{A general formula for bound-continuous transition form factors is derived. It is shown that these form factors can be represented in the form of finite sum of terms with simple analytical structure.}
hep-ph0201037.tex
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\begin{center}
{\bf
ANALYTIC FORM FACTORS OF HYDROGENLIKE ATOMS FOR DISCRETE-CONTINUUM TRANSITIONS.
II. GENERAL CASE\\}
\vspace*{1cm}
O.Voskresenskaya$^{1,2,3}$\footnote{E-mail:
Olga.Voskresenskaja@mpi-hd.mpg.de; voskr@cv.jinr.dubna.su}\\
\vspace*{.25cm}
{\it$^1$ Institut f\"ur Theoretische Physik der Universit\"at,
69120 Heidelberg, Germany\\
\it$^2$ Max-Planck-Institut f\"ur Kernphysik, Postfach 103980,
69029 Heidelberg, Germany\\
$^3$ Joint Institute for Nuclear Research,
141980 Dubna, Moscow Region, Russia\\}
\end{center}

\vspace*{.25cm}
\begin{abstract}
{\small A general formula for bound-continuous transition form factors
is derived. It is shown that these form factors can be represented
in the form of finite sum of terms with simple analytical structure.}

\end{abstract}

\vspace*{.5cm}
In the previous paper \cite{voskr1} it have been shown that the form factors
of transitions from $nS\; (n00)$ - states of hydrogenlike atoms \cite{faust}
to the state of continuous spectra with definite value of relative momenta
$\vec p$ may be expressed in the terms of the classical polynomials
in a rather simple way.
Below this result is generalized for the case of  transition
from arbitrary initial bound states.

The transition form factors are defined as follows:
\begin{equation}
S_{fi}(\vec q)=\int \psi_f^{\ast}(\vec r)e^{i\vec q\vec r}
\psi_i(\vec r)d^3r
\label{e1}\, ,
\end{equation}
Here, $\psi_{i(f)}$ are the wave functions of initial (final) states.

According to \cite{sommer} (see also \cite{land}), the final state wave
function must be choose in the form
\begin{equation}
\psi_i(\vec r)\equiv\psi_{n00}(\vec r)
\label{e2}\, .
\end{equation}
For arbitrary initial bound state
\begin{equation}
\psi_i(\vec r)=\psi_{nlm}(\vec r)=
Y_{lm}\left(\frac{\vec r}{r}\right)R_{nl}(r)\,,
\label{351}
\end{equation}
\begin{eqnarray}
R_{nl}(r)&=&\frac{2\omega^{\frac{3}{2}}}{\Gamma(2l+2)}
\left[\frac{\Gamma(n+l+1)}{n\Gamma(n-l)}\right]^{\frac{1}{2}}
\cdot(2\omega r)^l
\cdot\Phi(-n+l+1,2l+2;2\omega r)\cdot \exp(-\omega r)\nonumber
\end{eqnarray}
$$~~~~=2\omega^{\frac{3}{2}}
\left[\frac{\Gamma(n-l)}{n\Gamma(n+l+1)}\right]^{\frac{1}{2}}
\cdot(2\omega r)^l
\cdot L^{2l+1}_{n-l-1}(2\omega r)\cdot \exp(-\omega r)\,,$$
$$
\omega=\mu\alpha/n\,,
$$
where $L^{2l+1}_{n-l-1}$ are the associated Laguerre polynomials.

\newpage
Making use of the recurrence relations \cite{ryzh}
\begin{equation}
L_k^{\lambda+1}(x)=\frac{1}{x}\left[(k+\lambda+1)L_{k-1}^{\lambda}(x)
-(k+1)L_{k}^{\lambda}(x)\right]
\label{e4}
\end{equation}
and the representation of the Laguerre polynomials
in terms of the generating function
\begin{equation}
L_k^{\lambda}(x)=\Delta_z^{(k)}\left[(1-z)^{-(\lambda+1)}\exp
\left(\frac{xz}{z-1}\right)\right]
\label{e5}\, ,
\end{equation}
where operator $\Delta_z^{(k)}$ is defined as follows
\begin{equation}
\Delta_z^{(k)}\left[f(z)\right]=\frac{1}{k!}
\left.\left(\frac{d^k}{dz^k}f(z)\right)\right\vert_{z=0}
\label{e6}\, ,
\end{equation}
let us rewrite the radial part of initial state wave
function in the form
\begin{equation}
R_{nl}=\frac{\omega^{\frac{1}{2}}}{r}
\left[\frac{\Gamma(n-l)}{n\Gamma(n+l+1)}\right]^{\frac{1}{2}}
\cdot(2\omega r)^l
\cdot\left[(n+l)\Delta_z^{(n-l-2)}-(n-l)\Delta_z^{(n-l-1)}\right]
\label{e7}
\end{equation}
$$\times\left[(1-z)^{-(l+1)}\exp\left(-\omega(z)r\right)\right]\,,$$
\begin{equation}
\omega(z)=\omega\cdot(1+z)(1-z)^{-1}
\label{e11}
\end{equation}
more convenient for the further calculations.

Then it is not difficult to see that transition form factors
(\ref{e1}) may be represent as a linear combination of the quantities
\begin{equation}
I_{lm}^{j}=\Delta_z^{(j)}\left[(1-z)^{-(2l+1)}
J_{lm}(\vec q,\vec p,z)\right]
\label{eq:352}\, ,
\end{equation}
\begin{eqnarray}
J_{lm}(\vec q,\vec p,z)&=&\int \frac{d^3r}{r}Y_{lm}
\left(\frac{\vec r}{r}\right)
\Phi\left[i\xi,1;i(pr+\vec p\vec r)\right]
\label{eq:353}
\end{eqnarray}
$$\times\exp[i(\vec q-\vec p)\vec r-\omega(z)\cdot r]\cdot(2\omega r)^l\cdot
\exp\left[-\omega(z)r\right]\,,$$
$$j=n-l-2,\;n-l-1\,.$$

In order to calculate (\ref{eq:353}), it is useful to represent the
hypergeometrical function in (\ref{351}) in the form
\begin{eqnarray}
\Phi\left[i\xi,1;i(pr+\vec p\vec r)\right]&=&-\frac{1}{2\pi i}
\oint\limits_{C}^{} (-t)^{i\xi-1}(1-t)^{-i\xi}
\cdot\exp[i\cdot t(pr+\vec p\vec r)]dt\,.
\label{eq:354}
\end{eqnarray}

Using the following relations
\begin{equation}
\label{eq:355}
\exp(i\vec\tau\vec r)=4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{l}
i^{l}Y_{lm}\left(\frac{\vec \tau}{\tau}\right)
Y^{\ast}_{lm}\left(\frac{\vec r}{r}\right)j_{l}(\tau r)\,,
\end{equation}
\begin{equation}
\label{eq:356}
j_{l}(x)=\sqrt{\frac{\pi}{2x}}J_{l+\frac{1}{2}}(x)\,,
\end{equation}
\begin{equation}
\label{eq:357}
\int\limits_{0}^{\infty}r^{l+\frac{1}{2}}J_{l+\frac{1}{2}}(\tau r)
e^{-\bar\omega\cdot r}dr=
\frac{(2\tau)^{l+\frac{1}{2}}\Gamma(l+1)}
{\sqrt{\pi}(\tau^2+\bar \omega^2)^{l+1}}\,,
\end{equation}
where
\begin{equation}
\label{eq:358}
\vec\tau=\vec q-\vec p(1-t),\quad \bar\omega=\omega(z)-ip\cdot t\,,
\end{equation}
after simple calculations we find
\begin{eqnarray}
J_{lm}(\vec q,\vec p,z)&=&-\frac{\Gamma(l+1)}{2\pi i}
\oint\limits_{C}^{ }dt(-t)^{i\xi-1}(1-t)^{-i\xi}
\cdot\frac{4\pi(4i\omega)^lY_{lm}\left(\vec \tau/\tau\right)\tau^l}
{(\tau^2+\bar \omega^2)^{l+1}}\,.
\label{eq:359}
\end{eqnarray}

It is easy to check that
$$\tau^2+\bar \omega^2=a(1-t)+c\cdot t\,,$$
\begin{equation}
\label{eq:362}
a=\omega^2(z)+\vec\Delta^2\,,\quad c=\left[\omega(z)-ip\right]^2+q^2\,.
\end{equation}

Further, according to \cite{war}, we get
\begin{eqnarray}
\label{eq:360}
Y_{lm}\left(\frac{\vec \tau}{\tau}\right)\tau^l
=\sum_{l_1=0}^{l}q^{l_1}(-p)^{l-l_1}
\end{eqnarray}
$$\times\left[\frac{4\pi\Gamma(2l+2)}
{\Gamma(2l_1+2)\Gamma(2l-2l_1+2)}\right]^{\frac{1}{2}}
\cdot (1-t)^{l-l_1}\left[Y_{l_1}\left(\frac{\vec q}{q}\right)\otimes
Y_{l-l_1}\left(\frac{\vec p}{p}\right)\right]_{lm}\,,$$
\begin{eqnarray}
\label{eq:361}
\left[Y_{l_1}\left(\frac{\vec q}{q}\right)\otimes
Y_{l-l_1}\left(\frac{\vec p}{p}\right)\right]_{lm}
=\sum_{m_1+m_2=m}C^{lm}_{l_1m_1(l-l_1)m_2}
\cdot Y_{l_1m_1}\left(\frac{\vec q}{q}\right)\cdot
Y_{(l-l_1)m_2}\left(\frac{\vec p}{p}\right)\,.
\end{eqnarray}
Taking into account (\ref{eq:362}) and (\ref{eq:360}), it is easy
to see that (\ref{eq:359}) is the superposition of the quantities
\begin{equation}
-\frac{1}{2\pi i}\oint\limits_{C}^{} \frac{t^{i\xi-1}(1-t)^{-i\xi+l-l_1}}
{[a(1-t)+ct]^{l+1}}
\end{equation}
\begin{eqnarray}
&&=a^{-(l+1)}\frac{\Gamma(1-i\xi+l-l_1)}{\Gamma(1-i\xi)}
F\left(i\xi,l+1;l-l_1+1;1-c/a\right)\nonumber\\
&&=a^{i\xi-l-1}c^{-i\xi}\frac{\Gamma(1-i\xi+l-l_1)}
{\Gamma(1-i\xi)}
F\left(i\xi,-l_1;l-l_1+1;1-a/c\right)\nonumber\\
&&=a^{i\xi-l-1}c^{-i\xi}\frac{\Gamma(l-l_1+1)\Gamma(l+1-i\xi)}
{\Gamma(l+1)\Gamma(1-i\xi)}F(i\xi,-l_1;i\xi-l;a/c)\nonumber\\
&&=\sum_{s=0}^{l_1}(-1)^{l-s}
\frac{\Gamma(i\xi+s)\Gamma(l_1+1)}
{\Gamma(l_1-s+1)\Gamma(i\xi-l+s)\Gamma(s+1)\Gamma(l+1)}
a^{i\xi+s-l-1}c^{-s-i\xi}\nonumber\\
&&=(1-z)^{2l+2}\sum_{s=0}^{l_1}(-1)^{l-s}
\frac{\Gamma(i\xi+s)\Gamma(l_1+1)}
{\Gamma(l_1-s+1)\Gamma(i\xi-l+s)\Gamma(s+1)\Gamma(l+1)}\nonumber
\label{eq:363}
\end{eqnarray}
$$\times D_1^{i\xi+s-l-1}D_2^{-s-i\xi}\,,$$
where $D_{1,2}$ are defined as follows:
%\newpage
\begin{equation}
D_1=(1+z^2)(\omega^2+\vec\Delta^2)-2z(\vec\Delta^2-\omega^2)\,,
\label{e3}
\end{equation}
$$D_2=(\omega-ip)^2+q^2-2z(q^2-p^2-\omega^2)+z^2[(\omega+ip)^2+q^2]\,.$$

The further calculations are the same as in \cite{voskr1}.

Omitting the simple but cumbersome algebra, let us present the final
expression for transition form factors:
\begin{eqnarray}
S_{\vec p,nlm}(\vec q)&=&4\pi\cdot 2^{2l}i^l\omega^{l+\frac{1}{2}}
\left[\frac{\Gamma(n-l)}{n\Gamma(n+l+1)}\right]^{\frac{1}{2}}\nonumber\\
\label{eq:364}
\end{eqnarray}
$$
\times\sum_{s=0}^{l}G_{lms}(\vec p,\vec q)H_{nls}(\vec p,\vec q)
\cdot(\omega^2+\Delta ^2)^{i\xi+s-l-1}[(\omega-ip)^2+q^2]^{-s-i\xi}\,;
$$
\begin{eqnarray}
\label{eq:365}
G_{lms}(\vec p,\vec q)&=&(-1)^{l-s}
\frac{\Gamma(i\xi+s)}{\Gamma(i\xi-l+s)\Gamma(s+1)}
\end{eqnarray}
$$\times\sum_{l_1=s}^{l}
\left[\frac{4\pi\Gamma(2l+2)}{\Gamma(2l_1+2)\Gamma(2l-2l_1+2)}\right]^
{\frac{1}{2}}
\cdot\frac{\Gamma(l_1+1)}{\Gamma(l_1-s+1)}q^{l_1}(-p)^{l-l_1}$$
$$\times\left[Y_{l_1}\left(\frac{\vec q}{q}\right)\otimes
Y_{l-l_1}\left(\frac{\vec p}{p}\right)\right]_{lm}\,;$$
\begin{eqnarray}
\label{eq:366}
H_{nls}(\vec p,\vec q)&=&(n+l)F_{n_1ls}(\vec p,\vec q)-
(n-l)F_{n_2ls}(\vec p,\vec q)\,;
\end{eqnarray}
$$n_1=n-l-1\,,\quad n_2=n-l-2\,;$$
\begin{eqnarray}
\label{eq:367}
F_{n_{1(2)}ls}(\vec p,\vec q)&=&
\frac{\Gamma(l-s+\frac{1}{2}-i\xi)}{\Gamma(2l-2s+1-2i\xi)}
\sum_{k=0}^{n_{1(2)}}w^kC_k^{(i\xi+s)}(v)
\end{eqnarray}

$$\times\frac{\Gamma(n_{1(2)}-k+2l-2s+1-2i\xi)}
{\Gamma(n_{1(2)}-k+l-s+\frac{1}{2}-i\xi)}
\cdot P_{n_{1(2)}-k}^{(l-s-\frac{1}{2}-i\xi,l-s+\frac{1}{2}-i\xi)}(u)\,.$$\\

Thus, the form factors for transition from arbitrary bound states of
hydrogenlike atoms to the ``$\vec p-state$''
of continuous spectra are represented as the superposition of finite
number of terms with simple analytical structure and can be evaluated
numerically with arbitrary degree of accuracy.

Eqs. (\ref{eq:364})-(\ref{eq:367}) are the generalization of the
results of \cite{MASS33,voskr1}.
\vspace*{.5cm}
\begin{center}

{\large \bf Acknowledgments}

\end{center}

\vspace*{.5cm}


I would like to thank the Institute for Theoretical Physics
at Heidelberg University and the Max-Planck-Institut f\"ur Kernphysik,
where this work was carried out, for the hospitality.
I also thank Alexander Tarasov for stimulating discussions.
\newpage


\begin{thebibliography}\\

\bibitem{voskr1}  O.~Voskresenskaya, arXiv:hep-ph/0111216.

\bibitem{faust} R.N. Faustov, {\it Sov. J. Particles and Nuclei},
{\bf 3} (1972) 119;  R. Coombles et al., {\it Phys. Rev. Lett.},
{\bf 37} (1976) 249; S.H. Aronson et al., {\it Phys. Rev. Lett.}, {\bf 48}
(1982) 1078; J.Kapusta and A.Mocsy, arXiv:nucl-th/9812013.

\bibitem{sommer}  A.~Sommerfeld, {\it Ann. Phys.} {\bf 11} (1931)
257.

\bibitem{land} L.D. Landau and E.M. Lifshitz, {\it Quantum Mechanics}
(Nauka Publication, Moscow, 1974).

\bibitem{ryzh} I.S. Gradshtein and I.M. Ryzhik, {\it Tables of Integrals,
Series and Products} (Nauka Publication, Moscow, 1971);
{\it Handbook of Mathematical Functions}, Eds.
M.~Abramowitz and I.A. Stegun (National Bureau of Standards,
Applied Mathematics Series, 1964).

\bibitem{war} D.A. Varshalovich, A.H. Moskalev and V.K. Khersonsky,
{\it Quantum theory of angular momentum}
(Nauka Publication, Moscow, 1975); A.R. Edmonds, {\it Angular momentum
in quantum mechanics} (Prinston, 1957).

\bibitem{MASS33} H.S.W. Massey and C.B.O. Mohr, Proc. R. Soc. London,
{\bf A 140} (1933) 613; K. Omidvar, {\it Phys. Rev.}, {\bf 140} (1965) A25.

\end{thebibliography}






\end{document}

\bibitem{faust} R.N. Faustov, {\it Sov. J. Particles and Nuclei},
{\bf 3} (1972) 119;  R. Coombles et al., {\it Phys. Rev. Lett.},
{\bf 37} (1976) 249; S.H. Aronson et al., {\it Phys. Rev. Lett.}, {\bf 48}
(1982) 1078; J.Kapusta and A.Mocsy, nucl-th/9812013.

%\bibitem{Baym93} G.Baym, G.Friedman, R.Huges and B.Jacak,
{\it Phys. Rev.}, {\bf D48} (1993) R3957.

The author is grateful to Professor J\"org H\"uffner for invitation and
hospitality at the Institute for Theoretical Physics, Heidelberg University,
where this work was done, and to Alexander Tarasov for helpful discussions.







hep-ph0201037.extracted_bib
\begin{thebibliography}\\

\bibitem{voskr1}  O.~Voskresenskaya, arXiv:hep-ph/0111216.

\bibitem{faust} R.N. Faustov, {\it Sov. J. Particles and Nuclei},
{\bf 3} (1972) 119;  R. Coombles et al., {\it Phys. Rev. Lett.},
{\bf 37} (1976) 249; S.H. Aronson et al., {\it Phys. Rev. Lett.}, {\bf 48}
(1982) 1078; J.Kapusta and A.Mocsy, arXiv:nucl-th/9812013.

\bibitem{sommer}  A.~Sommerfeld, {\it Ann. Phys.} {\bf 11} (1931)
257.

\bibitem{land} L.D. Landau and E.M. Lifshitz, {\it Quantum Mechanics}
(Nauka Publication, Moscow, 1974).

\bibitem{ryzh} I.S. Gradshtein and I.M. Ryzhik, {\it Tables of Integrals,
Series and Products} (Nauka Publication, Moscow, 1971);
{\it Handbook of Mathematical Functions}, Eds.
M.~Abramowitz and I.A. Stegun (National Bureau of Standards,
Applied Mathematics Series, 1964).

\bibitem{war} D.A. Varshalovich, A.H. Moskalev and V.K. Khersonsky,
{\it Quantum theory of angular momentum}
(Nauka Publication, Moscow, 1975); A.R. Edmonds, {\it Angular momentum
in quantum mechanics} (Prinston, 1957).

\bibitem{MASS33} H.S.W. Massey and C.B.O. Mohr, Proc. R. Soc. London,
{\bf A 140} (1933) 613; K. Omidvar, {\it Phys. Rev.}, {\bf 140} (1965) A25.

\end{thebibliography}