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One of the features of the unconventional $s_\pm$ state in iron-basedsuperconductors is possibility to transform to the $s_{++}$ state with theincrease of the nonmagnetic disorder. Detection of such a transition wouldprove the existence of the $s_\pm$ state. Here we study the temperaturedependence of the London magnetic penetration depth within the two-band modelfor the $s_\pm$ and $s_{++}$ superconductors. By solving Eliashberg equationsaccounting for the spin-fluctuation mediated pairing and nonmagnetic impuritiesin the $T$-matrix approximation, we have derived a set of specific signaturesof the $s_\pm \to s_{++}$ transition: (1) sharp change in the behavior of thepenetration depth $\lambda_{L}$ as a function of the impurity scattering rateat low temperatures; (2) before the transition, the slope of $\Delta\lambda_{L}(T) = \lambda_{L}(T)-\lambda_{L}(0)$ increases as a function oftemperature, and after the transition this value decreases; (3) the sharp jumpin the inverse square of the penetration depth as a function of the impurityscattering rate, $\lambda_{L}^{-2}(\Gamma_a)$, at the transition; (4) changefrom the single-gap behavior in the vicinity of the transition to the two-gapbehavior upon increase of the impurity scattering rate in the superfluiddensity $\rho_{s}(T)$.

Decisive proofs of the $s_\pm \to s_{++}$ transition in the temperature dependence of the magnetic penetration depth