%%%
% On higher order QED corrections to muon decay spectrum
%%%
% Corresponding Author:
% ---------------------
% Andrej B. Arbuzov
% Avadh Bhatia Phys. Lab.
% University of Alberta
% Edmonton, AB
% Canada  T6G 2J1
%
% Tel:    (780) 492 41 23 (office)
% Tel:    (780) 492 52 86 (secretary)
% FAX:    (780) 492 07 14
% E-mail: aarbuzov@Phys.UAlberta.CA
%%%
\documentclass[letterpaper]{JHEP3}
\usepackage{epsfig}

\title{Higher order QED corrections to muon decay spectrum}

\author{Andrej Arbuzov\thanks{On leave of absence from JINR, Dubna.}
\\
Department of Physics, University of Alberta,
Edmonton, AB\ \  T6G 2J1, Canada \\
E-mail: \email{aarbuzov@phys.ualberta.ca}
}

\abstract{Radiative corrections to 
polarized muon decay spectrum of the second and higher
orders in $\alpha_{\mathrm{QED}}$ are considered. 
Leading and next--to--leading logarithmic approximations 
are used. Exponentiation of soft photons is discussed.
The present theoretical uncertainty of the spectrum description
is estimated.}
%{\sc PACS:}~
%12.20.-m  Quantum electrodynamics 
%12.20.Ds  Specific calculations  
%13.35.Bv  Decays of muons  
%13.40.Ks  Electromagnetic corrections to strong- 
%          and weak-interaction processes  

\keywords{smo.lde.emp}
%%\keywords{Standard Model,Weak Decays
%%         ,Electromagnetic Processes and Properties.}

\preprint \\ Alberta-Thy-10-02}

\begin{document}

%%% Some Simple Private Definitions
\def\TWIST{$\mathcal{TWIST}\ $}
\def\DD{{\mathcal D}}
\def\NS{\mathrm{NS}}
\def\SS{\mathrm{S}}
\def\LL{\mathrm{LL}}
\def\NLL{\mathrm{NLL}}
\def\MSbar{$\overline{\mathrm{MS}}\ $}
\def\Li#1#2{{\mathrm{Li}}_{#1}\left(#2\right)}
\def\Sot#1{{\mathrm{S}}_{1,2}\left(#1\right)}
\def\ba{\begin{eqnarray}}
\def\ea{\end{eqnarray}}
\def\dd{{\mathrm d}}
\def\la{\mathrel{\mathpalette\fun <}}
\def\ga{\mathrel{\mathpalette\fun >}}
\def\fun#1#2{\lower3.6pt\vbox{\baselineskip0pt\lineskip.9pt
  \ialign{$\mathsurround=0pt#1\hfil##\hfil$\crcr#2\crcr\sim\crcr}}}
\def\order#1{{\mathcal O}\left(#1\right)}
%%%


\section{\label{Sec:Int}Introduction}

Accurate measurements of the muon properties were providing substantial
information for the development of the elementary particle physics during 
many years. Nowadays precision experiments with muons serve as one of the
basements of the Standard Model (SM) and give a possibility to look for 
{\em new physics}~\cite{Fetscher:2000th,Kuno:2001jp}.

In this paper we discuss the present theoretical precision
of the polarized muon decay spectrum description.
The study is motivated by the experiment 
\TWIST\cite{Rodning:2001js,Quraan:2000vq}, which
is currently running at Canada's National Laboratory TRIUMF
and going to measure the spectrum with the accuracy level 
of about $10^{-4}$. 
That will make a serious test of the space--time structure 
of the weak interaction. The experiment is able to put stringent limits
on a bunch of parameters in models beyond SM, {\it e.g.}, on the mass 
and the mixing angle of a possible right--handed $W$-boson.
To confront the experimental results with SM,
adequately accurate theoretical predictions should
be provided. This requires to 
calculate radiative corrections within the perturbative
Quantum Electrodynamics (QED). Here we will concentrate on the 
effect of higher order leading logarithmic (LL) 
and next--to--leading logarithmic (NLL) corrections. 


\section{\label{Sec:Pre}Preliminaries}

Within the Standard Model, the differential 
distribution of electrons (averaged over electron spin states) 
in the polarized muon decay reads
\ba \label{general}
&& \frac{\dd^2\Gamma^{\mu^{\mp}\to 
e^{\mp}\nu\bar{\nu}}}{\dd x\dd c} 
= \Gamma_0 \left( F(x) \pm cP_{\mu} G(x) \right), \qquad
\Gamma_0 = \frac{G_F^2 m_\mu^5}{192\pi^3}\, , 
\nonumber \\ && 
c = \cos\theta, \qquad 
x = \frac{2m_{\mu}E_e}{m_\mu^2+m_e^2}, \qquad
x_0 \leq x \leq 1, \qquad
x_0 = \frac{2m_{\mu}m_e}{m_\mu^2+m_e^2},
\ea
where 
$m_\mu$ and $m_e$ are the muon and electron masses;
$G_F$ is the Fermi coupling constant;
$\theta$ is the angle between the muon polarization vector $\vec{P}_{\mu}$
and the electron (or positron) momentum;
$E_e$ and $x$ are the energy and the energy fraction of $e^{\pm}$.
Within the Standard Model the muon decay
happens due to the weak interaction of leptons and $W$-bosons. The Fermi
model corresponds to the limiting case of the infinite $W$-boson mass.
If the Fermi coupling constant $G_F$ is defined according to 
Refs.~\cite{Groom:2000in,Marciano:1988vm}, 
the first order effect in the muon and $W$-boson mass ratio gives
\ba
\Gamma_0 \longrightarrow \Gamma_0 
\biggl( 1 + \frac{3}{5}\;\frac{m_\mu^2}{m_W^2} \biggr).
\ea
In studies of the muon decay spectrum, however, it is natural 
to use the constant directly defined from very precise experiments
on the muon lifetime, as discussed in Ref.~\cite{vanRitbergen:2000fi}.

Functions $F(x)$ and $G(x)$ describe the isotropic and anisotropic
parts of the spectrum, respectively. Within perturbative QED, they 
can be expanded in series in the fine structure constant $\alpha$:
\ba
F(x) = f_{\mathrm{Born}}(x) + \frac{\alpha}{2\pi}f_1(x)
+ \biggl(\frac{\alpha}{2\pi}\biggr)^2f_2(x)
+ \biggl(\frac{\alpha}{2\pi}\biggr)^3f_3(x)
+ \order{\alpha^4},
\ea
and in the same way for $G(x)$. The Born--level functions 
$f_{\mathrm{Born}}$ and $g_{\mathrm{Born}}$ are well known
including small terms suppressed by the $m_e/m_\mu$ mass 
ratio~\cite{Fetscher:2000th}: 
\ba
f_{\mathrm{Born}}(x) &=& 6x\biggl(1+\frac{m_e^2}{m_\mu^2}\biggr)^4 
\sqrt{1-\frac{m_e^2}{E_e^2}}\; \biggl[ x(1-x) 
+ \frac{2}{9}\rho ( 4x^2 - 3x - x_0^2 )
+ \eta  x_0(1-x) \biggr],
\nonumber \\
g_{\mathrm{Born}}(x) &=& -2x^2 \xi \biggl(1+\frac{m_e^2}{m_\mu^2}\biggr)^4 
\biggl(1-\frac{m_e^2}{E_e^2}\biggr) \biggl[ 1 - x
+ \frac{2}{3}\delta \biggl(4x - 3 - \frac{m_e}{m_\mu}x_0\biggr) \biggr],
\ea
where $\rho$, $\eta$, $\xi$, and $\delta$ are the so--called Michel 
parameters~\cite{mich1,mich2,mich3}, which in the Standard Model
become $\rho=3/4$, $\eta=0$, $\xi=1$, and $\delta=3/4$.
By fitting the values of the parameters from the experimental data
on the electron spectrum in muon decay and comparing them
with the SM predictions, the \TWIST experiment is going 
to look for effects of non--standard interactions.

The first order corrections $f_1(x)$ and $g_1(x)$ are also known
with an exact account of the dependence on the electron 
mass~\cite{Behrends:1956mb,Kinoshita:1959ru,Arbuzov:2001ui}. 
Starting from $\order{\alpha}$, radiative corrections to
the electron spectrum contain so--called mass singularities
in the form of the large logarithm 
$L\equiv\ln(m^2_\mu/m_e^2)\approx 10.66$.
As demonstrated in Ref.~\cite{Arbuzov:2002pp} 
(see also Table~\ref{table:1} below), the terms of the order 
$\order{\alpha L}$ give the bulk of the first order correction.
This is our reason to look first for the terms enhanced by the 
large logarithm in higher orders.
Note that these enhanced terms (excluding the 
ones related to the running of the QED coupling constant)
cancel out in the expression for the muon decay width at any order 
in $\alpha$ in accordance with the Kinoshita--Lee--Nauenberg 
theorem~\cite{Kinoshita:1962ur,Lee:1964is}.

The second order corrections to the muon decay width were
calculated in Ref.~\cite{vanRitbergen:1998yd}. 
At this order for the differential decay spectrum, we know 
only the leading logarithmic corrections~\cite{Arbuzov:2002pp} 
and the isotropic part in the next--to--leading logarithmic
approximation~\cite{Arbuzov:2002cn}. The corresponding anisotropic part
will be given below. The third order LL corrections
will be presented as well.


\section{\label{Sec:FFA}The Fragmentation Function Approach}

Here I will describe briefly the application of the renormalization 
group method to the calculation of the leading and next--to--leading
radiative corrections to polarized muon decay spectrum. For the
detailed foundation of the procedure and 
an extended discussion look in Ref.~\cite{Arbuzov:2002cn}.

The factorizations theorems, proved for QCD~\cite{Ellis:qj},
can be easily translated to QED. In particular, by means of
factorization, one can represent the differential
spectrum of electrons as a convolution:
\ba \label{master}
\frac{\dd^2\Gamma}{\dd x\dd c}(x,c,m_\mu,m_e) =
\sum\limits_{j=e,\gamma}^{}
\int\limits_{x}^{1}\frac{\dd z}{z}\;
\frac{\dd^2\hat{\Gamma}_j}{\dd z\dd c}(z,c,m_\mu,\mu_f)
\DD_j\biggl(\frac{x}{z},\mu_f,m_e\biggr),
\ea
where $\dd^2\hat{\Gamma}_j/(\dd z\dd c)$
is the differential distribution for the muon decay with production of 
a massless electron $(j=e)$ or a photon $(j=\gamma)$
with the energy fraction $z$ ~$(z=2E_j/m_\mu$, where $E_j$ is 
the energy of the relevant particle). 
We will use here
the \MSbar factorization scheme~\cite{Bardeen:1978yd} with the 
scale $\mu_f$ to subtract the collinear singularities from the 
differential distributions. The fragmentation function
$\DD_j(x/z,\mu_f,m_e)$ describes the conversion
of the massless parton $j$ into a massive physical electron. 

The spectrum of a massless 
parton can be expanded in a perturbative series:
\ba \label{dhGj}
\frac{1}{\Gamma_0}\frac{\dd^2\hat{\Gamma}_j}{\dd z\dd c}(z,c,m_\mu,\mu_f)
= \hat{A}^{(0)}_j(z,c)
+ \frac{\bar{\alpha}(\mu_f)}{2\pi}\hat{A}^{(1)}_j(z,c)
+ \biggl(\frac{\bar{\alpha}(\mu_f)}{2\pi}\biggr)^2\hat{A}^{(2)}_j(z,c)
+ \order{\alpha^3},
\ea
where $\bar{\alpha}(\mu_f)$ is the
\MSbar renormalized coupling constant, it will be converted 
further into the traditional QED on--shell coupling constant 
$\alpha \approx 1/137.036$.
The lowest order spectrum of massless electrons reads 
\ba 
\hat{A}^{(0)}_e(z,c) = f_0(z) \pm cP_{\mu} g_0(z), \qquad
f_0(z) = z^2(3 - 2z), \qquad
g_0(z) = z^2(1 - 2z).
\ea
Here and in what follows the sign before $c$ should be chosen 
according to the charge of the decaying muon (plus for 
$\mu^-$ decay and vice versa).
The $\order{\alpha}$ correction to the massless electron spectrum is
\ba \label{hAe}
\hat{A}^{(1)}_e(z,c) &=& \hat{f}^{(1)}_e(z) 
\pm cP_{\mu} \hat{g}^{(1)}_e(z),
\\ \label{hge}
\hat{g}^{(1)}_e(z) &=& \biggl( 2z^2(1-2z)\ln\frac{1-z}{z}
- \frac{1}{6} - 4z^2 + \frac{8}{3}z^3 \biggr)\ln\frac{m_\mu^2}{\mu_f^2}
\nonumber \\
&+& 2z^2(1-2z)\biggl( \ln^2(1-z) - 4\Li{2}{1-z}
- \ln z\ln(1-z)  - 2\ln^2z \biggr)
\nonumber \\
&+& \biggl( \frac{11}{3} - \frac{4}{3z} - 6z 
- \frac{17}{3}z^2 + \frac{34}{3}z^3 \biggr)\ln(1-z)
+ \biggl( - \frac{1}{3} - 6z^2 - 6z^3 \biggr)\ln z
\nonumber \\
&-& \frac{7}{6} + 3z + \frac{7}{6}z^2 + 3z^3.
\ea
For the auxiliary photon spectrum with collinear singularities 
subtracted according to the \MSbar prescription, we have
\ba \label{hAg}
\hat{A}^{(0)}_{\gamma}(z,c) &=& 0, \qquad
\hat{A}^{(1)}_{\gamma}(z,c) = \hat{f}^{(1)}_{\gamma}(z) 
\pm cP_{\mu} \hat{g}^{(1)}_{\gamma}(z),
\\ \label{hgg}
\hat{g}^{(1)}_{\gamma}(z) &=& 
\biggl( \frac{1}{3} - \frac{1}{3z} - \frac{2}{3}z^2 
+ \frac{2}{3}z^3 \biggr)
\biggl(\ln\frac{m_\mu^2}{\mu_f^2} + \ln(1-z)\biggr)
\nonumber \\
&+& \biggl( \frac{2}{3} - \frac{2}{3z} \biggr)\ln z
- \frac{2}{3} + \frac{2}{3z} + \frac{11}{12}z 
- \frac{2}{3}z^2 - \frac{1}{4}z^3.
\ea
The isotropic parts of the first order corrections to the 
auxiliary massless parton distributions,
$\hat{f}^{(1)}_e(z)$ and $\hat{f}^{(1)}_{\gamma}(z)$,  
are given by Eqs.~(7,8) in Ref.~\cite{Arbuzov:2002cn}.
By the choice of the factorization parameter value,
$\mu_f\sim m_\mu$, we avoid calculation of the unknown 
functions $\hat{A}^{(2)}_j$, since then they can not give 
rise to any LL or NLL correction.

The fragmentation functions $\DD_j$ can be obtained by
solving the Dokshitzer--Gribov--Lipatov--Altarelli--Parisi (DGLAP) 
evolution equations for QED,
\ba
\label{DGLAP}
\frac{\dd\DD_i(x,\mu_f,m_e)}{\dd \ln\mu_f^2} = \sum\limits_{j=e,\gamma}
\int\limits_{x}^{1}\frac{\dd z}{z}\; 
P_{ji}(z,\bar{\alpha}(\mu_f))
\DD_j\biggl(\frac{x}{z}\, ,\mu_f,m_e\biggr),
\ea
where $P_{ji}$ are perturbative splitting functions,
\ba
P_{ji}(z,\bar{\alpha}(\mu_f)) = 
\frac{\bar{\alpha}(\mu_f)}{2\pi}P_{ji}^{(0)}(z)
+ \biggl(\frac{\bar{\alpha}(\mu_f)}{2\pi}\biggr)^2P^{(1)}_{ji}(z) 
+ \order{\alpha^3}.
\ea
The initial conditions, which are required to solve the DGLAP
equations by iterations, can be borrowed from QCD 
studies~\cite{Mele:1990cw}:
\ba \label{Deini}
\DD^{\mathrm{ini}}_e(x,\mu_0,m_e) &=& \delta(1-x)
+ \frac{\bar{\alpha}(\mu_0)}{2\pi} d_1(x,\mu_0,m_e) + \order{\alpha^2}, 
\nonumber \\
d_1(x,\mu_0,m_e) &=& \biggl[ \frac{1+x^2}{1-x}
\biggl( \ln\frac{\mu_0^2}{m_e^2} - 2\ln(1-x) - 1 \biggr) 
\biggr]_+,
\\ \label{Dgini}
\DD^{\mathrm{ini}}_\gamma(x,\mu_0,m_e) &=& \frac{\bar{\alpha}(\mu_0)}{2\pi}
\ln\frac{\mu_0^2}{m_e^2}
P^{(0)}_{\gamma e}(x) + \order{\alpha^2}.
\ea
The lowest order splitting functions relevant for our work are
\ba
P_{ee}^{(0)}(x)= \biggl[\frac{1+x^2}{1-x}\biggr]_+,
\qquad
P^{(0)}_{e\gamma}(x) = \frac{1 + (1-x)^2}{x}\, ,
\qquad
P^{(0)}_{\gamma e}(x) = x^2 + (1-x)^2.
\ea
The {\em plus} prescription works as usually:
\ba
&& \int\limits_{x_{\mathrm{min}}}^1\dd x\; [V(x)]_+W(x) =
\int\limits_{0}^1\dd x\; V(x) [W(x)\Theta(x-x_{\mathrm{min}})-W(1)], 
\\ \nonumber && 
%\qquad
\Theta(x) = 
\left\{\begin{array}{l} 1 \quad {\mathrm{for}} \quad x\geq 0 \\
0 \quad {\mathrm{for}} \quad x < 0
\end{array}\right. .
\ea

In a measurement of the muon decay spectrum, 
events with more than one electron in the final state require a special 
treatment.
Starting from the second order in $\alpha$, we have a certain
contribution due to emission of real and virtual $e^+e^-$ pairs.
Presumably a Monte Carlo event generator is needed to simulate
the process of muon decay with pair production.
Nevertheless, we will calculate the corresponding effect 
in a simple assumption, that an event with two electrons in the final state
is treated as a pair of simultaneous muon decays. In order
to have the possibility to drop the pair contributions (if they
are taken into account in a Monte Carlo program), we decompose our results
according to the classes of the corresponding Feynman diagrams
in the same way as in Ref.~\cite{Berends:1987ab}. 
Moreover, the decomposition will help us to demonstrate the
cancellation of the mass singularities in the integrated decay width.
Our results for pure pair corrections can serve further also as 
a benchmark for the Monte Carlo program. 
So the next--to--leading electron splitting function 
can be decomposed into four parts:
\ba
P^{(1)}_{ee}(x) = P^{(1,\gamma)}_{ee}(x)
+ P^{(1,\NS)}_{ee}(x)
+ P^{(1,\SS)}_{ee}(x)
+ P^{(1,\mathrm{int})}_{ee}(x),
\ea
where $P^{(1,\gamma)}_{ee}(x)$ is provided by the set of Feynman
diagrams with pure photonic corrections;
$P^{(1,\NS)}_{ee}(x)$ is related to the corrections due to non--singlet
real and virtual pairs;
$P^{(1,\SS)}_{ee}(x)$ stands for the singlet pair production
contribution; and $P^{(1,\mathrm{int})}_{ee}(x)$ describes
the interference of real singlet and non--singlet pairs. 
By extracting the appropriate color structures from the known
QCD results~\cite{Curci:1980uw,Floratos:1981hs,Furmanski:1980cm,Ellis:1996nn},
the explicit expressions for these functions were 
given in Ref.~\cite{Arbuzov:2002cn}.
%%%
\label{def:sns}
%%%
Here and in what follows, in the language of Feynman diagrams, 
the situation when the registered electron is connected by a solid
fermion line with the genuine electroweak decay vertex is
called {\em non--singlet}. The case, when the observed electron
belongs to a pair produced via a virtual photon, is called {\em singlet}.

The relation between the \MSbar and the on--shell coupling constants 
reads~\cite{vanRitbergen:2000fi}
\ba
\bar{\alpha}(\mu_f) = \alpha + \frac{\alpha^2}{3\pi}\ln\frac{\mu_f^2}{m_e^2} 
+ \frac{\alpha^3}{4\pi^2} \ln\frac{\mu_f^2}{m_e^2} 
+ \frac{15\alpha^3}{16\pi^2} + \order{\alpha^4}.
\ea

It is convenient to choose the renormalization scale to be
\ba
\mu_0 = m_e.
\ea

Now we have everything for solving the DGLAP equations~(\ref{DGLAP}).
Using iterations for the electron fragmentation function decomposed 
into four parts, we get 
\ba \label{De}
\DD_e(x,\mu_f,m_e) &=& \DD_e^{(\gamma)}(x)
+ \DD_e^{(\NS)}(x)
+ \DD_e^{(\SS)}(x)
+ \DD_e^{(\mathrm{int})}(x),
\\ \label{Deg}
\DD_e^{(\gamma)}(x) &=& \delta(1-x) 
+ \frac{\alpha}{2\pi}d_1(x,\mu_0,m_e) 
+ \frac{\alpha}{2\pi}L_{f}P^{(0)}_{ee}(x) 
\nonumber \\ 
&+& \biggl(\frac{\alpha}{2\pi}\biggr)^2
\biggl(\frac{1}{2}L_{f}^2P^{(0)}_{ee}\otimes P^{(0)}_{ee}(x)
+ L_{f}P^{(0)}_{ee}\otimes d_1(x,\mu_0,m_e)
+ L_{f}P^{(1,\gamma)}_{ee}(x) \biggr) 
\nonumber \\ 
&+& \biggl(\frac{\alpha}{2\pi}\biggr)^3
\frac{1}{6}L_{f}^3P^{(0)}_{ee}\otimes P^{(0)}_{ee}\otimes P^{(0)}_{ee}(x),
\\ \label{DeNS}
\DD_e^{(\NS)}(x) &=&
\biggl(\frac{\alpha}{2\pi}\biggr)^2
\biggl( \frac{1}{3}L_{f}^2P^{(0)}_{ee}(x)
+ L_{f}P^{(1,\NS)}_{ee}(x) \biggr) 
\nonumber \\ 
&+& \biggl(\frac{\alpha}{2\pi}\biggr)^3L_{f}^3
\biggl(\frac{1}{3}P^{(0)}_{ee}\otimes P^{(0)}_{ee}(x)
+ \frac{4}{27}P^{(0)}_{ee}(x) \biggr),
\\ \label{DeS}
\DD_e^{(\SS)}(x) &=&
\biggl(\frac{\alpha}{2\pi}\biggr)^2
\biggl( \frac{1}{2}L_{f}^2P^{(0)}_{e\gamma}\otimes P^{(0)}_{\gamma e}(x)
+ L_{f}P^{(1,\SS)}_{ee}(x) \biggr) 
\nonumber \\ 
&+& \biggl(\frac{\alpha}{2\pi}\biggr)^3L_{f}^3
\biggl( \frac{1}{3}P^{(0)}_{e\gamma}\otimes P^{(0)}_{\gamma e}
\otimes P^{(0)}_{ee}(x) 
- \frac{1}{9}P^{(0)}_{e\gamma}\otimes P^{(0)}_{\gamma e}(x) \biggr),
\\ \label{Deint}
\DD_e^{(\mathrm{int})}(x) &=& 
\biggl(\frac{\alpha}{2\pi}\biggr)^2L_{f}P^{(1,\mathrm{int})}_{ee}(x),
\qquad
L_f \equiv \ln\frac{\mu_f^2}{m_e^2}\, ,
\ea
where we systematically omitted terms of the following orders:
$\order{\alpha^2L_f^0}$, $\order{\alpha^3L_f^2}$, $\order{\alpha^4}$,
and higher.
The photon fragmentation function at the lowest order, 
\ba \label{Dg}
\DD_\gamma(x,\mu_f,m_e) = \frac{\alpha}{2\pi}
L_{f}P^{(0)}_{e\gamma}(x) + \order{\alpha^2},
\ea
is sufficient for our purposes.
The convolution operation is defined in the standard way:
\ba
\label{conv}
A\otimes B(x) = \int\limits^1_0\dd z
\int\limits^1_0\dd z'\; \delta(x-zz')A(z)B(z') 
=\int\limits^1_x\frac{\dd z}{z}\; A(z)B\biggl(\frac{x}{z}\biggr).
\ea

In principle, the leading logarithmic terms for the QED 
fragmentation function are known up to the fifth order in 
$\alpha$~\cite{Arbuzov:1999cq}. But, as will be seen from the numerical
results, keeping the third order ones is enough for the moment.

The explicit expressions for the functions, which appear in the
$\order{\alpha^2}$ terms of the function $\DD_e$,
are given in Ref.~\cite{Arbuzov:2002cn}. In the third order we have
two more functions~\cite{Skrzypek:1992vk}:
\ba \label{P3}
&& P^{(0)}_{ee}\otimes P^{(0)}_{ee}\otimes P^{(0)}_{ee}(x) =
\delta(1-x)\biggl( 16\zeta_3 - \frac{81}{4} \biggr)
+ \biggl[ 24\frac{1+x^2}{1-x}\biggl( \frac{1}{2}\ln^2(1-x)
+ \frac{3}{4}\ln(1-x) 
\nonumber \\ && \qquad
+ \frac{9}{32} - \frac{1}{2}\zeta_2 \biggr) \biggr]_+
+ 24\frac{1+x^2}{1-x}\biggl( \frac{1}{12}\ln^2x 
- \frac{1}{2}\ln{x}\ln(1-x)
- \frac{3}{8}\ln{x} \biggr)  
\nonumber \\ && \qquad
+ 6(1+x)\ln{x}\ln(1-x)
- 12(1-x)\ln(1-x)  
+ \frac{3}{2}(5-3x)\ln{x}
- 3(1-x) 
\nonumber \\ && \qquad
- \frac{3}{2}(1+x)\ln^2x + 6(1+x)\Li{2}{1-x},
\\ \label{PR}
&& P^{(0)}_{e\gamma}\otimes P^{(0)}_{\gamma e}
\otimes P^{(0)}_{ee}(x) = (1+x)\biggl(4\ln(1-x)\ln x 
- \ln^2x + 4\Li{2}{1-x} \biggr)
\nonumber \\ && \qquad
+ \frac{2}{3}(3x + 4x^2)\ln x
+ \frac{2}{3}\biggl( \frac{4}{x} + 3 - 3x - 4x^2 \biggr)\ln(1-x)
- \frac{23}{6}(1-x),
\ea
where
\ba
\zeta_n \equiv \sum_{k=1}^{\infty}\frac{1}{k^n}\, ,\qquad 
\zeta_2 = \frac{\pi^2}{6}\, ,\qquad
\Li{2}{x} \equiv - \int\limits_{0}^{x}\dd y\;\frac{\ln(1-y)}{y}\, .
\ea

By convolution of the fragmentation 
functions [Eqs.~(\ref{De}) and (\ref{Dg})]
with the differential distributions~(\ref{dhGj}), we
receive higher order corrections to the electron spectrum,
which will be presented in Sect.~\ref{Sec:Results}. 
In the results we can fix (see discussion in Ref.~\cite{Arbuzov:2002cn})
the factorization scale
\ba
\mu_f = m_{\mu}
\ea
and get $L_f\to L$.


\section{\label{Sec:Exp}Exponentiation}

Looking at the end point of the energy spectrum $(x\to 1)$ of
unpolarized muon decay, 
one can see that the first order correction becomes there negative
and very large, making the result senseless. An extensive
discussion of the phenomenon can be found 
in Refs.~\cite{Kinoshita:1959ru,Marciano:1975kw}.
The divergence is a clear signal to look beyond the first order
approximation. In fact, the Yennie--Frautschi--Suura 
theorem~\cite{Yennie:1961ad}
allows to make a re--summation of the dangerous terms
and to convert them into a definitely positive exponent.
The exponentiation procedure is not unique, it permits to
involve also some terms convergent at $x\to 1$. The main 
rule here is to keep unchanged the known perturbative results.
In our case, the exponentiation is allowed to add 
several terms of the following types:
$\order{\alpha^2L^0}$, $\order{\alpha^nL^m}$
with $n\geq 3,\ 0\leq m < n$, and 
$\order{\alpha^nL^n}$, $n\geq 4$. One has to keep in mind 
that the exponentiated sub--leading corrections can be treated
as an approximation to the complete sub--leading contribution
only in the region close to the end of the spectrum. 
 
Let us consider two ways of exponentiation. In the first case
one starts from the corrected cross section and tries to 
perform a re--summation of the known terms, which are
divergent at $x\to 1$, by converting them into an exponent:
\ba \label{exp:ah}
\frac{F(x)}{f_0(x)}\bigg|_{x\to 1} &\approx& 1 
+ \frac{\alpha}{\pi}(L-2)\ln(1-x)
+ \ldots \ \longrightarrow \
\mathrm{exp}\left\{\frac{\alpha}{\pi}(L-2)\ln(1-x)
\right\}.
\ea
This is a kind of the so--called {\it ad hoc} exponentiation.
The effect (see Table~\ref{table:1}) of this approach can be represented 
by the relative contribution of {\it new} terms
generated by the exponent,
\ba \label{delta:ah}
\delta^{\mathrm{exp}}_{\mathrm{a.h.}}(x) 
&=& \mathrm{exp}\left\{\frac{\alpha}{\pi}(L-2)\ln(1-x)\right\}
- 1 - \frac{\alpha}{\pi}(L-2)\ln(1-x)
\nonumber \\
&-& \frac{1}{2}\biggl(\frac{\alpha}{\pi}\biggr)^2
\bigl( L^2 - 4L \bigr)\ln^2(1-x)
- \frac{1}{6}\biggl(\frac{\alpha}{\pi}L\ln(1-x)\biggr)^3.
\ea
The most significant term above is
of the order $\alpha^2\ln^2(1-x)$ and gives a numerically
important contribution at large $x$. Note that all the subtracted terms 
do appear in the perturbative results.

The next step should be to check that in higher orders the exponent
doesn't contradict the known or anticipated results. 
The above procedure in the case of muon decay can be 
criticized~\cite{Roos:1971mj}, because the higher leading logarithmic terms
represent a mass singularity: one can not guarantee that all the large
logarithms, coming from $\delta^{\mathrm{exp}}_{\mathrm{a.h.}}(x)$,
disappear after the integration over the energy fraction.
Nevertheless the {\it ad hoc} exponentiation is not supposed to produce
a complete result. The region of its applications is limited:
it deals just with the terms, which are the most
important in the end of the spectrum. 

There is another way of exponentiation, which avoids the problem
of improper mass singularities. One can use the exponentiated
representation of the electron structure (fragmentation) function 
suggested in Ref.~\cite{Kuraev:1985hb}, which obeys the proper 
normalization:
\ba \label{normd}
\int\limits_0^1\DD^{(\gamma)}_{\mathrm{exp}}(x)\dd x = 1.
\ea 
For computations I used an advanced formula from 
Ref.~\cite{Cacciari:1992pz},
where I substituted $(L-1)$, which was natural for $e^+e^-$ annihilation,
by $(L-2)$. This substitution has not been obvious from 
the beginning, it followed from the experience of the above {\it ad hoc}
exponentiation. 
The relevant electron structure function is
taken within the leading logarithm approximation for pure 
photonic corrections with terms
up to $\order{\alpha^3L^3}$ and supplied with 
an exponentiation of some terms in higher orders. 
Convolution with the Born--level functions gives
\ba \label{delta:SF}
F_{\mathrm{SF}}^{\mathrm{exp}}(x) =
\DD^{(\gamma)}_{\mathrm{exp}} \otimes f_0(x), \qquad
G_{\mathrm{SF}}^{\mathrm{exp}}(x) =
\DD^{(\gamma)}_{\mathrm{exp}} \otimes g_0(x).
\ea
A subtraction of the known terms in the lower orders $(n\leq 3)$
of the perturbative
expansion, similar to Eq.~(\ref{delta:ah}), is used to receive the value of
the relative contribution $\delta^{\mathrm{exp}}_{\mathrm{SF}}(x)$.
The latter contains also terms of the order $\order{\alpha^2L^0}$,
which are not singular at $x\to 1$. This simulation of non--logarithmic
second order corrections is not well justified. 
In my opinion, the {\it ad hoc}
exponentiation in our case is more reliable and honest.
Numerical results (see Table~\ref{table:1}) of the 
two approaches are close to each other in the large-$x$ region. 

Simultaneous exponentiation of photonic and pair corrections can be
constructed as well. But it was criticized~\cite{Arbuzov:rt}, since soft 
pairs (contrary to soft photons) have a non--zero production 
threshold, which can't be taken into account by exponentiation properly.


\section{\label{Sec:Results}Results}

To the best of our present knowledge, we can write now function
$F(x)$ from the master formula~(\ref{general}) as follows:
\ba \label{Ffinal}
F(x) &=& f_{\mathrm{Born}}(x) + \frac{\alpha}{2\pi}f_1(x)
+ \biggl(\frac{\alpha}{2\pi}\biggr)^2\biggl\{
\biggl[ f_2^{(0,\gamma)}(x) + \frac{2}{3}f_2^{(0,\NS)}(x) 
+ f_2^{(0,\SS)}(x)\biggr] \frac{L^2}{2}
\nonumber \\
&+& \biggl[ f_2^{(1,\gamma)}(x) + f_2^{(1,\NS)}(x)
+ f_2^{(1,\SS)}(x) + f_2^{(1,\mathrm{int})}(x) \biggr] L 
%%+ f_2^{(\mathrm{hadr})}(x) 
\biggr\} 
\nonumber \\
&+& \biggl(\frac{\alpha}{2\pi}\biggr)^3\biggl[ 
f_3^{(0,\gamma)}(x) + f_3^{(0,\NS)}(x)
+ f_3^{(0,\SS)}(x) \biggr] \frac{L^3}{6}
+ \Delta f_{\mathrm{exp}}(x)
\nonumber \\
&+& \order{\alpha^2L^0,\alpha^3L^2,\alpha^4L^4}. 
\ea
Function $G(x)$ takes the same form with the substitution
$f_i\rightarrow g_i$. Effects due to virtual hadronic~\cite{Davydychev:2001ee},
$\mu^+\mu^-$, and $\tau^+\tau^-$ pairs 
are not shown explicitly, since they are of the order $\order{\alpha^2L^0}$.
The effect of exponentiation is given by 
\ba
\Delta f_{\mathrm{exp}}(x) = f_0(x)\delta^{\mathrm{exp}}_{\mathrm{a.h.}}(x),
\qquad
\Delta g_{\mathrm{exp}}(x) = g_0(x)\delta^{\mathrm{exp}}_{\mathrm{a.h.}}(x).
\ea


\subsection{Analytical Results}

In the second order for the NLL corrections to
the anisotropic part of the electron energy distribution, we have
\ba \label{g2nllg}
g_2^{(1,\gamma)}(x) &=&  4x^2(1-2x)\biggl( 
\Li{3}{1-x} + \Sot{1-x} - 2\Li{2}{1-x}\ln(1-x) 
\nonumber \\ 
&+& \ln x\ln^2(1-x) - 3\ln^2x\ln(1-x)
+ \ln^3x - \zeta_2\ln x + \frac{3}{2}\zeta_3 \biggr)
\nonumber \\ 
&+& \biggl( \frac{14}{3} - \frac{8}{3x} - 6x + 24x^2 
- \frac{92}{3}x^3 \biggr)\Li{2}{1-x} 
+ \biggl(  - 5 + 6x - \frac{86}{3}x^2 \biggr)\ln x \ln(1-x) 
\nonumber \\
&+& \biggl( 8 - \frac{8}{3x} - 12x + \frac{20}{3}x^2 + 8x^3 \biggr)\ln^2(1-x) 
+ \biggl( \frac{5}{12} + 18x^2 - \frac{70}{3}x^3 \biggr)\ln^2x 
\nonumber \\
&+& \biggl( - \frac{13}{3} + \frac{37}{3}x + \frac{50}{3}x^2 
- \frac{32}{3}x^3 \biggr)\ln(1-x)
+ \biggl( \frac{25}{12} - \frac{59}{6}x + 6x^2 + \frac{32}{9}x^3 \biggr)\ln x 
\nonumber \\ 
&+& \biggl( - 8 + \frac{8}{3x} + 12x - \frac{29}{3}x^2 - 2x^3 \biggr)\zeta_2 
+ \frac{817}{216} - \frac{91}{12}x + \frac{62}{3}x^2 - \frac{607}{54}x^3,
\\ \label{g2nllns}
g_{2}^{(1,\NS)}(x) &=& 
4x^2(1-2x)\biggl( - \Li{2}{1-x} - \frac{1}{3}\ln x\ln(1-x)
+ \frac{1}{3}\ln^2(1-x) 
\nonumber \\
&-& \frac{1}{2}\ln^2x - \frac{1}{3}\zeta_2 \biggr)
+ \biggl( \frac{22}{9} - \frac{8}{9x} - 4x - 6x^2 + 12x^3 \biggr)\ln(1-x) 
\nonumber \\
&+& \biggl( - \frac{1}{9} + \frac{8}{9}x^2 - \frac{76}{9}x^3 \biggr)\ln x 
- \frac{7}{18} + \frac{5}{3}x + \frac{86}{9}x^2 - \frac{20}{3}x^3,
\\ \label{g2nlls}
g_2^{(1,\SS)}(x) &=&
\biggl( \Li{2}{1-x} + \ln x\ln(1-x) \biggr)
\biggl( \frac{4}{3}x^2 - \frac{1}{3} \biggr)
+ \biggl( \frac{4}{3}x^2 - \frac{1}{2} \biggr)\ln^2x 
\nonumber \\
&+& \biggl( - \frac{1}{9} - \frac{2}{9x} + x + \frac{2}{9}x^2 
- \frac{8}{9}x^3 \biggr)\ln(1-x)
+ \biggl( \frac{5}{9} - \frac{4}{9x} + \frac{5}{2}x 
+ \frac{5}{9}x^2 \biggr)\ln x
\nonumber \\
&+& \frac{1}{3x} + \frac{4}{3} - \frac{7}{18}x - \frac{43}{18}x^2 
+ \frac{10}{9}x^3,
\\ \label{g2nlli}
g_2^{(1,\mathrm{int})}(x) &=&
4x^2(1-2x)\biggl( \Li{3}{1-x} - 2\Sot{1-x} - \Li{2}{1-x}\ln x \biggr)
\nonumber \\
&+& \biggl( - \frac{1}{3} - 14x^2 + \frac{52}{3}x^3 \biggr)\Li{2}{1-x}
+ \biggl(  - 3x^2 + \frac{26}{3}x^3 \biggr)\ln^2 x
\nonumber \\
&+& \biggl(  \frac{1}{3} + \frac{1}{3}x - \frac{28}{3}x^2 \biggr)\ln x 
+ \frac{10}{9} - \frac{1}{3}x - \frac{37}{3}x^2 + \frac{104}{9}x^3.
\ea
The polylogarithm functions are defined as
\ba
\Li{3}{x} \equiv \int\limits_{0}^{x}\dd y\;\frac{\Li{2}{y}}{y}\, , \qquad
\Sot{x} \equiv \frac{1}{2}\int\limits_{0}^{x}
\dd y\;\frac{\ln^2(1-y)}{y}\, .
\ea
The $\order{\alpha^2L^2}$ corrections $f_{2}^{(0,j)}(x)$ and
$g_{2}^{(0,j)}(x)$ $(j=\gamma,\, \NS,\, \SS)$ can be found in
Ref.~\cite{Arbuzov:2002pp}.
Explicit expressions for the second order next--to--leading
corrections to the isotropic part of the spectrum
$(f_{2}^{(1,i)}(x),\ i=\gamma,\, \NS,\, \SS,\, \mathrm{int})$
are given in Ref.~\cite{Arbuzov:2002cn}. 

The third order LL photonic contributions read
\ba \label{f3g}
f_{3}^{(0,\gamma)}(x) &=& 8x^2(3-2x)\Psi(x)
+ \bigl( 10 + 24x - 48x^2 + 32x^3 \bigr)\ln^2(1-x)
\nonumber \\
&+& \biggl( \frac{5}{12} + x - 8x^2 + 16x^3 \biggr)\ln^2x
+ \bigl( - 5 - 12x + 48x^2 - 64x^3\bigr)\ln x\ln(1-x)
\nonumber \\ 
&+& \bigl(5+12x-32x^3\bigr)\Li{2}{1-x}
+ \bigl(-10-24x+48x^2-32x^3\bigr)\zeta_2
\nonumber \\ 
&+& \biggl( -\frac{13}{18} - \frac{21}{2}x + \frac{64}{3}x^3 \biggr)\ln x
+ \biggl( \frac{11}{6} + 17x + 16x^2 - \frac{64}{3}x^3 \biggr)\ln(1-x)
\nonumber \\ 
&+& \frac{569}{216} + \frac{4}{3}x - \frac{16}{3}x^2 + \frac{128}{27}x^3,
\\ \label{g3g} 
g_{3}^{(0,\gamma)}(x) &=& 8x^2(1-2x)\Psi(x)
+ \bigl( - 2 - 48x^2 + 32x^3 \bigr)\ln^2(1-x)
\nonumber \\
&+& \biggl( - \frac{1}{12} - 8x^2 + 16x^3 \biggr)\ln^2x
+ \bigl( 1 + 48x^2 - 64x^3\bigr)\ln x\ln(1-x)
\nonumber \\ 
&+& \bigl(-1-32x^3\bigr)\Li{2}{1-x}
+ \bigl(2+48x^2-32x^3\bigr)\zeta_2
\nonumber \\ 
&+& \biggl( \frac{5}{18} + \frac{5}{2}x + \frac{64}{3}x^3 \biggr)\ln x
+ \biggl( - \frac{7}{6} - 7x + 16x^2 - \frac{64}{3}x^3 \biggr)\ln(1-x)
\nonumber \\ 
&-& \frac{133}{216} - \frac{13}{6}x - \frac{16}{3}x^2 + \frac{128}{27}x^3,
\\ \nonumber 
\Psi(x) &\equiv& 3\Li{3}{1-x} - 2\Sot{1-x} 
+ \ln^3(1-x) - \frac{1}{6}\ln^3x + \frac{3}{2}\ln^2x \ln(1-x)
\\ \nonumber 
&-& 3 \ln x\ln^2(1-x) - 3\Li{2}{1-x}\ln(1-x) + 2\zeta_3
- 3\zeta_2\ln\frac{1-x}{x}\, .
\ea
And the third order LL pair corrections\footnote{Strictly speaking,
we have here pair and photonic corrections simultaneously.} are
\ba \label{f3ns}
f_3^{(0,\NS)}(x) &=& 8x^2(3-2x) \Phi(x)
+ \biggl( \frac{20}{3} + 16x - \frac{80}{3}x^2 
+ \frac{160}{9}x^3 \biggr)\ln(1-x)
\nonumber \\
&+& \biggl( - \frac{5}{3} - 4x + \frac{32}{3}x^2 
- \frac{160}{9}x^3 \biggr)\ln x
+ \frac{73}{54} + \frac{67}{9}x + \frac{16}{9}x^2 - \frac{128}{27}x^3,
\\ \label{g3ns}
g_3^{(0,\NS)}(x) &=& 
8x^2(1-2x) \Phi(x)
+ \biggl( - \frac{4}{3} - \frac{272}{9}x^2 + \frac{160}{9}x^3 \biggr)\ln(1-x)
\nonumber \\
&+& \biggl( \frac{1}{3} + \frac{128}{9}x^2 - \frac{160}{9}x^3 \biggr)\ln x
- \frac{29}{54} - \frac{7}{3}x + \frac{16}{9}x^2 -\frac{128}{27}x^3,
\\ \label{f3s}
f_3^{(0,\SS)}(x) &=& \biggl( \frac{5}{3} + 4x + 4x^2 \biggr)\biggl(
4\Li{2}{1-x} + 4\ln x\ln(1-x) - \ln^2x \biggr) - 4x^2\ln^2x
\nonumber \\
&+&  \biggl( \frac{68}{9} + \frac{8}{3x} + 12x - \frac{56}{3}x^2
- \frac{32}{9}x^3 \biggr)\ln(1-x)
+ \biggl( - \frac{29}{9} - \frac{14}{3}x + 16x^2 
\nonumber \\
&+& \frac{32}{9}x^3 \biggr)\ln x
- \frac{287}{27} - \frac{4}{9x} - \frac{13}{9}x 
+ \frac{86}{9}x^2 + \frac{80}{27}x^3,
\\ \label{g3s}
g_3^{(0,\SS)}(x) &=& \biggl( \frac{4}{3}x^2 - \frac{1}{3} \biggr)
\biggl( 4\Li{2}{1-x} + 4\ln x\ln(1-x) - \ln^2x \biggr)
- \frac{4}{3}x^2\ln^2x
\nonumber \\
&+& \biggl(  - \frac{4}{9} - \frac{8}{9x} + 4x 
+ \frac{8}{9}x^2  - \frac{32}{9}x^3 \biggr)\ln(1-x)
\nonumber \\
&+& \biggl( \frac{1}{9} - 2x - \frac{16}{9}x^2 
+ \frac{32}{9}x^3 \biggr)\ln x
+ \frac{31}{27} + \frac{4}{27x} - \frac{35}{9}x 
- \frac{10}{27}x^2 + \frac{80}{27}x^3, 
\\ \nonumber 
\Phi(x) &\equiv& \frac{1}{2}\ln^2 x + \ln^2(1-x)
- 2\ln x\ln(1-x) - \Li{2}{1-x} - \zeta_2.
\ea

Functions 
$g_2^{(1,\SS)}$ and $g_2^{(1,\mathrm{int})}$ shown above as well as 
$f_2^{(1,\SS)}$ and $f_2^{(1,\mathrm{int})}$, which are given 
in Ref.~\cite{Arbuzov:2002cn}, agree with
the results of my calculations starting directly from Feynman
diagrams and using methods described in Ref.~\cite{Arbuzov:1995cn}.


\subsection{Cancellation of Mass Singularities}

An important check of the results is to demonstrate the cancellation
of mass singularities in the total decay width. At first
we should note that a naive integration of the electron spectrum
gives rather the counting rate of electrons than the total muon
decay width, since the number of the final state electrons can exceed
the number of decaying muons because of real $e^+e^-$ pair emission.
In other words, we should avoid the double counting of electrons in
the contributions due to real pair emission. For this purpose,
we can keep the non--singlet pair contributions and drop the 
singlet ones (see their definition on page~\pageref{def:sns}). 
But the functions $f_2^{(1,\mathrm{int})}$ and
$g_2^{(1,\mathrm{int})}$ contain the double counting too. To resolve this
problem we can use the splitting function~\cite{Altarelli:1979kv}
\ba
P^{(1,\mathrm{int})}_{\bar{e}e}(x) = 
2\frac{1+x^2}{1+x}\biggl( 2\Li{2}{1+x} + \frac{1}{2}\ln^2x
- 3\zeta_2 \biggr) + 2(1+x)\ln x + 4(1-x),
\ea
which describes the transition of an electron into a positron
just in the relevant class of Feynman diagrams. So the corresponding
contribution can be constructed by convolution with the lowest order
functions $f_0$ and $g_0$, integrated over the positron energy
fraction, and then subtracted.

The cancellation of the mass singularities in the LL contributions
due to photons and non--singlet pairs is rather trivial:
\ba
\int\limits_{0}^{1} \dd x\; f_{n}^{(0,j)}(x) =
\int\limits_{0}^{1} \dd x\; g_{n}^{(0,j)}(x) = 0, \qquad
j = \NS,\ \gamma, \qquad n = 1,2,3,\ldots
\ea
It is guaranteed by the normalization conditions of the corresponding
LL fragmentation functions.
Because of the problem with the interference contribution, discussed 
above, the cancellation of the singularities in the second order
next--to--leading photonic and S-NS pair interference corrections 
to the integrated muon decay width 
occurs in a bit more sophisticated manner.
It can be seen from the equality
\ba \label{iintf}
\int\limits_{0}^{1}\dd x\;
( f_2^{(1,\gamma)}(x) + f_2^{(1,\mathrm{int})}(x) )
= \int\limits_{0}^{1}\dd x\;
P^{(1,\mathrm{int})}_{\bar{e}e}(\bullet)\otimes f_0(x) =
\frac{13}{8} - \frac{3}{2}\zeta_2 + \zeta_3.
\ea

Let us look now at the integral of the second order
non--singlet pair correction. It is known~\cite{vanRitbergen:1998yd}, 
that the integrated contribution of this correction contains 
large logarithms due to the running of the coupling constant:
\ba
\biggl(\frac{\alpha}{2\pi}\biggr)^2L\int\limits_{0}^{1}\dd x\; 
f_2^{(1,\NS)}(x) &=& 
\frac{\Delta\alpha(m_\mu)}{2\pi} \biggl[
\int\limits_{0}^{1}\dd x\; f_1(x) \biggr] \bigg|_{m_e\to 0},
\ea
where
\ba
\Delta\alpha(m_\mu) = \frac{\alpha^2}{3\pi}L 
= \alpha(m_\mu) - \alpha + \order{\alpha^2L^0}.
\ea
To demonstrate that using our results, note first that the relevant
function consists of two parts:
\ba
f_2^{(1,\NS)}(x) = f_0\otimes P^{(1,\NS)}(x) 
+ \frac{2}{3}\hat{f}_1(x).
\ea
One can check that
\ba
\int\limits_{0}^{1}\dd x\;
P^{(1,\NS)}_{ee}(\bullet)\otimes f_0(x) = 0.
\ea
It remains now to recognize that
\ba
\biggl[ \int\limits_{0}^{1}\dd x\; f_1(x) \biggr] \bigg|_{m_e\to 0} 
= \int\limits_{0}^{1}\dd x\; \hat{f}_1(x),
\ea
which can be verified easily. Thus we checked successfully 
an important property of our analytical results.

The integral of anisotropic contributions to the decay spectrum 
can be treated in the same way.
They don't contribute to the total decay width at all, 
because they vanish after the integration over the angle. 
Nevertheless, the cancellation of mass
singularities can be observed also in the forward--backward asymmetry 
of the decay, which is not affected by isotropic functions 
on the contrary. 
In particular, we have an equality analogous to Eq.~(\ref{iintf}):
\ba \label{iintg}
\int\limits_{0}^{1}\dd x\;
( g_2^{(1,\gamma)} + g_2^{(1,\mathrm{int})} )
= \int\limits_{0}^{1}\dd x\;
P^{(1,\mathrm{int})}_{\bar{e}e}(\bullet)\otimes g_0(x) =
- \frac{13}{24} + \frac{1}{2}\zeta_2 - \frac{1}{3}\zeta_3.
\ea


\subsection{Numerical Results}

Numerical results for the pure photonic corrections are presented in 
Table~\ref{table:1},
where we give the values of different contributions normalized by
the Born distribution in the following way:
\ba \label{delta_g}
&& \delta_1 = \frac{\alpha}{2\pi}\;\frac{f_1(x) \pm cP_{\mu}g_1(x)}
{f_{\mathrm{Born}}(x) \pm cP_{\mu}g_{\mathrm{Born}}(x)}\, ,
\nonumber \\
&& \delta_n^{(0,\gamma)} = \frac{L^n}{n!}
\biggl(\frac{\alpha}{2\pi}\biggr)^n\frac{f_n^{(0,\gamma)}(x) 
\pm cP_{\mu}g_n^{(0,\gamma)}(x)}
{f_{\mathrm{Born}}(x) \pm cP_{\mu}g_{\mathrm{Born}}(x)}\, ,\qquad
n = 1,2,3\; ,
\nonumber \\
&& \delta_2^{(1,\gamma)} = L\biggl(\frac{\alpha}{2\pi}\biggr)^2
\frac{f_2^{(1,\gamma)}(x) \pm cP_{\mu}g_2^{(1,\gamma)}(x)}
{f_{\mathrm{Born}}(x) \pm cP_{\mu}g_{\mathrm{Born}}(x)}\, .
\ea
%--------------------------Table~1-----------------------
\TABLE[ht]{
\label{table:1}
\begin{tabular}{|r|r|r|r|r|r|r|r|} \hline
\multicolumn{1}{|c|}{$x$} & 
\multicolumn{1}{c|}{$10^4\delta_1$} & 
\multicolumn{1}{c|}{$10^4\delta_1^{(0,\gamma)}$} &
\multicolumn{1}{c|}{$10^4\delta_2^{(0,\gamma)}$} & 
\multicolumn{1}{c|}{$10^4\delta_2^{(1,\gamma)}$} & 
\multicolumn{1}{c|}{$10^4\delta_3^{(0,\gamma)}$} &
\multicolumn{1}{c|}{$10^4\delta^{\mathrm{exp}}_{\mathrm{a.h.}}$} &
\multicolumn{1}{c|}{$10^4\delta^{\mathrm{exp}}_{\mathrm{SF}}$} 
\\ \hline
0.05 &   4590.1 &  10325.0 &  184.96 &$-$247.63 & $-$0.43 & 0.00 &    6.90 \\
0.1  &   1715.1 &   3257.7 &   33.18 & $-$37.79 & $-$0.35 & 0.00 &    1.39 \\
0.2  &    674.0 &   1106.2 & $-$1.28 &     0.34 & $-$0.15 & 0.01 &    0.05 \\
0.3  &    364.0 &    549.1 & $-$6.58 &     4.48 & $-$0.05 & 0.01 & $-$0.20 \\
0.5  &     64.1 &     82.6 & $-$6.50 &     3.73 &    0.05 & 0.06 & $-$0.26 \\
0.7  & $-$160.3 & $-$214.9 & $-$1.28 &     0.70 &    0.07 & 0.18 & $-$0.09 \\
0.9  & $-$470.3 & $-$592.1 &   15.19 &  $-$5.97 & $-$0.14 & 0.72 &    0.62 \\
0.99 & $-$971.9 &$-$1198.8 &   69.84 & $-$26.08 & $-$1.95 & 3.47 &    3.70 \\
0.999&$-$1439.8 &$-$1772.5 &  155.10 & $-$57.86 & $-$6.64 & 9.17 &    9.75 \\
\hline
\end{tabular}
\caption{Photonic corrections to electron spectrum 
versus $x$ for $c=1$ and $P_{\mu}=1$.}
}
%--------------------------Table~1-----------------------
One can see that the first order LL correction $\delta_1^{(0,\gamma)}$
(look for $f_1^{(0,\gamma)}(x)$ and $g_1^{(0,\gamma)}(x)$ in 
Ref.~\cite{Arbuzov:2002pp})
provides the bulk of the effect, especially in the region of intermediate
and large $x$-values. Convergence of the corresponding series in $L$
in the second order corrections doesn't look so good: the NLL contribution
is only about two times less than the LL one.

There is a trick, which allows to make a better approximation in
the region of small $x$. Looking closely at the argument of
the large logarithm during the actual calculations of integrals
over the phase space of real photons, one can notice that it is 
rather $x^2m_\mu^2/m_e^2$, than simply $m_\mu^2/m_e^2$. So the
modification $L \to L + 2\ln x$ can be done in our formulae.
This will move some terms from the sub--leading corrections
into the leading ones. I checked that the trick does really help
to improve the agreement in the first order between 
$\delta_1^{(0,\gamma)}$ and the full $\delta_1$. 
But, as far as the \TWIST experiment is 
interested in the region $x \geq 0.3$ (the event distribution
is peaked at large $x$-values in any case), I don't apply it here.

Let us define the relative contributions of pair corrections as
\ba
&& \delta_2^{(0,e^+e^-)} = \frac{L^2}{6}\biggl(\frac{\alpha}{2\pi}\biggr)^2
\frac{ 2f_2^{(0,\NS)}(x) + 3f_2^{(0,\SS)}(x)
\pm cP_{\mu}(2g_2^{(0,\NS)}(x)+3g_2^{(0,\SS)}(x))}
{f_{\mathrm{Born}}(x) \pm cP_{\mu}g_{\mathrm{Born}}}\, ,
\nonumber \\
&& \delta_2^{(1,e^+e^-)} = L
\biggl(\frac{\alpha}{2\pi}\biggr)^2
\sum\limits_{i=\NS,\SS,\mathrm{int}}^{}
\frac{f_2^{(1,i)}(x) \pm cP_{\mu}g_2^{(1,i)}(x)}
{f_{\mathrm{Born}}(x) \pm cP_{\mu}g_{\mathrm{Born}}(x)}\, ,
\nonumber \\
&& \delta_3^{(0,e^+e^-)} = \frac{L^3}{6} 
\biggl(\frac{\alpha}{2\pi}\biggr)^3
\sum\limits_{j=\NS,\SS}^{}
\frac{f_3^{(0,j)}(x) \pm cP_{\mu}g_3^{(0,j)}(x)}
{f_{\mathrm{Born}}(x) \pm cP_{\mu}g_{\mathrm{Born}}(x)}\, .
\ea
Their behaviour is shown in Figure~\ref{figure:1}.
%--------------------------Figure~1-----------------------
\FIGURE[ht]{
\epsfig{file=mu_pair.eps,width=10cm,height=6cm}
\caption{Pair corrections versus $x$ for $c=1$ and $P_{\mu}=1$.}
\label{figure:1}
}
%--------------------------Figure~1-----------------------
One can see that the next--to--leading pair corrections
have the same order of magnitude as the leading ones.
This feature has been observed earlier in pair corrections
to other processes~\cite{Arbuzov:rt}. 
The functions, which describe the LL pair effect, have
numerically small coefficients and are less divergent 
(at $x\to 1$ and $x\to 0$)
than the NLL ones. This means that the calculations of the
non--logarithmic terms in the second order pair corrections 
is desirable, although the pair corrections are typically
less than the photonic ones (at the same order in $\alpha$).
 

\section{\label{Sec:Con}Conclusions}

To estimate the theoretical uncertainty in the description
of the polarized muon decay spectrum by Eqs.~(\ref{general}) 
and (\ref{Ffinal}), we should consider the contributions,
which have been omitted in our calculations. They are: 
the $\order{\alpha^2}$ order terms, which are not enhanced by the 
large logarithm; sub--leading contributions in the third order
$\order{\alpha^3L^m}$, $m\leq 2$; and all the leading and sub--leading
effects in the forth and higher orders $(\order{\alpha^nL^m}$ where
$n \geq 4$, $m\leq n)$ except those ones, which are taken
into account by exponentiation. The possible contribution to
the uncertainty from strong interactions is negligible in our case,
since it is suppressed at least by $(\alpha/\pi)^2$, and the lowest order
contribution of hadronic virtual pairs was found in 
Ref.~\cite{Davydychev:2001ee} to be small itself.

An estimate of the omitted contributions by a simple counting of
powers of the fine--structure constant and the large logarithm is
not very safe, because there could be some extra enhancement factors,
like numerically large constant coefficients or powers of $\ln(1-x)$
(the latter is partially taken into account by exponentiation). 
I suggest to estimate the omitted terms 
by a linear extrapolation of the known expansions in $\alpha$ and $L$.
Namely, 
\ba
\delta_2^{(2)} \sim \delta_2^{(1)} 
\frac{\delta_2^{(1)}}{\delta_2^{(0)}}\, ,
\qquad
\delta_3^{(1)} \sim \delta_2^{(1)}\frac{\delta_3^{(0)}}{\delta_2^{(0)}}\, ,
\qquad
\delta_4^{(0)} \sim \delta_3^{(0)}\frac{\delta_3^{(0)}}{\delta_2^{(0)}}\, ,
\ea
where $\delta_2^{(2)}$ denotes the contribution of
the second order terms, which are not enhanced by any 
large logarithm; $\delta_3^{(1)}$ is the third order next--to--leading
correction (which can be calculated using the fragmentation function
method described above); $\delta_4^{(0)}$ stands for the fourth
order LL effect.
This estimate of the theoretical error, can be applied to any
particular set of experimental conditions to derive the actual
uncertainty. In principle, the latter can depend
on various cuts and details of particle registration and event selection
(see discussion in Ref.~\cite{Arbuzov:2002pp}). 
The approach to estimate the uncertainty in the muon
decay spectrum description works well for the main part of the
kinematical domain. But, if one studies separately the extreme region 
$x \ll 1$ (or $1-x \ll 1)$, a special investigation of the convergence
properties of our perturbative expansions in $\alpha$ and $L$
should be performed.
Evaluation of the uncertainties for any concrete experiment
can be done using the analytical results and applying
specific conditions of particle registration and data
fitting. 

The new results of Ref.~\cite{Arbuzov:2002cn} and the present paper
reduce the theoretical uncertainty in the description of the polarized
muon decay spectrum. For the {\em quasi--realistic} experimental 
setup described in Ref.~\cite{Arbuzov:2002pp}, we can obtain now
about 1.5 times better precision, so that, for instance, the
theoretical uncertainty for the Michel parameter $\rho$ becomes
$2\cdot 10^{-4}$ instead of $3\cdot 10^{-4}$ obtained in 
Ref.~\cite{Arbuzov:2002pp}.
Nevertheless, this is still worse then the experimental
precision $1\cdot 10^{-4}$ planned at 
\TWIST~\cite{Rodning:2001js,Quraan:2000vq}.
Assuming that a theoretical precision of about
one third (or less) of the experimental one would not spoil
results of an experiment, we see a challenge for further 
investigations. 
A calculation of the $\order{\alpha^2}$
contributions, which are not enhanced
by the large logarithm, is required to ameliorate
the theoretical precision. This calculation is difficult, 
but possible by means of the standard methods. 

The formulae for higher order corrections (with simple substitutions) 
are valid also for the decays of $\tau$-lepton:
$\tau\to\mu\nu_{\tau}\bar{\nu}_{\mu}$ and 
$\tau\to e\nu_{\tau}\bar{\nu}_{e}$.


\acknowledgments
This research was supported by the Natural Sciences 
and Engineering Research Council of Canada.
I am grateful to A.~Czarnecki, A.~Gaponenko, and K.~Melnikov 
for fruitful discussions.


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\end{document}



