Within the framework of the statistical model the spreading width due
to an arbitrary K-body force has been calculated for a compound
nuclear reaction of an A-body nucleus. The spreading width is the
adequate measure for symmetry nonconservation in complex many-body
systems [2, 28]. It measures the extent to which the states are
smeared out due to the presence of the symmetry breaking force. It
also appears as the damping width of a simple configuration -- such
as an isobaric analog state or a giant resonance -- into the complex
compound nuclear configurations. Definite numbers for a particular
interaction can be given if eq. (4.6) is complemented by the
calculation of the spectral average of the interaction in K-body
space. Obvious applications are the electromagnetic and weak forces,
responsible for the breaking of isospin and parity,
respectively. Numerical studies in that direction are currently under
progress. For the case of isospin breaking a weak dependence of the
spreading width on energy and mass number is known [2]. In the
present study the energy or temperature dependence has been related to
the body rank of the underlying interaction. The variation of
with temperature can be compared with the
results of other authors. Assuming that the spread of a one-exciton
configuration (a 1p 0h state) is proportional to the square of the
excitation energy [29, 30], Lauritzen et al. [31]
have concluded a 
-dependence for the spreading width in
general. Since the decay of the one-exciton configuration is caused by
, we indeed
find from eq. (4.4) its damping to be proportional to 
and eq. (4.7) indeed indicates a -dependence of the
spreading width in A-body space (for transitions caused by the
strong interaction only). Studying the coupling of surface modes to
single particle motion, Esbensen and Bertsch [32] found the
``elementary damping'' to be proportional to E, which in the
framework of Lauritzen et al. corresponds to a 
-dependence
of the spreading width. There is nothing analogous in the present
results since we did not consider collective motion in the present
paper. De Blasio et al. [33] report the damping of giant
resonances to be independent of the nuclear temperature. Again the
reason is that the statistical damping of the present study is
different from the damping mechanism considered there. This is also
true for the study of spreading properties of isobaric analogue and
Gamov-Teller resonances by Colò et al. [34]. This
paper indicates, however, that the statistical damping must be taken
into account in order to explain the widths of the resonances.
Frequently the treatment of parity violation in CN reactions is restricted to the k=1 part of the weak interaction [35, 36, 37, 38, 39]. Eq. (4.8) indicates, however, that the contribution of the operators with exciton rank two should be included in the calculation of a local mean square matrix element.
The average 
in K-body space is by definition taken over the
bound K-body states only. This is not a complete system of
states. The result may therefore depend on the mean field that
generates the one-body states from which the K-body states are built
up. Consider for instance K=2 and an interaction which is
proportional to a delta function and independent of A. Nevertheless
the two-body matrix elements of this interaction decrease with
increasing volume of the system, i.e. with increasing A. We may
therefore not rule out a variation of with mass number.
Consequently, eqs. (4.6) and (4.7) might not exhibit
the complete A-dependence of the spreading width, which is expected
to be weak.
Finally we point out the limitations of our method. The Hartree Fock
method and the particle hole formalism rely on a basis of product
states. Such an independent particle model cannot describe collective
phenomena in nuclei. The present results therefore must be modified if
applied to reactions that involve collective excitations. A second
problem are the effects of the symmetries respected by the
interaction under study. These effects will of course manifest
themselves automatically when the spectral average in the K-body
space is calculated. The use of partial level densities for the
actors that contain all states at a given energy irrespective of
further quantum numbers, however, neglects possible local
effects of the respected symmetries. We give an example to illustrate
this complication. Consider the weak potential (k=1) in a
nucleus. This operator connects many-body states that differ by only
one single-body configuration. Parity violation and conservation of
total angular momentum demand that the single-body states differ in
l but not in j. Since single-body states with 
and 
only exist in different shells, they are separated by
a relatively large energy interval. Consequently, the present local
average suppresses these contributions to the spreading width. The
two-exciton part of the weak interaction on the other hand is not
subject to this ``local selection rule'' because it simultaneously
changes two single-exciton configurations. This fact was first pointed
out by Lewenkopf and Weidenmüller [40]. They estimated that
the k=2 part of the weak interaction dominates the local mean square
matrix element. In section ivB it was found that potential and
scattering contribute about equally to the spreading width without
taking the local effects of the symmetries into account. Sufficiently
elaborate expressions for the single-exciton level densities of the
actors before and after the interaction would only overlap in a small
energy range, and consequently the convolution of
with 
would
result in a smaller contribution of the potential to the strength.