Violation of Lorentz Invariance in High-Energy γ Rays

Transcription

1 Violation of Lorentz Invariance in High-Energy γ Rays M. E. Peskin July, 2005

2 There is a huge literature on Lorentz-Invarance violation. I am no expert on the subject, but hopefully I can give you some insight into a few parts of this literature relevant to GLAST physics. Some useful general references on Lorentz violation are: T. Jacobsen, S. Liberati, D. Mattingly, hep-ph/ , astro-ph/ G. Amelino-Camelia, C. Lämmerzahl, A. Macias, H. Müller, gr-qc/ R. Bluhm, hep-ph/ In this talk, I will mainly be concerned with the dynamics of photons.

3 The first systematic approach to Lorentz violation was introduced by Kostelecky: Write the most general effective Lagrangian with the gauge invariances of the Standard Model by allowing small violation of Lorentz invariance: L = 1 4 (F mn) kmnpq F mn F pq 1 2M hmnpqr F mn p F qr + The dimension-5 term was discussed by Myers and Pospelov, hep-ph/ Assume rotational invariance (isotropy of space); then should be constructed from a single vector k, h, u m = (1, 0, 0, 0) in the preferred frame

4 This gives L = 1 4 (F mn) 2 κ 2 (F 0n) 2 ξ 2M (F 0n 0 F 0n ) + κ simply shifts the speed of light. If we rescale time so that the speed of light is c = 1, κ is scaled out. We are left with one parameter. ξ However, once we have set c=1, we have fixed the scale of time. Electrons, muons, etc. may have different asymptotic speeds at high energy. (More about this later.)

5 ξ The term violates CPT. This is permitted, because the proof of the CPT theorem requires Lorentz invariance. CPT violation allows the photon helicities to have different dispersion relations: ω ± = k ± (R,L polarizations) Later, we will find a theory that gives another modification of the photon dispersion relation ω ± = k + ξ 2M k2 α 2M k2 ±1 These two effects can be searched for with high-energy gammas from distant sources. Amelino-Camelia et al, astro-ph/ (Nature, 1998)

6 In the rest of this talk, I will define Then γ ξ, α M = m Pl = GeV c ± = dω dk = 1 + (α ± ξ) k m Pl by choosing For -ray bursts at 1 Gpc, comparing 1 GeV and 10 GeV s: γ a time delay of 1 msec a polarization plane rotation by π/2 α = ξ = Already, non-observation of polarization rotation of synchrotron radiation from radio galaxies (Carroll, Field, Jackiw, PRD 41, 1231 (1990) gives ξ < with GHz photons. γ Observing polarization that is not averaged out already puts a strong constraint.

7 Return to the question: does the speed of light depend on the type of particle? Coleman and Glashow, hep-ph/ define, for particles a and b : δ ab = c 2 a c 2 b δ γe < 10 7 SLAC timing to 1 psec δ γe < no γ e + e in vacuum at 50 TeV δ pγ < no Cerenkov radiation from cosmic ray protons in vacuum However, even smaller values of are interesting, in relation to the the problem of cosmic rays above the GZK cutoff. δ

8 The problem posed by GZK is p + γ CMB + p + π 0 E = p + m2 2p E p = p + m2 p 2p so, conventionally, this reaction is allowed for With variable c: E a = c a p + m2 a 2p then the condition above becomes: and the reaction is forbidden for 2ω = m2 m2 p 2p 2ω = 1 2 δ p + m2 m2 p 2p δ p

9 Actually, there are many ways to monkey with dispersion relations to achieve this effect. For example, with we find, for α < 0 ω = p +, the criterion α p 2 + m2 2m Pl 2p 2ω = m2 m2 p 2p + α ωp m Pl For CMB photons, so this requires ω 0 = m Pl α 10 7, or, better, m Pl M = GeV Gonzales-Mestres (physics/ ) and Amelino-Camelia and Piran ( ) proposed similar scenarios for evading the GZK cutoff.

10 This motivates looking more deeply at the origin of the term. α

11 I would like to begin from very deep considerations of how Lorentz invariance might be generalized. Look first at the algebra of 3-d rotations, SU(2): [J 3, J ± ] = ±J ± [J +, J ] = 2J 3 We can modify these commutation relations to [J 3, J ± ] = ±J ± [J +, J ] = sinh 2βJ 3 sinh β This is called a quantum deformation of SU(2). The requirement for a quantum deformation is that the resulting commutation relations satisfy the Jacobi identity. In the simple case of SU(2), we can have any function on the right-hand side of the second relation. f(j 3 )

12 The deformed SU(2) algebra gives a new definition of rotations. In this deformed system, rotations do not act linearly on vectors, so there is apparently some distortion along the ˆ3 axis. Nevertheless, there is no preferred orientation. The invariant product J 2 is given by J 2 = J J + + sinh2 β(j ) sinh2 1 2 β sinh 2 β

13 With this idea, we can look for quantum deformations of the Poincaré group. Here is an example, discussed by Magueijo and Smolin, gr-qc/ : Rotations act as usual on boosts and translations: [J i, J j ] = iɛ ijk J k [J i, K j ] = iɛ ijk K k [J i, P j ] = iɛ ijk P k [K i, K j ] = iɛ ijk J k [K i, P j ] = i (δ ij P 0 1κ ) P i P j and but [K i, P 0 ] = i (1 P 0 κ ) P i taking κ gives the standard Poincaré algebra. Other deformed versions of the Poincaré algebra are reviewed in Kowalski-Glikman, hep-th/

14 The generators of the deformed algebra commute with M 2 = (P 0 ) 2 (P i ) 2 ( ) 1 P 0 2 κ M 2 = 0 gives the dispersion relation of light. This has the form of Lorentz violation with P 0 κ is a maximum or limiting energy. Every frame sees the same limit, and every observed agrees that all particles have P 0 < κ α m Pl = 1 κ Indeed, there is no preferred frame. The price of having a invariant maximum energy is that Lorentz transformations must act nonlinearly on P. This structure is call Doubly Special Relativity.

15 People who canonically quantize gravity like the idea that a modification of space-time structure can give an invariant maximum energy of the order of. Smolin (hep-th/ ) has gone further, to argue that in a broad class of quantum gravity Hamiltonians that includes loop quantum gravity, one necessarily finds Doubly Special Relativity rather than conventional local Lorentz invariance. The value of α is not predicted, but one expects. Violation of the GZK cutoff is not predicted, since it is not exactly that is conserved in particle collisions. P µ Smolin writes: α 1 m Pl The predictions of quantum gravity are falsifiable by the upcoming AUGER and GLAST experiments.

16 Superstring theory is a different kind of candidate theory of quantum gravity. Superstring theory has a different set of quantization rules, and these are exactly Lorentz invariant in the conventional sense. Lorentz violation is possible in string theory if certain tensor fields acquire vacuum expectation values. But, for the typical compactifications of string theory used to build unified models of particle physics, the theory predicts α = 0.

17 So, GLAST could potentially discriminate between the broad classes of models proposed for quantum gravity. Can it meet this challenge?