By Meredith Goldstein, Globe Staff The story of how the London pop band Dirty Vegas moved to Massachusetts begins with “Days Go By,” the mega hit that made the group famous back in 2002. The song started as a guitar ballad that Dirty Vegas frontman Steve Smith wrote about an ex-girlfriend he couldn’t get out of his mind. When Smith played the song for his bandmates Ben Harris and Paul Harris (no relation), they liked it. But since they were a dance-music trio, they sped up the pace and produced “Days Go By” with electronic beats and distorted vocals. It worked. The track was huge on the London dance scene and caught the attention of the car company Mitsubishi, which asked the group if it could use the song in a commercial that would feature attractive hipsters dancing to the tune while riding down a city highway in an Eclipse. “They’re like, we have this car. We want every 25-year-old to go out and buy our car,” Smith said. “There was a young director, younger than us. We thought, this is the direction to go in.” Suddenly, thanks to the commercial and a video for the song on MTV and