%Title: Relativistic Kinetic Vertex in Positronium

%Authors: S.M. Zebarjad, M. Haghighat

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\begin{document}

\noindent
\begin{titlepage}
\vspace{12pt}
\begin{center}
\begin{large}

             {\bf Relativistic Kinetic Vertex in Positronium \\}

\end{large}
\vspace{22pt} {\bf  S.M. Zebarjad $^{a,c}$ \footnote{E-mail :
zebarjad@physics.susc.ac.ir} and\,\,} {\bf M. Haghighat$^{b,c}$
\footnote{E-mail : mansour@cc.iut.ac.ir}}

\vspace{12pt} {\it
\vspace{8pt}
$^a$ Physics Department,  Shiraz University, Shiraz 71454,  Iran, \\
$^b$ Physics Department,  Isfahan University of Technology, Isfahan 84154,  Iran, \\
$^c$Institute for Studies in Theoretical Physics and Mathematics\\
             (IPM), Tehran 19395, Iran.\\

\vspace{0.3cm}}
\end{center}

\vspace{2cm} \abstract{We present Spinless Salpeter wave-function
instead of Schrodinger wave-function
 in positronium to show that this make NRQED calculation easier. we also
discuss that
 the singularity of the wave-function at the origin is necessary
to cancel the ultraviolet divergence in NRQED calculations. }
\end{titlepage}
\null

%\parskip 15pt \baselineskip =20pt
\section{Introduction}
A modern method to calculate a non-relativistic bound state
problem is based on the Effective Field Theory (NRQED) which was
proposed by Caswell and Lapage\cite{caswell}. This is the
advantage of this method that provides a set of systematic rules
(power-counting) that allows  an easy identification of all terms
that contribute to a certain order in the bound state
calculations.

In NRQED calculation, Relativistic Kinetic Vertex needs more
attention respect to the other interaction\cite{hfs6pra}. In this
paper, we consider this vertex in unperturbed part of Hamiltonian
as well as the Coulomb interaction to obtain positronium energy
correction. The program of this paper is as follows: In Sec. 2 we
present the Spinless Salpeter equation with Coulomb potential. In
Sec. 3 and 4  we describe the contribution of Relativistic Vertex
Correction to the energy spectrum of positronium at the order of
$\alpha^4$ and $\alpha^6$.

\section{Spinless Salpeter Equation}
The equation for two fermions which is justified from field theory
is the Bethe-Salpeter equation but because of certain
difficulties, several other equations incorporating relativistic
kinematics have also been used\cite{lich}. An approach which seems
promising is the Spinless Salpeter equation to obtain  spectrum
of a two-body system. We now consider the Spinless Salpeter
Equation\cite{sal}  for positronium as follows:

\begin{equation}
\label{SS}\bigg\{ {\nabla}^2+\frac{(E-({\bf r}))^2}{4}-(m+S({\bf
r}))^2 \bigg\} \Psi( {\bf {r})=0 }
\end{equation}
where m is the mass of the particle, $S({\bf r})$ is the scalar
potential and $V({\bf {r})}$ is the time component of a 4-vector
potential. In non-relativistic limit, Eq. (\ref{SS}) leads to a
Schrodinger equation with an effective potential as $S+V$. In fact
this equation gives all relativistic kinetic corrections to the
Schrodinger equation with such  effective potential. For a pure
vector potential, in the Coulomb case, we have

\begin{equation}
V({\bf r})=\frac{-\alpha}{r}\,\,\,\,\,\,\,\,\,\,S({\bf r})=0 ,
\end{equation}
One should note, Eq. (\ref{SS}) is similar to the generalized
Klein-Gordon(GKG) Equation\cite{lich} with a vector and a scalar
potential. By substitution , $\Psi=\frac{U}{r}$ the radial
equation results in :

\begin{equation}
\label{SS1}\bigg\{{\frac{d^2}{{d\rho^2}}}+ {\frac{2}{\rho}}-\frac{%
(\ell+\kappa)(\ell+\kappa+1)}{\rho^2} -\frac{1}{(n+\kappa)^2}\bigg\}%
U(\rho)=0,
\end{equation}
where $$\kappa=-(\ell-1/2)+\sqrt{(\ell+1/2)^2-\alpha^2/4}\,\,\,\,
\&\,\,\,\, \rho=E\alpha/4r$$ The eigenvalues and eigenfunctions of
Eq.(\ref{SS1}) are respectively\cite{fac}\cite{grei}:

\begin{eqnarray}
E_{n,\ell}=2m\bigg[1+\frac{\alpha^2}{4(n+\kappa)^2}\bigg]^{-1/2}  \label{energy} \\
U_{n+\kappa}^{\kappa}=N\bigg(\frac{2}{n+\kappa}\bigg)^{\frac{1}{2}+\kappa+n}
\Gamma^{-\frac{1}{3}}(2n+2\kappa+1)\,\,\,\rho^{n+\kappa}\exp{(\frac{-\rho}{n+\kappa}})
\end{eqnarray}
where $N$ is the normalization factor.\\ We can now expand
$\frac{\Psi_{GKG}}{\Psi_{Sch}}$ in terms of $\alpha$ (when $n=1$,
$\ell=0$):

\begin{equation}
\Psi_{GKG}({\bf r})=\Psi_{Sch}({\bf r})\bigg\{1+\alpha^2\,\bigg(
-\frac{1}{4}\gamma+\frac{5}{8}-\frac{1}{4}\ln{(mr \alpha)}\bigg)\dots%
\bigg\} \label{GKG}
\end{equation}
where $\Psi_{Sch}({\bf r})$ is the ground state wave-function:

\begin{equation}
\Psi_{Sch}({\bf r})=\frac{1}{\sqrt{\pi}} \bigg(\frac{m\alpha}{2}
\bigg)^{3/2}\exp^{-m \alpha /2}. \label{schr}
\end{equation}
 The Fourier transformation of Eq. (\ref{GKG}) is:

\begin{eqnarray}
\psi_{GKG}({\bf p})&=&\psi({\bf p})_{Sch}+\alpha^2  \varphi({\bf
p}), \label{KG2}
\end{eqnarray}
where $ \psi_{Sch}({\bf p})$ and $\varphi ({\bf p})$ are:

\begin{eqnarray}
%\widetilde
\psi_{Sch}({\bf p})&=&\frac{8\pi^{1/2}(m\alpha/2)^{5/2}}{[p^2+(m\alpha/2)^2]^2},\\
\varphi({\bf p})&=&\nonumber\\&-&\frac{(m\alpha/2)^{3/2}
\sqrt{\pi}}{((m\alpha/2)^{2}+p^2)^2p}\bigg\{\arctan(\frac{2p}{m\alpha})
\bigg(p^2-(m\alpha/2)^2\bigg) \nonumber \\
&      &\quad  \quad \quad \quad \quad \quad\quad \quad
+p(m\alpha/2)\ln\bigg(\frac{(m\alpha/2)^{2}+p^2}{m^2\alpha^2}\bigg)\nonumber \\
&      &\quad  \quad \quad \quad \quad \quad\quad \quad
+3p(m\alpha/2)\bigg\}
\end{eqnarray}

We are now ready to calculate positronium energy corrections at the order of $%
\alpha^4$ and  $\alpha^6$, using GKG wavefunction, Eq.
(\ref{GKG}).

\section{$\alpha^4$ Positronium Energy Corrections}
The power-counting of NRQED\cite{patrickPC} specifies all the
diagrams contributing to the energy shift at the order of
$\alpha^4$. These diagrams are all shown in Fig. (\ref{a4}) which
can be calculated \cite{mythesis} to
give the well-known positronium energy correction at the order of $\alpha^4$%
\cite{Itzykson}.

\begin{figure}[h]
\centerline{\epsfxsize=6in\epsffile{snrqe1.eps}} \caption{ The
whole bound state diagrams contributing to positronium energy
corrections at the order of  $\alpha^4$.  } \label{a4}
\end{figure}

  Contribution of Relativistic Kinetic Corrections at this order are shown in
Figs. (\ref{a4}(i,j)) which reads:

\begin{eqnarray}
\Delta E_{i}+\Delta
E_{j}&=&2\int\frac{d^3pd^3p'}{(2\pi)^6}\psi^*({\bf p'}) \bigg[
\frac{-p^4(2\pi)^3  \delta({\bf p}-{\bf
p'})}{8m^3}\bigg]\psi(\bf{p}) \nonumber \\
&=&\frac{-m\alpha^4}{16n^3(\ell+1/2)}+\frac{3m \alpha^4}{64n^4}
\label{ra4}
\end{eqnarray}

To obtain positronium energy correction at the order of $\alpha^4$
using GKG equation, we should omit Figs. (\ref{a4}(i,j)) and
calculate the remaining diagrams in Fig. (\ref{a4}) using GKG
wavefunction. That is obviously leads to the previous result at
the order of $\alpha^4$ and also extra pieces which start at
higher order. These terms are irrelevant to the order of our
interest. Meanwhile the result of Eq. (\ref{ra4}) can be obtained
by expanding the Eq. (\ref{energy}) in terms of  $\alpha$.

\section{$\alpha^6$ Positronium Energy Correction}
The full calculation of the positronium energy correction at the
order of $\alpha^6$, using NRQED has not yet been done. In this
paper, we focus on the
positronium HFS at the order of $\alpha^6$ coming from one-photon annihilation%
\cite{hfs6}\cite{hfs6pra}. First we concentrate on the NRQED
method. Since the single photon  annihilation of electron-positron
occurs  in $S=1$ state, one should calculate the diagrams which
contain spin-1 Four-Fermion Vertex. All the NRQED diagrams which
contribute to HFS at the order of $\alpha^6$ can be identified
using the NRQED power-counting rules\cite{patrickPC}. These are
shown in Figs.(\ref{a6a}),(\ref{a6b}) which are completely
calculated in \cite{mythesis} and \cite{hfs6pra}. Diagrams
(\ref{a6a}(k,l,m)) which contain Relativistic Vertex Corrections
result in\cite{mythesis}:

\begin{eqnarray}
Fig(\ref{a6a},k)+Fig(\ref{a6a},l)+Fig(\ref{a6a},m)=\frac{m\alpha
^6}8\ln \bigg( \frac \Lambda {m\alpha }\bigg)+\frac 1{32}m\alpha
^6+\frac {m\alpha^5}{ 4\pi} \frac{\Lambda} m \label{5}
\end{eqnarray}

\begin{figure}
\centerline{\epsfxsize=6in\epsffile{thesis9.eps}} \caption{All
the bound state diagrams which contribute to $\alpha^6$ except
the diagrams with the Double Annihilation Interaction. }
\label{a6a}
\end{figure}

\begin{figure}
\centerline{\epsfxsize=6in\epsffile{thesis7.eps}} \caption{The
whole Double Annihilation bound state diagrams.} \label{a6b}
\end{figure}
Since our goal is to obtain the final result using GKG
wave-function, we should omit Figs.(\ref{a6a}(k,l,m)) and
calculate the remaining diagrams in Fig(\ref{a6a}) and
Fig.(\ref{a6b}) using the GKG wave function. Straightforward
calculations shows that we  have the previous result for these
diagrams and also extra pieces at higher order than $\alpha^6$.
At first glance, it seems that there is no way to get the value
of Figs. (\ref{a6a}k,l,m) which we have omitted, but if we
consider the diagrams which contributed to HFS at the order of
$\alpha^4$, using the Schrodinger wave-function, we get some
pieces at the higher order. That is basically due to the fact
that we  should replace $\psi_{Sch}$ with $\psi_{GKG}$ in Fig.
(\ref{a4},k) and also using the $p/m$ expansion of relativistic
propagator.  For example, in case of Fig. (\ref{a4},k), one has
the left hand side of Fig. (\ref{KG1}).

The only diagram at the order of $\alpha^4$ coming from one photon
annihilation is shown in Fig. (\ref{a4}(k)).   It is easy to show
that  Fig. (\ref{KG1}(a)) leads to the previous result at the
order $\alpha^4$, while the  Figs. (\ref{KG1}(c,e,f)) contributes
to the higher order than $\alpha^6$. The only remaining diagrams
relevant to our calculation are the Figs. (\ref{KG1}(b,d)):

\begin{eqnarray}
Fig(4(b))&=&\frac{4\pi \alpha }{m^2}\int
\frac{d^3p^{\prime}}{(2\pi )^3}
\psi_{Sch} ({\bf p^{\prime }})\int \frac{d^3p}{(2\pi )^3}\alpha ^2{\varphi }(%
{\bf p})=\frac{m\alpha ^6}{8}\ln \bigg(\frac {\Lambda} {m\alpha
}\bigg)+\frac {5}{32}m\alpha ^6 \nonumber\\
Fig(4(d))&=&-\frac 1{8}m\alpha ^6+\frac {m\alpha^5}{ 4\pi}
\frac{\Lambda} m \label{2}
\end{eqnarray}

The sum of Fig(4(b)) and Fig(4(d)) is just equal to Eq.(\ref{5}).

\section*{Summary}
In this paper, we have shown Eq. (\ref{SS}) correctly predicts
energy spectrum of a non-relativistic two body system such as
positronium. In this way we ignore the Relativistic Vertex
Correction in bound state and this make calculation much easier.
On the other hand, although the GKG wave-function is singular at
the origin but this is a crucial need to cancel UV divergent in
bound state system.

\section*{Acknowledgement}
The authors gratefully acknowledge the financial support of Shiraz
University research council and  IPM.

\begin{figure}
\centerline{\epsfxsize=5in\epsffile{mansour14.eps}} \caption{}
\label{KG1}
\end{figure}
\newpage

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\raggedright

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\end{thebibliography}



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