Asymptotic normalization coefficients in nuclear reactions and nuclear astrophysics

Abstract

The asymptotic normalization coefficient (ANC) is a fundamental nuclear characteristic of bound states and resonances. The ANC plays an important role in low-energy elastic scattering, transfer and radiative capture reactions, and provides a powerful indirect method in nuclear astrophysics. In this comprehensive review the main properties of the ANC and the role of the ANC in nuclear reactions and nuclear astrophysics are discussed. Also different experimental and theoretical methods of determining the ANCs are addressed.

Appendices

Appendices

Appendix A: Coulomb elastic scattering, Coulomb scattering wave functions and Coulomb S-matrix

Let us consider the scattering of two point-like charged particles interacting via the pure Coulomb potential

$$\begin{aligned} V^{C}(r)= \frac{Z_{a}\,Z_{A}\,e^{2}}{r}, \end{aligned}$$

(A.1)

$Z_{i}\,e$ is the charge of particle i.

The Coulomb potential does not satisfy the condition

$$\begin{aligned} \lim \limits _{r \rightarrow \infty } r\,V^{C}(r)=0, \end{aligned}$$

(A.2)

what generates the Coulomb singularities in the k plane. The Coulomb potential between the point-like nuclei also does not satisfy the condition

$$\begin{aligned} \int \limits _{0}^{R} dr\,V^{C}(r) < \infty . \end{aligned}$$

(A.3)

It causes quite complicated behavior of the scattering wave function and the Coulomb elastic scattering S-matrix at $k=0$, which will be excluded from our consideration.

The Coulomb solutions of the Schrödinger equation at the origin ( $r=0$) can be singular or regular. We start our discussion from the singular solutions. The Coulomb singular solutions $f_{l}^{C(\pm )}(k,r)$ are expressed in terms of the Whittaker functions $W_{\mp \,i\,\eta , l+1/2}(\mp \,2\,i\,k\,r)$:

$$\begin{aligned} f_{l}^{C(\pm )}(k,r)=e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,W_{\mp \,i\,\eta , l+1/2}(\mp \,i\,2\,k\,r), \end{aligned}$$

(A.4)

where

$$\begin{aligned} \mathrm{sign}(x)= \Biggl \{ \begin{array}{ll} 1 &{} \quad \text{ for }\ x>0, \\ 0 &{} \quad \text{ for }\ x=0, \\ -1 &{} \quad \text{ for }\ x<0. \end{array} \end{aligned}$$

(A.5)

Here we also introduced the Coulomb (Sommerfeld) parameter

$$\begin{aligned} \eta = \frac{Z_{a}\,Z_{A}\,\alpha \,\mu }{k}, \end{aligned}$$

(A.6)

through which the Coulomb interaction manifests itself. It is worth noting that Eq. (A.6) is written in the systems of units in which $\hbar =c=1$. In this system of units the fine structure constant $\alpha =1/137$. Note that $\lim _{k \rightarrow \infty }\,\eta =0\,$ and $\lim _{k \rightarrow 0}\,\eta =\infty .$ Thus, the Coulomb interaction dominates at small k. Typically, one can neglect the Coulomb effects in nuclear processes at $\eta \ll 1$.

The Whittaker functions are defined by the integral representations:

$$\begin{aligned} {W_{ \mp \,i\,\eta ,l + 1/2}}( \pm z) = \frac{{{e^{ \mp \frac{z}{2}}}{{( \pm z)}^{ - l}}}}{{\Gamma (l + 1 \pm i\eta )}}\int \limits _0^\infty {dt\,{e^{ - t}}{{\Bigg (1 \pm \frac{z}{t}\Bigg )}^{l \mp i\eta }}\,{t^{2l}}}, \end{aligned}$$

(A.7)

$\Gamma (l+1\,\pm \,i\,\eta )$ are the Gamma-functions and $z=-2\,i\,k\,r$.

To determine both Whittaker functions $W_{\mp \,i\eta ,\,l+1/2}(\mp \,2\,i\,k\,r)$ we use the definitions:

$$\begin{aligned} -z= \Biggl \{ \begin{array}{ll} e^{-i\,\pi }\,z &{}\quad \mathrm{arg}z \in [0,\,\pi ], \\ e^{i\,\pi }\,z &{}\quad \mathrm{arg}z \in [-\pi ,\,0]. \end{array} \end{aligned}$$

(A.8)

From Eq. (A.8) follows that $-\,\pi \le \mathrm{arg} (\pm \,z) \le \pi$ what is required by the above definition of the Whittaker functions.

The Whittaker function $\,W_{-i\,\eta ,\,l+1/2}(-\,2\,i\,k\,r)\,$ is an analytic function everywhere in the open k plane except for the branching point singularity at $k=0$ and a discontinuity along the negative imaginary axis. Correspondingly, $\,W_{i\,\eta ,\,l+1/2}(\,2\,i\,k\,r)\,$ has the branching point singularity at $k=0$ and a discontinuity along the positive imaginary axis in the k plane. Thus, the linear combination of two Whittaker functions $\frac{{(2l)!}}{{\Gamma (l + 1 + \,i\,\eta )}}{W_{i\,\eta ,\,l + 1/2}}(\,2\,i\,k\,r)\, + \,{e^{ - i\pi (l + 1)}}\frac{{(2l)!}}{{\Gamma (l + 1 - \,i\,\eta )}}{W_{ - i\,\eta ,\,l + 1/2}}( - \,2\,i\,k\,r)$ has a discontinuity along the entire imaginary axis in the k plane, which splits this linear combination into two different analytical functions; one is in the left half-plane ($\mathrm{Re} k <0$) and the other is in the right half-plane ($\mathrm{Re}k >0$). The factor $e^{\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}$ removes this discontinuity making the linear combination of the Whittaker functions analytic in the entire open k plane in which the point $k=0$ is excluded, see Eq. (A.43) below.

The asymptotic behavior of the Whittaker functions is

$$\begin{aligned}&W_{\mp \,i\,\eta ,\,l+1/2}(\mp \,2\,i\,k\,r) {\mathop {\approx }\limits ^{r \rightarrow \infty }} e^{\pm i\,k\,r}\,(\mp 2\,i\,k\,r)^{\mp i\,\eta }. \end{aligned}$$

(A.9)

The Jost functions are defined as

$$\begin{aligned}&{{\mathcal {F}}}_{l}^{C(\pm )}(k) = \lim \limits _{r \rightarrow 0}\,(\mp \,2\,i\,k\,r)^{l}\, f_{l}^{C(\pm )}(k,r) \nonumber \\&\quad = e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,\lim \limits _{r \rightarrow 0}\,(\mp \,2\,i\,k\,r)^{l}\,W_{ \mp \,i\,\eta ,l + 1/2}(\mp \,2\,i\,k\,r) \nonumber \\&\quad = e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,\frac{\Gamma (2l+2)}{\Gamma (l+1 \pm \,i\,\eta )}. \end{aligned}$$

(A.10)

It is useful to introduce the normalized to unity at $|k| \rightarrow \infty$ Jost functions:

$$\begin{aligned}&{f}_{l}^{C(\pm )}(k) = e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,\frac{\Gamma (l+1)}{\Gamma (l+1\,\pm \,i\,\eta )}, \end{aligned}$$

(A.11)

$$\begin{aligned}&({f}_{l}^{C(\pm )}(k))^{2} = e^{\mp \,2\,i\,\sigma _{l}^{C}}\,\big |f_{l}^{C(\pm )}(k)\big |^{2}, \end{aligned}$$

(A.12)

and

$$\begin{aligned}&\big |{f}_{l}^{C(+)}(k)\big |^{-2}= \nu _{l}(\eta ^{2})\,\frac{2\,\pi \,\eta }{e^{2\,\pi \,\eta } - 1}, \end{aligned}$$

(A.13)

$$\begin{aligned}&\nu _{l}(\eta ^{2})= \displaystyle \prod _{n=1}^{l}\,\Big [1 + \frac{\eta ^{2}}{n^{2}}\Big ], \quad l>0, \end{aligned}$$

(A.14)

$\nu _{0}(\eta ^{2}) =1$. To derive Eq. (A.13) we have used the relationship, see Eqs. 8.331 and 8.332 [201],

$$\begin{aligned} \Big (\frac{(\Gamma (l+1 \pm i\,\eta )}{\Gamma (l+1)}\Big )^{2}\,e^{-\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}=e^{\pm \,2\,i\,\sigma _{l}^{C}}\,\nu _{l}(\eta ^{2})\,\frac{2\,\pi \,\eta }{e^{2\,\pi \,\eta } - 1 }. \end{aligned}$$

(A.15)

The Coulomb scattering S-matrix is

$$\begin{aligned} {{\mathbb {S}}}_{l}^{C}(k) = \frac{{{\mathcal {F}}}_{l}^{C(-)}(k)}{{{\mathcal {F}}}_{l}^{C(+)}(k)} = e^{2\,i\,\sigma _{l}^{C}} = \,\frac{\Gamma (l+1+i\,\eta )}{\Gamma (l+1-i\,\eta )}, \end{aligned}$$

(A.16)

$\sigma _{l}^{C}$ is the Coulomb scattering phase shift in the partial wave l. The gamma-functions $\Gamma (l+1\pm \,i\,\eta )$ are analytic functions in the entire k plane except for the simple poles at $l+1 \pm i\,\eta = -n$, where $n= 0,1,2\ldots$ Hence $\,\Gamma (l+1 \pm i\,\eta )\,$ has simple poles located in the $\,k\,$ plane at points $\,k = \mp \, i({\,Z_{a}\,Z_{A}\,\mu /137})/(l+1+n).\,$ For the repulsive case, $Z_{a}\,Z_{A} >0,\,$ $\,\Gamma (l+1 + \,i\,\eta )$ has poles located along the negative imaginary axis in the k plane. For nuclear processes these poles are unphysical. The poles of $\Gamma (l+1-i\,\eta )$, which are zeroes of the ${{\mathbb {S}}}_{l}^{C}(k)$, are located along the positive imaginary axis in the k plane.

Often in the literature another Jost solutions are used, which are called ingoing and outgoing solutions:

$$\begin{aligned}&I_{l}(k,\,r)= e^{i\,\pi \,l/2}\,f_{l}^{C(-)}(k,r)\nonumber \\&\quad =e^{i\,\pi \,l/2}\,e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,W_{i\,\eta , l+1/2}(2\,i\,k\,r) \end{aligned}$$

(A.17)

$$\begin{aligned}&\quad = i(-1)^{l}(2\,k\,r)^{l+1}\,e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,\nonumber \\&\qquad \times e^{-i\,\,k\,r}\,U(l+1-i\,\eta , 2\,l+2, 2\,i\,k\,r), \end{aligned}$$

(A.18)

$$\begin{aligned}&O_{l}(k,\,r)= e^{-i\,\pi \,l/2}\,f_{l}^{C(+)}(k,r)\nonumber \\&\quad = e^{-i\,\pi \,l/2}\, e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,W_{-\,i\,\eta , l+1/2}(-2\,i\,k\,r) \end{aligned}$$

(A.19)

$$\begin{aligned}&\quad = i\,(-1)^{l+1}(2\,k\,r)^{l+1}\,e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,\nonumber \\&\qquad \times e^{i\,\,k\,r}\,U(l+1+i\,\eta , 2\,l+2, -2\,i\,k\,r). \end{aligned}$$

(A.20)

Here we took into account that at real k $\,I_{l}(k,\,r)= O^{*}_{l}(k,\,r)$. $\,U(l+1+i\,\eta , 2\,l+2, -2\,i\,k\,r)$ is the confluent hypergeometric function of the second kind or Tricomi function.

The asymptotic behavior of $I_{l}(k,\,r)$ and $O_{l}(k,\,r)$ is:

$$\begin{aligned}&I_{l}(k,\,r) {\mathop {\approx }\limits ^{r \rightarrow \infty }} e^{-i[k\,r- \eta \,\ln (2\,k\,r) - \pi \,l/2] }, \end{aligned}$$

(A.21)

$$\begin{aligned}&O_{l}(k,\,r) {\mathop {\approx }\limits ^{r \rightarrow \infty }} e^{i[k\,r- \eta \,\ln (2\,k\,r) - \pi \,l/2] }. \end{aligned}$$

(A.22)

Popular Coulomb wave functions are regular $F_{l}(k,\,r)$ and singular (at the origin) $G_{l}(k,\,r)$ solutions:

$$\begin{aligned}&F_{l}(k,\,r)= \frac{ e^{ i\,\sigma _{ l }^{C}}\,O_{l}(k,\,r ) - e^{-i\,\sigma _{ l }^{C}}\,I_{l}(k,\,r ) }{2\,i} \end{aligned}$$

(A.23)

$$\begin{aligned}&\quad = (-2k\,r)^{l+1}\,e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,\mathrm{Re}\nonumber \\&\qquad \times \big [e^{i(k\,r + \sigma _{l}^{C})}\,U(l+1+i\,\eta , 2\,l+2, -2\,i\,k\,r)\big ] \end{aligned}$$

(A.24)

$$\begin{aligned}&\quad = e^{-\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\,\frac{ |\Gamma (l+1+ i\,\eta )|}{2\,\Gamma (2\,l+2)}\,(2\,k\,r)^{l+1}\,e^{i\,k\,r}\, \nonumber \\&\qquad \times {}_{1}F_{1}(l+1+i\,\eta , 2\,l+2; -2\,i\,k\,r), \end{aligned}$$

(A.25)

$$\begin{aligned}&G_{l}(k,\,r)= \frac{ e^{i\,\sigma _{ l }^{C}}\,O_{l}(k,\,r ) + e^{-i\,\sigma _{ l }^{C}}\,I_{l}(k,\,r ) }{2} \end{aligned}$$

(A.26)

$$\begin{aligned}&\quad = (-2\,k\,r)^{l+1}\,e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\, \mathrm{Im}\big [e^{i\,(k\,r + \sigma _{l}^{C})}\,\nonumber \\&\qquad \times U(l+1+i\,\eta , 2\,l+2, -2\,i\,k\,r)\big ], \end{aligned}$$

(A.27)

$$\begin{aligned}&I_{l}(k,\,r ) = e^{i\,\sigma _{l}^{C}}\,\big [G_{l}(k,\,r)- i\,F_{l}(k,\,r)\big ], \end{aligned}$$

(A.28)

$$\begin{aligned}&O_{l}(k,\,r ) = e^{-i\,\sigma _{l}^{C}}\,\big [G_{l}(k,\,r) + i\,F_{l}(k,\,r)\big ]. \end{aligned}$$

(A.29)

At the origin ($r \rightarrow 0$) $F_{l}(k,\,r)$ and $G_{l}(k,\,r)$ behave as

$$\begin{aligned}&F_{l}(k,\,r) {\mathop {=}\limits ^{r \rightarrow 0}} {{\mathcal {C}}}_{l}\,\,(k\,r)^{l+1}\,\,\Big [ 1+ \frac{\eta \,k\,r}{l+1} + O_{l}((k\,r)^{2})\Big ], \end{aligned}$$

(A.30)

$$\begin{aligned}&G_{0}(k,\,r) {\mathop {=}\limits ^{r \rightarrow 0}} {{\mathcal {C}}}_{0}^{-1}\,\Big [1+ 2\,\eta \,k\,r\,\ln (k\,r)\Big ] + O_{0}(k\,r), \end{aligned}$$

(A.31)

$$\begin{aligned}&G_{l}(k,\,r) {\mathop {=}\limits ^{r \rightarrow 0}} {{\mathcal {C}}}_{l}^{-1}\,\frac{(k\,r)^{-l}}{2\,l+1}\,\Big [1+ \eta \,l^{-1}O_{l}(k\,r)\Big ]. \end{aligned}$$

(A.32)

For $k \rightarrow 0$ so that $\eta>> k\,r$

$$\begin{aligned}&F_{l}(k,\,r) {\mathop {\approx }\limits ^{k \rightarrow 0}} \frac{ (2\,l+1)!\, {{\mathcal {C}}}_{l}}{(2\,\eta )^{l+1}}\,(2\,\eta \,k\,r)^{1/2}\, I_{2\,l+1}[2\,(2\,\eta \,k\,r)^{1/2}], \end{aligned}$$

(A.33)

$$\begin{aligned}&G_{l}(k,\,r) {\mathop {\approx }\limits ^{k \rightarrow 0}} \frac{2\,(2\,\eta )^{l}}{(2\,l+1)!\, {{\mathcal {C}}}_{l}}\,(2\,\eta \,k\,r)^{1/2}\, K_{2\,l+1}[2\,(2\,\eta \,k\,r)^{1/2}]. \end{aligned}$$

(A.34)

$I_{2\,l+1}(z)$ and $K_{2\,l+1}[2\,(2\,\eta \,k\,r)^{1/2}]$ are Bessel functions of the integer order $2\,l+1$. For Eqs. (A.30)–(A.32) the reader is referred to [202], see SF.3, page 485, and for Eqs. (A.33) and (A.34) the reader is referred to [203], Eqs. 14.6.7.

$$\begin{aligned}&{{\mathcal {C}}}_{l}= \lim \limits _{r \rightarrow 0} \frac{F_{l}(k,\,r)}{(k\,r)^{l+1} }\, =\, \frac{1}{(2\,l+1)!!}\,{{\mathcal {C}}}_{0}\,[\nu _{l}(\eta ^{2})]^{1/2}, \end{aligned}$$

(A.35)

$$\begin{aligned}&{{\mathcal {C}}}_{0}^{2}= \frac{2\,\pi \,\eta }{e^{2\,\pi \,\eta } -1}. \end{aligned}$$

(A.36)

$\nu _{l}(\eta ^{2})$ is defined in (A.14).

In view of Eq. (A.13), Eq. (A.35) can be written in the following form:

$$\begin{aligned} {{\mathcal {C}}}_{l}^{2} = \frac{1}{[(2\,l+1)!!]^{2}}\,|f_{l}^{C(+)}(k)|^{-2}. \end{aligned}$$

(A.37)

${{\mathcal {C}}}_{l}$ is the amplitude of the regular Coulomb solution $F_{l}(k,\,r)$ at $r \rightarrow 0$. Hence, ${{\mathcal {C}}}_{l}^{2}$ is called the Coulomb penetration factor in the partial wave l because it determines the probability for a particle to be found at $r=0$.¹ Note that ${{\mathcal {C}}}_{l}^{2}$ differs from the Coulomb penetrability factor, which appears in the R-matrix approach:

$$\begin{aligned} P_{l}(k)= \frac{k\,R_{ch}}{F_{l}^{2}(k,\,R_{ch}) + G_{l}^{2}(k,\,R_{ch}) }. \end{aligned}$$

(A.38)

$R_{ch}$ is the R-matrix channel radius, which can be taken as the sum of the radii of the interacting particles or as the radius of the strong interaction.

Taking into account Eq. (A.35) one can rewrite $F_{l}(k,\,r)$ as

$$\begin{aligned}&F_{l}(k,\,r)= {{\mathcal {C}}}_{l}\,(k\,r)^{l+1}\,e^{i\,k\,r}\,{}_{1}F_{1}(l+1+ i\,\eta , 2\,l+2; -2\,i\,k\,r), \end{aligned}$$

(A.39)

where $\,{}_1F_{1}(l+ 1 + i\,\eta , \,2\,l + 2; \, -2\,i\,k\,r)\,$ is the Kummer confluent hypergeometric function of the first order.

Note that at $r=R_{ch}$ and $\,k \rightarrow 0\,$ in view of Eqs. (A.33) and (A.34) one gets

$$\begin{aligned} \frac{ F_{l}(k,\,r) }{ G_{l}(k,\,r) } {\mathop {\approx }\limits ^{k \rightarrow 0}} \frac{ {{\mathcal {C}}}_{l}^{2}\,\eta \,[( 2\,l+1)!]^{2} }{2\,(2\,\eta )^{2\,l+1}}\,\frac{I_{2\,l+1}[2\,(2\,\eta \,k\,R_{ch})^{1/2}]}{K_{2\,l+1}[2\,(2\,\eta \,k\,R_{ch})^{1/2}]}. \end{aligned}$$

(A.40)

From Eqs. (A.21)–(A.23) and (A.26) follows the asymptotic behavior of $F_{l}(k,\,r)$ and $G_{l}(k,\,r)$:

$$\begin{aligned}&F_{l}(k,\,r) {\mathop {\approx }\limits ^{r \rightarrow \infty }} \sin [k\,r- \eta \,\ln (2\,k\,r) - \pi \,l/2 + \sigma _{l}^{C} ], \end{aligned}$$

(A.41)

$$\begin{aligned}&G_{l}(k,\,r) {\mathop {\approx }\limits ^{r \rightarrow \infty }} \cos [k\,r- \eta \,\ln (2\,k\,r) - \pi \,l/2 + \sigma _{l}^{C} ]. \end{aligned}$$

(A.42)

The wave function $F_{l}(k,\,r)$ itself is not analytical function because it contains $|\Gamma (l+1+i\,\eta )|$. The partial Coulomb wave function, which appears in the partial wave expansion of the Coulomb scattering wave function $\psi _{\mathbf{k }}^{C(+)}(\mathbf{r })$, is $e^{i\,\sigma _{l}^{C}}\,F_{l}(k,\,r)/(k\,r)$:

$$\begin{aligned}&e^{i\,\sigma _{l}^{C}}\,F_{l}(k,\,r) = e^{- \pi \,\eta /2\,\mathrm{sign}\mathrm{Re}k }\,\frac{\Gamma (l + 1+ i\,\eta )}{2\,\Gamma (2\,l+ 2)}\,(2\,k\,r)^{l + 1} \nonumber \\&\quad \times e^{i\,k\,r}\, {}_1F_{1}(l + 1 + i\,\eta , 2\,l + 2; -2\,i\,k\,r). \end{aligned}$$

(A.43)

It is analytic function in the complex k plane except the simple poles at $l+1 \pm i\,\eta = -n$ of $\,\Gamma (l+1 +\,i\,\eta )\,$ function, where $n= 0,1,2\ldots .$

Following [101], we construct another regular Coulomb solution called also a physical solution of the Coulomb Schrödinger equation. It is a regular at the origin ($r=0$) and analytic in the entire complex k plane (except the point $k=0$ and the poles of $\,\Gamma (l+1+i\,\eta )\,$ scattering solution of the Schrödinger equation:

$$\begin{aligned}&{\varphi }_{l}^{C(+)}(k,r)= e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\Big [\,f_{l}^{C(-)}(k,r) \nonumber \\&\qquad + e^{-i\,\pi \,(l+1)}{{\mathbb {S}}}_{l}^{C}(k)\,f_{l}^{C(+)}(k,r) \Big ] \end{aligned}$$

(A.44)

$$\begin{aligned}&\quad =e^{\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\Big [\,W_{\,i\,\eta , l+1/2}(2\,i\,k\,r) \nonumber \\&\qquad + e^{-i\,\pi \,(l+1)}{{\mathbb {S}}}_{l}^{C}(k)\,W_{-\,i\,\eta , l+1/2}(-2\,i\,k\,r) \Big ] \end{aligned}$$

(A.45)

$$\begin{aligned}&\quad = (-2\,i\,k\,r)^{l+1}\,\frac{\Gamma (l+1+i\,\eta )}{\Gamma (2\,l+2)}\,\nonumber \\&\qquad \times e^{i\,k\,r}\,{}_1F_{1}(l + 1 + i\eta ,\, 2\,l + 2;\, -2\,i\,k\,r) \end{aligned}$$

(A.46)

$$\begin{aligned}&\quad =2\, e^{\frac{1}{2}\,\pi \,\eta \,\mathrm{sign}\mathrm{Re}k}\, e^{-i\,\pi \,(l+1)/2}\,e^{i\,\sigma _{l}^{C}}\,F_{l}(k,\,r). \end{aligned}$$

(A.47)

Summary of analytical properties of the Coulomb wave functions:

• 1. Singular at the origin ($r=0$) Whittaker function $\,{W_{ - \,i\,\eta ,l + 1/2}}(- 2\,i\,k\,r)\,$ ( $\,{W_{ \,i\,\eta ,l + 1/2}}(2\,i\,k\,r))\,$ is analytic function in the open k plane except the branching point singularity at $k=0$ and a discontinuity along the negative (positive) imaginary axis.

• 2. The physical solution ${\varphi }_{l}^{C}(k,r)$ is analytic function in the entire k plane except the point $k=0$ and the simple poles along the imaginary negative axis in the k plane generated by $\Gamma ( l+1 +i\,\eta )$.

Appendix B: Nuclear-Coulomb scattering

Having established the analytical structure of the regular Coulomb scattering solution we proceed now to the scattering of two point-like particles interacting via the Coulomb plus nuclear potentials. During scattering process charged particles emit infinite number of photons. It requires a redefinition of the scattering S-matrix in the limit of zero photon mass. The redefined S-matrix contains in the k-plane the infrared (Coulomb) singularities. In the potential theory these singularities are caused by the infinite-range Coulomb potential. The Coulomb singularities can be explicitly singled out.

The obtained renormalized S-matrix on the physical sheet (upper half k plane) does not contain the Coulomb singularities and behaves like the one for short-range interactions. This renormalized S-matrix has bound-state poles on the positive imaginary axis in the k plane and the resonances in the third and fourth quadrants in the k plane, see Fig. 1. In what follows, we discuss the analytical structure of the elastic scattering in the presence of the Coulomb interaction, its renormalization and determine the residues in the bound-state and resonant poles of the elastic scattering.

Let us consider the scattering of two charged spinless particles a and A (we disregard the spins because they don’t affect the Coulomb renormalization) interacting via the local, energy-independent potential

$$\begin{aligned} V(r)= V^{C}(r) + V^{N}(r), \end{aligned}$$

(B.1)

where the Coulomb potential $V^{C}(r)$ and the nuclear potential $V^{N}(r)$ have been already introduced.

A reduced regular radial solution $\varphi _{l}(k,\,r)$ satisfies the Schrödinger equation

$$\begin{aligned}&\Bigg (E + \frac{1}{2\,\mu }\,\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}- V^{centr}_{l}(r) - V^{C}(r) \Bigg )\,\varphi _{l}(k,r) \nonumber \\&\quad = V^{N}(r)\varphi _{l}(k,r), \end{aligned}$$

(B.2)

where E is the relative kinetic energy of the interacting particles, $\,V^{centr}_{l}(r)\,$ is the centrifugal potential.

A singular radial solution $f_{l}^{(+)}(k,r)$ ($f_{l}^{(-)}(k,r)$) corresponding to asymptotically outgoing wave (incident wave) satisfies a similar equation for $r >0$. A regular solution is a linear combination of singular solutions $f_{l}^{(+)}(k,r)$ and $f_{l}^{(-)}(k,r)$. At $r \rightarrow 0$ a regular solution behaves as $\sim r^{l+1}$. Singular solutions at $r \rightarrow 0$ behaves as $\sim r^{-l}$. Regular and singular solutions of the Schrödinger equation represent solutions belonging to two different types, which are linearly independent. Regular solutions can differ only by the overall normalization factor.

The Shrödinger equation (B.2) is a differential equation which is the same for regular and singular solutions. But equivalent integral equations for two types of solutions are different and are given by Volterra type integral equations [101]:

$$\begin{aligned}&\varphi _{l}(k,r)= { \varphi }_{l}^{C}(k,r) - \int \limits _{0}^{r}\,\mathrm{d}r'\, K^{(+)}(r,r')\,\varphi _{l}(k,r'), \end{aligned}$$

(B.3)

$$\begin{aligned}&f_{l}^{(+)}(k,r)= f_{l}^{C(+)}(k,r) + \int \limits _{r}^{\infty }\,\mathrm{d}r'\, K^{(+)}(r,r')\,f_{l}^{(+)}(k,r'). \end{aligned}$$

(B.4)

Similar equations can be obtained using $f_{l}^{(-)}(k)$ rather than $f_{l}^{(+)}(k)$.

The kernel of these integral equations is

$$\begin{aligned} K(r,r')&= \frac{1}{{{\mathcal {W}}}^{C}(k)}\,\Bigg [ f_{l}^{C(+)}(k,r)\,{\varphi }_{l}^{C}(k,r') \nonumber \\&\quad - f_{l}^{C(+)}(k,r')\,{\varphi }_{l}^{C}(k,r) \Bigg ]\,V^{N}(r'), \end{aligned}$$

(B.5)

where the Coulomb Wronskian ${{\mathcal {W}}}^{C}(k)=-2\,i\,k$, see Eq. (C.17).

One can introduce another regular scattering wave function, which differs only by normalization from the wave function $\varphi _{l}(k,r)$ determined by Eq. (B.3):

$$\begin{aligned} {{\hat{\varphi }}}_{l}(k,r)= \Big [{{\mathcal {F}}}_{l}^{(+)}(k)\,f_{l}^{(-)}(k,r) - (-1)^{l}\,{{\mathcal {F}}}_{l}^{(-)}(k)\,f_{l}^{(+)}(k,r) \Big ]. \end{aligned}$$

(B.6)

The Jost functions are determined by

$$\begin{aligned} {{\mathcal {F}}}_{l}^{(\pm )}(k) = \lim \limits _{r \rightarrow 0}\,(\mp \, 2\,i\,k\,r)^{l}\,f_{l}^{(\pm )}(k,r). \end{aligned}$$

(B.7)

The Wronskian of the regular and singular solutions for the Coulomb plus nuclear potential does not depend on the radius and is given by:

$$\begin{aligned} {{\mathcal {W}}}(k)&= f_{l}^{(+)}(k,r)\,\frac{\mathrm{d}{\varphi }_{l}(k,r)}{\mathrm{d}r} - \frac{\mathrm{d}f_{l}^{(+)}(k,r)}{\mathrm{d}r}\,{\varphi }_{l}(k,r) \nonumber \\&= {{\mathcal {W}}}^{C}(k)\,\Bigg [1 + \frac{1}{{{\mathcal {W}}}^{C}(k)}\,\int \limits _{0}^{\infty }\,\mathrm{d}r\,{ \varphi }_{l}^{C}(k,r)\,V^{N}(r)\,f_{l}(k,r)\,\Bigg ]. \end{aligned}$$

(B.8)

Appendix C: ANC of bound state in terms of source term and Wronskian

In this Appendix we derive the expression for the ANC in terms of the source term and then transform it into the Wronskian form. First, we derive the expression for the ANC using the Pinkston–Satchler equation for the radial overlap function containing the source term [13, 79,80,81]. Note that the overlap function is not an eigenfunction of any Hermitian Hamiltonian. To derive the equation for the overlap function containing the source term, we start from the Schrödinger equation for the bound state of the parent nucleus $B=(a\,A)$:

$$\begin{aligned} (-\varepsilon _{B} - {{\widehat{T}}}_{A} - {{\widehat{T}}_a} - {{\widehat{T}}_{aA}} - \,V_{a} - \,V_{A} - V_{aA} ){\phi _B}({\xi _a},{\xi _A};\,\mathbf{r }_{aA}) = 0. \end{aligned}$$

(C.1)

Here, ${{\widehat{T}}}_i$ is the internal motion kinetic energy operator of nucleus i, $\,{{\widehat{T}}_{aA}}$ is the kinetic energy operator of the relative motion of nuclei a and A, $V_{i}$ is the internal potential of nucleus i and $V_{aA}$ is the interaction potential between a and A, $\varepsilon _{B}$ is the binding energy of nucleus B.

Multiplying the Schrödinger equation from the left by

$$\begin{aligned}&\,{\left( \begin{aligned} A \\ a \\ \end{aligned} \right) ^{1/2}}\,\sum \limits _{{m_{{j_B}}}\,{m_{{l_B}}}\,M_{a}M_{A}}\,\langle {J_A}{M_A}\,\,{j_B}{m_{{j_B}}}|{J_B}{M_B} \rangle \nonumber \\&\quad \times \langle {J_a}{M_a}\,{l_B}{m_{{l_B}}}|{j_B}{m_{{j_B}}} \rangle \,{Y^{*}_{{l_B}{m_{{l_B}}}}}({{\widehat{\mathbf{r }}}}_{aA})\,\phi _{A}^{*}(\xi _{A})\,\phi _{a}^{*}(\xi _{a}) \end{aligned}$$

(C.2)

and taking into account Eq. (106) we get the equation for the radial overlap function with the source term [81]

$$\begin{aligned}&\Big (-\varepsilon _{aA} - {{\widehat{T}}}_{r_{aA}}- V^{centr}_{l_{B}} - U_{aA}^{C} \Big )\,r_{aA}\,I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}(r_{aA}) \nonumber \\&\quad = r_{aA}\,Q_{l_{B}j_{B}J_{a}J_{A}J_{B}}(r_{aA}). \end{aligned}$$

(C.3)

Here, ${{\hat{T}}}_{r_{aA}}$ is the radial relative kinetic energy operator of the particles a and A, $\,V_{l_{B}}^{centr}$ is the centrifugal barrier for the relative motion of a and A with the orbital momentum $l_{B}$, $\,Q_{l_{B}j_{B}J_{a}J_{A}J_{B}}(r_{aA})$ is the source term

$$\begin{aligned}&Q_{l_{B}j_{B}J_{a}J_{A}J_{B}}(r_{aA}) \nonumber \\&\quad = \sum \limits _{m_{j_B} \,m_{l_B}\,M_{a}\,M_{A}} { \langle {J_A}{M_A}\,\,{j_B}{m_{{j_B}}}|{J_B}{M_B} \rangle \langle {J_a}{M_a}\,{l_B}{m_{{l_B}}}|{j_B}{m_{{j_B}}} \rangle }\nonumber \\&\qquad \times \,{\left( \begin{aligned}A \\ a \\ \end{aligned} \right) ^{1/2}}\,\int \,\mathrm{d}\,\Omega _{\mathbf{r }_{aA}}\,\langle \phi _{a}(\xi _{a})\,\phi _{A}(\xi _{A})\nonumber \\&\qquad \times |V_{aA} - U_{aA}^{C}|Y_{{l_B}{m_{{l_B}}}}^{*}({{\widehat{\mathbf{r }}}}_{aA})\phi _{B}(\xi _{a},\xi _{A};\mathbf{r }_{aA})\rangle . \end{aligned}$$

(C.4)

The integration in the matrix element in Eq. (C.4) is carried out over all the internal coordinates $\xi _{a}$ and $\xi _{A}$ of nuclei a and A, respectively. Note that we replaced the antisymmetrization operator ${{\hat{A}}}_{aA}$ in Eq. (C.2) by ${\left( \begin{aligned}A \\ a \\ \end{aligned} \right) ^{1/2}}$ because the operator $-\varepsilon _{aA} - {{\widehat{T}}_A} - {{\widehat{T}}_a} - {{\widehat{T}}_{aA}} - \,V_{a} - \,V_{A} - V_{aA}$ in Eq. (C.1) is symmetric over interchange of nucleons of a and A, while $\phi _{B}$ is antisymmetric. For charged particles it is convenient to single out the channel Coulomb interaction $U_{aA}^{C}(r_{aA})$ between the center of mass of nuclei a and A:

$$\begin{aligned} I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}(r_{aA})&= \,\frac{1}{r_{aA}}\,\int \limits _{0}^{\infty }\,\mathrm{d}r'_{aA}\,r'_{aA}\,G_{l_{B}}^{C}(r_{aA},\,r'_{aA}; i\,\kappa _{aA}) \nonumber \\&\quad \times Q_{l_{B}j_{B}J_{a}J_{A}J_{B}}(r'_{aA}), \end{aligned}$$

(C.5)

where $G_{l_{B}}^{C}(r_{aA},\,r'_{aA}; i\,\kappa _{aA})\,$ is the partial Coulomb Green function.

Owing to the presence of the short-range potential operator $V_{aA} - U_{aA}^{C}$ (potential $V_{aA}$ is the sum of the nuclear $V^{N}_{aA}$ and the Coulomb $V_{aA}^{C}$ potentials and subtraction of $U_{aA}^{C}$ removes the long-range term from $V_{aA}$) the source term is also a short-range function and we approximate Eq. (C.5) by

$$\begin{aligned} I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}(r_{aA})&= \,\frac{1}{r_{aA}}\,\int \limits _{0}^{R_{ch}}\,\mathrm{d}r'_{aA}\,r'_{aA}\,G_{l_{B}}^{C}(r_{aA},\,r'_{aA}; i\,\kappa _{aA}) \nonumber \\&\quad \times Q_{l_{B}j_{B}J_{a}J_{A}J_{B}}(r'_{aA}), \end{aligned}$$

(C.6)

where $R_{ch}$ is the channel radius for the system $a-A$.

The partial Coulomb two-body Green function at $k_{aA}>0$ is given by [100]

$$\begin{aligned}&G_{l_{B}}^{C}(r_{aA},r'_{aA}; k_{aA}) = -i\,\frac{\mu _{aA}}{k_{aA}}\, e^{-\frac{1}{2}\,\pi \,\eta _{aA}\,\mathrm{sign}\mathrm{Re}k_{aA}}\, \nonumber \\&\quad \times \,\varphi _{l_{B}}^{C}(k_{aA},\,r_{aA<})\,f_{l_{B}}^{C(+)}(k_{aA},\,r_{>}). \end{aligned}$$

(C.7)

$\eta _{aA}$ and $k_{aA}$ are the $a-A$ Coulomb parameter and relative momentum. Also, $\,r_{aA<}= \mathrm{min}\,\{r_{aA},r_{aA}' \}$ and $\,r_{aA>}= \mathrm{max}\,\{r_{aA},r_{aA}' \}$.

From the analytic properties of $\varphi _{l_{B}}^{C}(k_{aA},\,r_{aA})$ and $e^{-\frac{1}{2}\,\pi \,\eta _{aA}\,\mathrm{sign}\mathrm{Re}k_{aA}}\,f_{l_{B}}^{C(+)}(k_{aA},\,r)\,$, see the end of Appendix A, follows that the Green function $G_{l_{B}}^{C}(r_{aA},r_{aA}'; k_{aA})$ has a discontinuity along the negative imaginary axis in the $k_{aA}$ plane. Extrapolating the Coulomb Green function $G_{l_{B}}^{C}(r_{aA},r_{aA}'; k_{aA})$ to the point $k_{aA}= i\,\kappa _{aA}$ corresponding to the bound-state we get

$$\begin{aligned}&G_{l_{B}}^{C}(r_{aA},r'_{aA}; i\,\kappa _{aA}) = - \,\frac{\mu _{aA}}{\kappa _{aA}}\, e^{i\,\pi \,\eta _{aA}^{bs}/2} \nonumber \\&\quad \times \varphi _{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA<})\,f_{l_{B}}^{C(+)}(i\,\kappa _{aA},r_{aA>}). \end{aligned}$$

(C.8)

Substituting this equation into Eq. (C.6) and taking into account Eqs. (107) and (A.47) we get the ANC expressed in terms of the source $Q_{l_{B}j_{B}J_{a}J_{A}J_{B}}(r_{aA})$ [13, 81]:

$$\begin{aligned}&C_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B} = - \,\frac{\mu _{aA}}{\kappa _{aA}}\,{{\mathcal {R}}}_{1}\,\nonumber \\&\quad \times \int \limits _{0}^{R_{ch}}\,\mathrm{d}r_{aA}\,{{\tilde{\varphi }}}_{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA})\,Q_{l_{B}j_{B}J_{a}J_{A}J_{B}}(r_{aA}). \end{aligned}$$

(C.9)

Now we transform the radial integral in Eq. (C.9) into the Wronskian at $r_{aA}= R_{ch}$. The philosophy of this transformation is the same as in the surface integral formalism [31, 93, 115] only applied here for bound states.

To this end in Eq. (C.9) we replace the source term $Q_{l_{B}j_{B}J_{a}J_{A }J_{B}}(r_{aA})$ with the left side of Eq. (C.3):

$$\begin{aligned}&\tilde{C}_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B} = - \,\frac{\mu _{aA}}{\kappa _{aA}}\,\int \limits _{0}^{R_{ch}}\,\mathrm{d}r_{aA}\,{{\tilde{\varphi }}}_{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA})\, \nonumber \\&\quad \times \Big (-\varepsilon _{aA} - {\overrightarrow{{\widehat{T}}}}_{r_{aA}}- V^{centr}_{l_{B}} - U_{aA}^{C} \Big )\,r_{aA}\,I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}(r_{aA}). \end{aligned}$$

(C.10)

Here the kinetic energy operator ${\overrightarrow{{\widehat{T}}}}_{r_{aA}}$ acts on the radial overlap function $I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}(r_{aA})$.

In view of the equation

$$\begin{aligned}&{{\tilde{\varphi }}}_{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA}) \,\Big (-\varepsilon _{aA} - V^{centr}_{l_{B}} - U_{aA}^{C} \Big ) \nonumber \\&\quad = {{\tilde{\varphi }}}_{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA}) \,{\overleftarrow{{\widehat{T}}}}_{r_{aA}} \end{aligned}$$

(C.11)

one can rewrite Eq. (C.9)

$$\begin{aligned}&\tilde{C}_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B} = \frac{1}{2\,\kappa _{aA}}\, \int \limits _{0}^{R_{ch}}\,\mathrm{d}r_{aA}\,\Bigg [{{\tilde{\varphi }}}_{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA})\nonumber \\&\quad \times \,\Bigg ( \frac{{{{\overleftarrow{d} }^2}}}{{d{r_{aA}^2}}} -\frac{{{{\overrightarrow{d} }^2}}}{{d{r_{aA}^2}}} \Bigg )\,r_{aA}\,I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}(r_{aA})\Bigg ]. \end{aligned}$$

(C.12)

Taking into account the definition of the Wronskian

$$\begin{aligned} {{\mathcal {W}}}[f(r),\,g(r)) ]= f(r)\,\frac{ \mathrm{d}g(r)}{\mathrm{d}r} -g(r)\,\frac{\mathrm{d}f(r)}{\mathrm{d}r} \end{aligned}$$

(C.13)

one can express the ANC in terms of the Wronskian:

$$\begin{aligned}&\tilde{C}_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B} = \,\frac{1}{2\,\kappa _{aA}}\, {{\mathcal {W}}}[r_{aA}\,I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^B({r_{aA}}),\nonumber \\&\qquad \quad {{\tilde{\varphi }}}_{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA}) ]\Big |_{r_{aA}=R_{ch}} \end{aligned}$$

(C.14)

$$\begin{aligned}&\quad = \frac{1}{2\,\kappa _{aA}}\,\Bigg ( r_{aA}\,I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^B({r_{aA}})\,\frac{\mathrm{d}{{\tilde{\varphi }}}_{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA})}{\mathrm{d}r_{aA}}\nonumber \\&\qquad -{{\tilde{\varphi }}}_{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA})\,\frac{\mathrm{d}\big [r_{aA}\,I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^B({r_{aA}})\big ]}{\mathrm{d}r_{aA}} \Bigg )\,\Big |_{r_{aA}=R_{ch}}. \end{aligned}$$

(C.15)

Note that Eqs. (C.14) and (C.15) have been derived by transforming the internal integral into the Wronskian at the channel radius $R_{ch}$. At too small radii $R_{ch}$ the Wronskian ${{\mathcal {W}}}[r_{aA}\,I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^B({r_{aA}}),\,{{\tilde{\varphi }}}_{l_{B}}^{C}(i\,\kappa _{aA},\,r_{aA}) ]\Big |_{r_{aA}=R_{ch}}$ depends on the channel radius but the sensitivity to the radius decreases as $\,R_{ch} \rightarrow R_{aA}^{N},\,$ where $\,R_{aA}^{N}$ is the $a-A$ nuclear interaction radius. At $\,R_{ch} > R_{aA}^{N}$ one can replace the radial overlap function by its asymptotic term given by Eq. (107). Then Eq. (C.15) turns into identity: $\,\tilde{C}_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B} \equiv \tilde{C}_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}$. Here we present this proof.

We consider such a large $R_{ch}$ that $r_{aA}I_{aA\,\,{l_B}\,{j_B}\,J_{B}}^B({r_{aA}})$ can be replaced by its asymptotic term $C_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}\,W_{-\eta _{aA}^{bs},\,l_{B}+1/2}(2\,\kappa _{aA}\,r_{aA})$, see Eq. (107). Replacing ${{\tilde{\varphi }}}_{l_{B}}(i\,\kappa _{aA},\,r_{aA})$ with $\varphi _{l_{B}}(i\,\kappa _{aA},\,r_{aA})$ using Eq. (122) and taking into account that ${{\mathcal {W}}}[f_{l_{B}}^{C(+)}(-\kappa _{aA},\,r_{aA}),\,f_{l_{B}}^{C(+)}(-\kappa _{aA},\,r_{aA}) ] =0$ we arrive at

$$\begin{aligned} \tilde{C}_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}&= \,\tilde{C}_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}\,\frac{1}{2\,\kappa _{aA}}\,{{\mathcal {W}}}^{C}(i\,\kappa _{aA} ) \nonumber \\&\equiv \tilde{C}_{aA\,\,{l_B}\,{j_B}\,J_{B}}^{B}. \end{aligned}$$

(C.16)

Here we use the Coulomb Wronskian ${{\mathcal {W}}}^{C}(i\,\kappa _{aA} )$. It is defined as

$$\begin{aligned} {{\mathcal {W}}}^{C}(k_{aA})&= {{\mathcal {W}}}\big [f_{l_{B}}^{C(+)}(i\,k_{aA},\,r_{aA}),\,f_{l_{B}}^{C(-)}(i\,k_{aA},\,r_{aA}) \big ] \nonumber \\&= -2\,i\,k_{aA}, \end{aligned}$$

(C.17)

which does not depend on the radius. Hence the easiest way to calculate it is to consider asymptotic behavior of the Jost functions $\,f_{l_{B}}^{C(\pm )}(i\,k_{aA},\,r_{aA})$, see Eq. (A.9). For $k_{aA}= i\,\kappa _{aA}$ the Coulomb Wronskian becomes

$$\begin{aligned} {{\mathcal {W}}}^{C}(i\,\kappa _{aA})&= {{\mathcal {W}}}\big [f_{l_{B}}^{C(+)}(-\kappa _{aA},\,r_{aA}),\,f_{l_{B}}^{C(-)}(-\kappa _{aA},\,r_{aA}) \big ] \nonumber \\&= 2\,\kappa _{aA}. \end{aligned}$$

(C.18)

Appendix D: Modified expression for the surface term

As we have discussed in Sect. 8, the dominant contribution to the transfer reaction amplitude comes from the surface and the external prior terms. Let us rewrite the surface term in the prior form in which the independent Jacobi variables are ${\mathbf{r}}_{xA}$ and ${\mathbf{r}}_{sF}$:

$$\begin{aligned}&{{\mathcal {M}}}_{S}^{DW(prior)} = C_{xA\,l_{B}\,{j_{B}\,J_{B} }\,}^B \,\frac{{R_{ch}^2}}{{2{\mu _{xA}}}}\,\nonumber \\&\quad \times \sum \limits _{m_{l_{B}}\,\,m_{j_{B}}\,M_{x}}\,\langle J_{A}\, M_{A}\,\, j_{B}\, m_{j_{B}}| J_{B}\,M_{B}\rangle \,\langle J_{x}\,M_{x}\,\,l_{B}\,m_{l_{B}}|j_{B}\,m_{j_{B}}\rangle \nonumber \\&\quad \times \langle J_{s}\,M_{s}\,\,J_{x}\,M_{x}| J_{a}\,M_{a}\rangle \,\int {d{{\mathbf{r}}_{sF}}} \chi _{ - {{\mathbf{k}}_{sF}}}^{C( + )}({{\mathbf{r}}_{sF}})\nonumber \\&\quad \times \int d {\Omega _{{{\mathbf{r}}_{xA}}}} Y_{l_{B}m_{l_{B}}}^{*}({{\hat{\mathbf{r }}}}_{xA}) \nonumber \\&\quad \times \left[ \varphi _{a}({\mathbf{r}}_{sx})\,\chi _{ {\mathbf{k}}_{\mathbf{aA}} }^{( + )}({\mathbf{r}}_{aA})\frac{{\partial W_{-\eta ^{bs}_{xA}, l_{B}+1/2}(2\,\kappa _{xA}\,r_{xA})/r_{xA} } }{{\partial {r_{xA}}}}\right. \nonumber \\&\quad \left. - \frac{W_{-\eta ^{bs}_{xA}, l_{B}+1/2}(2\,\kappa _{xA}\,r_{xA})}{r_{xA}} \frac{{\partial {\varphi _{a}}({{\mathbf{r}}_{sx}})\chi _{{{\mathbf{k}}_{{\mathbf{aA}}}}}^{( + )}({{\mathbf{r}}_{aA}})}}{{\partial {r_{xA}}}}\right] \Big |_{{r_{xA}} = {R_{ch}}}. \end{aligned}$$

(D.1)

The external prior DWBA part of the amplitude, similar to the surface term, is also parametrized in terms of the ANC $C_{xA\,l_{B}\,{j_{B}\,J_{B} }\,}^B$ and reduces to

$$\begin{aligned}&{{\mathcal {M}}}_{ext}^{DW(prior)}= C_{xA\, \, {l_{B}}\,{j_{B}\,J_{B} }\,}^B\nonumber \\&\quad \times \,\sum \limits _{m_{l_{B}}\,\,m_{j_{B}}\,M_{x}}\, \langle J_{A}\, M_{A}\,\, j_{B}\, m_{j_{B}}| J_{B}\,M_{B}\rangle \langle J_{x}\,M_{x}\,\,l_{B}\,m_{l_{B}}|j_{B}\,m_{j_{B}}\rangle \nonumber \\&\quad \times \langle J_{s}\,M_{s}\,\,J_{x}\,M_{x}| J_{a}\,M_{a}\rangle \nonumber \\&\quad \times \big <\chi _{\mathbf{k }_{sF}}^{(-)}\,\frac{ W_{ - \eta _{xA}^{bs},\,{l_B} + 1/2 }(2\,\kappa _{xA}\,r_{xA})}{r_{xA} }\,Y_{l_{B}m_{l_{B}}}({{\hat{\mathbf{r }}}}_{xA})\nonumber \\&\quad \times \big | V_{sA}^{C} + V_{xA}^{C} - U_{aA}^{C} \big | \varphi _{sx}\,\chi _{\mathbf{k }_{aA}}^{(+)}\big>\Big |_{r_{xA} > R_{ch}}. \end{aligned}$$

(D.2)

Owing to the strong absorption in the nuclear interior caused by the optical potentials, the internal DWBA post form, as have been discussed in Sect. 8, is small compared to the surface and external prior amplitude.

We transform now the surface matrix element into zero-range DWBA amplitude. To this end we use

$$\begin{aligned} \mathbf{r }_{aA}= \mathbf{r }_{xA} + \frac{m_{s}}{m_{sx}}\,\mathbf{r }_{sx}, \quad \mathbf{r }_{sF}= \frac{m_{A}}{m_{xA}}\,\mathbf{r }_{xA} +\mathbf{r }_{sx}. \end{aligned}$$

(D.3)

Rewriting the wave functions $\chi _{{\mathbf{k }_{aA}}}^{( + )}({\mathbf{r _\mathbf{aA }}})\,$ and $\chi _{ - {\mathbf{k }_{sF}}}^{( + )}({\mathbf{r }_{sF}})$ in the momentum space we get

$$\begin{aligned}&{{\mathcal {M}}}_{S}^{DW(prior)} = \frac{{R_{ch}^2}}{{2{\mu _{xA}}}}\, C_{xA\, \, {l_{B}}\,{j_{B}\,J_{B} }\,}^B\,\nonumber \\&\quad \times \sum \limits _{m_{l_{B}}\,\,m_{j_{B}}\,M_{x}}\,\langle J_{A}\, M_{A}\,\, j_{B}\, m_{j_{B}}| J_{B}\,M_{B}\rangle \langle J_{x}\,M_{x}\,\,l_{B}\,m_{l_{B}}|j_{B}\,m_{j_{B}}\rangle \nonumber \\&\quad \times \langle J_{s}\,M_{s}\,\,J_{x}\,M_{x}| J_{a}\,M_{a}\rangle \ \int {d{\mathbf{r }_{sF}}}\, \int \,\frac{ \mathrm{d}\mathbf{p }_{sF} }{(2\,\pi )^{3} }\, \nonumber \\&\quad \times \int \,\frac{\mathrm{d}\mathbf{p }_{aA}}{(2\,\pi )^{3}}\, \chi _{-{\mathbf{k }_{sF}}}^{( + )}({\mathbf{p }_{sF}})\,\chi _{{\mathbf{k }_{aA}}}^{( + )}({\mathbf{p _\mathbf{aA }}})\,\varphi _{sx}(\mathbf{r }_{sx}) {e^{-i\mathbf{p }_{sx} \cdot {\mathbf{r }_{sx}}}}\nonumber \\&\quad \times \int d{\Omega _{\mathbf{r }_{xA}}}\,Y_{l_{B}m_{l_{B}} }^{*}({{\hat{\mathbf{r }}}}_{xA})\nonumber \\&\quad \times \left[ e^{ i\mathbf{p }_{xA} \cdot \mathbf{r }_{xA} }\,\frac{ {\partial W_{-\eta ^{bs}_{xA}, l_{B}+1/2}(2\,\kappa _{xA}\,r_{xA})/r_{xA} } }{{\partial {r_{xA}}}}\right. \nonumber \\&\quad \left. - \frac{ W_{-\eta ^{bs}_{xA}, l_{B}+1/2 }(2\,\kappa _{xA}\,r_{xA}) }{r_{xA}} \frac{{\partial e^{ i\mathbf{p }_{xA} \cdot \mathbf{r }_{xA} }}}{{\partial {r_{xA}}}}\right] , \end{aligned}$$

(D.4)

where

$$\begin{aligned}&\mathbf{p }_{xA}= {\mathbf{p }_{aA}} - \frac{{{m_A}}}{{{m_F}}}{\mathbf{p }_{sF}} , \qquad \nonumber \\&\mathbf{p }_{sx}={\mathbf{p }_{sF}} - \frac{{{m_s}}}{{{m_a}}}\,{\mathbf{p }_{aA}}. \end{aligned}$$

(D.5)

Taking into account the fact that $r_{xA} = R_{ch}$ is larger than the nuclear interaction radius we replace the relative momentum $\mathbf{p }_{xA}$ with the momentum $\mathbf{q }_{xA}= {\mathbf{k }_{aA}} - \frac{{{m_A}}}{{{m_F}}}{\mathbf{k }_{sF}}$, which is expressed in terms of on-the-energy-shell momenta ${\mathbf{k }_{aA}}$ and ${\mathbf{k }_{sF}}$. However, the $x-A$ relative momentum $\mathbf{q }_{xA}$ is off-the-energy-shell because the transferred particle x is virtual. Returning now to the coordinate-space representation for ${{\mathcal {M}}}_{S}^{DW(prior)}$ we get

$$\begin{aligned}&{{\mathcal {M}}}_{S}^{DW(prior)} = \frac{{R_{ch}^2}}{{2{\mu _{xA}}}}\, C_{xA\, \, {l_{B}}\,{j_{B}\,J_{B} }\,}^B\,\nonumber \\&\quad \times \sum \limits _{ m_{l_{B}}\,\,m_{j_{B}}\,M_{x}}\,\langle J_{A}\, M_{A}\,\, j_{B}\, m_{j_{B}}| J_{B}\,M_{B}\rangle \int d{\Omega _{\mathbf{r }_{xA}}}\,Y_{l_{B}m_{l_{B}} }^{*}({{\hat{\mathbf{r }}}}_{xA}) \nonumber \\&\quad \times \langle J_{x}\,M_{x}\,\,l_{B}\,m_{l_{B}}|j_{B}\,m_{j_{B}}\rangle \nonumber \\&\quad \times \langle J_{s}\,M_{s}\,\,J_{x}\,M_{x}| J_{a}\,M_{a}\rangle {M}^{DWZR(prior)} \nonumber \\&\quad \times \left[ e^{ - i\mathbf{q }_{xA} \cdot \mathbf{r }_{xA} }\,\frac{ {\partial W_{-\eta ^{bs}_{xA}, l_{B}+1/2}(2\,\kappa _{xA}\,r_{xA})/r_{xA} } }{{\partial {r_{xA}}}}\right. \nonumber \\&\quad \left. - \frac{ W_{-\eta ^{bs}_{xA}, l_{B}+1/2 }(2\,\kappa _{xA}\,r_{xA}) }{r_{xA}}\frac{{\partial e^{ - i\mathbf{q }_{xA} \cdot \mathbf{r }_{xA} }}}{{\partial {r_{xA}}}}\right] . \end{aligned}$$

(D.6)

Here,

$$\begin{aligned} {M}^{DWZR(prior)}= \int {d{\mathbf{r }_{sx}}}\, \chi _{ - {\mathbf{k }_{(sF)}}}^{C( + )}({\mathbf{r }_{sx}})\,{\varphi _{a}}({\mathbf{r }_{sx}}) \chi _{\mathbf{k _\mathbf{aA }}}^{C( + )}\left( \frac{{{m_s}}}{{{m_a}}}\mathbf{r }_{sx}\right) \end{aligned}$$

(D.7)

is the DWBA amplitude, which does not depend on the overlap function and $V_{xA}$ potential. This equation looks like the zero-range DWBA (ZRDWBA). However, in contrast to the standard zero-range approximation, Eq. (D.7) can be used for arbitrary value of the orbital momentum of the bound state (xA).

Integrating in Eq. (D.6) over $\,\Omega _{\mathbf{r }_{xA}}\,$ we arrive at the surface term of the DWBA reaction amplitude singled out using the surface-integral formalism:

$$\begin{aligned}&{{\mathcal {M}}}_{S}^{DW(prior)} = \frac{2\,\pi }{\mu _{xA}}\,i^{-l_{B}}\,C_{xA\,l_{B}\,{j_{B}\,J_{B}}\,}^B\,{W}_{-\eta ^{bs}_{xA}, l_{B}+1/2}\nonumber \\&\quad \times (2\,\kappa _{xA}\,R_{ch}) \,j_{{l_{B}}}({q_{xA}}{R_{ch}})\, {{\mathbb {W}}}_{l_{xA}}\,{Y_{{l_{B}},{m_{{l_{B}}}}}}({\widehat{\mathbf{q}}_{xA}})\, \nonumber \\&\quad \times \sum \limits _{m_{l_{B}}\,\,m_{j_{B}}\,M_{x}}\,\langle J_{A}\, M_{A}\,\, j_{B}\, m_{j_{B}}| J_{B}\,M_{B}\rangle \,\langle J_{x}\,M_{x}\,\,l_{B}\,m_{l_{B}}|j_{B}\,m_{j_{B}}\rangle \nonumber \\&\quad \times \langle J_{s}\,M_{s}\,\,J_{x}\,M_{x}| J_{a}\,M_{a}\rangle \,{M}^{DWZR(prior)}. \end{aligned}$$

(D.8)

The off-shell factor ${{\mathbb {W}}}_{l_{xA}}$ is written as

$$\begin{aligned}&{{\mathbb {W}}}_{l_{xA}} = \left[ \Big (B_{xA} -1 \Big ) - R_{ch}\,\frac{\partial {\ln \,[j_{l_{xA}}({q_{xA}}{r_{xA}})]}}{{\partial {r_{xA}}}}\right] {\Big |_{{r_{xA}} = {R_{ch}}}}, \end{aligned}$$

(D.9)

where $B_{xA}$ is defined in Eq. (197) and $j_{l_{xA}}$ is the spherical Bessel function.

Equation (D.8) is a very important result. It contains the off-shell factor ${{\mathbb {W}}}_{l_{xA}}$ reflecting the virtual character of the transferred particle x. It also contains the boundary conditions expressed in terms of the logarithmic derivatives and generalizes the R-matrix method for binary reactions. Thus in the surface-integral formalism the dominant contribution to the DWBA amplitude of the transfer reaction in the prior form is given by the sum of the dominant surface and external terms and the final expression for the prior-form DWBA amplitude, which can be used for the analysis of the transfer reaction.

Appendix E: Notations and abbreviations

Here we present some important notations.

• $m_{i}$: mass of particle i.

• $\mu _{ij}$: reduced mass of particles i and j.

• $\mathbf{k }_{ij}$ ( $\mathbf{p }_{ij}$): on-shell (off-shell) relative momentum of particles i and j.

• $\kappa _{ij}$: (ij) bound-state wave number.

• $\lambda _{N}$: nucleon Compton wave length.

• $m_{au}$: atomic mass unit.

• $\alpha$: fine structure constant.

• $E_{ij}$: relative kinetic energy of particles i and j.

• $\varepsilon _{ij}$: binding energy of the bound state (ij).

• $I_{aA}^{B}$: overlap function of the bound-state wave functions of $B,\,a$ and A.

• $\varphi _{l}(k,r),\,$ $\,\varphi _{l}^{N}(k,r)\,$ and $\,\varphi _{l}^{C}(k,r)$: regular at the origin nuclear+Coulomb, pure nuclear and pure Coulomb reduced radial scattering wave functions, respectively.

• ${{\tilde{\varphi }}}_{l}^{C}(k,r)$: Coulomb-renormalized reduced regular Coulomb radial scattering wave function.

• $f^{(\pm )}_{l}(k,r),\,$ $\,f^{N(\pm )}_{l}(k,r)$ and $f^{C(\pm )}_{l}(k,r)$: singular at the origin nuclear+Coulomb, pure nuclear and pure Coulomb scattering solutions (Jost solutions), respectively.

• ${{\mathcal {F}}}_{l}^{(\pm )}(k),\,$ $\, {{\mathcal {F}}}_{l}^{N(\pm )}(k)\,$ and $\,{{\mathcal {F}}}_{l}^{C(\pm )}(k)$: nuclear + Coulomb, pure nuclear and pure Coulomb Jost functions, respectively.

• $f^{(\pm )}_{l}(k),\,$ $\,f^{N(\pm )}_{l}(k)$ and $f^{C(\pm )}_{l}(k)$: normalized nuclear+Coulomb, pure nuclear and Coulomb Jost functions, respectively.

• $W_{\lambda ,\nu }$: Whittaker function.

• ${{\mathbb {S}}},\,$ $\,{{\mathbb {S}}}^{N},\,$ and $\,{{\mathbb {S}}}^{C} \,$: nuclear + Coulomb, pure nuclear and pure Coulomb scattering S-matrices, respectively.

• $A_{l}$: residue of the S-matrix in the bound-state pole.

• $G_{l}$: nuclear vertex constant.

• $C_{l}$: asymptotic normalization coefficient.

• ${{\tilde{C}}}_{l}$: Coulomb-renormalized asymptotic normalization coefficient.

• ${\mathtt S}$: spectroscopic factor (SF).

• S: astrophysical S-factor.

• ${{\mathcal {M}}}$: reaction amplitude.

• ${{\mathcal {T}}}_{l},\,$ $\,{{\mathcal {T}}}_{l}^{N} \,$ and $\,{{\mathcal {T}}}_{l}^{C},\,$: nuclear+Coulomb, pure nuclear and pure Coulomb scattering amplitudes, respectively.

• ${{\mathcal {T}}}_{l}^{CN}$: Coulomb-modified nuclear scattering amplitude.

• $\mathcal{{{\tilde{T}}}}_{l}^{CN}$: Coulomb-renormalized, Coulomb-modified nuclear scattering amplitude.

• $K_{l}$: effective-range function.

• ${{\mathcal {C}}}_{l}$: Coulomb penetrability factor.

• $P_{l}$: penetrability factor in R-matrix.

• $a_{B}$: Bohr radius.

• ${{\mathcal {W}}}$ (${{\mathcal {W}}}^{C}$): Wronskian (Coulomb Wronskian).

• ${{\mathcal {L}}}$: ratio of mirror ANCs in terms of Wronskians.

• $B_{ij}$: logarithmic boundary condition.

• ${{\mathbb {W}}}_{l}$: off-shell factor.

• ANC: asymptotic normalization coefficient.

• SF: spectroscopic factor.

• DWBA: distorted-wave Born approximation.

• ADWA: adiabatic distorted wave approximation.

• CDCC: continuum-discretized coupled channels.