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## MACRO constraints on violation of Lorentz invariance

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**MACRO constraints on violation of Lorentz invariance**M. Cozzi Bologna University - INFN Neutrino Oscillation Workshop Conca Specchiulla (Otranto) September 9-16, 2006**Outline**• Violation of Lorentz Invariance (VLI) • Test of VLI with neutrino oscillations • MACRO results on mass-induced n oscillations • Search for a VLI contribution in neutrino oscillations • Results and conclusions M. Cozzi**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. Vi(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 M. Cozzi**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 M. Cozzi**“Pure” mass-induced neutrino oscillations**• In the 2 family approximation, we have • 2 mass eigenstates and with masses m2 and m3 • 2 flavor eigenstates and • The mixing between the 2 basis is described by the θ23 angle: • If the states are not degenerate (Dm2 ≡ m22- m32 ≠ 0) and the mixing angle q23 ≠ 0, then the probability that a flavor “survives” after a distance L is: Note the L/E dependence M. Cozzi**“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 (Dv≡ v2- v3≠ 0) and the mixing angle qv23 ≡ qv≠ 0, then the survival oscillation probability has the form: Note the L·E dependence M. Cozzi**Mixed scenario**• When both mass-induced and VLI-induced oscillations are simultaneously considered: • where • 2Q=atan(a1/a2) • W=√a12+ a22 oscillation “length” oscillation “strength” h = generic phase connecting mass and velocity eigenstates M. Cozzi**Notes:**• In the “pure” cases, probabilities do not depend on the sign of Dv, Dm2 and mixing angles while in the “mixed” case relative signs are important. Domain of variability: • Dm2 ≥ 0 0 ≤ qm ≤ p/2 • Dv ≥ 0 -p/4 ≤ qv ≤ p/4 • Formally, VLI-induced oscillations are equivalent to oscillations induced by Violation of the Equivalence Principle (VEP) after the substitution:Dv/2↔ |f|Dgwhere f is the gravitational potential and Dg 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) M. Cozzi**Energy dependence for P(νμνμ) assuming L=10000 km,**Dm2 = 0.0023 eV2 and qm=p/4 Black line: no VLI Mixed scenario: VLI with sin2θv>0 VLI with sin2θv<0 M. Cozzi**MACRO results on mass-induced neutrino oscillations**M. Cozzi**Topologies of n-induced events**3 horizontal layers ot Liquid scintillators nm 14 horizontal planes of limited streamer tubes m 7 Rock absorbers ~ 25 Xo m nm m m 50/yr Internal Upgoing (IU) 35/yr Internal Downgoing (ID) + 35/yr Upgoing Stopping (UGS) 180/yr Up-throughgoing nm nm M. Cozzi <En(GeV)> 50 4.2 3.5**Neutrino events detected by MACRO**M. Cozzi**Used for this analysis**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 • 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) M. Cozzi**L/En distribution**DATA/MC(no oscillation) as a function of reconstructed L/E: Internal Upgoing 300 Throughgoing events M. Cozzi**Final MACRO results**• The analysis was based on ratios (reduced systematic errors at few % level): Eur. Phys. J. C36 (2004) 357 • Angular distribution R1= N(cosq<-0.7)/N(cosq>-0.4) • Energy spectrum R2= N(low En)/N(high En) • Low energy R3= N(ID+UGS)/N(IU) • Null hypothesis ruled out by PNH~5s • If the absolute flux information is added (assuming Bartol96 correct within 17%): PNH~ 6s • Best fit parameters for nm↔nt oscillations (global fit of all MACRO neutrino data): • Dm2=0.0023 eV2 • sin22qm=1 M. Cozzi**90% CL allowed region**Based on the “shapes” of the distributions (14 bins) D Including normalization (Bartol flux with 17% sys. err.) q M. Cozzi**Search for a VLI contributionusing MACRO data**Assuming standard mass-induced neutrino oscillations as the leading mechanism for flavor transitions and VLI as a subdominant effect. M. Cozzi**A subsample of 300 upthroughgoing muons (with energy**estimated via MCS) are particularly favorable: <En> ≈ 50 GeV(as they are uptroughgoing) <L> ≈ 10000 km(due to analysis cuts) Golden events for VLI studies! Good sensitivity expected from the relative abundances of low and high energy events Dv= 2 x 10-25 qv=p/4 M. Cozzi**Analysis strategy**Optimized with MC • Divide the MCS sample (300 events) in two sub-samples: • Low energy sample: Erec < 28 GeV → Nlow= 44 evts • High energy sample: Erec > 142 GeV → Nhigh= 35 evts • Define the statistics: and (in the first step) fix mass-induced oscillation parameters Dm2=0.0023 eV2 and sin22qm=1 (MACRO values) and assume eih real • assume 16% systematic error on the ratio Nlow/Nhigh (mainly due to the spectrum slope of primary cosmic rays) • Scan the (Dv, qv) plane and compute χ2 in each point (Feldman & Cousins prescription) M. Cozzi**Results of the analysis - I**χ2 not improved in any point of the (Dv, qv) plane: Original cuts 90% C.L. limits Optimized cuts Neutrino flux used in MC: “new Honda” - PRD70 (2004) 043008 M. Cozzi**Results of the analysis - II**• Changing Dm2 around the best-fit point with Dm2± 30%, the limit moves up/down by at most a factor 2 • Allowing Dm2 to vary inside ±30%, qm± 20% and any value for the phase h and marginalizing in qv (-π/4≤ qv ≤ π/4 ): |Dv|< 3 x 10-25 VLI VEP f|Dg|< 1.5 x 10-25 M. Cozzi**Results of the analysis - III**• A different and complementary analysis has been performed: • Select the central region of the energy spectrum 25 GeV < Enrec < 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 m2 M. Cozzi**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 |Dv|< 3 x 10-25 at 90% C.L.(or f|Dg|< 1.5 x 10-25 for the VEP case) M. Cozzi