The decays (5.2) and (5.3) involve the same
form factors as displayed in Eq. (5.12). We denote by
,
and
the current matrix elements of
the processes (5.1)-(5.3). These are related by
isospin symmetry
,
This relation also holds for the individual form factors, which
may be decomposed into a symmetric and an antisymmetric part under
. Because of Bose
symmetry and of the
rule of the
relevant weak currents, one has
where
and is defined in Eq. (5.12).
The isospin relation for the decay rates is
Isospin violating contributions affect the matrix elements and phase
space, as a result of which this relation is modified. In
order to illustrate the (substantial) effects from asymmetries in
phase space, we take
constant form factors
F,G and set R=0,H=0. Eq. (5.26) then reads (with
physical
masses for and with
MeV in
)
in the electron mode and
in the muon mode.