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The recent analyses of the Fermi Large Area Telescope data show an extendedGeV $\gamma$-ray excess on top of the expected diffuse background in theGalactic center region, which can be explained with annihilating dark matter ora population of millisecond pulsars (MSPs). We propose to observe the very highenergy $\gamma$-rays for distinguishing the MSP scenario from the dark matterscenario. The GeV $\gamma$-ray MSPs should release most energy to therelativistic $e^{\pm}$ 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 suchemission is detectable with the next generation very high energy $\gamma$-rayobservatory, the Cherenkov Telescope Array (CTA), under reasonable modelparameters. It is essential to search for the multi-wavelength counterparts tothe GeV $\gamma$-ray excess for solving this mystery in the high energyuniverse.

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