Neutrino masses and unitarity of the leptonic mixing...

Carla BiggioUniversidad Autónoma de Madrid

Neutrino masses and unitarity of the leptonic mixing matrix

Padova, 20/XI/2006

Carla BiggioUniversidad Autónoma de Madrid

Neutrino masses and unitarity of the leptonic mixing matrix

Padova, 20/XI/2006

1. ν physics in the “standard” scenario:• in the SM: exactly massless• experimentally: oscillations → ν are massive

leptonic flavour mixing• ν masses beyond the SM: see-saw mechanisms→ low energy effects of high energy models:

unitarity violations arise…

2. ν physics without unitarity• oscillation probabilities and decay rates• determining the elements of leptonic mixing matrix

Outline:

ν in the SM: exactly massless

..~ chlHY RL +νν

( )LRRLDm νννν +No Dirac mass:

• No right-handed νR → no

• No scalar triplets Δ → no

• SM is renormalizable → no dim5

• U(1)B-L non-anomalous global symmetry → no radiatively generated


..~ chlHY RL +νν

..chllY Lc

L +ΔΔ

( )( ) ..~~ † chlHHlY Lc

L +∗βααβ

( )LRRLDm νννν +

( )cLLL

cLMm νννν +

No Dirac mass:


No Majorana mass:

• No scalar triplets Δ → no

• SM is renormalizable → no dim5

• U(1)B-L non-anomalous global symmetry → no radiatively generated


..~ chlHY RL +νν

..chllY Lc

L +ΔΔ

( )( ) ..~~ † chlHHlY Lc

L +∗βααβ

( )LRRLDm νννν +

( )cLLL

cLMm νννν +

No Dirac mass:


No Majorana mass:

Massless neutrinos → neither mixing nor CP violation in the leptonic sector

But experimentally…

Experiments on SOLAR, ATMOSPHERIC and TERRESTRIAL νhave shown that ν oscillate

→ ν are massive and there is flavour mixing in the lepton sector too


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−=

−−

−−

132313231223121323121223

132313231223121323122312

1313121312

ccescsscesccsscsesssccessccsescscc

Uii

ii

i

δδ

δδ

δ



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×

Φ

Φ

2

1

0000001

i

i

ee

m1, m2, m3

Majorana phases


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−=

−−

−−

132313231223121323121223

132313231223121323122312

1313121312


Uii

ii

i

δδ

δδ

δ

m1, m2, m3



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×

Φ

Φ

2

1

0000001

i

i

ee

Oscillation experimentsθ12 , θ23 , Δm2

12 , |Δm223|

Majorana phases


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−=

−−

−−

132313231223121323121223

132313231223121323122312

1313121312


Uii

ii

i

δδ

δδ

δ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛×

Φ

Φ

2

1

0000001

i

i

ee

m1, m2, m3



Oscillation experimentsθ12 , θ23 , Δm2

12 , |Δm223|

0νββ decay

Majorana phases

( )( )21.0

26.0232

223

2

2521

22

2

1104.2

09.011092.7+−

−

−

⋅=−=Δ

±⋅=−=Δ

eVmmm

eVmmm

atm

sun

Fogli et al. 05

• Sign (Δm223) ?

m

NORMAL INVERTED

ν3

ν2ν1

ν1

ν2

ν3

Mass spectrum: present status

( )( )21.0

26.0232

223

2

2521

22

2

1104.2

09.011092.7+−

−

−

⋅=−=Δ

±⋅=−=Δ

eVmmm

eVmmm

atm

sun

Fogli et al. 05

• Sign (Δm223) ?

• Absolute mass scale?

m

NORMAL INVERTED

ν3

ν2ν1

ν1

ν2

ν3


( )( )21.0

26.0232

223

2

2521

22

2

1104.2

09.011092.7+−

−

−

⋅=−=Δ

±⋅=−=Δ

eVmmm

eVmmm

atm

sun

eVmUm ieiie 2.2 2 <= ∑

eVmUm ieiiee 6.14.0 2 −<= ∑

Fogli et al. 05

• Sign (Δm223) ?

• Absolute mass scale?

m

NORMAL INVERTED

ν3

ν2ν1

ν1

ν2

ν3

• Tritium β decay

• 0νββ decay

• Cosmology ∑⎪⎩

⎪⎨

⎧

−−−

<68.042.07.10.11.36.1

ii mCMB

CMB + LSS

CMB + LSS + Lyman-α

Mainz, Troitsk

Heidelberg-Moscow

modeldependent


Mixing angles: present status

047.0sin

38.023.0sin

68.034.0sin

132

122

232

≤

−=

−=

θ

θ

θ

Maltoni, Schwetz, Tortola, Valle 04

Mixing angles: present status

047.0sin

38.023.0sin

68.034.0sin

132

122

232

≤

−=

−=

θ

θ

θ

FUTURE:- Dirac phase δ → appearance experiments- Majorana phases φ1 , φ2 → 0νββ decay

Maltoni, Schwetz, Tortola, Valle 04

ν masses beyond the SM

• right-handed νR →

Why ν are so light?Why νR does not acquire large Majorana mass?

.. .. ~ chmchlHY RLDRL +→+ νννν

1. Adding “light” fields




• scalar triplet Δ →

Why <Δ> « <H> ?


( ) ( )cLLL

cLM

cLLL

cL mllllY νννν +→+ΔΔ





• scalar triplet Δ →

Why <Δ> « <H> ?

→ maybe there are better solutions…


( ) ( )cLLL

cLM

cLLL

cL mllllY νννν +→+ΔΔ



2. Adding heavy fields

Heavy fields manifest in the low energy effective theory (SM)via higher dimensional operators




Dimension 5 operator:

†* ~~ HHllM L

cL βα

αβλ

lα

lβ Mαβλ





†* ~~ HHllM L

cL βα

αβλβα

αβ ννλ

Lc

LM v2→

lα

lβ Mαβλ





This mass term violates lepton number→ neutrinos must be Majorana fermions

It’s unique → very special role of ν masses:lowest-order effect of higher energy physics

→

lα

lβ Mαβλ

†* ~~ HHllM L

cL βα

αβλβα

αβ ννλ

Lc

LM v2


Tree-level realizations

Type I See-SawMinkowski, Gell-Mann, Ramond, Slansky, Yanagida, Glashow, Mohapatra, Senjanovic

NR fermion singlet

NR

NR



μ

Type II See-SawMagg, Wetterich, Lazarides, Shafi, Mohapatra, Senjanovic, Schecter, Valle

Δ scalar triplet


NR fermion singlet

NR

NR



μ

Type II See-SawMagg, Wetterich, Lazarides, Shafi, Mohapatra, Senjanovic, Schecter, Valle

Δ scalar triplet

Type III See-SawMa, Hambye et al., …

tR fermion triplettR

tRMt

Yt

Yt


NR fermion singlet

NR

NR


• radiative mechanisms: ex.) 1 loop:

Other realizations



Other realizations

• SUSY models with R-parity violation



Other realizations


• (models with large extra dimensions)

SMRνψ ⊃Dirac mass suppressed by (2πR)d/2



Other realizations


• (models with large extra dimensions)

• …

SMRνψ ⊃Dirac mass suppressed by (2πR)d/2

Low-energy effects of see-saw: type I

( ) ( )cRRR

cRLRRLRRSM NMNMNNlHYNNYHlNNiLL ∗+−+−∂/+=

21~~ ††

νν

...11 62

5 +Λ

+Λ

+= == ddSM

eff LLLLIntegrate out NR


...11 62

5 +Λ

+Λ

+= == ddSM


( ) ( )cRRR


21~~ ††

νν

d=5 operatorit gives mass to ν

( ) ..~~41 ††* chHHlY

MYl LL +⎟

⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡− τητ β

αβννα

rr

Mm

2v≈ν


...11 62

5 +Λ

+Λ

+= == ddSM


( ) ( )βαβ

ννα LL lHYM

YiHl ††2

~1~⎥⎦⎤

⎢⎣⎡∂/

d=5 operatorit gives mass to ν

d=6 operatorit renormalises kinetic energy

Broncano, Gavela, Jenkins 02

( ) ( )cRRR


21~~ ††

νν

( ) ..~~41 ††* chHHlY

MYl LL +⎟

⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡− τητ β

αβννα

rr

Mm

2v≈ν


( )

( ) ( ) .....cos

..2

..21

++−+−

++−∂/=

+ chPZgchPlWg

chMKiL

LW

L

c

αμ

αμαμ

αμ

βαβαβαβα

νγνθ

νγ

νννν

αβνναβαβ δ ⎟

⎠⎞

⎜⎝⎛+= †

2

2 12v Y

MYK

αβνναβ

η⎟⎠⎞

⎜⎝⎛= †*

2

2v Y

MYM

After EWSB:


( )

( ) ( ) .....cos

..2

..21

++−+−

++−∂/=

+ chPZgchPlWg

chMKiL

LW

L

c

αμ

αμαμ

αμ

βαβαβαβα

νγνθ

νγ

νννν


⎠⎞

⎜⎝⎛+= †

2

2 12v Y

MYK

αβνναβ

η⎟⎠⎞

⎜⎝⎛= †*

2

2v Y

MYM

After EWSB:

Mαβ → diagonalized → unitary transformation

Kαβ → diagonalized and normalized → unitary transf. + rescaling


( )

( ) ( ) .....cos

..2

..21

++−+−

++−∂/=

+ chPZgchPlWg

chMKiL

LW

L

c

αμ

αμαμ

αμ

βαβαβαβα

νγνθ

νγ

νννν


⎠⎞

⎜⎝⎛+= †

2

2 12v Y

MYK

αβνναβ

η⎟⎠⎞

⎜⎝⎛= †*

2

2v Y

MYM

After EWSB:

( ) ( ) ( ) .....)(cos22

1 † ++−−−∂/= + chNNPZgNPlWgmiL jijLiW

iiLiiiic

ii νγνθ

νγνννν μμα

μαμ

Mαβ → diagonalized → unitary transformation

Kαβ → diagonalized and normalized → unitary transf. + rescaling

iiN νν αα = N non-unitary

Low-energy effects of see-saw: type II & IIIAbada, CB, Bonnet, Gavela, in progress

type II (scalar triplet)

d=5 2

2v

Δ

≈M

m μν


type II (scalar triplet)

d=5

d=6

2

2v

Δ

≈M

m μν

• 4 Higgs interaction• 6 Higgs interaction• 4 fermions interaction

no correction to kinetic terms

→ no deviations from unitarity


type II (scalar triplet) type III (fermion triplet)

d=5

d=6

2

2v

Δ

≈M

m μν

tMm

2v≈ν




Low-energy effects of see-saw: type II & III

corrections to

• kinetic terms for neutrinos• kinetic terms for charged

fermions• interactions with gauge bosons

→ deviations from unitarity

Abada, CB, Bonnet, Gavela, in progress

type II (scalar triplet) type III (fermion triplet)

d=5

d=6

2

2v

Δ

≈M

m μν

tMm

2v≈ν




Low energy effects of a model with xdimDe Gouvea, Giudice, Strumia, Tobe 01

( )[ ]∫∫ +−+/= ..~44 chHYlLdxDidydxS RLSM βαβαααδ νψψ


⎟⎠⎞

⎜⎝⎛ ⋅

= ∑ Rynix

Ryx

n n

rrrrr

r r exp)()2(

1),( ,2/ αδα ψπ

ψDevelope in KK modes and perform ∫ δdy



⎟⎠⎞

⎜⎝⎛ ⋅

= ∑ Rynix

Ryx

n n

rrrrr

r r exp)()2(

1),( ,2/ αδα ψπ



∫ ∑⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛ ⋅

−∂/+=n RnLnnSM chH

RY

lR

niLdxS r rrr

rr

..~)2( 2/

4βδ

αβααα ν

πψγψ

Dirac mass 2/

2

)2(v

δν πλR

m ≈


⎟⎠⎞

⎜⎝⎛ ⋅

= ∑ Rynix

Ryx

n n

rrrrr

r r exp)()2(

1),( ,2/ αδα ψπ



∫ ∑⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛ ⋅

−∂/+=n RnLnnSM chH

RY

lR

niLdxS r rrr

rr

..~)2( 2/

4βδ

αβααα ν

πψγψ

Dirac mass 2/

2

)2(v

δν πλR

m ≈

Integrate out the heavy modes: ( ) [ ] ( )βαβαδδ LL

d lHYYiHlL ††26 ~~ ∂/Λ∝ −=

→ deviations from unitarity

A general statement…

We have unitarity violation whenever we integrate out heavy fermions:



...112 +

/−−=

−/ MDi

MMDi

v

v



...112 +

/−−=

−/ MDi

MMDi

v

v

It connects fermions withopposite chirality → mass term



...112 +

/−−=

−/ MDi

MMDi

v

vThere’s a γμ: it connects fermionswith the same chirality → correction to the kinetic terms




...112 +

/−−=

−/ MDi

MMDi

v

vThere’s a γμ: it connects fermionswith the same chirality → correction to the kinetic terms


The propagator of a scalar field does not contain γμ → if it generates neutrino mass, it cannot correct the kinetic term

Neutrino physics without unitarity

• Unitarity violations arise in models for ν masses with heavy fermions

• They can arise from other new physics




→ worthwhile studying neutrino physics relaxing the hypothesisof unitarity of the leptonic mixing matrix




Antusch, CB, Fernández-Martínez, Gavela, López-PavónJHEP 0610:084,2006 [hep-ph/0607020]

we constrain the matrix elements

• first only with present oscillation experiments and • then by combining them with other electroweak data

→ worthwhile studying neutrino physics relaxing the hypothesisof unitarity of the leptonic mixing matrix

Minimal Unitarity Violation Scheme

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−=

−−

−−

132313231223121323121223

132313231223121323122312

1313121312


Uii

ii

i

δδ

δδ

δ

( ) ( ) ( ) .....cos22

1++−−−∂/= + chPZgUPlWgmiL iLi

WiiLiiii

cii νγν

θνγνννν μ

μαμ

αμ


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−=

−−

−−

132313231223121323121223

132313231223121323122312

1313121312


Uii

ii

i

δδ

δδ

δ

( ) ( ) ( ) .....)(cos22


iiLiiiic

ii νγνθ

νγνννν μμα

μαμ

( ) ( ) ( ) .....cos22


WiiLiiii

cii νγν

θνγνννν μ

μαμ

αμ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

321

321

321

τττ

μμμ

NNNNNNNNN

Neee


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−=

−−

−−

132313231223121323121223

132313231223121323122312

1313121312


Uii

ii

i

δδ

δδ

δ

( ) ( ) ( ) .....)(cos22


iiLiiiic

ii νγνθ

νγνννν μμα

μαμ

( ) ( ) ( ) .....cos22


WiiLiiii

cii νγν

θνγνννν μ

μαμ

αμ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

321

321

321

τττ

μμμ

NNNNNNNNN

Neee

This is completely general and model independent

• 3 light ν• all unitarity violation from NP @ E > ΛSM

Unique assumptions:

The effective Lagrangian in the MUV

( ) ( ) ( ) .....)(cos22


iiLiiiic

ii νγνθ

νγνννν μμα

μαμ


unchanged

⇓

( ) ( ) ( ) .....)(cos22


iiLiiiic

ii νγνθ

νγνννν μμα

μαμ

ijji δνν =


unchanged

⇓

Non-unitarity effects appear in the interaction:

( ) ii

iii

i NNNN

ννν αα

αα

α ∑∑ ∗∗ ≡= ~1†

⇓

( ) αβαβαβ δνν ≠= ∗ tNN ~~

( ) ( ) ( ) .....)(cos22


iiLiiiic

ii νγνθ

νγνννν μμα

μαμ

ijji δνν =


unchanged

⇓

Non-unitarity effects appear in the interaction:

( ) ii

iii

i NNNN

ννν αα

αα

α ∑∑ ∗∗ ≡= ~1†

⇓

( ) αβαβαβ δνν ≠= ∗ tNN ~~

( ) ( ) ( ) .....)(cos22


iiLiiiic

ii νγνθ

νγνννν μμα

μαμ

ijji δνν =

This affects both electroweak decays and oscillation probabilities…

ν oscillations in vacuum

iijj

freeiji

freei EHH

dtdi νννν === ∑ˆ• mass basis 0)(

i

itiEi et νν −=


ββ

αβαα ννν ∑== freefree EHdtdi ˆ• flavour basis

iijj

freeiji

freei EHH


i

itiEi et νν −=

( ) 1** ~~ −= βααβ j

freeiji

free NHNEwith tj

freeiji

free NHNH βααβ~~*=≠


ββ


iijj

freeiji

freei EHH


i

itiEi et νν −=

( ) 1** ~~ −= βααβ j

freeiji

free NHNEwith tj

freeiji


( ) ( ) ( )ββαα

βα

αβ ††

2*

,NNNN

NeNLEP i

iLiP

ii∑

=


ββ


( ) ( ) ( )ββαα

βα

αβ ††

2*

,NNNN

NeNLEP i

iLiP

ii∑

=

Zero-distance effect:

iijj

freeiji

freei EHH


i

itiEi et νν −=

( )( )

( ) ( )ββαα

αβαβ ††

2†

0,NNNN

NNEP =

( ) 1** ~~ −= βααβ j

freeiji

free NHNEwith tj

freeiji


ν oscillations in matter

2 families

∑−=−α

αα νγννγν 00int

212 nFeeeF nGnGL

VCC VNC

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

μμ νν

νν e

NC

NCCCte

VVV

UE

EU

dtdi

00

00

2

1*

ν oscillations in matter

2 families

∑−=−α

αα νγννγν 00int

212 nFeeeF nGnGL

VCC VNC

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

μμ νν

νν e

NC

NCCCte

VVV

UE

EU

dtdi

00

00

2

1*

( )( )( ) ( )

( ) ( )

( ) ( )( ) ( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−

−−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −

μμμμ

μμ

μμμ

μ νν

νν e

NCeee

NCCC

eee

NCeeNCCCe

NNVNNNNNNVV

NNNNNN

VNNVVN

EE

Ndtdi

†††

†

††

††

1*

2

1*

00

1. non-diagonal elements 2. NC effects do not disappear

N elements from oscillations: e-row

( ) ( ) ( ) ( )232

32

22

14

3

222

21 cos2ˆ Δ++++≅→ eeeeeeee NNNNNNP νν

Only disappearance exps → informations only on |Nαi|2

CHOOZ: Δ12≈0

K2K (νμ→νμ ): Δ23

1. Degeneracy

2.

cannot be disentangled

23

22

21 eee NNN ↔+

22

21 , ee NN

ELmijij 22Δ=Δ

UNITARITY


( ) ( )122

22

14

34

24

1 cos2ˆ Δ+++≅→ eeeeeee NNNNNP νν

⎪⎩

⎪⎨⎧

≈

≈+

01

23

22

21

e

ee

NNN

→ first degeneracy solved

KamLAND: Δ23>>1

KamLAND+CHOOZ+K2KKamLAND+CHOOZ+K2K


( ) ( )122

22

14

34

24

1 cos2ˆ Δ+++≅→ eeeeeee NNNNNP νν

⎪⎩

⎪⎨⎧

≈

≈+

01

23

22

21

e

ee

NNN

KamLAND: Δ23>>1

SNO:

( ) 22

21 9.01.0ˆ

eeee NNP +≅→νν

→ first degeneracy solved

→ all |Nei|2 determined

KamLAND+CHOOZ+K2K

SNO

N elements from oscillations: μ-row

( ) ( ) ( ) ( )23

2

3

2

2

2

1

4

3

22

2

2

1 cos2ˆ Δ++++≅→ μμμμμμμμ νν NNNNNNP

1. Degeneracy

2.

cannot be disentangled

2

3

2

2

2

1 μμμ NNN ↔+

2

2

2

1 , μμ NN

Atmospheric + K2K: Δ12≈0

UNITARITY

N elements from oscillations only

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−

<−−=

81.056.074.044.053.020.082.058.073.042.052.019.0

20.061.047.088.079.0U

González-García 04

with unitarityOSCILLATIONS

without unitarityOSCILLATIONS

3σ

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−=+

<−−=

???86.057.086.057.0

34.066.045.089.075.0][ 2/12

22

1 μμ NNN

…adding near detectors…

Test of zero-distance effect: ( ) ( ) αβαβαβ δ≠=2†0, NNEP

• KARMEN: (NN†)μe <0.05

• NOMAD: (NN†)μτ <0.09 (NN†)eτ <0.013• MINOS: (NN†)μμ =1±0.05

• BUGEY: (NN†)ee =1±0.04

…adding near detectors…

Test of zero-distance effect: ( ) ( ) αβαβαβ δ≠=2†0, NNEP

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−−

<−−=

???85.057.081.022.069.000.0

27.066.045.089.075.0N

→ also all |Nμi|2 determined

• KARMEN: (NN†)μe <0.05

• NOMAD: (NN†)μτ <0.09 (NN†)eτ <0.013• MINOS: (NN†)μμ =1±0.05

• BUGEY: (NN†)ee =1±0.04

Electroweak decays

( )ijNN † ≈νi

Z

νj

iNα ≈W νi

lα

( )ααα†2 NNN SM

iiSM Γ=Γ=Γ ∑

( )∑Γ=Γij

ijSM NN2†

Electroweak decays

νiZ

νj

iNα ≈W νi

lα

( )ααα†2 NNN SM

iiSM Γ=Γ=Γ ∑

With decays we can only constrain (NN†) and (N†N) ,we cannot extract the matrix elements

→ we need oscillations!

Different from quark sector…

( )ijNN † ≈ ( )∑Γ=Γij

ijSM NN2†

(NN†) from decays

( )( ) ( )μμ

αα††

†

NNNN

NN

ee

→• W decays

( )( ) ( )μμ

αβαβ

††

†

NNNN

NN

ee

∑→• Invisible Z

• Universality tests( )( )ββ

αα†

†

NNNN

→

Infos on(NN†)αα

(NN†) from decays

( )( ) ( )μμ

αα††

†

NNNN

NN

ee

→• W decays

( )( ) ( )μμ

αβαβ

††

†

NNNN

NN

ee

∑→• Invisible Z

• Universality tests( )( )ββ

αα†

†

NNNN

→

Infos on(NN†)αα

GF is measured in μ-decay

jν

W¯

μ

e

νiNμi

N*ej

( ) ( )μμμ

π††

3

52

192NNNN

mGee

F=Γ ( ) ( )μμ††

2exp,2

NNNNG

Gee

FF =

(NN†) from decays

• Rare leptons decays Infos on (NN†)αβW

νi

lα lβ

γ

Nαi N*βi

(NN†) from decays

• Rare leptons decays Infos on (NN†)αβ

SM → GIM suppression: 54

3,22

21 10

323) −

=

∗ <Δ

=→ ∑i W

iii M

mUUlBr(l βαβα παγ

( )( ) ( )ββαα

αβ††

2†

NNNN

NN→

Now → no suppression:→ constant term leading

W

νi

lα lβ

γ

Nαi N*βi

(NN†) from decays

• Rare leptons decays Infos on (NN†)αβ

• μ-e conversion in nuclei• μ → e+e-e

SM → GIM suppression: 54

3,22

21 10

323) −

=

∗ <Δ

=→ ∑i W

iii M

mUUlBr(l βαβα παγ

( )( ) ( )ββαα

αβ††

2†

NNNN

NN→

Now → no suppression:→ constant term leading

W

νi

lα lβ

γ

Nαi N*βi

(NN†) and (N†N) from decays

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

±⋅<⋅<⋅<±⋅<⋅<⋅<±

≈−−

−−

−−

005.0003.1103.1106.1103.1005.0003.1102.7106.1102.7005.0002.1

22

25

25

†NNExperimentally

(NN†) and (N†N) from decays

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

±⋅<⋅<⋅<±⋅<⋅<⋅<±

≈−−

−−

−−

005.0003.1103.1106.1103.1005.0003.1102.7106.1102.7005.0002.1

22

25

25

†NNExperimentally

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

±<<<±<<<±

≈032.000.1032.0032.0

032.0032.000.1032.0032.0032.0032.000.1

†NNEstimation

(the most conservative)

→ N is unitary at % level

†2† with 1 εεε =+== HNN

'1 1 †† εε +=+= VVNN 03.0'2

≈≤ ∑αβ

αβεε ij

HVN =

N elements from oscillations & decays

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−

<−−=

81.056.074.044.053.020.082.058.073.042.052.019.0

20.061.047.088.079.0U

González-García 04

with unitarityOSCILLATIONS

without unitarityOSCILLATIONS

+DECAYS3σ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−

<−−=

82.054.075.036.056.013.082.057.073.042.054.019.0

20.065.045.089.076.0N

In the future…

MEASUREMENT OF MATRIX ELEMENTS

• |Ne3|2 , μ-row → MINOS, T2K, Superbeams, NUFACTs…• τ-row → high energies: NUFACTs• phases → appearance experiments: NUFACTs, β-beams

In the future…



Rare leptons decays

• μ→eγ

• τ→eγ

• τ→μγ

TESTS OF UNITARITY

( ) 016.0† <τeNN

( ) 013.0† <μτNN

( ) 5† 102.7 −⋅<μeNN

PRESENT

In the future…



Rare leptons decays

• μ→eγ

• τ→eγ

• τ→μγ

TESTS OF UNITARITY

( ) 016.0† <τeNN

( ) 013.0† <μτNN

( ) 5† 102.7 −⋅<μeNN

PRESENT FUTURE

~ 10-6 MEG

~ 10-7 NUFACT

In the future…



Rare leptons decays

• μ→eγ

• τ→eγ

• τ→μγ

TESTS OF UNITARITY

( ) 016.0† <τeNN

( ) 013.0† <μτNN

( ) 5† 102.7 −⋅<μeNN

PRESENT FUTURE

~ 10-6 MEG

~ 10-7 NUFACT

ZERO-DISTANCE EFFECT40Kt Iron calorimeter @ NUFACT

• νe→νμ

4Kt OPERA-like @ NUFACT

• νe→ντ

• νμ→ντ ( ) 3† 106.2 −⋅<μτNN

( ) 3† 109.2 −⋅<τeNN

( ) 4† 103.2 −⋅<μeNN

Conclusions

• ν oscillations are an indication of ν masses

• ν masses require new physics beyond the SM

• model for ν masses involving heavy fermions produce violationsof unitarity at low energy

• unitarity violation can be a general effect of new physics

→ we studied ν physics relaxing the hypothesis of unitarity

If we don’t assume unitarity for the leptonic mixing matrix

• Present oscillation experiments alone can only measure the e-row and part of the μ-row

• EW decays probes unitarity at % level

• Combining oscillations and EW decays we obtain values forthe leptonic mixing matrix comparable with the ones obtainedwith the unitary analysis

Neutrino masses and unitarity of the leptonic mixing...

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