Do recent results on neutrino oscillations falsify the Standard Model?!

Do recent results on neutrino oscillations falsify the Standard Model?!

Jan T. Sobczyk

Institute for Theoretical Physics Wrocªaw University

October 11, 2010

Plan of the seminar

1 Introduction

2 Basic theoretical scheme for neutrino oscillations 3 Resume' of old (!!!) experimental results

4 Open questions

5 How many mass eigenstates?

6 MINOS anomaly

7 Conclusions

Current Section

1 Introduction

2 Basic theoretical scheme for neutrino oscillations 3 Resume' of old (!!!) experimental results

4 Open questions

5 How many mass eigenstates?

6 MINOS anomaly

7 Conclusions

Motivation for this seminar:

The investigation of neutrino oscillations started with the Davis solar neutrino Homestake experiment.

After SuperKamiokande reported atmospheric neutrino

oscillations signal several new experiments have been launched.

For many years the situation was boring: all the results could be accomodated in the Standard Model.

There are two new oscillation experimental results which if

conrmed demostrate that the Standard Model in incomplete.

Current Section

1 Introduction

2 Basic theoretical scheme for neutrino oscillations 3 Resume' of old (!!!) experimental results

4 Open questions

5 How many mass eigenstates?

6 MINOS anomaly

7 Conclusions

We need a theoretical scheme in order to understand the data.

We assume that there are three families with Dirac (for oscillation analysis they can be Majorana as well) neutrinos. The states with well dened avour are linear combinations of states with well dened mass:

|ν _l >= X

m

U _lm |ν _m > .

In the textbook derivation of the vacuum oscillation formula, we

assume that neutrino has well dened momentum ~p = (p, 0, 0) and

thus various mass states have dierent energies (and velocities!).

Because E _m ≈ p + ^M _2p

:

|ν _l ( x, t) >= X

m

U _lm |ν _m ( 0) > e ^{− i(E}

^t−px)

≈ e ^ip(x−t) X

m

U _lm |ν _m ( 0) > e ^{− i}

,

P(ν _l → ν _k , x) = | < ν _k ( x, t)|ν _l ( 0, 0) > | ²

P(ν _l → ν _k ; L) = X

m

| U _km | ² | U _lm | ²

+2 X

m>m

| U _km U _lm ^∗ U _km ^∗

U _lm

| cos

 L(M _m ² − M _m ²

)

2p − Φ _k,l;m,m

 ,

where Φ _k,l;m,m

= arg U _km U _lm ^∗ U _km ^∗

U _lm

. P(¯ν _l → ¯ ν _k ; L) = X

m

| U _km | ² | U _lm | ²

+ 2 X

m>m

| U _km U _lm ^∗ U _km ^∗

U _lm

| cos

 L(M _m ² − M _m ²

)

2p + Φ _k,l;m,m



,

Oscillations occur only if neutrinos are massive.

We restrict to two families only. A convenient way to discuss the oscillations::

i d dt

 ν ₁ ν ₂



= ˆ H

 ν ₁ ν ₂

 ,

H = ˆ

 E ₁ 0 0 E ₂



≈ p + ^M _2E

0 0 p + ^M _2E

!

=



p + M ₁ ² + M ₂ ² 4E

  1 0 0 1



− ∆ M ₁₂ ² 4E

 1 0 0 −1



.

Only the non-diagonal part is relevent for oscillations.

The Hamiltonian in the avour basis ˆH _f

 ν ν

ν µ



= U

 ν ₁ ν ₂



, U =

 cos Θ sin Θ

− sin Θ cos Θ

 ,

H ˆ _f = U ˆHU ^{− 1} =



p + M ₁ ² + M ₂ ² 4E

  1 0 0 1



− ∆ M ² 4E

 cos 2Θ − sin 2Θ

− sin 2Θ − cos 2Θ



The non-diagonal part gives rise to the oscillation pattern:

P(ν e → ν _µ ; L) = sin ² 2Θ sin ² L∆M ²

4E .

The wave packets analysis leads to the same results and we accept the standard theory.

But we must add matter eects which are very importants.

Before we do that, we present the typical oscillation analysis.

For a long time the best estimation of Θ ₁₃ was coming from the reactor neutrinos CHOOZ experiment.

No oscillations were seen with the accuracy of 5%:

sin ² 2Θ sin ² 1.27 ∆ M ² [ GeV ² ] L[km]

E[GeV ] < 0.05.

The experiment is characterized by:

L ∼ = 1 km, E ∼ = 5 · 10 ^{− 3} GeV.

On the left below we see the contour:

sin ² 2Θ sin ² 1.27 ∆ M ² [ GeV ² ] L[km]

E[GeV ] = 0.05.

The excluded region is on the right from the curve. Because neutrinos are not monoenergetic only few oscillation minima and maxima can be seen.

On the right below we show the excluded region from the actual

analysis.

Each experiment is sensitive to some values of ∆M ² . From the basic oscillation formula

min(∆M ² ) ∼ 2 < E >

L .

Typically, for each experiment one works out an approximation with a dominant 2D oscillation pattern.

There has been also a lot of research on the full 3D oscillation

parameters pattern.

In matter neutrinos are subject to scattering and absorption. The main eect is elastic forward scattering with coherently summed scattered waves. As a result, a refraction index does appear:

n α − 1 = X

j

f α ^j ( 0) · N j

k ² ,

α = e, µ, τ, f α ^j (ϑ) is the amplitude of ν α scattering in the angle ϑ

on j component of the matter with density N _j .

The refraction indices change the phase velocities of neutrino waves. If matter is nonsymmetric with respect to neutrino falavour states, the additional phase dierence appears:

∆φ = k(n e − n µ ) · t = X

j

∆ f ^j (0) · N _j k t.

The eect comes from dierent ν e and ν µ interactions with electrons.

∆φ =

√ 2G F N e t.

It is useful to introduce the eective potential (or strictly speaking the dierence of potentials):

V = √

2G F N e .

In the matter the Hamiltonian becomes:

H ˆ _f ^matt =



p + M ₁ ² + M ₂ ² 4E

  1 0 0 1



− ∆ M ² 4E

 cos 2Θ − sin 2Θ

− sin 2Θ − cos 2Θ

 + V

2

 1 0 0 1

 + V

2

 1 0 0 −1

 . The non-diagonal part, which is responsible for oscillations, can be written as (η = _{∆ M} ^{V /2}

/ 4E )

∆ M ² 4E

 −( cos 2Θ − η) sin 2Θ sin 2Θ cos 2Θ − η



=

= ∆ M _matt ² 4E

 − cos 2Θ _matt sin 2Θ _matt sin 2Θ matt cos 2Θ matt



,

where

sin 2Θ matt = sin 2Θ

q sin ² 2Θ + (cos 2Θ − η) ² ,

∆ M _matt ² = ∆ M ² q

sin ² 2Θ + (cos 2Θ − η) ² .

It shouls be clear that in the matter we get the identical oscillation formula, however with dierent parameters

Θ → Θ _matt , ∆ M ² → ∆ M _matt ² .

From the CPT invariance

P(ν _x → ν _y ; L) = P(¯ν _y → ¯ ν _x ; L) and in particular

P(ν _x → ν _x ; L) = P(¯ν _x → ¯ ν _x ; L)

The study of disappearance in vacuum tells us nothing about CP violation.

The proper measure of CP asymmetry:

A ⁽ _CP ^l

^l) = P(ν _l → ν _l

; L) − P(¯ν _l → ¯ ν _l

; L)

A ^{( l}

^l) = 4 X

(U U ^∗ U ^∗ U ) sin M _m ² − M _m ²

L.

A ^(µ _CP ^e) = −A ^(τ _CP ^e) = A ^{(τ µ)} _CP

= 4J _CP



sin M ₃ ² − M ₂ ²

2p L + sin M ₂ ² − M ₁ ²

2p L + sin M ₁ ² − M ₃ ²

2p L

 .

J _CP = = U µ 3 U _e3 ^∗ U _e2 U _µ ^∗ ₂

If any two masses are equal there is no CP violation!

Matter eects have important impact on CP violation-like eects.

The matter is not C invariant (it contains e ⁻ and not e ⁺ ).

For antineutrinos the eective potential changes sign:

N e → − N e .

With the matter eects there can be dierent ∆M _matt ² for

neutrinos and for antineutrinos.

Current Section

1 Introduction

2 Basic theoretical scheme for neutrino oscillations 3 Resume' of old (!!!) experimental results

4 Open questions

5 How many mass eigenstates?

6 MINOS anomaly

7 Conclusions

We assume three families of Dirac/Majorana massive neutrinos.

The general form of the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix is:

U =





1 0 0

0 c 23 s 23

0 −s ₂₃ c ₂₃



 ×





c 13 0 s 13 e ^{− iδ}

0 1 0

− s ₁₃ e ^iδ 0 c ₁₃





×





c _{12 12} 0

− s ₁₂ c ₁₂ 0

0 0 1



 ×





e ^iα

^{/ 2} 0 0 0 e ^iα

^{/ 2} 0

0 0 1



 .

Th last factor (phases α 1,2 ) is present for Majorana neutrinos only.

The oscillation formula contains three mixing angles: Θ 12 , Θ 13 , Θ ₂₃ and two independent dierences of squares of masses

∆ ² _jk = M _j ² − M _k ² . α _1,2 do not enter the formula.

The conventional ordering of avours is (ν e , ν µ , ν τ ) .

Atmospheric neutrinos (later on conrmed in K2K, MINOS long baseline experiments):

|∆ ² ₃₁ | ∼ = 2.4 · 10 ^{− 3} eV ² , Θ ₂₃ ∼ = 39 − 51 ^o ,

Solar neutrinos (later on conrmed in KAMLAND reactor neutrino experiment)

∆ ² ₂₁ ∼ = 7.6 · 10 ^{− 5} eV ² , Θ ₁₂ ∼ = 34 ^o ,

Finally from CHOZZ reactor neutrino experiment

Θ ₁₃ < 11 ^o .

There is a very interesting global data analysis which includes the most recent results from the KAMLAND [arXiv 1009.4771 (hep-exp)].

Perhaps we already know the value of Θ ₁₃ ?!

Note that sin ² Θ ₁₃ ∼ 0.017

translates to sin ² 2Θ 13 ∼ 0.068.

Current Section

1 Introduction

2 Basic theoretical scheme for neutrino oscillations 3 Resume' of old (!!!) experimental results

4 Open questions

5 How many mass eigenstates?

6 MINOS anomaly

7 Conclusions

Open questions:

Θ ₁₃ ?

absolute mass scale?

mass hierarchy?

Dirac/Majorana?

how many families?

There is a variety of approaches and it is dicult to predict which one will be most successfull.

direct measurement in tritium β decay

< m β >= q

P j | U _ej | ² m ² _j < 2eV

KATRIN will be sensitive to m _j ∼ 0.35 eV.

cosmology, from WMAP and large scale structure P

j m j < ( 0.4 − 1) eV 0ν2β decay

< m > depends on the Majorana phases α 1,2

supernova

SN1987A allowed for the bound of 12 eV.

There are two options for the mass hierarchy:

[from B. Kayser]

Current Section

1 Introduction

2 Basic theoretical scheme for neutrino oscillations 3 Resume' of old (!!!) experimental results

4 Open questions

5 How many mass eigenstates?

6 MINOS anomaly

7 Conclusions

In the Standard Model there are three light neutrino

avour states.

In 1995 LSND reported a puzzling oscillation signal

A part of the allowed region was excluded by Burgey and KARMEN experiments.

∆ M ² > 0.2 eV ² . 3.8σ eect.

If conrmed > 3 neutrino

mass states are necessary!

MiniBooNE experiment has been set up at FermiLab in order to investigate the same L/E region.

[from G. Garvey]

With the neutrino ux no oscillation signal was observed (the

integrated ν µ ux from 6.46E20 POT).

The very recent antineutrino ux data (from 5.67E20 POT) seem

to conrm the LSND signal!

It is interesting to put together LSND and MiniBooNE antineutrino

data points:

Global analysis of electron anti-neutrino data: Giunti & Laveder, arXiv:1010.1395

[hep-ph].

The data from the

LSND, MinBooNE,

KARMEN and reactor

experiments are in

excellent agreement.

Georgia Karagiorgi theory:

3 active + 1 sterile scheme cannot account for an apparent CP violation, in the 2-families approximation (the leading eect):

P(ν µ → ν _e , L) = 4 | U _e4 | ² | U µ 4 | ² sin ² ( 1.27∆m ₄₁ ² L/E), P(ν µ → ν _e , L) = P(¯ν µ → ¯ ν _e , L).

With an extra sterile neutrino i.e. in the 3+2 scheme we get P(ν µ → ν _e , L) = 4 | U e4 | ² | U µ 4 | ² sin ² ( 1.27∆m ₄₁ ² L/E)

+ 4 | U e5 | ² | U µ 5 | ² sin ² ( 1.27∆m ² ₅₁ L/E)

+ 4 | U e4 | | U µ4 | | U e5 | | U µ5 | sin(1.27∆m ₄₁ ² L/E) sin(1.27∆m ² ₅₁ L/E)

× cos(1.27∆ 54 L/E − φ 54 .

For antineutrinos φ ₅₄ → −φ ₅₄ .

[from G. Koragiorgi]

Akhmedov & Schwetz theory, arXiv:1007.4171 [hep-ph].

In addition to the standard CC interaction there is a term:

L _NSI = − 2 √

2G _F X

αβ

 ^{f ,f} _αβ

( L, R) ¯fP _L,R γ ^µ f ⁰

¯l α P _L γ µ ν _β
+ h.c.

In the presence of FSI a neutrino produced/detected along with a charged lepton l α in a process (f , f ⁰ ) ≡ X is a linear combination of

avour eigenstates:

| ν _α ^X >= C α ^X



| ν _α > + X

β

 ^X _αβ | ν _β >



 ,

where C α ^X is the normalization constant.

Akhmedov & Schwetz theory, arXiv:1007.4171 [hep-ph] (cont).

In addition to the standard avour states there exist also a fourth light sterile neutrino ν _s . As usual

| ν _α >= X

j

U α j | ν _j > .

The 3+1 scheme is assumed i.e. ∆m ₄₁ ² ∼ 1 eV ² . It is possible to achieve

P(ν α → ν _β , L) 6= P(¯ν α → ¯ ν _β , L).

MiniBooNE plans to double (almost) the statistics to

∼ 10E20 POT.

Also new experiments are planned: uBooNE and BooNE.

In Europe there is an idea to put the ICARUS detector near CERN

on an o-axis beam.

Current Section

1 Introduction

2 Basic theoretical scheme for neutrino oscillations 3 Resume' of old (!!!) experimental results

4 Open questions

5 How many mass eigenstates?

6 MINOS anomaly

7 Conclusions

The aim of the MINOS experiment is to investigate the oscillation region of the atmospheric neutrinos.

Recently the antineutrino data were published with unexpected results.

On the plot there are ts for the oscillation

parameters determined by the measurements of P(ν µ → ν _µ ) and P(¯ν µ → ¯ ν µ ) .

The MSW eect with

standard interactions

cannot explain the results.

Kopp, Machado, Parke theory (arXiv:1009.0014 [hep-ph]) Non-standard NC interactions are considered:

L _NSI = − 2 √

2G F ε _αβ [¯ f γ ^µ Pf ][¯ν α γ µ P L ν _β ].

The authors conne to two family ν µ and ν τ system.

The eective Hamiltonian is:

H _e = − ∆ M ² 4E

 cos 2Θ − sin 2Θ

− sin 2Θ − cos 2Θ

 +

+ A 2E

 ε µµ ε µτ

ε ^∗ _µτ ε _{τ τ}

 ,

√

The survival probability is:

P(ν µ → ν _m u) = 1 − | ∆ M ² sin 2Θ + 2ε µτ A | ²

∆ M _N ⁴ sin ²  ∆ M _N ² L 4E

 ,

∆ M _N ² = q

(∆ M ² cos 2Θ + ε τ τ A) ² + | ∆ M ² sin 2Θ + 2ε µτ A | ² .

For antineutrinos ε µτ → ε ^∗ _µτ and A → −A.

Thus P(ν µ → ν _µ ) 6= P(¯ν µ → ¯ ν _µ ) .

Heeck, Redejohann theory (arXiv:1007.2655 [hep-ph]) Extra U(1) gauge symmetry is introduced L µ − L τ is gauged and the theory is anomaly free

L = L _SM + L _Z

+ L _mix

L _Z

= − 1

4 Z ˆ ⁰ µν Z ˆ ^{0 µν} + 1

2 M ˆ ⁰² _Z Z ˆ ⁰ µ Z ˆ ^{0 µ} − ˆ g ⁰ j ^0µ Z ˆ ⁰ µ , j ^0µ = ¯ µγ ^µ µ + ¯ ν _µ γ ^µ P _L ν _µ − ¯ τ γ ^m uτ − ¯ν τ γ ^µ P _L ν _τ .

The term ¹ ₂ M ˆ ⁰² _Z Z ˆ ⁰ µ Z ˆ ^{0 µ} is generated by an unspecied Higgs sector.

sin χ

MINOS will have more data and better precison:

[from P. Vahle]

Current Section

1 Introduction

2 Basic theoretical scheme for neutrino oscillations 3 Resume' of old (!!!) experimental results

4 Open questions

5 How many mass eigenstates?

6 MINOS anomaly

7 Conclusions

Conclusions:

The recent MiniBooNE and MINOS antineutrino oscillation results can open a window to a physics beyond SM.

Good time for theorists: models with extra sterile neutrinos and/or extra interactions are testable

It is important to have better statistics results and also

conrmations from other experiments.