in the
denominator of the scattering amplitudes [35]. Thus, we rewrite the amplitudes
as

$\displaystyle\tilde{A}_{l}^{(rsN)}$ $\displaystyle=$
$\displaystyle-\frac{1}{p}\frac{\frac{1}{2}\Gamma_{(li)}(E)}{E-E_{R(li)}+R_{(li)}(E)+i\frac{1}{2}\Gamma_{(li)}(E)}\,,$
(21)

with

$\displaystyle\Gamma_{(li)}(E)$ $\displaystyle=$
$\displaystyle\Gamma_{R(li)}\frac{pC_{\eta}^{2}W_{l}(p)}{p_{r}C_{\eta_{r}}^{2}W_{l}(p_{r})}\,,$
(22) $\displaystyle R_{(li)}(E)$ $\displaystyle=$ $\displaystyle
a_{(li)}(E-E_{R(li)})^{2}+b_{(li)}(E-E_{R(li)})^{3}\,,$ (23)

where

$\displaystyle a_{(li)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}Z_{R(li)}\left(2P_{l}^{(rsN)}\mu^{2}-48Q_{l}^{(rsN)}\mu^{3}E_{R(li)}+2\kappa
Re\left.\frac{\partial^{2}H_{l}}{\partial
E^{2}}\right|_{E=E_{R(li)}}\right)\,,$ (24) $\displaystyle b_{(li)}$
$\displaystyle=$
$\displaystyle\frac{1}{6}Z_{R(li)}\left(-48Q_{l}^{(rsN)}\mu^{3}+2\kappa\left.Re\frac{\partial^{3}H_{l}}{\partial
E^{3}}\right|_{E=E_{R(li)}}\right)\,,$ (25) $\displaystyle Z_{R(li)}^{-1}$
$\displaystyle=$ $\displaystyle Re\left.\frac{\partial}{\partial
E}D_{l}^{(rsN)}(E)\right|_{E=E_{R(li)}}\,,\ \ \ \
Z_{R(li)}=\frac{\Gamma_{R(li)}}{2p_{r}W_{l}(p_{r})C_{\eta_{r}}^{2}}\,.$ (26)

In the above equations, $E_{R(li)}$ and $\Gamma_{R(li)}$ are the energies and
the widths of the resonant $l^{\pi}_{i-th}$ states of 16O, and $p_{r}$ are the
resonant momenta, $p_{r}=\sqrt{2\mu E_{R(li)}}$, which also appear in
$\eta_{r}$ as $\eta_{r}=\kappa/p_{r}$. We note that the expression of Eq. (21)
resembles that of the Breit-Wigner formula, but it is derived from the
expression of the effective range expansion in Eq. (20); it has the additional
terms, the corrections of the higher order terms being proportional to
$(E-E_{R(li)})^{2}$ and $(E-E_{R(li)})^{3}$ in the function $R_{(li)}(E)$ in
Eq. (23); though the coefficients $a_{(li)}$ and $b_{(li)}$ are functions of
the effective range parameters, $P_{l}^{(rsN)}$ and $Q_{l}^{(rsN)}$, we treat
$a_{(li)}$ and $b_{li)}$ as independent free parameters for the sake of
simplicity.

Using the expression of the amplitudes in Eqs. (7) and (21), we obtain an
expression of the $S$ matrices in Eq. (1) as

$\displaystyle e^{2i\delta_{l}}$ $\displaystyle=$
$\displaystyle\frac{K_{l}(p)-2\kappa
ReH_{l}(p)+ipC_{\eta}^{2}W_{l}(p)}{K_{l}(p)-2\kappa
ReH_{l}(p)-ipC_{\eta}^{2}W_{l}(p)}\prod_{i}\frac{E-E_{R(li)}+R_{(li)}(E)-i\frac{1}{2}\Gamma_{(li)}(E)}{E-E_{R(li)}+R_{(li)}(E)+i\frac{1}{2}\Gamma_{(li)}(E)}\,,$
(27)

where the part for the bound states in terms of $K_{l}(p)-2\kappa
ReH_{l}(p)\pm ipC_{\eta}^{2}W_{l}(p)$ appear for $l=0,1,2,3$, and we also
include it for $l=6$ as a background contribution from low energy. The part of
the resonant states in terms of $E-E_{R(li)}+R_{(li)}(E)\mp
i\frac{1}{2}\Gamma_{(li)}(E)$ are for those appearing in the energy range of
the phase shift data, and we also include it for each of the partial waves as
a background contribution from high energy.

4\. Numerical results

We construct an $S$ matrix of the elastic scattering for each of the partial
waves, $l=0,1,2,3,4,5,6$, and fit parameters in the $S$ matrices to the phase
shift data at $2.6~{}\textrm{MeV}<E_{\alpha}<6.62~{}\textrm{MeV}$ reported by
Tischhauser et al. [9], by means of a Markov chain Monte Carlo (MCMC) program
[36]. We will see that curves calculated by using the fitted parameters
reproduce the phase shift data very well.

4.1 Phase shift for $l=0$ channel

We consider three states, $0_{2}^{+}$, $0_{3}^{+}$, $0_{4}^{+}$ states of 16O
for the $S$ matrix of elastic $\alpha$-12C scattering for $l=0$, where
$0_{2}^{+}$ is the sub-threshold bound state, $0_{3}^{+}$ is the resonant
state appearing in the phase shift data at $E_{\alpha}=6.52$ MeV, and
$0_{4}^{+}$ is the resonant state as a background contribution from high
energy appearing at $E_{\alpha}=9.16$ MeV. Thus, we have an expression of the
$S$ matrix for $l=0$ as

$\displaystyle e^{2i\delta_{0}}$ $\displaystyle=$
$\displaystyle\frac{K_{0}(p)-2\kappa
ReH_{0}(p)+ipC_{\eta}^{2}}{K_{0}(p)-2\kappa
ReH_{0}(p)-ipC_{\eta}^{2}}\prod_{i=3}^{4}\frac{E-E_{R(0i)}+R_{(0i)}(E)-i\frac{1}{2}\Gamma_{(0i)}(E)}{E-E_{R(0i)}+R_{(0i)}(E)+i\frac{1}{2}\Gamma_{(0i)}(E)}\,,$
(28)

with

$\displaystyle K_{0}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}r_{0}(\gamma_{0}^{2}+p^{2})+\frac{1}{4}P_{0}(\gamma_{0}^{4}-p^{4})+Q_{0}(\gamma_{0}^{6}+p^{6})+2\kappa
H_{0}(i\gamma_{0})\,,$ (29) $\displaystyle\Gamma_{(0i)}(E)$ $\displaystyle=$
$\displaystyle\Gamma_{R(0i)}\frac{pC_{\eta}^{2}W_{0}(p)}{p_{r}C_{\eta_{r}}^{2}W_{0}(p_{r})}\,,$
(30) $\displaystyle R_{(0i)}(E)$ $\displaystyle=$ $\displaystyle
a_{(0i)}(E-E_{R(0i)})^{2}+b_{(0i)}(E-E_{R(0i)})^{3}\,,\ \ \ i=3,4$ (31)

where $\gamma_{0}=\sqrt{2\mu B_{0}}$ and $p_{r}=\sqrt{2\mu E_{(0i)}}$. Thus,
we have 7 parameters to fit the data,

$\displaystyle\theta_{0}$ $\displaystyle=$
$\displaystyle\\{r_{0},P_{0},Q_{0},E_{R(03)},\Gamma_{R(03)},a_{(04)},b_{(04)}\\}\,,$
(32)

where two parameters $a_{(03)}$ and $b_{(03)}$ are set to be zero,
$a_{(03)}=b_{(03)}=0$, because they are not sensitive to the parameter fit. We
use the experimental values for $E_{R(04)}$ and $\Gamma_{R(04)}$,
$E_{R(04)}^{(exp)}=6.870(15)$ MeV and $\Gamma_{R(04)}^{(exp)}=185(35)$ keV
[37], because they are not covered by the phase shift data, to which one
cannot fit them.

$l^{\pi}_{i-th}$ | $p^{0}~{}\textrm{order}$ | $p^{2}$ | $p^{4}$ | $p^{6}$ | $p^{8}$
---|---|---|---|---|---
$0_{2}^{+}$ | $a_{0}(\textrm{fm})$ |  $r_{0}$(fm) |  $P_{0}$($\textrm{fm}^{3}$) |  $Q_{0}$($\textrm{rm}^{5}$) |
| — | 0.26847(1) | $-0.0363(4)$ | 0.0011(1) |
$0_{3}^{+}$ |  $E_{R(03)}$(MeV) |  $\Gamma_{R(03)}$(keV) |  |  |
| 4.8884(1) | 1.34(3) |  |  |
$0_{4}^{+}$ | $E_{R(04)}(\textrm{MeV})$ | $\Gamma_{R(04)}(\textrm{keV})$ |  $a_{(04)}$($\textrm{MeV}^{-1}$) |  $b_{(04)}$($\textrm{MeV}^{-2}$) |
| — | — | 0.75(1) | 0.18(1) |
$1_{1}^{-}$, $1_{2}^{-}$ | $a_{1}(\textrm{fm}^{3})$ |  $r_{1}$($\textrm{fm}^{-1}$) |  $P_{1}$(fm) |  $Q_{1}$($\textrm{fm}^{3}$) |
| — | 0.415314(7) | $-0.57428(7)$ | 0.02032(2) |
$1_{3}^{-}$ | $E_{R(13)}(\textrm{MeV})$ | $\Gamma_{R(13)}(\textrm{keV})$ |  $a_{(13)}$($\textrm{MeV}^{-1}$) |  $b_{(13)}$($\textrm{MeV}^{-2}$) |
| — | — | 0.43(25) | 3.8(7) |
$2_{1}^{+}$ | $a_{2}(\textrm{fm}^{5})$ |  $r_{2}$($\textrm{fm}^{-3}$) |  $P_{2}$($\textrm{fm}^{-1}$) |  $Q_{2}$(fm) |
| — | 0.149(4) | $-1.19(5)$ | 0.081(16) |
$2_{2}^{+}$ |  $E_{R(22)}$(MeV) |  $\Gamma_{R(22)}$(keV) |  |  |
| 2.68308(5) | 0.75(2) |  |  |
$2_{3}^{+}$ |  $E_{R(23)}$(MeV) |  $\Gamma_{R(23)}$(keV) |  $a_{(23)}$($\textrm{MeV}^{-1}$) |  $b_{(23)}$($\textrm{MeV}^{-2}$) |
| 4.3545(2) | 74.61(3) | 0.46(12) | 0.49(9) |
$2_{4}^{+}$ | $E_{R(24)}(\textrm{MeV)}$ | $\Gamma_{R(24)}(\textrm{keV})$ |  |  |
| — | — |  |  |
$3_{1}^{-}$, $3_{2}^{-}$ | $a_{3}(\textrm{fm}^{7})$ |  $r_{3}$($\textrm{fm}^{-5}$) |  $P_{3}$($\textrm{fm}^{-3}$) |  $Q_{3}$($\textrm{fm}^{-1}$) |  $R_{3}$(fm)
| — | 0.0335(2) | $-0.446(9)$ | 0.311(5) | $-0.152(3)$
$3_{3}^{-}$ | $E_{R(33)}(\textrm{MeV})$ | $\Gamma_{R(33)}(\textrm{keV})$ | $a_{(33)}(\textrm{MeV}^{-1})$ | $b_{(33)}(\textrm{MeV}^{-2})$ |
| — | — | 32(32) | $3.2(32)\times 10^{2}$ |
$4_{1}^{+}$ |  $E_{R(41)}$(MeV) |  $\Gamma_{R(41)}$(keV) |  $a_{(41)}$($\textrm{MeV}^{-1}$) |  $b_{(41)}$($\textrm{MeV}^{-2}$) |
| 3.19606(1) | 25.91(1) | 0.740(3) | 0.304(5) |
$4_{2}^{+}$ |  $E_{R(42)}$(MeV) |  $\Gamma_{R(42)}$(keV) |  |  |
| 3.93655(2) | 0.425(4) |  |  |
$4_{3}^{+}$ | $E_{R(41)}(\textrm{MeV})$ | $\Gamma_{R(41)}(\textrm{keV})$ |  $a_{(43)}$($\textrm{MeV}^{-1}$) |  $b_{(43)}$($\textrm{MeV}^{-2}$) |
| — | — | 0.889(6) | 0.216(3) |
$5_{1}^{-}$ | $E_{R(51)}(\textrm{MeV})$ | $\Gamma_{R(51)}(\textrm{keV})$ |  $a_{(51)}$($\textrm{MeV}^{-1}$) |  $b_{(51)}$($\textrm{MeV}^{-2}$) |
| — | — | 0.572(6) | 0.104(2) |
(bg) |  |  $r_{6}$($\textrm{fm}^{-11}$) |  $P_{6}$($\textrm{fm}^{-9}$) |  |
|  | $-0.3(2)$ | 2(1) |  |
$6_{1}^{+}$ | $E_{R(61)}(\textrm{MeV})$ | $\Gamma_{R(61)}(\textrm{keV)}$ |  $a_{(61)}$($\textrm{MeV}^{-1}$) |  $b_{(61)}$($\textrm{MeV}^{-2}$) |
| — | — | 0.8(1) | 0.18(4) |
Table 2:  Fitted values of the parameters in the amplitudes of the sub-threshold and resonant states of 16O (and a background contribution for $l=6$), which are listed in table 1, in the $S$ matrices of elastic $\alpha$-12C scattering for $l=0,1,2,3,4,5,6$. The parameters whose values are not shown in the table are fixed by using the experimental data. Parameters not shown in the table are not included in the parameter fit.  $l$ | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
$\chi^{2}/N$ | 0.013 | 0.089 | 0.66 | 0.87 | 0.47 | 0.094 | 0.026
Table 3: Values of $\chi^{2}/N$ for $l=0,1,2,3,4,5,6$ for the parameter fit where $N=252$ for $l=0,1,2,3,4$, $N=243$ for $l=5$, and $N=186$ for $l=6$.  $l_{i-th}^{\pi}$ | $0_{2}^{+}$ | $1_{1}^{-}$ | $2_{1}^{+}$ | $3_{1}^{-}$
---|---|---|---|---
$|C_{b}|(\textrm{fm}^{-1/2})$ | 370(25) | $1.727(3)\times 10^{14}$ | $3.1(6)\times 10^{4}$ | 113(8)
Table 4: Values of ANC, $|C_{b}|$, for the sub-threshold $0_{2}^{+}$,
$1_{1}^{-}$, $2_{1}^{+}$, $3_{1}^{-}$ states of 16O.

In Table 2, we show fitted values and their errors of the 7 parameters in the
amplitudes of the $0_{2}^{+}$, $0_{3}^{+}$, $0_{4}^{+}$ states of 16O in the
$S$ matrix of elastic $\alpha$-12C scattering for $l=0$ in Eq. (28) where we
find a small value of $\chi^{2}/N$, $\chi^{2}/N=0.013$, for the parameter fit,
as shown in Table 3. The fitted values of $E_{R(03)}$ and $\Gamma_{R(03)}$
agree with their experimental values, $E_{R(03)}^{(exp)}=4.887(2)$ MeV and
$\Gamma_{R(03)}^{(exp)}=1.5(5)$ keV [37]. Those of the parameters, such as
$P_{0}$, $a_{(03)}$, and $b_{(03)}$, at higher order have large errors. One
may notice that the errors of the effective range parameters, $r_{0}$,
$P_{0}$, $Q_{0}$ become larger in order as the orders of the $p^{2}$ expansion
increase. This may indicate that the perturbative expansion in the effective
range parameters works well.

In Table 4, we show a value of the ANC, $|C_{b}|$, of the $0_{2}^{+}$ state of
16O; we obtain $|C_{b}|=370(25)~{}\textrm{fm}^{-1/2}$, which is smaller than
our previous estimates, $|C_{b}|=443(3)~{}\textrm{fm}^{-1/2}$ [24] and
$|C_{b}|=(6.4\textrm{--}7.4)\times 10^{2}~{}\textrm{fm}^{-1/2}$ [38]. We note
that the values of ANC of the $0_{2}^{+}$ state of 16O reported in the
literature are still scattered: from the $R$-matrix analysis, the reported
values of ANC are $|C_{b}|=44^{+270}_{-40}~{}\textrm{fm}^{-1/2}$ [39],
1800$~{}\textrm{fm}^{-1/2}$ [40], and 1560$~{}\textrm{fm}^{-1/2}$ [5], from
the $\alpha$ transfer reaction, 1560(100)$~{}\textrm{fm}^{-1/2}$ [41], and
from those fitting the phase shift data using the square-well potential,
3218.46$~{}\textrm{fm}^{-1/2}$ [42] and 886 – 1139$~{}\textrm{fm}^{-1/2}$
[43], and using the so-called $\Delta$-method, 406$~{}\textrm{fm}^{-1/2}$ [44]
and 293$~{}\textrm{fm}^{-1/2}$ [45]. So, this would be an interesting issue to
investigate in the future.

Figure 3:  Phase shift $\delta_{0}$ of elastic $\alpha$-12C scattering for
$s$-wave channel as a function of $E_{\alpha}$ calculated by using the fitted
values of the parameters. Experimental data are included in the figure as
well.

In Fig. 3, we plot a curve of the phase shift for $l=0$, $\delta_{0}$, by
using the fitted values of the parameters obtained in Table 2. The phase shift
data are also displayed in the figure. We find that the fitted curve agrees
well with the phase shift data.

4.2 Phase shift for $l=1$ channel

We consider three states, $1_{1}^{-}$, $1_{2}^{-}$, $1_{3}^{-}$ states of 16O
to construct the $S$ matrix of elastic $\alpha$-12C scattering for $l=1$ where
$1_{1}^{-}$ is the sub-threshold bound state, $1_{2}^{-}$ is the resonant
state appearing in the phase shift data at $E_{\alpha}=3.23$ MeV, and
$1_{3}^{-}$ is a resonant state as a background contribution from high energy
appearing at $E_{\alpha}=7.04$ MeV. Because the resonant $1_{2}^{-}$ state can
be described by the effective range parameters for the sub-threshold bound
$1_{1}^{-}$ state, as discussed in Ref. [26], we include the $1_{2}^{-}$ state
in the amplitude of the sub-threshold $1_{1}^{-}$ state. Thus, we have an
expression of the $S$ matrix for $l=1$ as

$\displaystyle e^{2i\delta_{1}}$ $\displaystyle=$
$\displaystyle\frac{K_{1}(p)-2\kappa
ReH_{1}(p)+ipC_{\eta}^{2}W_{1}(p)}{K_{1}(p)-2\kappa
ReH_{1}(p)-ipC_{\eta}^{2}W_{1}(p)}\frac{E-E_{R(13)}+R_{(13)}(E)-i\frac{1}{2}\Gamma_{(13)}(E)}{E-E_{R(13)}+R_{(13)}(E)+i\frac{1}{2}\Gamma_{(13)}(E)}\,,$
(33)

with

$\displaystyle K_{1}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}r_{1}(\gamma_{1}^{2}+p^{2})+\frac{1}{4}P_{1}(\gamma_{1}^{4}-p^{4})+Q_{1}(\gamma_{1}^{6}+p^{6})+2\kappa
H_{1}(i\gamma_{1})\,,$ (34) $\displaystyle\Gamma_{(13)}(E)$ $\displaystyle=$
$\displaystyle\Gamma_{R(13)}\frac{pC_{\eta}^{2}W_{1}(p)}{p_{r}C_{\eta_{r}}^{2}W_{1}(p_{r})}\,,$
(35) $\displaystyle R_{(13)}(E)$ $\displaystyle=$ $\displaystyle
a_{(13)}(E-E_{R(13)})^{2}+b_{(13)}(E-E_{R(13)})^{3}\,,$ (36)

where $\gamma_{1}=\sqrt{2\mu B_{1}}$ and $p_{r}=\sqrt{2\mu E_{R(13)}}$. We
have 5 parameters to fit the data as

$\displaystyle\theta_{1}=\\{r_{1},P_{1},Q_{1},a_{(13)},b_{(13)}\\}\,,$ (37)

and we use the experimental values for $E_{R(13)}$ and $\Gamma_{R(13)}$,
$E_{R(13)}^{(exp)}=5.278(2)$ MeV and $\Gamma_{R(13)}^{(exp)}=91(6)$ keV [37],
for the background contribution from high energy.

In Table 2, we show fitted values and their errors of the 5 parameters in the
amplitudes of the $1_{1}^{-}$, $1_{2}^{-}$, $1_{3}^{-}$ states of 16O in the
$S$ matrix of elastic $\alpha$-12C scattering for $l=1$ in Eq. (33) where we
find a small value of $\chi^{2}/N$, $\chi^{2}/N=0.089$, for the parameter fit,
as shown in Table 3. We also find relatively large error bars of the
coefficients $a_{(13)}$ and $b_{(13)}$ at the high order while all errors of
the effective range coefficients, $r_{1}$, $P_{1}$, $Q_{1}$, turn out to be
small presumably because those parameters need to fit the resonant $1_{2}^{-}$
state of 16O appearing in the phase shift data. In Table 4, we show the value
of the ANC of the sub-threshold $1_{1}^{-}$ state of 16O; we have
$|C_{b}|=1.727(3)\times 10^{14}~{}\textrm{fm}^{-1/2}$ which agrees well to our
previous estimate, $|C_{b}|=(1.6\textrm{--}1.9)\times
10^{14}~{}\textrm{fm}^{-1/2}$ [24].

Figure 4:  Phase shift $\delta_{1}$ of elastic $\alpha$-12C scattering for
$p$-wave channel as a function of $E_{\alpha}$ calculated by using the fitted
values of the parameters. Experimental data are included in the figure as
well.

In Fig. 4, we plot a curve of the phase shift for $l=1$, $\delta_{1}$, by
using the fitted values of the parameters obtained in Table 2. The phase shift
data are also displayed in the figure. We find that the fitted curve agrees
well with the phase shift data. As discussed in Ref. [26], the phase shift
data up to $E_{\alpha}=6$ MeV can be described by the three effective range
parameters only, and the tail of the phase shift data at the high-energy side
is now described by the background contribution of the resonant $1_{3}^{-}$
state of 16O from high energy.

4.3 Phase shift for $l=2$ channel

We include the $2_{1}^{+}$, $2_{2}^{+}$, $2_{3}^{+}$, $2_{4}^{+}$ states of
16O to construct an $S$ matrix for $l=2$ where $2_{1}^{+}$ is the sub-
threshold bound state, $2_{2}^{+}$ and $2_{3}^{+}$ are the resonant states
appearing in the phase shift data at $E_{\alpha}=3.58$ MeV and
$E_{\alpha}=5.81$ MeV, respectively, and $2_{4}^{+}$ is a resonant state as a
background contribution from high energy appearing at $E_{\alpha}=7.81$ MeV.
Thus, we have an expression of the $S$ matrix for $l=2$ as

$\displaystyle e^{2i\delta_{2}}$ $\displaystyle=$
$\displaystyle\frac{K_{2}(p)+2\kappa
ReH_{2}(p)+ipC_{\eta}^{2}W_{2}(p)}{K_{2}(p)+2\kappa
ReH_{2}(p)-ipC_{\eta}^{2}W_{2}(p)}$ (38)
$\displaystyle\times\prod_{i=2}^{4}\frac{E-E_{R(2i)}+R_{(2i)}(E)-i\frac{1}{2}\Gamma_{(2i)}(E)}{E-E_{R(2i)}+R_{(2i)}(E)+i\frac{1}{2}\Gamma_{(2i)}(E)}\,,$

with

$\displaystyle K_{2}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}r_{2}(\gamma_{2}^{2}+p^{2})+\frac{1}{4}P_{2}(\gamma_{2}^{4}-p^{4})+Q_{2}(\gamma_{2}^{6}+p^{6})+2\kappa
H_{2}(i\gamma_{0})\,,$ (39) $\displaystyle\Gamma_{(2i)}(E)$ $\displaystyle=$
$\displaystyle\Gamma_{R(2i)}\frac{pC_{\eta}^{2}W_{2}(p)}{p_{r}C_{\eta_{r}}^{2}W_{2}(p_{r})}\,,$
(40) $\displaystyle R_{(2i)}(E)$ $\displaystyle=$ $\displaystyle
a_{(2i)}(E-E_{R(2i)})^{2}+b_{(2i)}(E-E_{R(2i)})^{3}\,,\ \ \ i=2,3,4$ (41)

where $\gamma_{2}=\sqrt{2\mu B_{2}}$ and $p_{r}=\sqrt{2\mu E_{(2i)}}$. We have
9 parameters to fit the data as

$\displaystyle\theta_{2}$ $\displaystyle=$
$\displaystyle\\{r_{2},P_{2},Q_{2},E_{R(22)},\Gamma_{R(22)},E_{R(23)},\Gamma_{R(23)},a_{(23)},b_{(23)}\\}\,,$
(42)

where four parameters, $a_{(22)}$, $b_{(22)}$, $a_{(24)}$, $b_{(24)}$, are set
to be zero, $a_{(22)}=b_{(22)}=a_{(24)}=b_{(24)}=0$, because they are not
sensitive to the parameter fit while we use the experimental values for
$E_{R(24)}$ and $\Gamma_{R(24)}$, $E_{R(24)}^{(exp)}=5.858(10)$ MeV and
$\Gamma_{R(24)}^{(exp)}=150(10)$ keV [37], for the background contribution
from high energy.

In Table 2, we show fitted values and their errors of the 9 parameters in the
amplitudes of the $2_{1}^{+}$, $2_{2}^{+}$, $2_{3}^{+}$, $2_{4}^{+}$ states of
16O in the $S$ matrix of elastic $\alpha$-12C scattering for $l=2$ in Eq. (38)
where we find a small value of $\chi^{2}/N$, $\chi^{2}/N=0.66$, for the
parameter fit, as shown in Table 3. (Those values in the table are taken from
Table 2 in Ref. [29].) The fitted values of $E_{R(22)}$, $\Gamma_{R(22)}$,
$E_{R(23)}$, and $\Gamma_{R(23)}$ agree with their experimental values,
$E_{R(22)}^{(exp)}=2.68255(50)$ MeV, $\Gamma_{R(22)}^{(exp)}=0.625(100)$ keV,
$E_{R(23)}^{(exp)}=4.358(4)$ MeV, $\Gamma_{R(23)}^{(exp)}=71(3)$ keV [37]
while the fitted values of the parameters, such as $Q_{2}$, $a_{(23)}$, and
$b_{(23)}$, at higher order have large errors. In Table 4, we show the value
of the ANC of the sub-threshold $2_{1}^{+}$ state of 16O; we have
$|C_{b}|=3.1(6)\times 10^{4}~{}\textrm{fm}^{-1/2}$, which is larger than our
previous estimate, $|C_{b}|=(2.1\textrm{--}2.4)\times
10^{4}~{}\textrm{fm}^{-1/2}$ [24]. In addition, we also argued that the value
of the ANC of the $2_{1}^{+}$ state of 16O is sensitive to conditions imposed
on the effective range parameters, $r_{2}$, $P_{2}$, $Q_{2}$, at very low
energy region, $0\leq E_{\alpha}\leq 2.6~{}\textrm{MeV}$, where no phase shift
data are reported; we can even reproduce the large ANC value, $|C_{b}|\simeq
10\times 10^{4}~{}\textrm{fm}^{-1/2}$ reported from the $\alpha$ transfer
experiments [46, 47, 48, 41]. For more details, refer to Ref. [29].

Figure 5:  Phase shift $\delta_{2}$ of elastic $\alpha$-12C scattering for
$d$-wave channel as a function of $E_{\alpha}$ calculated by using the fitted
values of the parameters. Experimental data are included in the figure as
well.

In Fig. 5, we plot a curve of the phase shift for $l=2$, $\delta_{2}$, by
using the fitted values of the parameters obtained in Table 2. The phase shift
data are also displayed in the figure. We find that the fitted curve agrees
well with the phase shift data.

4.4 Phase shift for $l=3$ channel

We consider $3_{1}^{-}$, $3_{2}^{-}$, $3_{3}^{-}$ states of 16O to construct
an $S$ matrix of elastic $\alpha$-12C scattering for $l=3$ where $3_{1}^{-}$
is the sub-threshold bound state, $3_{2}^{-}$ is the resonant state appearing
in the phase shift data at $E_{\alpha}=5.92$ MeV and $3_{3}^{-}$ is a resonant
state as a background contribution from high energy appearing at
$E_{\alpha}=7.96$ MeV. Because the resonant $3_{2}^{-}$ state can be described
by the effective range parameters as well, as we have seen in the case of
$l=1$, we include the $3_{2}^{-}$ state in the amplitude of the sub-threshold
$3_{1}^{-}$ state. Thus, we have an expression of the $S$ matrix for $l=3$ as

$\displaystyle e^{2i\delta_{3}}$ $\displaystyle=$
$\displaystyle\frac{K_{3}(p)-2\kappa
ReH_{3}(p)+ipC_{\eta}^{2}W_{3}(p)}{K_{3}(p)-2\kappa
ReH_{3}(p)-ipC_{\eta}^{2}W_{3}(p)}\frac{E-E_{R(33)}+R_{(33)}(E)-i\frac{1}{2}\Gamma_{(33)}(E)}{E-E_{R(33)}+R_{(33)}(E)+i\frac{1}{2}\Gamma_{(33)}(E)}\,,$
(43)

with

$\displaystyle K_{3}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}r_{3}(\gamma_{3}^{2}+p^{2})+\frac{1}{4}P_{3}(\gamma_{3}^{4}-p^{4})+Q_{3}(\gamma_{3}^{6}+p^{6})+R_{3}(\gamma_{3}^{8}-p^{8})$
(44) $\displaystyle+2\kappa H_{3}(i\gamma_{3})\,,$
$\displaystyle\Gamma_{(33)}(E)$ $\displaystyle=$
$\displaystyle\Gamma_{R(33)}\frac{pC_{\eta}^{2}W_{3}(p)}{p_{r}C_{\eta_{r}}^{2}W_{3}(p_{r})}\,,$
(45) $\displaystyle R_{(33)}(E)$ $\displaystyle=$ $\displaystyle
a_{(33)}(E-E_{R(33)})^{2}+b_{(33)}(E-E_{R(33)})^{3}\,,$ (46)

where $\gamma_{3}=\sqrt{2\mu B_{3}}$ and $p_{r}=\sqrt{2\mu E_{R(33)}}$. We
have 6 parameters to fit the data as

$\displaystyle\theta_{3}=\\{r_{3},P_{3},Q_{3},R_{3},a_{(33)},b_{(33)}\\}\,,$
(47)

where we impose a condition to the parameter $R_{3}$ for the parameter fit,
$\tilde{R}_{3}<R_{3}$; $\tilde{R}_{3}$ is the contribution from the Coulomb
self-energy term, $-2\kappa H_{3}(p)$, to the $R_{3}$ term, and we have
$\tilde{R}_{3}=-17101/(90720\kappa)$ [24]. When $\tilde{R}_{3}>R_{3}$, a
spurious bound state appears below the sub-threshold $3_{1}^{-}$ state of 16O.
In addition, we use the experimental values for $E_{R(33)}$ and
$\Gamma_{R(33)}$, $E_{R(33)}^{(exp)}=5.967(10)$ MeV and
$\Gamma_{R(33)}^{(exp)}=110(30)$ keV [37], for the background contribution
from high energy. In Table 2, we show fitted values and their errors of the 6
parameters in the amplitudes of the $3_{1}^{+}$, $3_{2}^{-}$, $3_{3}^{-}$
states of 16O in the $S$ matrix of elastic $\alpha$-12C scattering for $l=3$
in Eq. (43) where we find a value of $\chi^{2}/N$, $\chi^{2}/N=0.87$, for the
parameter fit, as shown in Table 3. The fitted values of the parameters
$a_{(33)}$ and $b_{(33)}$ have large error bars and are not still fitted well
because they are insensitive in the parameter fit. While, if we exclude them
from the parameter fit, we have a large value of $\chi^{2}/N$,
$\chi^{2}/N=1.84$. In Table 4, we show a value of the ANC of the $3_{1}^{-}$
state of 16O; we have $|C_{b}|=113(8)~{}\textrm{fm}^{-1/2}$ which agrees well
with our previous estimate, $|C_{b}|=(1.2\textrm{--}1.5)\times
10^{2}~{}\textrm{fm}^{-1/2}$ [24].

Figure 6:  Phase shift $\delta_{3}$ of elastic $\alpha$-12C scattering for
$f$-wave channel as a function of $E_{\alpha}$ calculated by using the fitted
values of the parameters. Experimental data are included in the figure as
well.

In Fig. 6, we plot a curve of the phase shift for $l=3$, $\delta_{3}$, by
using the fitted values of the parameters obtained in Table 2. The phase shift
data are also displayed in the figure. We find that the fitted curve agrees
well with the phase shift data.

4.5 Phase shift for $l=4$ channel

We consider $4_{1}^{+}$, $4_{2}^{+}$, $4_{3}^{+}$ states of 16O to construct
the $S$ matrix of elastic $\alpha$-12C scattering for $l=4$ where $4_{1}^{+}$
and $4_{2}^{+}$ are the resonant states appearing in the phase shift data at
$E_{\alpha}=4.26$ MeV and $E_{\alpha}=5.25$ MeV, respectively, and $4_{3}^{+}$
is a resonant state as a background state from high energy appearing at
$E_{\alpha}=8.94$ MeV. Thus, we have an expression of the $S$ matrix for $l=4$
as

$\displaystyle e^{2i\delta_{4}}$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{3}\frac{E-E_{R(4i)}+R_{(4i)}(E)-i\frac{1}{2}\Gamma_{(4i)}(E)}{E-E_{R(4i)}+R_{(4i)}(E)+i\frac{1}{2}\Gamma_{(4i)}(E)}\,,$
(48)

with

$\displaystyle\Gamma_{(4i)}(E)$ $\displaystyle=$
$\displaystyle\Gamma_{R(4i)}\frac{pC_{\eta}^{2}W_{4}(p)}{p_{r}C_{\eta_{r}}^{2}W_{4}(p_{r})}\,,$
(49) $\displaystyle R_{(4i)}(E)$ $\displaystyle=$ $\displaystyle
a_{(4i)}(E-E_{R(4i)})^{2}+b_{(4i)}(E-E_{R(4i)})^{3}\,,\ \ \ i=1,2,3$ (50)

where $p_{r}=\sqrt{2\mu E_{(4i)}}$. We have 8 parameters to fit to the data as

$\displaystyle\theta_{4}$ $\displaystyle=$
$\displaystyle\\{E_{R(41)},\Gamma_{R(41)},a_{(41)},b_{(41)},E_{R(42)},\Gamma_{R(42)},a_{(43)},b_{(43)}\\}\,,$
(51)

where two parameters $a_{(42)}$ and $b_{(42)}$ are set to be zero,
$a_{(42)}=b_{(42)}=0$, because of their insensitivity for the parameter fit,
and we use the experimental values for $E_{R(43)}$ and $\Gamma_{R(43)}$,
$E_{R(33)}^{(exp)}=6.707(20)$ MeV and $\Gamma_{R(33)}^{(exp)}=89(2)$ keV [37],
for the background contribution from high energy.

In Table 2, we show fitted values and their errors of the 8 parameters in the
amplitudes of the $4_{1}^{+}$, $4_{2}^{+}$, $4_{3}^{+}$ states of 16O in the
$S$ matrix of elastic $\alpha$-12C scattering for $l=4$ in Eq. (48) where we
find a small value of $\chi^{2}/N$, $\chi^{2}/N=0.47$, for the parameter fit,
as shown in Table 3. In addition, the fitted values of $E_{R(41)}$,
$\Gamma_{R(41)}$, $E_{R(42)}$, and $\Gamma_{R(42)}$ agree with their
experimental values, $E_{R(41)}^{(exp)}=3.194(3)$ MeV,
$\Gamma_{R(41)}^{(exp)}=26(3)$ keV, $E_{R(42)}^{(exp)}=3.9347(17)$ MeV, and
$\Gamma_{R(42)}^{(exp)}=0.28(5)$ keV [37].

Figure 7:  Phase shift $\delta_{4}$ of elastic $\alpha$-12C scattering for
$g$-wave channel as a function of $E_{\alpha}$ calculated by using the fitted
values of the parameters. Experimental data are included in the figure as
well.

In Fig. 7, we plot a curve of the phase shift for $l=4$, $\delta_{4}$, by
using the fitted values of the parameters obtained in Table 2. The phase shift
data are also displayed in the figure. We find that the fitted curve agrees
well with the phase shift data.

4.6 Phase shift for $l=5$ channel

We consider $5_{1}^{-}$ state of 16O to construct an $S$ matrix of the elastic
$\alpha$-12C scattering for $l=5$ where $5_{1}^{-}$ is a resonant state as a
background contribution from high energy appearing at $E_{\alpha}=10.00$ MeV.
Thus, we have an expression of the $S$ matrix for $l=5$ as

$\displaystyle e^{2i\delta_{5}}$ $\displaystyle=$
$\displaystyle\frac{E-E_{R(51)}+R_{(51)}(E)-i\frac{1}{2}\Gamma_{(51)}(E)}{E-E_{R(51)}+R_{(51)}(E)+i\frac{1}{2}\Gamma_{(51)}(E)}\,,$
(52)

with

$\displaystyle\Gamma_{(51)}(E)$ $\displaystyle=$
$\displaystyle\Gamma_{R(51)}\frac{pC_{\eta}^{2}W_{5}(p)}{p_{r}C_{\eta_{r}}^{2}W_{5}(p_{r})}\,,$
(53) $\displaystyle R_{(51)}(E)$ $\displaystyle=$ $\displaystyle
a_{(51)}(E-E_{R(51)})^{2}+b_{(51)}(E-E_{R(51)})^{3}\,,$ (54)

where $p_{r}=\sqrt{2\mu E_{(51)}}$. We have 2 parameters to fit the data as

$\displaystyle\theta_{4}$ $\displaystyle=$
$\displaystyle\\{a_{(51)},b_{(51)}\\}\,,$ (55)

where we use the experimental values for $E_{R(51)}$ and $\Gamma_{R(51)}$,
$E_{R(51)}^{(exp)}=7.498(2)$ MeV and $\Gamma_{R(51)}^{(exp)}=670(15)$ keV
[37], for the background contribution from high energy.

In Table 2, we show fitted values and their errors of the 2 parameters in the
amplitudes of the $5_{1}^{-}$ state of 16O in the $S$ matrix of elastic
$\alpha$-12C scattering for $l=5$ in Eq. (52) where we find a very small value
of $\chi^{2}/N$, $\chi^{2}/N=0.094$, for the parameter fit, as shown in Table
3.

Figure 8:  Phase shift $\delta_{5}$ of elastic $\alpha$-12C scattering for
$h$-wave channel as a function of $E_{\alpha}$ calculated by using the fitted
values of the parameters. Experimental data are included in the figure as
well.

In Fig. 8, we plot a curve of the phase shift for $l=5$, $\delta_{5}$, by
using the fitted values of the parameters obtained in Table 2. The phase shift
data are also displayed in the figure. We find that the fitted curve agrees
well with the phase shift data.

4.7 Phase shift for $l=6$ channel

We include a background contribution from low energy and the $6_{1}^{+}$ state
of 16O to construct an $S$ matrix of the elastic $\alpha$-12C scattering for
$l=6$ where $6_{1}^{+}$ is a resonant state as a background contribution from
high energy appearing at $E_{\alpha}=10.20$ MeV. Thus, we have an expression
of the $S$ matrix for $l=6$ as

$\displaystyle e^{2i\delta_{6}}$ $\displaystyle=$
$\displaystyle\frac{K_{6}(p)-2\kappa
ReH_{6}(p)+ipC_{\eta}^{2}W_{6}(p)}{K_{6}(p)-2\kappa
ReH_{6}(p)-ipC_{\eta}^{2}W_{6}(p)}\frac{E-E_{R(61)}+R_{(61)}(E)-i\frac{1}{2}\Gamma_{(61)}(E)}{E-E_{R(61)}+R_{(61)}(E)+i\frac{1}{2}\Gamma_{(61)}(E)}\,,$
(56)

with

$\displaystyle K_{6}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}r_{6}p^{2}-\frac{1}{4}P_{6}p^{4}\,,$ (57)
$\displaystyle\Gamma_{(61)}(E)$ $\displaystyle=$
$\displaystyle\Gamma_{R(61)}\frac{pC_{\eta}^{2}W_{6}(p)}{p_{r}C_{\eta_{r}}^{2}W_{6}(p_{r})}\,,$
(58) $\displaystyle R_{(61)}(E)$ $\displaystyle=$ $\displaystyle
a_{(61)}(E-E_{R(61)})^{2}+b_{(61)}(E-E_{R(61)})^{3}\,,$ (59)

where $p_{r}=\sqrt{2\mu E_{R(61)}}$. We have four parameters to fit the data
as

$\displaystyle\theta_{3}=\\{r_{6},P_{6},a_{(61)},b_{(61)}\\}\,,$ (60)

where the leading effective range parameter, $-1/a_{6}$, is put to be zero,
$1/a_{6}=0$, and two effective range parameters, $r_{6}$ and $P_{6}$, are
included while we use the experimental values for $E_{R(61)}$ and
$\Gamma_{R(61)}$, $E_{R(61)}^{(exp)}=7.6534(16)$ MeV and
$\Gamma_{R(61)}^{(exp)}=70(8)$ keV [37], for the background contribution from
high energy.

In Table 2, we show fitted values and their errors of the four parameters in
the amplitudes of the background contribution from low energy and the
$6_{1}^{+}$ state of 16O in the $S$ matrix of elastic $\alpha$-12C scattering
for $l=6$ in Eq. (56) where we find a very small value of $\chi^{2}/N$,
$\chi^{2}/N=0.026$, for the parameter fit, as shown in Table 3.

Figure 9:  Phase shift $\delta_{6}$ of elastic $\alpha$-12C scattering for
$i$-wave channel as a function of $E_{\alpha}$ calculated by using the fitted
values of the parameters. Experimental data are included in the figure as
well.

In Fig. 9, we plot a curve of the phase shift for $l=6$, $\delta_{6}$, by
using the fitted values of the parameters obtained in Table 2. The phase shift
data are also displayed in the figure. We find that the fitted curve agrees
well with the phase shift data.

5\. Results and discussion

In the present work, we studied the expression of the $S$ matrices of elastic
$\alpha$-12C scattering at low energies for $l=0,1,2,3,4,5,6$ in EFT. The $S$
matrices are constructed as the summation of the phase shifts; the parts of
the phase shifts are obtained from the elastic scattering amplitudes of the
sub-threshold and resonant states of 16O. Those amplitudes are calculated from
the effective Lagrangian and obtained in terms of the effective range
parameters up to $p^{6}$ order for $l=0,1,2,4,5,6$ and up to $p^{8}$ order for
$l=3$ due to the modification of the counting rules for the effective range
parameters discussed in Refs. [26, 29]. We include the sub-threshold states
for $l=0,1,2,3$ and a background contribution from low energy for $l=6$, one
resonant state for $l=0,1,3$ and two resonant states for $l=2,4$, which appear
in the energy range of the phase shift data,
$2.6~{}\textrm{MeV}<E_{\alpha}<6.62~{}\textrm{MeV}$, and a resonant state
which appears at $6.62~{}\textrm{MeV}<E_{\alpha}$, as background contributions
from high energy for all the partial wave states. Those states included in the
present study are summarized in Table 1. Then, we fit the parameters to the
phase shift data while the resonant energies and widths of the background
contributions from high energy are fixed by using the experimental data of the
resonant states at $6.62~{}\textrm{MeV}<E_{\alpha}$, and some parameters which
are insensitive to the parameter fit are suppressed. We find that the
parameters are fitted very well where the $\chi^{2}/N$ values are less than
one for all the cases, as summarized in Table 3, and thus, the phase shifts
are well described within the theory.

One may regard this work as merely a simple parameter fit to the phase shift
data while our aim is to extrapolate the radiative capture rate down to the
Gamow peak energy, $E_{G}=0.3~{}\textrm{MeV}$; one may see that our result is
comparable to that worked out by the $R$-matrix analysis. One may also see
that the parameterization based on the effective range expansion in the $S$
matrices in Eq. (27) is simple and transparent. In addition, the
electromagnetic and weak interactions are straightforwardly introduced in the
theory. As mentioned in the introduction, we employed the EFT to the study of
the $E1$ transition of the 12C($\alpha$,$\gamma$)16O reaction [26] and the
$\beta$-delayed $\alpha$-emission from 16N [7].444 The dressed propagators of
16O shown in Fig. 1, $D_{l}^{-1}(p)$ for $l=1$ and $l=3$, whose parameters are
fitted to the phase shift data, are used as the building blocks of reaction
amplitudes of the $E1$ transition of 12C($\alpha$,$\gamma$)16O and the
$\beta$-delayed $\alpha$-emission of 16N.  Recently, we reported the first
application of EFT to the study of the radiative proton capture on 15N [49,
50], which is an important reaction in the CNO cycle. Thus, constructing an
EFT would be another theoretical method, as an alternative to the $R$-matrix
analysis, for the studies of nuclear reactions for stellar evolution.

Acknowledgements

We would like to thank S. W. Hong, T.-S. Park, and C. H. Hyun for discussions.
We also thank the reviewers for the discussions and for providing the data
files of wavefunctions and phase shifts calculated from a Woods-Saxon
potential. This work was supported by the National Research Foundation of
Korea (NRF) grant funded by the Korean government (MSIT) (No. 2019R1F1A1040362
and 2022R1F1A1070060).

Appendix

In the present appendix, we perform a test calculation by employing a Woods-
Saxon potential. The data, wavefunctions and phase shifts, are generated by
using the model potential, the values of ANCs are calculated from the
wavefunctions, and the scattering phase shifts are generated by using the
potential. We, then, fit the effective range parameters to the phase shift
data generated from the potential, and calculate the ANCs by using the fitted
values of parameters in Eq. (18). We note that the relation between the two
methods, the potential model and the effective range expansion, to deduce the
ANCs is not obvious. The comparison of the ANCs from the potential model and
the fit of effective range parameters may indicate some degree of the model
dependence in the two methods.

Figure 10: Wavefunction for $0^{+}$ state of 16O calculated by using a Woods-
Saxon potential, which is normalized as $\int_{0}^{\infty}u_{0}(r)^{2}dr=1$.
Figure 11: Wavefunction for $2^{+}$ state of 16O calculated by using a Woods-
Saxon potential, which is normalized as $\int_{0}^{\infty}u_{2}(r)^{2}dr=1$.

In Figs. 10 and 11, we display the wavefunctions of the $0^{+}$ and $2^{+}$
states of 16O, respectively, from a Woods-Saxon potential. The geometry of
Woods-Saxon potential is standard; $R_{0}=1.3$ fm; $R=R_{0}A^{1/3}=2.976$ fm
and $a=0.7$ fm. The wavefunctions of bound states are obtained by adjusting
the depth of potential. For the $0^{+}$ state, $E_{x}=6049.4$ keV and
$B_{0}=1112.5$ keV, and one has $V_{0}=131.1$ MeV, and for the $2^{+}$ state,
$E_{x}=6917.1$ keV and $B_{2}=244.8$ keV, and one has $V_{0}=130.8$ MeV. The
wavefunctions are normalized as $\int_{0}^{\infty}u_{l}(r)^{2}dr=1$ with
$l=0,2$. (The data files of wavefunctions and phase shifts are provided by a
reviewer.)

The values of ANCs may be obtained by using the relations

$\displaystyle
u_{0}(r)\sim|C_{b}|_{0}W_{-\kappa/\gamma_{0},\frac{1}{2}}(2\gamma_{0}r)\,,\ \
\ \
u_{2}(r)\sim|C_{b}|_{2}W_{-\kappa/\gamma_{2},\frac{5}{2}}(2\gamma_{2}r)\,,$
(61)

at the outside of the potential range, $R<r$, where $W_{\nu,\mu}(z)$ is the
Whittaker function, and $\gamma_{0}$ and $\gamma_{2}$ are the binding momenta,
$\gamma_{0}=\sqrt{2\mu B_{0}}$ and $\gamma_{2}=\sqrt{2\mu B_{2}}$. Thus, we
have the ANCs from the wavefunctions as

$\displaystyle|C_{b}|_{0}=2.6\times 10^{3}\ \ \textrm{fm}^{-1/2}\,,\ \ \ \
|C_{b}|_{2}=1.9\times 10^{5}\ \ \textrm{fm}^{-1/2}\,.$ (62)

It is good to see that the ANCs are straightforwardly obtained from the
wavefunctions generated from the potential model after fitting the depth of
potential to the binding energies. While one may wonder about the sensitivity
of ANCs to all the three parameters of potential and the number of nodes of
wavefunctions inside the potential range. In addition, the ANCs from the
potential model depend on the normalization condition of wavefunctions; the
reliability of two-body description of the 16-body nucleon system inside the
potential range is questionable.

Figure 12:  Phase shift $\delta_{0}$ of elastic $\alpha$-12C scattering for
$l=0$ as a function of the $\alpha$ energy $E_{\alpha}$ in the lab mass frame,
calculated by using a Woods-Saxon potential.  Figure 13:  Phase shift
$\delta_{2}$ of elastic $\alpha$-12C scattering for $l=2$ as a function of the
$\alpha$ energy $E_{\alpha}$ in the lab frame, calculated by using a Woods-
Saxon potential.

In Figs. 12 and 13, we display the phase shifts of elastic $\alpha$-12C
scattering for $l=0$ and 2, respectively, calculated from the Woods-Saxon
potential with the fixed values of $V_{0}$ mentioned above. Now we carry out a
test for our fitting method to deduce the ANCs from the phase shift data: the
effective range parameters are fitted to the phase shift data in the figures,
and we calculate the ANCs by using Eq. (18). When fitting the phase shift
data, we include a constant error to the phase shift data,
$\Delta\delta_{0,2}=0.20$ degree, which is a typical size of the error in the
phase shift data for $l=0$ from the experiment.

$E_{\alpha,max}$ (MeV) | 4.0 | 4.5 | 5.0 | 5.5
---|---|---|---|---
$|C_{b}|_{0}$ ($\textrm{fm}^{-1/2}$) | $1.1\times 10^{3}$ | $2.2\times 10^{3}$ | $3.7\times 10^{3}$ | $5.8\times 10^{3}$
$\chi^{2}/N$ | 0.05 | 0.16 | 0.43 | 1.18
Table 5:  ANC of $0^{+}$ state, $|C_{b}|_{0}$, as a function of the maximum
energy of the data, $E_{\alpha,max}$ (MeV), deduced from the phase shift data
in Fig. 12. Values of $\chi^{2}/N$ are also displayed in the table.

In Table 5, we display the values of ANC for the $0^{+}$ state as a function
of the maximum energy, $E_{\alpha,max}=4.0$, 4.5, 5.0, 5.5 MeV, of the phase
shift data in Fig. 12. We find that the shapes of the curves of the phase
shift are different (the curve from the effective range parameters is stiffer
than that from the potential model), and it leads to the energy dependence of
the fit. 555 The different energy dependence in the phase shifts possibly came
out due to the use of an energy-independent potential model, where one may
notice that the calculated phase shifts do not agree with the experimental
data. The use of an energy-dependent potential could improve the present
situation.  One can see that the values of ANC in the table have a significant
range, $(1.1\textrm{--}5.8)\times 10^{3}$ fm-1/2 with
$\chi^{2}/N=0.05\textrm{--}1.18$, while the ANC from the potential model,
$2.6\times 10^{3}$ fm-1/2, can be found within the range of ANC from the fit.
For the $2^{+}$ state, using all the phase shift data in Fig. 13, we have

$\displaystyle|C_{b}|_{2}=3.4(54.4)\times 10^{5}~{}\textrm{fm}^{-1/2}\,,$ (63)

with $\chi^{2}/N=0.13$. The center value of ANC is 1.8 times larger than that
of the potential model in Eq. (61) while the ANC from fit has a large error
bar. As discussed in Ref. [29], when the center value of $dD/dp^{2}$ term at
$p=i\gamma_{2}$ almost vanishes in the denominator in Eq. (18), the error of
ANC is enhanced.

In this appendix, we performed a test calculation to study whether the ANCs
obtained from the wavefunctions of a potential model can be reproduced by
fitting the effective range parameters to the phase shift generated by the
model potential. We find that the ANCs obtained from the fitted values of
effective range parameters agree with those obtained from the wavefunction of
the potential model within the uncertainties discussed above; aside from the
large uncertainties, the center values of ANCs agree within a factor of 2. The
large uncertainties may stem from the formula of ANC in Eq. (18). When the
slope of the inverse of the propagator, $D_{l}(p)$, becomes small at the
binding momentum $p=i\gamma_{l}$, the ANC becomes large. In other words, the
center value of ANC becomes sensitive to the values of effective range
parameters while the errors of ANC are enhanced. Therefore, because of the
difficulty discussed above, the present cases to deduce the ANCs of $0^{+}$
and $2^{+}$ state from the phase shift data may not be ideal: the uncertainty
of a factor of 2 may be involved between the two methods.

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