PMNS Matrix Elements Without Assuming Unitarity

PMNS Matrix Elements Without Assuming Unitarity

Enrique Fernández MartínezUniversidad Autónoma de Madrid

hep-ph/0607020

In collaboration with S. Antusch, C. Biggio, M.B. Gavela and J. López Pavón

NOW06, September 9-16, 2006, Otranto

Thanks also to C. Peña Garay

Motivations

• masses and mixing → evidence of New Physics beyond the SM

• Typical explanations (see-saw) involve NP at higher energies

• This NP often induces deviations from unitarity of the PMNS at low energy

We will analyze the present constraints on the mixing matrix without assuming unitarity

13 12 12 13 13

12 23 23 13 12 12 23 23 13 12 23 13

23 12 23 13 12 23 12 23 13 12 23 13

i

i i

i i

c c s c s e

U s c s s c e c c s s s e s c

s s c s c e s c c s s e c c

1 2 3

1 2 3

1 2 3

e e eN N N

N N N N

N N N

The general idea

Effective Lagrangian

.....cos

..2

..2

1 chPZ

gchPlW

gchMKiL L

WL

c

• 3 light • deviations from unitarity from NP at high energy


.....cos

..2

..2

1 chPZ

gchPlW

gchMKiL L

WL

c


iiN

Diagonal mass and canonical kinetic terms:

unitary transformation + rescaling

.....)(cos22

1 † chNNPZg

NPlWg

miL jijLiW

iiLiiiic

ii

N non-unitary


.....cos

..2

..2

1 chPZ

gchPlW

gchMKiL L

WL

c


iiN

Diagonal mass and canonical kinetic terms:

unitary transformation + rescaling

.....)(cos22

1 † chNNPZg

NPlWg

miL jijLiW

iiLiiiic

ii

N non-unitaryunchanged

ijji

The effects of non-unitarity…

… appear in the interactions

†2

NNN SMi

iSM ij

ijSM NN2†

This affects electroweak processes…

W -i

iN l

Z

i

ijNN † j

The effects of non-unitarity…

… appear in the interactions

†2

NNN SMi

iSM ij

ijSM NN2†

This affects electroweak processes…

ii

iii

i

ii

NNN

~12 tNN

~~

… and oscillation probabilities…

W -i

iN l

Z

i

ijNN † j

oscillations in vacuum

iijj

freeiji

freei EHH

dt

di ˆ• mass basis

0)(i

itiEi et


freefree EHdt

di ˆ• flavour basis

iijj

freeiji

freei EHH

dt

di ˆ• mass basis

0)(i

itiEi et

1** ~~ j

freeiji

free NHNEwith tj

freeiji

free NHNH ~~*≠


freefree EHdt


iijj

freeiji

freei EHH

dt

di ˆ• mass basis

0)(i

itiEi et

1** ~~ j

freeiji

free NHNEwith tj

freeiji

free NHNH ~~*≠

††

2

*

,NNNN

NeN

LEP ii

LiPi

i


freefree EHdt


††

2

*

,NNNN

NeN

LEP ii

LiPi

i

Zero-distance effect:

iijj

freeiji

freei EHH

dt

di ˆ• mass basis

0)(i

itiEi et

1** ~~ j

freeiji

free NHNEwith tj

freeiji

free NHNH ~~*≠

††

2†

0,NNNN

NNEP

oscillations in matter

2 families

00int

2

12 nFeeeF nGnGL

VCC VNC

e

NC

NCCCte

V

VVU

E

EU

dt

di

0

0

0

0

2

1*

oscillations in matter

2 families

00int

2

12 nFeeeF nGnGL

VCC VNC

e

NC

NCCCte

V

VVU

E

EU

dt

di

0

0

0

0

2

1*

e

NCeee

NCCC

eee

NCeeNCCCe

NNVNNNN

NNVV

NNNN

NNVNNVV

NE

EN

dt

di

†††

†

††

††

1*

2

1*

0

0

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

N elements from oscillations: e-row

23

2

3

2

2

2

1

4

3

22

2

2

1 cos2ˆ eeeeeeee NNNNNNP

Only disappearance exps → information only on |Ni|2

CHOOZ: Δ12≈0

K2K (→ ): Δ23

1. Degeneracy

2. cannot be disentangled

2

3

2

2

2

1 eee NNN

2

2

2

1 , ee NN

ELmijij 22

UNITARITY


12

2

2

2

1

4

3

4

2

4

1 cos2ˆ eeeeeee NNNNNP

0

12

3

2

2

2

1

e

ee

N

NN

KamLAND: Δ23>>1

→ first degeneracy solved

KamLAND+CHOOZ+K2K


12

2

2

2

1

4

3

4

2

4

1 cos2ˆ eeeeeee NNNNNP

0

12

3

2

2

2

1

e

ee

N

NN

KamLAND: Δ23>>1

SNO:

2

2

2

1 9.01.0ˆeeee NNP

→ first degeneracy solved

→ all |Nei|2 determined

KamLAND+CHOOZ+K2K

SNO

N elements from oscillations: -row

23

2

3

2

2

2

1

4

3

22

2

2

1 cos2ˆ NNNNNNP

1. Degeneracy

2. cannot be disentangled

2

3

2

2

2

1 NNN

2

2

2

1 , NN

Atmospheric + K2K: Δ12≈0

UNITARITY

N elements from oscillations only

81.056.074.044.053.020.0

82.058.073.042.052.019.0

20.061.047.088.079.0

U

González-García 04

with unitarityOSCILLATIONS

without unitarityOSCILLATIONS

???

86.057.086.057.0

34.066.045.089.075.0

][ 2/12

2

2

1 NNN

3

(NN†) from decays

• Rare leptons decays

††

2†

NNNN

NN

W

il l

Info on (NN†)

Wi

l

††

†

NNNN

NN

ee

iZ

j

††

†

NNNN

NN

ee

• W decays

• Invisible Z

• Universality tests

†

†

NN

NN

Info on(NN†)

(NN†) and (N†N) from decays

005.0003.1103.1106.1

103.1005.0003.1102.7

106.1102.7005.0002.1

22

25

25

†NNExperimentally

(NN†) and (N†N) from decays

005.0003.1103.1106.1

103.1005.0003.1102.7

106.1102.7005.0002.1

22

25

25

†NNExperimentally

032.000.1032.0032.0

032.0032.000.1032.0

032.0032.0032.000.1†NN

→ N is unitary at % level

†2† with 1 HNN

'1 1 †† VVNN 03.0'2

ij

HVN

N elements from oscillations & decays

81.056.074.044.053.020.0

82.058.073.042.052.019.0

20.061.047.088.079.0

U

González-García 04

with unitarityOSCILLATIONS

without unitarityOSCILLATIONS

+DECAYS

82.054.075.036.056.013.0

82.057.073.042.054.019.0

20.065.045.089.076.0

N

3

In the future…

MEASUREMENT OF MATRIX ELEMENTS

• |Ne|2 , -row → MINOS, T2K, Superbeams, NUFACT…• -row → high energies: NUFACT• phases → appearance experiments: NUFACTs, -beams

In the future…

Rare leptons decays

• →e

• →e

• →

TESTS OF UNITARITY

016.0† eNN

013.0† NN

5† 102.7 eNN

PRESENT



In the future…

PRESENT FUTURE

~ 10-6 MEG

~ 10-7 NUFACT

Rare leptons decays

• →e

• →e

• →

TESTS OF UNITARITY

016.0† eNN

013.0† NN

5† 102.7 eNN



In the future…

ZERO-DISTANCE EFFECT 40Kt Iron calorimeter near NUFACT

• e→

4Kt OPERA-like near NUFACT

• e→

• → 3† 106.2 NN

3† 109.2 eNN

4† 103.2 eNN

PRESENT FUTURE

~ 10-6 MEG

~ 10-7 NUFACT

Rare leptons decays

• →e

• →e

• →

TESTS OF UNITARITY

016.0† eNN

013.0† NN

5† 102.7 eNN



Conclusions

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

• Present oscillation experiments alone can only measure half the elements

• EW decays confirms unitarity at % level

• Combining oscillations and EW decays, bounds for all the elements can be found comparable with the ones obtained with the unitary analysis

Future experiments can:

- improve the present measurements on the e- and -rows- give information on the -row and on phases (appearance exps)- test unitarity by constraining the zero-distance effect with a near detector

Back-up slides

…adding near detectors…

Test of zero-distance effect: 2†0, NNEP

???

85.057.081.022.069.000.0

27.066.045.089.075.0

N

→ also all |Ni|2 determined

• KARMEN: (NN†)e 0.05

• NOMAD: (NN†)0.09 (NN†)e0.013• MINOS: (NN†)1±0.05

• BUGEY: (NN†)ee 1±0.04

Non-unitarity from see-saw

...11 6

25

dd

SMeff LLLL

HHlYM

Yl LL

~~

4

1 *

Integrate out NR

LL lHYM

YiHl

~1~

2

d=5 operatorit gives mass to

d=6 operatorit renormalises kinetic energy

Broncano, Gavela, Jenkins 02

cRRR

cRLRRLRRSM NNNNMlHNNHlYNNiLL

2

1~~

Number of events

)()(),()(

~ EENNLEPNNdE

EddEn SM

SM

ev

)()(),()(

~ EELEPdE

EddEnev

2

*,ˆ i

iLiP

i NeNLEP i

Exceptions:• measured flux• leptonic production mechanism• detection via NC

NNdE

d

dE

d SM

~

NNSM~ produced and detected in CC

(NN†) from decays: GF

Wi

l

NNNN

NN

ee

iZ

j

NNNN

NN

ee

ijij

• W decays

• Invisible Z

• Universality tests

NN

NN

Info on(NN†)aa

GF is measured in -decay

j

Wˉ

e

iNi

N*ej

j

eji

iF NNmG 22

3

52

192

jej

ii

FF

NN

GG 22

2exp,2

CHOOZ

10-3

Quarks are detected in the final state → we can directly measure |Vab|

ex: |Vus| from K0 →- e+ e

uK0 -

d d

sW+

Vus*

i

e+ Uei

→ ∑i|Uei|2 =1 if U unitary

→ Measurements of VCKM elements relies on UPMNS unitarity

Unitarity in the quark sector

With Vab we check unitarity conditions: ex: |Vud|2+|Vus|2+|Vub|2 -1 = -0.0008±0.0011

Quarks are detected in the final state → we can directly measure |Vab|

ex: |Vus| from K0 →- e+ e

→ Measurements of VCKM elements relies on UPMNS unitarity

Unitarity in the quark sector

With Vab we check unitarity conditions: ex: |Vud|2+|Vus|2+|Vub|2 -1 = -0.0008±0.0011

• decays → only (NN†) and (N†N)• N elements → we need oscillations• to study the unitarity of N: no assumptions on VCKM

With leptons:

uK0 -

d d

sW+

Vus*

e

e+

PMNS Matrix Elements Without Assuming Unitarity

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