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For a two-body decay of the form $$A\to B+B$$ if the interaction strength controlling the decay is $\lambda$, the Feynman amplitude $\mathcal{M}$ will contain a factor of $\lambda$ from the vertex factor $(-i\lambda)$. Therefore, the zero-temperature decay rate $\Gamma_{\rm T=0}$ will be given by $$\Gamma_{\rm T=0}\sim |\mathcal{M}|^2\sim \lambda^2.$$ If the same decay occurs at a temperature $T\neq0$, what can be said about the $T-$dependence of the decay rate (without a detailed calculation of thermal field theory) at least for temperatures $T\gg M_A$? I am looking for some plausibility arguments based on dimensional analysis or otherwise.

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For a Yukawa interaction of the type $$\mathscr{H}_{int}=-\lambda \bar{\psi}_B\psi_B\phi_A,\tag{1}$$ where $\phi$ is a scalar field of mass $M_A$ and $\psi$ is a fermionic field of mass $M_B$ (with $M_A>2M_B$), is responsible for a decay of the form $$A\to B+\bar{B}.\tag{2}$$ The exact zero-temperature decay rate at the tree-level is given by $$\Gamma_{\rm T=0}=\frac{\lambda^2M_A}{8\pi^2}\Big(1-\frac{4M_B^2}{M_A^2}\Big)^2.\tag{3}$$ In the limit where $M_B$ mass can be neglected, the relevant parameters of the problem are $\lambda$ and $M_A$. Hence, from dimensional analysis, $$\Gamma_{\rm T=0}=\lambda^2M_A f(M_B/M_A)\tag{4}$$ where $f$ is a function of dimensionless quantity $M_B/M_A$. Now at finite temperature $T$, where both $M_A,M_B$ can be neglected, dimensional analysis tells that $$\Gamma_{\rm T\neq 0}\sim \lambda^2T.\tag{5}$$ If someone wants to refine $(5)$ when $M_A,M_B$ are not neglected that would be even more interesting.

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