%Title: e+e- -> pi+pi-e+e-: a potential background for sigma(e+e- -> pi+pi-) measurement via radiative return method
%Author: Henryk Czyz, Elzbieta Nowak
%hep-ph/0310335
\documentclass{appolb}
\usepackage{epsfig}
% epsfig package included for placing EPS figures in the text
%------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% BEGINNING OF TEXT %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
% \eqsec % uncomment this line to get equations numbered by (sec.num)
\title{$e^+e^- \to \pi^+\pi^- e^+e^-$ : a potential background \\
for $\sigma (e^+e^- \to \pi^+\pi^-)$ measurement \\
via radiative return method
\thanks{Presented by E. Nowak
at XXVII International Conference of Theoretical Physics,
`Matter To The Deepest', Ustro{\'n}, 15-21 September 2003, Poland.
Work supported in part by
EC 5-th Framework Program under contract
HPRN-CT-2002-00311 (EURIDICE network) and
Polish State Committee for Scientific Research
(KBN) under contract 2 P03B 017 24.
% you can use '\\' to break lines
}}
\author{Henryk Czy\.z and El\.zbieta Nowak
\address{Institute of Physics, University of Silesia,
PL-40007 Katowice, Poland.}
}
\maketitle
\vspace{-5 cm}
\hfill{\bf TTP03-30}
\vspace{+5 cm}
\begin{abstract}
A Monte Carlo generator (EKHARA) has been constructed to simulate the
reaction $e^+e^- \to \pi^+\pi^- e^+e^-$
based on initial and final state emission of a $e^+e^-$ pair
from $e^+e^-\to\pi^+\pi^-$ production diagram.
A detailed study of the process, as a potential background
for $\sigma (e^+e^- \to \pi^+\pi^-)$ measurement
via radiative return method, is presented for
$\Phi$- and $B$- factory energies.
\end{abstract}
\PACS{13.40.Ks,13.66.Bc}
\section{Introduction}
The radiative return method \cite{Binner:1999bt} (look \cite{cg:ustron}
for a short introduction) is a powerful tool in the
measurement of $\sigma (e^+e^- \to \ {\mathrm{hadrons}})$, crucial
for predictions of the hadronic contributions to $a_\mu$, the anomalous
magnetic moment of the muon, and to the running of the electromagnetic
coupling from its value at low energy up to $M_Z$ (for recent reviews look
\cite{Davier:2003,Nyffeler:2003,Jegetc.}).
Due to a complicated experimental setup,
the use of Monte Carlo (MC) event generators
\cite{Binner:1999bt,Czyz:2000wh,Rodrigo:2001kf,Czyz:2002np,Czyz:PH03},
which includes various radiative corrections \cite{Rodrigo:2001jr,Kuhn:2002xg}
is indispensable. Some more extensive analysis of that subject can be found
also in \cite{Kuhn:2001,Rodrigo:2001cc,Rodrigo:2002hk}.
The most important hadronic mode, i.e. $\pi^+\pi^-$, is currently measured
by KLOE \cite{Achim:radcor02,KLOE:2003,Juliet,SdiFalco:Ustron}, and BaBar
\cite{Blinov} by means of radiative return method. This measurement
can suffer from
a background from the process $e^+e^- \to \pi^+\pi^- e^+e^-$,
as suggested in \cite{Hoefer:2001mx}, for at least
two reasons: 1. At present KLOE measures only pions (+ missing momenta)
in the final state and for that particular measurement there is no
difference between photon(s) and pair production. 2. The $e^+e^-$
pair can escape detection, being lost outside a detector,
e.g. in the beam pipe,
or having energy below a detection threshold.
Again a Monte Carlo study is
unavoidable, if one likes to know the actual value of the pair production
contribution in a given experimental setup, as the analytical, completely
inclusive, calculations might lead to misleading results.
\section{Monte Carlo program EKHARA and its tests}
\begin{figure}[ht]
\begin{center}
%\hskip-0.5cm
\epsfig{file=1zm.eps,width=3.3cm,height=2.9cm}
\hskip+0.5cm
\epsfig{file=5zm.eps,width=3.2cm,height=2.9cm}
\hskip+0.3cm
\epsfig{file=8zm.eps,width=2.2cm,height=3cm}
\hskip+0.5cm
\epsfig{file=10zm.eps,width=2cm,height=3cm}
\end{center}
\caption{Diagrams contributing to the process
\(e^+(p_1)e^-(p_2) \to \pi^+(\pi_1)\pi^-(\pi_2)e^+(q_1)e^-(q_2)\):
initial state electron pair emission (a),
final state electron pair emission (b),
pion pair emission from t--channel Bhabha process (c) and
$\gamma^*\gamma^*$ pion pair production (d).
}
\label{f00}
\end{figure}
In Fig.\ref{f00} different types of diagrams
contributing to process $ e^+e^- \to \pi^+\pi^-e^+e^- $
are shown schematically.
In the present version of the Monte Carlo program
we include only two gauge invariant sets of diagrams from
Fig.\ref{f00}a and \ref{f00}b.
The former represents initial state radiation (ISR),
and the latter final state radiation (FSR), of an $e^+e^- $ pair
from $ e^+e^- \to \pi^+\pi^-$ production diagram.
We use scalar QED (sQED) to model FSR $e^+e^- $ pair emission
and $\rho$ dominance model for $\gamma^*(\rho^*)\pi\pi$ coupling
(see \cite{Binner:1999bt} for details).
The diagrams from Fig.\ref{f00}c, representing
pion pair emission from t--channel Bhabha process, together with
s-channel Bhabha pion pair emission (not shown in Fig.\ref{f00}),
will be included in the new version \cite{CN_new}
of the presented generator, completing the discussion of this paper.
The contribution from $\gamma^*\gamma^*$ pion pair production process
(Fig.\ref{f00}d)
is negligible for DA$\Phi$NE energy \cite{Juliet}, and as its interference
with other diagrams does not contribute to the cross section
integrated over charge symmetric cuts, these contributions are not
relevant, at least for $\Phi$--factories.
For parametrisation of the phase space we use the following variables:
$Q^2=(\pi_1+\pi_2)^2$--invariant mass of $\pi^+\pi^-$ system,
$k_1^2=(q_1+q_2)^2$--invariant mass of $e^+e^-$ system, polar and azimuthal
angles of $\vec{Q}$ momentum,
defined in the centre-of-mass (CM) frame of initial $e^+e^-$ pair,
polar and azimuthal angles of $\vec\pi_1$ momentum,
defined in $Q$- rest frame and
polar and azimuthal angles of $\vec q_1$ momentum,
defined in $k_1$- rest frame. All four vectors are boosted
into the initial $e^+e^-$ CM frame after being generated and all
necessary cuts can be applied at this stage of the generation.
Multi--channel variance reduction method was used to improve efficiency of the
generator and all details
will be given in a separate publication \cite{CN_new}.
We have performed a number of 'standard' tests
to ensure that the written FORTRAN code is correct.
Gauge invariance of the sum of the amplitudes was checked analytically
separately for set of diagrams from Fig.\ref{f00}a and \ref{f00}b.
We use helicity amplitudes in EKHARA to calculate square
of the matrix element describing the $ e^+e^- \to \pi^+\pi^-e^+e^- $
process, but as a cross check, we have used also the standard trace
technique to calculate
the square of the matrix element, summed over polarisations
of initial and final leptons.
Both results were compared numerically scanning
the physical phase space, and the biggest relative difference
between the two results found was at the level of $10^{-9}$.
It was necessary to use quadrupole precision of the real numbers
for the trace technique result, as one can observe severe
cancellations between various terms.
The phase space volume calculated
by Monte Carlo program was cross checked
with the Gauss integration and
the relative difference at the level of $10^{-5}$ was well within
the errors of the MC result, which were of the same order.
\begin{figure}[ht]
\begin{center}
\hskip-0.2cm
\epsfig{file=roznicaMCteor16_2d_10_8exact.eps,width=6.4cm,height=5.8cm}
\hskip-0.2cm
\epsfig{file=roznicaMCteor16_2d_10_8_exs_me.eps,width=6.4cm,height=5.8cm}
\end{center}
\caption{EKHARA results compared with analytical results of \cite{KKKS88}
(see text for details). The errors come from MC integration.
}
\label{f0}
\end{figure}
Inclusive analytical formulae from \cite{KKKS88} provide additional,
nontrivial tests of the implementation of the contributions from
Fig.\ref{f00}a.
Formula (23) from \cite{KKKS88} provides $Q^2$ differential cross section
(other variables are integrated over the whole allowed range) valid for large
$Q^2$. In Fig.\ref{f0}a, we compare
the values of the integrals, over 10 equally spaced
intervals of $Q^2$, obtained by means of MC program and one-dimensional
Gauss integration of the above mentioned analytical formula. The
Gauss routine, which we use, guarantees precision of 12 significant digits.
One observes a relatively good agreement for values of $Q^2 \sim s$
and worse for $Q^2$ nearby $\pi^+\pi^-$ production threshold, as expected
from the applicability of the analytical formula.
EKHARA results agree much better (see Fig.\ref{f0}b) with known analytically
doubly differential cross section $\frac{d\sigma}{dQ^2 dk_1^2}$
\cite{BFK,KKKS88}, integrated over the whole allowed range
of $k_1^2$ and 10 equally spaced intervals of $Q^2$. The exact analytical
result was integrated numerically, using recursively one-dimensional
8-point Gauss procedure and dividing the region of integration
into pieces small enough to guarantee the overall accuracy of
10 significant digits. From Fig.\ref{f0}b it is clear that a technical
precision of EKHARA of the order of 0.1\% was achieved.
\section{Monte Carlo Data Analysis}
\begin{figure}[ht]
\begin{center}
\hskip-0.2cm
\epsfig{file=stosunek_2d-5d_16_50_4_7.eps,width=6.4cm,height=5.8cm}
\hskip-0.2cm
\epsfig{file=stosunek_2d-5d_16_50_c3_4_7.eps,width=6.4cm,height=5.8cm}
\end{center}
\caption{Comparisons of ISR and FSR contributions
to $e^+e^- \to \pi^+\pi^- e^+e^-$ cross section at DA$\Phi$NE energy.
}
\label{f1}
\end{figure}
For ISR of $e^+e^-$ pair ( Fig.\ref{f00}a),
a factorisation similar to photon emission
holds \cite{KKKS88} and adding ISR pair production to ISR photon
production results just in a change of
the radiator function, thus radiative return method
still can be used \cite{cg:ustron}.
On the other hand,
FSR of $e^+e^-$ pair ( Fig.\ref{f00}b) is model dependent, the same
way as it is the emission of a real photon, and
the question of its relative, to ISR, contribution to the cross section
is as important as for the photon emission.
One can observe, that $e^+e^-$ pair emission resembles a lot photon emission,
with big contributions of FSR for inclusive configurations (Fig.\ref{f1}a)
of a $\Phi$--factory,
which can be easily reduced, by suitable cuts, to a negligible level
(Fig.\ref{f1}b). Moreover, the cuts which reduce photon FSR
reduces as well the $e^+e^-$ pair FSR.
In addition, analogously to photon
FSR, $e^+e^-$ pair FSR is completely negligible at $B$--factories.
\begin{figure}[ht]
\begin{center}
\hskip-0.2cm
\epsfig{file=stosunek_ea_16_50_4_7.eps,width=6.4cm,height=5.8cm}
\hskip-0.2cm
\epsfig{file=stosunek_ea_16_50_c4_4_7.eps,width=6.4cm,height=5.8cm}
\end{center}
\caption{ Ratio of differential cross sections
of the process $ e^+e^- \to \pi^+\pi^-e^+e^- $ (pair)
and $ e^+e^-\to \pi^+\pi^- + {\mathrm{photon(s)}} $ (ph).
}
\label{f2}
\end{figure}
The most relevant information, how big is the contribution
of the process $ e^+e^- \to \pi^+\pi^-e^+e^- $ in comparison to
the main process used in the radiative return method, mainly
$ e^+e^-\to \pi^+\pi^- + {\mathrm{photon(s)}} $, is presented in
Fig.\ref{f2} for DA$\Phi$NE energy, both without any cuts (Fig.\ref{f2}a),
and with cuts resembling KLOE event selection
\cite{KLOE:2003,SdiFalco:Ustron} (Fig.\ref{f2}b). The results of
$ e^+e^-\to \pi^+\pi^- + {\mathrm{photon(s)}} $ cross section were
obtained using PHOKHARA 3.0 MC generator \cite{Czyz:PH03} and in
the following, whenever we refer
to $ e^+e^-\to \pi^+\pi^- + {\mathrm{photon(s)}} $ cross section
we mean cross section obtained using PHOKHARA 3.0. As one can see
from Fig.\ref{f2}, the contribution
of the $e^+e^-$ pair production is below 0.5\%, independently on the cuts.
It is $Q^2$ dependant, being big for
low $Q^2$ values.
Even if it is small,
this 0.5\% contribution can become
important, when aiming at the precision below, or of the order of 1\%, for the
$ e^+e^-\to \pi^+\pi^-$ cross section measurement.
At $B$--factories, the relative contribution of the pair production
might be as big as 0.9\% (Fig.\ref{f4}a) and it is again $Q^2$ and
cut dependent (Fig.\ref{f4}b).
\begin{figure}[ht]
\begin{center}
\hskip-0.2cm
\epsfig{file=stosunek_ea_16input10_20_8_7.eps,width=6.4cm,height=5.8cm}
\hskip-0.2cm
\epsfig{file=stosunek_ea_16input10_50_c5_8_7.eps,width=6.4cm,height=5.8cm}
\end{center}
\vskip-0.4cm
\caption{Ratio of differential cross sections
of the process $ e^+e^- \to \pi^+\pi^-e^+e^- $ (pair)
and $ e^+e^-\to \pi^+\pi^- + {\mathrm{photon(s)}} $ (ph).
}
\label{f4}
\end{figure}
\begin{figure}[hb]
\begin{center}
\hskip-0.2cm
\epsfig{file=stos_5d.c6-5d.c5_16_50_4_7.eps,width=6.4cm,height=5.8cm}
\hskip-0.2cm
\epsfig{file=stos_5d.c10-5d.c4_16_50_4_7.eps,width=6.4cm,height=5.8cm}
\end{center}
\vskip-0.4cm
\caption{Non-reducible pair production background
at DA$\Phi$NE energy : (a) no cuts in $ ( \frac{d\sigma}{dQ^2})_{tot}$;
(b) for both $ (\frac{d\sigma}{dQ^2})_{c2} $ and
$ ( \frac{d\sigma}{dQ^2})_{tot}$ the following cuts are imposed:
$ 50^\circ < \theta_{\pi^{\pm}} < 130^\circ $,
$ \theta_{\pi^+ + \pi^-} < 15^\circ $
or $ \theta_{\pi^+ + \pi^-} > 165^\circ $ and $M_{tr}$ cut.
For (a) and (b) c2 denotes additional cuts:
$ (\theta_{e^+} < 20^\circ $ or $ \theta_{e^+} > 160^\circ) $
and $ ( \theta_{e^-} < 20^\circ $ or $ \theta_{e^-} > 160^\circ) $.
}
\label{f3}
\end{figure}
As stated already, ISR of electron pairs can be treated in a similar way as
ISR of photons resulting in the change of radiator function in
the radiative return method. However, one can alternatively try to treat it
as a background to the process with photon(s) emission.
In this case, the most natural way of reducing that
background is to veto the electron (positron) in the final state.
In Fig.\ref{f3} we show an example of such a procedure performed for
$\Phi$--factory energy. We assume here that an electron or positron
can be seen, and the event rejected, if its angle with respect to
the beam axis is bigger then 20$^\circ$. Fig.\ref{f3}a shows that
up to 50\% of the events pass the rejection procedure, when no other
cuts are applied. However, in the case of KLOE event selection, which requires
that the $e^+e^-$ pair is emitted along the beam axis, one
rejects only a small fraction of these
events (Fig.\ref{f3}b).
\begin{figure}[ht]
\begin{center}
\hskip-0.2cm
\epsfig{file=stos_5d.c8-5d.c7_16input10_50_8_7.eps,width=6.4cm,height=5.8cm}
\hskip-0.2cm
\epsfig{file=stos_5d.c6-5d.c5_16input10_50_8_7.eps,width=6.4cm,height=5.8cm}
\end{center}
\caption{
Non-reducible pair production background
at $B$--factory energy : (a) no cuts in $ ( \frac{d\sigma}{dQ^2})_{tot}$;
(b) for both $ (\frac{d\sigma}{dQ^2})_{c2} $ and
$ ( \frac{d\sigma}{dQ^2})_{tot}$ cuts on pion angles are imposed:
$ 50^\circ < \theta_{\pi^{\pm}} < 130^\circ $.
For (a) and (b) c2 denotes additional cuts:
$ (\theta_{e^+} < 20^\circ $ or $ \theta_{e^+} > 160^\circ) $
and $ ( \theta_{e^-} < 20^\circ $ or $ \theta_{e^-} > 160^\circ) $.
}
\label{f5}
\end{figure}
The situation is completely different at $B$--factories, where one can
almost completely reduce the background coming from $e^+e^-$ pairs
( Fig.\ref{f5}), by rejecting the events with at least one charged
lepton, electron or positron, in the detector.
\section{Conclusions}
We have constructed the Monte Carlo generator EKHARA,
which simulates the
reaction $e^+e^- \to \pi^+\pi^- e^+e^-$
based on initial and final state emission of a $e^+e^-$ pair
from $e^+e^-\to\pi^+\pi^-$ production diagram.
A detailed study of this process, as a potential background
for $\sigma (e^+e^- \to \pi^+\pi^-)$ measurement
via radiative return method,
shows that it can become important, when
the experimental precision will reach 1\%, or better.
{\bf Acknowledgements:}
We thank J.H. K{\"u}hn for discussion and
we are grateful for the support and the kind hospitality of
the Institut f{\"u}r Theoretische Teilchenphysik
of the Universit\"at Karlsruhe.
\vskip 1 cm
\begin{thebibliography}{99}
\bibitem{Binner:1999bt}
S.~Binner, J.~H.~K\"uhn and K.~Melnikov,
%``Measuring sigma(e+ e- --> hadrons) using tagged photon,''
Phys.\ Lett.\ B {\bf 459} (1999) 279
[hep-ph/9902399].
%%CITATION = HEP-PH 9902399;%%
\bibitem{cg:ustron}
H.~Czy{\.z} and A.~Grzeli{\'n}ska, these proceedings.
\bibitem{Davier:2003}
M.~Davier, S.~Eidelman, A.~H\"ocker and Z.~Zhang,
hep-ph/0308213.
%Updated estimate of the muon magnetic moment using revised
%results from e+ e- annihilation.
%%CITATION = HEP-PH 0308213;%%
\bibitem{Nyffeler:2003}
A.~Nyffeler, hep-ph/0305135 and these proceedings.
%
%Theoretical status of the muon g-2
%%CITATION = HEP-PH 0305135;%%
\bibitem{Jegetc.}
F.~Jegerlehner, hep-ph/0310234.
%"Theoretical precision in estimates of the hadronic
% contributions to (g-2)_mu and alpha_QED(M_Z)",
%%CITATION = HEP-PH 0310234;%%
\bibitem{Czyz:2000wh}
H.~Czy\.z and J.~H.~K\"uhn,
%``Four pion final states with tagged photons at electron positron colliders,''
Eur.\ Phys.\ J.\ C {\bf 18} (2001) 497
[hep-ph/0008262].
%%CITATION = HEP-PH 0008262;%%
\bibitem{Rodrigo:2001kf}
G.~Rodrigo, H.~Czy\.z, J.H.~K\"uhn and M.~Szopa,
%``Radiative return at NLO and the measurement of the hadronic cross-section in electron positron annihilation,''
Eur.\ Phys.\ J.\ C {\bf 24} (2002) 71
[hep-ph/0112184].
%%CITATION = HEP-PH 0112184;%%
\bibitem{Czyz:2002np}
H.~Czy{\.z}, A.~Grzeli{\'n}ska, J.~H.~K{\"u}hn and G.~Rodrigo,
%``The radiative return at Phi- and B-factories: Small-angle photon emission at next to leading order,''
Eur.\ Phys.\ J.\ C {\bf 27} (2003) 563
[arXiv:hep-ph/0212225].
%%CITATION = HEP-PH 0212225;%%
\bibitem{Czyz:PH03}
H.~Czy{\.z}, A.~Grzeli{\'n}ska, J.~H.~K{\"u}hn and G.~Rodrigo,
%``The radiative return at Phi- and B-factories:
% FSR at next-to-leading order,''
hep-ph/0308312.
%%CITATION = HEP-PH 0308312;%%
\bibitem{Rodrigo:2001jr}
G.~Rodrigo, A.~Gehrmann-De Ridder, M.~Guilleaume and J.~H.~K\"uhn,
%``NLO QED corrections to ISR in e+ e- annihilation and the measurement of sigma(e+ e- $\to$ hadrons) using tagged photons,''
Eur.\ Phys.\ J.\ C {\bf 22} (2001) 81
[hep-ph/0106132].
%%CITATION = HEP-PH 0106132;%%
\bibitem{Kuhn:2002xg}
J.~H.~K\"uhn and G.~Rodrigo,
%``The radiative return at small angles: Virtual corrections,''
Eur.\ Phys.\ J.\ C {\bf 25} (2002) 215
[hep-ph/0204283].
%%CITATION = HEP-PH 0204283;%%
\bibitem{Kuhn:2001}
J.~H.~K{\"u}hn
%"Measuring sigma(e+ e- --> hadrons) with tagged photons at
% electron positron colliders"
Nucl. Phys. Proc. Suppl. {\bf 98} (2001) 289 [hep-ph/0101100].
%%CITATION = HEP-PH 0101100;%%
\bibitem{Rodrigo:2001cc}
G.~Rodrigo,
%``Radiative return at NLO and the measurement of the hadronic cross section,''
Acta Phys.\ Polon.\ B {\bf 32} (2001) 3833 [hep-ph/0111151].
%%CITATION = HEP-PH 0111151;%%
\bibitem{Rodrigo:2002hk}
G.~Rodrigo, H.~Czy\.z and J.~H.~K\"uhn,
%``Radiative return at NLO: The PHO\-KHA\-RA Monte Carlo generator,''
hep-ph/0205097; Nucl.Phys.Proc.Suppl.123(2003)167 [hep-ph/0210287];
Nucl.Phys.Proc.Suppl. {\bf 116} (2003) 249 [hep-ph/0211186].
%%CITATION = HEP-PH 0205097;%%
%%CITATION = HEP-PH 0210287;%%
%%CITATION = HEP-PH 0211186;%%
\bibitem{Achim:radcor02}
A.~Denig {\it et al.} [KLOE Collaboration],
Nucl. Phys. Proc. Suppl. {\bf 116} (2003)243
[hep-ex/0211024].
%"Measuring the hadronic cross section via radiative return"
%%CITATION = HEP-EX 0211024;%%
\bibitem{KLOE:2003}
A.~Aloisio {\it et al.} [KLOE Collaboration], hep-ex/0307051.
%Determination of sigma(e+ e- --> pi+ pi-) from radiative
% processes at DAPHNE",
%%CITATION = HEP-EX 0307051;%%
\bibitem{Juliet}
J.~Lee-Franzini, talk at Lepton Moments International Symposium,
Cape Cod (June 2003),
http://g2pc1.bu.edu/leptonmom/program.html .
\bibitem{SdiFalco:Ustron} S. di Falco (KLOE), these proceedings.
\bibitem{Blinov}
A.~Blinov, talk at International Conference
``New trends in high-energy physics'' Alushta, Crimea (May 2003)
http://www.slac.stanford.edu/~ablinov/
\bibitem{Hoefer:2001mx}
A.~Hoefer, J.~Gluza and F.~Jegerlehner, Eur.Phys.J.{\bf C24} (2002) 51,
%``Pion pair production with higher order radiative corrections
% in low energy e+ e- collisions,''
[hep-ph/0107154].
%%CITATION = HEP-PH 0107154;%%
\bibitem{CN_new} H.~Czy\.z and E.~Nowak, in preparation.
\bibitem{KKKS88}
B.A.~Kniehl, M.~Krawczyk, J.H.~K\"uhn, R.G.~Stuart, Phys. Lett. {\bf B209}
(1988) 337.
% HADRONIC CONTRIBUTIONS TO O (alpha**2) RADIATIVE
% CORRECTIONS IN e+ e- ANNIHILATION
%%CITATION = PHLTA,B209,337;%%
\bibitem{BFK}
V.N.~Baier, V.S.~Fadin, V.A.~Khoze
Sov.Phys.JETP {\bf 23} (1966)104.
% "Electromagnetic pair production"
%%CITATION = SPHJA,23,104;%%
\end{thebibliography}
\end{document}