Testing the millisecond pulsar scenario of the Galactic center gamma-ray excess with very high energy gamma-rays

The recent analyses of the Fermi Large Area Telescope data show an extended GeV $\gamma$-ray excess on top of the expected diffuse background in the Galactic center region, which can be explained with annihilating dark matter or a population of millisecond pulsars (MSPs). We propose to observe the very high energy $\gamma$-rays for distinguishing the MSP scenario from the dark matter scenario. The GeV $\gamma$-ray MSPs should release most energy to the relativistic $e^{\pm}$ wind, which will diffuse in the Galaxy and radiate TeV $\gamma$-rays through inverse Compton scattering and bremsstrahlung processes. By calculating the spectrum and spatial distribution, we show that such emission is detectable with the next generation very high energy $\gamma$-ray observatory, the Cherenkov Telescope Array (CTA), under reasonable model parameters. It is essential to search for the multi-wavelength counterparts to the GeV $\gamma$-ray excess for solving this mystery in the high energy universe.