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Role of vertex corrections in the $T$-linear resistivity at the Kondo breakdown quantum critical point

The Kondo breakdown scenario has been claimed to allow the $T$-linear resistivity in the vicinity of the Kondo breakdown quantum critical point, two cornerstones of which are the dynamical exponent $z = 3$ quantum criticality for hybridization fluctuations in three dimensions and irrelevance of vertex corrections for transport due to the presence of localized electrons. We revisit the issue of vertex corrections in electrical transport coefficients. Assuming that two kinds of bosonic degrees of freedom, hybridization excitations and gauge fluctuations, are in equilibrium, we derive coupled quantum Boltzmann equations for two kinds of fermions, conduction electrons and spinons. We reveal that vertex corrections play a certain role, changing the $T$-linear behavior into $T^{5/3}$ in three dimensions. However, the $T^{5/3}$ regime turns out to be narrow, and the $T$-linear resistivity is still expected in most temperature ranges at the Kondo breakdown quantum critical point in spite of the presence of vertex corrections. We justify our evaluation, showing that the Hall coefficient is not renormalized to remain as the Fermi-liquid value at the Kondo breakdown quantum critical point.