The Billerica and Bedford Railroad
The Billerica and Bedford was an odd little railroad. Oddly enough, it was first conceived to serve a community that did not want a railroad to begin with. But once that town, Billerica, saw the prosperity and success that the Middlesex Central Railroad brought forth to the town of Bedford to its south (see the The Lexington Branch), Billerica wanted to jump on board, so to speak, but with minimal finances and resources up front. In 1877, the answer came the form of another oddity: a narrow 2-foot gauge passenger railroad that ran north from Bedford (at a junction with the Middlesex Central), through Billerica, to a connection with the Boston and Lowell Railroad to the north.
Despite the initial fanfare and hope, the B&B was soon financially ailing and by 1879, the line was taken up. (The infrastructure, including locomotives and rolling stock, were moved to Farmington, Maine for use on the Big Sandy Railroad.) The Boston and Lowell Railroad, seeing further opportunity to capitalize on passenger ridership, built a standard-gauge line along the former B&B, which continued to see passenger and freight service until 1962, when it was abandoned.
Little remains of the former B&B today. The passenger station at Bedford, which once served both the Middlesex Central and the B&B, still stands today and serves as a reminder of Bedford's railroad heritage. Other structures of the B&B at Bedford included an engine house, which is now a bakery.
I live in Billerica. I think it's worth mentioning that the rail line is actually still maintained up till just past Floyd street (I physically saw them doing the maintenance myself spring of 2012. There was actually a bent rail that was replaced on one of the lines), and the former railyard, where what look like engine repair shops are now abandoned, does see some activity. There is a lumber company, fedex, and a one or two other small things still in service.
Your story fails to mention that the Boston & Lowell's standard line deviated to the west of Billerica Center and did not follow the narrow line's more steep graded ROW to the east. Your map also does not show this original narrow ROW going east of Billerica Center.