B1957+20 is a millisecond pulsar located in a black-widow-type compact binary system with a low-mass stellar companion. The interaction of the pulsar wind with the companion star wind and/or the interstellar plasma is expected to create plausible conditions for acceleration of electrons to TeV energies and subsequent production of very high-energy γ-rays in the inverse Compton process. We performed extensive observations with the Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC) telescopes of B1957+20. We interpret results in the framework of a few different models, namely emission from the vicinity of the millisecond pulsar, the interaction of the pulsar and stellar companion wind region or bow shock nebula. No significant steady very high-energy γ-ray emission was found. We derived a 95 per cent confidence level upper limit of 3.0 × 10−12 cm−2 s−1 on the average γ-ray emission from the binary system above 200 GeV. The upper limits obtained with the MAGIC constrain, for the first time, different models of the high-energy emission in B1957+20. In particular, in the inner mixed wind nebula model with mono-energetic injection of electrons, the acceleration efficiency of electrons is constrained to be below ∼2–10 per cent of the pulsar spin-down power. For the pulsar emission, the obtained upper limits for each emission peak are well above the exponential cut-off fits to the Fermi-LAT data, extrapolated to energies above 50 GeV. The MAGIC upper limits can rule out a simple power-law tail extension through the sub-TeV energy range for the main peak seen at radio frequencies.
