1. Tests of Lorentz Invariance using Atmospheric Neutrinos and AMANDA-II John Kelley for the IceCube Collaboration University of Wisconsin, Madison Workshop on Exotic Physics with Neutrino Telescopes Uppsala, Sweden September 21, 2006

2. Atmospheric Neutrinos as a New Physics Probe Mass-induced oscillations on solid ground — but subdominant effects at high energies? For E > 100 GeV and m 10 11 Oscillations are a sensitive quantum-mechanical probe: small differences in energy  large change in observable spectrum Eidelman et al.: “It would be surprising if further surprises were not in store…”

3. New Physics Effects Violation of Lorentz invariance (VLI) in string theory or loop quantum gravity* Violations of the equivalence principle (different gravitational coupling) † Interaction of particles with space- time foam  quantum decoherence of pure states ‡ * see e.g. Carroll et al., PRL 87 14 (2001), Colladay and Kosteleck ý, PRD 58 116002 (1998) † see e.g. Gasperini, PRD 39 3606 (1989) ‡ see e.g. Hawking, Commun. Math. Phys. 87 (1982), Ellis et al., Nucl. Phys. B241 (1984) c -  1 c -  2

4. VLI Phenomenology Modification of dispersion relation * : Different maximum attainable velocities c a (MAVs) for different particles:  E ~ (  c/c)E For neutrinos: MAV eigenstates not necessarily flavor or mass eigenstates  mixing  VLI oscillations * Glashow and Coleman, PRD 59 116008 (1999)

5. Conventional+VLI Oscillations For atmospheric, conventional oscillations turn off above ~50 GeV (L/E dependence) VLI oscillations turn on at high energy (n=1 above; L E dependence), depending on size of  c/c, and distort the zenith angle / energy spectrum González-García, Halzen, and Maltoni, hep-ph/0502223

6.  Survival Probability  c/c = 10 -27

7. AMANDA-II Data Sample 2000-2004 sky map Livetime: 1001 days 4282 events (up-going) No point sources or HE diffuse flux yet: clean atmospheric neutrino sample! Adding 2005 data: > 5000 events

8. Data Analysis detector MC detector MC Zenith angle: angular resolution ~2º But energy reconstruction more difficult

9. Energy-correlated Observable N ch (number of optical modules hit): stable observable, but acts more like an energy threshold Other methods exist: dE/dx estimates, neural networks…

10. Observable Space (MC)  c/c = 10 -25 No New Physics Differences in both shape (near vertical) and normalization

11. VLI: AMANDA-II Sensitivity Median Sensitivity  c/c (sin(2  ) = 1) 90%: 3.2  10 -27 95%: 3.6  10 -27 99%: 5.1  10 -27 2000-05 livetime simulated MACRO limit*: 2.5  10 -26 (90%) Super-K + K2K limit † : 2.0  10 -27 (90%) *Battistoni et al., hep-ex/0503015 † González-García & Maltoni, PRD 70 033010 (2004) MC test analysis in 1-D observable (N ch ) and parameter space (  c/c) (Feldman-Cousins likelihood ratio method)

12. Summary and Outlook Preliminary sensitivity to VLI effects competitive with existing limits Expect improvements as analysis technique is refined Other new physics effects (e.g. quantum decoherence) are also being tested!

13. Extra Slides

14. Quantum Decoherence Phenomenology Modify propagation through density matrix formalism: Solve DEs for neutrino system, get oscillation probability*: *for more details, please see Morgan et al., astro-ph/0412628 dissipative term

15. QD Parameters Various proposals for how parameters depend on energy: simplest preserves Lorentz invariance recoiling D-branes!

16.  Survival Probability (  model) a =  = 4  10 -32 (E 2 / 2)

17. Binned Likelihood Test Poisson probability Product over bins Test Statistic: LLH

18. Testing the Parameter Space Given a measurement, want to determine values of parameters {  i } that are allowed / excluded at some confidence level  c/c sin(2  ) allowed excluded

19. Feldman-Cousins Recipe For each point in parameter space {  i }, sample many times from parent Monte Carlo distribution (MC “experiments”) For each MC experiment, calculate likelihood ratio:  L = LLH at parent {  i } - minimum LLH at some {  i,best } For each point {  i }, find  L crit at which, say, 90% of the MC experiments have a lower  L (FC ordering principle) Once you have the data, compare  L data to  L crit at each point to determine exclusion region Primary advantage over  2 global scan technique: proper coverage Feldman & Cousins, PRD 57 7 (1998)

20. 1-D Examples all normalized to data sin(2  ) = 1

21. Systematic Errors Atmospheric production uncertainties Detector effects (OM sensitivity) Ice Properties Can be treated as nuisance parameters: minimize LLH with respect to them Or, can simulate as fluctuations in MC experiments Normalization is already included! (free parameter — could also constrain)

22. VLI: Sensitivity using N ch Median Sensitivity  c/c (sin(2  ) = 1) 90%: 3.2  10 -27 95%: 3.6  10 -27 99%: 5.1  10 -27 2000-05 livetime simulated allowed excluded

23. Decoherence Sensitivity (Using Nch,  model) Norm. constrained ±30% Normalization free

24. Decoherence Sensitivity Median Sensitivity  a,  (GeV -1 ) 90%: 3.7  10 -31 95%: 5.8  10 -31 99%: 1.6  10 -30 ANTARES (3 yr sens, 90%) * : 10 -44 GeV -1 * Morgan et al., astro-ph/0412618 ‡ Lisi, Marrone, and Montanino, PRL 85 6 (2000) SuperK limit (90%) ‡ : 0.9  10 -27 GeV -1 Almost 4 orders of magnitude improvement! (E 2 energy dependence)