1

MACRO constraints on MACRO constraints on violation of Lorentz violation of Lorentz

invarianceinvarianceM. Cozzi

Bologna University - INFN

Neutrino Oscillation WorkshopConca Specchiulla (Otranto)

September 9-16, 2006

NOW 2006NOW 2006 M. CozziM. Cozzi 2

OutlineOutline

Violation of Lorentz Invariance (VLI)Test of VLI with neutrino oscillationsMACRO results on mass-induced oscillationsSearch for a VLI contribution in neutrino oscillationsResults and conclusions

NOW 2006NOW 2006 M. CozziM. Cozzi 3

Violation of the Lorentz Violation of the Lorentz InvarianceInvariance

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 cutoffVLI from photon stabilityVLI from radioactive muon decayVLI from hadronic physics

Here we consider only those violation of Lorentz Invariance conserving CPT

NOW 2006NOW 2006 M. CozziM. Cozzi 4

Test of Lorentz invariance with Test of Lorentz invariance with neutrino oscillationsneutrino oscillations

The CPT-conserving Lorentz violations lead to neutrino oscillations even if neutrinos are masslessHowever, observable neutrino oscillations may result from a combination of effects involving neutrino masses and VLIGiven the very small neutrino mass ( eV), neutrinos are ultra relativistic particlesSearches for neutrino oscillations can provide a sensitive test of Lorentz invariance

1m

NOW 2006NOW 2006 M. CozziM. Cozzi 5

““Pure” mass-induced neutrino Pure” mass-induced neutrino oscillationsoscillations

In the 2 family approximation, we have2 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 (m2 ≡ m22- m3

2 ≠ 0) and the mixing angle ≠ 0, then the probability that a flavor “survives” after a distance L is:

m23

m3

m23

m2

m23

m3

m23

m2

cossin

sincos

E/Lm27.1sin2sin1P 22m

2

m2

m3

Note the L/E dependence

NOW 2006NOW 2006 M. CozziM. Cozzi 6

““Pure” VLI-induced neutrino Pure” VLI-induced neutrino oscillationsoscillations

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 ≡ v2- v3 ≠ 0)

and the mixing angle v≡ v≠ 0, then the survival

oscillation probability has the form:

v23

v3

v23

v2

v23

v3

v23

v2

cossin

sincos

ELv1054.2sin2sin1P 182v

2

v2

v3

Note the L·E dependence

NOW 2006NOW 2006 M. CozziM. Cozzi 7

Mixed scenarioMixed scenarioWhen both mass-induced and VLI-induced oscillations are simultaneously considered:

where2=atan(a1/a2)

=√a12+ a2

2

22 sin2sin1 P

)LE 2 cos v2·10 L/Ecos2 m(1.27 a

e LE 2 sin v2·10 L/Esin2 m1.27 a

v18

m2

2

iv

18m

21

oscillation“strength”

oscillation“length”

= generic phase connecting mass and velocity eigenstates

NOW 2006NOW 2006 M. CozziM. Cozzi 8

Notes:Notes:In the “pure” cases, probabilities do not depend on the sign of v, m2 and mixing angles while in the “mixed” case relative signs are important. Domain of variability:

m2 ≥ 0 0 ≤ m ≤ /2v ≥ 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)

NOW 2006NOW 2006 M. CozziM. Cozzi 9

Energy dependence for P(Energy dependence for P(ννμμννμμ) assuming ) assuming L=10000 km, L=10000 km, mm2 2 = 0.0023 eV= 0.0023 eV2 2 and and mm==/4/4

252 10 ,sin 2 0vv 252 10 ,sin 2 0.3vv

252 10 ,sin 2 0.7vv 252 10 ,sin 2 1vv

Black line: no VLI

Mixed scenario:

VLI with sin2θv>0

VLI with sin2θv <0

NOW 2006NOW 2006 M. CozziM. Cozzi 10

MACRO results on mass-induced MACRO results on mass-induced neutrino oscillationsneutrino oscillations

NOW 2006NOW 2006 M. CozziM. Cozzi 11

7 Rock absorbers~ 25 Xo

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

<E(GeV)>50 4.2 3.550 4.2 3.5

Topologies of Topologies of -induced -induced eventsevents

50/yr Internal Upgoing (IU)

NOW 2006NOW 2006 M. CozziM. Cozzi 12

Neutrino events Neutrino events detected by MACROdetected by MACRO

Data samples No-osc Expected (MC)Topologies Measure

d

Up Throughgoing 857 1169

Internal Up 157 285

Int. Down + Up stop

262 375

50E GeV

3.5E GeV

4.2E GeV

NOW 2006NOW 2006 M. CozziM. Cozzi 13

Upthroughgoing muonsUpthroughgoing muonsAbsolute flux

Even if new MCs are strongly improved, there are still problems connected with CR fit → large sys. err.

Zenith angle deformationExcellent resolution (2% for HE)Very powerful observable (shape known to within 5%)

Energy spectrum deformationEnergy estimate through MCS in the rock absorber of the

detector (sub-sample of upthroughgoing events) PLB 566 (2003) 35PLB 566 (2003) 35

Extremely powerful, but poorer shape knowledge (12% error point-to-point)

Used for this analysis

NOW 2006NOW 2006 M. CozziM. Cozzi 14

L/EL/E distribution distributionDATA/MC(no oscillation) as a function

of reconstructed L/E:

Internal Upgoing

300 Throughgoing events

NOW 2006NOW 2006 M. CozziM. Cozzi 15

The analysis was based on ratios (reduced systematic errors at few % level): Eur. Phys. J. C36 (2004) 357

Angular distribution R1= N(cos<-0.7)/N(cos>-0.4)

Energy spectrum R2= N(low E)/N(high E)

Low energy R3= N(ID+UGS)/N(IU)

Null hypothesis ruled out by PNH~5If the absolute flux information is added (assuming Bartol96 correct within 17%): PNH~ 6Best fit parameters for ↔ oscillations (global fit of all MACRO neutrino data):m2=0.0023 eV2

sin22m=1

Final MACRO resultsFinal MACRO results

NOW 2006NOW 2006 M. CozziM. Cozzi 16

90% CL allowed region90% CL allowed region

Based on the “shapes” of the distributions (14 bins)

Including normalization (Bartol flux with 17% sys. err.)

NOW 2006NOW 2006 M. CozziM. Cozzi 17

Search for a VLI contributionSearch for a VLI contributionusing MACRO datausing MACRO data

Assuming standard mass-induced neutrino oscillations as the leading mechanism for flavor transitions and VLI as a subdominant effect.

NOW 2006NOW 2006 M. CozziM. Cozzi 18

A subsample of 300 upthroughgoing muons (with energy estimated via MCS) are particularly favorable:<E> ≈ 50 GeV (as they are uptroughgoing)<L> ≈ 10000 km (due to analysis cuts)

Golden events for VLI studies!

v= 2 x 10-25

v=/4

Good sensitivity expected from the

relative abundancesof low and high energy events

NOW 2006NOW 2006 M. CozziM. Cozzi 19

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 evtsDefine the statistics:

and (in the first step) fix mass-induced oscillation parameters m2=0.0023 eV2 and sin22m=1 (MACRO values) and assume ei realassume 16% systematic error on the ratio Nlow/Nhigh (mainly due to the spectrum slope of primary cosmic rays)Scan the (v, v) plane and compute χ2 in each point (Feldman & Cousins prescription)

Analysis strategyAnalysis strategy

Optimizedwith MC

22

22 2

, ; ,MChigh i i v m

i low stat syst

N N v m

NOW 2006NOW 2006 M. CozziM. Cozzi 20

Results of the analysis - IResults of the analysis - I

Original cuts

Optimized cuts

χ2 not improved in any point of

the (v, v) plane:

90% C.L. limits

Neutrino flux used in MC: “new Honda” - PRD70 (2004) 043008

NOW 2006NOW 2006 M. CozziM. Cozzi 21

Results of the analysis - IIResults of the analysis - IIChanging m2 around the best-fit point with m2± 30%, the limit moves up/down by at most a factor 2Allowing m2 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

NOW 2006NOW 2006 M. CozziM. Cozzi 22

Results of the analysis - IIIResults of the analysis - IIIA 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 m2

NOW 2006NOW 2006 M. CozziM. Cozzi 23

ConclusionsConclusionsWe 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 levelWe 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)