text:
\section{INTRODUCTION}

Even if electromagnetic phenomena rank among the best understood in
particle physics, their merging with hadronic processes often brings
ambiguities that cannot be resolved on a purely theoretical
basis. It is generally believed, however, that many of the reactions
in which photons and dileptons are produced can be described as
bremsstrahlung from incoming and outgoing charged particles, at least
within a limited kinematic range.

It is well known that the cross section for production of photons
with very low energies is uniquely determined by the cross section of
the corresponding nonradiative reaction
\cite{low,adlerdot,burkroll,brown,bellroy,duca}. In the dilepton
sector, the situation is less clear. Several different bremsstrahlung
formulas have been proposed
\cite{rueckl,craithom,goshawee,galkap,galkap89,bpp,clgore1,hagaem2},
which, as will be demonstrated in this work, do not agree with one
another very well. Some of these formulas fail to satisfy constraints
implied by general principles such as Lorentz covariance and gauge
invariance. Besides the formulas cited above, a few additional
formulas exist that have been designed for specific kinematic regions
\cite{farrfrau,craigie}. They will not be considered here.

The purpose of this work is to present a set of consistent formulas
for various distributions in dilepton bremsstrahlung from pseudoscalar
mesons and unpolarized spin-one-half fermions. We consider the terms
that are proportional to the square of the nonradiative matrix element
(leading term approximation) and its derivatives (sub-leading
approximation). We first derive the formula for the most general
quantity, namely the double differential cross section in the momenta
of leptons. The correct form for it has not been yet known even in the
leading term approximation. Then we arrive at the cross section in
dilepton mass and momentum. Its leading part differs only by
higher-order terms in the dilepton four-momentum from one of the
already existing formulas \cite{bpp}.

To stress the importance of using the correct virtual bremsstrahlung
formalism, we compare our formulas to those already existing in the
literature. The comparison is made first on general grounds and is
followed by an application of the formulas to a simple physical
process. All formulas are scrutinized from the point of view of
Lorentz covariance \cite{einstein} and from what we will call a global
variable test.

The global variable test is based on the finding \cite{baier,invint}
that in a one-photon approximation, gauge invariance leads to the
following relation for the inclusive\footnote{We will omit the word
inclusive in what follows. All our cross sections for photon or
dilepton production are semi-inclusive (integrated over the final
state hadron momenta in a reaction with given number and types
of final state particles). Some of the relations are more general
and hold also for inclusive cross sections, which are given as sums
of the semi-inclusive cross sections over all possible reactions with
chosen initial particles [as, e.g., (\ref{global}), which is valid
even for exclusive cross sections]. In some cases, the sum over all
possible reactions should be supplemented to make the relation
inclusive.} differential cross section in global dilepton quantities
$M$ (dilepton mass) and ${\bf q}$ (dilepton momentum)
\begin{equation}
\label{global}
\frac{d^4\sigma^{e^+e^-}}{dM^2d^3q} =
{\cal T}(M^2)\
\left(\frac{d^3\sigma^{\gamma^*}}{d^3q}\right)_M
\end{equation}
with the function ${\cal T}$ given by
\begin{equation}
\label{tmsq}
{\cal T}(M^2) =
\frac{\alpha}{3\pi}\frac{1}{M^2}
\left(1+\frac{2\mu^2}{M^2}\right)
\sqrt{1-\frac{4\mu^2}{M^2}}\ ,
\end{equation}
where $\alpha$ is the fine structure constant and $\mu$ the lepton
mass. The rightmost quantity in Eq.~(\ref{global}) is called the
cross section for virtual photon
production. It does not have a direct physical meaning, as it is not
experimentally accessible. Nevertheless, Eqs.~(\ref{global}) and
(\ref{tmsq}) can test the soundness of theoretical formulas,
because they show the only two places in which the lepton mass $\mu$
may and must (unless neglected) appear. From a technical point of
view, the virtual photon cross section is calculable more easily than
the dilepton cross section \cite{baier}. Of course, if one needs the
distribution in the momenta of leptons, a more involved approach is
unavoidable.

Furthermore, if an otherwise identical reaction exists in which a
photon is produced instead of a dilepton (this need not always be the
case, viz., $\pi^+\pi^-\rightarrow e^+e^-$), the relation
\begin{equation}
\label{limit}
\lim_{M\rightarrow 0}
\left(\frac{d^3\sigma^{\gamma^*}}{d^3q}\right)_M
=\frac{d^3\sigma^{\gamma}}{d^3q}
\end{equation}
must be fulfilled \cite{baier} with the differential cross section
for real photon production on the right-hand side.

Relations analogous to (\ref{global}) and (\ref{limit}) can also
be written for the corresponding quantities in other than two-body
reactions, namely for decays and processes with more than two
particles in the initial state \cite{lichard}.

In the next section, we derive the formulas appropriate for the
production of very soft (low mass, low momentum) dileptons via
virtual photon bremsstrahlung in reactions with charged pseudoscalar
particles. Section~\ref{fermions} deals with the virtual
bremsstrahlung from fermions.
In Sec.~\ref{survey}, we show the additional approximations which
lead to the various formulas that have appeared explicitly or as
a part of more complex expressions in the literature. All the
formulas are examined to see if they fulfill the Lorentz covariance
and global variable tests. In Sec.~\ref{model}, we introduce a simple
theoretical model of the process $\rho^0\rightarrow \pi^+\pi^-e^+e^-$
(which has not been experimentally investigated yet) based on
a successful description \cite{singer} of the recently observed
\cite{vasserman} decay $\rho^0\rightarrow \pi^+\pi^-\gamma$. The
former process will be a testing ground for various virtual
bremsstrahlung formulas in Sec.~\ref{bremsstrahlung} with the
theoretical distribution from Sec.~\ref{model} serving as
a reference. We summarize our main points and add a few comments in
Sec.~\ref{comments}. Some related issues
are discussed in the Appendices. In Appendix \ref{appmt} we
show how $M_T$-scaling transpires from the leading term virtual
bremsstrahlung formalism. Appendix \ref{appsingvirt} deals with
the ``exact" formula for the $\rho^0\rightarrow \pi^+\pi^-
\gamma^*$ branching ratio.
Appendix \ref{appleading}
addresses the issue of photon and dilepton event generators that
conserve energy and momentum.

\section{Dileptons from virtual
bremsstrahlung off pseudoscalar mesons}
\label{mesons}

In this section, we assume that all charged particles are pseudoscalar
mesons. The technically more involved case of virtual bremsstrahlung
from spin-one-half fermions will be treated in the next section.

\subsection{Leading term approximation}
\label{leading}
Let us consider a $2\rightarrow n$ hadronic reaction
\begin{equation}
\label{reaction1}
a+b\rightarrow 1+2+\cdots+n
\end{equation}
and denote its matrix element as
${\cal M}_0\equiv {\cal M}_0(p_a,p_b,p_1,\cdots,p_{n})$. Our aim is to
find the matrix element ${\cal M}$ of the reaction
\begin{equation}
\label{reaction2}
a+b\rightarrow 1+2+\cdots+n+l^++l^-,
\end{equation}
in which a soft lepton pair with the four-momentum $q=p_++p_-$
is produced in addition to $n$ hadrons. The dominant contribution
to the matrix element ${\cal M}$ comes from diagrams where the virtual
photon is attached to one of the external legs (see Fig.~\ref{me}).
The diagrams in which a virtual photon is radiated from internal lines
give subleading contributions. This is caused by the nonvanishing
virtuality contributions of the type $(p^2-m^2)$ to the denominators
of newly emerging propagators.  Such terms do not appear if one of the
two particles attached to the electromagnetic vertex is real
($p^2=m^2$). See below.

Using the Feynman rules of pseudoscalar electrodynamics
\cite{bjorken} we can immediately write down the contribution to the
matrix element ${\cal M}$ from radiation of an initial ($x=a,b$)
\begin{equation}
\label{minitial}
{\cal M}_x=-eQ_x{\cal M}_0\ \frac{(2p_x-q)^\mu}
{2\,p_x\!\cdot\!q-M^2}L_\mu
\end{equation}
and a final ($i=1,\cdots,n$)
\begin{equation}
\label{mfinal}
{\cal M}_i=eQ_i{\cal M}_0\ \frac{(2p_i+q)^\mu}
{2\,p_i\!\cdot\!q+M^2}L_\mu
\end{equation}
state particle. Above, $e$ is the positive elementary charge,
$Q_x$ ($Q_i$) is the charge of an initial (a final) particle, and $M$
is the dilepton mass ($q^2=M^2$). As is customary in this field, we
have supposed that the nonradiative matrix element does not change
when an incoming or outgoing momentum becomes ``slightly"
off-mass-shell; for example,
\begin{equation}
\label{offshell}
{\cal M}_0(p_a-q,p_b,p_1,\cdots,p_{n})=
{\cal M}_0(p_a,p_b,p_1,\cdots,p_{n})\equiv {\cal M}_0\ .
\end{equation}
The lepton part is given by
\begin{equation}
\label{leptonterm}
L_\mu=\frac{e}{M^2}\bar u^{(s_-)}(p_-)\gamma_\mu v^{(s_+)}(p_+)\ .
\end{equation}
With this choice of $L_\mu$, the matrix element for photon
production is obtained by the substitution
$L_\mu\rightarrow\epsilon_\mu$, where $\epsilon$ is the photon
polarization vector. Summing the contributions of individual diagrams,
we arrive at
\begin{equation}
\label{msum}
{\cal M}=e{\cal M}_0\ J\!\cdot\!L
\end{equation}
with
\begin{equation}
\label{j}
J^\mu=-\sum_{x=a,b} Q_x\frac{(2p_x-q)^\mu}{2\,p_x\!\cdot\!q-M^2}+
      \sum_{i=1}^{n} Q_i\frac{(2p_i+q)^\mu}{2\,p_i\!\cdot\!q+M^2}\ .
\end{equation}
The four-vector $J$ satisfies the important relation
\begin{equation}
\label{jq}
J\!\cdot\!q=-Q_a-Q_b+\sum_{i=1}^{n} Q_i=0\ ,
\end{equation}
which reflects charge conservation. Squaring the matrix
element~(\ref{msum}) and summing over the spins of leptons, we get
\begin{equation}
\label{summsq1}
\sum_{s_+,s_-}\left|{\cal M}\right|^2 = 4\pi\alpha\left|
{\cal M}_0\right|^2\ J^\mu J^\nu\ L_{\mu\nu}\ .
\end{equation}
The tensor $L_{\mu\nu}$ is defined by
\begin{equation}
\label{lmunudef}
L_{\mu\nu}=\sum_{s_+,s_-}\ L_\mu L_\nu^*\ .
\end{equation}
A straightforward calculation leads to
\begin{equation}
\label{lmunu}
L_{\mu\nu}=\frac{8\pi\alpha}{M^4}
\left(q_\mu q_\nu-l_\mu l_\nu-M^2 g_{\mu\nu}\right)\ ,
\end{equation}
where we have introduced the four-vector $l=p_+-p_-$ as the difference
of leptons' momenta. Using the above relation we can
rewrite~(\ref{summsq1}) in the form
\begin{equation}
\label{summsq2}
\sum_{s_+,s_-}\left|{\cal M}\right|^2 = \left|{\cal M}_0\right|^2
\frac{32\pi\alpha^2}{M^2}\left[-J^2-\frac{1}{M^2}
\left(l\!\cdot\!J\right)^2
\right].
\end{equation}
Inserting this into the relation for the unpolarized cross section
of the reaction (\ref{reaction2}) leads to
\begin{eqnarray}
\label{xsec}
d\sigma & = &\frac{1}{4E_aE_b|{\bf v}_a-{\bf v}_b|}
\left|{\cal M}_0\right|^2
(2\pi)^4\delta(p_a+p_b-\sum_i p_i-q)
\nonumber \\[5pt]
& & \times
\frac{32\pi\alpha^2}{M^2}\left[-J^2-\frac{1}{M^2}
\left(l\!\cdot\!J\right)^2
\right]\
\prod_{i=1}^n\frac{d^3p_i}{2E_i(2\pi)^3}
\prod_{l=+,-}\frac{d^3p_l}{2E_l(2\pi)^3}\ .
\end{eqnarray}
After neglecting the dilepton four-momentum in the argument of the
$\delta$-function and integrating over the momenta of final hadrons,
we get
\begin{equation}
\label{d6sig}
E_+E_-\frac{d^6\sigma^{e^+e^-}}{d^3p_+d^3p_-}=\frac{\alpha^2}
{8\pi^4}\frac{1}
{M^2}\int\left[-J^2-\frac{1}{M^2}(l\!\cdot\!J)^2\right]d\sigma_0\ ,
\end{equation}
where
\begin{equation}
\label{xsec0}
d\sigma_0 = \frac{1}{4E_aE_b|{\bf v}_a-{\bf v}_b|}
\left|{\cal M}_0\right|^2
(2\pi)^4\delta(p_a+p_b-\sum_i p_i)
\prod_{i=1}^n\frac{d^3p_i}{2E_i(2\pi)^3}
\end{equation}
is the infinitesimal cross section of the reaction (\ref{reaction1}).
Double differential cross section
$E_+E_-d^6\sigma^{e^+e^-}/d^3p_+d^3p_-$
is the most general quantity that characterizes the production of pairs
of unpolarized unlike-sign leptons. Knowing it, we can find any other
distribution (but not {\it vice versa}).

A little exercise from relativistic kinematics provides us with
the following general formula
\begin{equation}
\label{page45}
E\ \frac{d^6\sigma^{e^+e^-}}{dM^2d^3q\ d\tilde\Omega_+} =
\frac{1}{4}\sqrt{1-\frac{4\mu^2}{M^2}}\,\,
E_+E_-\frac{d^6\sigma^{e^+e^-}}{d^3p_+d^3p_-}\ ,
\end{equation}
where $E=E_++E_-$ is the dilepton energy and $d\tilde\Omega_+$ is the
solid angle element for positron momentum in the dilepton rest frame.
Using (\ref{d6sig}) and (\ref{page45}), we arrive at another formula
of our virtual bremsstrahlung formalism
\begin{equation}
\label{d6sigmix}
E\ \frac{d^6\sigma^{e^+e^-}}{dM^2d^3q\ d\tilde\Omega_+} =
\frac{\alpha^2}{32\pi^4}\frac{1}{M^2}
\sqrt{1-\frac{4\mu^2}{M^2}}
\int\left[-J^2-\frac{1}{M^2}(l\!\cdot\!J)^2\right]d\sigma_0\ .
\end{equation}
Finally, integrating over the positron angles in the dilepton rest
frame, we obtain the differential cross section in global dilepton
variables
\begin{equation}
\label{newglobal}
E\frac{d^4\sigma^{e^+e^-}}{dM^2d^3q} =
\frac{\alpha^2}{12\pi^3}\frac{1}{M^2}
\left(1+\frac{2\mu^2}{M^2}\right)
\sqrt{1-\frac{4\mu^2}{M^2}}
\int\left(-J^2\right)\ d\sigma_0\ .
\end{equation}
This is just formula (\ref{global}) with the virtual photon cross
section given by
\begin{equation}
\label{myvirtual}
E\ \left(\frac{d^3\sigma^{\gamma^*}}{d^3q}\right)_M
=\frac{\alpha}{4\pi^2}\int\left(-J^2\right)\ d\sigma_0\ .
\end{equation}
Sometimes it is advantageous to express $J^2$ in terms of
three-dimensional
vectors. This can easily be done \cite{bpp} using the relation
$J_0=E^{-1}\ {\bf J}\!\cdot\!{\bf q}$,
which follows from Eq.~(\ref{jq}). We thus get
\begin{equation}
\label{j2}
-J^2=\left({\bf J}\times{\bf n}\right)^2+\frac{M^2}{E^2}
\left({\bf J}\!\cdot\!{\bf n}\right)^2,
\end{equation}
where ${\bf n} = {\bf q}/|{\bf q}|$ is the unit vector in the dilepton
momentum direction.

To investigate the limit (\ref{limit}) of (\ref{myvirtual}),
it is sufficient to realize that due to (\ref{j2}), the part of
$J^\mu$ that is proportional to $q^\mu$ will not contribute in the
limit $M\rightarrow 0$. What remains can be written as

\begin{equation}
\label{myreal}
\omega\ \frac{d^3\sigma^\gamma}{d^3q}
=\frac{\alpha}{4\pi^2}\int\left(-J_R^2\right)\ d\sigma_0
\end{equation}
with $\omega=|{\bf q}|$ and
\begin{equation}
\label{jr}
J_R = \sum  Q^\prime_i\frac{p_i}{p_i\!\cdot\!q}
        =\frac{1}{\omega}\sum  \frac{Q^\prime_i}{E_i}\ \frac{p_i}
{1-{\bf v}_i\!\cdot\!{\bf n}}\ .
\end{equation}
To make the formula more compact, we have introduced the variable
$ Q^\prime_i$, which is identical with the charge of
final particles and acquires the opposite sign for initial particles.
The unspecified sum runs over all (initial and final) hadrons
and ${\bf v}_i={\bf p}_i/E_i$ is the velocity of the $i$th hadron.
Eqs.~(\ref{myreal}) and (\ref{jr}) combine to give the well known
leading term formula for real photon bremsstrahlung
\cite{jackson,berest}.

The central results of this subsection are the formulas (\ref{d6sig}),
(\ref{d6sigmix}), and (\ref{newglobal}) for various differential
cross sections of dilepton production via virtual photon
bremsstrahlung in the leading term approximation.

\subsection{Next-to-leading term approximation}
\label{nexttoleading}
When going beyond the next-to-leading order, the nonradiative matrix
element is no longer considered to be immune against the changes in
incoming and outgoing momenta. We write, instead of
Eq.~(\ref{offshell}),
\begin{equation}
\label{m0xq}
{\cal M}_0(p_x-q)={\cal M}_0-
q^\alpha\frac{\partial{\cal M}_0}{\partial p_x^\alpha}
\end{equation}
if one of the incoming momenta changes, and
\begin{equation}
\label{m0iq}
{\cal M}_0(p_i+q)={\cal M}_0+
q^\alpha\frac{\partial{\cal M}_0}{\partial p_i^\alpha}
\end{equation}
to account for a change in one of the outgoing momenta. In the above
two equations, we suppress the momenta that keep their ``nonradiative"
values. If we now simply incorporated the ``corrected" values
(\ref{m0xq}) and (\ref{m0iq}) into the expressions (\ref{minitial})
and (\ref{mfinal}) for the radiative matrix elements, and summed these
up, we would obtain a non-gauge-invariant quantity. After replacing
the four-vector $L_\mu$ by the virtual photon four-momentum,
it would not vanish. The reason is that we have not yet included the
``contact" terms, which are generated from the strong interaction
Lagrangian by the minimal electromagnetic interaction principle.
The latter says that the electromagnetic interaction terms appear as
a result of the substitution
\begin{equation}
\label{minelpr}
p^\alpha\rightarrow p^\alpha-eQg^{\alpha\mu}\ ,
\end{equation}
where the index $\mu$ is to be contracted with the real photon
polarization vector or the virtual photon propagator. To
find the contact terms in our case, we make a formal expansion of
${\cal M}_0(p^\alpha_i-eQ_ig^{\alpha\mu})$, where $p_i$ denotes any
of the incoming or outgoing momenta. We thus find that the contact
term associated with the $i$th hadron is
\begin{equation}
\label{contacti}
C_i^\mu=-eQ_i\frac{\partial{\cal M}_0}{\partial p_{i\mu}}\ .
\end{equation}
For the same reason as stated in the previous subsection, the
radiation from internal lines will not contribute in this
approximation either. Putting it all together, the radiative matrix
element comes out to be
\begin{equation}
\label{mek}
{\cal M}=e\ K\!\cdot\!L\ ,
\end{equation}
where $L$ is the four-vector defined by Eq.~(\ref{leptonterm}), and
\begin{equation}
\label{kmu}
K^\mu={\cal M}_0\ J^\mu+\sum_{i}Q_i\frac{\partial {\cal M}_0}
{\partial p_i^\alpha}\left(\frac{p_i^\mu q^\alpha}{p_i\!\cdot\!q}-
g^{\mu\alpha}\right).
\end{equation}
We retained only the leading and next-to-leading terms in $q$. The
sum runs over both incoming and outgoing hadrons and the four-vector
$J$ is given by Eq.~(\ref{j}). To get the cross section for producing
a pair of unpolarized leptons, we need the quantity
\begin{equation}
\label{summsqh}
\sum_{s_+,s_-}\left|{\cal M}\right|^2 = 4\pi\alpha
\  H^{\mu\nu} L_{\mu\nu}\ ,
\end{equation}
with symmetric tensor $L_{\mu\nu}$ defined by (\ref{lmunudef}), and
\begin{equation}
\label{hdef}
H^{\mu\nu}=\frac{1}{2}\left(K^\mu {K^*}^\nu+{K^*}^\mu K^\nu\right)\ .
\end{equation}
Keeping only the $q$-terms of the same order as before, we easily
obtain
\begin{eqnarray}
\label{hmunu}
H^{\mu\nu}&=&\left|{\cal M}_0\right|^2\ J^\mu J^\nu +\frac{1}{2}
\sum_{i,j}\frac{Q_i Q_j^\prime}{(p_i\!\cdot\!q)(p_j\!\cdot\!q)}
\frac{\partial\left|{\cal M}_0\right|^2}{\partial p_i^\beta}
p_{i\alpha}\nonumber \\
& &\times
\left[p_j^\mu\left(g^{\nu\alpha}q^\beta-g^{\nu\beta}q^\alpha\right)+
p_j^\nu\left(g^{\mu\alpha}q^\beta-g^{\mu\beta}q^\alpha\right)
\right]\ .
\end{eqnarray}
To simplify the formula, we have again used the convention
$Q_j^\prime=-Q_j$ for
the incoming hadrons and $Q_j^\prime=Q_j$ for the outgoing hadrons.
The differential cross section in lepton pair momenta now reads
\begin{eqnarray}
\label{d6sigsub}
E_+E_-\frac{d^6\sigma^{e^+e^-}}{d^3p_+d^3p_-}&=&\frac{\alpha^2}
{8\pi^4}\frac{1}{M^2}\biggl\{
\int\Bigl[-J^2-\frac{1}{M^2}(l\!\cdot\!J)^2\Bigr]\ d\sigma_0\ +
\  \sum_{i,j}\int\frac{Q_iQ_j^\prime}{M^2(p_i\!\cdot\!q)
(p_j\!\cdot\!q)}\biggr.\nonumber \\
& & \times \biggl.
\Bigl[p_i\!\cdot\!q\left(l\!\cdot\!p_j\ l^\beta+M^2 p_j^\beta\right)
-\left(l\!\cdot\!p_i\ l\!\cdot\!p_j+M^2 p_i\!\cdot\!p_j\right)q^\beta
\Bigr]\ d\sigma_{0,i\beta}\biggr\}.
\end{eqnarray}
The notation is same as we met in Eq.~(\ref{d6sig}), except for
\begin{equation}
\label{xsec0ibeta}
d\sigma_{0,i\beta} = \frac{1}{4E_aE_b|{\bf v}_a-{\bf v}_b|}
\frac{\partial\left|{\cal M}_0\right|^2}{\partial p^\beta_i}
(2\pi)^4\delta(p_a+p_b-\sum_k p_k)
\prod_{k=1}^n\frac{d^3p_k}{2E_k(2\pi)^3}\ .
\end{equation}

Using the same procedure as in subsection \ref{leading}, we arrive at
the differential cross section in global dilepton variables in the
form~(\ref{global}) with the virtual photon cross section given by
\begin{eqnarray}
\label{virtsubl}
E\ \left(\frac{d^3\sigma^{\gamma^*}}{d^3q}\right)_M
&=&\frac{\alpha}{4\pi^2}\biggr\{\int\left(-J^2\right)\ d\sigma_0\ +
\  \sum_{i,j}\int\frac{Q_iQ_j^\prime}{(p_i\!\cdot\!q)(p_j\!\cdot\!q)}
\biggr.\nonumber \\
& & \times \biggl.
\Bigl[(p_i\!\cdot\!q) p_j^\beta
-(p_i\!\cdot\!p_j)q^\beta\Bigr]\ d\sigma_{0,i\beta}\biggr\}.
\end{eqnarray}
We will comment on the zero photon mass limit of this equation
in Sec.~\ref{fermions}.

\subsection{Decays}
\label{decays}
The description of virtual bremsstrahlung in hadronic decays can be
achieved following the lines sketched for the two-body reactions.
We need only
to change the number of incoming particles to one and replace the
cross sections by decay widths. This follows from the similar
structure of the relations between cross section, or decay width, on
the one hand and the matrix element squared on the other. Another
important factor is the universality (with respect to the numbers of
incoming and outgoing particles) of Eqs.~(\ref{msum}) and (\ref{j}).

Let us consider the decay
\begin{equation}
\label{decay}
a\rightarrow 1+2+\cdots+n+l^++l^-.
\end{equation}
For its differential decay width we can write, in the leading term
approximation,
\begin{equation}
\label{d6decay}
E_+E_-\frac{d^6\Gamma^{e^+e^-}}{d^3p_+d^3p_-}=\frac{\alpha^2}
{8\pi^4}\frac{1}
{M^2}\int\left[-J^2-\frac{1}{M^2}(l\!\cdot\!J)^2\right]d\Gamma_0\ .
\end{equation}
The quantity
\begin{equation}
\label{gamma0}
d\Gamma_0 = \frac{1}{2m_a}\left|{\cal M}_0\right|^2
(2\pi)^4\delta(p_a-\sum_i p_i)
\prod_{i=1}^n\frac{d^3p_i}{2E_i(2\pi)^3}
\end{equation}
is the invariant decay width into an infinitesimal element of the
momentum space for the decay
\begin{equation}
\label{decay0}
a\rightarrow 1+2+\cdots+n\ .
\end{equation}
The other leading-term formulas can be easily modified as
well. The differential decay width in global dilepton quantities is
\begin{equation}
\label{globalwidth}
E\frac{d^4\Gamma^{e^+e^-}}{dM^2d^3q} =
{\cal T}(M^2)\
\frac{\alpha}{4\pi^2}\int\left(-J^2\right)\ d\Gamma_0\ .
\end{equation}
If a nonradiative decay (\ref{decay0}) contains only two particles
in the final state, we can proceed further to get
\begin{equation}
\label{restwidth}
\frac{1}{\Gamma_0}\frac{d^2\Gamma^{e^+e^-}}{dM^2dq^*} =
{\cal T}(M^2)\ \frac{\alpha}{4\pi^2}
\frac{{q^*}^2}{E^*}\int\left(-J^2\right)\ d\Omega_{{\bf q}^{*}}\ ,
\end{equation}
where the asterisk refers to quantities in the rest frame of the
parent particle and $q^* = |{\bf q}^{\,*}|$. For a hypothetical decay
into two particles and a virtual photon it means
\begin{equation}
\label{virtdecay}
\frac{1}{\Gamma_0}\left(\frac{d\Gamma^{\gamma^*}}{dq^*}\right)_M =
\frac{\alpha}{4\pi^2}
\frac{{q^*}^2}{E^*}\int\left(-J^2\right)\ d\Omega_{{\bf q}^{*}}\ .
\end{equation}

\section{Virtual bremsstrahlung from fermions}
\label{fermions}

It has been shown in \cite{bpp} that for virtual bremsstrahlung
from fermions, the terms proportional
to $p^\mu$ in the expression (\ref{j}) can be obtained in the same
wayas in the real photon bremsstrahlung off fermions (see, e.g.,
\cite{berest}). In order to obtain the form of $J$ analogous to
(\ref{j}), additional approximations are required. One has to discard
some terms while keeping others of the same order in $q$. The
justification of such a procedure is unclear.

One may hope that if the summing (or averaging) over spins of hadrons
is performed, the additional contributions will rearrange and the
formulas identical to those pertaining to the pseudoscalar case will
be restored. This would resemble a similar development in the
next-to-leading terms in real photon bremsstrahlung \cite{burkroll}.

To investigate such a possibility, let as assume that in the
nonradiative reaction (\ref{reaction1}), a charged fermion-antifermion
(e.g., proton-antiproton) pair is produced. To be more definite, we
assign the index 1 to the fermion ($Q_1=1$) and 2 to the antifermion
($Q_2=-1$). Now, the matrix element exhibits the form
\begin{equation}
\label{mef1}
{\cal M}_0=\bar u^{(s_1)}(p_1)\ \Gamma\ v^{(s_2)}(p_2)\ ,
\end{equation}
where $\Gamma\equiv\Gamma(p_a,p_b,p_1,\cdots,p_n)$ is a matrix in the
spinor space. To simplify notation, we will display its arguments
only if they differ from the values just shown.
Squaring the matrix element (\ref{mef1}) and summing over the spin
projections $s_1$ and $s_2$, we get
\begin{equation}
\label{mef1sq}
\overline{\left|{\cal M}_0\right|^2}={\rm Tr}\left[(\hat{p}_1+m)\Gamma
(\hat{p}_2-m)\Gamma^\prime\right]\ ,
\end{equation}
where $\Gamma^\prime=\gamma^0\Gamma^\dagger\gamma^0$. The quantity
(\ref{mef1sq}) determines the unpolarized cross section of the
nonradiative reaction (\ref{reaction1}) with the two spin-one-half
hadrons in the final state. For a later use, let us notice that
\begin{equation}
\label{dm0sqdp1}
\frac{\partial{\overline{\left|{\cal M}_0\right|^2}}}
{\partial p_1^\alpha}={\rm Tr}\left\{\left[
\Gamma^\prime\gamma_\alpha\Gamma+
\Gamma^\prime(\hat{p}_1+m)\frac{\partial\Gamma}{\partial p_1^\alpha}+
\frac{\partial\Gamma^\prime}{\partial p_1^\alpha}
(\hat{p}_1+m)\Gamma\right](\hat{p}_2-m)\right\}\ ,
\end{equation}
and
\begin{equation}
\label{dm0sqdp2}
\frac{\partial{\overline{\left|{\cal M}_0\right|^2}}}
{\partial p_2^\alpha}={\rm Tr}\left\{
(\hat{p}_1+m)\left[\Gamma\gamma_\alpha\Gamma^\prime+
\frac{\partial\Gamma}{\partial p_2^\alpha}
(\hat{p}_2-m)\Gamma^\prime+
\Gamma(\hat{p}_2-m)
\frac{\partial\Gamma^\prime}{\partial p_2^\alpha}
\right]\right\}\ .
\end{equation}

As a next step, let us consider the corresponding dilepton-producing
reaction (\ref{reaction2}). We will concentrate on virtual
bremsstrahlung from fermions and, for simplicity, take all the mesons
neutral. In addition, we neglect any anomalous electromagnetic
interactions. The changes in the strong interaction matrix element
will be incorporated by
\begin{equation}
\label{gamma12q}
\Gamma(p_i+q)=\Gamma+q^\alpha\frac{\partial\Gamma}
{\partial p_i^\alpha}\ .
\end{equation}
The contact electromagnetic interaction term associated with $i$th
fermion line leaving the strong interaction core comes out as
\begin{equation}
\label{contfer}
C_i^\mu=-eQ_i\frac{\partial\Gamma}{\partial p_{i\mu}}\ .
\end{equation}
Using the Feynman rules for spinor electrodynamics,
Dirac equation in momentum space, and the properties of the
$\gamma$-matrices, we find the matrix element of reaction
(\ref{reaction2})
\begin{equation}
\label{mef2}
{\cal M} = eL_\mu\ \bar u^{(s_1)}(p_1)\ K^\mu v^{(s_2)}(p_2)\ .
\end{equation}
The four-vector $L_\mu$ has the same meaning as before
[see (\ref{leptonterm})].
We have, neglecting higher than linear $q$-terms in the numerators,
\begin{equation}
\label{kfermion}
K^\mu=\Gamma J^\mu+
\frac{1}{4\ p_1\!\cdot\!q}\biglb[\gamma^\mu,\hat{q}\bigrb]\Gamma+
\frac{1}{4\ p_2\!\cdot\!q}\Gamma\biglb[\gamma^\mu,\hat{q}\bigrb]+
\left(g^{\mu\alpha}q^\beta-g^{\mu\beta}q^\alpha\right)
\sum_{i=1,2}Q_i\frac{p_{i,\alpha}}{p_i\!\cdot\!q}
\frac{\partial\Gamma}{\partial p_i^\beta} \ .
\end{equation}
In accordance with (\ref{j}), we denoted
\begin{equation}
\label{j12}
J^\mu=\frac{(2p_1+q)^\mu}{2\,p_1\!\cdot\!q+M^2}-
      \frac{(2p_2+q)^\mu}{2\,p_2\!\cdot\!q+M^2}\ .
\end{equation}
At this point we can clearly see the difference between pseudoscalar
and fermion cases. When we neglected the changes in the strong
interaction core as well as the contact terms in pseudoscalar case,
we obtained immediately the leading term approximation in the form
(\ref{msum}). For fermions, the extra terms
(those with commutators) prevent us from reaching the same goal.

The sum over the spins of the matrix element (\ref{mef2}) squared
assumes
the form
\begin{equation}
\label{mef2sq}
\sum_{s_1,s_2,s_+,s_-}\left|{\cal M}\right|^2=4\pi\alpha\ H^{\mu\nu}
L_{\mu\nu}\ .
\end{equation}
The tensor $L_{\mu\nu}$ is given by Eq.~(\ref{lmunu}). Up to the
leading and next-to-leading order in $q$,
\begin{eqnarray}
\label{hmunuftr}
H^{\mu\nu}&=&\overline{\left|{\cal M}_0\right|^2}\ J^\mu J^\nu+
\frac{1}{2}\Bigl\{
J^\mu\ {\rm Tr}\bigl[\left(\hat{p}_1+m\right)\Gamma
\left(\hat{p}_2-m\right){K^\prime}^\nu\bigr]\Bigr.\nonumber\\
& &+\Bigl. J^\mu\ {\rm Tr}\bigl[\left(\hat{p}_1+m\right)K^\nu
\left(\hat{p}_2-m\right)\Gamma^\prime\bigr]+\ \mu \leftrightarrow \nu
\Bigr\}\
\end{eqnarray}
with ${K^\prime}^\nu=\gamma_0(K^\nu)^\dagger\gamma_0$.
A straightforward manipulation guides us to an expression in parts of
which we are able to identify the right-hand sides of
Eqs.~(\ref{dm0sqdp1}) and (\ref{dm0sqdp2}). After replacing them by
corresponding derivatives we get
\begin{eqnarray}
\label{hmunuf}
H^{\mu\nu}&=&\overline{\left|{\cal M}_0\right|^2}\ J^\mu J^\nu +
\frac{1}{2}
\sum_{i,j}\frac{Q_i Q_j^\prime}{(p_i\!\cdot\!q)(p_j\!\cdot\!q)}
p_{i\alpha}\frac{\partial\overline{\left|{\cal M}_0\right|^2}}
{\partial p_i^\beta}\nonumber \\
& &\times\left[p_j^\mu\left(g^{\nu\alpha}q^\beta-g^{\nu\beta}q^\alpha
\right)+p_j^\nu\left(g^{\mu\alpha}q^\beta-g^{\mu\beta}q^\alpha\right)
\right]\ .
\end{eqnarray}
This is identical with what we would get from Eq.~(\ref{hmunu}) for
two outgoing charged mesons, with only one difference. A simple
square of nonradiative mesonic matrix element is replaced by the sum
over fermion spins. As an independent check of our result, we explored
also other situations (an incoming fermion-antifermion pair, one
incoming--one outgoing fermion/antifermion) and reached the same
conclusion. For the  initial state fermions, the sum is replaced by
the average. The generalization to more than one fermion pair is
obvious.

The tensor $H^{\mu\nu}$ is the central object of all bremsstrahlung
formulas. We have therefore proven that the dilepton production via
virtual bremsstrahlung off unpolarized spin-one-half fermions is
governed, in the leading and next-to-leading approximation, by the
same formulas as that off pseudoscalar mesons.

Especially, if we neglect the terms proportional to derivatives of
the unpolarized nonradiative matrix element squared, the tensor
$H^{\mu\nu}$ reduces to
\begin{equation}
\label{hmunufac}
H^{\mu\nu}=J^\mu J^\nu\ \overline{\left|{\cal M}_0\right|^2}
\end{equation}
with $J^\mu$ given by Eq.~(\ref{j}).
The key leading term approximation relations (\ref{d6sig}),
(\ref{d6sigmix}), and (\ref{newglobal}), which are of most practical
interest, are valid also for virtual bremsstrahlung from unpolarized
fermions.

It is a good check that our expression (\ref{virtsubl}) for the
virtual photon cross section meets, in the limit of zero photon mass,
the unpolarized photon cross section calculated from the Burnett and
Kroll~\cite{burkroll} matrix element.

\section{A survey of virtual bremsstrahlung formulas}
\label{survey}
\subsection{R\"{u}ckl formula}
In his work \cite{rueckl}, R\"{u}ckl suggested the
formula
\begin{equation}
\label{rueckl}
E_+E_-\frac{d^6\sigma^{e^+e^-}}{d^3p_+d^3p_-}=\frac{\alpha}{2\pi^2}
\frac{1}{M^2}\left(\omega\frac{d^3\sigma^\gamma}{d^3q}
\right)_{{\bf q}={\bf p}_++{\bf p}_-},
\end{equation}
which links the cross section for production of dileptons via virtual
photon
bremsstrahlung to the bremsstrahlung cross section for real photons.
The meaning of the symbols is as follows: ${\bf p}_+$ and ${\bf p}_-$
are the momenta of leptons, $E_\pm=({\bf p}_\pm^{\ 2}+\mu^2)^{1/2}$
are their energies, $M^2=(p_++p_-)^2$ is the dilepton mass squared,
${\bf q}$\,\, is the momentum of photon, $\omega=|{\bf q}|$ is its
energy,
and $\alpha$ is the fine-structure constant. Because of the vanishing
photon mass, it is impossible to satisfy the relation
$\omega=E_++E_-$ simultaneously with ${\bf q}={\bf p}_++{\bf p}_-$.

Let us investigate now how Eq.~(\ref{rueckl}) copes with general
principles we mentioned in the Introduction. The principle of
relativity \cite{einstein} requires that any meaningful formula must
be relativistically covariant. This implies that if we view the
bremsstrahlung process from another (primed) frame, we must find the
same relation among the transformed quantities as we did in the
original frame:
\begin{equation}
\label{ruecklprime}
E^\prime_+E^\prime_-\frac{d^6\sigma^{e^+e^-}}{d^3p^\prime_+d^3
p^\prime_-}=\frac{\alpha}{2\pi^2}\frac{1}{M^2}
\left(\omega^\prime\frac{d^3\sigma^\gamma}{d^3q^\prime}
\right)_{{\bf q}^\prime={\bf p}^\prime_++{\bf p}^\prime_-}.
\end{equation}
We have assumed for simplicity that the
velocities of colliding particles are collinear
and performed a boost along the collision axis (otherwise
$\sigma$'s also acquire primes). The left-hand sides of
Eqs.~(\ref{rueckl}) and (\ref{ruecklprime}) are obviously equal.
Relativistic covariance will thus be satisfied if and only if
\begin{equation}
\label{covar}
\left(\omega\frac{d^3\sigma^\gamma}{d^3q}
\right)_{{\bf q}={\bf p}_++{\bf p}_-}=
\left(\omega^\prime\frac{d^3\sigma^\gamma}{d^3q^\prime}
\right)_{{\bf q}^\prime={\bf p}^\prime_++{\bf p}^\prime_-}.
\end{equation}
But this condition cannot be fulfilled because in the new frame the
photon momentum differs from the sum of lepton's momenta. In fact,
for the longitudinal components of the corresponding vectors we have
\begin{eqnarray}
\label{longtrafo}
p^\prime_{\pm,L}&=&\gamma\left(p_{\pm,L}-\beta E_\pm\right)
\nonumber\\ q^\prime_L&=&\gamma\left(q_L-\beta \omega\right),
\end{eqnarray}
which leads to
\begin{equation}
\label{longmome}
q^\prime_L=p^\prime_{+,L}+p^\prime_{-,L}+\beta\gamma
\left(E_++E_--\omega\right)
\end{equation}
with a nonvanishing extra term on the right-hand side.

In order to apply the global variable test, let us first use
Eq.~(\ref{page45}) to cast the R\"{u}ckl formula in the form
\begin{equation}
\label{mixrueckl}
E\ \frac{d^6\sigma^{e^+e^-}}{dM^2d^3q\ d\Omega_+^*} =
\frac{\alpha}{8\pi^2}\frac{1}{M^2}\sqrt{1-\frac{4\mu^2}{M^2}}\,\,
\omega\frac{d^3\sigma^\gamma}{d^3q}\ .
\end{equation}
Integration over the positron momentum angles is simple because
nothing depends on them:
\begin{equation}
\label{globrueckl}
E\ \frac{d^4\sigma^{e^+e^-}}{dM^2d^3q} =
\frac{\alpha}{2\pi}\frac{1}{M^2}\sqrt{1-\frac{4\mu^2}{M^2}}\,\,
\omega\frac{d^3\sigma^\gamma}{d^3q}\ .
\end{equation}
This formula does not obviously have the form required
by Eq.~(\ref{global}).
If we nevertheless extract from it the virtual photon cross section,
which is defined by (\ref{global}), we arrive at
\begin{equation}
\label{virtrueckl}
\left(\frac{d^3\sigma^{\gamma^*}}{d^3q}\right)_M
=\frac{3\omega}{2\sqrt{\omega^2+M^2}}\frac{M^2}{M^2+2\mu^2}
\frac{d^3\sigma^{\gamma}}{d^3q}\ .
\end{equation}
The $M\rightarrow 0$ limit of this expression is zero. If we are not
so strict and require only $\mu\ll M\ll \omega$ (this is the
situation met, e.g., in the low-mass, high-transverse-momentum
dielectron production), we are left with another surprising relation
\begin{equation}
\label{1.5rueckl}
\left(\frac{d^3\sigma^{\gamma^*}}{d^3q}\right)_M
=\frac{3}{2}\frac{d^3\sigma^{\gamma}}{d^3q}\ .
\end{equation}
The presence of an incorrect numerical factor in the R\"{u}ckl
formalism was already noticed by Craigie \cite{craigie}, who
ascribed it to the unjustified omission of the term $l_\mu l_\nu$ in
the lepton tensor [see Eq.~(\ref{lmunu})].

It is clear that the principal problems we have discussed here are
not germane to the R\"{u}ckl bremsstrahlung formalism, but are common
to all the approaches where the dilepton cross section is assumed to
be proportional  to the photon cross section at the same momentum.

To obtain Eq.~(\ref{rueckl}) in our formalism, we must
(i) write $\ p_i\!\cdot\!q=|{\bf q}\,|E_i-{\bf p}_i\!\cdot\!{\bf q}\ $
instead of the correct
$\ p_i\!\cdot\!q=(M^2+{\bf q}^{\ 2})^{1/2}E_i-
{\bf p}_i\!\cdot\!{\bf q}\ $ in Eq.~(\ref{j});
(ii)
omit the term $l_\mu l_\nu$ in Eq.~(\ref{lmunu});
(iii)
neglect $q^\mu$ in the numerators and $M^2$ in the denominators
of Eq.~(\ref{j}).
The first approximation is fatal for Lorentz covariance, the second
one for the global variable test.

\subsection{Modifications of the R\protect{\"{u}}ckl formula}
\label{modifications}

The R\"{u}ckl formula (\ref{rueckl}) was often used in the
calculation of dilepton yield in experimental and theoretical works.
Several modifications of it have been put forward. As will be shown
later, they brought improvement  in a pragmatic sense, but were not
able to cure its principal drawbacks.

Gale and Kapusta \cite{galkap} replaced the photon energy squared
$q_0^2$ ($\omega^2$ in our notation) in the denominator in their
Eq.~(5) by a symmetrized combination $E(E^2-M^2)^{1/2}$, where
$E=E_++E_-$ is the dilepton energy. The quantity $(E^2-M^2)^{1/2}$
represents the dilepton momentum and as such must be equal to
the photon momentum $|{\bf q}|$, which is in turn equal to the
photon energy $q_0$. The replacement $q_0^2\rightarrow
E(E^2-M^2)^{1/2}$ is thus equivalent to multiplying the right-hand
side of Eq.~(\ref{rueckl}) by $\omega/E$. The same modification was
used by Haglin, Gale, and Emel'yanov in \cite{hagaem1} and also in
a part of the paper \cite{clresa91} by Cleymans, Redlich, and Satz.

In a subsequent paper \cite{galkap89}, Gale and Kapusta introduced a
factor which partially corrected the soft photon approximation for
processes with two particles in the final state. It accounts for the
shrinking of the Lorentz invariant phase space available to them,
which results from the emission of a dilepton with mass $M$ and
center-of-mass-system energy $E^*$. The
production of dileptons that would violate the energy-momentum
conservation is forbidden. The correction factor is given by
(we display it in a simpler form assuming equal masses $m$ for the
final-state hadrons)
\begin{equation}
\label{rgaka}
R(s,M,E^*)=\sqrt{\frac{s\left(s_2-4m^2\right)}
{s_2\left(s-4m^2\right)}}\ ,
\end{equation}
where $s$ is the invariant energy available for all final-state
particles and
\begin{equation}
\label{s2}
s_2=s+M^2-2E^*\sqrt{s}\ .
\end{equation}

In a paper \cite{hagaem2} of Haglin, Gale, and
Emel'yanov, the photon energy squared $q_0^2$ enters the
denominator of the quantity
$|\epsilon\!\cdot\!J|^2_{ab\rightarrow cd}$
[see their (3.6), (3.7) or (A9)]. It is forced to acquire the value
of $(E_++E_-)^2$ by the second $\delta$-function in their Eq.~(3.4).
It induces a multiplicative factor of $\omega^2/E^2$  on the
right-hand side of the R\"{u}ckl formula (\ref{rueckl}). The authors
also used the correction factor (\ref{rgaka}). Recently, Haglin and
Gale \cite{haga94} utilized the same modification of the R\"{u}ckl
formula to assess the bremsstrahlung contribution to the $e^+e^-$
invariant mass distribution in proton-proton and proton-neutron
collisions at the lab kinetic energy of 4.9 GeV. For the final
states with more than two hadrons they modified the phase-space
correction factor accordingly.

\subsection{Paper by Craigie and Thompson}
After a thorough discussion of the real photon bremsstrahlung, the
authors of~\cite{craithom} turned to dileptons. If they had really
done what they described verbally at the bottom of p.~129, they would
have obtained immediately a simple and correct formula for the double
differential
cross section in the momenta of leptons [our Eq.~(\ref{d6sig})] with
a little different Ansatz for the four-vector $J$ (same as was used
later in \cite{bpp}). Unfortunately, they instead wrote a cumbersome
and obviously
wrong formula (3.1) on p.~130. The incorrectness of the latter
can be seen, e.g., from the fact that the quantity Tr$\{\rho L\}$,
which enters it, depends on the momenta of hadrons via the four-vector
$J$ [see their Eq.~(3.2)]. But there is no integration over hadron
momenta in (3.1) [only that hidden in $2q_0d\sigma/d^3q$, given by
Eq.~(2.5)]. The left-hand side of formula (3.1) in \cite{craithom}
thus depends on hadron momenta, which makes it unusable for
evaluating the inclusive dilepton cross section.

The authors probably tried to express the double differential cross
section as proportional to the cross section for virtual photon
production. But, as we have  learnt, this is possible only for
the dilepton cross section in global dilepton variables [compare our
(\ref{d6sig}) and (\ref{newglobal})].

\subsection{Formula used by Goshaw {\it et al.}}
The experimentally observed production of very-low-energy
$e^+e^-$ pairs in 18 GeV/$c$ $\pi^\pm p$ collisions was reported
and compared to the expectations based on the leading-term
bremsstrahlung calculations in the paper \cite{goshawee}. The
authors used the formula
\begin{eqnarray}
\label{goshaw}
\frac{d\sigma}{d\mu\ d\omega\ d\Omega\ d\Omega^*} &=&\frac{\alpha^2}
{(2\pi)^4}\frac{\left[\mu^2-(2m)^2\right]^{1/2}
\left(\omega^2-\mu^2\right)^{1/2}}{\mu^2\omega^2}\nonumber\\[5pt]
&&\times\int d^3P_3\cdots d^3P_n\left[\frac{(J\!\cdot\!l)^2}
{\mu^2}-(J\!\cdot\!J)
\right]\frac{d\sigma_h}{d^3P_3\ d^3P_4\cdots d^3P_n}\ ,
\end{eqnarray}
where $\mu$, $\omega$, and $d\Omega$ are the dilepton mass, energy,
and infinitesimal solid angle, respectively, $d\Omega^*$ is the
infinitesimal solid angle of positron in the dilepton rest frame,
$l=P_+-P_-$, and
\begin{equation}
\label{jgoshaw1}
J=\omega\sum_{i=1}^n\frac{Q_i}{(P_++P_-)\!\cdot\!P_i}P_i\ .
\end{equation}
The charge quantum number of outgoing particles ($i=3,\cdots,n$) is
$Q_i$, of incoming ($i=1,2$) ones $(-Q_i)$. After changing the
notation used in \cite{goshawee} to ours, Eq.~(\ref{goshaw}) reads
\begin{equation}
\label{d6siggos}
E\ \frac{d^6\sigma^{e^+e^-}}{dM^2d^3q\ d\tilde\Omega_+} =
\frac{\alpha^2}{32\pi^4}\frac{1}{M^2}
\sqrt{1-\frac{4\mu^2}{M^2}}
\int\left[\frac{1}{M^2}(l\!\cdot\!J_G)^2-J_G^2\right]d\sigma_0\ ,
\end{equation}
where now
\begin{equation}
\label{jg}
J_G^\mu=\sum  Q^\prime_i\frac{p_i^\mu}{p_i\!\cdot\!q}\ .
\end{equation}
The formula (\ref{d6siggos}) differs from our Eq.~(\ref{d6sigmix})
by the sign of one of the terms in brackets. It is difficult to trace
the origin of this discrepancy. The derivation of the
formula (\ref{d6siggos}) has not been published, although it
was signalized in \cite{goshawee}. We suspect that
Eq.~(\ref{d6siggos}) is not correct, because after integrating it
over the positron momentum directions we get the formula
\begin{equation}
\label{globgosh}
E\ \frac{d^4\sigma^{e^+e^-}}{dM^2d^3q} =
\frac{\alpha^2}{6\pi^3}\frac{1}{M^2}
\left(1-\frac{\mu^2}{M^2}\right)
\sqrt{1-\frac{4\mu^2}{M^2}}\
\int (-J_G^2)\ d\sigma_0\ ,
\end{equation}
which does not comply with the global variable test, defined
by Eqs.~(\ref{global}) and (\ref{limit}). In this respect, the fact
that the authors used additional approximations to get their
Eq.~(\ref{jgoshaw1}) [compare (\ref{j}) and (\ref{jg})] seems to be
of lesser importance.

\subsection{Formula of Balek, Pi\v{s}\'{u}tov\'{a},
and Pi\v{s}\'{u}t}

In paper \cite{bpp} the formula analogous to our
(\ref{newglobal}) was written for a charged particle  scattering
on a neutral particle (or on a potential). Their formalism
satisfies both the Lorentz covariance and  global variable tests.
However, instead of the vector $J^\mu$ given by Eq.~(\ref{j}), the
following one was
chosen:
\begin{equation}
\label{pisutc}
C^\mu=\frac{p_1^\mu}{p_1\!\cdot\!q}-\frac{p_a^\mu}{p_a\!\cdot\!q}\ .
\end{equation}
Here, $p_a$ and $p_1$ are four-momenta of the charged particle
before and after the scattering, respectively.
In this case, Eq.~(\ref{j}) becomes
\begin{equation}
\label{jscatt}
J^\mu= \frac{(2p_1+q)^\mu}{2\,p_1\!\cdot\!q+M^2}-
       \frac{(2p_a-q)^\mu}{2\,p_a\!\cdot\!q-M^2}\ .
\end{equation}
To get the Ansatz (\ref{pisutc}) of Balek {\it et~al.} we have to
neglect $q^\mu$ in the numerators and $M^2$ in the denominators above.

For later convenience we write here the $n$-final-particle
generalization of the Balek {\it et~al.} formula in our notation.
The same conventions as used in (\ref{jr}) apply.
\begin{equation}
\label{bppn}
E\left(\frac{d^3\sigma^{\gamma^*}}{d^3q}\right)_M
=\frac{\alpha}{4\pi^2}\int\ -\left(\sum Q_i^\prime\frac{p_i^\mu}
{E_iE-{\bf p}_i\!\cdot\!{\bf q}\ } \right)^2d\sigma_0\ .
\end{equation}

We will return to the approximation of Balek, Pi\v{s}\'{u}tov\'{a}
and Pi\v{s}\'{u}t in Appendix \ref{appmt}, in connection with the
concept of transverse-mass scaling (see below).

\subsection{Formula of Cleymans, Goloviznin, and Redlich}
The authors of \cite{clgore1} used the following
classical-electrodynamics motivated expression for the energy per
unit of momentum radiated in the form of virtual photons with mass
$M$ if the charged particle changes its velocity from
${\bf v}_1$ to ${\bf v}_2$ (we switch from their notation to ours):
\begin{equation}
\label{clgore}
\frac{d^3I}{d^3q}=\frac{\alpha}{4\pi^2}\left|
\frac{{\bf n} \times{\bf v}_2}{E-{\bf v}_2\!\cdot\!{\bf q}}
-\frac{{\bf n} \times{\bf v}_1}{E-{\bf v}_1\!\cdot\!{\bf q}}
\right|^2.
\end{equation}
Our expression for this quantity stems from the relations
\begin{equation}
\label{enrelation}
\frac{d^3I}{d^3q}=E\frac{d^3N^{\gamma^*}}{d^3q}
               =\frac{1}{\sigma_0}E\left(\frac{d^3\sigma^{\gamma^*}}
                  {d^3q}\right)_M,
\end{equation}
where $N^{\gamma^*}$ is the mean number of virtual photons with mass
$M$ per a collision, in conjunction with Eqs.~(\ref{j2}) and
(\ref{myvirtual}).
Choosing the cross section $d\sigma_0$ that allows only the required
change of velocity, we get
\begin{equation}
\label{myclgore}
\frac{d^3I}{d^3q}=\frac{\alpha}{4\pi^2}\left[
\left({\bf n}\times{\bf J}\ \right)^2
+\frac{M^2}{E^2}\left({\bf n}\!\cdot\!{\bf J}\ \right)^2\right]
\end{equation}
with
\begin{equation}
\label{myvectorj}
{\bf J}=\frac{{\bf v}_2+{\bf q}/(2E_2)}
{E-{\bf v}_2\!\cdot\!{\bf q}+M^2/(2E_2)}-
\frac{{\bf v}_1-{\bf q}/(2E_1)}
{E-{\bf v}_1\!\cdot\!{\bf q}-M^2/(2E_1)}.
\end{equation}
Comparing (\ref{clgore}) with (\ref{myclgore}) and (\ref{myvectorj})
we can see that in \cite{clgore1} two additional approximations have
tacitly been made in contrast to our formalism:
(i)
The terms proportional to ${\bf q}\ $ in the numerators and those
proportional to $M^2$ in the denominators of Eq.~(\ref{myvectorj})
have been neglected. This is equivalent to the approximation made
in~\cite{bpp}.
(ii)
The second term in the brackets of (\ref{myclgore}) has not been
considered.
This approximation is more dangerous, since it makes the quantity
(\ref{clgore}) non-covariant under Lorentz transformations.

\subsection{Transverse mass scaling}

In this and next subsections we are going to report about two
attempts to relate the virtual photon cross section to the
experimentally accessible cross section for real photon production.

Farrar and Frautschi \cite{farrfrau} and Cobb {\it et al.} \cite{cobb}
used the concept of transverse mass scaling. They assumed that the
virtual photon cross section does not depend on three variables
(longitudinal momentum $q_L$, transverse momentum $q_T$, and virtual
photon mass $M$), but rather on only two [$q_L$ and transverse mass
$M_T=(q_T^2+M^2)^{1/2}$]. The condition (\ref{limit}) then leads to
the relation
\begin{equation}
\label{mtscaling}
\left(\frac{d^3\sigma^{\gamma^*}}{d^3q}\right)_M=
\frac{d^3\sigma^\gamma}{d^3p}\ ,
\end{equation}
where $p_L=q_L$ and $p_T=(q_T^2+M^2)^{1/2}$. Because both the energies
and longitudinal momenta of real and virtual photons are now equal,
the relation (\ref{mtscaling}) is covariant under longitudinal Lorentz
boosts. In our test process $\rho^0\rightarrow\pi^+\pi^-e^+e^-$,
this approach will be very successful.

The origin and limitations of the transverse-mass scaling from the
point of view of the leading term bremsstrahlung formalism are
discussed in Appendix \ref{appmt}.

\subsection{Real photon approximation}
\label{subsecrpa}

Blockus {\it et al.} \cite{blockus} conjectured that the formula
\begin{equation}
\label{blockus}
\left(E^*\frac{d^3\sigma^{\gamma^*}}{d^3q^*}\right)_M=
\frac{f(M,q_L^*,q_T)}{f(0,q_L^*,q_T)}\ \left(\omega^*
\frac{d^3\sigma^\gamma}{d^3q^*} \right)
\end{equation}
is valid in the center-of-mass frame (we use $q_L^*$ instead of their
$x=2q_L^*/\sqrt{s}$). The quantity $f$ is the structure function for
the production of a virtual photon of mass $M$ summed over photon
polarization states. The authors considered two options for their
ratio entering the right-hand side of Eq.~(\ref{blockus}):
(i) independent of $M$ (i.e., identically equal to 1), and
(ii) linearly dependent on $M$. None of these options seemed to be
excluded by their data integrated over the region of acceptance. In
our numerical comparison of various bremsstrahlung formulas we will
explore the former option
\begin{equation}
\label{rpa}
\left(E^*\frac{d^3\sigma^{\gamma^*}}{d^3q^*}\right)_M= \omega^*
\frac{d^3\sigma^\gamma}{d^3q^*}\ ,
\end{equation}
and will refer to it as the ``real photon approximation".

\section{A model of two-pion radiative decays of $\rho^0$}
\label{model}

As noted earlier, we will check the reliability of the various
leading term virtual bremsstrahlung formulas by applying them to the
strong-interaction decay $\rho^0\rightarrow \pi^+\pi^-$ in order to
get estimates of the differential decay width for
\begin{equation}
\label{rhotoee}
\rho^0\rightarrow \pi^+\pi^-e^+e^-.
\end{equation}
While a comparison of the bremsstrahlung formulas themselves is
instructive, additional insight may be gained by also comparing them
to results from a formalism that goes beyond the leading term
approximation. To our knowledge, nobody has investigated the
decay~(\ref{rhotoee}) theoretically yet, probably because the chance
to detect it experimentally is very meager. We present some estimates
in Subsection B.

The real photon counterpart of (\ref{rhotoee}), namely
$\rho^0\rightarrow \pi^+\pi^-\gamma$
has been dealt with by several authors \cite{singer,pascual,renard}.
In this work we will adopt Singer's approach \cite{singer}, which
agrees nicely with the experimental data (see below).
We will first recapitulate its main points and then generalize it to
the massive photon (dilepton) case.

To simplify notation, we introduce the following ratio of decay
widths
\begin{equation}
\label{bgamdef}
B^\gamma=\frac{\Gamma_{\rho^0\rightarrow \pi^+\pi^-\gamma}}
{\Gamma_{\rho^0\rightarrow \pi^+\pi^-}}\ .
\end{equation}
We will call it a branching ratio in spite of a small incorrectness
this introduces ($\Gamma_{\rho^0\rightarrow \pi^+\pi^-}$ is smaller
than the total width of $\rho^0$ by about 1 \%). The quantities
$B^{\gamma^*}$ and $B^{e^+e^-}$ will have analogous meaning.

\subsection{Decay {$\rho^0\rightarrow \pi^+\pi^-\gamma$}}

Strong interaction dynamics enters Singer's calculation through the
assumption about the $\rho^0\pi^+\pi^-$ vertex in the form
\begin{equation}
\label{vertex}
V^\alpha=f_{\rho\pi\pi}(p_1-p_2)^\alpha\ ,
\end{equation}
where $p_1$ ($p_2$) is the four-momentum of $\pi^+$ ($\pi^-$). For
external $\rho$'s, this is to be contracted with the polarization
vector $\epsilon_\alpha(\lambda_\rho)$. The coupling constant
$f_{\rho\pi\pi}$
can be fixed by utilizing the $\rho\rightarrow\pi\pi$ decay width.

When electromagnetic interactions are switched on by the minimal
interaction principle, the vertex (\ref{vertex}) generates
a contact term
\begin{equation}
\label{contact}
C^{\mu\alpha}=-2ef_{\rho\pi\pi}g^{\mu\alpha}\ ,
\end{equation}
which must be considered in addition to the usual vertices
of pseudoscalar electrodynamics (see Fig.~\ref{feynman}). It should
be contracted with both the $\rho$ and $\gamma$ lines which enter the
contact vertex together with the two pion lines. Combining
the resulting expressions for $\rho^0\rightarrow \pi^+\pi^-$
and $\rho^0\rightarrow \pi^+\pi^-\gamma$ decay widths
from~\cite{singer}, we obtain the formula
\begin{eqnarray}
\label{realsinger}
\frac{dB^\gamma}{d\omega^*}&=&
\frac{4\alpha}{\pi}\frac{1}
{\left(m_\rho^2-4m_\pi^2\right)^{3/2}}\frac{1}{\omega^*}
\biggl\{\omega_m^*\left(m_\rho^2-2m_\pi^2-2m_\rho\omega^*\right)
\ln\frac{1+\xi}{1-\xi}\biggr.\nonumber \\
& & \biggl. -m_\rho\xi
\left[\omega^*_m\left(m_\rho-2\omega^*\right)-2{\omega^*}^2\right]
\biggr\}\ ,
\end{eqnarray}
where $\omega^*$ is the photon energy in the $\rho^0$ rest frame and
\begin{equation}
\label{xi}
\xi=\sqrt\frac{2\left(\omega_m^*-\omega^*\right)}{m_\rho-2\omega^*}.
\end{equation}
The maximum value of $\omega^*$ is
$\omega_m^*=(m_\rho^2-4m_\pi^2)/(2m_\rho)$.

After integrating Eq.~(\ref{realsinger}) over photon energies greater
than 50 MeV we get the branching ratio of 1.12 \%, in nice agreement
with the experimental value $(0.99\pm0.16)$ \% \cite{dolinsky}. Also,
the distribution in photon energies (depicted in Fig.~\ref{realgam}
by solid curve)
agrees remarkably well with experiment \cite{vasserman}.

\subsection{Decay {$\rho^0\rightarrow \pi^+\pi^-e^+e^-$}}

As has already been stressed, in order to know the differential
decay width in global dilepton variables of the
decay~(\ref{rhotoee}), it is sufficient to calculate the differential
decay width for $\rho^0\rightarrow\pi^+\pi^-\gamma^*$ with a massive
photon. As was proven in \cite{baier}, the evaluation of the latter
is governed by the same Feynman rules as in the case of a real
photon. What changes are the kinematical relations which must
accommodate the non-vanishing photon mass. As a result, we obtain
\begin{eqnarray}
\label{virtsinger}
\left(\frac{dB^{\gamma^*}}{dq^*}\right)_M
&=&
\frac{4\alpha}{\pi}\frac{1}
{\left(m_\rho^2-4m_\pi^2\right)^{3/2}}
\frac{q^*}{{E^*}^2}\nonumber \\
&\times & \left\{\left[E_m^*\left(m_\rho^2-2m_\pi^2-2E^*m_\rho\right)
+M^2\left(E^*_m+\frac{m_\rho}{4}\right)\right]
\ln\frac{E^*+\xi q^*}{E^*-\xi q^*}\right.\nonumber \\
 &-&\left. E^*q^*\xi
\left[\frac{\left(m_\rho-2E_m^*\right)\left(2m_\rho E^*_m-M^2\right)}
{2\left({E^*}^2-\xi^2{q^*}^2\right)}-2m_\rho\right]\right\}\ .
\end{eqnarray}
Here,
\begin{equation}
\label{estm}
E_m^*=\frac{m_\rho^2+M^2-4m_\pi^2}{2m_\rho}
\end{equation}
is the maximum value of $E^*$, the massive photon energy in the
$\rho^0$-rest frame and
\begin{equation}
\label{ximsq}
\xi=\sqrt\frac{2m_\rho\left(E_m^*-E^*\right)}
{m_\rho^2+M^2-2m_\rho E^*}\ .
\end{equation}
The interested reader may find a few intermediate steps in Appendix
\ref{appsingvirt}.
Integrating the differential branching ratio for
$\rho^0\rightarrow\pi^+\pi^-e^+e^-$
\begin{equation}
\label{eesinger}
\frac{d^2B^{e^+e^-}}{dM^2dq^*}={\cal T}(M^2)
\left(\frac{dB^{\gamma^*}}{dq^*}\right)_M
\end{equation}
over the dilepton momenta and masses, we get
the value $1.5\times10^{-4}$ for $\rho^0\rightarrow \pi^+\pi^-e^+e^-$
and $4.9\times10^{-7}$ for $\rho^0\rightarrow \pi^+\pi^-\mu^+\mu^-$.
A low-dielectron-mass cut of 50 MeV/$c^2$ (100 MeV/$c^2$) reduces
the branching ratio to $1.1\times10^{-5}$ ($4.0\times10^{-6}$).

\section{Bremsstrahlung formulas for two-pion
radiative decays of $\rho^0$}
\label{bremsstrahlung}

Our goal here is to calculate the differential branching ratio
of $\rho^0\rightarrow \pi^+\pi^-e^+e^-$  within all approaches to
dilepton bremsstrahlung we found in the literature. Because some
approaches \cite{bpp,clgore1} do not provide the distribution in the
momenta of electrons, we will concentrate on a less general
distribution in global dilepton variables. For later use and
reference we first explore the simpler case of the $\rho^0$ decay
to a dipion and photon.

\subsection{Decay $\rho^0\rightarrow \pi^+\pi^-\gamma$}
In order to calculate the branching ratio in the leading term
bremsstrahlung approximation, we start with the formula
\begin{equation}
\label{eq11}
\frac{dB^\gamma}{d\omega^*}=
\frac{\alpha\omega^*}{4\pi^2}\int(-J_R^2)d\Omega_{{\bf q}^*}\ ,
\end{equation}
which is a real photon mutation of Eq.~(\ref{virtdecay}). Using Eqs.
(\ref{j2}) and (\ref{jr}), we can write the integrand in the form
\begin{equation}
\label{jrsqdec}
-J_R^2=\left(\frac{2v^*}{\omega^*}\right)^2\frac{\sin^2\alpha^*}
{\left(1-{v^*}^2\cos^2\alpha^*\right)^2}\ ,
\end{equation}
where
\begin{equation}
\label{speed}
v^*=\sqrt{1-\frac{4m_\pi^2}{m_\rho^2}}
\end{equation}
is the speed of pions and $\alpha^*$ the angle between the photon and
$\pi^+$ momenta in the $\rho^0$ rest frame. An elementary integration
gives us
\begin{equation}
\label{eq7}
\frac{dB^\gamma}{d\omega^*}=\frac{2\alpha}{\pi\omega^*}
\left(\frac{1+{v^*}^2}{2v^*}
\ln\frac{1+v^*}{1-v^*}-1\right).
\end{equation}
A numerical evaluation shows (Fig.~\ref{realgam}, dashed curve) that
the leading term bremsstrahlung formula (\ref{eq7}) exceeds the data,
especially at large momenta. This trend is understandable, as the
formula ignores the energy-momentum-conservation constraints. The
branching ratio for $\omega^*>50$ MeV is 2.05 \%, almost twice as
much as the experimental observation. For later reference, we also
present in Fig.~\ref{comp000}  the ratio of the
leading-term-bremsstrahlung
branching ratio (\ref{eq7}) to that calculated from Singer's model
(\ref{realsinger}) as a function of the photon momentum $q^*$
(which is, of course, equal to its energy $\omega^*$).

\subsection{Decay $\rho^0\rightarrow \pi^+\pi^-e^+e^-$}

Our aim here is to derive formulas for the differential branching
ratios in dilepton momentum and mass $d^2B^{e^+e^-}/dM^2dq^*$ in
the various formalisms. We will apply the same assumptions and
approximations which the various authors have used when deriving
their formulas for the differential cross section. For the
formalisms that satisfy the global variable test
(\cite{bpp,clgore1}, our formalism), it is sufficient to calculate
the branching ratio for the virtual photon production
\begin{equation}
\label{virtbrb}
\left(\frac{dB^{\gamma^*}}{dq^*}\right)_M =
\frac{\alpha}{4\pi^2}
\frac{{q^*}^2}{E^*}\int\left(-J^2\right)\ d\Omega_{{\bf q}^{*}}
\end{equation}
[cf. (\ref{virtdecay})] and then multiply it by the function
${\cal T}(M^2)$, given by Eq.~(\ref{tmsq}). In other cases, the
procedure will be less straightforward.

Applying our formula (\ref{j}) to the process considered here, we
can write
\begin{equation}
\label{myjvec}
{\bf J}=\frac{2m_\rho^{-1}}{(E^*+M^2/m_\rho)^2-
(v^*q^*\cos\theta^*)^2}
\left[(E^*m_\rho+M^2){\bf v}^{\ *}+
v^*q^*\cos\theta^*\ {\bf q}^{\ *}\right] \ .
\end{equation}
Now we insert this into (\ref{j2}) and calculate the integral which
enters the formula (\ref{virtbrb}). We finish with
\begin{eqnarray}
\label{mybremsb}
\frac{d^2B^{e^+e^-}}{dM^2dq^*}&=&\frac{2\alpha}{\pi}\frac{q^*}{{E^*}}
{\cal T}(M^2)
\left[\frac{1+{v^*}^2-M^2/m_\rho^2}{2v^*(E^*+M^2/m_\rho)}
\ln\frac{E^*+M^2/m_\rho+v^*q^*}{E^*+M^2/m_\rho-v^*q^*}\right.
\nonumber \\& &-\left.
\frac{(1-{v^*}^2-M^2/m_\rho^2)q^*}{{E^*}^2-(v^*q^*)^2+
(2E^*+M^2/m_\rho)M^2/m_\rho}\right]\ .
\end{eqnarray}
This is the leading term bremsstrahlung formula based on the
formalism we developed in Sec.~\ref{mesons}. The ratio of
(\ref{mybremsb}) to the ``exact" formula (\ref{eesinger})
is presented as a function of dielectron momentum $q^*$ for
several dielectron masses in Figs.~\ref{comp010} through
\ref{comp200} by a solid curve.

The corresponding formula in the Balek, Pi\v{s}\'{u}tov\'{a}, and
Pi\v{s}\'{u}t \cite{bpp} approximation is obtained along the same
lines. The only difference is in using their (\ref{pisutc}), which,
translated from the scattering case to the
$\rho^0\rightarrow\pi^+\pi^-$ decay, reads
\begin{equation}
\label{bppcvec}
{\bf J}=\frac{2E^*}{{E^*}^2-(v^*q^*\cos\theta^*)^2}{\bf v}^{\ *}\ ,
\end{equation}
instead of our (\ref{myjvec}). The result is
\begin{equation}
\label{bppbremsb}
\frac{d^2B^{e^+e^-}}{dM^2dq^*}=\frac{2\alpha}{\pi}\frac{q^*}
{{E^*}^2}{\cal T}(M^2)
\left[\frac{1+{v^*}^2}{2v^*}
\ln\frac{E^*+v^*q^*}{E^*-v^*q^*}-
\frac{\left(1-{v^*}^2\right)E^*q^*}{{E^*}^2-(q^*v^*)^2}\right]\ .
\end{equation}
It is a good check that for $M\ll m_\rho$, Eq.~(\ref{mybremsb})
tends to agree with (\ref{bppbremsb}).

When we combined the $M_T$-scaling hypothesis (\ref{mtscaling})
with the leading term bremsstrahlung formula for real photons
(\ref{myreal}), we got the same result as from the Balek~{\it et~al.}
approximation (\ref{bppbremsb}).
This prompted us to investigate the connection between
the two approaches in more detail in Appendix \ref{appmt}.

In order to calculate the differential branching ratio in the
Cleymans, Goloviznin, and Redlich approximation \cite{clgore1},
we also use (\ref{bppcvec}). In addition, we have to discard the dot
product in Eq.~(\ref{j2}). In this case we get
\begin{equation}
\label{cgrbremsb}
\frac{d^2B^{e^+e^-}}{dM^2dq^*}=\frac{2\alpha}{\pi}\frac{1}{E^*}
{\cal T}(M^2)
\left[\frac{{E^*}^2+(q^*v^*)^2}{2v^*E^*q^*}
\ln\frac{E^*+v^*q^*}{E^*-v^*q^*}-1\right]\ .
\end{equation}

To find the branching ratios in remaining approximations no additional
calculations are needed; we need only to combine the proper formulas.

The R\"{u}ckl approximation formula (\ref{globrueckl}) in terms of
branching ratios may be written as
\begin{equation}
\label{rueglobb}
E^*\frac{d^2B^{e^+e^-}}{dM^2dq^*}=
\frac{\alpha}{2\pi}\frac{1}{M^2}\sqrt{1-\frac{4\mu^2}{M^2}}\,\,
q^*\frac{dB^\gamma}{dq^*}.
\end{equation}
Merging this with (\ref{eq7}), we get
\begin{equation}
\label{ruebremsb}
\frac{d^2B^{e^+e^-}}{dM^2dq^*}=
\frac{\alpha^2}{\pi^2}\frac{1}{E^*M^2}
\sqrt{1-\frac{4\mu^2}{M^2}}
\left(\frac{1+{v^*}^2}{2v^*}\ln\frac{1+v^*}{1-v^*}-1\right)\ .
\end{equation}

The Gale and Kapusta \cite{galkap} modification of the
R\"{u}ckl formula consists in multiplying (\ref{ruebremsb})
by the factor $q^*/E^*$. To account for the later modification by
the same authors \cite{galkap89}, we must also include their
phase-space correction factor $R(m_\rho^2,M,E^*)$, see (\ref{rgaka}).

To adopt the Haglin, Gale and Emely'anov \cite{hagaem2} modification
of the R\"{u}ckl formula, we have to multiply (\ref{ruebremsb}) by
$R(m_\rho^2,M,E^*)(q^*/E^*)^2$.

When dealing with the Goshaw {\it et al.} \cite{goshawee} formalism,
we first rewrite Eq.~(\ref{globgosh}) by means of (\ref{virtbrb})
to the form
\begin{equation}
\label{goshb}
\frac{d^2B^{e^+e^-}}{dM^2dq^*}=\frac{2\alpha}{3\pi}\frac{1}{M^2}
\left(1-\frac{\mu^2}{M^2}\right)\sqrt{1-\frac{4\mu^2}{M^2}}
\left(\frac{dB^{\gamma^*}}{dq^*}\right)_M\ .
\end{equation}
The virtual-photon branching ratio on the right-hand side can be
taken from the Balek {\it et al.} formalism, because they use the
identical four-vector $J$ [compare (\ref{jg}) with (\ref{pisutc})].
It can easily be read off the Eq.~(\ref{bppbremsb}); one simply
ignores the function ${\cal T}(M^2)$. As a result, we get
\begin{eqnarray}
\label{gosbremsb}
\frac{d^2B^{e^+e^-}}{dM^2dq^*}&=&\frac{4\alpha^2}{3\pi^2}\frac{q^*}
{(E^*M)^2}
\left(1-\frac{\mu^2}{M^2}\right)\sqrt{1-\frac{4\mu^2}{M^2}}
\left[\frac{1+{v^*}^2}{2v^*}
\ln\frac{E^*+v^*q^*}{E^*-v^*q^*}\right.\nonumber \\
&-&\left.\frac{\left(1-{v^*}^2\right)E^*q^*}{{E^*}^2-(q^*v^*)^2}
\right]\ .
\end{eqnarray}

The real photon approximation (\ref{rpa})
combined with (\ref{eesinger}) and (\ref{eq7}) leads to
\begin{equation}
\label{fgabremsb}
\frac{d^2B^{e^+e^-}}{dM^2dq^*}=
\frac{2\alpha}{\pi E^*}{\cal T}(M^2)
\sqrt{1-\frac{4\mu^2}{M^2}}
\left(\frac{1+{v^*}^2}{2v^*}\ln\frac{1+v^*}{1-v^*}-1\right)\ .
\end{equation}

All the formulas we have derived here are normalized to the ``exact"
formula [Eqs.~(\ref{virtsinger}) and (\ref{eesinger})] and
visualized as functions of dielectron momentum at fixed dielectron
masses in Figs.~\ref{comp010} through \ref{comp200}.

\section{COMMENTS AND CONCLUSIONS}
\label{comments}

We based our formalism for dilepton production via virtual photon
bremsstrahlung on the following assumptions:
\begin{enumerate}
\item
Only the Feynman diagrams in which a virtual photon line is attached
to one of the external legs are important.
\item
The charged particles that participate in the process are
pseudoscalar mesons or unpolarized spin-one-half fermions.
\item
The dilepton four-momentum in the argument of the Dirac
$\delta$-function in Eq.~(\ref{xsec}) (or in similar relations for
decays or processes with more than two particles in the initial
state) may be neglected.
\item
The modifications of electromagnetic interactions of hadrons induced
by form factors and anomalous magnetic moments (in the case of
fermions) are negligible.
\end{enumerate}

To obtain the leading term approximation, we further neglected terms
that are proportional to the derivatives of the nonradiative matrix
element squared. Our notion of the leading term approximation is thus
based on the proportionality of the bremsstrahlung cross section to
that of the nonradiative reaction rather than on the order in dilepton
momentum $q$. In our leading term formulas, we use the four-vector $J$
that contains also subleading terms in $q$. The numerical comparison
with the ``exact" formula shows that it is beneficial.

In Sec.~\ref{survey}, we have pointed out the additional assumptions
which are required to reduce our formulas to the virtual
bremsstrahlung formulas that were derived or suggested previously. Of
these formulas, only one, namely that of Balek, Pi\v{s}\'{u}tov\'{a}
and Pi\v{s}\'{u}t \cite{bpp}, satisfied both the Lorentz covariance
and global variable tests.

The numerical analysis presented in Sec.~\ref{bremsstrahlung} shows
that all leading term virtual bremsstrahlung formulas (including
ours) overestimate the dilepton production, albeit to different
degrees. As a matter of fact, also the real photon
production is overestimated by the standard leading-term formula
(see Figs.~\ref{realgam} and \ref{comp000}).
But due to the more complex nature of virtual bremsstrahlung,
the situation with dileptons is more complicated.
Besides the excess in the high-momentum region, which is similar to
that for real photons, some formulas also overshoot the yield at small
momenta. These formulas relate dilepton production to
the real photon cross section rather than to the virtual one. Near
threshold, the differential cross section in virtual photon momentum
behaves like ${q^*}^2$ [in both the ``exact" formula
(\ref{virtsinger}) and Eq.~(\ref{mybremsb})], whereas for real photons
the behavior is ${q^*}^{-1}$ [see Eqs.~(\ref{realsinger}) and
(\ref{eq7})].

Energy-momentum conservation is not built into the leading term
bremsstrahlung formulas. The fact that they overestimate the dilepton
yield at high dilepton momenta is therefore not surprising. But we
will see that it is only a part of the story.

Gale and Kapusta \cite{galkap89} introduced the factor (\ref{rgaka}),
which accounts for the shrinking of the final-state phase space and
prevents violation of energy-momentum conservation. They, and also
the authors of
a later work \cite{hagaem2}, combined it with the R\"{u}ckl formula.
This procedure obscures the conclusions somewhat, because the
R\"{u}ckl formula contains a wrong numerical
factor. We therefore apply the correction factor (\ref{rgaka}) to
the leading term virtual bremsstrahlung formula (\ref{mybremsb}).
The results are displayed in Fig.~\ref{myform} after being normalized
to the ``exact" formula. Comparison with corresponding non-corrected
curves shows that only a part of the excess over the ``exact" formula
has been removed. The remaining excess should be ascribed to other
sources. It can be a modification of the nonradiative matrix element
combined with a destructive interference between radiation from the
external particles and from the contact term. Both these effects are
neglected in the leading-term approximation.

The bremsstrahlung calculations play an important role in assessing
the conventional sources of photons or dileptons in experiments
aimed at revealing anomalous production of electromagnetic probes as
a sign of new physics phenomena. For example, Haglin and Gale
\cite{haga94} have recently shown that bremsstrahlung is the
largest source of low-mass dielectrons in 4.9 GeV $pp$ collisions.
A quite good, even if not perfect, agreement with experimental data
\cite{dls} was achieved. The authors used the modified R\"{u}ckl
formula, discussed in \ref{modifications}. But our toy example showed
that this formula overestimates the dielectron yield by a factor of
2--3. This suggests that calculations using a more correct
bremsstrahlung formalism have to be performed before a definite
conclusion about the physics behind the Dilepton
Spectrometer collaboration data \cite{dls} is drawn.

To conclude, we have derived formulas for differential cross
sections of dilepton production via virtual photon bremsstrahlung
from pseudoscalar mesons and unpolarized fermions in the leading and
subleading approximations. These formulas satisfy the conditions
imposed by Lorentz covariance and gauge invariance.
From the practical point of view, the leading term formulas
(\ref{d6sig}), (\ref{d6sigmix}), and (\ref{newglobal}) are the most
important. They enable us to estimate the radiative cross sections on
the basis of the corresponding nonradiative cross section. Comparing
to previously published formulas, our formalism exhibits the best
agreement with
the exact formula when applied to a concrete physical process.

\acknowledgements

The author is indebted to V.~Balek, J.~Cleymans, C.~Gale,
K.~Haglin, J.~Kapusta, J.~Pi\v{s}\'{u}t, M.~Prakash, and
K.~Redlich for stimulating discussions and critical remarks.
M.~Prakash also read the manuscript very carefully and suggested
many improvements of the text. My interest in virtual
bremsstrahlung was aroused in discussions with J.~Anto\v{s},
C.~Fabjan, J.~Schukraft, and J.~Thompson, the members of the
HELIOS collaboration at CERN. H.~Eggers corrected several typos.
The stay at the State University of New York at Stony Brook
was supported by the U.S. Department of Energy under grant
No.~DE-FG02-88ER-40388. A part of the work was done during
a visit to the CERN Theory Division, the
hospitality of which is gratefully acknowledged.

redpajama_set_name: RedPajamaArXiv


text:
Can Lee's Young Shoulders Cope with the Weight of Expectation?
September 7, 2015 Steve Price *Features*, *South Korean National Football Team, *Youth*, Koreans in Europe, Opinion, U-16 South Korean Team, U-18 South Korean Team 0
Barcelona youth prospect Lee Seung-Woo attracting attention back in Korea.
Over the last week, large crowds of fans visited Suwon World Cup Stadium, the self-appointed 'home of Korean football' to watch the Korean Under-17's in action against Brazil, Croatia, and Nigeria. The majority of the fans present were there to see one player, and one player only, the pink haired Barcelona youth star Lee Seung-Woo. Lee's goal-scoring antics for the Barcelona youth team have not only caught the attention of La Masia's coaches, but have been making headlines on the other side of the world too. Koreans, hoping that Lee Seung-Woo can be the next superstar of the Korean national team are watching his every move with anticipation. Comparisons to Barcelona's star player, Lionel Messi, have been made, partly due to his position on the pitch and his relatively short stature, but also due to him beating Messi's scoring record at the same age for Barca's youth team. That Lee has been so impressive in his current age-group has created an expectation that he will end up being as good as Messi is once he reaches maturity, but such expectations fail to take into account the different ways that footballers develop. Arsene Wenger claims that from sixteen to eighteen, it is possible to see whether a talented individual can play as part of a team, and it isn't until a player is in his early twenties that you can tell whether a player has what it takes mentally to be a top player. Indeed, Lee was a footballing prodigy at the age of twelve, and absolutely owned the AFC Under-16 tournament that he played in, but he has made far less of an impact at Under-17 and Under-18 level.
Lee Seung-Woo warming up ahead of a match against Brazil U-17s
There have been countless other individuals who were world-beaters in their early-teens but who never quite made it professionally, so it is unfair on Lee Seung-Woo that people have such high expectations for him. In 2007, Bojan Krkic made his Barcelona debut, and at the time was hailed as the 'new Messi'. In that season, he actually scored as many league goals as Messi for Barcelona's first team, but in the end he didn't reach Messi's heights and still only has one international cap to his name. He is currently rebuilding his career with some impressive performances for Stoke City, a come-down from the heights of Barcelona and Roma, but still a relatively strong team in one of the world's strongest leagues. If Lee Seung-Woo reaches Krcic's level, Koreans should be pleased that he has still made it as a Premier League standard player, rather than feel disappointed that he hasn't won a Ballon D'or.
Lee Seung-Woo playing against Brazil U-17s
That is not to say that he won't necessarily make it as a top player, and indeed most of the sounds coming out of Barcelona suggest that they think he can play at the highest level themselves. The real test for Lee will come next year when he becomes eligible for competitive matches with Barcelona-B. Playing against fully grown opponents will help develop Lee's game, he struggled against Nigeria's larger defenders last week, and a year up against this sort of opposition will be a real test of him mentally and physically. If he can survive that test and make an impression on the Barcelona-B side then a first team debut might not be too far off.
by Steve Price
www.kleaguefootball.com
korea national team
About Steve Price 11 Articles
Writer on Korean, Asian, and global football. Runs the website www.kleaguefootball.com and writes for various other magazines and websites. Steve has also worked as FIFA's correspondent for the Korean national team.
[2018 WCQ] Lebanon vs South Korea preview
Suwon U17 Cup: Wakeup Call for Choi Jin Chul
foot coreen forum

redpajama_set_name: RedPajamaCommonCrawl


text:
Recoveries outnumber new coronavirus cases in Karnataka
Bengaluru, Jan 12 (IANS): With 797 patients discharged in a day, recoveries outnumbered 496 new Covid cases in Karnataka, according to the state health bulletin.
"Recoveries increased to 9,06,548, with 797 discharged during the day, while 496 new cases took the southern state's Covid tally to 9,28,055, including 9,344 active cases," said the state health bulletin on Monday.
With only four succumbing to the infection, the state's death toll rose to 12,144.
Bengaluru registered 282 new cases, taking its Covid tally to 3,92,863, including 5,974 active cases, while 3,82,540 recovered till date, with 374 in the last 24 hours.
The virus claimed only two lives, taking the city's death toll to 4,348 since the pandemic broke in the state on March 8.
Of the 196 patients in the intensive care units (ICUs) across the state, 82 are in Bengaluru hospitals and 19 in Kalaburagi district in the northern region.
Out of 86,903 tests conducted during the day, 2,956 were through rapid-antigen detection and 83,947 through RT-PCR method.
The state health officials screened 982 international passengers at the city airport on arrival from the UK, the US and other countries.
"Positivity rate for the day was 0.57 per cent and case fatality rate 0.80 per cent across the state," added the bulletin.
Title : Recoveries outnumber new coronavirus cases in Karnataka


text:
Leo Conlon (3 January 1901 – 29 January 1973) was  a former Australian rules footballer who played with Footscray in the Victorian Football League (VFL).

Notes

External links

1901 births
1973 deaths
Australian rules footballers from Victoria (Australia)
Western Bulldogs players
South Ballarat Football Club players

redpajama_set_name: RedPajamaWikipedia


text:
Q: Does mbstring normalize utf-8 strings? I noticed that mb_convert_encoding always returns valid utf-8, which is great for cleaning up user input, but I wonder whether it will return a normalized form of utf-8 or whether I would have to do that manually after with the \Normalizer class?

A: Reading the source, mb_convert_encoding does not appear to normalize. It appears to convert between encodings then to substitute illegal characters, and that is all.

redpajama_set_name: RedPajamaStackExchange


Your ticket has been passed to our development department and is now in the project process.
Our project management team will review your ticket to determine if it is a suitable candidate for development. If your ticket is approved then it will pass on to scheduling, development, testing and release. We will send further updates as your ticket progresses through the project process.


Daisy Opens a World of Wonder!
There's nothing like a good book!
Arthur is a beautiful Angora goat whose new owner is moving and has to find Arthur a new home. His new owner teaches Arthur to talk. Based on a real goat, Arthur's home is at Bethlehem Retreat Farm in beautiful southwestern Virginia. He has a great personality and is a delight to everyone who sees him. Bethlehem Retreat is a part of the House of Bethlehem Ministry & Missions, International. Lots of birds, rabbits, squirrels, llamas, chickens, deer, and other wild animals can be seen on the farm.
Be sure to look for the first book in the series, "Arthur the talking Goat". Arthur lives on a farm with his owners, Sam and Sally. Sam taught Arthur to talk but Arthur was still very sad because he had no animal friends. This book is about how a chicken, named Hazel the Hen, becomes Arthur's friend. Arthur and Hazel were real animals on the Bethlehem Retreat Farm in beautiful southwestern VA. Llamas and lots of other animals can be seen on the farm. Bethlehem Retreat is a part of the House of Bethlehem Ministry & Missions, International.
Have you ever gone to sleep at night with your hands in one position, woke up with them in a totally different position, and wondered how they got there? This rhyming children's bedtime story asks very important questions that will start young minds thinking. This delightful book will provide young children many happy moments before going to bed, as they decide what they will do with their little hands at night.
Copyright © 2019 Daisy Cromwell. All Rights Reserved.


Ahmadu Bello University Admission List for 2019 [UPDATED]. The list of candidates offered admission into the Ahmadu Bello University (ABU), Zaria, Kaduna State, DEGREE programmes for the 2018/2019 academic session is out.
First of all, Go to ABU admission portal at https://putme.abu.edu.ng/index.php.
Also, Successful candidates are to report on Monday 7th January, 2019 for the collection of their admission letters at A.B.U convocation square (main campus) at 9:00am daily.
Furthermore, Congratulations to all the candidates that made it to ABU admission list for the 2018/2019 academic session.


This is a specific type of balun which enables the transmission of HDTV video and digital audio using ordinary unshielded twisted pair (UTP) wire instead of coaxial (coax) cable. The word balun comes from combining the words "balanced" and "unbalanced" and a baluns function is to transform an un-balanced signal into a balanced signal.
When a video signal is transmitted through coax, the distance traveled by the signal is limited in part because the signal is in the form of an unbalanced signal which is susceptible to Electromagnetic / Radio Frequency interference or "NOISE". In order to minimize noise, coax cables incorporate shielding which increases its cost and bulk.
These baluns transform the signals into a balanced form in which each wire in the twisted pair transmits an identical signal but with opposite polarized magnetic fields. Any radio frequency interference will affect each signal equally and when combined, the interference is cancelled out allowing for much longer cable runs between components without signal degradation.
In combination with a properly designed balun, unshielded twisted pair wire can transmit audio and video signals for much longer distances than coax cables, with the advantage of lower cable costs and easier installation.
UTP cable is significantly less than expensive than coax. The cost per foot of UTP can be 60% less than the cost per foot of coax. In addition, standard UTP cable contains 4 pairs of wire so one foot of UTP cable can do four times the work of one foot of coax.
Cat 5/6 cables can be field-terminated, while HDMI cables cannot. Not only fairly easy not to mention the fact that Cat cables are easy to install and a lot less expenisve.
Lots of positive information has been written about using Cat 5e / Cat 6 to run your Audio and Video signals over. Despite the positive data, there have been issues. Here are some simple but important rules integrators should follow when installing any Cat 5 or 6 systems.
Always be aware of timing problems. This involves our old friends, inter and intra pair skew limitations. Cat 5e and Cat6 are twisted pairs and look the same. They are just as easy to pull, pinch, stretch, and distort during installation. In many cases, the damage comes easier than with HDMI cable. These distortions with the cable can cause server timing and ISI (Inter Symbol Interference) issues, which can cause sparkles or intermittent video. Be careful installing Cat 5e Cat 6. No Sharp Turns or right angle or 90 degree turns.
Use Cat 6 if you have the option to choose between the two. Studies have proven Cat 6 will offer almost twice the distance you would normally get with Cat 5e. Plus Cat 6 is built with an inner sheath that keeps each twisted pair separated and flexible meaning that if you do pull it the wrong way, each twisted pair inside can move easier.
Don't forget about the DDC serial channel Although video integrity is important in any of these converters, tests have proven that over Cat 5e and Cat 6 the data on this channel can be easily lost due to rise time issues. That means a high probability of EDID and HDCP failure.


text:
Q: passing sed argument with slash ( making a directory a string ?) I am trying to pass in assets/css/images to replace images in multiple files.
So I wrote this script
$list
$word
$neword
$i

echo "select files: "
read list
echo "input word to search for: "
read word
echo "input word to replace with: "
read neword

for i in $list
do
sed -i "s/${word}/${neword}/g" $i

    done
It works for words without a /, and I can see why, but I don't know how to get around it. Anyone?

A: You should do like this to escape / in sed
Try Following,
sed -i "s#${word}#${neword}#g" "$i"

or
sed -i "s~${word}~${neword}~g" "$i"

Edit: Added Quotes to Filename as Suggested by @tripleee for avoiding condition like filename contains in $i have spaces.


text:
Sports Car Market Magazine Names its 2021 "40 Under 40" Honorees
Posted on 5 months ago by Malcolm Price
PORTLAND, Ore. (Aug. 9, 2021) – Sports Car Market has named its top "40 Under 40" for 2021. This is the second time the magazine has recognized 40 car enthusiasts under 40 years old who are leaders in the collector-car world.
Hundreds of nominations were received from SCM readers and the final list of honorees was selected by the magazine's editorial staff. "We are proud to share their stories and look forward to seeing what each of these individuals will accomplish next," said SCM Executive Editor Jeff Sabatini.
The 2021 "40 Under 40" feature appears in the October issue of Sports Car Market Magazine (www.sportscarmarket.com), which ships to SCM Members, bookstores and newsstands in mid-August. SCM is sold at Barnes & Noble, Hudson News, and many other independent stores.
"This group has not only inherited interests, knowledge and skills from their elders, but have created their own automotive pursuits," said SCM Publisher Keith Martin. "They are helping to ensure that the future of the collector-car hobby is bright, most of all by sharing their enjoyment."
The 2021 class recognizes men and women ranging in age from 25 to 39 who have based their lives and careers around automobiles. The honorees include professionals from a variety of backgrounds, such as auction specialists, businesspersons, collector-car dealers, entrepreneurs, museum curators, restoration experts, and technicians. An inaugural group of honorees was named in 2017.
The 2021 SCM "40 Under 40":
Acabbo, Dana; York International Agency
Beyer, Manda; Manda Marie Upholstery LLC
Bodell, Michael; Petersen Automotive Museum
Brodfuehrer, Garek; Cornerstone Automotive Restorations LLC
Cassini, Caroline; The Market by Bonhams
Cervantes, Art; RADwood Inc.
Clary, Evan; Paul Russell and Company
Cristi, Roberta; WeatherTech Raceway Laguna Seca
Dietz, Robert; WOB Cars, Dietz MotorCraft
Dykman, Angus; Gooding & Company
Grundy, Sam; Grundy Insurance
Hale, Tom; Morton Street Partners
Hammer, Tabetha; America's Automotive Trust
Ibarra, Adrian; Ron Tonkin Gran Turismo
Kanter, Maddie; The Shop Club
Keller, Eric; Enthusiast Auto Group
Kittrell, Steven; Veloce Valuations, Veloce Media, Mohr Imports
Koscak, Keith; LBI Limited
Leonard, Blakley; The MB Market
Logan, Bennett; Logan Car Services, Jan B. Luehn
Lovett, Edward; Collecting Cars, Watch Collecting
Lucacel, Adam; Vintique Motors
Marzano, Michael; Mouse Motors
Massari, Adolfo; LBI Limited
Mastin, Andrew; LBI Limited
Maxon, Casey; Hagerty Drivers Foundation
Minoff, Eric; Bonhams
Moore, Derek E.; National Corvette Museum
Oliver, Brian; Automotive Restorations Inc.
Paruch, Austin; Paruch Automotive Craftsmanship LLC
Porter, Alex; Bring a Trailer
Roberts, Michael; F40 Motorsports
Santostefano, Adriana; Santostefano Auto Body
Santostefano, Sabrina; Santostefano Auto Body
Sierakowski, Jonathan; Sierakowski Classic Car Advisors
Simko, Orrie; Simko Motorcars LLC
Skinner, Jonathon; John Skinner (Manufacturing) Ltd.
Sweeney, Ed; Proper Noise Limited
Weaver, Alexander; RM Sotheby's
Wheaton, Clayton; Wheaton Motorwerks
About Keith Martin's Sports Car Market:Publisher Keith Martin founded the monthly Sports Car Market magazine, based in Portland, Oregon, in 1989, and it has developed into the authoritative voice of the collector car hobby.
Martin has been personally involved with the collector car hobby for more than 40 years. As a writer, publisher, television commentator and enthusiast, he is constantly on the go, meeting collectors and getting involved in their activities throughout the world.
His columns on collecting and reviews of exotic cars have appeared in The New York Times, and he's been an emcee at the Pebble Beach Concours d'Elegance, Concours of America, Concorso Italiano, Legends of the Autobahn, and many other events. He was honored with two of the most prestigious awards in the collector car hobby – The Lee Iacocca Award and the Edward Herrmann Award, and has been inducted into the Concorso Italiano Hall of Fame.
Martin has hosted numerous television specials, was the co-host of "What's My Car Worth" on the Velocity channel and served as Chairman of the Meguiar's Collector Car Person of the Year Award. He's on the Board of Trustees of the LeMay Museum and has served on the faculty of the Collier Museum Symposium on Connoisseurship.
A full bio on Keith Martin can be found HERE.
COURTESY JVS Enterprises & Keith Martin's Sports Car Market
Malcolm Price
View all posts byMalcolm Price | Website
Posted in Motor Sports
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text:
Q: Access floating-point return of assembly function our course exercise asks us to create a delta = b2 - 4ac function in GNU assembly, and access it from C. Since this is a course about compilers, and not about assembly, the professor chose to only demonstrate integer capabilities, and expects an integer function.
However I would like for personal learning to go a little beyond the call of duty, and create a usable function returning floating points rather than integers.
I came up with this C (dont't mind the globals, the purpose of the next exercise is to have a proper function with parameters):
# include <stdio.h>

extern float delta();
float a, b, c;

int main() {
    a = 3;
    b = 5;
    c = 4;
    printf("δ = %f\n", delta());
    return 0;
}

and this GNU GAS:
.globl a
.globl b
.globl c
.globl delta
.f4: .float 4.0     # constante 4

.text
delta:
    fld b           # b sur la pile
    fmul b          # b2 sur la pile
    fld .f4         # 4 sur la pile
    fmul a          # 4a sur la pile
    fmul c          # 4ac sur la pile
    fsubp           # b2 - 4ac sur la pile
ret

From Google I was led to believe that I should leave a floating point result at the top of the floating point stack, however this doesn't work, and the result printed in the C caller is invariably 0.0000000.
I must miss a very small something, but no amount of googling brings it up, can anyone point me in the right direction?
Thanks for your attention.

A: It does work for me. Make sure you are not accidentally using 64 bit mode, because calling convention is different there. That is, use gcc -g -m32 foo.c bar.s to compile.
That said, I also see some potential problems that should be fixed.

*

*Since your globals are defined in the C code, you should not use .globl in the assembly. If anything, you should use .extern, but GAS does not require that.

*You should not rely on default operand size. You should explicitly use s suffix for floats and l suffix for doubles when you have memory operands. For example, flds b to make sure it's loaded as a float.

*You should use fsubrp because the top of stack is your 4ac, so what you calculate using fsubp is 4ac-b^2.


text:
Time to strengthen protection of IDPs
Walter Kälin
kalin.pdf
Lack of resources and infrastructure, the volatile security situation and the absence of state structures pose serious threats to the human rights of returnees and IDPs.
In October 2005 I undertook a mission to study the situation of IDPs and returnees to the South. I realised that the Sudanese authorities, the international community and donors need to do much more to ensure the protection of the rights of the four million people displaced by conflict.
As stated in the Guiding Principles on Internal Displacement, the right of IDPs to choose freely whether to return, locally integrate or resettle elsewhere must be respected. However, it became clear from talking to IDPs in camps in and around Khartoum that many were badly or not at all informed about conditions in their places of origin. Most have unrealistic expectations of intact infrastructure, functioning basic services and opportunities to maintain a livelihood. Many lack information about how to obtain relevant documents.
The authorities in Khartoum State are implementing an urban planning programme which aims to give residents of irregular settlements a plot of land in the desert outskirts of the city. In principle, IDPs - like other Sudanese citizens - are eligible to benefit from a land allocation programme which has potential to provide a durable solution to their plight. However, I learned that there are great difficulties with its implementation. Many IDPs are not eligible and methods by which settlements are relocated give rise to human rights concerns. IDPs and NGOs reported that persons are forcibly removed from their previous settlements with inadequate or no prior consultation. During the removal, housing, personal belongings and infrastructure such as schools are destroyed. Fearing for their survival many IDPs relocated to inhospitable urban margins see return to the South as their only option, despite the fact they often lack the means to travel.
In order to provide returnees en route with limited and short-term assistance and protection, the humanitarian community has been planning to erect way stations. Unfortunately, these plans have not yet materialised, and most returnees lack any material support during their journeys. Some reported having been attacked, illegally taxed and looted on the road. As returns are already under way and expected to increase during the coming dry season, there is an urgent need for the international community to implement existing plans.
Despite the peace agreement, returnees fear for their safety due to militia activities, land mines and the widespread presence of armed civilians within communities . Upon arrival, many remain without shelter, sufficient food, clean drinking water and access to medical services. Parents whose children attended schools during their displacement in the North are fearful they will not be enrolled in already over-subscribed schools. The elderly and women, especially female-headed households, remain particularly vulnerable.
Information is scanty due to lack of monitoring outside the main towns and settlements. Protection activities of international agencies and NGOs are hampered by lack of funds and restriction of access. Rushed returns may create serious humanitarian problems and should not be promoted. The current capacities in the South to receive and quickly integrate large numbers of returnees are limited. In areas of mass arrivals, the humanitarian problems of IDP camps and irregular settlements in the North may simply be transferred to the South. As a result, tensions may arise within overstretched local communities due to competition over scarce resources and services. There is a danger that the international community may be forced to spend funds on emergency humanitarian assistance which should be invested in recovery and development projects.
Although return movements are still limited there is already evidence of cases where returnees see their expectations frustrated upon arrival and opt to return to Khartoum. Creating an environment in which returns can be sustainable requires:
removing militias from areas of return, disarming civilians and demining of access roads and agricultural land
action to address the major obstacles to return reported by IDPs – absence of clean drinking water, primary education, health service and household building material
UN agencies and NGOs to be resourced to undertake fast recovery programmes and provide such non-food items as seeds, tools, fishing nets and basic building materials
establishment of mechanisms for speedy resolution of land disputes
ensuring that returnees receive, with little or no charge, documents to enable them to access basic services.
Instead of aiming at perfection and getting lost in long-term planning, flexible and creative community-based solutions for providing basic infrastructure and health and education services should be found in order to bridge the period until full services can be set up.
Professor Walter Kälin is the Representative of the UN Secretary-General on the Human Rights of Internally Displaced Persons and co-director of the Brookings-Bern Project on Internal Displacement www.brook.edu/idp. Email: walter.kalin@oefre.unibe.ch
Book traversal links for Time to strengthen protection of IDPs
‹ Information, counselling and legal assistance for returning Sudanese IDPs
Challenges facing returnees in Sudan ›
African renaissance: towards a New Sudan
Joint Assessment Mission provides road-map for peace
Reflections on making peace
The role of the international community
The Comprehensive Peace Agreement: will it also be dishonoured?
Developmental transformation and peace consolidation in Southern Sudan
Reflections on the Joint Assessment Mission
Land and the Sudanese transition to peace
Standing up for the JAM partnerships
Perspectives on the Comprehensive Peace Agreement
Pledges versus commitments
Could peace unravel?
NGO coordination in South Sudan
Sustainable IDP/refugee return depends on collaborative approach
Promoting the rule of law in post-conflict Sudan
The reality of return: IDPs in Darfur
Assisting the return of displaced Dinka Bor
Epic trek of Mabia IDPs
IDPs and urban planning in Khartoum
Information, counselling and legal assistance for returning Sudanese IDPs
Challenges facing returnees in Sudan
Towards a culture of human rights in Darfur
Sudanese women waging peace
From Afghanistan to Sudan: how peace risks marginalising women
Sudanese women's role in peacemaking
Endangering peace by ignoring women
Separated children in south Sudan
Gender, education and peace in southern Sudan
Education crisis in south Sudan
A local approach to peace in Torit, south Sudan
The Sudanese peace process and the media
Critical juncture for peace, democracy and the environment
Sudan: world's greatest humanitarian transport challenge
Encounter with a fighter
The Mexico Plan of Action: protecting refugees through international solidarity
Statelessness: a forgotten human rights crisis
Addressing sexual and gender-based violence
North-east India's forgotten IDPs
The Zambia Initiative
IRIN marks its first decade
IDPs in the Balkans - the challenges of sustainable return
Training on the Guiding Principles in Aceh
Another talkshop? Or seeds of an effective Southern African IDP policy?