theories with compact extra dimensionshep.wisc.edu/~sheaff/pasi2012/lectures/burdman2.pdf ·...
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Theories with Compact Extra Dimensions
Gustavo Burdman
Instituto de Fısica – USP
PASI 2012 –UBA
Gustavo Burdman Theories with Compact Extra Dimensions
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Theories with Compact Extra dimensions
Part II
Generating Large Hierarchies with ED Theories
Warped Extra Dimensions
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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
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Warped Extra Dimensions
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).
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Warped Extra Dimensions
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
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Warped Extra Dimensions
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.
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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
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Warped Extra Dimensions
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−
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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. ?
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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.
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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.
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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)
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Gauge Fields in the 5D bulk
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).
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Gauge Fields in the 5D bulk
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.
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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 !
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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.
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Fermions in Warped Extra Dimensions
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.
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Fermions in Warped Extra Dimensions
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.
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Fermions in Warped Extra Dimensions
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.
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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.
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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)
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Yukawa Couplings
The Yukawa couplings as a function of cL, assuming cR = −0.4(i.e. TeV localized right-handed fermion):
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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
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Dynamical Origin of the Higgs Sector
What localizes the Higgs to/near the IR/TeV brane ?
Gauge-Higgs Unification
Zero-mode Fermion Condensation
Higgsless
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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
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Gauge-Higgs Unification in AdS5
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)
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Gauge-Higgs Unification in AdS5
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)
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Gauge-Higgs Unification in AdS5
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)
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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
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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
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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:
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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.
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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.
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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
Gustavo Burdman Theories with Compact Extra Dimensions
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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.
Gustavo Burdman Theories with Compact Extra Dimensions
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Single Top Production at High Invariant Mass (Aquino, GB, Eboli ’07)
c_
Utcgq
gt
_q
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������
Gtq
Signal: t + jet, very large invariant mass (> 1.5 TeV).
Use t → b`ν.
Typically few hundred fb−1.
Gustavo Burdman Theories with Compact Extra Dimensions
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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.
Gustavo Burdman Theories with Compact Extra Dimensions
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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)
Gustavo Burdman Theories with Compact Extra Dimensions