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\begin{document}
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\begin{flushright}
\\
{FTUAM-02-06}\\
{IFT-UAM/CSIC-02-04} \\

\end{flushright}
\vspace*{1cm}
\begin{center}
{\Large{\bf Corrections to the fluxes of a Neutrino Factory} }\\
\vspace{.5cm}
A. Broncano$^{\rm a,}$\footnote{alicia@delta.ft.uam.es},
O. Mena$^{\rm a,}$\footnote{mena@delta.ft.uam.es}
 
\vspace*{1cm}
$^{\rm a}$ Dept. de F\'{\i}sica Te\'orica, Univ. Aut\'onoma de
Madrid, 28049 Spain \\
\end{center}
\vspace{.3cm}
\begin{abstract}
\noindent

In view of their physics goals, future neutrino factories from muon decay aim at an overall flux precision of  ${\cal O}(1\%)$ or better. 
We analytically study the QED radiative corrections to the differential distributions of muon decay. The  ${\cal O}(1\%)$ corrections to
 the energy and angular distributions 
for electrons are obtained in the ``leading log'' approximation, while the exact corrections are considered for neutrinos. Kinematic 
uncertainties due to the divergence of the muon beam are considered as well. The resulting corrections to the neutrino flux turn out to be
 of order ${\cal O}(0.1\%)$, safely below the required precision.
 
\end{abstract}

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\section{Introduction}
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Results on neutrino oscillations from Superkamiokande \cite{SuperK} and SNO \cite{SNO} provide a compelling evidence for neutrino 
masses, constituting the first  strong indication of Physics beyond the Standard Model. Much is still unknown, though, regarding fundamental 
issues such as 
the absolute neutrino mass scale, the possible Majorana character of neutrino fields, the ordering of their mass eigenstates with respect 
to charged lepton eigenstates, or the possible existence of leptonic 
CP violation and its tantalizing relationship to baryogenesis. In this situation one could argue that the subject of lepton flavor 
physics is at its exciting infancy, and to obtain rough answers to those questions could be a sufficient goal at present, 
postponing any aim at a precise determination of the involved parameters.
Nevertheless, some of those questions prerequire precision: for instance the study of CP violation rests upon a precise knowledge 
of the angles in the neutrino mixing matrix. 

In a more general way and much as for the quark sector, it is necessary to know accurately the values of the 
masses and mixing parameters in the lepton sector, as a first step to unravel the flavor puzzle. 
And what does precision means, quantitatively?. For instance, with which precision is it desirable to determine the values of the leptonic 
mixing angles in order to discriminate between models for neutrino masses? Clearly no definite answer can be given to such question, but as an 
indication it has been argued \cite{Snowmass} that a $10\%-1\%$ precision in the knowledge of, say, $sin^2 2\theta_{atm}$ would result 
in significant advance\footnote{$\theta_{atm}$ denotes the mixing angle dominantly responsible for the atmospheric oscillations, denoted by 
$\theta_{23}$ in the by now standard parameterization \cite{Particle}}. It is not impossible to envisage such a precision.
In resume, we are simultaneously entering 
a discovery and a precision era in neutrino physics. With the bonus that the extraction of physical conclusions will not be necessarily hindered by large 
theoretical errors, as it happens in the quark sector due to QCD long distance contributions.

 A quest for precise physics answers evidently requires an effort in precision on the experimental conditions, and on the knowledge of the 
neutrino flux to start with.
Several experiments using neutrino beams from particle accelerators such as K2K, MINOS and OPERA \cite{exper} will take data in the next few years.  
Their reach will be limited by the use of
 conventional neutrino beams produced from a charged pion source. The decay $\pi^+\rightarrow \mu^+\nu_\mu\, (\pi^-\rightarrow \mu^-\bar\nu_\mu)$ 
produces a  $\nu_\mu$ beam with a ${\cal O}(1\%)$ component of $\nu_e$ from kaon decays. The $\nu_e$ contamination limits the precision of 
the flux measurements, resulting in an error of $7\%$ for K2K, while MINOS reduces it to $2\%$ \cite{exper}.
A further step forward could be provided by the so-called superbeams which, although based on the same traditional beams, can achieve 
better precision thanks to the much higher statistics. It has been argued, for example, that by working at energies below the threshold of kaon production, the $\nu_e$ flavor 
contamination could be reduced, with the overall figure of merit for precision in the flux measurements limited to  ${\cal O}(1\%)$ 
\cite{todos,pilar}.

A major advance should come from a neutrino factory from muon decays, aiming at both fundamental discoveries and ${\cal O}(1\%)$ 
precision measurements. Present projects consider the production of very intense muon sources of 
about $10^{20}$ muons per year \cite{Geer1}. Neutrino beams 
originate from the decay of high-momentum muons along the straight sections of a storage ring. The beam produced presents a precisely known neutrino 
content: $50\%$ muon neutrinos and $50\%$ electron antineutrinos if a $\mu^-$ beam is used, and $50\%$ muon antineutrinos 
and $50\%$ electron neutrinos if a $\mu^+$ beam is used. The resulting $\nu$ fluxes are expected to be known with a precision better than  
$1\%$ \cite{Blondel}. It is necessary to ensure that any possible corrections and sources of errors are controlled at that level.  
In this work, we study two effects:  the contribution of QED one-loop corrections to muon decay and the divergence 
of the muon beam. 

Radiative corrections to the electron differential distribution in $\mu^{-} \rightarrow e^{-} + \bar\nu_e + \nu_\mu $ were calculated long 
ago \cite{Beherends} resulting in a correction of ${\cal O}(1\%)$. In our work, we discuss the origin of this large effect by computing the 
electron energy and angular distribution within the ``leading log'' approximation, in which the ${\cal O}(\alpha)$ correction 
is restricted to the dominant terms, proportional to $\ln(\frac{m_\mu}{m_e})$. Electron angular distributions 
are used in neutrino factories to measure the muon polarization and, from this, the muon beam energy distribution \cite{Blondel2}.   

First order QED corrections to the neutrino angular and energy distribution are also computed. No large logarithmic corrections 
are present in this case, and the ${\cal O}(\alpha)$ contributions are exactly evaluated in the limit $m_e=0$. The correction to  the (massive) neutrino spectra  
from unpolarized muons has been calculated in \cite{Greub}. In our analysis we set $m_\nu=0$ and include muon polarization, relevant 
for neutrino factory measurements.

The second subject addressed in this paper is that of the  muon beam divergence, one of the basic properties that can bias the predicted neutrino 
spectra . We explore the error induced in the neutrino distributions at the far site due to the systematic uncertainty on the angular divergence, 
and compare ours results with previous ones in which this effect was not included \cite{golden}.   

The paper is organized as follows. In section 2 we recall the tree-level angular distributions. Section 3 discusses the leading logarithmic 
QED corrections. In section 4 the exact one-loop corrected formulae are given, with subsections 4.1 and 4.2 specializing in the contribution from virtual 
and real photons, respectively, and subsection 4.3 in the soft photon limit and cancellation of infrared divergences. Section 5 accounts for the corrections due to the beam divergence.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{General definitions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
In the muon rest-frame, the angular distributions of the neutrinos produced in 
the decay $\mu^- \rightarrow e^- + \nu_\mu  + \bar\nu_e$, Fig.~\ref{loops}a, are computed from the muon decay rate:
\be
d\Gamma_0=\frac{1}{2m_{\mu}}\,64\,{\rm G_F}^2\,\,|M_0(p_\mu;p_e,p_{\bar\nu_e},p_{\nu_\mu}) |^2\,d\Phi_3(p_\mu;p_e,p_{\bar\nu_e},p_{\nu_\mu}),
\ee
where $|M_0(p_\mu,p_e,p_{\bar\nu_e},p_{\nu_\mu}) |^2$ is the averaged squared amplitude obtained from the Feynmann diagram at tree level. 
For polarized muons:
\be
\label{tree_amp}
|M_0(p_\mu;p_e,p_{\bar\nu_e},p_{\nu_\mu}) |^2 = \left[(p_\mu - m_\mu s)\,p_{\bar\nu_e}\right](p_e p_{\nu_\mu}),
\ee 
where $s$ is the four-spin. For unpolarized muons $s=0$.

$d\Phi_3$ is the three-body phase-space. In general, the n-body phase space is defined by :
\be
d\Phi_n(P;p_1,...,p_n) = (2\pi)^4\,\delta (P-p_1-...-p_n) \,\prod_{i=1}^n                                                      \frac{d^3{\bf p_i}}{2p_i^0}\frac{1}{(2\pi)^3}. \ee

Differential distributions of decay products are obtained integrating over the phase space of the remaining decay particles,  
\bea 
\frac {d^2 N }{ dx\,d\cos{\theta} } = 
F^{(0)} (x) + J^{(0)}(x)\,{\cal P}_\mu \cos\theta, 
\label{treedistr} 
\eea
where $x$ denotes the scaled energy, $x=2E_{e,\nu}/m_{\mu}$ and 
${\cal P}_\mu$ is the average over polarization  of the initial state muon along the beam direction.
$\theta$ is the angle between three-momentum of the emitted particle and 
the muon spin direction and $m_{\mu}$ is the muon mass. 
The normalized functions $F^{(0)}$ and $J^{(0)}$, in the limit $m_e=0$, read \cite{gaisser}:
\bea
F_{e}^{(0)}(x) &=&  x^2(3-2x),
\qquad
J_{e}^{(0)}(x) = x^2(1-2x),
\label{tree_e}\nn\\
F_{\nu_\mu}^{(0)}(x) &=&  x^2(3-2x),
\qquad
J_{\nu_\mu}^{(0)}(x) = x^2(1-2x),
\label{tree_numu}\nn\\
F_{\bar\nu_e}^{(0)}(x) &=& 6x^2(1-x),
\qquad
J_{\bar\nu_e}^{(0)}(x) = 6x^2(1-x).
\label{tree_nue}
\eea

The QED radiative corrections to the above formulae will be computed in what follows using the Fermi effective lagrangian for four Fermi weak corrections, 
as all pure QED divergences do cancel in this 
approximation. There are three diagrams containing a photon loop: the exchange of the virtual photon between the muon and the 
electron legs, Fig.~\ref{loops}b, and lepton propagator corrections, Figs.~\ref{loops}c,~\ref{loops}d. They correct the invariant amplitude of 
the muon decay as follows:
\be
-i\ M = \frac{G_F}{\sqrt{2}}\{\bar{u}(p_e)\Gamma_\sigma {u}(p_\mu) \}
                              \{\bar{u}(p_{\nu_\mu})\gamma^\sigma(1-\gamma_5) {v}(p_{\bar\nu_e}) \}.
\ee
$\Gamma_\sigma$ is the corrected $\mu-e$ vertex:
\be
\label{gam_sig}
\Gamma_\sigma= \gamma_\sigma(1-\gamma_5)+ \Gamma_\sigma^a+\Gamma_\sigma^{c,d},
\ee
where $\Gamma_\sigma^b$ results from the diagram in Fig.~\ref{loops}b and $\Gamma_\sigma^{c,d}$ from those  
in Figs.~\ref{loops}c,~\ref{loops}d. Ultraviolet (UV) divergences cancel when the three diagrams are summed up. Their infrared 
singularities (IR) are cancelled
with the soft contributions of the bremsstrahlung diagrams in Figs.~\ref{loops}e,~\ref{loops}f. The overall effect is expected to be of ${\cal O}(\frac{\alpha}{\pi})\sim {\cal O}(0.1\%)$, although in some cases, the correction can be enhanced up to the percent level 
by logarithmic factors. This will be discussed in the next section.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Leading logarithms in QED radiative corrections }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the $e^-$ properties in muon decay, the most important QED radiative corrections stem from the physics of collinear photons. Electrons 
are much lighter than muons and radiate more. As a consequence, the leading radiative effects come from processes where photons are attached
 to electrons. Indeed, the emission of multiple collinear photons from the electron leg gives contributions of the form 
$(\frac{\alpha}{\pi}\ln(\frac {s}{m_e}))^n$, where $s$ is the typical energy scale of the process, i.e. $m_\mu$ for muon decay.  
In the $m_e=0$ limit, these logarithms can be potentially large.
% and radiative corrections to e^- distribution can be restricted to the ``leading log'' approximation. 

Although the exact corrections to the $e^-$ differential distributions have been computed at ${\cal O}(\alpha)$, we will in practice consider 
just the terms in $\frac{\alpha}{\pi}\ln(\frac {m_\mu}{m_e})$: for the electrodynamic corrections considered here this is a highly accurate approximation.
However, the correction to the neutrino distributions turns out to be independent of $m_e$ and no mass singular logarithms are present as it occurs for the muon lifetime. In both cases, 
the integration over the part affected by QED corrections, namely, the photon-electron system, cancels all the terms involving mass 
singularities \cite{KLN}. It must be remarked that this only occurs once the virtual and bremsstrahlung contributions at ${\cal O}(\alpha)$ are added.

In the next subsection we discuss the origin of the ``leading logs''. As we shall see, the relation between the observed decay width and the 
bare differential decay width $d\Gamma_B$ takes the form
\be
\label{decayw}
d\Gamma_{obs}\,=\, d\Gamma_B + \frac {\alpha}{2\pi}\ln\left(\frac{m_{\mu}^2} {m_e^2} \right)\,F[d\Gamma_B] +  \frac {\alpha}{2\pi} G,
\ee
where $F$ is an explicit functional of $[d\Gamma_B]$. The coefficient $G$ does not contain logarithms of the form $\frac{\alpha}{\pi}\ln(\frac {m_\mu}{m_e})$.
The ``leading log'' approximation seriously deteriorates at the edges of the kinematically allowed regions where singularity appears.  
This behavior is a general feature of QED corrections and is 
related to photon emission when the photon is simultaneously collinear and soft.  In subsection \ref{endpoint}, we show that this divergence takes the form of double singular logarithms and we discuss their exponentiation which gets rid of it. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{``Leading log'' approximation to the $e^-$  distribution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Muons at a neutrino factory are completely polarized in their own rest frame. Boosting the system to the laboratory frame, the muon momentum and spin loose their collinearity and the longitudinal polarization is generally less than 1. For the typical energies of the neutrino factory, the average muon polarization is ${\cal P}_\mu \sim 0.2 $ \cite{Blondel3}. The distribution 
of electrons at the neutrino factory serves as a muon polarimeter because
the dependence of the angle between the muon spin direction and the electron trajectory is proportional to the muon polarization. Using this property, the 
 muon polarization is expected to be measured with a precision of  ${\cal O}(1\%)$ \cite{Blondel2}. Any 
possible source of error at this level must be taken into account. In this subsection we consider the leading correction introduced 
in the electron distribution by QED radiative corrections.

Mass singularities arise from the collinear emission of photons by a lepton in an external leg . For muon decay, it occurs in the 
limit $m_e\rightarrow 0$  when the electron emits a collinear photon, Fig.~\ref{loops}f. In this case, the propagator of the 
electron $1/(p_e k)$ vanishes, where $p_e$ ($k$) is the electron (photon) momentum.  To compute the leading logarithm it suffices to set 
$m_e^2=0$ and $k^2=0$ in the numerator 
of the bremsstrahlung amplitude and take $k$ to be proportional to $p_e$: 
\be
p_e^\mu=z P^\mu ,\qquad
k^\mu=(1-z)P^\mu, 
\ee
where $0<z<1$ since $P=p_e+k$ is the momentum of the virtual electron before photon radiation.  

Hence, the squared amplitude of the collinear emission process reads:
\be
\label{ampl_LL}
|M_L|^2= |M_0(p_\mu,z p_e, p_{\bar\nu_e},p_{\nu_\mu})|^2 \,\frac{\alpha}{2\pi}\,\frac{1}{(p_e k)}\frac{1+z^2}{1-z},
\ee
where $|M_0(p_\mu,zp_e,p_{\bar\nu_e},p_{\nu_\mu})|^2$ is the tree level amplitude given in eq.~(\ref{tree_amp}), with an emitted virtual 
electron of momentum $zp_e=P$.
To obtain the electron distribution in the muon rest frame, we must integrate eq.~(\ref{ampl_LL}) over the neutrinos and photon phase space: 
\bea
\label{lead}
\int\frac{d^3{\bf k}}{(2 k^0)}\int|M_L|^2\,
             \frac{d^3{\bf p_{\bar\nu_e}}}{2 p_{\bar\nu_e}^0}
             \frac{d^3{\bf p_{\nu_\mu}}}{2 p{\nu_\mu}^0}
&=&  \frac{\alpha}{(2\pi)^2}\int\frac{d^3{\bf k}}{(2 k^0)}\,A(z)
      \nn \\
&=&  \frac{\alpha}{2\pi}\,
     \int^{1}_{\varphi_{\rm min}}\frac{{\rm d}\cos\varphi}
     {(1- \omega\cos\varphi)}, 
     \int^{1}_{x}\frac{{\rm d}z}{z}\,\frac{1+z^2}{1-z} \,A(z)\nn \\  
&=& \frac{\alpha}{2\pi} \ln\left(\frac{m_\mu}{m_e^2}\right)
     \int^{1}_{x}\frac{{\rm d}z}{z}\,\frac{1+z^2}{1-z} \,A(z)\,\,,
\eea
where $\omega=|{\bf p_e}|/E_e$, $\varphi$ is the angle between ${\bf k}$ and  ${\bf p_e}$ and 
\be
A(z)\equiv \,\frac{m_\mu^2}{4}\,x^2\,z \left[(3 -2\,z\,x)-s\cos{\theta}(1 -2\,z\,x)\right]
\ee
is the integrated amplitude with $x=2E_{e}/m_{\mu}$ . The lower 
limit $z>x$ arises from the requirement that the maximum energy of the virtual
electron be larger than the detected electron energy $E_{e}$. 

In addition to the mass singularity $m_e\rightarrow0$, eq.~(\ref{lead}) contains a singularity for $z=1$, i.e. when the photon radiated 
has zero energy (soft photon). This is an infrared singularity which must be canceled by virtual photon corrections where no photon is radiated. 
Hence, the photon exchange diagrams subtract from eq.~(\ref{lead}) a term proportional to $\delta(1-z)$ \cite{Yndurain}. Working in the physical gauge where 
the photon enjoys only transverse polarizations, the $\mu-e$ vertex correction, Fig.~\ref{loops}b, does not give ``leading log'' 
contributions \cite{Derujula}. The factor preceding the delta function is extracted only from the electron 
self-energy diagram, Fig.~\ref{loops}d, since $m_e \ll m_\mu$. Taking into account the virtual photon terms, the corrected electron 
distribution has the following form:
\bea
\label{lead2}
\frac {d^2 N_{e}^{\rm L L}}{ dx\,d\cos{\theta} } &=& \frac {d^2 N_{e}^{(0)}}{ dx\,d\cos{\theta} }\nn\\ 
& + & \frac{\alpha}{2\pi} \ln\left(\frac{m_\mu^2}{m_e^2}\right)
 \int^{1}_{x}\frac{{\rm d}z}{z}\,\left(\frac{1+z^2}{(1-z)_+} +\,\frac{3}{2}\delta(1-z)\right)\,A(z),
\eea
where the function $1/(1-z)_+$ is defined as having no singularities \cite{Peskin}:
\bea
\frac{1}{(1-z)_+}\equiv \lim_{\epsilon\rightarrow 0}
                 \Bigg\{\frac{1}{(1-z)}\theta(1-z-\epsilon)-
                 \delta(1-z)\int^{1-\epsilon}_{0}{\rm d}y\,\frac{1}{1-y}\Bigg\}\,\,\,.
\eea
The factor $-3/2$ of the delta function can also be worked out considering that the muon lifetime does not contain mass logarithms. Therefore, when the corrected electron distribution is 
integrated over $\theta$ and $x$, the term of ${\cal O}(\alpha)$ must give zero contribution to the lifetime. 
Imposing that condition on the collinear photon emission results, eq.~(\ref{lead}), we obtain the same factor preceding the delta function. 

The leading correction follows then from integrating eq.~(\ref{lead2}) over $z$ and averaging over the muon spin:
\bea
\frac {d^2 N_{\nu_{\mu}} }{ dx\,d\cos{\theta} } &=&  
F_{e}^{(0)} (x) + J_{e}^{(0)}\, {\cal P}_\mu(x) \cos \theta\nn\\
& + & \frac{\alpha}{2\pi} \ln\left(\frac{m_\mu^2}{m_e^2}\right)                      [F_{e}^{(L L)}(x)+ J_{e}^{(LL)}(x)\,{\cal P}_\mu \cos\theta] ,
\eea
where $F_{e}^{(0)}(x),J_{e}^{(0)}(x)$ are given in (\ref{tree_e}) and 
\bea
\label{LL}
F_{e}^{(LL)} (x) &=& 2F_{e}^{(0)}(x)\ln \left( \frac{1-x}{x} \right)+2x(1-2x^2)+\frac{5}{6}+\frac{8 z^3 }{3}, \nn \\
J_{e}^{(LL)} (x) &=& 2J_{e}^{(0)}(x)\ln \left( \frac{1-x}{x}\right )-4x^2-\frac{1}{6}+\frac{8z^3}{3}.
\eea

The ``leading log'' contributions will arise solely from the larger terms in eq.~(\ref{LL}). Discarding all other terms, it follows a correction proportional to the tree level distributions
\bea
\label{SL}
\frac {d^2 N_{e} }{ dx\,d\cos{\theta} } =  
[F_{e}^{(0)} (x) + J_{e}^{(0)}\, {\cal P}_\mu(x) \cos \theta] 
\left( 1 + \frac{\alpha}{\pi}\,\psi(x)\right),
\eea
where
\bea
\label{psi}
\psi(x) \equiv \ln\left(\frac{m^2_\mu}{m^2_e}\right)\ln\left(\frac{1-x}{x}\right)
\eea
contains the leading logarithms and diverges for $x\rightarrow 1$. 

Fig.~\ref{ele} compares the tree level forward distribution with those corrected within the ``leading log'' approximation and with the exact ${\cal O}(\alpha)$ formulae \cite{Beherends}, given in appendix C. The effect is in 
both cases of ${\cal O}(1\%)$, except at the upper part of the allowed energy range where it is larger, see Fig.~\ref{ele_end}. At the kinematical edge, the ``leading log'' deteriorates, since we have not considered the function $G$, defined as having no mass singularities, which also diverges for $x\rightarrow 1$.
In the next subsection we will study the origin and cancellation of the end-point singularity.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\label{endpoint} End-point singularity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The end-point singularity originates when the phase space for the emission of real photons shrinks to zero and does not compensate the 
IR infinities of the virtual photons. Hence, it is essentially an IR problem and a signal of a failure of 
the perturbative treatment at the end-point region $x\rightarrow 1$. We summarize here the solution to this problem discussed in the literature \cite{Matson} for the corrected $e^-$ distribution. After that, we study the form of the 
divergence for the neutrino distributions, following as a guideline the solution found for the electron.
 
For $x \rightarrow 1$ and $m_e \rightarrow 0$, the exactly corrected electron differential distribution,  diverge as two singular logarithms:
\be
\label{ende}
\frac {d^2 N_{e}}{ dx\,d\cos{\theta} } =  [F_{e}^{(0)} (x) + J_{e}^{(0)}\, {\cal P}_\mu(x) \cos \theta]\left(1 + \frac{\alpha}{\pi}\ln(1-x)\right)\left[\ln\left(\frac{m_\mu^2}{m_e^2}\right)-2\right]\,\,.
\ee
The IR divergences stem from soft-photons on the limit $k\rightarrow 0$. Then, the solution proposed to control the double divergence of eq.~(\ref{ende}) is to consider multiple soft-photon emission. In \cite{Yennie} it was shown that 
the contribution of soft photons at all orders in $\alpha$ gives rise to the exponentiation of the double singularity: 
\be
\frac {d^2 N_{e} }{ dx\,d\cos{\theta} } =  [F_{e}^{(0)} (x) + J_{e}^{(0)}\, {\cal P}_\mu(x) \cos \theta]\,e^{\frac{\alpha}{\pi}\ln(1-x)\left[\ln\left(\frac{m_\mu^2}{m_e^2}\right)-2\right]}
\ee
The exponentiation method can only be applied very close to the end-point of the $e^-$ spectrum. The application to the whole range $0<x<1$ results in terms of ${\cal O }(\alpha^2)$ and higher when the exponential is expanded, which would go beyond  the  ``leading log'' approximation considered here.  

Consider now the procedure for the neutrino distributions. We shall see explicitly in the next section how, when the soft photon limit is applied to the ${\cal O}(\alpha)$ diagrams in Figs.~\ref{loops}b-\ref{loops}d, only IR singular terms remain for both virtual and photon emission. These two sets of IR divergences cancel each other, and a safe IR correction results. 

We anticipate that the soft-photon correction for both $\nu_\mu$ and $\bar\nu_e$ is proportional to the tree level amplitude and, in the limits $x\rightarrow 1$ and $m_e\rightarrow 0$, it has the form:
\be
\frac {d^2 N_{\nu}}{ dx\,d\cos{\theta} } =   [F_{\nu}^{(0)} (x) + J_{\nu}^{(0)}\, {\cal P}_\mu(x) \cos \theta]\left(1 - \frac{\alpha}{2\pi}\ln^2(1-x)\right).
\ee
Then, at the end-point, for each soft virtual photon and each soft real photon we get a $\ln^2(1-x)$ term, which multiplies the tree level amplitude. 
If there are $n$ soft virtual photons and $n$ soft real photons, there are $n$ double logarithms with an additional symmetry factor of $1/n!$. 
Therefore, the correction to the neutrino distribution at all orders in $\alpha$ is obtained summing over $n$:
\be
\frac {d^2 N_{\nu} }{ dx\,d\cos{\theta} } =   [F_{\nu}^{(0)} (x) + J_{\nu}^{(0)}\, {\cal P}_\mu(x) \cos \theta]\,e^{-\frac{\alpha}{2\pi}\ln^2(1-x)}.
\ee

As a result of the evaluation of infrared divergences at all orders, double logarithms are exponentiated, which ensures 
a non-divergent behavior of the distributions. The exponentiation is only valid for a small region $x\rightarrow 1$. For lower $x$, we 
must include all the terms of the exact corrections, computed in the next section.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Exact QED Corrections $(m_e=0,m_\nu=0)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The radiative corrections to the neutrino angular and energy distributions 
are expected to be of ${\cal O}(\frac{\alpha}{\pi})\sim {\cal O}(0.1\%)$, that is, once it is integrated over the variables affected by QED corrections, 
we do not expect any ``leading-log'' correction  of the form $\frac{\alpha}{\pi}\ln(\frac {m_\mu}{m_e})$. The absence of QED corrections 
in the neutrino legs also ensures that no terms 
in $\log (\frac {m_\mu}{m_\nu})$ will appear. Hence, we set $m_\nu=0$ from the beginning of the calculation but retain $m_e\neq 0$ until loop and photon emission diagrams are summed up, which cancels IR and mass singularities. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Photon loops}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The computation of the diagrams in Figs.~\ref{loops}b-\ref{loops}d, is summarized in Appendix A. The cancellation of UV divergences is detailed there. The remaining terms
contain IR divergences that will be regularized here by a finite photon mass $\lambda$. As a result of adding up the contributions from all virtual 
photon diagrams, 
the $\mu-e$ vertex (\ref{gam_sig}) is corrected at order $\alpha$ as follows:
\bea
\label{vertex}
\Gamma_\sigma &=& (1-\gamma_5)
             - \frac{\alpha}{2\pi}\,[\, 
             g_{\rm{L}}^{\rm{S}}\,\gamma_\sigma\,(1-\gamma_5)
            +g_{\rm{R}}^{\rm{S}}\,\gamma_\sigma\,(1+\gamma_5)\nn\\
            &+&g_{\rm{L}}^{\rm{V}}\,p_{1\sigma}\,(1-\gamma_5)
            +g_{\rm{R}}^{\rm{V}}\,p_{2\sigma}\,(1+\gamma_5)\,],
\eea
where $g_{\,\rm{L,R}}^{\rm{S,V}}$ are listed in the appendix A. The function $g_{\rm{L}}^S$ contains the IR divergences 
while the rest of the ``$g$'' functions are finite.

The interference of loop diagrams with the tree level amplitude
 yields the $\cal{O}(\alpha)$ correction to the decay rate 
\be
d\Gamma_v=\frac{1}{2m_{\mu}}\,64\,{\rm G_F}^2\,\,|M_V|^2\,d\Phi_3(p_\mu;p_e,p_{\bar\nu_e},p_{\nu_\mu}),
\ee
where $|M_V|^2$ for unpolarized muons is given by:
\be
\label{ampl_virtual}
|M_V|^2= |M_0|^2\, -\,\frac{\alpha}{2\pi}\left[H_0|M_0|^2+H_1(p_\mu p_{\bar\nu_e})(p_\mu p_{\nu_\mu})+H_2(p_e p_{\bar\nu_e})(p_e p_{\nu_\mu})+H_3(p_{\bar\nu_e}p_{\nu_\mu})\right],
\ee
and the ``H'' functions are defined as
\be
H_0  =   2\,g_{\rm{L}}^{\rm{S}}\;,\qquad
H_1  =   2\,m_e\,g_{\rm{L}}^{\rm{V}}\;,\qquad
H_2  =   2\,m_\mu\,g_{\rm{R}}^{\rm{V}}\;,\qquad
H_3  =   \frac{m_{\mu}m_e}{2} \,g_{\rm{R}}^{\rm{S}}.
\ee

In eq.~(\ref{ampl_virtual}) we have followed the same notation
used in \cite{Jezabek1} for the QCD corrections to the lepton spectra in the 
decay $t\rightarrow b+l+\nu_l$.  
We use their techniques to parameterize the three body phase 
space $d\Phi_3(p_\mu;p_e,p_{\bar\nu_e},p_{\nu_\mu})$ in terms of the Euler angles $(\alpha,\theta,\gamma)$ describing the orientation 
of the outgoing particles, the scaled neutrino energy $x_\nu=E_\nu/m_{\mu}$  and the variable $y=\frac{(p_{\bar\nu_e}+p_{\nu_\mu})^2}{m_{\mu}^2} $: 
\be
\label{Dalitz}
d\Phi_3(x,y)\,=\, \frac{1}{(4m_\mu)^2}\,\frac{1}{(2\pi)^5}\,d x\, dy\,d\alpha\,d\gamma\, d\cos{\theta}.
\ee
The functions ``H'' and the scalar products appearing in eq.~(\ref{ampl_virtual}) are functions of $y$, $x_{\bar\nu_e}$, $x_{\nu_\mu}$. For instance, 
for $m_e=0$ the tree level amplitude reads:
\bea
|M_0(x_{\bar\nu_e},y)|^2 &=& \frac{(m_\mu)^2}{4}\,x_{\bar\nu_e}\,\left(1-x_{\bar\nu_e}\right),\nn \\
|M_0(x_{\nu_\mu),y}|^2 &=&  \frac{(m_\mu)^2}{4}\,(x_{\nu_\mu}-y)\left(1-y-x_{\nu_\mu}\right),
\eea
where the following relation has been used:
\be
1=x_{\bar\nu_e}+x_{\nu_\mu}-y.
\ee


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Real photon emission}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The contribution from real photon emission, Figs.~\ref{loops}e,~~\ref{loops}f, is given by 
\be
d\Gamma_r=\frac{1}{2m_{\mu}}\,64\,{\rm G_F}^2\,\,|M_R|^2\,d\Phi_4(p_\mu;p_e,p_{\bar\nu_e},p_{\nu_\mu},k),
\ee
where the amplitude $|M_R|^2$ has the following expression:
\be
\label{Mreal}
|M_R|^2= \frac{\alpha}{2\pi}\,\bigg[\frac{A}{(p_\mu k)^2}+\frac{B}{(p_e k)^2}-\frac{C}{(p_\mu k)(p_e k)}\bigg],
\ee
and the numerators for unpolarized muons read:
\bea
\label{ampl_real}
A & = & p_\mu^2 \,[(p_\mu p_{\bar\nu_e})(p_e p_{\nu_\mu})-(kp_{\bar\nu_e})(p_e p_{\nu_\mu})-(p_\mu k)(kp_{\bar\nu_e})(p_e p_{\nu_\mu})] \nn \\ 
B & = & p_e^2 \,[(p_\mu p_{\bar\nu_e})(p_e p_{\nu_\mu})+(p_\mu p_{\bar\nu_e})(kp_{\nu_\mu})-(p_\mu p_{\bar\nu_e})(p_e k)(kp_{\nu_\mu})] \nn \\ 
C & = & (p_\mu p_e)\,[2(p_\mu p_{\bar\nu_e})(p_e p_{\nu_\mu})+(p_\mu p_{\bar\nu_e})(kp_{\nu_\mu})-(kp_{\bar\nu_e})(p_e p_{\nu_\mu})]  \nn \\      & 
+ & (p_e k)[(p_\mu p_{\bar\nu_e})(p_e p_{\nu_\mu})+(p_\mu p_{\bar\nu_e})(p_\mu p_{\nu_\mu})-(p_\mu p_{\bar\nu_e})(kp_{\nu_\mu})]  \nn \\         & - & (p_\mu k)[(p_\mu p_{\bar\nu_e})(p_e p_{\nu_\mu})+(p_e p_{\bar\nu_e})(p_e p_{\nu_\mu})-(kp_{\bar\nu_e})(p_e p_{\nu_\mu})].
\eea
Terms of order $k^2$  are not included in eq.~(\ref{ampl_real}), since they vanish in the limit of massless photons. 

To obtain the exactly corrected neutrino spectra, we must integrate over the photon and electron momenta. The following decomposition 
of the four-body phase space \cite{Particle} is useful:
\bea
d\Phi_4 (p_\mu;p_e,p_{\bar\nu_e},p_{\nu_\mu},k)=m_{\mu}^2\, dM^2\, d\Phi_3 (p_\mu;P,p_{\bar\nu_e},p_{\nu_\mu})\,d\Phi_2 (P;p_e,k),
\eea
where $M^2 \equiv P^2$ is the invariant mass of the electron-photon system. The integration over $[dM^2 d\Phi_2(P;p_e,k)]$ is a difficult task and we follow the method detailed in \cite{Jezabek1} to perform it. The amplitude can then be written as a function of $x$ and $y$  and the integration over the remaining decay particles is performed parametrazing $d\Phi_3 (p_\mu;P,p_{\bar\nu_e},p_{\nu_\mu})$ as in eq.~(\ref{Dalitz}).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Soft photon limit and IR cancellation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Before continuing with the discussion of the exact corrections, let us consider their soft photon limit, i.e. $k \rightarrow 0$, 
in order to illustrate the discussion about the end-point singularities mentioned in the previous section. As stated there, only IR 
singular terms of virtual and real photon diagrams remain in this limit. When soft virtual and soft real photon contributions are added up, all ${\cal O}(\alpha)$ IR singularities are canceled.

In this limit,  the ${\cal O}(k)$ terms in the virtual photon diagrams, Figs.~\ref{loops}b-\ref{loops}d, are neglected and eq.~(\ref{vertex}) reduces to:
\be
\label{soft}
\Gamma_\sigma^{S P}=\gamma_\sigma(1-\gamma_5)\,(1\,-\,\frac{\alpha}{2\pi}\, 
             g_{\rm{L}}^{\rm{S}}),
\ee
where $g_{\rm{L}}^{\rm{S}}$ contains all IR divergent term of the form:
\be
\label{IR}
\ln\left(\frac{\lambda^2}{m_e m_\mu} \right),
\ee

The corrected squared amplitude in eq.(\ref{ampl_virtual}) is simplified to :
\be
\label{soft_ampl_virtual}
|M_V^{\rm S P}|^2=  \left(1-\,\frac{\alpha}{2\pi} \, H_0(x,y)\right)\,|M_0|^2, 
\ee

Consider now the diagrams containing real photon emission, Figs.~\ref{loops}e,\ref{loops}f . Only terms of order  ${\cal O}(k^{-2})$ remain in the soft photon limit.  They contain all IR divergent contributions 
from bremsstrahlung . The squared amplitude in eq.(\ref{ampl_real}) reduces to:  
\be
\label{soft_ampl_real}
|M_r^{\rm S P}|^2= 32\,\frac{\alpha}{2\pi}\,\bigg[\frac{p_\mu^2}{(p_\mu k)^2}+\frac{p_e^2}{(p_e k)^2}-\frac{2(p_\mu p_e)}{(p_\mu k)(p_e k)}\bigg]|M_0|^2.
\label{IRreal}
\ee

The divergences in eq.~(\ref{soft_ampl_virtual}) cancel when added with the soft bremsstrahlung part. However, eq.~(\ref{soft_ampl_real}) must be previously integrated over the photon-electron phase space in order to reduce the contribution to a three-body 
problem. The integral is performed introducing a finite photon mass $\lambda$ resulting in a correction:
\be
\label{soft_ampl_real_2}
\frac{\alpha}{2\pi} \, J_0(x,y)\, |M_0(p_\mu,P, p_{\bar\nu_e},p_{\nu_\mu})|^2,
\ee
where now the IR singularities are contained in the function $J_0(x,y)$ and 
which exactly cancel those in eq.(\ref{IR}). By the same token, all logarithmic singularities appearing in the $m_e\rightarrow 1$ limit do cancel. 

The resulting function in  $\lambda$-independent:
\be
\label{cancelation}
K_0(x,y)\equiv H_0(x,y)+J_0(x,y).
\ee 

In all generality the function $K_0(x,y)$ has a rather long expression. It takes a simple form, though, in the massless electron limit:
\be
\label{K0}
K_0(x,y)= \frac{\pi^2}{3}+2{\rm L}(x)+2{\rm L}(y/x)+\ln^2\left(\frac{1-y/x}{1-x}\right),
\ee
where $L(x)$ is the Spence function defined in Appendix A. 
Double logarithms in (\ref{K0}) are exposed once infrared singularities are canceled. After the integration 
over the Euler angles and $y$, the corrected distributions for both $\nu_\mu$ and $\bar\nu_e$, in the soft photon and 
massless electron limits, read:
\bea
\frac {d^2 N_{\nu}^{S P} }{ dx\,d\cos{\theta} } =   F_{\nu}^{(0)} (x)[1 - \frac{\alpha}{2\pi}k(x)],
\eea
where
\be
k(x)\equiv\int dy\, K_0(x,y) = 2{\rm L}(x) + 2 \pi^2/3 + \ln^2(1-x).
\ee
As anticipated above, the function $k(x)$ behaves as  $\ln^2(1-x)$. We see thus explicitly how the end-point singularity appears 
in the soft  and collinear limits to the complete ${\cal O} (\alpha)$  corrections. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Corrected expressions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%In the previous subsection, it is shown how IR divergences arise from soft virtual photon and real photon 
%terms and how they cancel when real diagrams are added to virtual diagrams after the integration of the former over the electron-photon phase space. 

Exactly corrected neutrino distributions are obtained considering all 
terms of the ${\cal O}(\alpha)$ corrected amplitudes, eq.~(\ref{ampl_virtual}) and eq.~(\ref{ampl_real}). 
Both can be summed up after the integration of eq.~(\ref{ampl_virtual}) over  $[dM d\Phi_2(P;p_e,k)]$, which simplifies real photon emission from a four body-problem to a three-body problem. The corrected energy and angular distributions are then obtained integrating $d\Phi_3(p_\mu;p_e,p_{\bar\nu_e},p_{\nu_\mu})$ over the Euler angles $0<\alpha,\gamma<2\pi$ and over $0<y<1$. 

QED corrections for polarized muons are calculated identically to those of unpolarized ones
 with the replacement $p_\mu\rightarrow p_\mu-sm_{\mu}$ in the amplitudes \cite{Jezabek2}, where $s$ is the muon four-spin. 
The integration over the phase space is slightly modified. For example, 
for $\bar\nu_e$ distribution, the Euler angles $\alpha$, $\gamma$ and $\theta$ are chosen to give the orientation of 
${\bf p_{\bar\nu_e}}$ and the electron three-momentum, ${\bf P}\equiv{\bf p_e}$ for virtual photons and  ${\bf P}\equiv {\bf p_e + k}$ and ${\bf p_{\bar\nu_e}}$ with respect to a system of coordinates where the muon spin $\bf s$ lies along the z-axis. 
As in the tree level expressions, $\theta$ is defined as the angle between $\bf s$ and neutrino 
momentum $\bf p_{\bar\nu_e}$. Defining the three-vector ${\bf w}={\bf p_{\bar\nu_e}}\times{\bf s}$, $\alpha$ is chosen as the 
angle between ${\bf w}$ and the y-axis and $\gamma$ between ${\bf w}$ and ${\bf P_t}$, the projection of ${\bf P}$ onto the 
plane perpendicular to ${\bf p_{\bar\nu_e}}$, see Fig.~\ref{angles}. With this choice, the scalar products which appear in 
$|M_V|^2$ and $|M_R|^2$ do not depend on $\gamma$. However, the integration over $\alpha$ is not trivial because $(sP)$ and 
$(sp_{\nu_\mu})$ are $\alpha$-dependent. We follow the results obtained in \cite{Jezabek2} for the QCD contributions.

The corrected neutrino spectra including all finite terms in the limit $m_e=0$ \cite{Jezabek3} are: 
\bea
\frac {d^2 N_{\nu_{\mu}} }{ dx\,d\cos{\theta} } =  
F_{\nu_\mu}^{(0)} (x) + J_{\nu_\mu}^{(0)}\, {\cal P}_\mu(x) \cos \theta - \frac{\alpha}{2\pi}( F_{\nu_{\mu}}^{(1)}(x)+  
J_{\nu_\mu}^{(1)}(x)\,{\cal P}_\mu \cos \theta)]\,\,,\nn \\ \nn\\
\frac {d^2 N_{\bar\nu_e}}{ dx\,d\cos{\theta} } =    F_{\bar\nu_e}^{(0)} (x) + J_{\bar\nu_e}^{(0)}(x)\,{\cal P}_\mu\cos \theta - \frac{\alpha}{2\pi}( F_{\bar\nu_e}^{(1)}(x) + J_{\bar\nu_e}^{(1)}(x)\,{\cal P}_\mu\cos \theta)]\,\,\\ \nn  \eea
where $F_{{\bar\nu_e},\nu_\mu}^{(0)}$-$J_{{\bar\nu_e},\nu_\mu}^{(0)}$ are given in eq.~(\ref{tree_numu}), 
and the one-loop corrections are given by:
\bea
\label{correct}
F_{\nu_{\mu}}^{(1)}(x) & = &  F_{\nu_\mu}^{(0)}(x) k(x)
                 + \frac{1}{6}(41-36x+42x^2-16x^3)\ln(1-x)\nn \\
                 & + & \frac{1}{12}x(82-153x+86x^2)\,\,, \nn \\
J_{\nu_{\mu}}^{(1)}(x) & = &  J_{\nu_\mu}^{(0)}(x) k(x)
                 + \frac{1}{6}(11-36x+14x^2-16x^3-4/x)\ln(1-x)\nn \\
                 & + & \frac{1}{12}(-8+18x-103x^2+78x^3)\,\,, \nn \\
F_{\bar\nu_e}^{(1)}(x)     & = & F_{\bar\nu_e}^{(0)}(x) k(x) 
                 +  (1-x)\Big[(5+8x+8x^2)\ln(1-x)\nn \\
                 & + & \frac{1}{2}x(10-19x)\Big] \,\,,\nn \\\nn \\
J_{\bar\nu_e}^{(1)}(x)     & = & J_{\bar\nu_e}^{(0)}(x)k(x)
                 + (1-x)\Big[(-3+12x+8x^2+4/x)\ln(1-x) \nn \\
                 & + & \frac{1}{2}(8-2x-15x^2)\Big]\,\,.
\eea


As expected from the discussion of the former subsection, the function $k(x)$ appears in eq.~(\ref{correct}) multiplying the tree level functions $F_{\nu}^{(0)}$-$J_{\nu}^{(0)}$. When double logarithms are exponentiated, the singular term in $\ln(1-x)$ is wash-out as well.

Fig.~\ref{numu} and Fig.~\ref{nue} compare the corrected and the tree level forward $\nu_\mu$ and $\nu_e$ distributions, respectively. The relative correction is shown in Figs.~\ref{numucor},~\ref{nuecor} for a typical range of the neutrino energy. In both cases they are of ${\cal O}(0.1\%)$, well below the order of the expected precision in the knowledge of the beam parameters. 

In the laboratory frame, neutrino fluxes are boosted along the muon momentum direction. The formulae of the corrected distributions 
in that frame are given in Appendix B. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Muon-beam divergence}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We study below the systematic uncertainty in the neutrino distributions produced by the muon beam divergence. For the sake of illustration, the 
quantitative results will be given for a $30$ GeV unpolarized muon beam decaying in a long straight section pointing to a far detector 
located at $2810$ km.

The natural decay angle of the forward neutrino beam in the laboratory frame is deduced from the relation 
between the rest and laboratory frames. In the rest frame, half of the neutrinos are 
emitted within the cone $\theta\leq\pi/2$. In the laboratory frame:
\bea
\cos \theta^{'} = \frac{\cos\theta +\beta}{1 +\beta\cos\theta},
\eea 
where  $ \beta=\sqrt{1-\gamma^{-2}}$ is the muon velocity in the laboratory frame. Therefore, half of neutrinos are emitted 
within the cone subtended by the decay angle $\theta^{'} \leq 1/\gamma$. For instance, for $30$ GeV muons $1/\gamma=m_\mu/E_\mu=3$ mrad.   

For the beam and baseline illustrated here, a $10$ kt detector and one year of data taking \cite{new}, 
 the statistical error on the neutrino flux 
 is of the order of  O($0.4\%$). It is then convenient to restrain the uncertainty induced by the muon beam divergence below that level.
To achieve this, the direction of the beam must be carefully monitored within the decay straight section by placing beam position 
monitors at its ends.  The angular divergence of the parent muon beam is then small compared to the natural decay angle of the neutrino 
beam $\theta^{'}\sim 1/\gamma$, see Fig.~\ref{divergence}, aiming at present to a divergence of  ${\cal O}(0.1 /\gamma)$. 
It implies that the neutrino beam will be collinear, within the limits set by the decay kinematics.

In our calculations we parameterize this beam  focalization by a gaussian distribution with standard deviation $\sigma\sim 0.1/\gamma$ 
(i.e. $0.3$ mrad for $30$ GeV muon beam) \cite{geer}, which suppresses the flux of neutrinos as they separate from the straight direction.
The divergence is introduced analytically by considering that the muon direction opens an angle $\alpha$ with respect to the z-axis, defined 
as the direction pointing towards the far detector at a distance $L$, see Fig.~\ref{divergence}. The neutrino distributions in the rest frame, eq.~(\ref{treedistr}), are Lorentz boosted along the z-axis. 
The rest-frame basis $(x,\cos{\theta})$ is 
transformed to the lab-frame basis $(z, \cos{\theta}^{'})$, where $z=E_{\nu}/E_{\mu}$ and $\theta^{'}$ is the angle between the neutrino 
beam and the z-axis.
Using the parameters $\beta=\sqrt{1-\gamma^{-2}}$, the boosted distributions read:
\bea
\frac{d^2 N_{\bar\nu_\mu, \nu_\mu}}{dz d\Omega} &=& 
   \frac{4 n_\mu}{\pi L^2 m_\mu^6} \, E_\mu^4 z^2 (1 - \beta (\sin \varphi^{'} \sin \alpha \sin \theta^{'} + \cos \alpha \cos \theta^{'}))\nn\\
 & &   \times \left\{\left[3 m_\mu^2 - 4 z E_\mu^2 (1-\beta (\sin \varphi^{'} \sin \alpha \sin \theta^{'} + \cos \alpha \cos \theta^{'}))\right] 
\right. \nn \\\nn \\
    & & \left. \mp \, {\cal P}_\mu 
   \left[m_\mu^2 - 4 z (1 - \beta(\sin \varphi^{'} \sin \alpha \sin \theta^{'} + \cos \alpha \cos \theta^{'}))\right] \right\},\, \nn \\\nn \\
\frac{d^2 N_{\nu_e, \bar\nu_e}}{dz d\Omega} &=& 
   \frac{24 n_\mu}{\pi L^2 m_\mu^6} \, E_\mu^4 z^2 (1 - \beta(\sin \varphi^{'} \sin \alpha \sin \theta^{'} + \cos \alpha \cos \theta^{'}))\nn\\
& &   \times \left\{ \left[ m_\mu^2 - 2 z E_\mu^2 (1-\beta(\sin \varphi^{'} \sin \alpha \sin \theta^{'} + \cos \alpha \cos \theta^{'}))\right] \right. \nn \\\nn \\
   & & \left. \mp \, {\cal P}_\mu 
   \left[m_\mu^2 - 4 z (1 - \beta(\sin \varphi^{'} \sin \alpha \sin \theta^{'} + \cos \alpha \cos \theta^{'}))\right] \right\}.
\eea  

The above expressions are integrated on $\alpha$, weighted with the gaussian factor
\be 
\frac{e^\frac{-\alpha^{2}}{2\sigma^{2}}}{\sqrt{2\pi \sigma^{2}}}.
\ee

For unpolarized muons ($P_{\mu}=0$) (for different muon polarizations we obtain similar results), it results:
\bea
\nn\\
\frac{d^2 N_{\bar\nu_\mu, \nu_\mu}}{dz d\Omega} &=& 
   \frac{4 n_\mu}{\pi L^2 m_\mu^6} \, E_\mu^4 z^2 \, \left\{3 m_\mu^2 \left( 1 -\beta e^{\frac{-\sigma^{2}}{2}} \cos \theta^{'} \right) \right.\nn\\
 & &\left.- 4 z E_\mu^2 \left(1- 2\beta e^{\frac{-\sigma^{2}}{2}} \cos \theta^{'} 
 \right.\right.\nn \\ 
 & &\left.\left.+\beta^{2}\left(\frac{1 - e^{-2 \sigma^{2}}}{2} \right)\sin^{2}\theta^{'} \sin^{2}\varphi^{'} +\beta^{2}\left(\frac{1 + e^{-2 \sigma^{2}}}{2} \right)\cos^{2}\theta^{'} \right)\right\},\nn\\\nn\\\nn\\\nn\\\nn\\
\frac{d^2 N_{\nu_e \bar\nu_e}}{dz d\Omega} &=& 
   \frac{24 n_\mu}{\pi L^2 m_\mu^6} \, E_\mu^4 z^2 \, \left\{m_\mu^2 \left( 1 -\beta e^{\frac{-\sigma^{2}}{2}} \cos \theta^{'} \right) \right.\nn\\
& &\left.- 2 z E_\mu^2 \left(1- 2\beta e^{\frac{-\sigma^{2}}{2}} \cos \theta^{'}   \right.\right.\nn \\ 
 & &\left.\left.+\beta^{2}\left(\frac{1 - e^{-2 \sigma^{2}}}{2} \right)\sin^{2}\theta^{'} \sin^{2}\varphi^{'}+\beta^{2}\left(\frac{1 + e^{-2 \sigma^{2}}}{2} \right)\cos^{2}\theta^{'} \right)\right\}.
\eea 
Setting $\theta^{'}=0$, the expression of forward neutrino fluxes reads:
\bea
\frac{d^2 N_{\bar\nu_\mu, \nu_\mu}}{dz d\Omega} &=& 
   \frac{4 n_\mu}{\pi L^2 m_\mu^6} \, E_\mu^4 z^2 \, \left\{3 m_\mu^2 \left( 1 -\beta e^{\frac{-\sigma^{2}}{2}} \right) \right.\nn\\
 & &\left.- 4 z E_\mu^2 \left(1- 2\beta e^{\frac{-\sigma^{2}}{2}} 
 +\beta^{2}\left(\frac{1 + e^{-2 \sigma^{2}}}{2} \right)\right)\right\},\nn\\\nn \\
\frac{d^2 N_{\nu_e \bar\nu_e}}{dz d\Omega} &=& 
   \frac{24 n_\mu}{\pi L^2 m_\mu^6} \, E_\mu^4 z^2 \, \left\{m_\mu^2 \left( 1 -\beta e^{\frac{-\sigma^{2}}{2}} \right) \right.\nn\\
& &\left.- 2 z E_\mu^2 \left(1- 2\beta e^{\frac{-\sigma^{2}}{2}} +\beta^{2}\left(\frac{1 + e^{-2 \sigma^{2}}}{2} \right)\right)\right\}.
\eea  
 
Figs.~\ref{spectrum_mu},~\ref{spectrum_e} show the numerical results for the neutrino and antineutrino spectra in a medium baseline ($2810$ km). We compare 
the distribution where the muon beam is aligned with the detector direction (no beam divergence) with the distribution where the muon-beam divergence is included. In the former, 
neutrino beams are averaged over an angle $\theta^{'}$ of $0.1$ mrad at the far detector \cite{golden}. 

Neutrino and antineutrino fluxes are shown in Figs.~\ref{fluxomu},~\ref{fluxoe} for the same localization of the detector. As it was the case for the spectra, the divergence of the  beam is more important for the electron neutrino case. Our formulae predict a similar flux correction 
than previous numerical estimations \cite{geer}. For instance, a $10\%$ uncertainty in the muon beam divergence 
would lead to a flux uncertainty of $0.3\%$.  We obtain,
\bea
\frac{\frac{\Delta d N_{\nu}}{d E}}{\frac{d N_{\nu}}{d E}}\sim 0.03 \frac{\Delta \alpha}{\alpha}.
\eea   
If the muon beam divergence is constrained by lattice design to be less than $0.05/\gamma$, the loss of flux will be negligible \cite{griego}.   
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

A neutrino factory from muon decay aims at a precision  better than ${\cal O}(1\%)$ in the knowledge of the resulting 
intense neutrino fluxes.
We have considered here the QED corrected angular and energy flux distributions in muon decay. 
The tree-level electron distribution get corrections of ${\cal O}(1\%)$, resulting 
from large terms of the form $\frac{\alpha}{\pi}\ln(\frac{m_\mu}{m_e})$, that we have computed in the ``leading log'' approximation which turns out to 
be very accurate for this problem. The corresponding formulae are simple and have been presented both in the rest and laboratory frames.
 The radiative corrections to the neutrino distributions are also evaluated in this work, including muon polarization effects. 
In this case there are no leading log corrections and the exact ones must are considered (for $m_e=0$ and $m_\nu=0$). 

Both electron and neutrino one-loop corrected distributions diverge at the upper edge of the kinematical allowed region.
This results from a failure in the cancellation of infrared divergences from virtual photons by real photons. 
Applying the soft photon limit to the exact calculations, we have isolated the end-point divergent term for the neutrino 
distributions which takes the form of $\ln^2(1-x)$. In order to control this one-loop singularity, the double logarithmic-contribution is exponentiated,   
encompassing the contributions from all orders of perturbation theory.
 All in all, the exact neutrino distributions get corrections of ${\cal O}(0.1)\%$, safely below the 
expected precision in the flux measurements.

We have also  studied carefully another source of precision uncertainty: the influence of the muon beam divergence on the  
neutrino spectra at the far site. 
The challenge in designing the neutrino production section, where the muons decay, is to constrain the muon beam divergence 
to a value smaller than the natural cone of forward going neutrinos in the laboratory frame, ($\sim 1/\gamma$).  
At present, the long straight sections under discussion aim at an angular muon beam divergence of the order of $0.1/\gamma$, typically less than one mrad. 
We have included the angular divergence in the computation through a gaussian distribution. 
The induced uncertainties on the neutrino spectra turn out to be a safe ${\cal O}(0.1 \%)$.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Acknowledgments}

We thank M.B. Gavela, P. Hern\'andez and A. De R\'ujula for their physics suggestions and useful discussions. We thank as well F.J Yndur\'ain for illuminating conversations. We are further indebted to  A. Blondel, F. Dydak, J.J.~G\'omez-C\'adenas. A.B acknowledges M.E.C.D for financial support by FPU grant AP2001-0521 and O.M acknowledges C.A.M for financial support by a FPI grant. The work has been partially supported by CICYT FPA2000-0980 project.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Note added in proof}

While this paper was being written, a preprint appeared \cite{LL} where the  ${\cal O}(\alpha)$ and ${\cal O}(\alpha^2)$ ``leading log'' corrections to the positron energy and angular distribution of muon decay were computed. Their final results for the  ${\cal O}(\alpha)$ corrections agree with ours.


\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\Large{\bf Appendix A: QED loop corrections}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}

{\bf A.1 Vertex correction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.1cm}

The vertex correction diagram, Fig.~\ref{loops}b, has the expression:
\be  
\Gamma_\sigma^b= i e^2\,\int\frac{d^4k}{(2\pi)^4}\,\,\frac{1}{k^2}\,\,\gamma_\lambda\, \frac{ p_e\fsls-k\fsl+m_e}{(p_e-k)^2-m_e^2}\,\gamma_\sigma(1-\gamma_5) \frac{ p_\mu\fsls-k\fsl+m_\mu}{(p_\mu-k)^2-m_\mu^2}\,\gamma^\lambda.
\ee
where $k$ is the four-momentum of the virtual photon 

With some algebra can be rewritten as: 
\be
\label{vertexcorr}  
\Gamma_\sigma^b= i e^2 \int\frac{d^4k}{(2\pi)^4}\,\,\frac{N_\sigma}{k^2\,[(p_e-k)^2-m_e^2]\,[(p_\mu-k)^2-m_\mu^2]},
\ee
where
\bea
N_\sigma &=& 4(p_\mu p_e)\gamma_\sigma(1-\gamma_5)-2\,[\,p_\mu\fsls\gamma^\rho\gamma_\sigma(1-\gamma_5)\nn \\
&+&\gamma_\sigma(1-\gamma_5)\gamma^\rho p_e\fsls\,]\,k_\rho+\gamma_\lambda \gamma_\rho\gamma_\sigma(1-\gamma_5)\gamma_\tau\gamma^\lambda k^\rho k^\tau.
\eea

The IR divergent part is contained in the first term of $N_\sigma$ (independent of $k$) which is regularized by introducing a finite photon mass $\lambda$. The last terms (quadratic in $k$) is UV divergent.

After integration over the photon momentum, 
\bea
\label{V}
\Gamma_\sigma^b  =   - \frac{\alpha}{2\pi}\,&[& 
            (g_{\rm{IR}}^{b}+g_{\rm{UV}}^{b})\,\gamma_\sigma\,(1-\gamma_5)
            +g_{\rm{R}}^{\rm{S}}\,\gamma_\sigma\,(1+\gamma_5)\nn \\
            &+& g_{\rm{L}}^{\rm{V}}\,p_{1\sigma}\,(1-\gamma_5)
            + g_{\rm{R}}^{\rm{V}}\,p_{2\sigma}\,(1+\gamma_5)\,],
\eea
where
\bea
\label{IRvert} 
g_{\rm{IR}}^{b} & = &  \coth{\phi} \left  [   
          {\rm L}\left( \frac {2\sinh{\phi}} {e^{\omega}-e^{-\phi}} \right)            - {\rm L}\left( \frac {2\sinh{\phi}} {e^{\phi}-e^{-\omega}} \right)            + (\omega-\phi)
            \ln{\left( \frac {2\sinh {\left( \frac{\omega-\phi}{2} \right)}}                              {2\sinh {\left( \frac{\phi+\omega}{2} \right)}}                 \right)}
                    \right.  \nn \\  
        &\,& \qquad  \left. \,\,+\,\phi\left(\omega
                  -\ln{ \left( \frac{\lambda^2}{m_e^2} \right) } \right)
                      \right], \nn\\
\label{UVvert} 
g_{\rm{UV}}^{b}  & = & \frac {\phi\sinh{\phi}-\omega\sinh{\omega}}
                      {2(\cosh{\omega}-\cosh{\phi})} + \frac{1}{2}\left(\omega                      -  \ln{ \left( \frac{\Lambda^2}{m_e^2} \right) } \right)
                     -  \frac{3}{2}, \nn \\
g_{\rm{R}}^{\rm{S}}& = &\frac{- \phi}{\sinh{\phi}},\nn\\
g_{\rm{L}}^{\rm{V}}& = & \frac{1}{2m_{\mu}\sinh{\phi}} \left[ \phi - 
            \frac {\omega\sinh{\phi}-\phi\sinh{\omega}}
                  {\cosh{\omega}-\cosh{\phi}} \right],\nn\\
g_{\rm{R}}^{\rm{V}}& = & \frac{1}{2m_{e}\sinh{\phi}} \left[ \phi + 
            \frac {\omega\sinh{\phi}-\phi\sinh{\omega}}
                  {\cosh{\omega}-\cosh{\phi}} \right],
\eea

where, following \cite{Beherends},
\be
\cosh{\phi}=\frac{(p_\mu p_e)}{m_{\mu}m_e}\qquad
e^{\omega}=\frac{m_\mu}{m_e}.
\ee
L(x) is the Spence function 
\be
L(x)\equiv -\int_0^x\,dt\,\frac{\ln|1-t|}{t}.
\ee

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf A.2 Propagator corrections}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.1cm}
 
Consider diagrams in Figs.~\ref{loops}c,d for $i=\mu,e$, respectively:
\be
\Sigma(p_i) = i e^2\,\int\frac{d^4k}{(2\pi)^4}\,\,\frac{1}{k^2}\,\,\gamma_\lambda\, \frac{ p_i\fsls-k\fsl+m_i}{(p_i-k)^2-m_i^2}\,\gamma^\lambda
\ee

The contribution from self-energy diagrams to the muon-electron vertex is
\be
\label{prop}
\Gamma^{\rm b,c}_\sigma= \frac{1}{2}\left[Z(p_\mu)+Z(p_e)\right]\gamma_\sigma(1-\gamma_5),
\ee
where 
\be
Z(p_i)=\frac{\partial\Sigma(p_i)}{\partial p_i\fsls}\Bigg|_{{p_i\fsls}=m_i}.
\ee

$Z(p_i)$ is both UV and IR divergent:
\be
Z(p_i)= -\frac{\alpha}{2\pi}\,(h_{\rm{IR}}^{(i)}+h_{\rm{UV}}^{(i)}),
\ee
where the functions ``h'' read 
\bea
h_{\rm{UV}}^{(i)}& = &  \frac{3}{2} - \frac{1}{2}
                    \ln{ \left( \frac{\Lambda^2}{m_i^2} \right) },\\
h_{\rm{IR}}^{(i)} & = &  2 + \ln{\left( \frac{\lambda^2}{m_i^2} \right) }.\\
\eea

Adding both muon and electron self-energies, eq.~(\ref{prop}) has the following expression:
\be
\Gamma^{c,d}_\sigma= 
       -\frac{\alpha}{2\pi}(h_{\rm{UV}}^{c,d}+h_{\rm{IR}}^{c,d}),
\ee
where, now,
\bea
\label{UVself}
h_{\rm{UV}}^{c,d} & = & - \frac{1}{2}\left(\omega -
                    \ln{ \left( \frac{\Lambda^2}{m_e^2} \right) } \right)
                 +  \frac{3}{2},\\
h_{\rm{IR}}^{c,d} & = &  \left(\omega
                    - \ln{ \left( \frac{\lambda^2}{m_e^2} \right) } \right)+2. 
\eea
Adding eq.~(\ref{UVvert}) and eq.~(\ref{UVself}) the UV divergences are exactly cancelled. The IR divergences, when combined with eq.~(\ref{V}) gives rise to the term 
\bea
\label{gls}
g_{\rm{L}}^{\rm{S}} & = & - \coth{\phi} \left[ 
        \phi  
      - {\rm L}\left( \frac {2\sinh{\phi}} {e^{\omega}-e^{-\phi}} \right)            + {\rm L}\left( \frac {2\sinh{\phi}} {e^{\phi}-e^{-\omega}} \right)            - (\omega-\phi)
        \ln{\left( \frac {2\sinh {\left( \frac{\omega-\phi}{2} \right)}}                              {2\sinh {\left( \frac{\phi+\omega}{2} \right)}}                 \right)}
                       \right] \nn, \\ 
     & + & \frac {\phi\sinh{\phi}-\omega\sinh{\omega}}
                 {2(\cosh{\omega}-\cosh{\phi})} + 2  
                -(1-\phi\coth{\phi}) \left(\omega
                  -\ln{ \left( \frac{\lambda^2}{m_e^2} \right) } \right).
\eea
which multiplies $\gamma_\sigma(1-\gamma_5)$ in the ${\cal O}(\alpha)$ correction to $\mu-e$ vertex of eq.~(\ref{vertex}).


\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\Large{\bf Appendix B: QED corrected distributions
\vspace{0.1cm}
in the laboratory frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}

In order to obtain the neutrino distributions in the laboratory frame, a Lorentz boost is performed in the direction of the muon velocity towards the detector at a distance $L$. The rest-frame basis $(x,\cos{\theta})$ is transformed to the lab-frame basis $(z, \cos{\theta}^{'})$, where $z=E_{\nu}/E_{\mu}$ is the scaled energy at the laboratory frame and $\theta^{'}$ is the angle between the neutrino beam and the direction of the muon beam \cite{golden}. The muon beam divergence is set to zero. Using the parameters $\gamma=E_\mu/m_\mu$ and $\beta=\sqrt{1-\gamma^{-2}}$, the boosted distributions read:
\bea
\frac {d^2 N_{\nu_{\mu}} }{ dz d\cos{\theta}^{'} }& = &  \,F_{\nu_\mu}^{(0)} (z,\theta^{'}) + {\cal P}_\mu\, J_{\nu_\mu}^{(0)}(z,\theta^{'}) \cos \theta^{'}  \nn\\\nn\\
&- &\frac{\alpha}{2\pi}\left[\tilde F_{\nu_{\mu}}^{(1)}(z,\theta^{'})
+ {\cal P}_\mu\, \tilde J_{\nu_\mu}^{(1)}(z,\theta^{'}) \cos \theta^{'}\right]\,,\nn \\ \nn\\
%
\frac {d^2 N_{\bar\nu_e} }{ dz d\cos{\theta}^{'}} & = & \,F_{\bar\nu_e}^{(0)}(z,\theta^{'}) + {\cal P}_\mu\, J_{\bar\nu_e}^{(0)}(z,\theta^{'}) \cos \theta^{'}nn \\\nn\\
&- &\frac{\alpha}{2\pi}\left[\tilde F_{\bar\nu_e}^{(1)}(z,\theta^{'})
+ {\cal P}_\mu\,\tilde J_{\bar\nu_e}^{(1)}(z,\theta^{'})\cos \theta^{'}\right]\,, 
\eea
where
\bea
F_{\nu_{\mu}}^{(0)}(z,\theta^{'}) & = & 8 \, \frac{E_\mu^4}{m_\mu^6}\,  z^2\, (1-\beta\cos{\theta}^{'})(3m_{\mu}^2-4E_{\mu}^2 z(1-\beta\cos{\theta}^{'})),\nn \\ 
\nn \\ 
%
J_{\nu_{\mu}}^{(0)}(z,\theta^{'}) & = & 8  \, \frac{E_{\mu}^4}{m_\mu^6}\, z^2\, (1-\beta\cos{\theta}^{'})(m_{\mu}^2-4E_{\mu}^2 z(1-\beta\cos{\theta}^{'})), \nn\\ \nn  \\
%
F_{\bar\nu_e}^{(0)}(z,\theta^{'}) & = & 48 \frac{E_{\mu}^4}{m_{\mu}^6}\,z^2\,                             (1-\beta\cos{\theta}^{'})(m_{\mu}^2-2E_{\mu}^2z(1-\beta\cos{\theta}^{'})),\nn\\ \nn \\
%
J_{\bar\nu_e}^{(0)}(z,\theta^{'}) & = & 48 \frac{E_{\mu}^4}{m_{\mu}^6}\,z^2\,                                (1-\beta\cos{\theta}^{'})(m_{\mu}^2-2E_{\mu}^2z(1-\beta\cos{\theta}^{'})),\nn\\ \nn \\ 
%
F_{\nu_{\mu}}^{(1)}(z,\theta^{'})  & = & 
        F_{\nu_\mu}^{(0)}(z,\theta^{'}) k(z,\theta^{'})
        +\frac{1}{3\J}\Bigg\{\left[41-36\x \right. \nn \\
        & + & \left.42\x^2-16\x^3\right]\nn \\
        & \times & \ln{\xm}\nn \\
        & + & \frac{1}{2}\x\left[82-153\x \right. \nn  \\
        & + & \left. 86\x^2\right] \Bigg\},\nn \\ \nn \\\nn \\ \nn \\
%
J_{\nu_{\mu}}^{(1)}(z,\theta^{'})  & = & 
        J_{\nu_\mu}^{(0)}(z,\theta^{'}) k(z,\theta^{'})
        +\frac{1}{3\J}\Bigg\{\left[11- 36\x \right. \nn \\
        & + & \left.14\x^2-16\x^3 \right. \nn \\
        & + & \left.4\x^{-1} \right]       
        \times  \ln{\xm}\nn \\
        & + & \frac{1}{2}\left[-8+18\x \right. \nn \\
        & - & \left. 103\x^2 + 78\x^3 \right] \Bigg\},\nn \\ \nn \\\
%
F_{\bar\nu_e}^{(1)}(z,\theta^{'})  & = & 
        F_{\nu_e}^{(0)}(z,\theta^{'}) k(z,\theta^{'}) 
          +  \frac{2\xm}{\J}\nn \\
        & \times &\Bigg\{\left[5+ 8\x \right. \nn \\
        & + & \left. 8\x^2\right]\ln{\xm}\nn \\
        & + & \frac{1}{2}\,\x\left[10-19\x\right]
                                       \Bigg\},\nn \\ \nn \\      
%
J_{\bar\nu_e}^{(1)}(z,\theta^{'}) & = & 					       J_{\nu_e}^{(0)}(z,\theta^{'}) k(z,\theta^{'})
        +  \frac{2\xm}{\J}\nn \\
        &\times & \Bigg\{\left[-3+ 12\x + 8\x^2 \right. \nn \\
        & + & \left. 4\x^{-1} \right]\,\ln{\xm}\nn \\
        & + & \frac{1}{2}\,\left[8-2\x - 15\x^2 \right]\Bigg\},\nn \\ \nn \\    %
k(z,\theta^{'}) & = & \ln^2{\xm}+ 2{\rm L}\x  + \frac{2\pi^2}{3}
\eea
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\Large{\bf Appendix C: QED corrected $e^-$ distributions
\vspace{0.1cm}
in the laboratory frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}

The ``leading log'' corrected distributions in eq.~(\ref{SL}) are boosted to the lab frame, using the same parameterization than Appendix B:
\bea
\frac {d^2 N_e} { dz d\cos{\theta}^{'} }& = & \left\{\,F_e^{(0)} (z,\theta^{'}) + {\cal P}_\mu\, J_e^{(0)}(z,\theta^{'}) \cos \theta^{'}\right\}\nn\\ \nn\\
&\times& \left( 1 - \frac{\alpha}{2\pi}\,\psi (z,\theta^{'})\right), 
\eea
where 
\bea
\label{treeboost}
F_e^{(0)}(z,\theta^{'}) & = & 8 \, \frac{E_\mu^4}{m_\mu^6}\,  z^2\, (1-\beta\cos{\theta}^{'})(3m_{\mu}^2-4E_{\mu}^2 z(1-\beta\cos{\theta}^{'})),  \nn  \\\nn\\
%
J_e^{(0)}(z,\theta^{'}) & = & 8  \, \frac{E_{\mu}^4}{m_\mu^6}\, z^2\, (1-\beta\cos{\theta}^{'})(m_{\mu}^2-4E_{\mu}^2 z(1-\beta\cos{\theta}^{'})),
\eea
The ``leading log'  correction is contained in
\bea
\psi (z,\theta^{'}) & = &  2\ln\left(\frac{m_\mu}{m_e^2}\right)
                        \left[\ln\left(\frac{\xm}{\x}\right)\right. \nn\\\nn\\
                     & + & \left.\frac{5}{6}+2\x-4\x^2\right.\nn\\\nn\\
                     & + &  \left.\frac{8\x^3}{3}\right].  
\eea

We will also boost the ${\cal O}(\alpha)$ exactly corrected electron distributions,
 computed in \cite{Beherends}. In the muon rest frame they have the following expression:
\bea
\label{edist}
\frac {d^2 N_e} { dx d\cos{\theta}}& = &F_e^{(0)} (x) + {\cal P}_\mu\, J_e^{(0)}(x) \cos \theta\nn \\\nn\\
&- & \frac{\alpha}{2\pi}\left[\tilde F_{\nu_{\mu}}^{(1)}(x)
+ {\cal P}_\mu\, \tilde J_e^{(1)}(x) \cos \theta\right]\,, 
\eea
where $F_e^{(0)}(x)$ and $J_e^{(0)}(x)$ are the tree level results, and
\bea
F_e^{(1)}(x) & = & 2 F_e^{(0)}(x)\,h(x)
               + 6(1-x)x^2\ln(x)\nn \\\nn\\
             & + &\frac{(1-x)}{3}\bigg\{\left(5+17x-34x^2\right)
             \left[\ln\left(\frac{m_\mu}{m_e}\right)
               +\ln{(x)}\right]
         -  22x+34x^2 \bigg\}, \nn \\\nn \\\nn\\
%
J_e^{(1)}(x)  & = & 
        2 J_e^{(0)}(x)\,h(x) + 2\left(1-3x\right)x^2\ln(1-x)\nn \\\nn\\
        & - & \frac{(1-x)}{3}\bigg\{\left(1+x+34x^2\right)
        \left[\ln\left(\frac{m_\mu}{m_e}\right)+                                         \ln(x)\right] 
         +  3-7x-32x^2\nn\\
         &+&  4\frac{(1-x)^2}{x}
          \ln{(1-x)} \bigg\} ,\nn \\ \nn\\ 
%
h(x) & = &  2{\rm L}(x) - \frac{\pi^2}{3}-2
                 +  \ln\left(\frac{m_\mu}{m_e}\right)
               \left[\frac{3}{2}+ 2\ln{\left(\frac{1-x}{x}\right)}\right]
                \nn \\ \nn \\       
                & - &  2\ln^2{(x)}+\ln{(x)}  
                  +  3\ln{(x)}\ln{(1-x)} 
                 - \ln{(1-x)}- \frac{\ln(1-x)}{x}.
\eea

After the boost, the corrected electron distribution has the form:
\bea
\frac {d^2 N_e} { dz d\cos{\theta}^{'} }& = & F_e^{(0)} (z,\theta^{'}) + {\cal P}_\mu\, J_e^{(0)}(z,\theta^{'}) \cos \theta^{'}\nn \\\nn \\
&- &\frac{\alpha}{2\pi}\left[\tilde F_{\nu_{\mu}}^{(1)}(z,\theta^{'})
+ {\cal P}_\mu\, \tilde J_e^{(1)}(z,\theta^{'}) \cos \theta^{'}\right]\,, 
\eea
where $F_e^{(0)}(z,\theta^{'}),J_e^{(0)}(z,\theta^{'})$ are given in eq.~(\ref{treeboost})

\bea
F_e^{(1)}(z,\theta^{'})  & = & 
        2 F_e^{(0)}(z,\theta^{'})\,h(z,\theta^{'})\nn \\
         & + & \frac{12\xm}\x^2{\J}\nn \\\nn \\
         & \times &  \ln{\x}   \nn \\\nn \\
         & + & \frac{2\xm}{3\J}\Bigg\{\left(5+17\x\right.\nn \\\nn \\
         & - & \left.34\x^2\right)
         \left[\ln\left(\frac{m_\mu}{m_e}\right)
               +\ln{\x}\right]\nn \\\nn \\
        & - & 22\x+34\x^2 \Bigg\} ,\nn \\  \nn \\\nn \\\nn \\\nn \\
%
J_e^{(1)}(z,\theta^{'})  & = & 
        2 J_e^{(0)}(z,\theta^{'})\,h(z,\theta^{'})\nn\\
        & + &\frac{4\left(1-3\x\right)\x^2\ln{\x}}{\J}\nn \\\nn \\
        & - & \frac{2\left(1-\x\right)}{3\J}\Bigg\{\left(1+\x\right.\nn \\\nn \\
        & + & \left.  34\x^2\right)
        \left[\ln\left(\frac{m_\mu}{m_e}\right)+                                         \ln{\x}\right]\nn \\\nn \\
        & + & 3-7\x-32\x^2\nn \\\nn \\
        & + & 4\frac{\xm^2}{\x}
          \ln{\xm} \Bigg\} ,\nn \\ \nn  \\\nn  \\
%
h(z,\theta^{'}) & = &  2{\rm L}\x - 2 - \frac{\pi^2}{3} \nn \\
                & + & \ln\left(\frac{m_\mu}{m_e}\right)
                \left[\frac{3}{2}+ 2\ln{\xm}\right.\nn \\\nn \\
                & - & \left.2\ln{\x}\right]\nn \\  \nn \\      
                & - & 2\ln^2{\x}+\ln{\x}\nn \\  \nn \\
                & + & 3\ln{\x}\ln{\xm}\nn \\ \nn \\
                & - & \left(1+\x^{-1}\right)\ln{\xm}.
\eea


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\epsfig{file=loops.ps}
\caption{\footnotesize QED Radiative corrections to muon decay: (a) Tree level diagram, (b) Vertex correction, (c) Muon propagator correction, (d) Electron propagator correction, (e) Muon leg bremsstrahlung, (d) Electron leg bremsstrahlung}
\label{loops}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=ele.ps, height=8cm, width=12cm}
\caption{\footnotesize Electron energy distribution in the forward direction for $E_\mu=30$ GeV and ${\cal P}_\mu$=0.2. The tree level results (solid line) are shown together with the ``leading log''  (dotted line) and exact one-loop corrected ones (dashed-line)}
\label{ele}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=ele_end.ps, height=8cm, width=12cm}
\caption{\footnotesize End-point behavior of Fig.~\ref{ele}.}
\label{ele_end}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[ht]
%\centering
%\epsfig{file=corele.ps, height=8cm,width=12cm}
%\caption{\footnotesize  ${\cal O}(\alpha)$ exact and ``leading-log'' corrections to $e^-$ distributions as a function of the $e^-$ energy.}
%\label{elecor}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=angles.eps, height=12cm,width=12cm}
\caption{\footnotesize The three Euler angles $\alpha,\gamma$ and $\theta$ which describe the orientation of the outgoing neutrino with momentum ${\bf p_\nu}$ and the electron with momentum ${\bf P}$ with respect to the muon spin ${\bf s}$ directed along the z-axis}
\label{angles}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=numu.ps, height=8cm,width=12cm}
\caption{\footnotesize Zeroth order and ${\cal O}(\alpha)$ corrected $\nu_\mu$ forward distributions. Parent muon parameters as in Fig.~\ref{ele}.}
\label{numu}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=cornumu.ps, height=8cm,width=12cm}
\caption{\footnotesize  ${\cal O}(\alpha)$ relative correction to the $\nu_\mu$ distribution in Fig.~\ref{numu} for a typical range of $E_{\nu_\mu}$.}
\label{numucor}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=nue.ps, height=8cm,width=12cm}
\caption{\footnotesize Zeroth order and ${\cal O}(\alpha)$ corrected $\bar\nu_e$ forward distributions. Parent muon parameters as in Fig.~\ref{ele}.}
\label{nue}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=cornue.ps, height=8cm,width=12cm}
\caption{\footnotesize  ${\cal O}(\alpha)$ relative correction to $\bar\nu_e$ distribution in Fig.~\ref{nue} for a typical range of $E_{\bar\nu_e}$.}
\label{nuecor}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=divergence.eps,height=8cm, width=10 cm} 
\caption{\footnotesize Muon divergence in the laboratory frame}
\label{divergence}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centering
\epsfig{file=espectromu.eps, height=15cm,width=10cm}
\caption{\footnotesize $\nu_\mu$ and $\bar\nu_\mu$ differential distributions. The solid lines represent the spectra obtained by averaging over an angular divergence of $0.1$ mrad and the dashed lines the spectra including muon beam divergence. The distributions are plotted in the forward direction $\cos\theta=0$ pointing towards a detector located 2810 km from a the neutrino source
of unpolarized positive or negative muons circulating in the storage ring with energies of $30$ GeV.}
\label{spectrum_mu}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centering
\epsfig{file=espectroe.eps, height=15cm,width=10cm}
\caption{\footnotesize $\nu_e$ and $\bar\nu_e$ differential distributions. The solid lines represent the spectra obtained by averaging over an angular divergence of $0.1$ mrad and the dashed lines the spectra including muon beam divergence. The distributions are plotted with the same parameters as of fig(\ref{spectrum_mu}).}
\label{spectrum_e}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=fluxomu.eps, height=15cm,width=10cm}
\caption{\footnotesize $\nu_\mu$ and $\bar\nu_\mu$ fluxes. The solid lines represent the spectra obtained by averaging over an angular divergence of $0.1$ mrad and the dashed lines the spectra including muon beam divergence. The distributions are plotted with the same parameters as of fig(\ref{spectrum_mu}).}
\label{fluxomu}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\centering
\epsfig{file=fluxoe.eps, height=15cm,width=10cm}
\caption{\footnotesize $\nu_e$ and $\bar\nu_e$ fluxes. The solid lines represent the spectra obtained by averaging over an angular divergence of $0.1$ mrad and the dashed lines the spectra including muon beam divergence. The distributions are plotted with the same parameters as of fig(\ref{spectrum_mu}).}
\label{fluxoe}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\end{document}








