Fermion Mass Hierarchy from the Soft Wall DAVID DIEGO Work done in - - PowerPoint PPT Presentation

Fermion Mass Hierarchy from the Soft Wall

DAVID DIEGO

Work done in collaboration with Antonio Delgado

hep-ph/09051095

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 1 / 12

Outline

1

GENERAL INTRODUCTION

2

THE MODEL

3

EQUATIONS OF MOTION

4

PHENOMENOLOGICAL PREDICTIONS ON FERMION MASSES

5

CONCLUSIONS

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 2 / 12

GENERAL INTRODUCTION

Warped extra dimensions may provide an alternative explanation for the hierarchy between gravity and ElectroWeak scales.

Randall and Sundrum (1999) Arkani-Hamed et al. (1998)

More recently, an alternative to the hard wall termination of RS models have been proposed and given the name of Soft Wall models. In these models one has a non compact extra dimension with UV (4d) boundary and the effective metric is not AdS. Instead it decays faster enough to make the extra dimension of finite length. The departure from the AdS behavior is associated with a smooth Vacuum Expectation Value acquired by some dilaton field.

Karch, Katz, Son and Stephanov (2006) Falkowski and Pérez-Victoria (2008) Batell, Gherghetta and Sword (2008) Delgado and DD (2009)

The former motivation for the soft wall models was the appearance of linear KK excitations (mn ∼ n) which may provide a description of mesons spectrum in QCD, AdS/QCD.

Csaki and Reece (2007) Gursoy and Kiritsis (2008) Gursoy, Kiritsis and Nitti (2008) Batell and Gherghetta (2008)

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 3 / 12

THE MODEL

Field Content

Field SU(2)L × U(1)Y

• ff-shell dynamical d.o.f.

Ψi

L

2 ⊗ 1 8 Ψ1,2

R

1 ⊗ 1 8 Hi 2 ⊗ 1 8 AM 3 ⊕ 1 4 × 4 gMN 1 ⊗ 1 φ 1 ⊗ 1

The Action

S = Z

Σ

√g e−φ » i 2 ¯ Ψi

LγMDL MΨi L + i

2 ¯ Ψs

RγMDR MΨs R + M ¯

Ψi

LΨi L + M ¯

Ψs

RΨs R

+ λ1 ǫij ¯ Ψi

LΨ1 R H∗ j + λ2 Hi ¯

Ψi

LΨ2 R + h.c.

+gMN (DMH)† DNH − m2

hH†H +

1 4g2

5

gMRgNS tr {FMNFRS} # − Z

∂Σ

√g e−φ » λ0R2 “ H2 − v2 ”2 − e5

5

1 2 “ ¯ Ψi

LΨi L − ¯

Ψs

RΨs R

”– , (1) Σ = R4 × [z0, ∞), z0 > 0 , φ = µ2z2 , gMN = f 2 ηMN, f = R

z

and z0 ∼ R

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 4 / 12

THE MODEL

In addition ... γM = eM

A γA with γA =

` γµ, −iγ5´ , γ5 = „ 1 −1 « and eM

A = 1 f δM A .

AM = g5Aα

MTα,

ˆ Tα, Tβ ˜ = if γ

αβTγ,

DM, DR

M, DL M

gauge and Lorentz covariant derivatives. m2

h = a (a − 4)

R2 − 2a µ2 R2 z2 , λ1,2 and g5 are the 5d Yukawa and gauge coupling constants, respectively. λ0 and a are dimensionless constants while [v0] = E3/2. The lowest modes of the above (dynamical) fields are to be identified with the Standard Model

• nes.
• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 5 / 12

EQUATIONS OF MOTION

Equations of Motion for the Higgs VEV

H(x, z) = „ H(z) « , H(z) ∈ R H′′ + „ 3 f ′ f − φ′ « H′ − f 2 a (a − 4) R2 − 2a µ2 R2 z2 ! H = 0 , H′ − 2λ0R2 “ H2 − v2 ” H ˛ ˛ ˛

z0

= 0 , H = 8 > < > : “

z z0

”a q

a 2z0λ0R2 + v2

The scalar part of the action (1) can be written as

Z

∂Σ

√g e−φλ0R2 “ H4 − v4 ” ,

decreasing around H = 0 for λ0 < 0 = ⇒ the non trivial solution is the minimum of the action. We choose a = 2. Indeed this is the lowest order polynomial behavior for which the induced potential is binding enough for the fermionic solutions to be normalizable.

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 6 / 12

EQUATIONS OF MOTION

Fermionic Equations of motion

Ψi

L ≡ ΨL

Ψa

R ≡ ΨR ,

λ1,2 ≡ λ ieM

A γADR MΨL − 1

2 e5

AγAφ′ΨL + M ΨL + λ H(z) ΨR ,

ieM

A γADR MΨR − 1

2 e5

AγAφ′ΨR + M ΨR + λ H(z) ΨL = 0 ,

“ 1 ± γ5” ΨL,R ˛ ˛ ˛

z0

= 0 .

Smallest (Dirac) Eigenmass

m2R2 ≈ 2 1 Γ ` |α| − 1

2

´ (ζ0)|α|+ 1

2 ,

α = M R , ζ0 = λ √ R s 1 λ0 + v2

0 R3 ,

for ζ0 ≪ 1 and |α| > 1/2.

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 7 / 12

EQUATIONS OF MOTION

Higgs Wave Functions

H(x, z) = H(z) + ˜ H(x, z) , ˜ H − ∂2

5 ˜

H − „ 3 f ′ f − φ′ « ∂5 ˜ H + f 2 a (a − 4) R2 − 2a µ2 R2 z2 ! ˜ H = 0 , ∂5 ˜ H − 2λ0k2 »„˛ ˛ ˛H + ˜ H ˛ ˛ ˛

2 − v2

« ˜ H + ˛ ˛ ˛˜ H ˛ ˛ ˛

2 H +

“ ˜ H + ˜ H∗” H –

z0

= 0 , ˜ H = −mH

2 ˜

H .

Smallest Eigenmass (Higgs Mass)

mH

2 ≈ 2µ2

1 |ln (µR)| .

Then we have to take µ ∼TeV.

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 8 / 12

PHENOMENOLOGICAL PREDICTIONS ON FERMION MASSES

Smallest Fermion Mass vs Effective Yukawa Couplings We can estimate the order of magnitude of the lightest fermionic masses as

(mfR)2 ∼ 2 Γ ` |αf| − 1

2

´ 2 4 v u u t |αf| − 3

2

|αf| − 1

2

yf 4d 2 |ln (µR)| 3 5

˛ ˛ ˛αf ˛ ˛ ˛+ 1 2

.

f represents the fermion flavor. yf 4d is the effective 4d Yukawa coupling.

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 9 / 12

PHENOMENOLOGICAL PREDICTIONS ON FERMION MASSES

R−1 ∼ 1019GeV

10 12 14 16 18 20 22 Α 15 20 25 30 log10mR

Figure: y4d 1 (dashed line), y4d ∼ 0.1 (solid line).

8 > > > > < > > > > : y4d 1  mtR ∼ 10−17 , |αt| ≃ 13 mνR ∼ 10−28 , |αν| ≃ 20 y4d ∼ 0.1  |αt| ≃ 10 |αν| ≃ 15

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 10 / 12

PHENOMENOLOGICAL PREDICTIONS ON FERMION MASSES

R−1 ∼ 104GeV

10 15 20 Α 5 10 15 log10mR

Figure: y4d 1 (dashed line), y4d ∼ 0.1 (solid line).

8 > > > > < > > > > : y4d 1  mtR ∼ 10−2 , |αt| ≃ 5 mνR ∼ 10−13 , |αν| ≃ 18 y4d ∼ 0.1  |αt| ≃ 2 |αν| ≃ 12

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 11 / 12

CONCLUSIONS

We have worked out a soft wall model for ElectroWeak physics where all the matter and gauge content propagate in the 5d bulk. By computing the effective 4d Yukawa couplings we find that the fermion physical masses depend as a power law of the former, the exponent being the corresponding 5d bulk masses. This non universal power law behavior allows us to reproduce the hierarchy of the Standard Model fermion masses (from top quark to the neutrinos) with a non hierarchical 5d bulk masses. A more realistic study including the CKM matrices and the ElectroWeak constraints, although their inclusion would not substantially change our results, as well as the underlying gravity model inducing the dilaton VEV and its stability should be address.

• D. Diego (University of Notre Dame)

Fermion Mass Hierarchy· · · May, 2009 12 / 12