Supersymmetrizing Massive Gravity · Supersymmetrizing Massive Gravity O. Malaeb Department of...

Supersymmetrizing Massive Gravity

O. Malaeb

Department of PhysicsAmerican University of Beirut

Seventh Aegean Summer School, 2013

based on arXiv:1303.3580

O. Malaeb (American University of Beirut) Supersymmetrizing Massive Gravity 2013 1 / 18

Outline

MotivationForming matter actionCoupling to SupergravityMain resultsSummary

Motivation The Massive Gravity Model

MOTIVATION

Forming massive gravity via Higgs mechanism (by Chamseddineand Mukhanov)Four scalar fields with global Lorentz symmetry coupled to gravityScalar fields taking a vacuum expectation value breakingdiffeomorphism invariance spontaneouslyResult massive graviton

[A. Chamseddine and V. Mukhanov, Journal of High Energy Physics, 08, (2010), 11.]

−→ Generalize this to the supersymmetric case

MOTIVATION

OUR WORK Forming the Matter Action

FOUR CHIRAL SUPERFIELDS

Consider four N = 1 chiral superfields with global Lorentzsymmetry

D .αΦA (

x , θ, θ)

Given by

ΦA = ϕA + i(θσµθ)∂µϕA −14θθθθ∂µ∂

µϕA +√

2θψA

− i√2θθ

(∂µψAσ

+ θθFA

FOUR CHIRAL SUPERFIELDS

Consider four N = 1 chiral superfields with global Lorentzsymmetry

D .αΦA (

x , θ, θ)

Given by

ΦA = ϕA + i(θσµθ)∂µϕA −14θθθθ∂µ∂

µϕA +√

2θψA

− i√2θθ

(∂µψAσ

+ θθFA

BASIC FIELD

In bosonic case: HAB = gµν∂µφA∂νφB

Our case: Define HABC as the basic field

HABC = DαΦA(σB)ααDαΦ∗C = DΦAσBDΦ∗

Work instead with HABC = HABC − DxAσBDx∗C to avoid including

higher order terms

BASIC FIELD

higher order terms

BASIC FIELD

higher order terms

FORMING OUR ACTION

Using this HABC construct D-type terms to form the mattermultiplets (HABCHBCA,..)Couple to the Supergravity Lagrangian using the rules of tensorcalculus.

LS.G = − e2κ2 R(e,w)− e

3|u|2 +

AµAµ − 12φµRµ (2)

Lagrangian field content:spin-2 field, eaµspin-3/2 field, φµauxiliary fields u, Aµ

FORMING OUR ACTION

LS.G = − e2κ2 R(e,w)− e

3|u|2 +

FORMING OUR ACTION

LS.G = − e2κ2 R(e,w)− e

3|u|2 +

COUPLING VETOR MULTIPLET TOSUPERGRAVITY

Vector multipletV = (C, ξ,M,Vµ, λ,D) (3)

D-type multiplet action formula

e−1LD = D +iκ2φµγ

5γµλ− κ

3(uM∗ + u∗M) +

8εµνρσφµγνφρξφσ

+23κVµ

(Aµ +

ie−1εµρστ φργτφσ

)− i

3e−1ξγ5γµRµ

− 23κ2Ce−1LS.G. + e−1LS.G. (4)

[P. Nath and R. Arnowitt and A. Chamseddine: Applied N=1 Supergravity, World Scientific

Publishing, (1984)]

[E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Nuclear Physics B, 212,

(1983), 413]

COUPLING VETOR MULTIPLET TOSUPERGRAVITY

Vector multipletV = (C, ξ,M,Vµ, λ,D) (3)

D-type multiplet action formula

e−1LD = D +iκ2φµγ

5γµλ− κ

3(uM∗ + u∗M) +

8εµνρσφµγνφρξφσ

+23κVµ

(Aµ +

ie−1εµρστ φργτφσ

)− i

3e−1ξγ5γµRµ

− 23κ2Ce−1LS.G. + e−1LS.G. (4)

[P. Nath and R. Arnowitt and A. Chamseddine: Applied N=1 Supergravity, World Scientific

Publishing, (1984)]

[E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Nuclear Physics B, 212,

(1983), 413]

WHAT WE WANT?

Expanding the fields around the vacuum solution

ϕA = xA + χA, eµA = δµA + eµA, (5)

An action with the following required conditionsA Fierz-Pauli term for the vierbeins (eµ

A eAµ − e2)

No linear vierbein termMaxwell form for the χA fields

l(∂µχA∂

µχA∗ − ∂AχA∂Bχ

B∗) (6)

where l is a constantGhost free: no terms like

∂µχA∂µχA, or ∂Aχ

A∂BχB (7)

Massive gravitinos

WHAT WE WANT?

A eAµ − e2)

l(∂µχA∂

B∗) (6)

A∂BχB (7)

Massive gravitinos

WHAT WE WANT?

A eAµ − e2)

l(∂µχA∂

B∗) (6)

A∂BχB (7)

Massive gravitinos

WHAT WE WANT?

A eAµ − e2)

l(∂µχA∂

B∗) (6)

A∂BχB (7)

Massive gravitinos

WHAT WE WANT?

A eAµ − e2)

l(∂µχA∂

B∗) (6)

A∂BχB (7)

Massive gravitinos

WHAT WE WANT?

A eAµ − e2)

l(∂µχA∂

B∗) (6)

A∂BχB (7)

Massive gravitinos

WHAT WE WANT?

A eAµ − e2)

l(∂µχA∂

B∗) (6)

A∂BχB (7)

Massive gravitinos

MATTER ACTION

First four conditions well satisfied with three D-type terms:HABCHBCAHABBHCCAHABH∗

AB where HAB = DΦADΦB

Gravitino massive by adding two F-type terms and coupling themto supergravity:

D2(DΦAσ

ABDΦB)

D2(DΦADΦADΦ∗

BDΦB∗)

MATTER ACTION

First four conditions well satisfied with three D-type terms:HABCHBCAHABBHCCAHABH∗

AB where HAB = DΦADΦB

Gravitino massive by adding two F-type terms and coupling themto supergravity:

D2(DΦAσ

ABDΦB)

D2(DΦADΦADΦ∗

BDΦB∗)

COUPLING CHIRAL MULTIPLET TOSUPERGRAVITY

Chiral multiplet (F-type)

F = (z,XL,h) (8)

Complex scalar field zLeft-handed Weyl spinors XLComplex auxiliary field h

Action formula

e−1LF = h + κuz + κφµγµχ+ iκ2φµγ

µνφvRz + h.c. (9)

COUPLING CHIRAL MULTIPLET TOSUPERGRAVITY

Chiral multiplet (F-type)

F = (z,XL,h) (8)

Complex scalar field zLeft-handed Weyl spinors XLComplex auxiliary field h

Action formula

e−1LF = h + κuz + κφµγµχ+ iκ2φµγ

µνφvRz + h.c. (9)

Our Results Main Results

MAIN RESULTS

Global supersymmetry promoted to a local one using the rules oftensor calculusScalar components of the chiral multiplets ϕA acquired a vacuumexpectation value

→ Diffeomorphism invariance and local supersymmetry brokenspontaneously

MAIN RESULTS

Global Lorentz index A identified with the space-time Lorentzindex

Scalar fields χA → vectorsChiral spinors ψA → spin-3/2 Rarita-Schwinger fields

Spectrum of the model in the broken phase:A massive spin-2 fieldTwo massive spin-3/2 fields with different massesA massive vector

MAIN RESULTS

Global Lorentz index A identified with the space-time Lorentzindex

Scalar fields χA → vectorsChiral spinors ψA → spin-3/2 Rarita-Schwinger fields

Spectrum of the model in the broken phase:A massive spin-2 fieldTwo massive spin-3/2 fields with different massesA massive vector

DEGREES OF FREEDOM: BEFORE COUPLING

Counting degrees of freedom: supergravity coupled to a N = 1supersymmetry model similar to the Wess-Zumino model

Before the coupling, supergravity containsone massless spin-2 graviton (two bosonic dof)one massless spin-3/2 gravitino (two fermionic dof)

Counting degrees of freedom: supergravity coupled to a N = 1supersymmetry model similar to the Wess-Zumino model

Before the coupling, supergravity containsone massless spin-2 graviton (two bosonic dof)one massless spin-3/2 gravitino (two fermionic dof)

N = 1 supersymmetry model:Four spin-0 particles, ϕA, with only six degrees of freedom (3 times2) since ϕ0 decouples due to Fierz-Pauli choiceSix fermionic degrees of freedom forming a multiplet

Overall eight fermionic degrees of freedom and eight bosonicdegrees of freedom

DEGREES OF FREEDOM: AFTER COUPLING

After coupling to supergravity: N = 1 massive representationBosonic degrees of freedom

A massive spin-2 particle, with five degrees of freedomA massive vector field (spin-1 particle), three degrees of freedom

Overall eight bosonic degrees of freedom

Fermionic degrees of freedomTwo massive spin-3/2 particles, φµ and ψA with four degrees offreedom each

Overall eight fermionic degrees of freedomSame number of degrees of freedom as before coupling

Summary

SUMMARY

Constructed a supersymmetric theory of massive gravityCoupled a pure Supergravity multiplet to a matter multipletLeft with two massive spin-3/2 particles, φµ and ψA withcompletely different massesSimilar to the N = 2 supersymmetry in which we have twogravitinos, but there they have the same mass.

Summary

SUMMARY

Summary

SUMMARY

Summary

SUMMARY

Summary

SUMMARY

Supersymmetry completely broken exactly at the same scale asthe diffeomorphism breakingBefore diffeomorphism breaking:spin-1/2 and not spin-3/2→ two gravitinos with different massesφµ: genuine gravitinoψA: identified with a gravitino after the breaking

Thank You for Your Attention!

Summary

SUMMARY

Summary

SUMMARY

Summary

SUMMARY

Supersymmetrizing Massive Gravity · Supersymmetrizing Massive Gravity O. Malaeb Department of...

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