Cluster effective field theory and nuclear reactions

The radiative $\alpha$ capture on ¹²C, ¹²C($\alpha$, $\gamma$)¹⁶O, is one of the fundamental reactions in nuclear astrophysics, which determines the ratio ¹²C/¹⁶O produced in helium burning f-rmp84 . The reaction rate, equivalently the astrophysical $S$-factor, of the process at the Gamow peak energy, $T_{G}=0.3$ MeV ⁵⁵5 See the footnote 6. , however, cannot experimentally be determined due to the Coulomb barrier. A theoretical model is necessary to be employed in order to extrapolate the cross section down to $T_{G}$ by fitting model parameters to available experimental data measured at a few MeV or larger. During the last half century, a lot of experimental and theoretical studies for the process have been carried out. For reviews, see, e.g., Refs. bb-npa06 ; detal-17 ; chk-epja15 and references therein.

In constructing a model for the radiative capture process, one needs to take into account the excited states of ¹⁶O bb-npa06 , particularly, two excited bound states for $l^{\pi}_{n\mbox{-}th}=1_{1}^{-}$ and $2_{1}^{+}$ just below the $\alpha$-¹²C breakup threshold at $T=-0.045$ and $-0.24$ MeV ⁶⁶6 The kinetic energy $T$ denotes that of the $\alpha$-¹²C system in the center of mass frame. , respectively, as well as $1_{2}^{-}$ and $2_{2}^{+}$ resonant (second excited) states at $T=2.42$ and $2.68$ MeV, respectively. Thus the capture reaction to the ground state of ¹⁶O at $T_{G}$ is expected to be $E1$ and $E2$ transitions dominant due to the subthreshold $1_{1}^{-}$ and $2_{1}^{+}$ states. While the resonant $1_{2}^{-}$ and $2_{2}^{+}$ states play a dominant role in the available experimental data at low energies, typically $1\leq T\leq 3$ MeV. A minor contribution to the $S$-factor appears from so-called cascade transitions in which the initial $\alpha$ and ¹²C form an excited state of ¹⁶O emitting a photon, and it subsequently decays to the ground state of ¹⁶O. Experimental data pertaining to processes for nuclear astrophysics are compiled, known as NACRE-II compilation xetal-npa13 , in which the $S$-factor of the ¹²C($\alpha$,$\gamma$)¹⁶O reaction is estimated employing a potential model, and reported uncertainty of the process is less than 20 %.

Theoretical frameworks employed for the study are categorized mainly into two detal-17 : the cluster models using generalized coordinate method dbh-npa84 or potential model lk-npa85 and the phenomenological models using the parameterization of Breit-Wigner, $R$-matrix lt-rmp58 , or $K$-matrix hdz-npa76 . A recent trend of the study is to rely on intensive numerical analysis, in which a larger amount of the experimental data relevant to the study are accumulated, such as those from ¹²C($\alpha$,$\gamma$)¹⁶O, radiative proton capture on ¹⁵N to the ground state of ¹⁶O, ¹⁵N($p$,$\gamma_{0}$)¹⁶O, $\beta$ delayed $\alpha$ emission from ¹⁶N, ¹⁶N($\beta\alpha$)¹²C, elastic $\alpha$-¹²C scattering, ¹²C($\alpha$,$\alpha$)¹²C, and $\alpha$ transfer reactions, ¹²C(⁶Li,$d$)¹⁶O, ¹²C(⁷Li,$t$)¹⁶O, ⁶Li(¹²C,$d$)¹⁶O, and ⁷Li(¹²C,$t$)¹⁶O, up to relatively high energies, $T\simeq 7$ MeV, and a significant number of parameters of the models are fitted to the data by using computing power detal-17 ; xetal-npa13 ; aetal-prc15 . In the following, we discuss an alternative approach to estimate the $S$-factor at $T_{G}$; we discuss counting rules of the EFT at $T_{G}$ and display an effective Lagrangian for the reactions.

In the study of the radiative $\alpha$ capture on ¹²C at $T_{G}=0.3$ MeV employing an EFT, at such a low energy, one may regard the ground states of $\alpha$ and ¹²C as point-like particles whereas the first excited states of $\alpha$ and ¹²C are chosen as irrelevant degrees of freedom, by which a large momentum scale of the theory is determined. An effective Lagrangian for the process is constructed by using two spinless scalar fields for $\alpha$ and ¹²C, and terms of the Lagrangian are expanded in terms of the number of derivatives. The expansion parameter of the theory is $Q/\Lambda_{H}\sim 1/3$ where $Q$ denotes a typical momentum scale $Q\sim k_{G}$: $k_{G}$ is the Gamow peak momentum, $k_{G}=\sqrt{2\mu T_{G}}\simeq 41$ MeV, where $\mu$ is the reduced mass of $\alpha$ and ¹²C.⁷⁷7 A typical length scale of the $\alpha$-¹²C system is, thus, $k_{G}^{-1}\simeq 4.8$ fm. This is comparable to the sum of the radii of $\alpha$ and ¹²C obtained from a nuclear radius formula, $r_{A}=1.2A^{1/3}$ fm; $r_{\alpha}+r_{{}^{12}C}=1.2(4^{1/3}+12^{1/3})=4.65$ fm. Meanwhile, the emitted photon carries away almost all of the released energy, $E_{\gamma}^{\prime}\simeq 7.64$ MeV, and the length scale of the photon is larger than the other length scales, $E_{\gamma}^{\prime-1}\simeq 26$ fm. Thus, the photon may recognize the nuclear system as to be point-like. $\Lambda_{H}$ denotes a large momentum scale $\Lambda_{H}\simeq\sqrt{2\mu_{4}T_{(4)}}$ or $\sqrt{2\mu_{12}T_{(12)}}\sim 150$ MeV where $\mu_{4}$ is the reduced mass of one and three-nucleon system and $\mu_{12}$ is that of four and eight-nucleon system. $T_{(4)}$ and $T_{(12)}$ are the first excited energies of $\alpha$ and ¹²C, respectively: $T_{(4)}=20.21$ MeV for $0^{+}_{2}$ state of ⁴He and $T_{(12)}=4.44$ MeV for $2^{+}_{1}$ state of ¹²C. By including the terms up to next-to-next-to-leading order (NNLO), for example, one may obtain about 10% theoretical uncertainty for the process.

Coefficients appearing in an effective Lagrangian are fixed by using the experimental data which are measured at significantly higher energies than $T_{G}$. In the following sections, we employ the data from three processes, elastic $\alpha$-¹²C scattering measured at $T\simeq 2-5$ MeV, $S_{E1}$ factor for ¹²C($\alpha$,$\gamma$)¹⁶O at $T\simeq 1-3$ MeV, and $\beta$ delayed $\alpha$ emission from ¹⁶N at $T\simeq 0.8-3.2$ MeV. Thus, the perturbative scheme of the theory may not be reliable at the energies where those experimental data are measured. We first fix some parameters of the dressed ¹⁶O propagators, parameterized in terms of effective range expansion, by using the binding energies of the excited ¹⁶O states and fit the other parameters of the propagators to the phase shift data of the elastic scattering measured at $T\simeq 2-5$ MeV. Because of non-perturbative nature of a propagator we treat it as a non-perturbative quantity; the dressed ¹⁶O propagator commonly appears in the three processes, and we keep it as a non-perturbative one and expand reaction amplitudes perturbatively around it. For the radiative capture process, two additional parameters for the $E1$ transition amplitude of the radiative $\alpha$ capture process are fitted to the $S_{E1}$ factor data measured at $T\simeq 1-3$ MeV, and a value of the $S_{E1}$ factor is estimated at $T_{G}=0.3$ MeV.

For the $\beta$ delayed $\alpha$ emission from ¹⁶N, parameters of the decay amplitudes are fitted to the experimental data measured at $T\simeq 0.8-3.2$ MeV. This study would explore a validity of the present approach. One reason is that the $\beta$ delayed $\alpha$ emission data are covered with the small energy region compared to that for the elastic scattering data. The other reason is, as mentioned before, to test a different parameterization for the dressed ¹⁶O propagator compared to that in the $R$-matrix or $K$-matrix analysis. In the conventional $R$-matrix analysis, the subthreshold $1_{1}^{-}$ state and the broad resonant $1_{2}^{-}$ state of ¹⁶O are represented by the Breit-Wigner formula and are linearly combined in the reaction matrix along with a background contribution; a secondary maximum of the $\beta$ delayed $\alpha$ emission data is known to be important to constrain an interference pattern among those levels bd-npa88 ; jfhk-prc90 ; hfk-prc91 . In the present approach, the interference between the $1^{-}$ subthreshold and resonant states doesn’t exist because the subthreshold $1_{1}^{-}$ state and the broad resonant $1_{2}^{-}$ state of ¹⁶O is represented by a single dressed ¹⁶O propagator (in terms of effective range expansion); instead, we will see that the secondary peak can be reproduced by an interference between the amplitudes from a non-pole diagram and a pole diagram.

An effective Lagrangian for the present study is written as a-epja16 ; a-18

where ${\cal L}_{ES}$ is the Lagrangian for the elastic scattering process a-epja16 , ${\cal L}_{RC(+)}$ is for additional terms for the radiative $\alpha$ capture process a-18 , and ${\cal L}_{BD(+)}$ is for additional terms for the $\beta$-delayed $\alpha$ emission from ¹⁶N.

The Lagrangian ${\cal L}_{ES}$ may be written by using the composite ¹⁶O fields consisting of $\alpha$ and ¹²C as a-epja16 ; a-jkps18 ; bs-npa01 ; ah-prc05

where $\phi_{\alpha}$ ($m_{\alpha}$) and $\phi_{C}$ ($m_{C}$) are 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. The dots denote higher order terms. $d_{(l)}$ represent $\alpha$ and ¹²C composite fields of angular momentum $l=0,1,2,3$: $d_{(0)}$ for $l=0$, $d_{(1)i}$ for $l=1$, $d_{(2)ij}$ for $l=2$, and $d_{(3)ijk}$ for $l=3$ where those spin states are represented by the subscripts $i$, $ij$, and $ijk$ as Cartesian tensors j-prb70 . (We have suppressed those indices for the Cartesian tensors in the Lagrangian in Eq. (2).) Those composite fields are introduced to make an expansion around the unitary limit. $C_{n}^{(l)}$ are coupling constants for the propagation of the $\alpha$-¹²C composite fields for the $l$ channels, and are related to effective range parameters along with common multiplicative factors $1/y_{(l)}^{2}$. As we will discuss later, because of a modification of the counting rules for the elastic $\alpha$-¹²C scattering we include the terms up to $n=3$ for $l=0,1,2$ and those up to $n=4$ for $l=3$. $O_{l}$ are projection operators by which the $\alpha$-¹²C system is projected to the $l$-th partial wave states. Thus one has

The Lagrangian ${\cal L}_{RC(+)}$ for additional terms to the study for the $E1$ transition of the radiative $\alpha$ capture reaction may be written down as a-18

with

where $m_{O}$ is the mass of ¹⁶O in the ground state. We note that because the ¹⁶O ground state appears only in the final state, $\phi_{O}$ ($\phi_{O}^{\dagger}$) is introduced as a source field for the ¹⁶O ground state in the final (initial) state. In the first term in Eq. (4), for example, $\phi_{O}$ ($\phi_{O}^{\dagger}$) destroys (creates) the ¹⁶O ground state, and $(\phi_{\alpha}\phi_{C})^{\dagger}$ [($\phi_{\alpha}\phi_{C}$)] fields create (destroy) a $s$-wave $\alpha$-¹²C state. The transition rate between the ¹⁶O ground state and the $s$-wave $\alpha$-¹²C state is parameterized by the coupling constant $y^{(0)}$, which is fixed by using experimental data. A contact interaction, the $h^{(1)}$ term, is introduced to renormalize divergence from loop diagrams for the radiative $\alpha$ capture reaction.

A weak decay amplitude is described in terms of the $V-A$ type current-current interaction for semi-leptonic decay process, and its interaction Hamiltonian density is given as

where $G_{F}$ is the Fermi constant, $J^{(n)}_{\mu}(x)$ is a nuclear current to be constructed by using the external vector and axial-vector fields in the effective Lagrangian while $J^{(l)\mu}(x)$ is a lepton current. In the energy-momentum space it is given as

where $q$ is the momentum transfer, $q^{\mu}=p^{\prime\mu}-p^{\mu}$.

The Lagrangian ${\cal L}_{BD(+)}$ is for additional terms for the study of the $\beta$ delayed $\alpha$ emission from ¹⁶N. We construct ${\cal L}_{BD(+)}$ for interactions where the initial $2_{1}^{-}$ ground state of ¹⁶N decaying to the $1_{1}^{-}$ and $3_{1}^{-}$ states of ¹⁶O and the $\alpha$-¹²C breakup state for $l=1$ and 3 channels through the Gamow-Teller transition, which may be written as

where $a_{i}$ is the external axial-vector field, which generates an axial nuclear current for the Gamow-Teller transition, and $\phi_{N,ij}$ is the source field for the initial $2_{1}^{-}$ ground state of ¹⁶N. $C_{a}^{(l=1)}$, $C_{b}^{(l=1)}$, $C_{a}^{(l=3)}$, and $C_{b}^{(l=3)}$ are coefficients for $l=1$ and $l=3$ at LO, and $D_{a}^{(l=1)}$ and $D_{b}^{(l=1)}$ are those for $l=1$ at NNLO; the indices of the squared operators, $O_{1}^{2}$ and ${\cal O}^{(1)2}$, for the terms at NNLO are suppressed in the above equation.

The differential cross section of the elastic $\alpha$-¹²C scattering (for two spin-0 charged particles) in terms of the phase shifts is given by (see, e.g., Ref. lt-rmp58 )

where $f(\theta)$ is the scattering amplitude including both pure Coulomb part and Coulomb modified strong interaction part, $\theta$ is the scattering angle, $k$ is the absolute relative momentum, and $\eta=\kappa/k$. $\kappa$ is the inverse of the Bohr radius, $\kappa=Z_{\alpha}Z_{C}\mu\alpha_{E}$ where $Z_{\alpha}$ and $Z_{C}$ are the number of protons in $\alpha$ and ¹²C, respectively, and $\alpha_{E}$ is the fine structure constant. In addition, $\omega_{l}$ is the Coulomb scattering phase, $\omega_{l}(=\sigma_{l}-\sigma_{0})=\sum\arctan(\eta/s)$ for $s=1$ to $l$; $\sigma_{l}$ are the Coulomb phase shifts, $\sigma_{l}=\arg\Gamma(1+l+i\eta)$ with $l=0,1,2,\cdots$, and

$\delta_{l}$ are real scattering phase shifts. $P_{l}(x)$ are the Legendre polynomials. The scattering amplitudes are represented in terms of $\delta_{l}$ as lt-rmp58  ⁸⁸8 One can see the relation between $U_{l}$ and $A_{l}$ through the relation $\displaystyle\frac{1}{2}i\left[\exp\left(2i\omega_{l}\right)-U_{l}\right]=\exp\left(2i\omega_{l}\right)\frac{1}{\cot\delta_{l}-i}\,.$

In the EFT, the amplitudes of the elastic scattering are calculated from diagrams depicted in Figs. 1 and 2. We obtain the scattering amplitudes for $l$-th partial wave states as a-epja16 ; aetal-prc07 ; a-epja07

with

and

where $\psi(z)$ is the digamma function. Note that the function, $-2\kappa H_{l}(k)$, the Coulomb self-energy term, in the denominator of the amplitude is obtained from the Coulomb bubble diagram for the dressed propagator of ¹⁶O in Fig. 1, and the factor, $e^{2i\sigma_{l}}W_{l}(\eta)C_{\eta}^{2}$, in the numerator is from the initial and final state Coulomb interactions between $\alpha$ and ¹²C in Fig. 2. In Appendix A, we display the renormalized dressed three-point vertices for the initial and final Coulomb interactions and the renormalized dressed composite ¹⁶O propagators for $l=0,1,2,3$.

The function $K_{l}(k)$ represents the interaction due to the short range nuclear force (compared with the long range Coulomb force), which is obtained in terms of the effective range parameters as

One can find that the expression obtained in Eq. (12) reproduces well the previous results reported in Refs. klh-jpg13 ; hot-npb73 ; h-jmp77 ; scb-prc10 .

By comparing the two expressions of the amplitudes $A_{l}$ in Eqs. (11) and (12), one has a relation between the phase shift and the effective range parameters in the denominator of the scattering amplitudes, $D_{l}(k)$, as

where

To estimate the ANC, $|C_{b}|$, for the $0_{2}^{+}$, $1_{1}^{-}$, $2_{1}^{+}$, $3_{1}^{-}$ states of ¹⁶O, we employ the definition of $|C_{b}|$ ir-prc84 ,

where $\eta_{b}=\kappa/k_{b}$; $k_{b}=i\gamma_{l}$, and $\gamma_{l}$ are binding momenta which will be given in the next subsection.

We now discuss a modification of the counting rules a-prc18 , based on an observation in Eq. (16). The term on the left-hand-side of Eq. (16) is significantly suppressed by the factor $C_{\eta}^{2}$ at low energies. Meanwhile, the Coulomb self-energy term, $-2\kappa H_{l}(k)$, on the right-hand-side of Eq. (16) turns out to be two orders of magnitude larger than the term on the left-hand-side of the equation. Thus, we introduce terms in $K_{l}(k)$ as counter terms at LO, by which those unnaturally large terms from the Coulomb self-energy term are subtracted. We will discuss it in detail below.

The effective range parameters in $K_{l}(k)$ are expanded in powers of $k^{2}$ whereas the real part of the function $H_{l}(k)$ can be expanded in powers of $k^{2}$ as well. For the function $H(\eta)$ in $H_{l}(k)$, one has

where $\kappa$ is the inverse of the Bohr radius, $\kappa\simeq 245$ MeV, and regarded as another large scale of the theory. Thus, the right-hand-side of equation, $ReD_{l}(k)$, in Eq. (16) can be expanded as a power series of $k^{2}$ for both $K_{l}(k)$ and $2\kappa ReH_{l}(k)$. Meanwhile, the left-hand-side of Eq. (16) is suppressed by the factor $C_{\eta}^{2}$, due to the Gamow factor $P=\exp(-2\pi\eta)$.

In the case of the $s$-wave, for example, the reported phase shift at the smallest energy, $T_{\alpha}=2.6$ MeV, where $T_{\alpha}$ is the kinetic energy of $\alpha$ in the laboratory frame ⁹⁹9 The kinetic energies, $T$ in the center of mass frame and $T_{\alpha}$ in the laboratory frame, are related by a relation $T=\frac{3}{4}T_{\alpha}$. , is $\delta_{0}=-1.893^{\circ}$ tetal-prc09 . The factor $C_{\eta}^{2}$ becomes $C_{\eta}^{2}\simeq 6\times 10^{-6}$ at $k=104$ MeV which corresponds to $T_{\alpha}=2.6$ MeV, and the left-hand-side of Eq. (16) numerically becomes $C_{\eta}^{2}k\cot\delta_{0}=-0.019$ MeV. The function $2\kappa ReH_{0}(k)$ is expanded as

at $k=104$ MeV. The numerical values in the third line of Eq. (20) correspond to the terms appearing in the second line of the equation in order. One can see that the power series converges well, but the first and second terms are two and one order of magnitude larger than a value estimated by using the experimental data in the left-hand-side of Eq. (16), $-$0.019 MeV. Those terms are unnaturally large, and thus it is necessary to introduce a new renormalization method, in which the counter terms remove the unnaturally large terms and make the terms in a natural size. In other words, we assume that fitting polynomial functions are represented as a natural power series at the low energy region, and to maintain such polynomial functions, large cancellations for the first and second terms with the $r_{l}$ and $P_{l}$ effective terms, respectively, are expected. So we include the three effective range parameters, $r_{l}$, $P_{l}$, and $Q_{l}$, for the $l=0$ channel as the counter terms. The same tendency can be seen in the $l=1,2$ channels whereas one needs to include four effective range parameters for the $l=3$ channel. Thus, we employ the three effective range parameters, $r_{l}$, $P_{l}$, $Q_{l}$ for the $l=0,1,2$ channels and the four effective range parameters, $r_{l}$, $P_{l}$, $Q_{l}$, $R_{l}$ for the $l=3$ channel when fitting the parameters to the phase shift data.

At the binding energies of the excited $0_{2}^{+}$, $1_{1}^{-}$, $2_{1}^{+}$, $3_{1}^{-}$ states of ¹⁶O, the amplitudes should have a pole at $k_{b}=i\gamma_{l}$ where $\gamma_{l}$ are the binding momenta, $\gamma_{l}=\sqrt{2\mu B_{l}}$; $B_{l}$ denote the binding energies of $l_{i\textrm{-}th}^{\pi}$ excited states of ¹⁶O. Thus the denominator of the scattering amplitude, $D_{l}(k)$, should vanish at $k_{b}$. Using this condition, a first effective range parameter, $a_{l}$, is related to other effective range parameters as

and the remaining effective range parameters are fixed by using the phase shift data of the elastic scattering.¹⁰¹⁰10 In Ref. a-jkps18 , we have also studied an inclusion of the ground $0_{1}^{+}$ state of ¹⁶O in the parameter fitting for the elastic $\alpha$-¹²C scattering at low energies.

The effective range parameters, $r_{l}$, $P_{l}$, $Q_{l}$ for $l=0,1,2$ and $r_{l}$, $P_{l}$, $Q_{l}$, $R_{l}$ for $l=3$ are fitted, employing the standard $\chi^{2}$ fit ¹¹¹¹11 We employ a python package, emceefmetal-12 , for the fitting. , to the experimental phase shift data of the elastic $\alpha$-¹²C scattering reported by Tischhauser et al. tetal-prc09 ; epaps . ¹²¹²12 Input data for the parameter fitting are the phase shifts of the elastic $\alpha$-¹²C scattering for $l=0,1,2,3$ tetal-prc09 ; epaps , which have been generated from the $R$-matrix analysis of the elastic scattering data tetal-prl02 . In the input data files epaps , there are four column data: the first column is the alpha energy, the second one the phase shift as derived from the globalized Monte Carlo simulations, the third one is the same phase shift randomized by the error from the Monte Carlo simulations, and the fourth one is the error of the phase shifts from the Monte Carlo simulations. We have used the second column of the phase shift data in our previous works a-epja16 ; a-prc18 and the third column of the phase shift data in our other works a-jkps18 ; ya-19 ; a-18 for fitting. The reported energy range of the data is $2.6\leq T_{\alpha}\leq 6.62$ MeV ($2\leq T\leq 5$ MeV.) tetal-prc09 . Though we have discussed that a large energy scale of the theory is that for the first excited $2_{1}^{+}$ state of ¹²C, $T_{(12)}=4.44$ MeV ($T_{\alpha(12)}=5.92$ MeV) in the previous section, other large scales can emerge from resonate energies of ¹⁶O. As suggested by Teichmann t-pr51 , below the resonance energies, the Breit-Wigner-type parameterization for the resonances can be expanded in powers of the energy, and one can obtain an expression of the amplitude in terms of the effective range expansion. Therefore, in the previous works a-epja16 ; a-prc18 , we choose the energies of the resonant states, $T_{\alpha}(0_{3}^{+})=6.52$ MeV, $T_{\alpha}(1_{2}^{-})=3.23$ MeV, $T_{\alpha}(2_{2}^{+})=3.58$ MeV, $T_{\alpha}(3_{2}^{-})=5.92$ MeV, as a large scale of the theory for each partial wave state. After fitting the parameters to the data below the resonant energies, we confirm large cancellations between the fitted parameters and those generated from the Coulomb self-energy terms, e.g., in Eq. (20) while the perturbative series of the effective range expansions at $T_{G}$ converges well as expected by the theory. See Table V in Ref. a-prc18 and Table 2 in Ref. a-jkps18 . The perturbative series, however, does not converge at the large, experimental energies. We will come back to the issue in the next section.

In our recent works ya-19 ; a-18 , we include broad resonant states in the fitting because one may expect that because a relatively deep pole for a broad resonance state in the complex energy plane is located at a distance from the real axis, it can possibly be represented by a polynomial function in terms of the effective range expansion. We find that the broad resonant $1_{2}^{-}$ and $3_{2}^{-}$ states with the decay widths $\Gamma(1_{2}^{-})=420\pm 20$ keV and $\Gamma(3_{2}^{-})=800\pm 100$ keV can be well incorporated into the fitting of the effective range parameters whereas the narrow resonant $0_{3}^{+}$ and $2_{2}^{+}$ states with the decay widths $\Gamma(0_{3}^{+})=1.5\pm 0.5$ keV and $\Gamma(2_{2}^{+})=0.625\pm 0.100$ keV cannot. In addition, a tail-like structure from a higher resonant $1_{3}^{-}$ state at $T_{\alpha}(1_{3}^{-})=7.96$ MeV with the width $\Gamma(1_{3}^{-})=110\pm 30$ keV cannot be incorporated into the fitting for the high energy region of the data at $T_{\alpha}\geq 6$ MeV for $l=1$ either. Thus we use the maximum energies, $T_{\alpha,max}=5.5$, 6.0, 3.2, and 6.62 MeV for $l=0,1,2$, and 3, respectively, of the data sets for the fitting where we assume that the resonant $2_{1}^{+}$ state of ¹²C, which could appear at $T_{\alpha(12)}=5.92$ MeV, would be negligible for $l=1$ and $l=3$.