1. 1 MACRO constraints on violation of Lorentz invariance M. Cozzi Bologna University - INFN Neutrino Oscillation Workshop Conca Specchiulla (Otranto) September 9-16, 2006

2. NOW 2006 M. Cozzi 2 Outline Violation of Lorentz Invariance (VLI) Test of VLI with neutrino oscillations MACRO results on mass-induced oscillations Search for a VLI contribution in neutrino oscillations Results and conclusions

3. NOW 2006 M. Cozzi 3 Violation of the Lorentz Invariance In general, when Violation of the Lorentz Invariance (VLI) perturbations are introduced in the Lagrangian, particles have different Maximum Attainable Velocities (MAVs), i.e. V i (p=∞)≠c Renewed interest in this field. Recent works on: VLI connected to the breakdown of GZK cutoff VLI from photon stability VLI from radioactive muon decay VLI from hadronic physics Here we consider only those violation of Lorentz Invariance conserving CPT

4. NOW 2006 M. Cozzi 4 Test of Lorentz invariance with neutrino oscillations The CPT-conserving Lorentz violations lead to neutrino oscillations even if neutrinos are massless However, observable neutrino oscillations may result from a combination of effects involving neutrino masses and VLI Given the very small neutrino mass ( eV), neutrinos are ultra relativistic particles Searches for neutrino oscillations can provide a sensitive test of Lorentz invariance

5. NOW 2006 M. Cozzi 5 “Pure” mass-induced neutrino oscillations In the 2 family approximation, we have 2 mass eigenstates and with masses m 2 and m 3 2 flavor eigenstates and The mixing between the 2 basis is described by the θ 23 angle: If the states are not degenerate (  m 2 ≡ m 2 2 - m 3 2 ≠ 0) and the mixing angle   ≠ 0, then the probability that a flavor “survives” after a distance L is: Note the L/E dependence

6. NOW 2006 M. Cozzi 6 “Pure” VLI-induced neutrino oscillations When VLI is considered, we introduce a new basis: the velocity basis: and (2 family approx) Velocity and flavor eigenstates are now connected by a new mixing angle: If neutrinos have different MAVs (  v ≡ v 2 - v 3 ≠ 0) and the mixing angle  v  ≡  v ≠ 0, then the survival oscillation probability has the form: Note the L·E dependence

7. NOW 2006 M. Cozzi 7 Mixed scenario When both mass-induced and VLI-induced oscillations are simultaneously considered: where 2  =atan(a 1 /a 2 )  =√a 1 2 + a 2 2 oscillation “strength” oscillation “length”  = generic phase connecting mass and velocity eigenstates

8. NOW 2006 M. Cozzi 8 Notes: In the “pure” cases, probabilities do not depend on the sign of  v,  m 2 and mixing angles while in the “mixed” case relative signs are important. Domain of variability:  m 2 ≥ 0 0 ≤  m ≤  /2  v ≥ 0  /4 ≤  v ≤  /4 Formally, VLI-induced oscillations are equivalent to oscillations induced by Violation of the Equivalence Principle (VEP) after the substitution:  v/2  ↔ |  |  where  is the gravitational potential and  is the difference of the neutrino coupling to the gravitational field. Due to the different (L,E) behavior, VLI effects are emphasized for large L and large E (  large L·E)

9. NOW 2006 M. Cozzi 9 Energy dependence for P(ν μ  ν μ ) assuming L=10000 km,  m 2 = 0.0023 eV 2 and  m =  /4 Black line: no VLI Mixed scenario: VLI with sin2θ v >0 VLI with sin2θ v <0

10. NOW 2006 M. Cozzi 10 MACRO results on mass-induced neutrino oscillations

11. NOW 2006 M. Cozzi 11  7 Rock absorbers ~ 25 X o 35/yr Internal Downgoing (ID) + 35/yr Upgoing Stopping (UGS) 180/yr Up- throughgoing 3 horizontal layers ot Liquid scintillators 14 horizontal planes of limited streamer tubes     50 4.2 3.5 Topologies of -induced events Topologies of -induced events    50/yr Internal Upgoing (IU)

12. NOW 2006 M. Cozzi 12 Neutrino events detected by MACRO Data samplesNo-osc Expected (MC) TopologiesMeasured Up Throughgoing 8571169 Internal Up 157285 Int. Down + Up stop 262375

13. NOW 2006 M. Cozzi 13 Upthroughgoing muons Absolute flux Even if new MCs are strongly improved, there are still problems connected with CR fit → large sys. err. Zenith angle deformation Excellent resolution (2% for HE) Very powerful observable (shape known to within 5%) Energy spectrum deformation PLB 566 (2003) 35 Energy estimate through MCS in the rock absorber of the detector (sub-sample of upthroughgoing events) PLB 566 (2003) 35 Extremely powerful, but poorer shape knowledge (12% error point-to-point) Used for this analysis

14. NOW 2006 M. Cozzi 14 L/E distribution DATA/MC(no oscillation) as a function of reconstructed L/E: Internal Upgoing 300 Throughgoing events

15. NOW 2006 M. Cozzi 15 The analysis was based on ratios (reduced systematic errors at few % level): Eur. Phys. J. C36 (2004) 357 Angular distribution R 1 = N(cos  -0.4) Energy spectrum R 2 = N(low E )/N(high E ) Low energy R 3 = N(ID+UGS)/N(IU) Null hypothesis ruled out by P NH ~ 5  If the absolute flux information is added (assuming Bartol96 correct within 17%): P NH ~ 6  Best fit parameters for  ↔  oscillations (global fit of all MACRO neutrino data):  m 2 =0.0023 eV 2 sin 2 2  m =1 Final MACRO results

16. NOW 2006 M. Cozzi 16 90% CL allowed region   Based on the “shapes” of the distributions (14 bins) Including normalization (Bartol flux with 17% sys. err.)

17. NOW 2006 M. Cozzi 17 Search for a VLI contribution using MACRO data Assuming standard mass-induced neutrino oscillations as the leading mechanism for flavor transitions and VLI as a subdominant effect.

18. NOW 2006 M. Cozzi 18 A subsample of 300 upthroughgoing muons (with energy estimated via MCS) are particularly favorable: ≈ 50 GeV (as they are uptroughgoing) ≈ 10000 km (due to analysis cuts) Golden events for VLI studies!  v= 2 x 10 -25  v =  /4 Good sensitivity expected from the relative abundances of low and high energy events

19. NOW 2006 M. Cozzi 19 Divide the MCS sample (300 events) in two sub-samples: Low energy sample: E rec < 28 GeV → N low = 44 evts High energy sample: E rec > 142 GeV → N high = 35 evts Define the statistics: and (in the first step) fix mass-induced oscillation parameters  m 2 =0.0023 eV 2 and sin 2 2  m =1 (MACRO values) and assume e i  real assume 16% systematic error on the ratio N low /N high (mainly due to the spectrum slope of primary cosmic rays) Scan the (  v,  v ) plane and compute χ 2 in each point (Feldman & Cousins prescription) Analysis strategy Optimized with MC

20. NOW 2006 M. Cozzi 20 Results of the analysis - I Original cuts Optimized cuts χ 2 not improved in any point of the (  v,  v ) plane: 90% C.L. limits Neutrino flux used in MC: “new Honda” - PRD70 (2004) 043008

21. NOW 2006 M. Cozzi 21 Results of the analysis - II Changing  m 2 around the best-fit point with  m 2 ± 30%, the limit moves up/down by at most a factor 2 Allowing  m 2 to vary inside ±30%,  m ± 20% and any value for the phase  and marginalizing in  v (-π/4≤  v ≤ π/4 ): |  v|< 3 x 10 -25  |  |< 1.5 x 10 -25 VLI VEP

22. NOW 2006 M. Cozzi 22 Results of the analysis - III A different and complementary analysis has been performed: Select the central region of the energy spectrum 25 GeV < E rec < 75 GeV (106 evts) Negative log-likelihood function was built event by event and fitted to the data. Mass-induced oscillation parameters inside the MACRO 90% C.L. region; VLI parameters free in the whole plane. Average  v < 10 -25, slowly varying with  m 2

23. NOW 2006 M. Cozzi 23 Conclusions We re-analyzed the energy distribution of MACRO neutrino data to include the possibility of exotic effects (Violation of the Lorentz Invariance) The inclusion of VLI effects does not improve the fit to the muon energy data → VLI effects excluded even at a sub- dominant level We obtained the limit on VLI parameter |  v|< 3 x 10 -25 at 90% C.L. (or  |  |< 1.5 x 10 -25 for the VEP case)