#
2. Motivation

In the last 50 years of the history of particle physics, the decays of the pion
have been the subject of intense theoretical and experimental research
activities. For example the decay
_{
}
(2.1)
was first discovered in 1958 and provided a confirmation of the V-A theory of
weak interactions. Pion decays are weak processes which are theoretically
calculable with high precision. In order to provide experimental information
about pions three pion production facilities have been built worldwide, LAMPF
(Los Alamos Meson Production Facility), TRIUMF (Tri-University Meson Facility),
and one facility at PSI (Paul Scherrer Insitute), which produce pion beams of
high intensities at low momenta.
Pions are spin zero bosons with negative parity; they form an isospin one
multiplet with
_{
},_{}
(2.2)
where *I*_{3} is the third component of the isospin *I*. The
charged pion has an average lifetime of *[tau]*_{ p } =
26.03 ns; its most probable decay mode is
_{
},
(2.3)
with an average decay rate of _{
}.

The process called pion beta decay is the decay
_{
},
(2.4)
with a * p *^{0}-meson and two leptons in the final
state, thus a so called semileptonic decay. Since the pion beta decay is a
transition between two members of an isospin triplet (2.2) it is analogous to
super-allowed Fermi transitions in nuclear beta decay with
_{
}
, (
2.5)
*J*^{ p } denoting spin *J* and parity
^{ p }. At present, there are nine precise measurements of
nuclear beta decay (a compilation is given in [Tow 95a]); one of them is
the decay ^{14}O to the first excited state of ^{14}N [Cla 73].
The total decay energy of ^{14}O is *E*_{F} =
2.32 MeV* *and the decay rate is
*[lambda]*_{F}=0.00971/s. A rough estimation on the pion beta
decay rate is obtained from the fifth-power law for beta decays
_{
}.
(2.6)
With _{
}
one gets *[lambda]*_{ p b } ~ 0.3/s, which leads to the very
small branching ratio
_{
}.
(2.7)
Super-allowed Fermi transitions are pure vector interactions, since the axial
vector contributions to the matrix elements of the transitions vanish due to
spin and parity of initial and final states (Eq. 2.5). Experimental
measurements of different super-allowed Fermi transitions show, that the
product *ft* of the Fermi integral function *f* and the decay half
life *t* is constant. This is explained by the CVC-Hypothesis, which
postulates the conservation of the weak vector current. Following the
CVC-Hypothesis, the *ft* values for super-allowed Fermi transitions are
given by
_{
},
(2.8)
where *K *is a product of fundamental constants (*K *=
8.1201·10^{-7}). The uncorrected matrix element *M*_{F}
for transitions which fulfill (2.5) is
_{
}*
*(2.9)
and the vector coupling constant *G*_{V} can be expressed by
_{
}.
(2.10)
The Fermi coupling constant *G*_{µ} of pure leptonic decays
like _{
}
is known with high precision from *µ*^{+} mean life
measurements. *V*_{ud} is an element of the
Cabbibo-Kobayashi-Maskava (CKM) mixing matrix V
_{
},
(2.11)
where (*d, s, b*) are the mass eigenstates and (*d', s', b'*) are the
weak eigenstates of the quarks with charge -1/3. The unitarity of the CKM
matrix *V* is tested most stringently in the top row
_{
}.
(2.12)
The unitarity, which must be satisfied if the Standard Model with 3 leptonic
generations is correct, is dominated by _{
}~0.975.
The term _{
}
is determined by analysis of semileptonic hyperon decays such as e.g. _{
};
its present value [PDG 94] is _{
}~0.220.
The third term contributes only weakly to the sum in (2.12), _{
}~0.004.

In order to determine *V*_{ud} from the nine accurately measured
super-allowed Fermi transitions Eq. (2.8) has to be modified:
_{
}
(2.13)
Radiative corrections *[Delta]*_{R} are composed of dominating,
*Z*-independent parts and of *Z*-dependent terms of the order
*Z a *^{2} and *Z*^{2} a ^{3}, where
*Z* is the atomic number of the nucleus and * a * is the
fine-structure constant (* a ~1/137)*. They are calculated for each of
the nine mentioned transitions and amount to ~3.5% at most. The nuclear
mismatch * d *_{c} corrects the fact that isospin symmetry
along the isomultiplet is violated by the coulomb and the nuclear force. The
estimates on * d *_{c} for the different nuclei are of the
order of some tenth of a percent but have model dependent divergences [Wil 94].
A recent re-analysis [Tow 95b] , based on the newest *Ft*-values, leads to
an unitarity test of
_{
}.
(
2.14)
This result indicates a violation of the unitarity condition for the three
generations by 2.1 standard deviations. Thus, the present determination of
*V*_{ud } from super-allowed Fermi transitions is not limited by
the experimental accuracy with which the relevant ft-values are determined nor
by the confidence on the various radiative corrections, but rather by the
uncertainty of the nuclear mismatch * d *_{c}.

Another independent determination of *V*_{ud}_{ }comes
from the decay of the neutron which does not require the
* d *_{c} correction but has the disadvantage of being a mixed
vector-axial transition:
_{
}.
(2.15)
The extraction of *V*_{ud} requires therefore the knowledge of the
neutron lifetime *[tau]*_{n} as well as the knowledge of
*G*_{A}/*G*_{V}, determined by the asymmetry in the
emission of electrons from polarized neutrons; both of these measurements are
very delicate.

From the mean lifetime of the neutron *[tau]*_{n}=887.0±2.0 s
together with the weighted average of the ratio
*G*_{A}/*G*_{V} (the results are compiled in
[Tow 95a]) individual values for the coupling constants can be deduced.

Again, the value of G_{V} obtained from neutron decay is used to test
the unitarity of the CKM-matrix and leads to [Tow 95b]

_{
},

which differs from unitarity in the opposite sense to the result from nuclear
beta decay (eq. 2.14).

*Figure
2.1:
From [Tow 95a]. Allowed regions from the analysis of super-allowed Fermi
transitions, neutron decay and CKM unitarity. The results are expressed in
values for the corrected coupling constants G'*_{V}=G_{V}(1+_{
})
and G'_{A}=G_{A}(1+_{
}),
where _{
}
are the nucleus independent radiative corrections.
The present situation in determining the unitarity sum of the CKM-matrix can be
summarized as follows:
* An analysis of the super-allowed Fermi transitions including the nuclear
mismatch * d *_{c} violates the unitarity of the CKM-matrix by
2.1 standard deviations.

* Neutron beta decay has contributions from axial currents, the experimental
data lead to a result 2 standard deviations above CKM-unitarity.

The present status of the analysis from super-allowed Fermi transitions and
neutron decay is displayed in Fig. 2.1.

Due to its similarity to super-allowed Fermi transitions and the absence of
nuclear and screening corrections, the pion beta decay presents another
independent determination of *V*_{ud}. However, the small
branching ratio of the decay (Eq. (2.7)) makes it a very difficult process to
study with a precision below 1%. The present determination of the pion beta
decay rate has a precision of 4% [McFa 85]. A measurement below 1% would be a
substantial improvement of the experimental tests of the CVC-Hypothesis and the
radiative corrections. At a level of 0.5% it provides important information for
determining *V*_{ud}.