Theories with Compact Extra Dimensionshep.wisc.edu/~sheaff/PASI2012/lectures/burdman2.pdf · Gustavo Burdman Theories with Compact Extra Dimensions. The Origin of Flavor In the SM,

Theories with Compact Extra Dimensions

Gustavo Burdman

Instituto de Fısica – USP

PASI 2012 –UBA

Gustavo Burdman Theories with Compact Extra Dimensions

Theories with Compact Extra dimensions

Part II

Generating Large Hierarchies with ED Theories

Warped Extra Dimensions



One compact extra dimension. Non-trivial metric inducessmall energy scale from Planck scale! (L. Randall, R. Sundrum).

AdS5

Planck

L

TeVk ke−k L

Geometry of extra dimension generates hierarchy exponentially!

ΛTeV ∼ MPlanck e−k L with k the curvature



Warped 5D metric in RS

ds2 = e−2κ|y | ηµνdxµdxν − dy2

Compactified on S1/Z2 with L = πR

πR

AdS5

eκκ π R−

0

κ

y

and k ∼< MP , AdS5 curvature.

For kR ' (11− 12)

−→ κ e−κπR ' O(TeV).



Solving the Hierarchy Problem:

If Higgs localized at y = πR

SH =

∫d4x

∫ πR

0dy√−g δ(y − πR)

[gµν∂

µH†∂νH

−λ(|H|2 − v2

0

)2]

Warp factors eky appear in gµν and√−g .

SH =

∫d4x

[e−2kπRηµν∂

µH†∂νH − e−4kπRλ(|H|2 − v2

0

)2]

Canonically normalize H ⇒ e−kπRH → H



What we get now is

SH =

∫d4x

[ηµν∂

µH†∂νH − λ(|H|2 − e−2kπRv2

0

)2]

⇒ If v0 ' MP , then “new”scale in exponentially suppressed

v = e−kπR v0

Choosing kR ' O(10) gets us v ' weak scale.


Solving the Hierarchy Problem in AdS5

Metric in extra dimension ⇒ small energy scale from MP

(Randall-Sundrum)

ds2 = e−2κ|y | ηµνdxµdxν − dy2

TeVPlanck

AdS5

MP

kL_e

MP

L

Corrections to mh OKIf Higgs close to TeV brane

Need Higgs IR localization



If only gravity propagates in the bulk (SM fields on TeV brane):=⇒ Kaluza-Klein graviton tower

Zero-mode graviton G (0) localized toward the Planck brane.This is why gravity is weak! G (0) couples to SM fields as1/M2

P

First few KK graviton excitations localized toward TeV brane→ They couple strongly (as (1/TeV)2 to fields there.E.g.: Drell-Yan at hadron colliders

(n)Ge+

e−

q

q−


The Origin of Flavor

In the SM, flavor comes from dim-4 ops. For instance for quarks

LY = Y Uij Q

i Huj + Y Dij Q

iHd j ,

and similarly for leptons. Masses arise after EWSB: 〈H〉 = v/√

2

mijD = Y D

ij

v√2

But what is the origin of the Yukawa couplings Y U , Y D , etc. ?


The Origin of Flavor

Building a theory of flavor in extensions of the SM is not easy.Scale of EWSB is TeV scale ⇒ separation of scales withflavor.

Supersymetry: Flavor is generated at very high scale, possiblyGUT scale. (Avoiding flavor vioation from running effects ishard!)Technicolor: TC gets strong at TeV ⇒ EWSB. Flavor comesfrom ETC gauge bosons, with M ' (100− 1000) TeV.

In all cases, is very hard to accommodate the spectrum offermion masses naturally.


Bulk Life in WED

In original proposal, only gravity propagates in 5D bulk.

Allowing gauge fields and matter to propagate in the bulkopens many possibilities: models of EWSB, flavor, GUTs, etc.

Bulk Randall-Sundrum models:

Choose the gauge symmetry in the bulk: The electroweak SMgauge group has to be enlarged to avoid large corrections toEWP observables:

SU(2)L × SU(2)R × U(1)X

Write theory in the bulk and expand in Kaluza-Klein modes toget the effective 4D description.


Gauge Fields in the 5D bulk

A gauge field AM(xµ, y) propagating in the extra dimensionalbulk:

SA = −1

4

∫d4xdy

√−g FMNFMN ,

with√g = det[gMN ] = e−4σ, σ ≡ k |y |.

Field strength defined as

FMN ≡ ∂MAN − ∂NAM + ig5[AM ,AN ]

The KK decomposition is

Aµ(x , y) =1√

2πR

∞∑n=0

A(n)µ (x)χ(n)(y)



χ(n)(y): wave-function of the n-th KK mode in the extradimension.

Imposing the EoM (Euler-Lagrange) on AM(xµ, y) we get

−∂y(e−2σ∂yχ

(n))

= m2nχ

(n)

They satisfy the ortho-normality condition

1

2πR

∫ πR

−πRdy χ(m)χ(n) = δmn

Solving this differential equation gives χ(n)(y).



The solutions are of the form

χ(n)(y) =eσ

Nn

[J1(

mn

κeσ) + αnY1(

mn

κeσ)]

where Nn is a normalization factor, and again σ = k|y |.In the interval [0, πR], χ(n) peaks ' πR for low values of n.⇒ first few KK modes of gauge bosons arelocalized near the TeV brane.


Gauge Fields in the bulk - KK Masses

The constants αn, and the KK masses are determined by theboundary conditions on the Planck and TeV branes.

The masses of the first few KK modes are

mn ' (n − O(1))× πκe−κπR

I.e. for appropriate choice of κR1st KK excitations are O(TeV).

To be expected: what is localized near the TeV brane hasTeV-like masses !


Fermions in Warped Extra Dimensions

The action for fermion fields Ψ(xµ, y) in the bulk

Sf =

∫d4x dy

√g

{i

2ΨγM

[DM −

←DM

]Ψ− sgn(y)Mf ΨΨ

}with the covariant derivative given by

DM ≡ ∂M +1

8[γα, γβ] VN

α VβN;M

and γM ≡ VMα γα, with VM

α = diag(eσ, eσ, eσ, eσ, 1) theinverse vierbein.

To be natural, we need Mf ' O(1)k , with k ' MPlanck, theonly scale.



Although Ψ(xµ, y) is not a chiral fermion, we can still write

ΨL,R ≡1

2(1∓ γ5)

Then the KK expansion is

ΨL,R(x , y) =1√

2πR

∑n=0

ψL,Rn (x)e2σf L,Rn (y)

with f L,Rn (y) the wave-function of the n-th KK fermion in theextra dimension.



Demanding that ψL,Rn (x) be usual 4D fermions, we obtain

(∂y −Mf ) f Ln (y) = −Mn eσ f Rn (y)

(∂y + Mf ) f Rn (y) = Mn eσ f Ln (y)

Zero mode fermions: M0 = 0. Solving the differential eqns.we get:

f R,L0 (y) =

√kπR (1± 2cL,R)

ekπR(1±2cL,R) − 1e±cL,R k y

where we defined Mf = cf k .

⇒ the bulk mass parameter cf ' O(1), defines thelocalization of the fermion in the extra dimension.



To see how the localization depends on c we need tonormalize the fermions canonically. Then the zero-modefermion wave-function is

F LZM(y) =

1√2πR

f L0 (0)e( 12−cL) ky

If cL > 1/2 ⇒ fermion localized near y = 0, Planck brane.If cL < 1/2 ⇒ fermion localized near y = πR, TeV brane.


Flavor Models in WED

O(1) flavor breaking in bulk can generate fermion masshierarchy:

Higgs

πR0

C<1/2

C=1/2

C>1/2

Fermions localized toward the TeV brane can have largerYukawas,Those localized toward the Planck brane have highlysuppressed ones.


Yukawa Couplings

For instance, assuming a TeV-brane localized Higgs, the Yukawacouplings are

SY =

∫d4x dy

√gλ5Dij

2M5Ψi (x , y)δ(y − πr)H(x)Ψj(x , y)

Then, the 4D Yukawa couplings are

Yij =

(λ5Dij k

M5

) √(1/2− cL)

ekπR(1−2cL) − 1

√(1/2− cR)

ekπR(1−2cR) − 1ekπR(1−cL−cR)


Yukawa Couplings

The Yukawa couplings as a function of cL, assuming cR = −0.4(i.e. TeV localized right-handed fermion):


The Bulk RS Picture

tb L t R

LightFermions

Higgs

orHiggless

SU(2)L SU(2)Rx x U(1)

0 L

Models ofEWSB and Flavor

EWPC: T OK, but S ' N/π at tree-level

MKK ∼> (2− 3) TeV

Z → bb require discrete symmetry (L↔ R)(Agashe, Contino, Da Rold, Pomarol)

Potentially important bounds and/or effectsfrom flavor violation


Dynamical Origin of the Higgs Sector

What localizes the Higgs to/near the IR/TeV brane ?

Gauge-Higgs Unification

Zero-mode Fermion Condensation

Higgsless


Gauge-Higgs Unification in AdS5

If there is a Higgs: what is its dynamical origin ?Or why is it localized towards the TeV brane ?

Gauge field in 5D has scalar A5

To extract H from A5 need to enlarge SM gauge symmetry.

E.g. SU(3)→ SU(2)× U(1) by boundary conditions:

Aµ :

(+,+) (+,+) (−,−)(+,+) (+,+) (−,−)

(−,−) (−,−) (+,+)

A5 :

(−,−) (−,−) (+,+)(−,−) (−,−) (+,+)

(+,+) (+,+) (−,−)

⇒ Higgs doublet from A5 = Aa

5ta



To build realistic models of EWSB from AdS5:

Isospin symmetry: need SO(4)× U(1)X in bulk⇒ SO(5)× U(1)X → SO(4)× U(1)X by BCs(Agashe, Contino, Pomarol)

Higgs is 4 of SO(4): 4 d.o.f. ↔ complex SU(2)L doublet

Gauge bosons and fermions in complete SO(5) multiplets

Implementing additional symmetry to protect Z → bb⇒ spectrum of KK fermions, lighter than KK gauge bosons.(Contino, Da Rold, Pomarol)



E.g.: Fermions can be

52/3 = (2, 2)2/3 ⊕ (1, 1)2/3

or102/3 = (2, 2)2/3 ⊕ (1, 3)2/3 ⊕ (3, 1)2/3

to satisfy custodial + L↔ R symmetry.

BCs ⇒ masses of KK fermions tend to be light (because topis heavy)



Signals:

Rich gauge boson spectrum, at few TeV

Light KK fermion spectrum: could be as light as 500 GeV

Very distinctive signals:

E.g. b-type KK fermion → tW⇒ 4W’s + 2b signals (Dennis, Servant, Unel, Tseng)Enhanced t1 pair production through KK gluon(Carena, Medina, Panes, Shah, Wagner)


Flavor Violation in AdS5 Models

KK Gauge Bosons couple stronger to heavier fermions

0 L

light fermions heavy fermions

KK Gauge Bosons

⇒ Tree-level flavor violation is hierarchical:Only important with the heavier generations


Flavor Violation: The Good

If ZM fermion is localized close enough to IR can interaction withKK gauge boson (e.g. KK gluon) be strong enough to condensefermions ?

Need 〈F F 〉 ⇒ F is ZM quark

Scale of condensation is MKK ' 1 TeV

⇒ mF ' (500− 600) GeV⇒ 4th Generation Condensation in AdS5


EWSB from Fourth-Generation in AdS5

Need 4th-generation strongly coupled to new interaction

If 4th-generation propagates in AdS5 bulk and is highlylocalized on the TeV brane (G.B., Da Rold)4th-generation quarks are strongly coupled to KK gluon:


EWSB from Fourth-Generation in AdS5

If gU > g crit.U , ⇒ 〈ULUR〉 6= 0

⇒ Solution to the gap equation:

This implies

Electroweak Symmetry Breaking

Dynamical mU

We can also write an effective theory at low energy for the Higgs.


Predictions for mU , mH

For MKK = 3 TeV, we get

mU ' 600 GeV

mh ' 900 GeV

Higgs naturally TeV-brane localized

Fermion masses (other than mU):From bulk Higher Dimensional Ops.


Two Higgs Doublets from 4G Condensation

If both U and D 4th gen. quarks condense⇒ Two-Higgs Doublet model (GB Haluch ’11)

PQ Symmetry ⇒ pseudo-scalar A is light:mA ' (25− 130) GeV.

Scalars are heavy (h,H,H±) in the (550− 700) GeV range

Phenomenology at LHC interesting (GB, Haluch, Matheus ’11)

But current LHC bounds on 4G quarks closing inmQ4 > 500 GeV


Flavor Violation: The Bad

Constraints from low energy/flavor physicsPotentially, deviations in

CP asymmetries in B decays

b → s`+`−

∆mBs

D0 − D0 mixing

But for MKK ∼> 2 TeV, and/or adjusting the down-quark rotationmatrices DL,R , these bounds can be evaded.


Single Top Production at High Invariant Mass (Aquino, GB, Eboli ’07)

c_

Utcgq

gt

_q

��

��

Gtq

Signal: t + jet, very large invariant mass (> 1.5 TeV).

Use t → b`ν.

Typically few hundred fb−1.


Flavor Violation: The Ugly

But one flavor observable is hard to get around: εK

Mixed Chirality operators

dRsLdLsR

have enhancement of(mK

ms

)2

η−51 ∼> 100

Bound from εK is MKK > 30 TeV !!

Need flavor symmetries to protect from this.


Future Avenues

Build AdS5 models with flavor symmetries protecting from εKbounds

Abandon AdS5 dogma:

Keep the good featuresAbandon 5D metric: Deconstruct AdS5.Resulting 4D Theories have much less flavor violation far fromthe continuum (GB, Fonseca, Lima ’12)


Documents

Theories with Compact Extra Dimensionshep.wisc.edu/~sheaff/PASI2012/lectures/burdman2.pdf · Gustavo Burdman Theories with Compact Extra Dimensions. The Origin of Flavor In the SM,