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A long-term optical - X-ray correlation in 4U 1957+11

[abridged] Three years of optical monitoring of the low-mass X-ray binary (LMXB) 4U 1957+11 is presented. The source was observed in V, R and i-bands using the Faulkes Telescopes North and South. The light curve is dominated by long-term variations which are correlated (at the > 3 sigma level) with the RXTE ASM soft X-ray flux. The variations span one magnitude in all three filters. We find no evidence for periodicities in our light curves, contrary to a previous short-timescale optical study in which the flux varied on a 9.3-hour sinusoidal period by a smaller amplitude. The optical spectral energy distribution is blue and typical of LMXBs in outburst, as is the power law index of the correlation beta = 0.5, where F_{nu,OPT} propto F_X^beta. The discrete cross-correlation function reveals a peak at an X-ray lag of 2 - 14 days, which could be the viscous timescale. However, adopting the least squares method the strongest correlation is at a lag of 0 +/- 4 days, consistent with X-ray reprocessing on the surface of the disc. We therefore constrain the optical lag behind X-ray to be between -14 and +4 days. In addition, we use the optical - X-ray luminosity diagram for LMXBs as a diagnostic tool to constrain the nature of the compact object in 4U 1957+11, since black hole (BH) and neutron star (NS) sources reside in different regions of this diagram. If the system contains a BH (as is the currently favoured hypothesis), its distance must exceed ~ 20 kpc for the optical and X-ray luminosities to be consistent with other soft state BH systems. For distances < 20 kpc, the data lie in a region of the diagram populated only by NS sources. 4U 1957+11 is unique: it is either the only BH LMXB to exist in an apparent persistent soft state, or it is a NS LMXB which behaves like a black hole.