discrete flavour symmetries in guts: the beauty and the...

Structure in the flavour sectorCombining Pati-Salam with S4

Conclusions

Discrete Flavour Symmetries in GUTs:the beauty and the beast

Reinier de Adelhart Toorop, Nikhef, Amsterdamwith Federica Bazzocchi and Luca Merlo.

arXiv: 1003.4502 [hep-ph]

Reinier de Adelhart Toorop Discrete Flavour Symmetries in GUTs


Conclusions

Outline

1 Structure in the flavour sector(Discrete) Flavour SymmetriesGrand Unified Symmetries

2 Combining Pati-Salam with S4

The beautyThe beastThe beauty and the beast

3 Conclusions



Conclusions

Observed structure in the flavour sector(Discrete) Flavour SymmetriesGrand Unified Symmetries

Structure in the flavour sector

There seems to be structure in the fermion masssector

Hierarchical masses,Quarks: moderate Cabibbo-mixing,Leptons: strong mixing.



Conclusions





|(VCKM)ij| = 0.97 0.23 0.00390.23 1.0 0.041

0.0081 0.038 1?

C. Amsler et al. Review of particle physics, July 2008



Conclusions





|(UPMNS)ij| =0.81 0.56 <0.220.39 0.59 0.680.38 0.55 0.70

M.C. Gonzales Garcia Nucl. Phys. A 827 (2009) 5C



Conclusions


Flavour SymmetriesThese structures might be explained by (discrete) flavoursymmetries

We charge the families under a symmetry group.

To make the Lagrangian invariant, we need to add Higgs-likefields ’flavons’.

Structure in the VEVs of the flavons leads to structure in thefermion masses.A very good introduction: G. Altarelli, Models of neutrino masses and mixings; hep-ph/0611117



Conclusions


Flavour Symmetries - a Simple Example

Consider charged Lepton masses in the SM.

Ignore family mixing for a slide or three.

−LM ;e = ye

(νee

)·(H1

H2

)ec+yµ

(νµµ

)·(H1

H2

)µc+yτ

(νττ

)·(H1

H2

)τ c.

After the Higgs gets its VEV

−LM ;e =(e µ τ

)yev yµvyτv

ecµcτ c

.

Whereye � yµ � yτ .



Conclusions


Flavour Symmetries - a Simple Example

Consider charged Lepton masses in the SM.

Ignore family mixing for a slide or three.

−LM ;e = ye

(νee

)·(H1

H2

)ec+yµ

(νµµ

)·(H1

H2

)µc+yτ

(νττ

)·(H1

H2

)τ c.

After the Higgs gets its VEV

−LM ;e =(e µ τ

)yev yµvyτv

ecµcτ c

.

Actually

ye : yµ : yτ ≈ λ4 : λ2 : 1, λ = 0.2.



Conclusions


Froggatt and Nielsen: not a coincidence.Assume there is an extra U(1) charge in the familysector.

all doublets ec µc τ c

0 4 2 0

The Lagrangian is no longer invariant

−LM ;e = ye

(νee

)·(H1

H2

)ec+yµ

(νµµ

)·(H1

H2

)µc+yτ

(νττ

)·(H1

H2

)τ c.

E.g. the first term’s U(1)fam-charges: 0 + 0 + 4 6= 0.



Conclusions


Solution: add an extra field, charged under this symmetry.

all doublets ec µc τ c θ0 4 2 0 −1

Use this to make the Lagrangian invariant

−LM ;e =1

M4ye

(νee

)·(H1

H2

)θ4ec+

1

M2yµ

(νµµ

)·(H1

H2

)θ2µc+yτ

(νττ

)·(H1

H2

)τ c

Let the flavon obtain a VEV < θ >= λM and we get

−LM ;e =(e µ τ

)yeλ4vyµλ

2vyτv

ec

µc

τ c.

Now with ye, yµ and yτ all O(1).



Conclusions


Flavour symmetries

Froggatt-Nielsen

Abelian symmetry group,Explains the mass hierarchy for one particle type (e.g. chargedleptons).

Non-Abelian Symmetry groups

Can also explain the mixing patterns.

Data TBM Prediction



Conclusions


Flavour Symmetries and Grand Unification

Family symmetries unify the three families.

Grand Unified Theories unify

The gauge couplings,The SM particles in fewer representations.

→



Conclusions








Conclusions






→[ (uR uG uBdR dG dB

)+(ucR ucG ucB

)+(dcR dcG dcB

)+

(νe

)+(ec)+(νc) ]

SM→

[ (uR uG uB dR dG dB ucR ucG ucB dcR dcG dcB ν e ec νc

) ]SO(10)



Conclusions






We like to build models that combine Family Physics andGrand Unification.

→Reinier de Adelhart Toorop Discrete Flavour Symmetries in GUTs


Conclusions


Grand Unified Symmetries

Grand Unified symmetries also give relations in the masssector

Often between different particle types.

In Pati-Salam we can have

bottom-tau unification

|mτ | = |mb|,

The Georgi-Jarlskog relation

|mµ| = 3|ms|.



Conclusions

MotivationThe beautyThe beastThe beauty and the beast

A model with Pati-Salam ×S4 Symmetry

Conclusion so far: we’d like to build a model with a GrandUnified and a Flavour symmetry.

We use

The Pati-Salam grand unified theory,The S4 flavour symmetry group.



Conclusions


Pati-Salam

We use Pati and Salam’sSU(4)c × SU(2)L × SU(2)R

Gauge: not quite unifying.All SM particles in two representations.This allows relations between quarks and leptons.

The type II seesaw can be dominant.



Conclusions


Pati-Salam



The type II seesaw can be dominant.

[ (uR uG uBuR dG dB

) (νe

) ]&

[ (ucR ucG ucB

) (νc)(

dcR dcG dcB) (

ec) ]



Conclusions


Pati-Salam



The type II seesaw can be dominant.This prevents unwanted correlations between quarks and neutrinos.

mIν = −mT

Dir

1MRR

mDir ∝M−1R ,

mIIν = MLL ∝M−1

3L .



Conclusions


The S4 flavour group

We choose the S4 flavour group.

It has 1, 2 and 3 dimensional representations.

It can give rise to the so-called bimaximal (BM) mixingpattern.

To appreciate this, we should compare with the more commontribimaximal (TBM) mixing.



Conclusions


Tribimaximal mixing

Many models use tribimaximal mixing to model the neutrinomixing P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B 530 (2002) 167

UTBM =

√

23

1√3

0− 1√

61√3− 1√

2

− 1√6

1√3

1√2

,

θ12 = 35◦, θ13 = 0◦, θ23 = 45◦.

This describes the data really well ...

Data TBM Prediction



Conclusions


TBM mixing might describe the neutrino data a bit too well.

Most models describe the quark mixing poorly at LeadingOrder∗.The Cabibbo angle arises only via NLO effects.Often these NLO effects influence the PMNS as well.

∗ LO is defined as having the lowest number of flavons possible; NLO has one extra flavon

Quarks

TBM LO

Typical TBM NLO



Conclusions


TBM mixing might describe the neutrino data a bit too well.

Most models describe the quark mixing poorly at LeadingOrder∗.The Cabibbo angle arises only via NLO effects.Often these NLO effects influence the PMNS as well.

∗ LO is defined as having the lowest number of flavons possible; NLO has one extra flavon

Neutrinos

TBM LO

Typical TBM NLO

δθ12 = −0.9θC , δθ13 = 0.5θC , soθ12 = 23◦, θ13 = 7◦, θ23 = 45◦, while δ = 0.



Conclusions


BiMaximal mixing

An alternative is BiMaximal mixing.

At leading order, two angles are maximal

UBM =

1√2

1√2

0−1

212 − 1√

2

−12

12

1√2.

And we can write UBM = as the product of two maximalrotations

UBM = R23(−π4

)×R12(π

4).



Conclusions


The leading order does not explain the data well.

Via quark lepton complementarity at NLO it might.G. Altarelli, F. Feruglio and L. Merlo, JHEP 0905 (2009) 020

Neutrinos

BM LO

Typical BM NLO

δθ12 = −0.7θC , δθ13 = 0.5θC , soθ12 = 36◦, θ13 = 7◦, θ23 = 45◦, while δ = 0.



Conclusions


The leading order does not explain the data well.

Via quark lepton complementarity at NLO it might.G. Altarelli, F. Feruglio and L. Merlo, JHEP 0905 (2009) 020

Quarks

BM LO

Typical BM NLO



Conclusions


“The beauty”

The model with the two symmetries can explain many features inthe flavour sector.



Conclusions


WishlistWe like to explain/predict

Quark and lepton masses:Bottom-tau unification |mτ | = |mb|;The Georgi-Jarlskog relation |mµ| = 3|ms|;Suppressed second family masses |ms||mb| ≈

|mµ||mτ | ≈ λ

2 (with

λ = 0.2);

A stronger suppressed charm quark |mc||mt| ≈ λ4.

Neutrino massesReproduce ∆m2

sol and ∆m2atm;

Predict the hierarchy type and the absolute mass scale.



Conclusions


Wishlist

We like to explain/predict

Quark mixing angles:reproduce the quark angles according toWolfenstein

θq12 ≈ λ,θq23 ≈ λ2,θq13 ≈ λ3.

Lepton mixing angles:Reproduce the solar and atmospheric mixingangles.In the BM scheme:

θl12 ∼ π2 − λ,

θl23 ∼ π2 − λ

2,Predict θl13.



Conclusions


Realisation

Model details

We put the lefthanded PS multiplets FL in a 3a of S4.We put the righthanded PS multiplets F c1,2,3in 1b, 1b and 1a of S4.We introduce two flavons (σ, χ) in the neutrino sectorAnd two (ϕ, ϕ′) for the charged particles.



Conclusions


Rotations at the Leading Order

Bimaximal mixing models have at leading order

VCKM = V †uVd = 1,UPMNS = V †e Vν = R23(−π4 )R12(π4 ).

This can be achieved if

M IIν is diagonalized by V = R12(π4 )

And MeM†e , MdM

†d and MuM

†u by Ve,d,u = R23(π4 ).



Conclusions


Neutrinos

In the neutrino sector we introduce a singlet σ and a triplet χ

χ ∝

001

.

This gives the type II seesaw contributions

FLFL∆L →

k0 0 00 k0 00 0 k0

, FLFL∆Lχ→

0 k1λ 0k1λ 0 00 0 0

.

That is indeed diagonalized by V = R12(π4 ).

The quasi-degenerate hierarchy is favoured.



Conclusions


Charged particles

For the charged particles we introduce a 3a triplet ϕ and a 3btriplet ϕ′

ϕ ∝

011

ϕ′ ∝

01−1

.

This leads to masses

FLF3cφϕ→

0 0 00 0 y0 0 y

, FLF2cρϕ′ →

0 0 00 x 00 −x 0

.

The square is indeed diagonalized by V = R23(π4 )

MM † =

0 0 00 y2 + x2 y2 − x2

0 y2 − x2 y2 + x2

.



Conclusions


Rotations at leading order

We indeed have at Leading order

M IIν is diagonalized by V = R12(π4 ),

And MeM†e , MdM

†d and MuM

†u by Ve,d,u = R23(π4 ).

So we indeed reproduce BiMaximal mixing


Remember that this does not describe the data well.



Conclusions


Rotations at leading order

We indeed have at Leading order

M IIν is diagonalized by V = R12(π4 ),

And MeM†e , MdM

†d and MuM

†u by Ve,d,u = R23(π4 ).

So we indeed reproduce BiMaximal mixing


Remember that this does not describe the data well.

Data BM LO



Conclusions


Next-to-Leading Order

At NLO M IIν is still diagonalized by V = R12(π4 ).

M IIν =

k0 k1λ k3λ3

k1λ k0 0k3λ

3 0 k0 + k2λ2

λ ≈ 0.2

The charged particle mass matrices now read

Mx =

0 xλ yλ0 x y0 −x y

.



Conclusions


Next-to-Leading Order

MM †-matrices are diagonalized by

Vx = R23(π

4)×R13(fλ)×R12(gxλ),

where f = fu = fd = fe and gu 6= gd = ge are O(1).

The CKM matrix is

VCKM = Vu†Vd = R12((gd − gu)λ).

And the PMNS matrix is

VPMNS = Ve†Vν = R23(−π

4)×R13(fλ)×R12(

π

4− geλ).



Conclusions


Mass hierarchies

We suppress the first and second generation via theFroggatt-Nielsen mechanism.

An SU(4)-singlet Higgs field φ couples at LO to the thirdgeneration.This ensures the bottom-tau unification.

An SU(4)-15-plet ρ couples at LO to the second generation.

ρ ∼

1 0 0 00 1 0 00 0 1 00 0 0 −3

This gives rise to the Georgi-Jarlskog relation.

If ρ is suppressed in the up-direction, we can have the extrasuppression of the charm (and the up).



Conclusions


Wishlist

The model is quite succesful.

Quark and lepton masses:

Bottom-tau unification |mτ | = |mb|; XThe Georgi-Jarlskog relation |mµ| = 3|ms|; XSuppressed second family masses |ms||mb| ≈

|mµ||mτ | ≈ λ

2; XA stronger suppressed charm quark |mc||mt| ≈ λ

4. XNeutrino masses in the quasi-degenerate hierarchy. XThe CKM matrix. XThe PMNS matrix with large reactor angle. X



Conclusions


“The beast”The interplay between the two types of symmetries strongly affectsthe model building.

The Higgs sector is very non-minimal

The gauge RGEs allow type II seesaw domination only in asmall part of parameter space

The Yukawa RGEs rule out most of parameter space for theneutrinos



Conclusions


The Higgs sector in minimal Pati-Salam

Higgses A B ∆R ∆R φ, φ′

PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1

Higgses ρ ∆L ∆L Σ, Σ′ ξ

PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

A breaks SU(4)C → SU(3)C × U(1)B−L.



Conclusions




PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1


PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

(∆R, ∆R) break SU(2)R × U(1)B−L → U(1)Y and set the scale for the type I

Seesaw.



Conclusions




PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1


PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

φ is the EW Higgs field and it breaks SU(2)L × U(1)Y → U(1)em.



Conclusions


The Higgs sector in non-minimal Pati-Salam


PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1


PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

ρ is a second Higgs field that can give the Georgi-Jarlskog relation.



Conclusions




PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1


PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

(∆L, ∆L) can be used in the type II Seesaw.



Conclusions




PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1


PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

σ helps in the type II Seesaw.



Conclusions


In our model we have


PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1


PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

A second φ field is needed to give the ρ field a vanishing component in the up

direction.



Conclusions




PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1


PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

We need an extra Z4 group to select the flavons and Higgs fields at the right

places...



Conclusions




PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1


PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

... and because of that a second (15, 1, 1)-field...



Conclusions




PS (15, 1, 1) (15, 1, 1) (10, 1, 3) (10, 1, 3) (1,2,2)

Z4 1 −1 −1 −1 1


PS (15, 2, 2) (10, 3, 1) (10, 3, 1) (1, 3, 3) (1,1,1)

Z4 −1 1 1 1, −1 −1

... And a second Σ plus a PS-singlet to couple them.



Conclusions


The gauge RGEs

Due to the extra fields and the demand that type II seesaw isdominating, the renormalisation group running of the gaugecouplings is strongly affected.



Conclusions


The gauge RGEsDue to the extra fields and the demand that type II seesaw isdominating, the renormalisation group running of the gaugecouplings is strongly affected.



Conclusions


The Yukawa RGEs

The entries of the mass matrices are also strongly modified bythe renormalisation group running.

This affects in particular the neutrinos.

The atmospheric angle θl23 gets corrections that are often toolarge.

The quasi-degenerate hierarchy is excluded.

The slightly less natural inverse and normal hierarchies are stillallowed.



Conclusions


Conclusion so far:

The “beauty” could explain a lot of phenomena in the flavonsector.

The “beast” excluded much of the available parameter space.

In the part of parameter space that is still allowed, we have aninteresting neutrino phenomenology.



Conclusions


The reactor angle

The inverse hierarchy The normal hierarchy

We expect the reactor angle to be in the range of upcomingexperiments like Daya Bay.



Conclusions


Neutrinoless double beta decay

In the inverse hierarchy, neutrinoless double beta decay is in thesensitivity of experiments like CUORE and EXO.



Conclusions

Conclusions

Discrete Flavour symmetries and GUTs are interesting and acombination of them even more so.

Our Pati-Salam × S4 model seemed quite realistic and viable.

A detailed analysis of the Higgs sector and the assumptionsbehind the model restricts the available parameter space.

Our construction is model dependant, but shares manyfeatures with other models.



Conclusions

Thank you for your attention


discrete flavour symmetries in guts: the beauty and the...

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