Constraints on the annihilation of heavy dark matter in dwarf spheroidal galaxies with gamma-ray observations

Electrons and positrons produced in dark matter annihilation can generate secondary emission through synchrotron and IC processes, and such secondary emission provides a possible means to detect DM particles with masses beyond the detector's energy band. The secondary emission of heavy dark matter (HDM) particles in the TeV-PeV mass range lies within the Fermi-LAT energy band. In this paper, we utilize the Fermi-LAT observations of dwarf spheroidal (dSph) galaxies to search for annihilation signals of HDM particles. We consider the propagation of $e^+/e^-$ produced by DM annihilation within the dSphs, derive the electron spectrum of the equilibrium state by solving the propagation equation, and then compute the gamma-ray signals produced by the $e^+/e^-$ population through the IC and synchrotron processes. Considering the spatial diffusion of electrons, the dSphs are modeled as extended sources in the analysis of Fermi-LAT data according to the expected spatial intensity distribution of the gamma rays. We do not detect any significant HDM signal. By assuming a magnetic field strength of $B=1\,{\rm \mu G}$ and a diffusion coefficient of $D_0 = 3\times10^{28}\,{\rm cm^{2}s^{-1}}$ of the dSphs, we place limits on the annihilation cross section for HDM particles. Our results are weaker than the previous limits given by the VERITAS and IceCube observations of dSphs, but extend the existing limits to higher DM masses. As a complement, we also search for the prompt $\gamma$-rays produced by DM annihilation and give limits on the cross section in the 10-$10^5$ GeV mass range. Consequently, in this paper we obtain the upper limits on the DM annihilation cross section for a very wide mass range from 10 GeV to 100 PeV in a unified framework of the Fermi-LAT data analysis.