Gamma-ray burst polarization reduction induced by the Lorentz invariance violation

It has been observed that photons in the prompt emission of some gamma-ray bursts (GRBs) are highly polarized. The high polarization is used by some authors to give a strict constraint on the Lorentz invariance violation (LIV). If the Lorentz invariance is broken, the polarization vector of a photon may rotate during its propagation. The rotation angle of polarization vector depends on both the photon energy and the distance of source. It is believed that if high polarization is observed, then the relative rotation angle (denoted by alpha) of polarization vector of the highest energy photon with respect to that of the lowest energy photon should be no more than pi/2. Otherwise, the net polarization will be severely suppressed, thus could not be as high as what was actually observed. In this paper, we will give a detailed calculation on the evolution of GRB polarization arising from LIV effect duration the propagation. It is shown that the polarization degree rapidly decrease as alpha increases, and reaches a local minimum at alpha approximate to pi, then increases until alpha approximate to 3 pi/2, after that decreases again until alpha approximate to 2 pi, etc. The polarization degree as a function of alpha oscillates with a quasi-period T a approximate to pi, while the oscillating amplitude gradually decreases to zero. Moreover, we find that a considerable amount (more than 60 per cent of the initial polarization) of polarization degree can be conserved when alpha approximate to pi/2. The polarization observation in a higher and wider energy band, a softer photon spectrum, and a higher redshift GRB is favourable in order to tightly constrain LIV effect.