Searches for Anisotropic Lorentz-Invariance Violation in the Photon Sector
aa r X i v : . [ a s t r o - ph . H E ] A ug Proceedings of the Seventh Meeting on CPT and Lorentz Symmetry (CPT’16), Indiana University, Bloomington, June 20-24, 2016 Searches for Anisotropic Lorentz-Invariance Violation in thePhoton Sector
F. Kislat and H. Krawczynski
Department of Physics and McDonnell Center for the Space SciencesWashington University in St. Louis, St. Louis, MO 63130, USA
Lorentz invariance, the fundamental symmetry of Einstein’s theory of SpecialRelativity, has been established and tested by many classical and modern ex-periments. However, many theories that unify the Standard Model of particlephysics and General Relativity predict a violation of Lorentz invariance at thePlanck scale. While this energy range cannot be reached by current exper-iments, minute deviations from Lorentz symmetry may be present at lowerenergies. Astrophysical experiments are very suitable to search for these devi-ations, since their effects accumulate as photons travel across large distances.In this paper, we describe astrophysical methods that we used to constrain thephoton dispersion and vacuum birefringence.
1. Introduction
The Standard-Model Extension (SME) is an effective field theory approachto describe effects of a more fundamental theory beyond the StandardModel by introducing additional terms to the Standard-Model lagrangian.In the photon sector, these additional terms result in the following disper-sion relation: E ( p ) ≃ (cid:18) − ς ± q(cid:0) ς (cid:1) + (cid:0) ς (cid:1) + (cid:0) ς (cid:1) (cid:19) p, (1)with ς = X djm p d − Y jm ( θ k , ϕ k ) c ( d )( I ) jm , (2) ς ± = ς ± ς = X djm p d − ∓ Y jm ( θ k , ϕ k ) (cid:16) k ( d )( E ) jm ∓ ik ( d )( B ) jm (cid:17) , (3) ς = X djm p d − Y jm ( θ k , ϕ k ) k ( d )( V ) jm . (4)Hence, ς results in an energy and direction-dependent photon dispersion,and ς ± and ς additionally introduce a polarization dependence and thus roceedings of the Seventh Meeting on CPT and Lorentz Symmetry (CPT’16), Indiana University, Bloomington, June 20-24, 2016 vacuum birefringence. Furthermore, Eqs. (2)-(4) imply a direction depen-dence of photon propagation. The coefficients c ( d )( I ) jm , k ( d )( E ) jm , and k ( d )( B ) jm are nonzero only for even d , while k ( d )( V ) jm are nonzero only for odd d , where d is the mass dimension of the corresponding operator.Astrophysical tests are highly sensitive even to tiny values of these co-efficients since their effects accumulate as photons travel across large dis-tances. Photon dispersion is tested by measuring arrival times of photonsfrom short transient events such as gamma-ray bursts at different wave-lengths; see, e.g., Ref. 2. Vacuum birefringence induced by ς ± and ς leadsto an energy-dependent rotation of the polarization vector, which can beobserved directly in spectropolarimetric measurements.Due to the anisotropic nature of Lorentz-invariance violation (LIV) inthe SME, searches utilizing a single source can only test a linear combi-nation of coefficients. For example, at d = 5 there are 16 real coefficientsdetermining the complex k (5)( V ) jm , and at d = 6 there are 25 real coefficientsof the c (6)( I ) jm . In this paper we present a search for anisotropic LIV usingoptical polarization measurements of active galactic nuclei (AGNs), and asearch for anisotropic nonbirefringent LIV using gamma-ray time-of-flightmeasurements of AGN flares. In both searches, we observe multiple as-trophysical sources and then constrain the coefficients k (5)( V ) jm and c (6)( I ) jm individually using a spherical decomposition of the results. Here we out-line the methods used, while the actual limits on the SME coefficients willbe published elsewhere. X-ray polarization measurements of gamma-raybursts have already been used to constrain the d = 5 coefficients in previ-ous studies. However, the statistical and sytematic errors on the reportedX-ray polarization properties are very large and the detections are still notfirmly established.
2. Optical polarimetry
The polarization angles of two photons observed at energies E and E emitted at redshift z k which initially have the same polarization angle willdiffer by ∆ ψ = ( E − E ) L (5) z k X j =0 ... m = − j...j Y jm ( θ k , ϕ k ) k (5)( V ) jm ≡ ( E − E ) ζ (5) k , (5)where we introduced the parameter ζ (5) k , assuming that d = 5 terms arethe dominant correction to the Standard-Model lagrangian. We measured roceedings of the Seventh Meeting on CPT and Lorentz Symmetry (CPT’16), Indiana University, Bloomington, June 20-24, 2016 ∆ ψ using multiple spectropolarimetric observations of 27 different AGNsin the northern hemisphere. Using a likelihood ratio test, we compared theobservations of distant ( z > .
6) to “nearby” ( z < .
4) sources in order toset upper limits on the LIV-induced rotation parameter ζ (5) k .Furthermore, we used spectrally integrated optical polarization mea-surements of 36 southern-hemisphere AGNs. The rotation of the polariza-tion angle leads to a partial cancellation of the net polarization. For eachobservation, we found the largest value of ζ (5) k that was still in agreementwith the observed polarization fraction, as an upper limit on a possible LIV.Both the spectropolarimetric and the spectrally integrated polarizationmeasurements resulted in strong upper limits on LIV in the photon sectorfor each source.
3. Gamma-ray time-of-flight measurements
While most coefficients of the SME are best constrained using polariza-tion measurements, the coefficients c ( d )( I ) jm with even d are nonbirefringent.They can only be constrained using time-of-flight measurements. We usedmeasurements of 500 MeV to 300 GeV gamma-ray lightcurves of 24 AGNsto derive direction-dependent constraints on the speed of light in vacuum. Using the
DisCan method, we found constraints on the quadratic energydependence of the speed of light introduced by the d = 6 SME coefficients.No significant energy dependence of the light-travel time was found,and we set upper limits γ (6) k on the LIV coefficients. They are not strongenough to constrain LIV at the Planck scale. However, they represent thefirst complete set of constraints in the d = 6 photon sector.
4. Anisotropic Lorentz-invariance violation
When constraining the rotation of the polarization direction or the energy-dependence of the photon velocity from an astrophysical source k , a limit γ k is placed on a linear combination of SME coefficients: (cid:12)(cid:12)(cid:12) X j =0 ... m = − j...j Y jm ( θ k , ϕ k ) k (5)( V ) jm (cid:12)(cid:12)(cid:12) < γ (5) k or (cid:12)(cid:12)(cid:12) X j =0 ... m = − j...j Y jm ( θ k , ϕ k ) c (6)( I ) jm (cid:12)(cid:12)(cid:12) < γ (6) k . (6)When observing N sources, we can rewrite these inequalities in matrix formin terms of the real parameters comprising the SME coefficients: H • v < γ , (7) roceedings of the Seventh Meeting on CPT and Lorentz Symmetry (CPT’16), Indiana University, Bloomington, June 20-24, 2016 where v is a vector of the components of the k (5)( V ) jm or c (6)( I ) jm , γ is a vectorof the values γ ( d ) k , and H is the M × N coefficient matrix (with M = 16 for d = 5, and M = 25 in the d = 6 case). We find limits on the componentsof v using Monte Carlo integration: we sampled 10 random vectors γ by drawing each components from a normal distribution with a standarddeviation chosen such that the values satisfy the confidence level of thelimits γ ( d ) k . We then solve the equality corresponding to Eq. (7): v = ( H T H ) − H T γ . (8)Each solution marks a point in the space of the SME coefficients. In thisway, we build the distribution of SME coefficients and find their 95% upperand lower bounds. The actual coefficient constraints will be published inRefs. 3 and 4.
5. Summary
In order to constrain the c (6)( I ) jm at the Planck scale, time-of-flight mea-surements of photons with energies of the order of 100 PeV would be nec-essary, which is currently not possible. On the other hand, it may bepossible with future gamma-ray instruments to measure gamma-ray po-larization at 100 MeV, which would allow us to constrain the coefficients k (6)( E ) jm and k (6)( B ) jm . The next step, however, will be to constrain the bire-fringent coefficients of mass dimension d = 4, which currently are not fullyconstrained. References