Context. Debris disks are observed around 10 to 20% of FGK main-sequence stars as infrared excess emission. They are important signposts for the presence of colliding planetesimals and therefore provide important information about the evolution of planetary systems. Direct imaging of such disks reveals their geometric structure and constrains their dust-particle properties. Aims. We present observations of the known edge-on debris disk around HIP 79977 (HD 146897) taken with the ZIMPOL differential polarimeter of the SPHERE instrument. We measure the observed polarization signal and investigate the diagnostic potential of such data with model simulations. Methods. SPHERE-ZIMPOL polarimetric data of the 15 Myr-old F star HIP 79977 (Upper Sco, 123 pc) were taken in the Very Broad Band (VBB) filter (lambda(c) = 735 nm, Delta lambda = 290 nm) with a spatial resolution of about 25 mas. Imaging polarimetry efficiently suppresses the residual speckle noise from the AO system and provides a differential signal with relatively small systematic measuring uncertainties. We measure the polarization flux along and perpendicular to the disk spine of the highly inclined disk for projected separations between 0 : 200 (25 AU) and 1 : 600 (200 AU). We perform model calculations for the polarized flux of an optically thin debris disk which are used to determine or constrain the disk parameters of HIP 79977. Results. We measure a polarized flux contrast ratio for the disk of (F-pol) disk/F-* = (5 : 5 +/- 0 : 9) x 10(-4) in the VBB filter. The surface brightness of the polarized flux reaches a maximum of SBmax = 16.2 mag arcsec(-2) at a separation of 0 : 200 -0 : 500 along the disk spine with a maximum surface brightness contrast of 7 : 64 mag arcsec(-2). The polarized flux has a minimum near the star < 0 : 200 because no or only little polarization is produced by forward or backward scattering in the disk section lying in front of or behind the star. The width of the disk perpendicular to the spine shows a systematic increase in FWHM from 0 : 1 (12 AU) to 0 : 3 -0.5, when going from a separation of 0 : 2 to > 1. This can be explained by a radial blow-out of small grains. The data are modelled as a circular dust belt with a well defined disk inclination i = 85(+/- 1 : 5)degrees and a radius between r(0) = 60 and 90 AU. The radial density dependence is described by (r/r(0))alpha with a steep (positive) power law index alpha = 5 inside r(0) and a more shallow (negative) index alpha = -2 : 5 outside r(0). The scattering asymmetry factor lies between g = 0.2 and 0.6 (forward scattering) adopting a scattering-angle dependence for the fractional polarization such as that for Rayleigh scattering. Conclusions. Polarimetric imaging with SPHERE-ZIMPOL of the edge-on debris disk around HIP 79977 provides accurate profiles for the polarized flux. Our data are qualitatively very similar to the case of AU Mic and they confirm that edge-on debris disks have a polarization minimum at a position near the star and a maximum near the projected separation of the main debris belt. The comparison of the polarized flux contrast ratio (F-pol)(disk)/F* with the fractional infrared excess provides strong constraints on the scattering albedo of the dust.