Coupled-channel treatment of ${\varvec{^7}}$Li${\varvec{(n,\gamma )^8}}$Li in effective field theory

Abstract

The E1 contribution to the capture reaction $^7\mathrm {Li}(n,\gamma )^8\mathrm {Li}$ is calculated at low energies. We employ a coupled-channel formalism to account for the $^7\mathrm {Li}^\star$ excited core contribution. We develop a halo effective field theory power counting where capture in the spin $S=2$ channel is enhanced over the $S=1$ channel. A next-to-leading order calculation is presented where the excited core contribution is shown to affect only the overall normalization of the cross section. The momentum dependence of the capture cross section, as a consequence, is the same in a theory with or without the excited $^7\mathrm {Li}^\star$ degree of freedom at this order of the calculation. The kinematical signature of the $^7\mathrm {Li}^\star$ core is negligible at momenta below 1 MeV and significant only beyond the $3^+$ resonance energy, though still compatible with a next-to-next-to-leading order correction. We compare our formalism with a previous halo effective field theory calculation [Zhang et al., Phys. Rev. C 89, 024613 (2014)] that also treated the $^7\mathrm {Li}^\star$ core as an explicit degree of freedom. Our formal expressions and analysis disagree with this earlier work in several aspects.

Appendix A: Projectors

The following are from Ref. [31] that we include for reference. For each partial wave we construct the corresponding projection operators from the relative core-nucleon velocity, the spin-1/2 Pauli matrices $\sigma _{i}$’s, and the following spin-1/2 to spin-3/2 transition matrices

$$\begin{aligned} S_1&=\frac{1}{\sqrt{6}}\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} -\sqrt{3} &{} 0 &{} 1 &{} 0\\ 0&{}-1&{}0&{}\sqrt{3} \end{array}\right) \,, \nonumber \\ S_2&= -\frac{i}{\sqrt{6}} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \sqrt{3} &{} 0 &{} 1 &{} 0\\ 0&{}1&{}0&{}\sqrt{3} \end{array}\right) \,, \nonumber \\ S_3&= \frac{2}{\sqrt{6}} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 1 &{} 0 &{} 0\\ 0&{}0&{}1&{}0 \end{array}\right) \,, \end{aligned}$$

(A1)

which satisfy

$$\begin{aligned} S_{i}S^{\dagger }_{j}&=\frac{2}{3}\delta _{ij}-\frac{i}{3}\epsilon _{ijk} \sigma _{k}\,, \nonumber \\ S_{i}^{\dagger }S_{j}&=\frac{3}{4}\delta _{ij}-\frac{1}{6}\big \{J_{i}^{(3/2)}, J_{j}^{(3/2)}\big \}+\frac{i}{3}\epsilon _{ijk}J_{k}^{(3/2)}\,, \end{aligned}$$

(A2)

where $J_{i}^{(3/2)}$’s are the generators of the spin-3/2. We construct the Clebsch-Gordan coefficient matrices

$$\begin{aligned} F_i&=-\frac{i\sqrt{3}}{2}\sigma _2 S_i\, ,&Q_{i j}&= -\frac{i}{\sqrt{8}}\sigma _2\big (\sigma _i S_i+\sigma _j S_i\big ), \end{aligned}$$

(A3)

for projections onto spin channels $S=1$ and $S=2$, respectively. Then in coordinate space the relevant projectors that appear in the Lagrangians involving the $^7$Li ground state in Eqs. (1), (2) are [30, 31]

$$\begin{aligned} P_i^{(^3S_1)}&= F_j\, , \nonumber \\ P_{ij}^{(^5S_2)}&= Q_{ij}\,, \nonumber \\ P^{(^3P_1)}_i&=\sqrt{\frac{3}{2} }F_x \left( \frac{\mathop {\nabla }\limits ^{{\rightarrow }}}{m_c}-\frac{\mathop {\nabla }\limits ^{{\leftarrow }}}{m_n}\right) _y \epsilon _{i x y}\,, \nonumber \\ P^{(^3P_2)}_{i j}&=\sqrt{3} F_x \left( \frac{\mathop {\nabla }\limits ^{{\rightarrow }}}{m_c}-\frac{\mathop {\nabla }\limits ^{{\leftarrow }}}{m_n}\right) _y\ R_{x y i j }\,, \nonumber \\ P^{(^5P_1)}_i&=\sqrt{\frac{9}{5}} Q_{i x}\left( \frac{\mathop {\nabla }\limits ^{{\rightarrow }}}{m_c}-\frac{\mathop {\nabla }\limits ^{{\leftarrow }}}{m_n}\right) _ x\,, \nonumber \\ P^{(^5P_2)}_{i j}&=\frac{1}{\sqrt{2}} Q_{x y} \left( \frac{\mathop {\nabla }\limits ^{{\rightarrow }}}{m_c}-\frac{\mathop {\nabla }\limits ^{{\leftarrow }}}{m_n}\right) _z\ T_{x y z i j}\, . \end{aligned}$$

(A4)

The tensors

$$\begin{aligned} R_{ijxy}&=\frac{1}{2}\left( \delta _{i x}\delta _{j y}+\delta _{i y}\delta _{j x} -\frac{2}{3}\delta _{i j}\delta _{x y}\right) , \nonumber \\ T_{xyz i j}&=\frac{1}{2}\Big (\epsilon _{x z i}\delta _{y j} +\epsilon _{x z j}\delta _{y i} +\epsilon _{y z i}\delta _{x j}+\epsilon _{y z j}\delta _{x i} \Big ), \end{aligned}$$

(A5)

ensures total angular momentum $J=2$ is picked.

The new projectors to describe the interactions in Eq. (2) with the excited $^7\mathrm {Li}^\star$ core are

$$\begin{aligned} P_i^{(^3S_1^\star )}&= -\frac{i}{\sqrt{2}}\sigma _2\sigma _i \, ,&\nonumber \\ P_i^{(^3P_1^\star )}&= -i\frac{\sqrt{3}}{2}\sigma _2 \sigma _x \left( \frac{\mathop {\nabla }\limits ^{{\rightarrow }}}{m_c}-\frac{\mathop {\nabla }\limits ^{{\leftarrow }}}{m_n}\right) _y \epsilon _{i x y}\, , \nonumber \\ P_{ij}^{(^3P_2^\star )}&= -i\sqrt{\frac{3}{2}}\sigma _2\sigma _x \left( \frac{\mathop {\nabla }\limits ^{{\rightarrow }}}{m_c}-\frac{\mathop {\nabla }\limits ^{{\leftarrow }}}{m_n}\right) _y R_{xyij}\, . \end{aligned}$$

(A6)

For the external states we introduce the photon vector ($\varepsilon ^{(\gamma )}_{i}$), excited state $^8$Li $1^+$ spin-1 ($\varepsilon _{j}$), and ground state $^8$Li $2^+$ spin-2 ($\varepsilon _{ij}$) polarizations, obeying the following polarization sums [58, 59],

$$\begin{aligned} \sum _\mathrm{pol.}\varepsilon ^{(\gamma )}_{i}\varepsilon ^{(\gamma )*}_{j}&= \delta _{ij}-\frac{k_ik_j}{k^2}\,, \nonumber \\ \sum _\mathrm{pol.\ ave.}\varepsilon _{i}\varepsilon ^{*}_{j}&=\frac{\delta _{ij}}{3}\,, \nonumber \\ \sum _\mathrm{pol.\ ave.}\varepsilon _{ij}\varepsilon ^{*}_{lm}&=\frac{R_{ijlm}}{5}\,. \end{aligned}$$

(A7)