[PPT] - New Solutions to the Hierarchy Problem What to Expect at the TeV PowerPoint Presentation

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New Solutions to the Hierarchy Problem

What to Expect at the TeV Scale

Gustavo Burdman Instituto de F´ ısica - USP Lishep 2006, Rio de Janeiro, March 27-30 2006

New Solutions to the Hierarchy Problem – p.1/39

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Outline

∗ The Hierarchy Problem: Why do we believe there will be new

physics at the TeV scale ?

∗ Theories with Extra Dimensions:

Large Extra Dimensions Universal Extra Dimension Warped Extra Dimensions

∗ Taming the hierarchy with global or discrete symmetries:

The Little Higgs The Twin Higgs

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Limitations of the Standard Model

Many questions unanswered: What is the origin of Fermion masses ?: In the Standard Model, ad hoc couplings of Higgs to fermions are adjusted to obtain

(me)/(mt) ∼ 10−6, mν ∼ < 1 eV

Do interactions Unify at high energies ?

SU(3) × SU(2)L × U(1) − → G ?

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Limitations of the Standard Model

What is the origin of the Baryon Asymmetry ? What is the Dark Matter ? Why is the Cosmological Constant so small ? What is the Dark Energy ?

. . .

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The Question at the TeV Scale

The Hierarchy Problem: Why is

MW (∼ 100 GeV) ≪ MP (∼ 1019 GeV)?

If Higgs elementary and the SM is valid up to MP then what generates

MW MP ≪ 1

This requires fine-tuning of the SM parameters. Quantum corrections naturally drive v to MP

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The Hierarchy Problem

Quantum corrections to mh =

√ 2λv:

m ∆

h 2 =

h h

⇒ ∆m2

h ∼ E2 UV ∼ M 2 P

⇒ We need

mbare

h

− Radiative Corrections

∼ mhphys.

(O(MP ) − O(MP ) ) ∼ 100GeV

In the Standard Model weak scale not naturally stable.

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The Hierarchy Problem

New physics at the TeV scale to stabilize the weak scale. Additional states cancel divergences due to symmetries (e.g. Supersymmetry) Higgs is composite and “comes apart” at scale Λ. Technicolor/Topcolor Little Higgs: Higgs as a Nambu-Goldstone Boson. Extra Spatial Dimensions: Large Extra Dimensions Universal Extra Dimensions Warped ED (Randall-Sundrum)

. . .

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Compact Extra Dimensions

Extra spatial dimensions with points periodically identified 1 Extra Dimension: equivalent to a circle

L 2L 3L x+2L x+L x+3L x

R

with R = L/2π. We identified the points

x ∼ x + L ∼ x + 2L ∼ x + 3L ∼ · · ·

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Large Extra Dimensions

Assume space has 3 + n dimensions. The extra n dimensions are compact and with radius R. All particles are confined to a 3-dimensional slice (“brane”). Gravity propagates in all 3 + n dimensions.

gravitons

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Large Extra Dimensions

(Arkhani-Hamed, Dimopoulos, Dvali ’98) Gravity appears weak (MP ≪ MW ), because it propagates in large extra dimensions... Its strength is diluted by the volume of the n extra dimensions. Fundamental scale is M∗ ∼ MW , not MP

M 2

P ∼ M n+2 ∗

Rn

There is no hierarchy problem: The fundamental scale of Gravity

M∗ ∼ 1 TeV

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Large Extra Dimensions

If we require M∗ = 1 TeV:

R ∼ 2 · 10−17 10

32 n cm

n = 1 = ⇒ R = 108 Km. Already excluded! n = 2 = ⇒ R ≃ 2 mm. Barely allowed by current gravity

experiments.

n > 2 = ⇒ R < 10−6 mm. This is fine.

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Large Extra Dimensions - Compactification

When field propagates in one extra dimension

PM = Pµ + P5

with µ = 0, 1, 2, 3, M = µ, 5. But XD is compact ⇒ P5 is quantized: periodicity ⇒ wavewlength has to be integer number of 2πR.

P5 = n R , (n = 0, 1, 2, 3, · · ·)

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Large Extra Dimensions - Compactification

If field has mass M

PMP M = PµP µ − P 2

5 = PµP µ − n2

R2

From the 4D point of view:

PµP µ = M 2 + n2 R2

E.g. for a photon (or graviton) M = 0. There is a “n = 0-mode” with zero mass (our photon/graviton), plus infinite excitations with masses n/R.

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Large Extra Dimensions

Compact extra dimensions ⇒ graviton excitations (Kaluza-Klein)

∆ m ∆ m ∆ m ∆ m 2 π R

Mass gap ∆m ∼ 1/R

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Large Extra Dimensions

E.g. for

n = 2 − → ∆m = 10−3 eV. n = 3 − → ∆m = 100 eV.

. . .

n = 7 − → ∆m = 100 MeV.

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Large Extra Dimensions - Phenomenology

Individual KK graviton couplings gravitationally suppressed (∼ 1/MP ). But for E ≫ 1/R → sum of KK mode results in

σ ∼ En M n+2

∗

.

Collider Processes:

(n) G (n) G

g g g E.g. Graviton production γ

q q −

Individual graviton decay rates ∼ 1/M 2

P , ⇒ ET signals at colliders.

Bounds on M∗ from LEP and Tevatron (1 − 10) TeV.

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Universal Extra Dimensions

(Appelquist, Cheng, Dobrescu ’01) If some SM fields propagate in the bulk ⇒ 1/R ∼

> 1 TeV .

But if we assume all fields can propagate in the extra dimensions. What is the allowed R ?

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Universal Extra Dimensions

For example, a scalar field Φ(x, y) in one extra dimension:

S[Φ(x, y)] = 1 2

d4x dy
∂MΦ∂MΦ − M 2Φ2

Periodic boundary conditions:

Φ(y) = Φ(y + 2πR)

Expand in Fourier modes:

Φ(x, y) = 1 √ πR

n=0
φn(x) cos

ny R

+ ˜

φn(x) sin ny R

φn(x) and ˜

φn(x) are 4D fields.

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Universal Extra Dimensions

Integrate over the compact dimension:

S4Deff.[φ, ˜ φ] = 2πR dy S[Φ]

with

S4Deff. =

n=0

1 2

dx

∂µφn∂µφn − m2

nφ2 n

+
n=0

1 2

dx

∂µ ˜ φn∂µ ˜ φn − m2

n ˜

φ2

n

with

m2

n = M 2 + n2

R2

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Universal Extra Dimensions

Momentum conservation in the extra dimensions At any vertex, PM, is conserved. Then 4D-momentum conservation ⇒ P5 is conserved. E.g.in (1) + (2) → (3)

p5

(1) + p5 (2) = p5 (3)

In terms of KK modes, this reads

±n1 ± n2 = ±n3 ⇒ KK-number conservation

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Universal Extra Dimensions

For instance,

1 Forbidden 1 1 OK

⇒

KK excitations must be pair produced This leads to Bounds on 1/R are lower / Distinctive phenomenology

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Universal Extra Dimensions - Fermions

The action for a bulk fermion in 5D:

SΨ =

d4x dy ¯

Ψ(x, y)

i∂MΓM − M
Ψ(x, y)
d4x dy ¯

Ψ(x, y) [i∂µΓµ − M] Ψ(x, y) − ¯ Ψ(x, y)γ5∂5Ψ(x, y)

Clifford algebra in 5D

{ΓM, ΓN} = 2ηMN

with Γµ = γµ and Γ5 = iΓ5.

⇒ Ψ(x, y) are 4-component Dirac spinors.

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Universal Extra Dimensions - Fermions

After “dimensional reduction” (integrating in y):

Sψ =

n=0
d4x
¯

ψn

i∂µγµ − M + i n

R

ψn
Zero mode (n = 0), is always a vector-like fermion!

But in the SM we need chiral fermions!

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Universal Extra Dimensions - Fermions

Chirality: Define

Ψ = ΨL + ΨR

And ask properties under y → −y reflections (“parity”):

γ5Ψ(−y) = ±Ψ(y)

Given that

γ5Ψ(−y) = −ΨL(−y) + ΨR(−y)

If we have

ΨR(−y) = ΨR(y) ΨL(−y) = −ΨL(y)

then ΨL(x, y) is odd, ΨR(x, y) is even under parity.

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Universal Extra Dimensions - Fermions

In this case, expanding in KK modes:

Ψ(x, y) = 1 √ πR

n=0
ψnR(x) cos

ny R

+ ˜

ψnL(x) sin ny R

So that the zero mode is Right-Handed !

Had we chosen γ5Ψ(−y) = −Ψ(y), i.e.

ΨR(−y) = −ΨR(y) ΨL(−y) = ΨL(y)

Then the zero mode would be Left-Handed.

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Universal Extra Dimensions

But how do we define “parity” in a circle ? Orbifold Compactification: Identify points opposite in the circle (y ∼ −y).

πR Z

2

S

1

y −y

Circle now reduced to segment, with “fixed points” at 0 and πR. Fields can be even or odd under y → −y. Bulk fermions have chiral zero modes (either LH or RH).

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Compact Extra Dimensions

Disclaimer: Theories with more than 4D are non-renormalizable.

1 ___ R Enp 4D Eff. Theory (~SM) Λ 5D Eff. Theory (KK)

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Universal Extra Dimensions

But Orbifolding breaks KK-number conservation ! Translation invariance broken in the y direction

⇒ p5 not conserved !

The presence of fixed points breaks KK number. By how much ?

πR

O

4D

Localized 4D operators at y = 0 and y = πR generate KK-number-violating interactions. E.g:

Sloc. =
d4x

πR dy i¯ Ψ(x, y)γµDµΨ(x, y)

δ(y)c1

Λ + δ(y − πR)c2 Λ

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Universal Extra Dimensions

UV physics might not operate differently in y = 0 and y = πR.

πR

Z2

If c1 = c2 ⇒ KK-number violating interactions still respect KK-parity. E.g. in (1) + (2) ↔ (3)

(−1)n1+n2+n3 = 1

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Universal Extra Dimensions

Conservation of KK parity ⇒ Can produce level 2 KK modes in s-channel.

1

2

Forbidden OK

Lightest KK Particle of level 1 (LKP) is stable

Forbidden

LKP(1)

Forbidden

LKP(1) 1’

⇒ LKP is Dark Matter candidate

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UED Phenomenology

Electroweak precision constraints:

1/R ∼ > 300 GeV for 5D 1/R ∼ > (400 − 600) GeV for 6D

Current direct searches give similar bounds.

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UED Phenomenology

Spectrum at each KK level is degenerate at tree level. Localized operators split the masses (one-loop generated). First KK mode in 5D model, with ci’s computed at one-loop.

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UED Phenomenology

Light KK modes ⇒ large cross sections. But, almost degenerate KK levels ⇒ little energy release. Best mode q¯

q → Q1Q1 → Z1Z1+ ET → 4ℓ+ ET (Cheng, Matchev,

Schmaltz ’02).

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UED Phenomenology

Reach using this golden mode q¯

q →→ 4ℓ+ ET

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UED Phenomenology

Production and Decay of Second KK Level: They couple to 2 zero modes through brane couplings (loop generated). (Datta, Kong, Matchev ’05)

2 ci R Λ

with ΛR ≫ 1 and ci ∼ O(1). But has to compete with 2 → 1 + 1

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UED 6D Phenomenology

Two Universal Extra Dimensions(Burdman, Dobrescu, Pontón ’06): Proton decay is adequately suppressed. Number of generations can only be multiple of 3. Very different phenomenology: Additional “adjoint” scalar: For each gauge boson

AM → Aµ, A5, A6.

In 5D, A5 eaten by KK modes of gauge bosons. In 6D, one combination survives in the spectrum. It decays almost exclusively to t¯

t.

Is likely the LKP ⇒ scalar dark matter. Mass of level 2 is

√ 2/R (instead of 2/R in 5D).

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UED 6D Phenomenology

Spectrum: mass splittings from loops in the bulk:

400 450 500 550 600 650 700

mass

1R 500 GeV

GΜ

1,0

WΜ

1,0

BΜ

1,0

GH

1,0

WH

1,0

BH

1,0

Q

3 1,0

Q

1,0

T

1,0

U

1,0

D

1,0

L

1,0

1,0

H1,0 600 650 700 750 800 850 900 950 1000 1050

mass

1R 500 GeV

GΜ

1,1

WΜ

1,1

BΜ

1,1

GH

1,1

WH

1,1

BH

1,1

Q

3 1,1

Q

1,1

T

1,1

U

1,1

D

1,1

L

1,1

1,1

H1,1

1st KK Level (1, 0) 2nd KK Level (1, 1)

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UED 6D Phenomenology

Typical diagram:

❅ ❅ ❅ ❅ ❅

①

✟✟✟✟✟✟

✦✦✦✦✦✦

❛❛❛❛❛❛ ✇

q q G(1,1)

µ

U (1,1)

−

jet jet G(1,1)

H

t t

❍❍❍❍❍❍

Other diagrams, with W (1,1)

µ

and B(1,1)

µ

also important.

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UED 6D Phenomenology

Large enhancement of t¯

t at large invariant mass.

Several resonances closely spaced. Tevatron reach: for 10fb−1 can see resonances up to 800 GeV. Bound in compactification scale

1/R ≤ 600 GeV

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