Cluster effective field theory and nuclear reactions

Abstract

An effective field theory (EFT) for a nuclear reaction at low energies is studied. The astrophysical S-factor of radiative $\alpha$ capture on $^{12}\hbox {C}$ at the Gamow-peak energy, $T_\mathrm{G} = 0.3$ MeV, is a fundamental quantity in nuclear astrophysics, and we construct an EFT for the reaction. To fix parameters appearing in the effective Lagrangian, the EFT is applied to the study for three reactions: elastic $\alpha$–$^{12}\hbox {C}$ scattering at low energies, the E1 transition of radiative $\alpha$ capture on $^{12}\hbox {C}$, and $\beta$ delayed $\alpha$ emission from $^{16}\hbox {N}$. We report an estimate of the $S_{E1}$-factor of the reaction through the E1 transition at $T_\mathrm{G}$ by employing the EFT. We also discuss applications of EFTs to nuclear reactions at low energies.

Appendix

The elastic scattering amplitudes are calculated from the renormalized dressed three-point vertices and the renormalized dressed composite $^{16}\hbox {O}$ propagators as

$$\begin{aligned} iA_l = -i\varGamma ^{(l)}(k')D^{(l)}(T)\varGamma ^{(l)}(k), \end{aligned}$$

(49)

where $k'=k$ and $T=k^2/(2\mu )$, and we have suppressed the indices from the Cartesian tensors.

We have the renormalized dressed three-point vertices for the initial and final Coulomb interaction for $l=0,1,2,3$ as

$$\begin{aligned} \varGamma ^{(l=0)}(k)= & {} y_{(0)}e^{i\sigma _0}C_\eta \,, \end{aligned}$$

(50)

$$\begin{aligned} \varGamma ^{(l=1)}_i(k)= & {} \frac{y_{(1)}}{\mu }k_ie^{i\sigma _1} \sqrt{1-\eta ^2}C_\eta \,, \end{aligned}$$

(51)

$$\begin{aligned} \varGamma ^{(l=2)}_{ij}(k)= & {} \frac{y_{(2)}}{\mu ^2}15e^{i\sigma _2}C_2\left( k_ik_j -\frac{1}{3}\delta _{ij}k^2 \right) \,, \end{aligned}$$

(52)

$$\begin{aligned} \varGamma ^{(l=3)}_{ijk}(k)= & {} \frac{y_{(3)}}{\mu ^3}105e^{i\sigma _3}C_3 \nonumber \\&\times \left[ k_ik_jk_k -\frac{1}{5}k^2\left( \delta _{ij}k_k + \delta _{ik}k_j + \delta _{jk}k_i \right) \right] ,\nonumber \\ \end{aligned}$$

(53)

where

$$\begin{aligned} C_2= & {} \frac{1}{30}C_\eta \sqrt{ (1+\eta ^2)(4+\eta ^2) }, \end{aligned}$$

(54)

$$\begin{aligned} C_3= & {} \frac{1}{630}C_\eta \sqrt{ (1+\eta ^2)(4+\eta ^2)(9+\eta ^2) }\,. \end{aligned}$$

(55)

We have the renormalized dressed composite $^{16}\hbox {O}$ propagators for $l=0,1,2,3$ as

$$\begin{aligned} D^{(l=0)}(T)= & {} \frac{2\pi }{\mu y_{(0)}^2}\frac{1}{-K_0(k)+2\kappa H_0(k)}\,, \end{aligned}$$

(56)

$$\begin{aligned} D^{(l=1)}_{i,x}(T)= & {} P^{(l=1)}_{i,x} \frac{6\pi \mu }{y_{(1)}^2} \frac{1}{-K_1(k)+2\kappa H_1(k)}, \end{aligned}$$

(57)

$$\begin{aligned} D^{(l=2)}_{ij,xy}(T)= & {} \frac{3}{2}P_{ij,xy}^{(l=2)}\frac{10\pi \mu ^3}{y_{(2)}^2} \frac{1}{-K_2(k)+2\kappa H_2(k)}, \end{aligned}$$

(58)

$$\begin{aligned} D^{(l=3)}_{ijk,xyz}(T)= & {} \frac{5}{2}P^{(l=3)}_{ijk,xyz} \frac{14\pi \mu ^5}{y_{(3)}^2} \frac{1}{-K_3(k)+2\kappa H_3(k)}, \end{aligned}$$

(59)

where $P_{i,x}^{(l=1)}$, $P^{(l=2)}_{ij,xy}$, and $P^{(l=3)}_{ijk,xyz}$ are the projection operators which satisfy the relation, $P=PP$, and we have

$$\begin{aligned} P^{(l=1)}_{i,x}= & {} \delta _{ix}, \end{aligned}$$

(60)

$$\begin{aligned} P^{(l=2)}_{ij,xy}= & {} \frac{1}{2}\left( \delta _{ix}\delta _{jy} + \delta _{iy}\delta _{jx} - \frac{2}{3}\delta _{ij}\delta _{xy} \right) , \end{aligned}$$

(61)

$$\begin{aligned} P^{(l=3)}_{ijk,xyz}= & {} \frac{1}{6}\left[ \frac{}{} \delta _{ix}\delta _{jy}\delta _{kz} + \text{5 } \text{ terms } \right. \nonumber \\&\left. -\frac{2}{5}\left( \delta _{ij}\delta _{kx}\delta _{yz} + \text{8 } \text{ terms } \right) \right] \,. \end{aligned}$$

(62)

In addition, the couplings, $y_{(l)}$ are redundant when one fixes them by using the effective range parameters, conventionally one may choose them as

$$\begin{aligned} y_{(0)}= & {} \sqrt{\frac{2\pi }{\mu }}, \ \ \ y_{(1)} = \sqrt{6\pi \mu }\,, \nonumber \\ y_{(2)}= & {} \sqrt{10\pi \mu ^3}, \ \ \ y_{(3)} = \sqrt{14\pi \mu ^5}. \end{aligned}$$

(63)