The 6D Standard Model - Purdue Universityweb.ics.purdue.edu/~markru/rakhi.pdfThe 6D Standard Model...

The 6D Standard ModelA tale of spinless adjoints

Rakhi Mahbubani

Fermilab

Burdman, Dobrescu and Ponton: hep-ph/0506334 and hep-ph/0601186

Dobrescu, Kong and RM: hep-ph/0703231

Dobrescu, Hooper, Kong and RM: arXiv:0706.3409

Purdue HEP seminar – p.1

A free lunch: gravity in 5DUnder infinitesimal diffeomorphisms xµ → xµ + ξµ fields transform

as

δhµν = ∂µξν + ∂νξµ

δhµ4 = ∂µξ4 + ∂4ξµ

δh44 = 2∂4ξ4



as

δh(0)µν = ∂µξ(0)

ν + ∂νξ(0)µ

δh(0)µ4 = ∂µξ

(0)4

δh(0)44 = 0

Get U(1) gauge theory from gravity for free!



as

δh(n)µν = ∂µξ(n)

ν + ∂νξ(n)µ

δh(n)µ4 = ∂µξ

(n)4 + ∂4ξ

(n)µ

δh(n)44 = 2∂4ξ

(n)4

For massive modes form diff invariant combinations.

Schematically:

h′(n)µν = h(n)

µν − 1

∂4

[

∂νh(n)µ4 + (µ ↔ ν)

]

+ 2∂µ∂ν

∂24

h(n)44

All components get ‘eaten’ to form longitudinal polarizations of mas-

sive graviton.


Gauge theories in 6DUnder infinitesimal gauge transformations in 6D massive modes

transform as:

δA(n)µ = ∂µξ(n)

δA(n)m = ∂mξ(n)

Now we can form 2 gauge-invariant combinations of fields:

A′(n)µ = A(n)

µ − 1

∂24 + ∂2

5

∂µ

(

∂4A(n)4 + ∂5A

(n)5

)

A(n)H = ∂5A

(n)4 − ∂4A

(n)5

From each 6D gauge field comes a tower of heavy 4D gauge fields

and a tower of heavy 4D scalar (or ‘spinless’) adjoints


Preview

Level-1 modes and decays in 5D


Preview

In 6D

6D spinless adjoints make for interesting and distinctivecollider and astrophysical phenomenology


Outline

Motivation

The 6DSM

Astrophysics of LKP

Hadron collider phenomenology of level-1 modes

Hadron collider phenomenology of level-2 modes

Conclusion


Why do we care about the 6DSM?

It has something for everyone!

For theorists:

The 6DSM provides a simple explanation for:

proton stability

neutrino masses [Appequist, Dobrescu and Ponton: hep-ph/0201131]

number of fermion generations [Dobrescu and Poppitz:

hep-ph/0102010]


Why do we care about the 6DSM?

For experimentalists:

In comparison with MUED [Cheng, Matchev and Schmaltz: hep-

ph/0205314] the 6DSM has

level-2 modes that are easier toproduce

level-2 modes that arekinematically forbidden fromdecaying to pairs of level-1modes → harder decayproducts

Spectacular collider signatures

M(2)6D = M

(2)5D

2√1 + 1

M(2)6D =

√2M

(1)6D


The 6DSM

All SM fields propagate in 2

extra dimensions compactified

on square of side πR = L

Adjacent sides of square identi-fied (‘chiral square’), conical sin-gularities at boundaries

x5

x4

Sides are

Sides are

identifiedL

L0identified

Reflection symm

etry

Description valid until cutoff scale Λ where ΛR = 10

Reflection symmetry → KK parity

Lagrangian for theory is

L = Lbulk +

(

δ(x4)δ(x5) + δ(L − x4)δ(L − x5)

)

L1 + δ(L − x5)L2


Solving for bulk fields

Lagrangian densities on adjacent sides of square equal→ bulk fields and first derivatives vary smoothly acrossboundary

Extra dimensional dependence of fields encoded in

functions f(j,k)n with n = 0, 1, 2, 3

KK numbers (j, k) are integers and j ≥ 1, k ≥ 0 orj = k = 0.

All (j, k) modes have tree-level mass√

j2 + k2/R

All fields in a tower including a zero mode have thefunctional form

f(1,0)0 (x4, x5) = cos

(x4

R

)

+ cos(x5

R

)


Solving for bulk fields (fermions)

Solving for fermions with identified boundary conditionsenforces different extra dimensional dependence for LHand RH fermions

→ Chiral fermions

Each 6D helicity contains both LH and RH fermions, butonly one of the 4D helicities will have a zero mode

Theory invariant under KK-parity

Φ(j,k) → (−1)(j+k) Φ(j,k)


Gauge and fermion KK towers

adjoint L R L R

Gauge bosons. .

. .

. . .

.

. . .

.

. . .

.

. . .

.

. . .

.

. . .

.

eaten

Fermions‘+’ chirality ‘−’ chirality

spinless

A(j,k)µ

√2

R

√5

R

U(j,k)−Q

(j,k)+

1R

2R

A(j,k)HA

(j,k)G

For each SM gauge boson, there is a spinless adjoint

Chiral square gives rise to chiral zero modes forfermions

All other modes in fermion towers vector-like


Localized terms

L = Lbulk +(

δ(x4)δ(x5)+ δ(L−x4)δ(L−x5))

L1 + δ(L−x5)L2

6D Lorentz invariance broken on boundaries → masssplitting

Boundary terms break KK number conservation(translational invariance in extra dimensions), butconserve KK parity, and can come from

physics above cutoffloops involving bulk interactions


Localized terms

L = Lbulk +(

δ(x4)δ(x5)+ δ(L−x4)δ(L−x5))

L1 + δ(L−x5)L2

6D Lorentz invariance broken on boundaries → masssplitting

Boundary terms break KK number conservation(translational invariance in extra dimensions), butconserve KK parity, and can come from

physics above cutoff assume negligibleloops involving bulk interactions

The 6DSM is predictive - only 2 parameters (R,mh)


Mass Corrections

700

650

600

550

500

450

400

W(1,0)

H

B(1,0)

H

B(1,0)µ

G(1,0)µ

Q(1,0)3+ T

(1,0)−

U(1,0)−

W (1,0)µ

tree-level one-loop

everything1

R

M(G

eV) D

(1,0)−

Q(1,0)+

L(1,0)+

G(1,0)

HE

(1,0)−

Mass degeneracy broken by one-loop generatedboundary terms [Ponton and Wang, hep-ph/0512304]


Mass Corrections (cont)

700

650

600

550

500

450

400

W(1,0)

H

B(1,0)

H

B(1,0)µ

G(1,0)µ

Q(1,0)3+ T

(1,0)−

U(1,0)−

W (1,0)µ

one-loop

M(G

eV) D

(1,0)−

Q(1,0)+

L(1,0)+

G(1,0)

HE

(1,0)−

Level-1 spinless adjoints get

negative contributions to

mass

Splittings allow cascade

decays → interesting

phenomenology

Hypercharge spinless adjoint

(B(1,0)H ) is LKP: “spinless

photon”


Calculating thermal relic density

Density of thermal relic is computed by solvingBoltzmann’s equation

dn

dt= −3Hn − 〈σvr〉

(

n2 − n2eq

)

Can find approximate solutions at early and late times

For temperatures T ≫ MB, n ∼ T 3

For T < MB, n =(

MB T2π

)3/2e−MB/T

As temperature decreases further, B(1,0)H freezes out


Calculating thermal relic density (cont.)

Freezout temperature roughly given by

〈σvr〉n|T=TF∼ H

where 〈σvr〉 = a + 6 b TF

MB+ · · · in non-relativistic limit

Matching early- and late- time solutions at freezout

gives current B(1,0)H density

ΩBHh2 ≈ #

MB/TF

a + 3 b TF /MB


Spinless photon dark matter

Spinless photon is stable and a good dark mattercandidate.

It annihilates to:

W+W− and ZZ pairs

B(1,0)H

B(1,0)H

W+, Z

W−, Z

h

hh pairs

+ +

B(1,0)H

B(1,0)H

B(1,0)H

B(1,0)H

B(1,0)H

B(1,0)H

h

h

h

h

h

h

h

B(1,0)H


Spinless photon dark matter

Spinless photon is stable and a good dark mattercandidate.

It annihilates to:

ff pairs

+ +

B(1,0)H

B(1,0)H

B(1,0)H

B(1,0)H

B(1,0)H

B(1,0)H

h

f

f

f

f

f

f

F(1,0)± F

(1,0)±


Fermion final states

Interactions between two spinless photons and twostandard model fermions given by

i

Λ2B

(1,0)H B

(1,0)H fγµ∂µf +

1

Λ2B

(1,0)H B

(1,0)H fγµγ5∂µf

Using equations of motion

mf

Λ2B

(1,0)H B

(1,0)H ff − imf

Λ2B

(1,0)H B

(1,0)H fγ5f

Annihilation and elastic scattering helicity suppressed


Diagonalizing fermion mass matrix

Because of mass suppression only top sector isimportant

EWSB induces off-diagonal terms in KK top massmatrix

(

T(1,0)−

T(1,0)+

)

(

− 1R (1 + ∆

−) mt(1 + δ1)

mt(1 + δ2)1R (1 + ∆+)

)(

T(1,0)−

T(1,0)+

)

Can diagonalize to find KK top mass eigenstates withmass

MT =

√

1

R2+ m2

t (1 + ∆)


Annihilation

Non-relativistic expansion of annihilation cross sectionis valid away from resonances.

In vicinity of s-channel higgs resonance, calculate relicdensity using micrOMEGAS


Relic Density


Direct Detection

Dark matter can be detected through elastic scatteringwith ordinary matter

This can be thought of as scattering off constituentquarks in the nucleus of some detector material

+ +

q

BH

q

BH

q

BH

Q(1,0)±

q

BH

q

BH

BH

q

h Q(1,0)±

Total B(1,0)H -nucleon cross-section is completely spin

independent:

σ =m2

N

4π(MB + mN )2

(

ZfBH

p + (A − Z)fBH

n

)2


Direct detection cross-section


Phenomenology of (1,0) modes

(1,0) particles pair produced at colliders, cascade decay

to B(1,0)H (LKP) which escapes detector → missing

energy signature.

Large multiplicity of production channels, productioncross sections calculated using compHEP

Colored particles produced with greatest abundance viae.g

+ +

g

g

g

g

g

gg

G(1,0)H

G(1,0)H

G(1,0)H

G(1,0)H

G(1,0)H

G(1,0)H

G(1,0)H


Production at LHC


Decays of (1,0) modes

Bosons that are lighter than KKfermions only decay via phase-space suppressed 3-body de-cays

•

•

A2 A1

F

f

f

One-loop 2-body decay width

for B(1,0)µ is of the same order

Gives rise to photons in finalstate

B(1,0)ν

B(1,0)H

F (j′,k′)

F (j,k)

F (j,k)

γµ

Branching fractions are calculable analytically withcertain approximations

Significant cross sections to final states with multiple leptons


Multi-lepton event

700

650

600

550

500

450

400

M[G

eV]

Q(1,0)+

L(1,0)+

W(1,0)

H

B(1,0)

H

W (1,0)µ

B(1,0)µ

gg

g

Q(1)

+

Q(1)

+

q ′

q

W(1)+

µ

W(1)3

H

B(1)

H

ℓℓ

B(1)

H

ℓ

ℓℓ

L(1)

+B

(1)

µ

ν


Lepton + photon event

700

650

600

550

500

450

400

$M$

[GeV

]

$Q_+^(1,0)$

$U_−^(1,0)$

$L_+^(1,0)$

$W^(1,0)_H$

$B^(1,0)_H$

$W^(1,0)_\mu$

$B^(1,0)_\mu$

qq

G(1)

H

Q(1)

+

Q(1)

−

qq

W(1)3

µB(1)

µ

B(1)

H

ℓ ℓ

B(1)

H γ

ℓ

N(1)

+W

(1)+

H

ν


Multiple leptons at the LHC

∼100 events with > 5ℓ for 1fb−1 of data!


Unusual signatures

Lepton + photon eventsUnusual combinations of lepton

charge

Preliminary analysis shows that around 90% of leptonsare hard enough to trigger on.


Limits from the Tevatron

Trileptonsearches1/R > 270GeV

Lepton +photonsearches1/R > 250GeV[Abulencia et al,

hep-ex/0702029]


Mass spectrum for (1,1) modes

600

650

700

750

800

850

900

950

1000

1050

mass

1R = 500 GeVGΜH1,1L

WΜH1,1L

BΜH1,1L

GHH1,1L

WHH1,1L

BHH1,1L

Q+3 H1,1L

Q+H1,1L T-

H1,1L

U-H1,1L

D-H1,1L

L+H1,1L

E-H1,1L

HH1,1L



Level-2 modes can be singly produced at colliders,decay to pairs of SM particles



Coupling to SM quarks generated at loop level

for vector modes: qγµT aqA(1,1)aµ

for spinless adjoints Λ−1AH qγµ∂µq



Coupling to SM quarks generated at loop level

for vector modes: qγµT aqA(1,1)aµ

for spinless adjoints Λ−1mqAH qq

Process AH → tt has ≈100% branching fractionCascades through spinless adjoints give rise to ttresonances

0.50 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5MR

10-1

100

101

102

103

104

105

sigma

0.50 1.0 1.5 2.0 2.5 3.0 3.5Mtt

10-1

100

101

102

103

104sigma

MV [TeV]

σ(p

p→

V(1

,1)

µ)

[fb]

M(tt) [TeV]

σ(p

p→

V(1

,1)

µ→

tt)

[fb]

LHC reach for 30 fb−1

G(1,1)µ

W (1,1)µ

B(1,1)µ

W(1,1)3H + B(1,1)

µ

B(1,1)H

G(1,1)µ + W (1,1)3

µ


Other interesting aspects

Pair production of level-2 gluons is large

Cross section from gluon-gluon initial state is modelindependent (results apply to e.g. topgluons, axigluons,techni-ρ)

Cross section from quark-quark initial state is large inthe presence of heavy quark

Mixing with heavy quark also provides mechanism tosuppress couplings to SM quarks

Produces di- di-jet resonances - backgrounds reducible


Pair production

Reach ≈ 350 GeV at the Tevatron, ≈ 1 TeV at the LHCfor 1 fb−1 of data


Summary

Emergence of spinless adjoints from gauge fields in 6Dgives rise to a wide variety of interestingphenomenology:

Relic abundance and elastic cross-section ofspinless photon B

(1,0)H helicity suppressed

Decay of level-1 modes gives long cascade decayswith large cross sections to multiple leptons andleptons plus photons

Decay of level-2 modes results in clusters of narrow,closely-spaced tt resonances


Summary

Accessibility at colliders is enhanced by smaller massdifferences between KK levels

Astrophysical constraints place upper limit oncompactification scale (caveat)Current Tevatron results for trilepton searches placelower limit on compactification scaleMore stringent constraints are expected fromTevatron Run II, including di-jet (and di- di-jet!),searches

We anxiously await the LHC to tell us more!


Backup slides


(1,0) top quark mixing

EWSB gives rise to mixing between 6D chirality ‘+’ and‘-’ modes

(

T(1,0)−

T(1,0)+

)

(

− 1R (1 + ∆

−) mt(1 + δ1)

mt(1 + δ2)1R (1 + ∆+)

)(

T(1,0)−

T(1,0)+

)

Eigenstates are degenerate in mass

MT =

√

1

R2+ m2

t (1 + ∆)


Zero mode

x5

x4

0

L

L


Spinless adjoint

x5

x4

0

L

L


Where do they come from?

88% 1st and 2nd generations

Q(1)Q(1)

gg

D(1)+U(1)+du G(1);HU(1)+U(1)+uu G(1);H other


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