Transcript Document
MACRO constraints on violation of Lorentz invariance
M. Cozzi Bologna University - INFN
Neutrino Oscillation Workshop
Conca Specchiulla (Otranto)
September 9-16, 2006
1
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 NOW 2006 M. Cozzi 2
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 NOW 2006 M. Cozzi 3
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 NOW 2006 M. Cozzi 4
“Pure” mass-induced neutrino oscillations
In the 2 family approximation, we have n 2 and m 3 n n The mixing between the 2 basis is described by the θ 23 angle: n n m 2 cos q m 23 n m 3 sin q m 23 n n m 2 sin q m 23 n m 3 cos q m 23 If the states are not degenerate ( D m 2 ≡ m 2 2 - m 3 2 ≠ 0) and the mixing angle q 23 ≠ 0, then the probability that a flavor “survives” after a distance L is: P n n 1 sin 2 2 q m sin 2 1 .
27 D m 2 L Note the L/E dependence / E n NOW 2006 M. Cozzi 5
“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: n n v 2 cos q v 23 n v 3 sin q v 23 n n v 2 sin q v 23 n v 3 cos q v 23 If neutrinos have different MAVs ( D v ≡ v 2 - v 3 ≠ 0) and the mixing angle q v 23 ≡ q v ≠ 0, then the survival oscillation probability has the form: P n n 1 sin 2 2 q v sin 2 2 .
54 10 18 D v L E n Note the L·E dependence NOW 2006 M. Cozzi 6
Mixed scenario
When both mass-induced and VLI-induced oscillations are simultaneously considered:
P
n n 1 sin 2 2 Q sin 2 W where 2 Q =atan(a 1 /a 2 ) W =√a 1 2 + a 2 2
oscillation “strength” oscillation “length”
a 1 1.27
D m 2 sin2 q m L/E n 2·10 18 D v sin 2 q v LE n e i a 2 1.27
( D m 2 cos2 q m L/E n 2·10 18 D v cos 2 q v LE n ) = generic phase connecting mass and velocity eigenstates NOW 2006 M. Cozzi 7
Notes:
In the “pure” cases, probabilities do not depend on the sign of D v, D m 2 and mixing angles while in the “mixed” case relative signs are important . Domain of variability: D m 2 ≥ 0 0 ≤ q m ≤ p /2 D v ≥ 0 p /4 ≤ q v ≤ p /4 Formally, VLI-induced oscillations are equivalent to oscillations induced by Violation of the Equivalence Principle where f (VEP) after the substitution: D v/2 ↔ | f | Dg is the gravitational potential and difference of the neutrino coupling to the gravitational field.
Dg is the Due to the different (L,E) behavior, VLI effects are emphasized for large L and large E ( large L·E) NOW 2006 M. Cozzi 8
Energy dependence for P(ν μ ν μ ) assuming L=10000 km, D m 2 = 0.0023 eV 2 and q m = p /4
v
25 , sin 2
v
0
v
25 , sin 2
v
0.3
v
25 , sin 2
v
0.7
v
Black line: no VLI
Mixed scenario:
VLI with sin2θ v >0 VLI with sin2θ v <0 25 , sin 2
v
1 NOW 2006 M. Cozzi 9
MACRO results on mass-induced neutrino oscillations
NOW 2006 M. Cozzi 10
Topologies of
n
-induced events
3 horizontal layers ot Liquid scintillators 14 horizontal planes of limited streamer tubes 7 Rock absorbers ~ 25 X o
n n 180/yr Up throughgoing NOW 2006 n 50/yr Internal Upgoing (IU) n M. Cozzi
50 4.2 3.5
35/yr Internal Downgoing (ID) + 35/yr Upgoing Stopping (UGS)
n
(GeV)>
NOW 2006
Neutrino events detected by MACRO
Data samples Topologies Measured No-osc Expected (MC)
Up Throughgoing
857 1169
Internal Up Int. Down + Up stop
157 262 285 375
E
E E
50
GeV
3.5
GeV
4.2
GeV
M. Cozzi 12
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) Used for this analysis NOW 2006 M. Cozzi 13
L/E
n
distribution
DATA/MC(no oscillation) as a function of reconstructed L/E:
Internal Upgoing
NOW 2006
300 Throughgoing events
M. Cozzi 14
Final MACRO results
The analysis was based on ratios (reduced systematic errors at few % level):
Eur. Phys. J. C36 (2004) 357
Angular distribution R 1 = N(cos q <-0.7)/N(cos q >-0.4) Energy spectrum R 2 = N(low E n )/N(high E n ) Low energy R 3 = N(ID+UGS)/N(IU) Null hypothesis ruled out by P NH ~5 s If the absolute flux information is added (assuming Bartol96 correct within 17%): P NH ~ 6 s Best fit parameters for n ↔ n oscillations (global fit of all MACRO neutrino data): D m 2 =0.0023 eV 2 sin 2 2 q m =1 NOW 2006 M. Cozzi 15
90% CL allowed region
Based on the “shapes” of the distributions (14 bins) NOW 2006 Including normalization (Bartol flux with 17% sys. err.) q M. Cozzi 16
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.
NOW 2006 M. Cozzi 17
A subsample of 300 upthroughgoing muons (with energy estimated via MCS) are particularly favorable:
n
> ≈ 50 GeV
(as they are uptroughgoing) (due to analysis cuts)
Golden events for VLI studies!
Good sensitivity expected from the relative abundances of low and high energy events D
v= 2 x 10 -25
q
v =
p
/4
NOW 2006 M. Cozzi 18
Analysis strategy
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: 2
high
N i
N i MC
s 2
stat
D s q 2
v syst
; D
m
2 , q
m
Optimized with MC
2 and (in the first step) fix mass-induced oscillation parameters D m 2 =0.0023 eV 2 and sin 2 2 q m =1 (MACRO values) and assume e i real assume 16% systematic error on the ratio N low /N high due to the spectrum slope of primary cosmic rays) Scan the ( D v, q v ) plane and compute χ 2 in each point (Feldman & Cousins prescription) (mainly NOW 2006 M. Cozzi 19
Results of the analysis - I
χ 2 not improved in any point of the ( D v, q v ) plane: Original cuts Optimized cuts
90% C.L. limits
Neutrino flux used in MC: “new Honda” - PRD70 (2004) 043008 NOW 2006 M. Cozzi 20
Results of the analysis - II
Changing D m 2 around the best-fit point with D m 2 ± 30%, the limit moves up/down by at most a factor 2 Allowing D m 2 to vary inside ±30%, q m ± 20% and any value for the phase and marginalizing in q v (-π/4≤ q v ≤ π/4 ): | D
v|< 3
x
10
-25 VLI f | Dg
|< 1.5
x
10
-25 M. Cozzi VEP 21 NOW 2006
Results of the analysis - III
A different and complementary analysis has been performed: Select the central region of the energy spectrum 25 GeV < E n 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 D v < 10 -25 , slowly varying with D m 2 NOW 2006 M. Cozzi 22
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 | D v|< 3 x 10 -25 at 90% C.L.
(or f | Dg |< 1.5 x 10 -25 for the VEP case) NOW 2006 M. Cozzi 23