Tests of Lorentz Invariance using Atmospheric Neutrinos and

AMANDA-IIJohn Kelley

for the IceCube CollaborationUniversity of Wisconsin, Madison

Workshop on Exotic Physics with Neutrino Telescopes

Uppsala, Sweden

September 21, 2006

Atmospheric Neutrinos as a New Physics Probe

• Mass-induced oscillations on solid ground — but subdominant effects at high energies?

• For E > 100 GeV and m < 1 eV, Lorentz > 1011

• Oscillations are a sensitive quantum-mechanical probe:small differences in energy large change in observable spectrum

Eidelman et al.: “It would be surprising if further surprises were not in store…”

New Physics Effects

• Violation of Lorentz invariance (VLI) in string theory or loop quantum gravity*

• Violations of the equivalence principle (different gravitational coupling)†

• Interaction of particles with space-time foam quantum decoherence of pure states‡

* see e.g. Carroll et al., PRL 87 14 (2001), Colladay and Kostelecký, PRD 58 116002 (1998)† see e.g. Gasperini, PRD 39 3606 (1989)

‡ see e.g. Hawking, Commun. Math. Phys. 87 (1982), Ellis et al., Nucl. Phys. B241 (1984)

c - 1

c - 2

VLI Phenomenology

• Modification of dispersion relation*:

• Different maximum attainable velocities ca (MAVs) for different particles: E ~ (c/c)E

• For neutrinos: MAV eigenstates not necessarily flavor or mass eigenstates mixing VLI oscillations

* Glashow and Coleman, PRD 59 116008 (1999)

Conventional+VLI Oscillations

• For atmospheric , conventional oscillations turn off above ~50 GeV (L/E dependence)

• VLI oscillations turn on at high energy (n=1 above; L E dependence), depending on size of c/c, and distort the zenith angle / energy spectrum

González-García, Halzen, and Maltoni, hep-ph/0502223

Survival Probability

c/c = 10-27

AMANDA-II Data Sample

2000-2004 sky mapLivetime: 1001 days

4282 events (up-going)

No point sources or HE diffuse flux yet:clean atmospheric neutrino sample!Adding 2005 data: > 5000 events

Data Analysis

detectorMC

Zenith angle: angular resolution ~2ºBut energy reconstruction more difficult

Energy-correlated Observable

Nch (number of optical modules hit): stable observable, but acts more like an energy threshold

Other methods exist: dE/dx estimates, neural networks…

Observable Space (MC)

c/c = 10-25 No New Physics

Differences in both shape (near vertical) and normalization

VLI: AMANDA-II Sensitivity

Median Sensitivity c/c (sin(2) = 1)• 90%: 3.2 10-27

• 95%: 3.6 10-27

• 99%: 5.1 10-27

2000-05 livetimesimulated

• MACRO limit*: 2.5 10-26 (90%)

• Super-K + K2K limit†: 2.0 10-27 (90%)

*Battistoni et al., hep-ex/0503015†González-García & Maltoni, PRD 70 033010 (2004)

MC test analysis in 1-D observable (Nch) and parameter space (c/c)(Feldman-Cousins likelihood ratio method)

Summary and Outlook

• Preliminary sensitivity to VLI effects competitive with existing limits

• Expect improvements as analysis technique is refined

• Other new physics effects (e.g. quantum decoherence) are also being tested!

Extra Slides

Quantum Decoherence Phenomenology

• Modify propagation through density matrix formalism:

• Solve DEs for neutrino system, get oscillation probability*:

*for more details, please see Morgan et al., astro-ph/0412628

dissipative term

QD Parameters

• Various proposals for how parameters depend on energy:

simplest preserves Lorentz invariance

recoiling D-branes!

Survival Probability ( model)

a = = 4 10-32 (E2 / 2)

Binned Likelihood Test

Poisson probability

Product over bins

Test Statistic: LLH

Testing the Parameter Space

Given a measurement, want to determine values of parameters {i} that are allowed / excluded

at some confidence level

c/c

sin(2)

allowed

excluded

Feldman-Cousins Recipe

• For each point in parameter space {i}, sample many times from parent Monte Carlo distribution (MC “experiments”)

• For each MC experiment, calculate likelihood ratio: L = LLH at parent {i} - minimum LLH at some {i,best}

• For each point {i}, find Lcrit at which, say, 90% of the MC experiments have a lower L (FC ordering principle)

• Once you have the data, compare Ldata to Lcrit at each point to determine exclusion region

• Primary advantage over 2 global scan technique: proper coverage

Feldman & Cousins, PRD 57 7 (1998)

1-D Examples

all normalized to datasin(2) = 1

Systematic Errors

• Atmospheric production uncertainties• Detector effects (OM sensitivity)• Ice Properties

Can be treated as nuisance parameters:minimize LLH with respect to them

Or, can simulate as fluctuations in MC experiments

Normalization is already included!(free parameter — could also constrain)

VLI: Sensitivity using Nch

Median Sensitivity c/c (sin(2) = 1)• 90%: 3.2 10-27

• 95%: 3.6 10-27

• 99%: 5.1 10-27

2000-05 livetimesimulated

allowed excluded

Decoherence Sensitivity(Using Nch, model)

Norm. constrained ±30%Normalization free

Decoherence Sensitivity

Median Sensitivitya, (GeV-1)

• 90%: 3.7 10-31

• 95%: 5.8 10-31

• 99%: 1.6 10-30ANTARES (3 yr sens, 90%)* : 10-44 GeV-1

* Morgan et al., astro-ph/0412618‡ Lisi, Marrone, and Montanino, PRL 85 6 (2000)

SuperK limit (90%)‡ : 0.9 10-27 GeV-1

Almost 4 orders of magnitude improvement!

(E2 energy dependence)