Grand Unified Theory with SU (5) Modular Symmetry A 4 Predicting - - PowerPoint PPT Presentation

Grand Unified Theory with Modular Symmetry

Predicting Fermion Mass Matrices

Horizon 2020

based on

1812.05620 w/ Francisco J. de Anda, Stephen F. King

Elena Perdomo

Tuesday Seminars

29 January 2019

SU(5)

A4

2

Outline

• Motivation
• Introduction to

✴ Orbifold with fixed points related by an A4 symmetry. ✴ Modular symmetry. How do you build invariants? Couplings

becoming modular forms giving rise to mass matrix structure.

• The model

✴ Fields on the branes. ✴ Fields on the bulk and boundary conditions. ✴ The Yukawa structure and alignments from modular forms. ✴ Mass matrices and predictions of the model.

T2/

Z2

The Standard Model

Why are there 3 families of each SM fermion field, in the same representation of the gauge group, differing only by their mass?

[King ’17]

The Flavour Problem

3

The Flavour Problem

Why such large hierarchy among fermion masses? Why is flavour mixing in the quark sector small compared to the lepton sector?

[Stone ‘12] [King ’17]

4

SUSY FLAVOUR GUTs

❖ SUSY: gauge coupling unification, ameliorates the

hierarchy problem.

❖ Grand Unified Theory: unifies fermions within each

family and reproduces an universal mass matrix structure.

❖ Family symmetry: “Horizontal” unification of SM

fermions.

5

❖ Extra dimensions: ❖ GUT breaking through boundary conditions. ❖ Higgs doublet-triplet splitting. ❖ Origin of the family symmetry after orbifolding. ❖ No need of extra fields and alignment superpotentials.

Neutrino Masses

How are neutrino masses generated and why are they so small? Type I Seesaw mechanism

mν = v2yνM −1

N yT ν

Lν = yνL†HN + MNN cN + h.c.

Include RH neutrinos which are singlets under the SM Large Majorana RHN masses

[Minkowski ’77] [Yanagida ’79] [Gell-Mann, Ramond, Slansky ’79] [Ramond ’98]

6

Are neutrino masses modular forms?

[Feruglio ’17] [Criado, Feruglio ’18] [Novichkov, Penedo, Petcov, Titov ’18] [Kobayashi, Omoto, Shimizu, Kauai, Tanimoto, Tatsyusgu ’18] [Penedo, Petcov ’18] [Kobayahsi, Tanak, Tatsuishi, ’18] [Novichkov, Penedo, Petcov, Titov ‘18]

7

Orbifold

Z2

T2/

ω = ei 2π

3

with twist

'

z = z + 1

z = z + ω

z = −z

Z2

T2

T2

❖ Two extra dimensions can be combined in z = x5 + ix6

❖

The orbifold leaves 4d invariant branes

❖

Theory in 6 dimensions, where the two extra dimensions are compactified on orbifold. T2/

Z2

8

Orbifold

Z2

T2/

ω = ei 2π

3

with twist

The spacetime (active) transformations S, T and U permute the branes, leaving invariant the set of 4 branes:

S : z → z + 1/2 or z → z + ω/2, T : z → ω2z, U : z → z∗ or z → −z∗

These transformations satisfy: A4 group S4 group

S2 = T 3 = (ST)3 = 1 U 2 = (SU)2 = (TU)2 = (STU)4 = 1

Any fixed field on the branes shall respect this symmetry.

S,U identify sides a,b,c while T rotates everything by identifying sides d.

Modular symmetries

9

Extra dimensions can give rise to discrete symmetries after compactification

'

T2

With identification

ω1

ω2

Our model

ω2 = ω = ei 2π

3

ω1 = 1

Modular symmetries

9

Extra dimensions can give rise to discrete symmetries after compactification

'

T2

With identification

ω1

ω2

Lattice (passive) invariant transformations ≡

do not change the torus Our model

ω2 = ω = ei 2π

3

ω1 = 1

Modular symmetries

9

Extra dimensions can give rise to discrete symmetries after compactification

'

T2

With identification

ω1

ω2

Lattice (passive) invariant transformations T-transformation

≡

do not change the torus Our model

✓ω2 1 ω ◆ ✓1 ω ◆ = ✓−1 − ω −ω ◆

ω0

2

ω0

1

ω2 = ω = ei 2π

3

ω1 = 1

Modular symmetries

9

Extra dimensions can give rise to discrete symmetries after compactification

'

T2

With identification

ω1

ω2

Lattice (passive) invariant transformations S-transformation T-transformation

≡

do not change the torus Our model

✓ω2 1 ω ◆ ✓1 ω ◆ = ✓−1 − ω −ω ◆

✓0 −1 1 ◆ ✓1 ω ◆ = ✓−ω 1 ◆

ω0

2

ω0

1

ω0

2

ω0

1

ω2 = ω = ei 2π

3

ω1 = 1

Modular symmetries

9

Extra dimensions can give rise to discrete symmetries after compactification

'

T2

With identification

ω1

ω2

Lattice (passive) invariant transformations S-transformation T-transformation

≡

do not change the torus Our model Modular group

✓ω2 1 ω ◆ ✓1 ω ◆ = ✓−1 − ω −ω ◆

✓0 −1 1 ◆ ✓1 ω ◆ = ✓−ω 1 ◆

ω0

2

ω0

1

ω0

2

ω0

1

Γ(3) ' A4

S2 = T 3 = (ST)3 = I

ω2 = ω = ei 2π

3

ω1 = 1

Modular symmetries

10

Modular group Γ(3) ' A4

S2 = T 3 = (ST)3 = I

The entire theory, including fields on the branes as well as in the bulk, must respect this symmetry, with fixed modulus τ = ω = ei 2π

3

Modular symmetries

10

Modular group Γ(3) ' A4

S2 = T 3 = (ST)3 = I

The entire theory, including fields on the branes as well as in the bulk, must respect this symmetry, with fixed modulus All the fields in the bulk will transform under the modular group, an invariant can be built from

τ = ω = ei 2π

3

L ⊃ f(ω)ψ(1) · · · ψ(n)

f(ω)

ψ(I)

: fields : coupling constant

Modular symmetries

10

Modular group Γ(3) ' A4

S2 = T 3 = (ST)3 = I

The entire theory, including fields on the branes as well as in the bulk, must respect this symmetry, with fixed modulus All the fields in the bulk will transform under the modular group, an invariant can be built from

τ = ω = ei 2π

3

ω → aω + b cω + d

L ⊃ f(ω)ψ(1) · · · ψ(n)

f(ω)

ψ(I)

: fields : coupling constant

f(ω) → (cω + d)kρf(ω)

ψ(I) → (cω + d)−kIρ(I)ψ(I)

Modular symmetries

10

Modular group Γ(3) ' A4

S2 = T 3 = (ST)3 = I

The entire theory, including fields on the branes as well as in the bulk, must respect this symmetry, with fixed modulus All the fields in the bulk will transform under the modular group, an invariant can be built from

τ = ω = ei 2π

3

ω → aω + b cω + d

L ⊃ f(ω)ψ(1) · · · ψ(n)

f(ω)

ψ(I)

: fields : coupling constant

f(ω) → (cω + d)kρf(ω)

Modular form with weight k

ψ(I) → (cω + d)−kIρ(I)ψ(I)

Modular symmetries

10

Modular group Γ(3) ' A4

S2 = T 3 = (ST)3 = I

The entire theory, including fields on the branes as well as in the bulk, must respect this symmetry, with fixed modulus All the fields in the bulk will transform under the modular group, an invariant can be built from

τ = ω = ei 2π

3

ω → aω + b cω + d

Modular weight, a real number

L ⊃ f(ω)ψ(1) · · · ψ(n)

f(ω)

ψ(I)

: fields : coupling constant

f(ω) → (cω + d)kρf(ω)

Modular form with weight k

ψ(I) → (cω + d)−kIρ(I)ψ(I)

Modular symmetries

10

Modular group Γ(3) ' A4

S2 = T 3 = (ST)3 = I

The entire theory, including fields on the branes as well as in the bulk, must respect this symmetry, with fixed modulus All the fields in the bulk will transform under the modular group, an invariant can be built from

τ = ω = ei 2π

3

ω → aω + b cω + d

Representation under the modular group

L ⊃ f(ω)ψ(1) · · · ψ(n)

f(ω)

ψ(I)

: fields : coupling constant

f(ω) → (cω + d)kρf(ω)

Modular form with weight k

ψ(I) → (cω + d)−kIρ(I)ψ(I)

Γ(3) ' A4

Modular symmetries

11

Modular group Γ(3) ' A4

S2 = T 3 = (ST)3 = I

The entire theory, including fields on the branes as well as in the bulk, must respect this symmetry, with fixed modulus All the fields in the bulk will transform under the modular group, an invariant can be built from

τ = ω = ei 2π

3

ω → aω + b cω + d

L ⊃ f(ω)ψ(1) · · · ψ(n)

f(ω)

ψ(I)

: fields : coupling constant

f(ω) → (cω + d)kρf(ω)

ψ(I) → (cω + d)−kIρ(I)ψ(I)

Invariant if:

• 1. The weight compensates the overall

weight of the fields:

• 2. The product of the representations

k = X

I

kI

ρ × ρ(1) × · · · × ρ(n)

contains a singlet k

12

Enhanced symmetry of the branes

• We want to make the connection with modular symmetries for two reasons:

❖ The modular symmetry is a symmetry of the lattice, and therefore, bulk

fields have to satisfy it as well.

❖ Yukawa couplings become modular forms and can help with understanding

the structure of the mass matrices.

• The orbilfod

T2/

Z2

leaves 4 invariant 4d branes which can be permuted under both an A4 and S4 symmetry

13

Orbifold

Z2

T2/

ω = ei 2π

3

with twist

The spacetime (active) transformations S, T and U permute the branes, leaving invariant the set of 4 branes:

S : z → z + 1/2 or z → z + ω/2, T : z → ω2z, U : z → z∗ or z → −z∗

These transformations satisfy: A4 group S4 group

S2 = T 3 = (ST)3 = 1 U 2 = (SU)2 = (TU)2 = (STU)4 = 1

Any fixed field on the branes shall respect this symmetry

14

Enhanced symmetry of the branes

• The orbilfod

T2/

Z2

leaves 4 invariant 4d branes which can be permuted under both an A4 and S4 symmetry

• We want to make the connection with modular symmetries for two reasons:

❖ The modular symmetry is a symmetry of the lattice, and therefore, bulk

fields have to satisfy it as well.

❖ Yukawa couplings become modular forms and can help with understanding

the structure of the mass matrices.

• The extra transformation U involves complex conjugation which would involve the

loss of holomorphicity and cannot be interpreted as a modular symmetry.

• We conclude that the symmetry of the branes is Γ(3) n Z2 ' A4 n Z2

where the Z2

symmetry is generated by C· U

with U

the generator from S4 and C

stands for complex conjugation of the complex coordinate z

15

The model

Fields located on the branes

The super fields located on the branes do not depend on the extra dimensions and therefore they must have weight zero.

16

The model

Fields on the bulk

• The two copies of each T, with different boundary conditions, allow different

masses for down quarks and charged leptons.

• The MSSM Higgs fields hu,hd are inside the H5,5, respectively.
• Two flavons that help to give structure to fermion masses

φ1,2

17

Boundary conditions: GUT and flavour breaking

The orbilfod action T2/

Z2

z = z + 1

z = z + ω

z = −z

Z2

T2

• n field space

ψ(x, −z) = Pψ(x, z)

P 2 = 1

The boundary conditions at each brane are chosen to be

P0 = I3 ⊗ I5, P1/2 = T1 ⊗ diag(−1, −1 − 1, 1, 1), Pω/2 = T2 ⊗ diag(−1, −1 − 1, 1, 1),

These conditions break A4 completely, fixing the alignments of the flavons to be

SU(5) → SU(3) × SU(2) × U(1)

hφ1i = v1   1  

hφ2i = v2   1 1  

They also break

18

Yukawa structure

WY = yN

s ξN c sN c s + yN a ξ ξ3

Λ3 N c

aN c a

+ yν

s

ξ ΛFH5N c

s + yν a

φ2ξ Λ2 FH5N c

a

+ ye

3

φ1 Λ FH5T +

3 + ye 2

φ1ξ Λ2 FH5T +

2 + ye 1

φ1ξ2 Λ3 FH5T +

1

+ yd

3

φ1 Λ FH5T −

3 + yd 2

φ1ξ Λ2 FH5T −

2 + yd 1

φ1ξ2 Λ3 FH5T −

1

+ yu

ijH5T + i T − j

ξ6−i−j Λ6−i−j ,

Majorana masses Dirac neutrino masses Charged leptons Down quarks Up quarks

18

Yukawa structure

WY = yN

s ξN c sN c s + yN a ξ ξ3

Λ3 N c

aN c a

+ yν

s

ξ ΛFH5N c

s + yν a

φ2ξ Λ2 FH5N c

a

+ ye

3

φ1 Λ FH5T +

3 + ye 2

φ1ξ Λ2 FH5T +

2 + ye 1

φ1ξ2 Λ3 FH5T +

1

+ yd

3

φ1 Λ FH5T −

3 + yd 2

φ1ξ Λ2 FH5T −

2 + yd 1

φ1ξ2 Λ3 FH5T −

1

+ yu

ijH5T + i T − j

ξ6−i−j Λ6−i−j ,

Majorana masses Dirac neutrino masses Charged leptons Down quarks Up quarks Modular forms with non trivial weights!

19

Mass Matrices

• The down quark and charged

lepton mass matrices are diagonal:

M d =vd   yd

1 ˜

ξ2 yd

2 ˜

ξ yd

3

  ˜ v1, M e =vd   ye

1 ˜

ξ2 ye

2 ˜

ξ ye

3

  ˜ v1,

• The fit to the observed masses is

straightforward.

• The hierarchy between the masses
• f the different families is

understood through the powers of ˜

ξ

19

Mass Matrices

• The down quark and charged

lepton mass matrices are diagonal:

M d =vd   yd

1 ˜

ξ2 yd

2 ˜

ξ yd

3

  ˜ v1, M e =vd   ye

1 ˜

ξ2 ye

2 ˜

ξ ye

3

  ˜ v1,

The up quark mass matrix is given by:

Mu = vu   yu

11 ˜

ξ4 yu

12 ˜

ξ3 yu

13 ˜

ξ2 yu

21 ˜

ξ3 yu

22 ˜

ξ2 y23 ˜ ξ yu

31 ˜

ξ2 yu

32 ˜

ξ yu

33

  ˜ v2,

• The fit to the observed masses is

straightforward.

• The hierarchy between the masses
• f the different families is

understood through the powers of ˜

ξ

• All contributions to quark mixing is

coming from the up sector.

• The smallest phenomenologically

viable choice for weight is γ = 7

20

Alignments from modular forms

A modular form has to satisfy

f(ω) → (cω + d)kρf(ω)

They can be constructed from the Dedekind Eta function

η(τ) = q

1 24

∞

Y

n=1

(1 − qn) , q ≡ ei2πτ.

The modular form must be a triplet

yν

s

The modular form can be a singlet or a triplet

yν

a

20

Alignments from modular forms

A modular form has to satisfy

f(ω) → (cω + d)kρf(ω)

They can be constructed from the Dedekind Eta function

η(τ) = q

1 24

∞

Y

n=1

(1 − qn) , q ≡ ei2πτ.

The modular form must be a triplet

yν

s

The modular form can be a singlet or a triplet

yν

a

Only alignments satisfying the extended symmetry A4 n Z2

Neutrino mass matrix defined from these modular forms!

21

Neutrino Mass Matrix

mν = v2yνM −1

N yT ν

After applying the type-I Seesaw Mechanism

mν = v2

u

hξi ˜ ξ2 yN

s

! α6(α6)T + ✓ v2

u

hξi ˜ v2

2

˜ ξyN

a

◆ β6(β6)T

where and are given by the modular forms

α6

β6

α6 = y   −1 2ω2 2ω   , β6 =   2y2 + y3(2ω2 − 2ω) y1 + y2(4ω + 1) − y3 y1 + y2(4ω2 + 1) + y3  

The effective parameters at low energy are {y, y1, y2, y3}

The symmetry fixes the parameters to be real while is purely imaginary.

Z2

{y, y1, y2}

y3

22

reflection symmetry

µ − τ

The neutrino mass matrix is reflection symmetric:

µ − τ

νe → ν∗

e,

νµ → ν∗

τ ,

ντ → ν∗

µ

symmetry leads to having non zero reactor angle, together with

θ13 6= 0, θ23 = 45, δl = ±90

µ − τ

maximal atmospheric mixing angle and maximal Dirac CP phase: C

• n

s e q u e n c e

• f

A

4

n Z

2

s y m m e t r y ! The parameters {y, y1, y2, y3}

will fit the rest of the PMNS observables

{θl

12, θl 13, ∆m2 21, ∆m2 31}

23

reflection symmetry

µ − τ

• The predictions of the model are θ23 = 45, δl = −90

• The latest results from NuFit4.0:

[Esteban, Gonzalez-Garcia, Hernandez- Cabezudo, Maltoni, Schwetz, 18’]

θ23/

= 49.6+1.0 1.2

δl/

= 215+40 29

• The predictions of the model still lie inside the region with χ2 = 5.48

3σ

• Furthermore, we predict normal mass ordering and a massless neutrino m1 = 0

since we only added 2 RH neutrinos.

24

Conclusions

• model in 6d, where the two extra dimensions are compactified on a
• rbifold, with a twist angle .
• First example of a GUT with a modular symmetry interpreted as a family symmetry.
• The fields on the branes respect an enhanced symmetry which leads to an

effective reflection symmetry at low energies, predicting maximal atmospheric mixing angle and maximal CP phase.

• The Yukawa couplings are determined by modular weights, leading to specific

alignments which fixes the Dirac mass matrix. (No need of extra fields to fix the alignments).

• GUT breaking and Higgs doublet-triplet splitting is done through boundary

conditions.

• The charged lepton and down-type quarks have diagonal and hierarchical Yukawa

matrices, with quark mixing due to a hierarchical up-quark matrix. T2/

Z2

SU(5)

ω = ei 2π

3

A4 n Z2

µ − τ