Trojan horse method as an indirect approach to study resonant reactions in nuclear astrophysics

Abstract

The primary goal of the Trojan horse method (THM) is to analyze resonant rearrangement reactions when the density of the resonance levels is low and statistical models cannot be applied. The main difficulty of the analysis is related with the facts that in the final state the THM reaction involves three particles and that the intermediate particle, which is transferred from the Trojan horse particle to the target nucleus to form a resonance state, is virtual. Another difficulty is associated with the Coulomb interaction between the particles, especially, taking into account that the goal of the THM is to study resonant rearrangement reactions at very low energies important for nuclear astrophysics. The exact theory of such reactions with three charged particles is very complicated and is not available. This is why different approximations are used to analyze THM reactions. In this review paper we describe a new approach based on a few-body formalism that provides a solid basis for deriving the THM reaction amplitude taking into account rescattering of the particles in the initial, intermediate and final states of the THM reaction. Since the THM uses a two-step reaction in which the first step is the transfer reaction populating a resonance state, we address the theory of the transfer reactions. The theory is based on the surface-integral approach and R-matrix formalism. We also discuss application of the THM to resonant reactions populating both resonances located on the second energy sheet and subthreshold resonances, which are subthreshold bound states located at negative energies close to thresholds. We consider the application of the THM to determine the astrophysical factors of resonant radiative-capture reactions at energies so low that direct measurements can hardly be performed due to the negligibly small penetrability factor in the entry channel of the reaction. We elucidated the main ideas of the THM and outline necessary conditions to perform the THM experiments.

Appendices

Appendices

Spectral decomposition of the two-channel Green’s function

To single out the resonance in the subsystem $\,F= x+A\,$ from the Green’s function G we follow Ref. [12] and rewrite it as

$$\begin{aligned} G= {G}_{s}\,\big (1 + {{{\overline{V}}}}_{xA}^{N}\,G \big ), \end{aligned}$$

(122)

where ${{{\overline{V}}}}_{xA}^{N}= V_{sx}^{N} + V_{sA}^{N}$ and

$$\begin{aligned} {G}_{s}(z) = \frac{1}{z - {{{\hat{T}}}}_{sF} - {{{\hat{H}}}}_{F} - {{{\overline{V}}}}_{xA}^{C}}. \end{aligned}$$

(123)

Note that ${{{\overline{V}}}}_{xA}^{C}=V_{sx}^{C}+V_{sA}^{C}$. Substituting Eq. (122) into Eq. (53) one gets

$$\begin{aligned} { M}= {\Big \langle }{{{{\overline{\Phi }}}}_{bB}^{C(-)}}{\bigg |}{\Big \langle }{X_f}{\bigg |}{V_{bB}\, G_{s}\,{ {{{\tilde{U}}}}}_{sA}}{\bigg |}X_{i}{\Big \rangle }{\bigg |}I_{x}^{a}\, {\Psi _{{{\mathbf{k}}}_{aA}}^{C(+)} }{\Big \rangle }, \end{aligned}$$

(124)

where the transition operator ${ {{{\tilde{U}}}}}_{sA}$ is

$$\begin{aligned} { {{{\tilde{U}}}}}_{sA} = {{{\overline{V}}}}_{sx}^{N} + {{{\overline{V}}}}_{xA}^{N}\,G\,{{{\overline{V}}}}_{sx}^{N}. \end{aligned}$$

(125)

Now we use approximation by replacing ${{{\overline{V}}}}_{xA}^{C}$ in Eq. (123) with $U_{sF}^{C}$ to get

$$\begin{aligned} {G}_{s} (z)&\approx \frac{1}{z - {{{\hat{T}}}}_{sF} - {{{\hat{H}}}}_{F} - U_{sF}^{C} }, \end{aligned}$$

(126)

where $U_{sF}^{C}$ is the channel Coulomb potential describing the interaction between the c.m. of nuclei s and F. Then the reaction amplitude M takes the form

$$\begin{aligned} { M}'= {\Big \langle }{{{{\overline{\Phi }}}}_{bB}^{C(-)}} {\bigg |}{\Big \langle }{X_f} {\bigg |}V_{bB}\,G_{s}\, {{{\tilde{U}}}}_{sA}{\bigg |}{X_i} {\Big \rangle }{\bigg |}I_{x}^{a} {\Psi _{{{\mathbf{k}}}_{aA}}^{C(+)}} {\Big \rangle }. \end{aligned}$$

(127)

Singling out the first step of the THM reaction $\,a+ A \rightarrow s + F^{*}\,$ we get

$$\begin{aligned} { M}'&= \bigg \langle {{{{\overline{\Phi }}}}_{bB}^{C(-)}} {\bigg |}\bigg \langle {X_{f}} {\bigg |}{V_{bB}} {\bigg |}{X_{f}} {\Big \rangle }\bigg \langle {X_{f}} {\bigg |}{ G_{s}} {\bigg |}{X_{i}} \bigg \rangle \nonumber \\&\quad \times \bigg \langle {X_{i}} {\bigg |}{{ {{{\tilde{U}}}}}_{sA}} {\bigg |}{X_i} \bigg \rangle {\bigg |}I_{x}^{a}\,{\Psi _{{{\mathbf{k}}}_{aA}}^{C(+)}} \bigg \rangle \nonumber \\&= \bigg \langle {{{{\overline{\Phi }}}}_{bB}^{C(-)}} {\bigg |}{{{{\tilde{V}}}}_{bB} {\bigg |}\bigg \langle {X_{f}} {\bigg |}{ G_{s}} {\bigg |}{X_{i}} \bigg \rangle {\bigg |}{{{\mathcal {U}}}}_{sA}} {\bigg |}I_{x}^{a}\, {\Psi _{{\mathbf{k}}_{aA}}^{C(+)}} \bigg \rangle , \end{aligned}$$

(128)

where the short-hand notations $\,{{{\tilde{V}}}}_{bB}\,= \bigg \langle {X_f}\big |{V_{bB}}{\bigg |}{X_f}\bigg \rangle$ and ${{{\mathcal {U}}}}_{sA}= {\Big \langle }{X_{i}}\big |{{ {{{\tilde{U}}}}}_{sA}}\big |{X_i}{\Big \rangle }$ are introduced. We assume that the potential $V_{bB}$ is spin-independent.

To single out a resonance state in the intermediate subsystem $\,F\,$ one can introduce the spectral decomposition of $\,G_{s} (z)$:

$$\begin{aligned}&\bigg \langle X_{f} \bigg | G_{s} {\bigg |}\,X_{i}\bigg \rangle \nonumber \\&\quad = \sum _{n}\,\int \,\frac{{\mathrm{d}} {\mathbf{k}}_{sF}}{(2\,\pi )^{3}}\,\frac{ {\bigg |}I_{f}^{F_{n}}\,\Psi _{{\mathbf{k}}_{sF} } ^{(-)} \bigg \rangle \bigg \langle \Psi _{{{\mathbf{k}}}_{sF} }^{(-)} I_{i}^{F_{n}} {\bigg |}}{E_{aA} + Q_{n} - k_{sF}^{2}/(2\,\mu _{sF}) + i0 } \nonumber \\&\qquad + \,\int \, \frac{{\mathrm{d}} {\mathbf{k}}_{bB}}{(2\,\pi )^{3}}\,\frac{{\mathrm{d}} {\mathbf{k}}_{sF}}{(2\,\pi )^{3}}\nonumber \\&\qquad \times \frac{ {\bigg |}\Psi _{{\mathbf{k}}_{bB};f}^{(-)} \, \Psi _{{{\mathbf{k}}}_{sF} }^{C(-)} \bigg \rangle \bigg \langle \Psi _{ {{\mathbf{k}}}_{sF} }^{C(-)}\,\Psi _{{\mathbf{k}}_{bB};i}^{(-)}{\bigg |}}{ E_{aA} - \varepsilon _{a} +Q_{if} - k_{bB}^{2}/(2\,\mu _{bB}) - k_{sF}^{2}/(2\,\mu _{sF}) + i0 }. \end{aligned}$$

(129)

Here we use the notion of the overlap function $I_{f}^{F_{n}}(r_{bB}) = \bigg \langle \varphi _{B}(\xi _{B})\,\varphi _{b}(\xi _{b}) {\bigg |}\varphi _{n}(\xi _{B},\xi _{b}; r_{bB}) \bigg \rangle$, which is the projection of the n-th bound-state of the many-body wave function $\varphi _{n}$ of $\,F$ on $X_{f}$. Integration in the matrix element is carried out over all the internal coordinates $\xi _{b}$ and $\xi _{B}$ of nuclei b and B. Hence, $I_{f}^{F_{n}}$ depends only on $r_{bB}$ (for non-zero orbital angular momenta it depends on ${{\mathbf{r}}}_{bB}$). Similar meaning has the second overlap function $I_{i}^{F_{n}}(r_{xA})= {\Big \langle }\varphi _{n}(\xi _{A},\xi _{x}; r_{xA}) {\bigg |}\varphi _{x}(\xi _{x})\,\varphi _{A}(\xi _{A}) {\Big \rangle }$ introduced in Eq. (129).

Usually the overlap functions are determined for bound states. But here we also introduce the overlap functions for the continuum states. In particular, we define

$$\begin{aligned} \Psi _{ {{\mathbf{k}}}_{bB};f}^{(-)}({{\mathbf{r}}}_{bB})&= \left\langle \varphi _{B}(\xi _{B})\,\varphi _{b}(\xi _{b}) {\bigg |}\Psi _{{{\mathbf{k}}}_{bB}}^{(-)}(\xi _{B},\xi _{b}; {{\mathbf{r}}}_{bB}) \right\rangle \end{aligned}$$

(130)

$$\begin{aligned} \Psi _{ {{\mathbf{k}}}_{bB};i}^{(-)*}({{\mathbf{r}}}_{xA})&= \left\langle \Psi _{{{\mathbf{k}}}_{bB}}^{(-)}(\xi _{A},\xi _{x}; {{\mathbf{r}}}_{xA}) {\bigg |}\varphi _{A}(\xi _{A})\,\varphi _{x}(\xi _{x}) \right\rangle \end{aligned}$$

(131)

to be the projections of the wave function $\Psi _{{\mathbf{k}}_{bB}}^{(-)}$ of the system F in the continuum on $X_{f}$ and $X_{i}$, respectively. We assume that the continuum wave function $\Psi _{{{\mathbf{k}}}_{bB}}^{(-)}$ has the incident wave in the channel $f=b+B$ with ${{\mathbf{k}}}_{bB}$ being the $b+B$ relative momentum.

Also in Eq. (129), $\;E_{aA}$ is the $a-A$ relative kinetic energy, $Q_{n}= m_{a} + m_{A} - m_{s} - m_{F_{n}}= \varepsilon _{F_{n}} - \varepsilon _{a}$, $\,\varepsilon _{F_{n}}\, = m_{x} + m_{A} - m_{F_{n}}$ is the binding energy of the bound state $\,F_{n}\,$ for the virtual decay $\,F_{n} \rightarrow x+A$, $\,\Psi _{{\mathbf{k}}_{sF} }^{C(-)}\,$ is the Coulomb scattering wave function of particles s and F with the relative momentum $\,{\mathbf{k}}_{sF}$, $\,\mu _{sF}\,$ is the reduced mass of particles s and F, $\;\varepsilon _{sx} = m_{s} + m_{x} - m_{a}$ is the binding energy of a, $\,m_{i}$ is the mass of particle i, $\,E_{aA}-\varepsilon _{a} + Q_{if}$ is the total kinetic energy of the three-body system $s+b +B$, $\,Q_{if}= m_{x} + m_{A} - m_{b} - m_{B}$, $m_{i}$ is the mass of particles i, $i=x+A$ and $f=b+B$ are the initial and final channels of the binary subreaction $x+ A \rightarrow b+B$.

Here, for simplicity, we consider the wave function $\, \Psi _{{\mathbf{k}}_{bB}}^{(-)}$ only in the external region. The internal region can be taken similarly using the R-matrix approach (see Appendix A in [9]). In the external region the wave function $\, \Psi _{{{\mathbf{k}}}_{bB}}^{(-)}$ with the incident wave in the channel $f=b+B$ becomes an external multichannel scattering wave function [9, 10]:

$$\begin{aligned} \Psi _{{{\mathbf{k}}}_{bB}}^{(-)}({{\mathbf{r}}}_{bB})&= -i\frac{1}{2\,k_{bB}}\sum \limits _c {\sqrt{\frac{{{v_f}}}{{{v_c}}}} } \frac{1}{{{r_c}}}{{ X}_c} \nonumber \\&\quad \times \big [I^*(k_{bB} ,\,r_{bB})\,\delta _{cf} - \,\,S^*_{cf}\,O^*({k_c},{r_c}) \big ] . \end{aligned}$$

(132)

Only the $l=0$ partial wave is taken into account. We recall that f stands for the channel $b+B$. The sum over c is taken over all open final channels $\,c\,$ coupled with the initial channel f. $\,X_{c}$ stands for the product of the bound-state wave functions of the fragments in the channel c, $\,v_{c}\,$ is the relative velocity of the nuclei in the channel c, $S_{c\,f}$ is the scattering S matrix for the transition $\, f \rightarrow c$, $\;O(k_{c},r_{c})$ is the Coulomb Jost singular solution of the Schrödinger equation with the outgoing-wave asymptotic behavior.

In the case under consideration we consider only two coupled channels, $\,i= x+A$ and $\,f=b+B$. In the external region the channels are decoupled and the overlap function $\Psi _{{\mathbf{k}}_{bB}; i}^{(-)}$ is written as

$$\begin{aligned} \Psi _{{{\mathbf{k}}}_{bB};i}^{(-)*}({{\mathbf{r}}}_{xA}) = \,i\frac{1}{{{2\,k_{bB}}{\,r_{xA}}}} {\sqrt{\frac{{{\mu _{xA}\,k_{bB}}}}{{{\mu _{bB}\,k_{xA}}}}} }\,S_{ f\, i}\,O({k_{xA}},{r_{xA}}). \end{aligned}$$

(133)

Equation (133) determines the projection of the external two-channel wave function $\Psi _{F}^{(-)}$ , which has an incident wave in the channel $\,f=b+B$, onto the channel $\,i=x+A$.

The second overlap function takes the form

$$\begin{aligned} \Psi _{{{\mathbf{k}}}_{bB};f}^{(-)}({{\mathbf{r}}}_{bB})&= -i\frac{{1 }}{{{2\,k_{bB}}{r_{bB}}}}\, \nonumber \\&\quad \times \big [ I^{*}({k_{bB}},{r_{bB}}) - \,\,S_{f\,f}^{*}\,O^{*}({k_{bB}},{r_{bB}}) \big ], \end{aligned}$$

(134)

where $I^{*}({k_{bB}},{r_{bB}})= O({k_{bB}},{r_{bB}})$.

We denote the second (continuum) term in Eq. (129) as $\bigg \langle X_{f} \bigg |{ G}_{s}^{{\mathrm{cont}}} {\bigg |}X_{i} \bigg \rangle$:

$$\begin{aligned}&\bigg \langle X_{f} \big |{ G}_{s}^{{\mathrm{cont}}} {\bigg |}X_{i} \bigg \rangle = \int \frac{{\mathrm{d}} {{\mathbf{k}}}_{bB}}{(2 \pi )^{3}}\nonumber \\&\qquad \times \frac{{\mathrm{d}} {\mathbf{k}}_{sF}}{(2 \pi )^{3}} \frac{ {\bigg |}\Psi _{ {\mathbf{k}}_{bB};f}^{(-)} \Psi _{{{\mathbf{k}}}_{sF} }^{C(-)} \bigg \rangle \bigg \langle \Psi _{{{\mathbf{k}}}_{sF} }^{C(-)} \Psi _{{{\mathbf{k}}}_{bB};i}^{(-)} {\bigg |}}{ E_{aA} - \varepsilon _{a} + Q_{if} - k_{bB}^{2}/(2 \mu _{bB}) - k_{sF}^{2}/(2 \mu _{sF}) + i0 }. \end{aligned}$$

(135)

The resonant term corresponding to the subsystem F can be singled out from Eq. (135). To this end

• 1. We first perform the integration over the solid angle $\Omega _{{{\mathbf{{ k}}}}_{bB}}$.

• 2. The S-matrix element $S_{i\,f}$ has a resonance pole on the second Riemann sheet at the $b-B$ relative energy $E_{{\mathrm{R}}(bB)} =E_{0(bB)} - i\Gamma /2$, where $\Gamma$ is the total resonance width. In the momentum plane this resonance pole occurs at the $b-B$ relative momentum $k_{{\mathrm{R}}(bB)}= k_{0(bB)} - i\,k_{I(bB)}$. We assume that the resonance is narrow: $\Gamma \ll E_{0(bB)}$ or $k_{I(0)} \ll k_{0(bB)}$.

• 3. When $k_{bB} \rightarrow k_{{\mathrm{R}}(bB)}$ the integration contour over $k_{bB}$ moves down to the fourth quadrant pinching the contour to the pole at $k_{{\mathrm{R}}(bB)}$. Taking the residue at the pole $\,E_{bB}= E_{{\mathrm{R}}(bB)}\,$ one can single out the contribution to ${\Big \langle }X_{f} \big |{ G}_{s}^{{\mathrm{cont}}} (z) {\bigg |}X_{i} {\Big \rangle }\,$ from the resonance term in the subsystem $\,F$.

Following all these steps we get the desired spectral decomposition of $G_{s}$ for two coupled channels [12]:

$$\begin{aligned} {\Big \langle }X_{f} \big | G_{s}^{{\mathrm{R}}} {\bigg |}X_{i} {\Big \rangle }&=- \frac{i}{4\,\pi }\,\int \,\frac{{\mathrm{d}} {\mathbf{k}}_{sF}}{(2\,\pi )^{3}}\nonumber \\&\quad \times \frac{ {\, {\bigg |}\phi _{\mathrm{R}(bB)}(r_{bB})\, \Psi _{{{\mathbf{k}}}_{sF} }^{C(-)} \bigg \rangle \bigg \langle \Psi _{{{\mathbf{k}}}_{sF} }^{C(-)} \, {{{\tilde{\phi }}}}_{\mathrm{R}(xA)}(r_{xA})} {\bigg |}}{ E_{aA} - \varepsilon _{a} + Q_{if} - E_{\mathrm{R}(bB)} - k_{sF}^{2}/(2\,\mu _{sF}) }, \end{aligned}$$

(136)

where

$$\begin{aligned} {{{\tilde{\phi }}}}_{{\mathrm{R}}(xA)}(r_{xA})&= e^{-i\,\delta ^{p}(k_{0(xA)})}\,\sqrt{ \frac{\mu _{xA}}{k_{{\mathrm{R}}(xA)} }\,\Gamma _{xA} }\,\frac{O^{*}({k_{{\mathrm{R}}(xA)}},{r_{xA}})}{r_{xA}}, \nonumber \\ \phi _{{\mathrm{R}}(bB)}(r_{bB})&=e^{i\,\delta ^{p}(k_{0(bB)})}\, \sqrt{ \frac{\mu _{bB}}{k_{{\mathrm{R}}(bB)} }\,\Gamma _{bB} }\,\frac{O({k_{\mathrm{R}(bB)}},{r_{bB}})}{r_{bB}} \end{aligned}$$

(137)

are the Gamow resonant wave functions in channels i and f. Note that $\, {{{\tilde{\phi }}}}_{{\mathrm{R}}(xA)}(r_{xA})\,$ is the Gamow wave function from the dual basis, $\Gamma _{xA}$ and $\Gamma _{bB}$ are the partial resonance widths in the initial channel i and final channel f, respectively, $\delta ^{p}(k_{0(xA)}$ and $\delta ^{p}(k_{0(bB)})$ are the potential (non-resonant) scattering phase shifts in the initial and final channels.

After deriving the expression for the resonance term in the spectral decomposition of ${\Big \langle }X_{f} \big | G_{s}^{{\mathrm{R}}} (z) \big | X_{i}{\Big \rangle }$ we can substitute it into Eq. (128) and derive an equation for the amplitude of the reaction $a+ A \rightarrow s+b+B$ with three charged particles in the final state, proceeding through an intermediate resonance in the subsystem $F=x+ A=b+B$. As mentioned above, the TH reaction amplitude is described by the two-step process: the first step is the transfer reaction populating the resonance state $a+ A \rightarrow s +F^{*}$ and the second step is the decay of the resonance into two-fragment channel $F^{*} \rightarrow b+B$ leading to the formation of the three-body final state, $s+ b+ B$. We derive below the expression for the TH reaction amplitude taking into account the Coulomb interactions in the intermediate and final states.

Substituting Eq. (136) into Eq. (128) and writing it in the momentum representation one gets

$$\begin{aligned} { M}'&= - \frac{i\,\mu _{sF}}{2\,\pi } \,\int \, \frac{{\mathrm{d}} {\mathbf{p}}_{B}}{(2\,\pi )^{3}}\,\frac{{\mathrm{d}} {\mathbf{p}}_{b}}{(2\,\pi )^{3}} {{{\overline{\Phi }}}}_{ (bB){{\mathbf{k}}}_{B}\, {{\mathbf{k}}}_{b} }^{(+)}\big ({{\mathbf{p}}}_{B},\,{{\mathbf{p}}}_{b}\big ) \nonumber \\&\quad \times W_{bB}\,\big (\mathbf{p}_{bB} \big )\, {{{\mathcal {J}}}}({\mathbf{p}}_{sF},\, {{\mathbf{k}}}_{aA}), \end{aligned}$$

(138)

where

$$\begin{aligned} {{{\mathcal {J}}}}({{\mathbf{p}}}_{sF},\, {{\mathbf{k}}}_{aA}) = \int \, \frac{{\mathrm{d}} {{\mathbf{k}}}_{sF}}{(2\,\pi )^{3}}\, \frac{ \Psi _{ {{\mathbf{k}}}_{sF} }^{C(-)}\big ({{\mathbf{p}}}_{sF}\big )\,{ M}_{(tr)}\big ({{\mathbf{k}}}_{sF} ,\,{{\mathbf{k}}}_{aA} \big ) }{ k_{{\mathrm{R}}}^{2} - k_{sF}^{2} }, \end{aligned}$$

(139)

and

$$\begin{aligned} { M}_{(tr) } \big ({{\mathbf{k}}}_{sF},\,{{\mathbf{k}}}_{aA}\big )= \bigg \langle \Psi _{ {{\mathbf{k}}}_{sF} }^{C(-)}\, {{{\tilde{\phi }}}}_{\mathrm{R}(xA)} {\bigg |}{{{\mathcal {U}}}}_{sA} {\bigg |}I_{x}^{a}\,\Psi _{ {\mathbf{k}}_{aA} }^{C(+)} \bigg \rangle \end{aligned}$$

(140)

is the amplitude of the transfer reaction $a + A \rightarrow s + F^{*}$ populating the resonance $F^{*}$. This amplitude can be approximated by $M_{{M_F}{M_s};{M_A}{M_a}}^{(prior)}({k_{0(sF)}}{{{{{\hat{\mathbf{k}}}}}}_{sF}},{{\mathbf{k}}_{aA}})$ defined in Eq. (42).

The expression for the form factor $W_{bB}\big ({\mathbf{p}}_{bB}\big )$ can be obtained using Eq. (36) from [17]:

$$\begin{aligned} W_{bB}\big (\mathbf{p}_{bB} )&= \int {d{\mathbf{r}_{bB}}} {e^{-i{\mathbf{p}_{bB}} \cdot {\mathbf{r}_{bB}}}}{{{\widetilde{V}}}_{bB}}({ r}_{bB}){ \varphi _{{\mathrm{R}}(bB)}}({r}_{pB}) \nonumber \\&={e^{i{\delta ^p}({k_{0(bB)}})}}\,{e^{\frac{{ - \pi {\eta _{\mathrm{R}(bB)}}}}{2}}}\,{\left( {\frac{{p_{bB}^2 - k_{\mathrm{R}(bB)}^2}}{{4k_{{\mathrm{R}}}^2}}} \right) ^{i{\eta _{R(bB)}}}}\, \nonumber \\&\quad \times \Gamma (1 - i{\eta _{\mathrm{R}(bB)}})\,\sqrt{\frac{\mu _{bB}\,\Gamma _{bB}}{k_{{\mathrm{R}}(bB)} }}\,g(p_{bB}^{2}), \end{aligned}$$

(141)

where $g(p_{bB}^{2})$ is the so-called reduced form factor, which satisfies $g(k_{{\mathrm{R}}(bB)}^{2})=1$. Also

$$\begin{aligned} k_{{\mathrm{R}}}^{2}/(2\,\mu _{sF}) = E_{aA} - \varepsilon _{a} + Q_{if} - E_{{\mathrm{R}}(bB)}. \end{aligned}$$

(142)

Note that the imaginary part ${\mathrm{Im}} (k_{{\mathrm{R}}}) >0$.

The above equations allow us to obtain the expression for the resonant THM reaction (16).

Derivation of the THM radiative capture amplitude

We analyze the THM radiative capture reaction (109) using the PWA. The derivation of the PWA for the transfer reaction is presented in [3] and in Sect. 3.5. For the case under consideration, we use only the surface term, which takes the form:

$$\begin{aligned}&M_{M_{a}\,M_{A}\,\tau }^{M_{s}\,M_{F_{\tau }^{(s)}}}({\mathbf{k}}_{sF_{\tau }},{{\mathbf{k}}}_{aA})\nonumber \\&\quad =\frac{\sqrt{\pi }}{\mu _{xA}}\,i^{l_{xA}}\,\varphi _{a}(p_{sx})\,\sqrt{2\,\mu _{xA} \,R_{xA}}\,\gamma _{\tau \,j_{xA}l_{xA}J_{F^{(s)}}} {{{{\tilde{\mathcal {W}}}}}}_{l_{xA}}\nonumber \\&\qquad \times \sum \limits _{M_{x}\,m_{j_{xA}}\,m_{l_{xA}}}\, \bigg \langle J_{s}\,M_{s}\,\,J_{x}\,M_{x} \bigg |J_{a}\,M_{a} \bigg \rangle \nonumber \\&\qquad \times \bigg \langle J_{x}\,M_{x}\,\,J_{A}\,M_{A} \bigg |j_{xA}\,m_{j_{xA}} \bigg \rangle \nonumber \\&\qquad \bigg \langle j_{xA}\,m_{j_{xA}}\,\,l_{xA}\,m_{l_{xA}} \bigg | J_{F^{(s)}}\,M_{F_{\tau }^{(s)}} \bigg \rangle \,Y_{l_{xA}\,m_{l_{xA}}}^{*}({{{{{{\hat{\mathbf{p}}}}}}}}_{xA}), \end{aligned}$$

(143)

where ${{{{\tilde{\mathcal {W}}}}}}_{l_{xA}}$ is given by

$$\begin{aligned} {{{{\tilde{\mathcal {W}}}}}}_{l_{xA}}&= \Bigg [ j_{{l_{xA}}}({p_{xA}}{r_{xA}})\Bigg [R_{ch}\,\frac{ {\partial {\ln [O_{{l_{xA}}}}({k_{xA}}{r_{xA}})]}}{{\partial {r_{xA}}}} -1\Bigg ] \nonumber \\&\quad - R_{ch}\,\frac{\partial {{\ln \,[j_{{l_{xA}}}}({p_{xA}}{r_{xA}})]}}{{\partial {r_{xA}}}}\Bigg ]{\Big |_{{r_{xA}} = {R_{ch}}}} \nonumber \\&\quad + 2\,\mu _{xA}\,\frac{Z_{x}\,Z_{A}}{\alpha }\,\int \,{\mathrm{d}}r_{xA}\,j_{l_{xA}}(p_{xA}r_{xA}) \nonumber \\&\quad \times \frac{O_{l_{xA}}(k_{xA}r_{xA})}{O_{l_{xA}}(k_{xA}R_{ch})}. \end{aligned}$$

(144)

Note that the argument of $O_{l_{xA}}$ is different from the one in Eq. (41). This is the generalization of Eq. (41) by including the integral giving the external term contribution. Momenta ${\mathbf{p}}_{xA}$ and ${{\mathbf{p}}}_{sx}$ are defined by Eqs. (36).

Note that $M_{M_{a}\,M_{A}\,\tau }^{M_{s}\,M_{F_{\tau }^{(s)}}}({\mathbf{k}}_{sF_{\tau }},{{\mathbf{k}}}_{aA})$ does not contain the hard-sphere scattering phase shift $\delta _{j_{xA}l_{xA}J_{F^{(s)}} }^{hs}$. Also, $\varphi _{a}(p_{sx})$ is the Fourier transform of the radial part of the $\,s$-wave bound-state wave function $\varphi _{a}(p_{sx})$ of the $a=(s\,x)$. Also, $\kappa _{ sx}=\sqrt{ 2\,\mu _{sx}\,\varepsilon _{sx}}\,$ is the wave number of the bound-state $\,a=(s\,x)$, $\,\varepsilon _{sx}\,$ is its binding energy for the virtual decay $a \rightarrow s+ x$. Since particles s and x are structureless, the spectroscopic factor of the bound state $a=(s\,x)$ is unity and we can use just the bound-state wave function $\varphi _{sx}$. Also $\,k_{s}$ and $E_{xA}$ are related by the energy conservation [3]:

$$\begin{aligned} E_{aA} - \varepsilon _{sx} = E_{xA} + k_{s}^{2}/(2\,\mu _{sF}), \end{aligned}$$

(145)

where ${{\mathbf{k}}}_{s}$ is the momentum of the spectator s in the c.m. of the THM reaction, $\mu _{sF}$ is the reduced mass of particles s and F.

Now we consider the amplitude $V_{\nu },\,\,\nu =1,2,$ describing the radiative decay of the intermediate resonance $F_{\nu } \rightarrow F + \gamma$ [106]:

$$\begin{aligned} V_{M_{F_{\nu }}^{(s)}\,\nu }^{M_{F}\,M\,\lambda }&= -\int \,{\mathrm{d}}{\mathbf{r}}_{xA} \nonumber \\&\quad \times {\Big \langle }I_{xA}^{F}({\mathbf{r}}_{xA})\,\big |{{{\hat{{{\mathbf{J}}}}}}}({{\mathbf{r}}}) \big |\Upsilon _{\nu }({{\mathbf{r}}}_{xA}) {\Big \rangle }\cdot {\mathbf{A}}^{*}_{\lambda \,{{\mathbf{k}}}_{\gamma }}({{\mathbf{r}}}), \end{aligned}$$

(146)

where $I_{xA}^{F}({{\mathbf{r}}}_{xA})$ is the overlap function of the bound-state wave functions of $\,x,$ $\,A$ and the ground state of $\,F=(x\,A)$. Again, for the point-like nuclei x and A the overlap function $I_{xA}^{F}({{\mathbf{r}}}_{xA})$ can be replaced by the single-particle bound-state wave function of (xA) in the ground state. Also ${{\mathbf{A}}}^{*}_{\lambda \,{\mathbf{k}}_{\gamma }}({{\mathbf{r}}})$ is the electromagnetic vector potential of the photon with helicity $\lambda =\pm 1$ and momentum ${{\mathbf{k}}}_{\gamma }$ at coordinate ${{\mathbf{r}}}_{xA}$, $\,{{{{{\hat{\mathbf{J}}}}}}}({{\mathbf{r}}})$ is the charge current density operator. Matrix element in Eq. (146) is written assuming that on the first stage of the reaction the excited state $F_{\nu },\,\nu =1,2,$ is populated, which subsequently decays to the ground state F.

Using the multipole expansion of the vector potential, leaving only the electric components with the lowest allowed multipolarities L and using the long wavelength approximation for ${{\hat{{{\mathbf{J}}}}}}({{\mathbf{r}}})$ we get (see Ref. [106] for details)

$$\begin{aligned} V_{M_{F_{\nu }^{(s)}}\,\nu }^{M_{F}\,M\,\lambda }&= \frac{\sqrt{2}}{4\,\pi }\,\sum \limits _{L}i^{-L}\,(-1)^{L+1}\sqrt{{{\hat{L}}}} \,k_{\gamma }^{L-1/2} \bigg [\gamma _{(\gamma )\,\nu \,J_{F}\,L}^{J_{F^{(s)}}^{L}} \bigg ]\, \nonumber \\&\quad \times \left[ D^{L}_{M\,\lambda }(\phi ,\,\theta ,0)\right] ^{*}\, {\Big \langle }J_{F}\,M_{F}\,\,L\,M \Big |J_{F^{(s)}}^{L}\,M_{F_{\nu }^{(s)}}^{L} {\Big \rangle }, \end{aligned}$$

(147)

where $\,\gamma _{(\gamma )\,\nu \,J_{F}\,L}^{J_{F^{(s)}}^{L}}$ is the formal R-matrix radiative width amplitude for the electric (EL) transition $\,J_{F^{(s)}}^{L} \rightarrow J_{F}$ given by the sum of the internal and external radiative width amplitudes, see Eqs (32) and (33) from [42], in which we singled out $\sqrt{2}\,k_{\gamma }^{L+1/2}$. Since now we take into account a few multipolarities L, we replace the previously introduced spin of the intermediate resonance $J_{F^{(s)}}$ by $J_{F^{(s)}}^{L}$, where the superscript L denotes the multipolarity of the EL transition to the ground state F. Replacement of $J_{F^{(s)}}$ by $J_{F^{(s)}}^{L}$ takes into account that the spins of the intermediate excited states are different for different multipolarities. Since we added the superscript L to the spin of the intermediate resonance we added the same superscript to its projection $M_{F_{\nu }^{(s)}}^{L}$. Also in Eq. (147) $\,M$ is the projection of the angular momentum L of the emitted photon (multipolarity of the electromagnetic transition). We remind that $V_{M_{F_{\nu }^{(s)}}\,\nu }^{M_{F}\,M\,\lambda }$ does not depend on the hard-sphere scattering phase shift.

The determined radiative width amplitude is related to the formal resonance radiative width by the standard equation

$$\begin{aligned} \Gamma _{(\gamma )\,\nu \,J_{F}\,L}^{J_{F^{(s)}}^{L}}= 2\,k_{\gamma }^{L+1/2}\,\left( \gamma _{(\gamma )\,\nu \,J_{F}\,L}^{J_{F^{(s)}}^{L}}\right) ^{2}. \end{aligned}$$

(148)

Note that the observable radiative width is related to the formal one by [see also Eq. (93)]

$$\begin{aligned} \left( {{{\tilde{\gamma }}}}_{(\gamma )\,\nu \,J_{F}\,L}^{J_{F^{(s)}}^{L}}\right) ^{2} = \frac{\left( \gamma _{(\gamma )\,\nu \,J_{F}\,L}^{J_{F^{(s)}}^{L}}\right) ^{2}}{ 1+ \gamma _{\nu \,j_{xA}l_{xA}J_{F^{(s)}}}^{2} \left[ \frac{\mathrm{d}S_{l_{xA}}(E_{xA})}{{\mathrm{d}}E_{xA}}\right] _{E_{xA}= E_{\nu }}}. \end{aligned}$$

(149)

We consider the two-level approach with $\nu =1$ ($\nu =2$) corresponding to the subthreshold resonance. Then $E_{\nu }=-\varepsilon _{xA}^{(s)}$ for $\nu =1$ and $E_{\nu }= E_{R}$ for $\nu =2$ with $E_{R}$ being the resonance energy corresponding to the level $\nu =2$. This observable radiative width is related to the observable resonance radiative width as

$$\begin{aligned} {{{\tilde{\Gamma }}}}_{(\gamma )\,\nu \,J_{F}\,L}^{J_{F^{(s)}}^{L}}= 2\,k_{\gamma }^{L+1/2}\,\left( {{{\tilde{\gamma }}}}_{(\gamma )\,\nu \,J_{F}\,L}^{J_{F^{(s)}}^{L}}\right) ^{2}. \end{aligned}$$

(150)

Substituting Eqs. (143) and (147) into Eq. (110) we get the expression for the indirect reaction amplitude

$$\begin{aligned} M_{M_{a}\,M_{A}}^{M_{s}\,\,M_{F}\,M\,\lambda }&=\frac{\varphi _{a}(p_{sx})}{2}\,\sqrt{\frac{R_{xA}}{\pi \,\mu _{xA}}} \,\sum \limits _{L}\,(-1)^{L+1}\,{{{\hat{L}}}}^{1/2}\,k_{\gamma }^{L-1/2}\nonumber \\&\quad \times \big [D^{L}_{M\,\lambda }(\phi ,\,\theta ,0)\big ]^{*}\,\sum \limits _{l_{xA}}\,i^{l_{xA} -L}\,{{{{\tilde{\mathcal {W}}}}}}_{l_{xA}} \nonumber \\&\quad \times \sum \limits _{\nu ,\,\tau =1}^{2}\,\gamma _{(\gamma )\,\nu \,J_{F}\,L}^{J_{F^{(s)}}^{L}}\,{{\mathbf{A}}}_{\nu \,\tau }^{L} \,\gamma _{\tau \,j_{xA}l_{xA}J_{F^{(s)}}^{L}}\nonumber \\&\quad \times \sum \limits _{ M_{F_{\nu }^{(s)}}^{L} }\, {\Big \langle }J_{F}\,M_{F}\,\,L\,M \big |J_{F^{(s)}}^{L}\,M_{F_{\nu }^{(s)}}^{L} {\Big \rangle }\nonumber \\&\quad \times \sum \limits _{m_{j_{xA}}\,m_{l_{xA}}\,M_{x} } {\Big \langle }j_{xA}\,m_{j_{xA}}\,\,l_{xA}\,m_{l_{xA}}\big | J_{F^{(s)}}^{L}\,M_{F_{\tau }^{(s)}}^{L} {\Big \rangle }\nonumber \\&\quad {\Big \langle }J_{x}\,M_{x}\,\,J_{s}\,M_{s} \big |J_{a}\,M_{a} {\Big \rangle }\nonumber \\&\quad \times {\Big \langle }J_{x}\,M_{x}\,\,J_{A}\,M_{A} \big | j_{xA}\,m_{j_{xA}} {\Big \rangle }\,Y_{l_{xA}\,m_{l_{xA}}}^{*} ({{{{{{\hat{\mathbf{p}}}}}}}}_{xA}). \end{aligned}$$

(151)

The amplitude $M_{M_{a}\,M_{A}}^{M_{s}\,\,M_{F}\,M\,\lambda }$ describes the indirect reaction proceeding through the intermediate resonances, which decay to the ground state $F=(x\,A)$ by emitting photons. Equation (151) is generalization of Eq. (110) by including the sum over multipolarities L corresponding to the radiative electric transitions from the intermediate resonances with the spins $J_{F^{(s)}}^{L}$ to the ground state F with the spin $J_{F}$. Note also that we assume that two levels contribute to each transition of multipole L. It requires the two-level generalized R-matrix approach. The generalization of Eq. (151) for three- or more-level cases is straightforward. In Eq. (151) the reaction part and radiative parts are interconnected by the R-matrix level matrix elements ${{\mathbf{A}}}_{\nu \,\tau }^{L}$.

The sum over $\nu$ and $\tau$ in Eq. (151) is the standard R-matrix term for the binary resonant radiative-capture reaction. However, we analyze the three-body reaction $a(x\,s) + A \rightarrow s +F +\gamma$ with the spectator s in the final state rather than the standard two-body radiative-capture reaction $x +A \rightarrow F+ \gamma$. This difference leads to the generalization of the standard R-matrix approach for the three-body reactions resulting in the appearance of the additional factor $\varphi _{a}(p_{sx})\,{{{\mathcal {\widetilde{W}}}}}_{l_{xA}}$ which should be familiar to the reader from Eq. (74). That is why we call the developed approach the generalized R-matrix method for the indirect resonant radiative-capture reactions.

We take the indirect reaction amplitude at fixed projections of the spins of the initial and final particles including the fixed projection M of the orbital momentum L of the emitted photon and fixed its chirality $\lambda$. For example, for the ${}^{12}\mathrm{C}(\alpha ,\,\gamma ){}^{16}{\mathrm{O}}$ reaction the electric dipole $\,E1$ ($L=1$) and quadrupole $\,E2$ ($L=2$) transitions do contribute and they interfere. In the long-wavelength approximation only minimal allowed $l_{xA}$ for given L does contribute. For example, for the case considered below $\,l_{f}=0$ $\,\,l_{xA}=L=1$ for the dipole and $\,l_{xA}=L=2$ for the quadrupole electric transitions do contribute. The dimension of the R-matrix level matrix ${{\mathbf{A}}}^{L}$ depends on the number of the levels taken into account for each L.

The indirect reaction amplitude depends on the off-shell momenta ${{\mathbf{p}}}_{sx}$ and ${{\mathbf{p}}}_{xA}$. Both off-shell momenta are expressed in terms of ${{\mathbf{k}}}_{a}$ and ${\mathbf{k}}_{s}$, see Sect. 2. Also the the indirect reaction amplitude depends on the momentum of the emitted photon ${\mathbf{k}}_{\gamma }$ whose direction is determined by the angles in the Wigner D-function.

In the center-of-mass of the reaction (16) neglecting the recoil effect of the nucleus F during the photon emission from the energy conservation we get

$$\begin{aligned}&E_{aA} + Q = E_{sF} + k_{\gamma }, \end{aligned}$$

(152)

$$\begin{aligned}&k_{\gamma } = E_{xA} + \varepsilon _{xA}, \end{aligned}$$

(153)

where $\,E_{sF}= k_{s}^{2}/(2\,\mu _{sF}),$ $\,Q=\varepsilon _{xA} - \varepsilon _{sx}$ and $\varepsilon _{xA}$ is the binding energy of the ground state of the nucleus F.

To estimate the recoil effect we take into account that in the center-of-mass of the reaction (16) the momentum conservation in the final state gives

$$\begin{aligned} {{\mathbf{k}}}^{'}_{F} = -{{\mathbf{k}}}_{\gamma } - {{\mathbf{k}}}_{s}, \end{aligned}$$

(154)

where ${{\mathbf{k}}}_{F}^{'}$ is the momentum of the final nucleus F after emitting the photon. Then the energy conservation leads to

$$\begin{aligned} E_{aA}- \varepsilon _{sx}&= \frac{k_{s}^{2}}{2\,\mu _{sF}} + E_{xA}= \frac{k_{s}^{2}}{2\,m_{s}} + \frac{ (k_{F}')^{2} }{2\,m_{F} } + k_{\gamma } \end{aligned}$$

(155)

$$\begin{aligned}&= \frac{k_{s}^{2}}{2\,\mu _{sF}} + 2\frac{k_{s}\,k_{\gamma }}{2\,m_{F}}\cos \theta ' + \frac{k_{\gamma }^{2}}{2\,m_{F}} + k_{\gamma }. \end{aligned}$$

(156)

We remind that we use the system of units in which $\hbar =c=1$, hence, $E_{\gamma }=k_{\gamma }$. Clearly, the term $\frac{k_{\gamma }^{2}}{2\,m_{F}} = E_{\gamma }\,\frac{E_{\gamma }}{2\,m_{F}}$ can be neglected because $E_{\gamma } \ll m_{F}$. The contribution of the term $2\frac{k_{s}\,k_{\gamma }}{2\,m_{F}}\cos \theta '$ depends on $\cos \theta ' = {{{{{{\hat{\mathbf{k}}}}}}}}_{s} \cdot {{\hat{{{\mathbf{k}}}}}}_{\gamma }$.

Neglecting the recoil effect of the nucleus F in Eq. (155) we can replace $k_{F}^{'}$ by $k_{s}$. Then $k_{\gamma }$ and $k_{s}$ are related by Eq. (152) while $k_{\gamma }$ and $E_{xA}$ are related by Eq. (153). If we would take into account the recoil effect then the relationship between $k_{\gamma }$ and $E_{xA}$ is more complicated than Eq. (153) and is given by

$$\begin{aligned} k_{\gamma } = \frac{E_{xA}}{\frac{k_{s}\,\cos \theta '}{m_{F}} +1}, \end{aligned}$$

(157)

where we neglected the extremely small term ${k_{\gamma }^{2}}/{(2\,m_{F})}$.

The expression for $p_{xA}$ is needed to calculate ${{{\tilde{\mathcal W}}}}_{l_{xA}}$. From the energy-momentum conservation law in the three-ray vertices $a \rightarrow s+x$ and $x+ A \rightarrow F^{(s)}$ of the diagram in Fig. 14 we get [3]

$$\begin{aligned} E_{xA}= \frac{p_{xA}^{2}}{2\,\mu _{xA}} - \frac{p_{sx}^{2}}{2\,\mu _{sx}} -\varepsilon _{s\,x}. \end{aligned}$$

(158)

In the QF kinematics $p_{sx}=0$ and

$$\begin{aligned} E_{xA}= \frac{p_{xA}^{2}}{2\,\mu _{xA}} - \varepsilon _{s\,x}. \end{aligned}$$

(159)

Thus always ${p_{xA}^{2}}/{(2\,\mu _{xA})} > E_{xA}$.