Propagation of a $K$-body force into $A$-body space

The calculation of the spreading width of a compound nuclear reaction caused by a symmetry breaking $K$-body force acting in an $A$-body system ($K << A$, usually $K=2$) involves the determination of the local average square matrix element in $A$-body space. This problem is reduced to finding the global mean square matrix element $v^2$ in $K$-body space. The result is a compact formula for the spreading width which contains $v^2$ as an input. Our method is based on the dilute gas approximation for excitons close to the Fermi edge. The relative strength of the contributions of operators with different exciton rank as well as the connection between the energy dependence of the spreading width and the body rank of the underlying interaction are established.