1. M. Giorgini University of Bologna, Italy, and INFN Limits on Lorentz invariance violation in atmospheric neutrino oscillations using MACRO data From Colliders to Cosmic Rays 7 – 13 September 2005, Prague, Czech Republic

2. Outline Mass-induced oscillations : MACRO atmospheric results Violation of Lorentz Invariance (VLI) VLI-induced oscillations Mixed oscillation scenario : MACRO results Conclusions

3. Mass-induced oscillations  23 ≡  m m m 3 2 es

4. Mass-induced atmospheric oscillations Strong evidence for mass-induced oscillations given by MACRO, SK, Soudan 2 : Deficit of muon events with respect to the predictions Distortion of the zenith distribution Energy spectrum First dip in the L/E distribution  ↔  oscillations favoured (> 99% C.L.) with respect to:  ↔ sterile (MACRO Coll., Phys. Lett. B517 (2001) 59 ; SK Coll., Phys. Rev. Lett. 85 (2000) 3999)  ↔ e (SK Coll., Phys. Rev. Lett. 93 (2004) 101801) decay, decoherence, CPT violation,… (A. Habig, 28 th ICRC, Japan, 2003) Violation of Lorentz Invariance (VLI) (Phys. Rev. D60 (1999) 053006 ; Phys. Rev. D70 (2004) 033010 ; hep-ph/0407087)

5. Mass-induced oscillations : MACRO results Category  E  Data MC no osc Upthr. 50 857 1169 IU 4.2 157 285 ID+UGS 3.5 262 375

6.  Upthroughgoing  (857 events) Absolute flux : new MC codes have problems with the new cosmic ray fit Zenith angle distribution : shape known within 5%

7. L/E distribution From the shape of the muon zenith distribution From the measurement of the muon energy using the  Multiple Coulomb Scattering (PLB566 (2003) 35). Upthr.  data (~300 events) IU  data 12% point-to-point syst. error MC predictions for    oscillations with the best MACRO parameters

8. Zenith distribution Final results (Eur. Phys. J. C36 (2004) 357) E estimate by MCS IU, ID and UGS  R 1 = N(cos  -0.4) R 2 = N(low E ) / N(high E ) R 3 = N(ID+UGS) / N(IU) { H.E. NO OSCILLATION HYPOTHESIS RULED OUT BY ~ 5  L.E. Adding the absolute flux information (Bartol96 correct within 17%) NO OSCILLATION HYPOTHESIS RULED OUT BY ~ 6  Only 3 ratios Best parameters for     m 2 = 2.3 10 -3 eV 2 ; sin 2 2  =1 3 ratios + 2 normalizations 90% C.L.

9. Violation of Lorentz Invariance (VLI) If VLI is introduced, particles could have different Maximum Attainable Velocities (MAVs) v i (p=∞) ≠ c 3 2

10. Mixed oscillation scenario

11. Mixed oscillations scenario While in the “pure” cases probabilities do not depend on the sign of  v,  m 2 and mixing angles, in the mixed scenario relative signs are important Domain of variability :  m 2 ≥ 0  v ≥ 0  0 ≤  m ≤  /4  -  /4 ≤  v ≤  /4 Oscillations induced by the Violation of the Equivalence Principle (VEP) may be treated similarly to VLI-induced oscillations Due to the L and E dependence, VLI effects are emphasized for large L and large E

12. Survival probability vs E (L=10000 km,  m 2 =2.3. 10 -3 eV 2,  m =  /4 +0.3 -0.3 +0.7 -0.7 +1 Main effect are mass-induced oscillations VLI is considered as a subdominant effect, at least for the accessible energies

13. Mixed scenario: MACRO data analyses We used the data with E reconstructed by Multiple Coulomb Scattering (~300 events). Phys. Lett. B566 (2003) 35 Two different techniques were used to estimate the upper limits of possible exotic contributions to atmospheric neutrino oscillations Conventional Feldman-Cousins analysis based on the χ 2 criterion Analysis based on the Maximum Likelihood function

14. χ 2 analysis η= 0 high > Cuts optimized with MC

15. Results (Phys. Lett. B615 (2005) 14) Neutrino flux used in MC: Honda et al., Phys Rev. D70 (2004) 043008

16. Likelihood analysis Minimization of the function: F = -2 ∑ ln f(E i,L i ;  m 2,  v,  m 23,  v 23 ) i f(x:a) = K × p MC × p(  →  ) Event by event analysis to exploit the full information

17. Analysis procedure We allowed the oscillation mass parameters to vary along the 90% C.L. contour of the final MACRO solution without normalization. For each point of the contour we performed maximum likelihood fits for the VLI parameters We used the events (106) with the most accurate energy reconstruction 25 GeV ≤ E ≤ 75 GeV

18. Results The 90% C.L. limits obtained from the convolution of the local 90% C.L. upper/lower limits.

19. Conclusions We re-analyzed the L and E distributions of MACRO neutrino data to include the possibility of exotic effects (Violation of Lorentz Invariance) Two different analyses were performed on 2 different data subsamples, both yielding |  v| upper limits of the order of 10 -25 Very large volume neutrino experiments could tell more in the next years…