Transcript Document

MACRO constraints on
violation of Lorentz
invariance
M. Cozzi
Bologna University - INFN
Neutrino Oscillation Workshop
Conca Specchiulla (Otranto)
September 9-16, 2006
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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
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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
VLI
VLI
VLI
connected to the breakdown of GZK cutoff
from photon stability
from radioactive muon decay
from hadronic physics
Here we consider only those violation of Lorentz
Invariance conserving CPT
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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 ( m  1 eV),
neutrinos are ultra relativistic particles
Searches for neutrino oscillations can provide a
sensitive test of Lorentz invariance
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“Pure” mass-induced neutrino
oscillations
In the 2 family approximation, we have
2 mass eigenstates n m2
2 flavor eigenstates n 
and n 3m
and n
with masses m2 and m3

The mixing between the 2 basis is described by the θ23
angle:
n   n m2 cosqm23  n 3m sin qm23
n    n m2 sin qm23  n 3m cosqm23
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:
Pn  n   1  sin2 2qm sin2 1.27Dm2 L / En 
Note the L/E dependence
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“Pure” VLI-induced neutrino
oscillations
When VLI is considered, we introduce a new basis:
v
v
the velocity basis: n 2 and n 3 (2 family approx)
Velocity and flavor eigenstates are now connected by
a new mixing angle:
v
v
n  n 2v cosq23
 n 3v sin q23
v
v
n    n 2v sin q23
 n 3v cosq23
If neutrinos have different MAVs (Dv ≡ v2- v3 ≠ 0)
and the mixing angle qv23 ≡ qv≠ 0, then the survival
oscillation probability has the form:
Pn  n   1  sin2 2qv sin2 2.54 1018 Dv L  En 
Note the L·E dependence
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Mixed scenario
When both mass-induced and VLI-induced
oscillations are simultaneously considered:
Pn  n    1  sin 2Q sin W
2
where
2Q=atan(a1/a2)
W=√a12+ a22
oscillation
“strength”
2
oscillation
“length”
2
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i

a

1.27
D
m
sin2
q
L/E

2·10
D
v
sin
2
q
LE
e
 1
m
n
v
n

2
18

a

1.27
(
D
m
cos2
q
L/E

2·10
Dv cos 2qv LE n )
m
n
 2
 = generic phase connecting mass and velocity eigenstates
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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
Dv ≥ 0
0 ≤ qm ≤ p/2
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|Dg
where 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)
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Energy dependence for P(νμνμ) assuming
L=10000 km, Dm2 = 0.0023 eV2 and qm=p/4
Dv  2 1025 ,sin 2v  0
Dv  2 1025 ,sin 2v  0.3
Black line: no VLI
Mixed scenario:
VLI with sin2θv>0
VLI with sin2θv <0
Dv  2 1025 ,sin 2v  0.7
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Dv  2 1025 ,sin 2v  1
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MACRO results on mass-induced
neutrino oscillations
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Topologies of n-induced events
3 horizontal
layers ot
Liquid
scintillators
n
14 horizontal
planes of
limited
streamer
tubes

7 Rock
absorbers
~ 25 Xo



180/yr Upthroughgoing
n
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n
50
50/yr
Internal
Upgoing (IU)
4.2
n
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3.5
35/yr Internal Downgoing (ID)
+
35/yr Upgoing Stopping (UGS)
<En(GeV)>
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Neutrino events
detected by MACRO
Data samples
Measured
No-osc
Expected
(MC)
Up Throughgoing
857
1169
Internal Up
157
285
Int. Down + Up stop
262
375
Topologies
E  50 GeV
E  3.5 GeV
E  4.2 GeV
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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
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L/En distribution
DATA/MC(no oscillation) as a function
of reconstructed L/E:
Internal Upgoing
300 Throughgoing events
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Final MACRO results
The analysis was based on ratios (reduced systematic
errors at few % level):
Eur. Phys. J. C36 (2004) 357
Angular distribution
Energy spectrum
Low energy
R1= N(cosq<-0.7)/N(cosq>-0.4)
R2= N(low En)/N(high En)
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 n↔n oscillations (global fit of
all MACRO neutrino data):
Dm2=0.0023 eV2
sin22qm=1
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90% CL allowed region
D
Based on the “shapes”
of the distributions (14 bins)
Including normalization
(Bartol flux with 17% sys. err.)
q
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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.
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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
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Dv= 2 x 10-25
qv=p/4
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Analysis strategy
Divide the MCS sample (300 events) in two sub-samples:
Low energy sample: Erec < 28 GeV → Nlow= 44 evts Optimized
High energy sample: Erec > 142 GeV → Nhigh= 35 evts with MC
Define the statistics:
2

 N   N MC Dv, q ; Dm 2 , q
i
i
v
m

2
  
2
2
s

s
i low 
stat
syst

high
 



and (in the first step) fix mass-induced oscillation parameters
Dm2=0.0023 eV2 and sin22qm=1 (MACRO values) and assume
ei 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)
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Results of the analysis - I
χ2 not improved
in any point of
the (Dv, qv) plane:
Original cuts
Optimized cuts
90% C.L. limits
Neutrino flux used in MC: “new Honda” - PRD70 (2004) 043008
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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  and marginalizing in qv
(-π/4≤ qv ≤ π/4 ):
|Dv|< 3 x
-25
10
f|Dg|< 1.5 x 10-25
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VLI
VEP
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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 Dv < 10-25,
slowly varying with Dm2
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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 subdominant 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)
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