Photon emission in the graphene under the action of a quasiconstant external electric field

Abstract

Following a nonperturbative formulation of strong-field QED developed in our earlier works, and using the Dirac model of the graphene, we construct a reduced QED$_{3,2}$ to describe one species of the Dirac fermions in the graphene interacting with an external electric field and photons. On this base, we consider the photon emission in this model and construct closed formulas for the total probabilities. Using the derived formulas, we study probabilities for the photon emission by an electron and for the photon emission accompanying the vacuum instability in the quasiconstant electric field that acts in the graphene plane during the time interval T. We study angular and polarization distribution of the emission as well as emission characteristics in a high-frequency and low-frequency approximations. We analyze the applicability of the presented calculations to the graphene physics in laboratory conditions. In fact, we are talking about a possible observation of the Schwinger effect in these conditions.

Appendices

Appendix A: Low-frequency approximation

Let us consider the probability densities, given by Eqs. (54), (55), and (57) in the range of low frequencies,

$$\begin{aligned} u_{0}\ll 1. \end{aligned}$$

(A1)

The ratio $\left| u_{x}\right| /u_{0}$ , given by Eq. (66), is very small such that

$$\begin{aligned} \rho \approx u_{0}\ll 1. \end{aligned}$$

(A2)

In this limit, the behavior of the function $I_{j^{\prime },j}(\rho )$, given by Eq. (71), can be found using properties of the confluent hypergeometric function $\Psi$; see Eqs. (6.8(2)) - (6.8(4)) from Ref. [38]. The only functions $I_{0,1}(\rho )$ and $I_{1,0}(\rho )$ grow as $\rho \rightarrow 0,$

$$\begin{aligned} \left| I_{0,1}(\rho )\right| \sim \left| I_{1,0}(\rho )\right| \sim \rho ^{-1}. \end{aligned}$$

(A3)

Thus, the leading contribution to the amplitude $M_{{\textbf{p}}^{\prime } {\textbf{p}}}^{+}$ given by Eq. (57) is due to the terms $Y_{00}\approx \mathcal {J}_{0,0}(\rho )\sim \rho ^{-1}$ and $Y_{11}\approx \mathcal {J} _{1,1}(\rho )\sim \rho ^{-1}$. The leading contribution to the amplitude $M_{ {\textbf{p}}^{\prime }{\textbf{p}}}^{0}$ given by Eq. (57) is due to the terms ${\tilde{Y}}_{01}\approx \mathcal {{\tilde{J}}}_{0,1}(\rho )\sim \rho ^{-1}$ and ${\tilde{Y}}_{10}\approx \mathcal {{\tilde{J}}}_{1,0}(\rho )\sim \rho ^{-1}$. Therefore, the both modules squares amplitudes square grow proportionally to $u_{0}^{-2}$,

$$\begin{aligned} \left. \left| M_{{\textbf{p}}^{\prime }{\textbf{p}}}^{0}\right| ^{2}\right| _{{\textbf{p}}^{\prime }=-{\textbf{p}}-\hbar {\textbf{k}}}\sim u_{0}^{-2},\;\left. \left| M_{{\textbf{p}}^{\prime }{\textbf{p}}}^{\pm }\right| ^{2}\right| _{{\textbf{p}}^{\prime }={\textbf{p}}-\hbar \textbf{ k}}\sim u_{0}^{-2}. \end{aligned}$$

(A4)

At the same time, the both probability densities (54) and (55) are divergent functions of the order $u_{0}^{-1}$ as $u_{0}\rightarrow 0$,

$$\begin{aligned} \frac{d\mathcal {P}\left( {\textbf{p}};\left. \textbf{K,}\vartheta \right| 0\right) }{du_{0}d\Omega }\sim \frac{\alpha }{\varepsilon }\left( \frac{v_{F} }{c}\right) ^{2}\frac{1}{u_{0}},\;\frac{d\mathcal {P}\left( \left. {\textbf{K}} ,\vartheta \right| \overset{\pm }{{\textbf{p}}}\right) }{du_{0}d\Omega } \sim \frac{\alpha }{\varepsilon }\left( \frac{v_{F}}{c}\right) ^{2}\frac{1}{ u_{0}}. \end{aligned}$$

(A5)

Such a behavior is an indication that the perturbative description of such soft photons, does not work. Namely for photons with frequencies less than a threshold frequency $u_{0}^{\textrm{soft}}$, $u_{0}\lesssim u_{0}^{\textrm{ soft}}$, in case when functions (A5) becomes of the order of unity. This makes it possible to evaluate the quantity $u_{0}^{\textrm{soft}}$,

$$\begin{aligned} u_{0}^{\textrm{soft}}\sim \frac{\alpha }{\varepsilon }\left( \frac{v_{F}}{c} \right) ^{2}. \end{aligned}$$

(A6)

The number of such soft photons may be big enough. However, the only physically measurable quantity is the emitted energy. This energy is a negligibly small in the domain $u_{0}\lesssim u_{0}^{\textrm{soft}}$. This case is called the infrared catastrophe whose nature is associated with the impossibility of separating a charged particle from its radiation field; see, e.g., section 98 in Ref. [41] and sections 46 and 50.3 in Ref. [42]. The case of the strong-electric field QED is considered in Ref. [43]. The infrared divergences of QED are essentially classical, and depend on the nature of the external current and on the experimental resolution. The infrared catastrophe is absent from the complete nonperturbative solution. Thus, one sees that the domain of the applicability of the perturbation theory is $u_{0}>u_{0}^{\textrm{soft}}$ and a contribution from the domain $u_{0}\lesssim u_{0}^{\textrm{soft}}$ is negligible.

In the case under consideration, the quantity $u_{0}^{\textrm{soft}}$ is very small, $u_{0}^{\textrm{soft}}\sim 10^{-7}$. It follows from estimation (98) and from restrictions on the parameter $u_{0}^{\textrm{IR}}$ given by Eqs. (72) and (97) that in the realistic values of the parameters $u_{0}^{\textrm{soft}}\ll u_{0}^{\textrm{IR}}$.

Appendix B: Fourier transformation of the product of two WPC functions

Integrals (64) can be reduced to a more simple form using the Nikishov’s representations given by Eq. (65)) for the hyperbolic coordinates $\rho$ and $\varphi$ (see Ref. [19, 20]). To demonstrate how it works, we note that the integrals represent particular cases of the more general integrals

$$\begin{aligned} J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho ,\varphi \right) =\int _{-\infty }^{+\infty }du\,f_{\Lambda ^{\prime }}^{\zeta ^{\prime }}(u-u_{x}/2)f_{\Lambda }^{\zeta }(u+u_{x}/2)e^{iu_{0}u}\,, \end{aligned}$$

(B7)

where $f_{\Lambda }^{\zeta }(z)$ are WPCF’s satisfying the differential equation

$$\begin{aligned} \left( \frac{d^{2}}{dz^{2}}+z^{2}+\Lambda \right) f_{\Lambda }^{\zeta }(z)=0\,, \end{aligned}$$

(B8)

and $u_{0}$ and $u_{x}$, defined by Eq. (56), are:

$$\begin{aligned} u_{0}=\rho \cosh \varphi ,\;u_{x}=\rho \sinh \varphi \;\textrm{if} \;u_{0}^{2}>u_{x}^{2}. \end{aligned}$$

(B9)

The functions $f_{\Lambda }^{\zeta }(z)$ with different values of $\zeta =\pm$ are solutions of Eq. (B8) with some complex parameters $\Lambda$. In particular,

$$\begin{aligned} J_{\Lambda ^{\prime }\Lambda }^{-+}\left( \rho ,\varphi \right)= & {} Y_{j^{\prime }j},\;\Lambda =\lambda +i\left( 2j-1\right) ,\;\Lambda ^{\prime }=\lambda ^{\prime }+i\left( 1-2j^{\prime }\right) , \nonumber \\ J_{\Lambda ^{\prime }\Lambda }^{--}\left( \rho ,\varphi \right)= & {} {\tilde{Y}} _{j^{\prime }j},\;\Lambda =\lambda +i\left( 1-2j\right) ,\;\Lambda ^{\prime }=\lambda ^{\prime }+i\left( 1-2j^{\prime }\right) . \end{aligned}$$

(B10)

Calculating the derivative of integral (B7) with respect to the hyperbolic angle $\varphi$, we find:

$$\begin{aligned} \frac{\partial J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho ,\varphi \right) }{\partial \varphi }=\, & {} W+\int _{-\infty }^{+\infty }iu_{x}f_{\Lambda ^{\prime }}^{\zeta ^{\prime }}(u-u_{x}/2)f_{\Lambda }^{\zeta }(u+u_{x}/2)e^{iu_{0}u}du\ , \\ W= \,& {} \frac{u_{0}}{2}\int _{-\infty }^{+\infty }\left[ f_{\Lambda ^{\prime }}^{\zeta ^{\prime }}(u-u_{x}/2)\left. \frac{\partial \,f_{\Lambda }^{\zeta }\left( z\right) }{\partial z}\right| _{z=u+u_{x}/2}\right. \\{} & {} \left. -\,\left. \frac{\partial \,f_{\Lambda ^{\prime }}^{\zeta ^{\prime }}\left( z\right) }{\partial z}\right| _{z=u-u_{x}/2}f_{\Lambda }^{\zeta }(u+u_{x}/2)\right] e^{iu_{0}u}du\ ,\ u_{x}=\frac{\partial u_{0}}{\partial \varphi }\ ,\ u_{0}=\frac{\partial u_{x}}{\partial \varphi }\ . \end{aligned}$$

Integrating by parts and neglecting boundary terms, we can transform W to the following form:

$$\begin{aligned} W= & {} \frac{i}{2}\int _{-\infty }^{+\infty }\left[ f_{\Lambda ^{\prime }}^{\zeta ^{\prime }}(u-u_{x}/2)\left. \frac{\partial \,^{2}f_{\Lambda }^{\zeta }\left( z\right) }{\partial z^{2}}\right| _{z=u+u_{x}/2}\right. \nonumber \\{} & {} \left. -\,\left. \frac{\partial ^{2}\,f_{\Lambda ^{\prime }}^{\zeta ^{\prime }}\left( z\right) }{\partial z^{2}}\right| _{z=u-u_{x}/2}f_{\Lambda }^{\zeta }(u+u_{x}/2)\right] e^{iu_{0}u}du\ . \end{aligned}$$

(B11)

Using Eq. (B8) in integral (B11), we find:

$$\begin{aligned} \frac{\partial J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho ,\varphi \right) }{\partial \varphi }=\frac{i}{2}\left( \Lambda ^{\prime }-\Lambda \right) J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho ,\varphi \right) . \end{aligned}$$

(B12)

Solutions of this equation are:

$$\begin{aligned} J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho ,\varphi \right) =e^{\frac{i}{2}\left( \Lambda ^{\prime }-\Lambda \right) \varphi }J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho ,0\right) . \end{aligned}$$

(B13)

We use the notation $J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho \right) =J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho ,0\right)$ in what follows. This function satisfies the differential Eq. (19)

$$\begin{aligned} \left[ \frac{d^{2}}{d\rho ^{2}}+\frac{1}{\rho }\frac{d}{d\rho }+\frac{\left( \Lambda -\Lambda ^{\prime }\right) ^{2}}{4\rho ^{2}}+\frac{\rho ^{2}}{4}- \frac{\Lambda +\Lambda ^{\prime }}{2}\right] J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho \right) =0\,. \end{aligned}$$

(B14)

This fact can be verified performing integrations by parts with account taken of Eq. (B8). The differential Eq. (B14) can be reduced to a confluent hypergeometric equation. Using two linearly independent solutions of such an equation, we find general solution of the differential Eq. (B14)

$$\begin{aligned}{} & {} J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho \right) =e^{-\eta /2}\left[ C_{1}\eta ^{i\beta }\Phi \left( \frac{i\Lambda }{2}+ \frac{1}{2},1+2i\beta ;\eta \right) +C_{2}\eta ^{-i\beta }\Phi \left( \frac{ i\Lambda ^{\prime }}{2}+\frac{1}{2},1-2i\beta ;\eta \right) \right] , \nonumber \\{} & {} \eta =-i\rho ^{2}/2,\;\beta =\left( \Lambda -\Lambda ^{\prime }\right) /4\;, \end{aligned}$$

(B15)

where the $C_{1}$ and $C_{2}$ are some undetermined coefficients, which must be fixed by appropriate boundary conditions, so that solution (B15) corresponds to the original integral (B7).

The confluent hypergeometric function $\Phi \left( a,c;\eta \right)$ is entire in $\eta$ and a, and is a meromorphic function of c. Note that $\Phi \left( a,c;0\right) =1$. WPCF’s are entire functions of $\Lambda$ and $\Lambda ^{\prime }$. One can see that the integrals $J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho \right)$ are entire functions of $\Lambda$ and $\Lambda ^{\prime }$ and meromorphic functions of $\Lambda -\Lambda ^{\prime }$. Then one can find a boundary condition $J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho \right)$ at $\rho \rightarrow 0$ for some convenient values of j and $j^{\prime }$. The remaining integrals $J_{\Lambda ^{\prime }\Lambda }^{\zeta ^{\prime }\zeta }\left( \rho \right)$ can be obtained extending domains of $\Lambda$ and $\Lambda ^{\prime }$ by an analytic continuation.

Let us start with ${\tilde{J}}_{0,0}(\rho )$ given by Eq. (68). This integral can be represented as a solution of Eq. (B14) where $\Lambda ^{\prime }=\lambda ^{\prime }+i$ and $\Lambda =\lambda +i$. The coefficients $C_{1}$ and $C_{2}$ in Eq. (B15) can be fixed by a comparison with the $\rho \rightarrow 0$ limit of integral (68). Let us first represent this integral as follows:

$$\begin{gathered} {\tilde{\mathcal{J}}}_{{0,0}} (\rho ) = F^{0} + F^{ + } + F^{ - } ,\;F^{ + } = \int_{0}^{\infty } {f^{ + } } \left( u \right)e^{{i\rho u}} du,\;F^{ - } = \int_{{ - \infty }}^{0} {f^{ - } } \left( u \right)e^{{i\rho u}} du, \hfill \\ F^{0} = \int_{0}^{\infty } f \left( u \right)\left[ {f\left( u \right) - f^{ + } \left( u \right)} \right]e^{{i\rho u}} du + \int_{{ - \infty }}^{0} f \left( u \right)\left[ {f\left( u \right) - f^{ - } \left( u \right)} \right]e^{{i\rho u}} du, \hfill \\ f\left( u \right) = D_{{ - \nu ^{\prime } }} [ - (1 + i)u]D_{{ - \nu }} [ - (1 + i)u], \hfill \\ \end{gathered}$$

(B16)

where $f^{\pm }\left( u\right) =\left. f\left( u\right) \right| _{u\rightarrow \pm \infty }$. It can be seen that function (B15) is reduced to the oscillations $C_{1}\eta ^{i\beta }+C_{2}\eta ^{-i\beta }$ as $\rho \rightarrow 0$. Then $\rho$-independent terms do not contribute to the integrals $F^{0}$ and $F^{\pm }$. Taking into account that $\lim _{\rho \rightarrow 0}F^{0}$ and $\lim _{\rho \rightarrow 0}F^{-}$ do not depend on $\rho ,$ one sees that the oscillation terms of $F^{+}$ are only essential. Using relations (8.2.(7)) and (8.4.(1)) from Ref. [44], one finds:

$$\begin{aligned} \mathcal {{\tilde{J}}}_{0,0}(\rho )= & {} \sqrt{\pi }e^{i\pi \left( \nu +\nu ^{\prime }-1\right) /4}\left[ e^{i\pi \nu /2}\frac{\Gamma \left( \nu -\nu ^{\prime }\right) }{\Gamma \left( \nu \right) }\left( \frac{\rho }{\sqrt{2}} \right) ^{\nu ^{\prime }-\nu }\right. \nonumber \\{} & {} \left. +e^{i\pi \nu ^{\prime }/2}\frac{\Gamma \left( \nu ^{\prime }-\nu \right) }{\Gamma \left( \nu ^{\prime }\right) }\left( \frac{\rho }{\sqrt{2}} \right) ^{\nu -\nu ^{\prime }}\;\textrm{as}\;\rho \rightarrow 0\right] . \end{aligned}$$

(B17)

Comparing Eqs. (B15) and (B17), we obtain:

$$\begin{aligned} C_{1}=\sqrt{\pi }e^{i\pi \left( \nu +\nu ^{\prime }-1/2\right) /2}\frac{ \Gamma \left( \nu ^{\prime }-\nu \right) }{\Gamma \left( \nu ^{\prime }\right) },\;C_{2}=\sqrt{\pi }e^{i\pi \left( \nu +\nu ^{\prime }-1/2\right) /2}\frac{\Gamma \left( \nu -\nu ^{\prime }\right) }{\Gamma \left( \nu \right) }. \end{aligned}$$

(B18)

Using relation (6.5.(7)) from Ref. [38], one can represent function given by Eqs. (B15) and (B18) as

$$\begin{aligned} \mathcal {{\tilde{J}}}_{0,0}(\rho )=\sqrt{\pi }e^{i\pi \left( \nu +\nu ^{\prime }-1/2\right) /2}e^{-\eta /2}\eta ^{\left( \nu -\nu ^{\prime }\right) /2}\Psi \left( \nu ,1+\nu -\nu ^{\prime };\eta \right) , \end{aligned}$$

(B19)

where $\Psi \left( \nu ,1+\nu -\nu ^{\prime };\eta \right)$ is the confluent hypergeometric function,

$$\begin{aligned} \Psi \left( \nu ,1+\nu -\nu ^{\prime };\eta \right)= & {} \frac{\Gamma \left( \nu ^{\prime }-\nu \right) }{\Gamma \left( \nu ^{\prime }\right) }\Phi \left( \nu ,1+\nu -\nu ^{\prime };\eta \right) \nonumber \\{} & {} +\frac{\Gamma \left( \nu -\nu ^{\prime }\right) }{\Gamma \left( \nu \right) }\eta ^{\nu ^{\prime }-\nu }\Phi \left( \nu ^{\prime },1+\nu ^{\prime }-\nu ;\eta \right) . \end{aligned}$$

(B20)

Using transformation $\nu \rightarrow \nu +j$ and $\nu ^{\prime }\rightarrow \nu ^{\prime }+j^{\prime }$ in Eq. (B19), one obtains the final form (70) for integral (68).

The integral $\mathcal {J}_{j^{\prime },j}(\rho )$ given by Eq. (67) can be represented as the solution of Eq. (B14) where $\Lambda ^{\prime }=\lambda ^{\prime }+i\left( 1-2j^{\prime }\right)$ and $\Lambda =\lambda +i\left( 2j-1\right)$. Using relation (8.2.(6)) from Ref. [44], we transform one of the WPCF’s in Eq. (67) to obtain convenient representations:

$$\begin{aligned}{} & {} \mathcal {J}_{j^{\prime },j}(\rho )=\frac{\Gamma \left( \nu -j+1\right) }{ \sqrt{2\pi }}\left[ e^{i\pi \left( \nu -j\right) /2}\mathcal {{\tilde{J}}} _{j^{\prime },1-j}(\rho )+e^{-i\pi \left( \nu -j\right) /2}\mathcal {J} _{j^{\prime },1-j}^{\prime }(\rho )\right] \ , \end{aligned}$$

(B21)

$$\begin{aligned}{} & {} \mathcal {J}_{j^{\prime },1-j}^{\prime }(\rho )=\int _{-\infty }^{\infty }D_{-\nu ^{\prime }-j^{\prime }}[-(1+i)u]D_{-\nu +j-1}[(1+i)u]e^{i\rho u}du\ , \end{aligned}$$

(B22)

where $\mathcal {{\tilde{J}}}_{j^{\prime },1-j}(\rho )$ is given by Eq. (70). The integral $\mathcal {J}_{j^{\prime },1-j}^{\prime }(\rho )$ is represented by function (B15) where some coefficients $C_{1}^{\prime }$ and $C_{2}^{\prime }$ can be fixed by the comparison with $\rho \rightarrow 0$ limit of the integral $\mathcal {J}_{j^{\prime },1-j}^{\prime }(\rho )$.

Let us start with $\mathcal {J}_{0,0}^{\prime }(\rho )$, where $\Lambda ^{\prime }=\lambda ^{\prime }+i$ and $\Lambda =\lambda +i$. In this case, it can be seen that function (B15) takes the form $C_{1}^{\prime }\eta ^{i\beta }+C_{2}^{\prime }\eta ^{-i\beta }$ as $\rho \rightarrow 0$. Hence all $\rho$-independent terms of $\mathcal {J}_{0,0}^{\prime }(\rho )$, given by Eq. (B22), can be ignored at $\rho \rightarrow 0$ limit and only the oscillation terms of following integrals

$$\begin{gathered} G^{ + } = \int_{0}^{\infty } {g^{ + } } \left( u \right)e^{{i\rho u}} du,\;F^{ - } = \int_{{ - \infty }}^{0} {g^{ - } } \left( u \right)e^{{i\rho u}} du,\; \hfill \\g^{ \pm } \left( u \right) = \left. {g\left( u \right)} \right|_{{u \to \pm \infty }} ,\; g\left( u \right) = D_{{ - \nu ^{\prime } }} [ - (1 + i)u]D_{{ - \nu }} [(1 + i)u] \hfill \\ \end{gathered}$$

(B23)

are essential. Using relations (8.2.(7)) and (8.4.(1)) from Ref. [44] , one finds:

$$\begin{aligned} \mathcal {J}_{0,0}^{\prime }(\rho )= & {} \sqrt{\pi }e^{i\pi \left( \nu ^{\prime }-\nu -1/2\right) /2}\left[ e^{i\pi \left( \nu -\nu ^{\prime }\right) /4} \frac{\Gamma \left( \nu -\nu ^{\prime }\right) }{\Gamma \left( \nu \right) } \left( \frac{\rho }{\sqrt{2}}\right) ^{\nu ^{\prime }-\nu }\right. \nonumber \\{} & {} \left. +e^{-i\pi \left( \nu -\nu ^{\prime }\right) /4}\frac{\Gamma \left( \nu ^{\prime }-\nu \right) }{\Gamma \left( \nu ^{\prime }\right) }\left( \frac{\rho }{\sqrt{2}}\right) ^{\nu -\nu ^{\prime }}\;\textrm{as}\;\rho \rightarrow 0\right] . \end{aligned}$$

(B24)

Comparing Eqs. (B15) and (B24) we obtain:

$$\begin{aligned} C_{1}^{\prime }=\sqrt{\pi }e^{i\pi \left( \nu ^{\prime }-\nu -1/2\right) /2} \frac{\Gamma \left( \nu ^{\prime }-\nu \right) }{\Gamma \left( \nu ^{\prime }\right) },\;C_{2}^{\prime }=\sqrt{\pi }e^{i\pi \left( \nu ^{\prime }-\nu -1/2\right) /2}\frac{\Gamma \left( \nu -\nu ^{\prime }\right) }{\Gamma \left( \nu \right) }. \end{aligned}$$

(B25)

Using relation (6.5.(7)) from Ref. [38], the function given by Eqs. (B15) and (B25) can be represented as:

$$\begin{aligned} \mathcal {J}_{0,0}^{\prime }(\rho )=\sqrt{\pi }e^{i\pi \left( \nu ^{\prime }-\nu -1/2\right) /2}e^{-\eta /2}\eta ^{\left( \nu -\nu ^{\prime }\right) /2}\Psi \left( \nu ,1+\nu -\nu ^{\prime };\eta \right) . \end{aligned}$$

(B26)

Using the transformations $\nu \rightarrow \nu +1-j$ and $\nu ^{\prime }\rightarrow \nu ^{\prime }+j^{\prime }$ in Eq. (B26), we obtain the following representation for integral (B22):

$$\begin{aligned} \mathcal {J}_{j^{\prime },1-j}^{\prime }(\rho )=e^{-i\pi \left( \nu -\nu ^{\prime }+1-j-j^{\prime }\right) /2}I_{j^{\prime },1-j}(\rho ), \end{aligned}$$

(B27)

where $I_{j^{\prime },j}(\rho )$ is given by Eq. (71). Substituting representations (70) and (B27) into Eq. (B21) we find the final form (69).