neutrino model building with the sme - indiana …lorentz/sme2012/diaz.pdf · iucss summer school...

IUCSS Summer School on the

Lorentz- and CPT-violating Standard-Model Extension

Neutrino model building with the SME

Jorge S. Diaz

Indiana University

June 8, 2012

Outline

1 Introduction to neutrino oscillations

2 Conventional description of neutrinooscillations

3 Lorentz-violating neutrinos

4 Building your own model of neutrinooscillations

5 Summary

Jorge S. Diaz (Indiana University) Neutrino model building with the SME June 8, 2012 2 / 49

Neutrinos: some history

Beta-decay puzzle Observed spectrum of energies violatesconservation of energy and momentum!

1930: “I have done a terrible thing.I have proposed a particle that cannot bedetected. It is something no theoristshould ever do.” Pauli.

1933: Fermi formulates theory of β decayand names the little neutral one.

1956: Reines and Cowan observe the firstelectron antineutrino (Project Poltergeist).


Neutrinos: some natural and artificial sources


NeutrinosProperties

Neutrinos are fundamental particlesin the Standard Model (SM).

They carry no electric charge(insensitive to electromagneticfields).

They interact only via the weakinteraction (barely interact withmatter).

They come in three active flavors:νe, νµ, ντ .

In the SM, neutrinos are massless.


Neutrinos

Question #1

The Sun produces electron neutrinos νe, whereas anuclear reactor produces electron antineutrinos νe.What is the transformation that relates neutrinosand antineutrinos?

νe?−→ νe

1 C2 P3 CP4 T

Right-handed neutrinos are not observed in nature, similarly left-handedantineutrinos have never been observed

(νe)LC−→ (νe)L

P−→ (νe)R


Neutrinos

The problem of the missing neutrinos

1960s: 66% of neutrinos from the Sunmissing: (

Φνe

)exp(

Φνe

)th

' 0.33

1980s: 40% of atmospheric neutrinosmissing: (

Φνµ/Φνe

)exp(

Φνµ/Φνe

)th

' 0.60


Neutrino OscillationsSolar neutrino problem: solved (

Φνe

)exp(

Φνe

)th

' 0.33

Atmospheric neutrino problem: solved

(Φνµ

/Φνe

)exp(

Φνµ/Φνe

)th

' 0.60


Conventional description: three-neutrino massive model

Phenomenological extension of the SM:

involves three massive neutrinos

h = U†

E1 0 00 E2 00 0 E3

U

energy-independent mixing: U(θ12, θ13, δ, θ23)neutrinos and antineutrinos are decoupled

6× 6 matrix→ HHH3νSM =(h 00 h∗

)oscillation probability

Pνa→νb(L) =

∑a′,b′

U∗a′aUa′bUb′aU∗b′b e

i(Ea′−Eb′ )L

Ea′ =√|ppp|2 +m2

a′

≈ |ppp|+ m2a′

2|ppp|

Energy spectrum (NH)



the mixing matrix is the product of three rotations

U =

c12 −s12 0s12 c12 00 0 1

︸︷︷︸

solar, LB reactor

c13 0 −s13 e−iδ

0 1 0s13 e

iδ 0 c13

︸︷︷︸

SB reactor, LB accelerator

1 0 00 c23 −s230 s23 c23

︸︷︷︸

LB accelerator, atmospheric

relevant survival probabilities

Pνe→νe ≈ 1− sin2 2θ12 sin2(

∆m221L

4E

)(LB reactor)

Pνe→νe ≈ 1− sin2 2θ13 sin2(

∆m231L

4E

)(SB reactor)

Pνµ→νµ ≈ 1− sin2 2θ23 sin2(

∆m231L

4E

)Five of the six parameters have been measured (arXiv:1205.4018)

∆m221 ' 7.62× 10−5 eV2, sin2 2θ12 ' 0.87, sin2 2θ23 ' 0.99

∆m231 ' 2.53× 10−3 eV2, sin2 2θ13 ' 0.10, δ : unknown



Question #2

Five of the six parameters have been measured (arXiv:1205.4018)

∆m221 ' 7.62× 10−5 eV2, sin2 2θ12 ' 0.87, sin2 2θ23 ' 0.99

∆m231 ' 2.53× 10−3 eV2, sin2 2θ13 ' 0.10, δ : unknown

Where is ∆m232?

The mass-squared differences are defined as

∆m221 = m2

2 −m21, ∆m2

31 = m23 −m2

1, ∆m232 = m2

3 −m22,

only two mass-squared differences are independent

∆m232 = ∆m2

31 −∆m221



KM plot

Kostelecky & Mewes, PRD 69, 016005 (2004)

energy and baseline coverage ofoscillation experiments

oscillation half-wavelengths (firstoscillation maximum)

λa′b′/2 = π/(Ea′ − Eb′)

Massive model:

λ21/2 = 2πE/∆m221

λ31/2 = 2πE/∆m231



This model successfully describes all established oscillation data

Atmospheric neutrinos: Super-Kamiokande Reactor antineutrinos: KamLAND

Solar neutrinos: SNO+Borexino+SK Accelerator neutrinos: MINOS+T2K+K2K

PRL 93, 101801 (2004) PRL 94, 081801 (2005)

PRD 82, 033006 (2010) PRL 106, 181801 (2011)


Neutrino Anomalies

LSND (2001)

Evidence of antineutrino oscillations atshort distances (L ∼ 30 m).PRD 64, 112007 (2001)

Gallium anomaly (2006)

Signal of antineutrino oscillations at shortdistances (L ∼ 1 m).PRC 73, 045805 (2006)

MiniBooNE (2007)

Oscillation signal at low energies(E ∼ 300 MeV).PRL 98, 231801 (2007)

MiniBooNE (2010)

Differences between neutrinos andantineutrinos.PRL 105, 181801 (2010)

MINOS (2010)

Differences between neutrinos andantineutrinos. Update: anomaly is gonePRL 107, 021801 (2010)

Reactor anomaly (2011)

Signal of antineutrino oscillations at shortdistances (L ∼ 10− 1000 m).PRD 83, 073006 (2011)

None of these results can be accommodated by the 3νSM!


Neutrino Anomalies

LSND (2001)


MiniBooNE (2007)


MINOS (2010)




Neutrino Anomalies



MiniBooNE (2010)






Conventional description: three-neutrino massive modelthe mixing matrix is the product of three rotations

U =

c12 −s12 0s12 c12 00 0 1

︸︷︷︸

solar, LB reactor

c13 0 −s13 e−iδ

0 1 0s13 e

iδ 0 c13

︸︷︷︸

SB reactor, LB accelerator

1 0 00 c23 −s230 s23 c23

︸︷︷︸

LB accelerator, atmospheric

Pνe→νe≈ 1− sin2 2θ12 sin2

(∆m2

21L4E

)(LB)

Pνe→νe≈ 1− sin2 2θ13 sin2

(∆m2

31L4E

)(SB)

Pνµ→νµ≈ 1− sin2 2θ23 sin2

(∆m2

31L4E

)

Question #3

Why are disappearanceexperiments insensitive toCP violation?(why is the CP phase δ absent

in the survival probabilities

Pνx→νx?)

Lorentz invariance ⇒ CPT invariance: Pνx→νx

CPT−→ Pνx→νx = Pνx→νx

Neutrino disappearance is a T-invariant process: Pνx→νx

T−→ Pνx→νx

CPT and T invariance requires CP invariance ⇒ CP violation is not observable.



Question #4

How can we search for CP violation in neutrinos?(How can we measure the CP-violating phase δ?)

1 Search for disappearance of antineutrinos instead ofneutrinos.

2 Search for oscillations between different flavors.

3 Search for neutrinos oscillating into antineutrinos.

4 The CP-violating phase δ is not observable inneutrinos.

Lorentz invariance ⇒ CPT invariance: Pνa→νb

CPT−→ Pνb→νa = Pνa→νb

Observation of CP violation requires a T-violating process:

Pνµ→νe

T−→ Pνe→νµ


Building a new model

using Lorentz violation


Approach

no Lorentz violationobserved in nature

⇓construct a generalframework (SME)

⇓perform genericsearches of LV

LSND (PRD 72, 076004 (2005))

MINOS (PRL 101, 151601 (2008))

MINOS (PRL 105, 151601 (2010))

IceCube (PRD 82, 112003 (2010))

MiniBooNE (arXiv:1109.3480)

MINOS (PRD 85, 031101 (2012))

signals of new physics(anomalous results)

⇓build a model

bicycle (2004)

tandem (2006)

“BMW” (2007)

puma (2010)

“fried chicken bicycle” (2011)

perturbed puma (2012)

←− see S. Mufson’s talk today!


Neutrino Anomalies

LSND (2001)




MiniBooNE (2007)


MiniBooNE (2010)


MINOS (2010)





Signals of new physics?


(suggested) criteria for building a model

Kostelecky’s cut: one crazy idea at a time.

model should be:

global (describe all data)

simple (optional)

contain few parameters (to be competitive)

make predictions −→ test the model


Building a global model

A global model must describe all established data

1 disappearance of solar neutrinos

2 oscillation of atmospheric neutrinos νµ → ντ

3 disappearance of reactor antineutrinos over hundreds of km.

Solar Atmospheric Reactor

Borexino, PRD 82, 033006 (2010) Super-Kamiokande, PRL 81, 1562 (1998) KamLAND, PRL 90, 021802 (2003)



A global model must describe all established data

1 disappearance of solar neutrinos2 oscillation of atmospheric neutrinos νµ → ντ

3 disappearance of reactor antineutrinos over hundreds of km.4 oscillation phase proportional to L/E at low and high energies

E = 2− 9 MeV

KamLAND, PRL 94, 081801 (2005)

E = 0.2− 100 GeV

Super-Kamiokande, PRL 93, 101801 (2004)



L/E oscillation phase arises from H ∝ E−1: positive slope in the KM plot

Model-independent description Massive model


Neutrinos in the SME Kostelecky & Mewes, PRD 69, 016005 (2004)Kostelecky & Mewes, PRD 85, 096005 (2012)

M.Mewes’ lecture III

6×6matrix

(a,b=e,µ,τ)

The effective hamiltonian in the SME has the form

HHHeff =

(|ppp|+ mmm2

2|ppp| 0

0 |ppp|+ mmm2∗

2|ppp|

)+

1|ppp|

(aeff − ceff −geff + Heff

−g†eff + H†eff −aT

eff − cTeff

)Neutrino 3× 3 block:

hhhνab = |ppp|δab +

mmm2ab

2|ppp|+∑djm

|ppp|d−3 Yjm(ppp)[(a(d)

eff )abjm − (c(d)

eff )abjm

]Novel effects

unconventional energy dependence (oscillation phase ∝ L/E, L, LE, . . .)

direction dependence (for j 6= 0)

local time dependence (for Earth-based experiments and m 6= 0)

CPT violation (differences between neutrinos and antineutrinos)

ν-ν mixing



Visualizing Lorentz violation in the KM plot

mass terms produce lines withpositive slope (HHH ∝ E−1)

positive powers of the energyproduce lines with negative slope

individual coefficients produce welldefined lines

combinations of coefficients producegeneral curves

can Lorentz violation producethe L/E phase without masses?

YES: Lorentz-violating seesawmechanism


Building a global model Kostelecky & Mewes, PRD 70, 031902 (2004)

LV seesaw mechanism: Bicycle model

HHHbicycle =

−2cE

a√2

a√2

a√2

0 0

a√2

0 0

E1 = −cE −√

(cE)2 + a2

E2 = 0E3 = −cE +

√(cE)2 + a2

very simple two-parameter model

neutrinos are massless

involves operators of dimensionsthree (a) and four (c)

hamiltonian is easily diagonalizable

oscillation phase at high energy:

∆32L = −cEL+ cEL

√1 +

a2

(cE)2

≈ −cEL+ cEL+a2

2cEL

≈ a2

2cL

E


Building a global model Kostelecky & Mewes, PRD 70, 031902 (2004)

LV seesaw mechanism: Bicycle model

very simple two-parameter model

neutrinos are massless

involves operators of dimensionsthree (a) and four (c)

hamiltonian is easily diagonalizable

oscillation phase at high energy:

∆32L = −cEL+ cEL

√1 +

a2

(cE)2

≈ −cEL+ cEL+a2

2cEL

≈ a2

2cL

E



Neutrinos in the SME

L/E behavior can be obtained byusing mass terms or the LV seesawmechanism

unconventional energy dependencein the SME gives us more freedom

use of high-dimension operatorscould explain anomalous signals


Neutrinos in the SME: global models

bicycle model Kostelecky & Mewes, PRD 70, 031902 (R) (2004)

tandem model Katori, Kostelecky, & Tayloe, PRD 74, 105009 (2006)

BMW model Barger, Marfatia, & Whisnant, PLB 653, 267 (2007)

puma model JSD & Kostelecky, PLB 700, 25 (2011); PRD 85, 016013 (2012)

isotropic bicycle model Barger, Liao, Marfatia, & Whisnant, PRD 84, 056014 (2011)

perturbed puma model Rong & Liu, CPL 29, 041402 (2012)

many others...



1 Give a hamiltonian HHH

2 diagonalize hamiltonian: determine eigenvalues (Ea′) and eigenvectors (va′)

:eigenvalues are given by the solutions of the cubic equation

(Ea′)3 − Tr(HHH)(Ea′)2 + 12

[(Tr(HHH))2 − Tr(HHH2)

](Ea′)− det(HHH) = 0

trick: set det(HHH) = 0one null eigenvalue: E0 = 0the other two eigenvalues are solutions of a quadratic equation: E±




2 diagonalize hamiltonian: determine eigenvalues (Ea′) and eigenvectors (va′):eigenvalues are given by the solutions of the cubic equation

(Ea′)3 − Tr(HHH)(Ea′)2 + 12

[(Tr(HHH))2 − Tr(HHH2)

](Ea′)− det(HHH) = 0

trick: set det(HHH) = 0one null eigenvalue: E0 = 0the other two eigenvalues are solutions of a quadratic equation: E±




2 diagonalize hamiltonian: determine eigenvalues (Ea′) and eigenvectors (va′)

3 construct mixing matrix

Ua′a =

v†1v†2v†3

a′=1,2,3; a=e,µ,τ

4 construct oscillation probabilities

Pνa→νb(L) =

∑a′,b′

U∗a′aUa′bUb′aU∗b′b e

i(Ea′−Eb′ )L

5 compare with established data


The puma modelJSD & Kostelecky, PLB 700, 25 (2011)JSD & Kostelecky, PRD 85, 016013 (2012)

Isotropic (j = 0)

Includes nonminimal terms

Three real parameters

Alternative to the 3νSM

Effective hamiltonian (neutrinos)

HHHνab = |ppp|δab +

mmm2ab

2|ppp|+∑djm

|ppp|d−3 Yjm(ppp)[(a(d)

eff )abjm − (c(d)

eff )abjm

](HHHpuma)ν

ab =mmm2

ab

2|ppp|+ |ppp|p−3 (a(p)

eff )ab00√

4π− |ppp|q−3 (c(q)eff )ab

00√4π



Isotropic (j = 0)





HHHpuma = A

1 1 11 1 11 1 1

+B

1 1 11 0 01 0 0

+ C

1 0 00 0 00 0 0

A = m2/2E, B = aE2, C = cE5.

“[this model] was discovered by a systematic hunt through the jungle of possibleSME-based models.”



Isotropic (j = 0)





HHHpuma = A

1 1 11 1 11 1 1

+B

1 1 11 0 01 0 0

+ C

1 0 00 0 00 0 0

A = m2/2E, B = aE2, C = cE5.

isotropic = “fried chicken” (models that everybody likes)

Beyonce’s rule: “If you like it then you should put a ring on it”



Hamiltonian

HHHpuma = A

1 1 11 1 11 1 1

+B

1 1 11 0 01 0 0

+ C

1 0 00 0 00 0 0

A = m2/2E, B = aE2, C = cE5

Eigenvalues

E0 = 0

E± =12

(3A+B + C ±

√(A−B − C)2 + 8(A+B)2

)LV seesaw mechanism at high energies

E−L ≈ L

2

(B + C −

√(B + C)2 + 8B2

)≈ L

2

(C − C

√1 + 8

B2

C2

)≈ L

2

(C − C − 4

B2

C

)= − 2a2

c

L

E



Puma model 3νSM

LB reactor

Pνe→νe ≈ 1− 0.89 sin2

(3m2L

4E

), Pνe→νe ≈ 1− 0.87 sin2

(∆m2

21L

4E

),

atmospheric

Pνµ→νµ≈ 1− 1.00 sin2

(a2L

cE

), Pνµ→νµ

≈ 1− 0.99 sin2

(∆m2

31L

4E

),

Match between the puma model and the 3νSM parameters requires

3m2 = ∆m221,

a2

c=

∆m231

4



Puma model

Established experimental results

Accelerator neutrinos X

Atmospheric neutrinos X

Reactor antineutrinos X

Solar neutrinos X

Anomalies

MiniBooNE low-energy excess X

MiniBooNE ν-ν difference X

LSND signal X∗



Consistency of the puma model with established data


The puma model

JSD & Kostelecky, PLB 700, 25 (2011)JSD & Kostelecky, PRD 85, 016013 (2012)

Predictions:

(sin2 2θ13)T2K > (sin2 2θ13)MINOS

T2K, PRL 107, 041801 (2011)

MINOS, PRL 107, 181802 (2011) XXX

(sin2 2θ13)reactors = 0 ×××

Daya Bay: sin2 2θ13 = 0.092± 0.018 PRL 108, 191802 (2012) (4.9σ)Double Chooz: sin2 2θ13 = 0.085± 0.051 PRL 108, 131801 (2012) (3.1σ)∗

RENO: sin2 2θ13 = 0.113± 0.023 PRL 108, 171803 (2012) (5.2σ)


Summary

Established data is well described by a model of massive neutrinos,

although there are hints that point to physics beyond the 3νSM.

Simple SME-based models show that neutrino data (including someanomalies) can be described using Lorentz violation.

The SME offers an excellent playground for a phenomenologist.

Many people have already built their models: build yours!(and give it a funny name).


Question #5

Three new experiments are beingdesigned to study conventional neutrinooscillations. The three experimentscombine different baselines and energies.

1 What will Experiment 1 measure?



4 What pair of experiments would yousupport?

5 What experiments are moresensitive to H ∝ aE0?

6 What experiments are moresensitive to H ∝ cE1?

1 Experiment 1: oscillations driven by ∆m221 = ∆m2

�.2 Experiment 2: oscillations driven by ∆m2

31 = ∆m2atm.

3 Experiment 3: null result expected.


Question #6

If the oscillation probability between neutrino flavorsνa, νb and antineutrinos νa, νb satisfy

Pνb→νa = Pνa→νb,

we then know that neutrinos do not violate CPTinvariance.

True

False

CPT invariance⇒ Pνb→νa= Pνa→νb

Pνb→νa= Pνa→νb

does not guarantee CPT invariance

Example: in the puma model a breaks CPT; however, the oscillation probability is

Pνµ→νµ≈ 1− sin2

(a2L

cE

)←− CPT invariant


Question #7

A hypothetical experiment measures thedisappearance of νa neutrinos and νa

antineutrinos. Using the equations below,the fit to the data indicates the oscillationparameters shown in the figure

Pνa→νa= 1− sin2 2θ sin2

(1.27∆m2L/E

),

Pνa→νa= 1− sin2 2θ sin2

(1.27∆m2L/E

).

How can we write the CPT-violatingquantity ∆m2 −∆m2 in terms of SMEcoefficients?

We can’t!

In realistic effective field theory ∆m2 = ∆m2 even in the presence of CPTviolation.If Pνa→νa

6= Pνa→νa, then there is a nonzero coefficient for CPT-odd Lorentz

violation.A different energy dependence is required in the data analysis.


Neutrino speed update (from last night!)

OPERA (2011): δt = 57.8± 7.8 ns. arXiv:1109.4897

MINOS (2007): δt = 126± 32± 64 ns. PRD 76, 072005 (2007)

2012:

OPERA: δt = 1.6± 1.1+6.1−3.7 ns.

Borexino: δt = 2.7± 1.2± 3.0 ns.

ICARUS: δt = 5.1± 1.1± 5.5 ns.

LVD: δt = 2.9± 0.6± 3.0 ns.

MINOS: δt = 18± 11± 29 ns.


ReferencesLorentz violation

- CPT violation and the Standard Model, D. Colladay and V.A. Kostelecky, PRD 55, 6760 (1997).- Lorentz-violating extension of the Standard Model, D. Colladay and V.A. Kostelecky, PRD 58, 116002 (1998).- Gravity, Lorentz violation, and the Standard Model, V.A. Kostelecky, PRD 69, 105009 (2004).- Overview of the Standard Model Extension: implications and phenomenology of Lorentz violation, R. Bluhm, Lect. Notes Phys 702, 191 (2006).

Lorentz-violating neutrino oscillations

- Lorentz and CPT violation in neutrinos, V.A. Kostelecky and M. Mewes, PRD 69, 016005 (2004).- Lorentz violation and short-baseline neutrino experiments., V.A. Kostelecky and M. Mewes, PRD 70, 076002 (2004).- Perturbative Lorentz and CPT violation for neutrino and antineutrino oscillations, J.S. Dıaz, V.A. Kostelecky, and M. Mewes, PRD 80, 076007 (2009).- Overview of Lorentz violation in neutrinos, J.S. Dıaz, arXiv:1109.4620 [hep-ph] (2011).- Neutrinos with Lorentz-violating operators of arbitrary dimension, V.A. Kostelecky and M. Mewes, PRD 85, 096005 (2012).

Global models of Lorentz-violating neutrinos

- Lorentz and CPT violation in the neutrino sector, V.A. Kostelecky and M. Mewes, PRD 70, 031902 (2004).- Global three-parameter model for neutrino oscillations using Lorentz violation, T. Katori, V.A. Kostelecky, and R. Tayloe, PRD 74, 105009 (2006).- Challenging Lorentz noninvariant neutrino oscillations without neutrino masses, V. Barger, D. Marfatia, and K. Whisnant, PLB 653, 267 (2007).- Lorentz noninvariant oscillations of massless neutrinos are excluded, V. Barger, J. Liao, D. Marfatia, and K. Whisnant, PRD 84, 056014 (2011).- Three-parameter Lorentz-violating texture for neutrino mixing, J.S. Dıaz and V.A. Kostelecky, PLB 700, 25 (2011).- Lorentz- and CPT-violating models for neutrino oscillations, J.S. Dıaz and V.A. Kostelecky, PRD 85, 096005 (2012).

Experimental searches of Lorentz-violating neutrinos

- Tests of Lorentz violation in muon antineutrino to electron antineutrino oscillations, LSND Collaboration, L.B. Auerbach et al., PRD 72, 076004 (2005).- Testing Lorentz invariance and CPT conservation with NuMI neutrinos in the MINOS near detector, MINOS Collaboration, P. Adamson et al.,

PRL 101, 151601 (2008).- A search for Lorentz invariance and CPT violation with the MINOS far detector, MINOS Collaboration, P. Adamson et al., PRL 105, 151601 (2010).- Search for a Lorentz-violating sidereal signal with atmospheric neutrinos in IceCube, IceCube Collaboration, R. Abbasi et al., PRD 82, 112003 (2010).- Test of Lorentz and CPT violation with short baseline neutrino oscillation excesses, MiniBooNE Collaboration, A.A. Aguilar-Arevalo et al.,

arXiv:1109.3480 [hep-ex] (2011).- Search for Lorentz invariance and CPT violation with muon antineutrinos in the MINOS near detector, MINOS Collaboration, P. Adamson et al.,

PRD 85, 031101 (2012).


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