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We analyze the applicability of the Fermi-golden-rule description ofquasiparticle relaxation in a closed diffusive quantum dot withelectron-electron interaction. Assuming that single-particle levels are alreadyresolved but the initial stage of quasiparticle disintegration can still bedescribed by a simple exponential decay, we calculate the average inelasticenergy relaxation rate of single-particle excitations and its mesoscopicfluctuations. The smallness of mesoscopic fluctuations can then be used as acriterion for the validity of the Fermi-golden-rule description. Technically,we implement the real-space Keldysh diagram technique, handling correlations inthe quasi-discrete spectrum non-perturbatively by means of the non-linearsupersymmetric sigma model. The unitary symmetry class is considered forsimplicity. Our approach is complementary to the lattice-model analysis of Fockspace: thought we are not able to describe many-body localization, we derivethe exact lowest-order expression for mesoscopic fluctuations of the relaxationrate, making no assumptions on the matrix elements of the interaction. It isshown that for the quasiparticle with the energy $\varepsilon$ on top of thethermal state with the temperature $T$, fluctuations of its energy width becomelarge and the Fermi-golden-rule description breaks down at$\max\{\varepsilon,T\}\sim\Delta\sqrt{g}$, where $\Delta$ is the mean levelspacing in the quantum dot, and $g$ is its dimensionless conductance.

Energy relaxation rate and its mesoscopic fluctuations in quantum dots