Compton Scattering in the Presence of Lorentz and CPT Violation

We examine the process of Compton scattering, in the presence of a Lorentzand CPTviolating modification to the structure of the electron. We calculate the complete tree-level contribution to the cross section; our result is valid to all orders in the Lorentz-violating parameter. We find a cross section that differs qualitatively from the Klein-Nishina result at small frequencies, and we also encounter a previously undescribed complication that will arise in the calculation of many Lorentz-violation cross sections: The Lorentz violation breaks the spin degeneracy of the external states, so we cannot use a closure relation to calculate the unpolarized cross section. 1 [email protected] Recently, there has been a great deal of interest in the possibility of there existing small CPTand Lorentz-violating corrections to the standard model [1, 2, 3, 4]. Such small corrections might arise from larger violations of Lorentz symmetry occurring at the Planck scale. The most general possible Lorentz-violating effective field theory has been described in detail, and its renormalizability has been studied. These results open the way for a wide variety of experiments that could test for the existence of Lorentz violation. There are already many experimental constraints on Lorentz-violating corrections to the standard model. The tests have included studies of matter-antimatter asymmetries for trapped charged particles [5, 6, 7, 8] and bound state systems [9, 10], frequency standard comparisons [11, 12, 13], measurements of neutral meson oscillations [14, 15, 16], polarization measurements on the light from distant galaxies [17, 18], and many others. However, although there have been a number of kinematical analyses of the astrophysical consequences of Lorentz violation in particle scattering [19, 20, 21, 22, 23, 24], there has as yet been very little investigation into the possible effects of Lorentz-violating dynamics in laboratory scattering experiments [25, 26, 27]. In this paper, we shall examine some of those effects. We shall examine the process of Compton scattering, in the presence of a particular Lorentzand CPT-violating modification of the electron sector. The study of Compton scattering has historically been very important to the development of quantum mechanics and quantum field theory [28, 29, 30] and in the future might provide an important test of Lorentz violation. The calculation of scattering cross sections in a Lorentz-violating theory involves a number of subtleties that are not present in the standard, Lorentz-invariant case. Different reference frames are no longer necessarily equivalent, and the correct definition of the particle flux becomes potentially ambiguous. However, with appropriate care, meaningful cross sections can be found, and a general theory for their calculation is given in [26]. Our analysis will also reveal an additional complication, not discussed in [26], that may arise in Lorentz-violating scattering processes. The various spin states of the scattered particles may have differing energies, and this can affect the velocity and phase space factors that appear in the cross section. As a result, it may become impossible to calculate an unpolarized cross section by the usual means. The Lagrange density for our theory is L = − 4 F Fμν + ψ̄(i6∂ −m− e6 A−6bγ5)ψ. (1) The action includes only the single Lorentz-violating coefficient b. This is the simplest perturbatively nontrivial form of Lorentz violation that can exist in the electron sector. The domain of validity of this Lagrange density extends all the way up the Planck scale [3]. Considering a theory with only a b term would not be reasonable for calculations beyond tree level; other Lorentz-violating terms would be radiatively generated at oneloop order [4]. We shall therefore consider only tree-level effects. However, although we 1 shall only be working to leading order in the electromagnetic coupling e, our results will be correct to all orders in b. In general, the spacetime direction of b is arbitrary. However, we shall choose b to be purely timelike, b = (B,~0 ), in the laboratory frame. It is a common practice to suppose that any Lorentz-violating coefficients have vanishing spatial components; this practice arises from the observation that the universe shows a very high degree of isotropy in the rest frame of the cosmic microwave background. In this case, considering only a timelike b will also substantially simplify our b-exact analysis of the theory. Since our Lorentz-violating Lagrange density (1) involves no changes to the electrons’ kinetic term, and there are no additional time derivatives not present in the Lorentzinvariant theory, the electrons may be quantized without any changes to the spinor representation [3, 26]. The exact electron propagator may be read off directly from the Lagrange density; it is S(l) = i 6 l −m−6bγ5 . (2) We may rationalize this expression and obtain [31, 32] S(l) = i (6 l +m−6bγ5)(l −m2 − b + [6 l,6b ]γ5) (l −m2 − b) + 4[lb − (l · b)] . (3) This modification of the propagator represents one of the ways in which the presence of b will affect the theory. However, before we can investigate how the modified propagator S(l) affects the dynamics of the scattering, we must examine the effects of the Lorentz violation on the kinematics. The coefficient b will affect the structure of the theory’s asymptotic states. The photon states are, of course, unaffected, but the incoming and outgoing spinors will be significantly modified. We must solve the free momentum-space Dirac equation, with the b term included, to determine the propagation modes of the electrons. Since the matrix γ5 features prominently in the theory, it is natural to use the Weyl chiral representation for the Dirac matrices: γ = [