Higgs Quadruplet for Type III Seesaw and Implications for →e and −e Conversion

Ren Bo Coauther: Koji Tsumura, Xiao-Gang He

arXiv:1107.5879

I. Introduction

A Higgs Quadruplet for Type III Seesaw Model

II. The electroweak constraints

III. Loop induced neutrino mass with just

one triplet lepton

IV. Some phenomenological implications 1. Neutrino mass and mixing

2. →e and −e Conversion

V. Conclusions

Outline

I. Introduction

In the minimal SM: Gauge group: (3) (2) (1)C L YSU SU U

0

(8,1,0), (1,3,0), (1,1,0)

1 2 1(3,2, ), (3,1, ), (3,1, ),6 3 3

1(1,2, ), (1,1, 1),2

1. (1,2, ), υ-vev of Higgs.22

LL R R

L

Ll R

L

G W B

UQ U D

D

L Ee

h

H h

Quark and charged lepton masses are from the following Yukawa couplings

, , .L R L R L RQ HU Q HD L HE

Nothing to pair up with In minimal SM, neutrinos are massless! Extensions needed: Give neutrino masses and small ones!

( ).L LL

For simplicity, we consider two flavor neutrino mixing and oscillation.

Two flavor oscillation

The oscillation probability for an appearance neutrino experiment:

The conversion and survival probability in realistic units:

Due to the smallness of (1,3) mixing, both solar & atmospheric neutrino oscillations are roughly the 2-flavor oscillation.

Type-I: SM + 3 right-handed Majorana ’s (Minkowski 77; Yanagida 79; Glashow 79; Gell-Mann, Ramond, Slanski 79; Mohapatra, Senjanovic 79)

Type-II: SM + 1 Higgs triplet (Magg, Wetterich 80; Schechter, Valle 80; Lazarides et al 80; Mohapatra, Senjanovic 80; Gelmini, Roncadelli 80)

Type-III: SM + 3 triplet fermions (Foot, Lew, He, Joshi 89)

A natural theoretical way to understand why 3 -masses are very small.

Type-III (Triplet) Seesaw: add one fermions triplet into the SM.

†1. .

2c

L e R L R R R RL L Y E L Y M H c

The Lagrangian of neutrino and charged lepton masses is

where the ‘c’ denote the charge conjugation and 0

0

2

2

R RR

R R

i

i

In the component we have

0 0

0 0

1 1 1( )( ) ( ) ,

2 2 2

.

jjL R Li Rij j L R L R L R L R

c c ii jj c c cR R Rij Rij R R R R R R

L L i e h iI i e h

where 12 21 11 221, 1, 0.

: (1,3,0)R

Now introduce the quadruplet Higgs representation : (1, 4, 1/ 2)

. .L RL L Y H c The quartet has component field

In tensor notation is total symmetric tensor with 3 indices

0 0( , , , ), where ( ) / 2.R Ii

ijk0

111 112 112 222

1 1, , , .

3 3

The neutrino and charged lepton mass matrices are given by the basis

0( , ) , ( , ) .c T TL R R Re

† †

0, ,

0e e

ETR R

M M MM M

M M M

where Dirac mass term / 2, / 2, / 2.e e eM iY M iY M Y

0 0 0 0 01 2 2 1( ) ( ).

3 33 3

jj kkL R Li Rjk ij k

L R R R L R R R

L L

i e i

A non-zero will modify the neutrino and charged lepton mass matrices

1 1 1 1, .

2 3 2 6e eM i Y i Y M Y Y

The tree level light neutrino mass matrix, defined by the neutrino mass is

1 * * 1 †* † †1 1 1 1( ) ( ).2 23 3

R Rm M M M Y Y M Y Y

1. .,

2c

m L LL m H c

The light neutrino mass matrix can be diagonalized by the PMNS mixing matrix V

ˆ ,Tm V m V

1 2 3ˆ ( , , )m diag m m m where is the diagonalized light neutrino mass matrix.

II. The electroweak constraints

The electroweak precision data constrain the VEV of Higgs representation. The Higgs representation with

isospin I and hypercharge Y will modify the parameter at tree level with, 2 2

2 2

( ( 1) ).

2

I I Y

Y

2 2 2

2 2 2 2

7 51 .

The experimental data is constrained to be less than 5.8GeV which is about 40 times smaller than that of the doublet Higgs VEV.

0.00290.00110.0004 (95%c.l.),

For our case of one doublet and quintuplet, we have

III. Loop induced neutrino mass with just one triplet lepton

2 2† † 25† †3

† †2 † †( ) ( ) ( ) ( ) [ ( ) . .]2

V M H c

where alpha denotes and index for SU(2) contractions.

The most general Higgs potential is given by

The summations of SU(2) index are written as

2 2 1 2 25

2 2 1 2 25

2 2 1 2 2

1 1 1( ) ,2 6 31 1 1

( ) ,2 6 31

( ) .2

R

I

m M

m M

m M

The above terms will generate a neutrino mass matrix proportional to for the first term and, for the second term. The masses of component fields in χ are given by neglecting the contribution from terms proportional to

At one loop level Majorana masses will be generated for light neutrinos. Just keep proportional to terms

3

* †y y * †y y

2

The Mass matrix for singly charged scalars can also be approximately given by

1

2

12 2 25 2*

1*1 2 2*

2 2 1 2 25 2

02 2 3, , ,01 1

( )2 32 3

M m

mM

where

1 1 52

2 2

cos sin 3, tan 2 .

sin cos

H

H

One-loop generation of neutrino mass.

Collecting the tree and loop contribution, the neutrino mass matrix as

* * * * 21 1 1 1 1( ) ( ) ( ) .

2 23 3ij tree loop ij i i j j i j

N N

M M M Y Y Y Y Y Ym m

The explicit dependence on is given

1 2

2 22 2* †

2 2 2 2 2

1 1 sin(2 )[ ] [ ( ) ( )]

8 3 3R Iloop

N EN N E E

m mm mM Y Y m I I m I I

m m m m

where and are masses of neutral and charged heavy leptons, and I(x) = x ln x/(1 − x).

5

Em Nm

IV. Some phenomenological implications

1. Neutrino masses and mixing

Mass squared differences of neutrino masses and neutrino mixing have been measured to good precision.

The best-fit values and allowed 1, 2 and 3 ranges for the mass-mixing parameters.

G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo, A. M. Rotunno, [arXiv:1106.6028 [hep-ph]].

2 22 2 2 2 2 22 1

2 1 3where 0, , 0( 0) corresponds to noraml(inverted) mass spectrum hierarchy.2

m mm m m m m m

In our model, for normal

hierarchy,

For inverted hierarchy,

2 2 2 2 21 2 12 3 310, , .m m m m m

2 2 2 2 2 23 1 31 2 21 310, , .m m m m m m

To the leading order, the mixing pattern can be approximated by the tribimaximal mixing matrix

2 10

3 31 1 1

.6 3 2

1 1 1

6 3 2

TBU

* * * * 21 1 1( ) ( )( ) .

2 2ij tree loop ij i j i j

N N

M M M Y Y Y Ym m

The light neutrino mass matrix can be made to fit data. In case the light neutrino mass can be written as ,

For normal hierarchy case, the Yukawa couplings can be

taken to the forms

then,

If the heavy neutrino mass is the order of 1TeV, we get

(0,1 2 , 1 2) , (1 3 ,1 3 ,1 3) .T TY y Y y 2 2 2 2

3 24 , .N Nm y m m y m

6 1/2 6 1/21.89 10 ( 1TeV) , 0.38 10 ( 1TeV) .N Ny m y m

For the inverted hierarchy,

then,

( 2 3, 1 6, 1 6) , (1 3,1 3,1 3) .T TY y Y y

2 2 2 21 24 , .N Nm y m m y m

6 1/2 6 1/21.78 10 ( 1TeV) , 0.90 10 ( 1TeV) .N Ny m y m

If the heavy neutrino mass is the order of 1TeV, we get

The parameter is proportional to the Higgs potential . If is small, quadruplet Yukawa coupling can be order of one.

5 y

5

Making perturbation to the above forms, one can get non-zero solutions, which is indicated by the results at T2K.

13

For normal mass hierarchy, and keep the same

( , , ) .TY Y Y Y y a b c

.Y

2 2 2

2 2 3

( 0.14,0,0) ,

4 5.23 10 ,

9.14 10 ,

T

N

N

Y y

y m eV

y m eV

with3 2

2 3

2 212 23

213

8.78 10 , 4.82 10 ,

sin 0.323,sin 0.44,

sin 0.025. within one error.

m m

For inverted mass hierarchy

2 2 2

2 2 2

( 0.0095, 0.1,0.1085) ,

4 4.81 10 ,

4.88 10 ,

T

N

N

Y y

y m eV

y m eV

2 21 2

2 212 23

213

4.80 10 , 4.88 10 ,

sin 0.306,sin 0.41,

sin 0.014. within one error.

m m

2. →e and −e conversionThe dominant contribution come at the one loop level due to possible large Yukawa coupling The effective Lagrangian is given by

( ) . .L L R R e q L e Lq

L A P A P F eQ q q P B H c

with being the electric charge of the q-quark, and qQ

.Y

1 1 2 2

2 2 2 22 2

2 2 2 2 2 2 2 2 2

2 2†

2 2 2

2

2 2 2

1 1 1 2{ ( ) ( ) ( ) ( )

32 6 3

1( ) 2 ( ) } ,

,

1 1 1{ [ ( )

16 6

R R I I

R R

N NE EL

E E

eR L

EL

s m c mm meA Y F F F F

m m m m m m m m

m mF F Y m

m m m

mA A

m

meB Y G

m m m

1 1 2 2

2 2 2 22

2 2 2 2 2 2

2 2†

2 2 2

2 2 2

3 4 3 4

3 2

2( )] [ ( ) ( )]

3

1( ) 2 ( )]} ,

where

5 2 ln 2 5 1 ln( ) , ( ) ,

12( 1) 2( 1) 12( 1) 2( 1)

7 36 45 16 6(3( )

I I

N NE

E E

s m c mmG G G

m m m m m

m mG G Y

m m m

z z z z z z z zF z F z

z z z z

z z z zG z

3 2 3

4 4

2) ln 11 18 9 2 6 ln, ( ) .

36( 1) 36( 1)

z z z z z zG z

z z

The LFV →eγ decay branching ratio is easily evaluated by

22 2

2 2

( ) 48( ) (| | | | ).

( ) L RF

eB e A A

e G m

The strength of − e conversion is measured by the quantity,

con ( ( , ) ( , )).

( ( , ) ( 1, 1))

AA v

e Acapt

A N Z e A N ZB

A N Z A N Z

2( ) ( ) ( ) ( )0

2 50 2

2

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )( ) 1 ,

( ) ( ) ( )

where

( ) | ( ) | ,192

2 , 2 , / ( 2 ).

A p p n ne LV LV

eR R

Fe A

capt

p nLV LV u LV d LV LV u LV d LV q q L F

B g V A g V AR A

B e A D A A D A

G mR A D A

g g g g g g g eQ B G

The current experimental upper limit is

The μ−e conversion for Au nuclei is

The near future MEG experimental

The μ−e conversion, Mu2E/COMET Al

PRISM Ti

137 10 .AueB

The relevant parameters for μ-e conversion processes.

13( ) 1 10 .B e

12( ) 2.4 10 .B e

161 10 .AleB

181 10 .TieB

The current and future experimental constraints on the quadruplet Yukawa coupling from and conversion. The mass of quadruplet scalar is taken as

e e 1 .m TeV

V. Conclusion1. The heavy neutrinos are contained in leptonic triplet seesaw III model.2. A quadruplet χ is introduced to get the new type of Yukawa couplings. Light neutrino masses can receive sizeable contribution from both the tree and loop level.3. The mass matrix obtained can be made consistent with experimental data on mixingparameters. Large Yukawa coupling may have observable effects on lepton flavor violating processes, such as,

→e and −e conversion.

Thank you for your attention!