  The nature of cosmic rays (CRs) and cosmic ray transport in galaxy clusters
is probed by a number of observations. Radio observations reveal the
synchrotron radiation of cosmic ray electrons (CRe) spiraling around cluster
magnetic fields. $\gamma$-ray observations reveal hadronic reactions of cosmic
ray protons (CRp) with ambient gas nuclei which produce pions. To date, no such
cluster-wide $\gamma$-ray signal has been measured, putting an upper limit on
the density of CRp present in clusters. But the presence of CRe implies some
source of CRp, and consequently there must be some CRp loss mechanism. In this
paper we quantify the observational constraints on this loss mechanism assuming
that losses are dominated by CR transport, ultimately deriving a minimum
diffusion coefficient of $\sim10^{31}$ cm$^2$ s$^{-1}$ in the Coma cluster.
This lower limit on transport may help illuminate some unknown properties of
the cluster field topology. Conversely, measurements of cluster field tangling
scales can constrain other model parameters, such as the relative acceleration
efficiency of protons to electrons. To be consistent with the Coma
observations, protons cannot be accelerated more than 15 times more efficiently
than electrons of the same energy.
