nLab gravitino

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theory (physics), model (physics)

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Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

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flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks (qq)
up-typeup quark (uu)charm quark (cc)top quark (tt)
down-typedown quark (dd)strange quark (ss)bottom quark (bb)
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chargedelectronmuontauon
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bound states:
mesonslight mesons:
pion (udu d)
ρ-meson (udu d)
ω-meson (udu d)
f1-meson
a1-meson
strange-mesons:
ϕ-meson (ss¯s \bar s),
kaon, K*-meson (usu s, dsd s)
eta-meson (uu+dd+ssu u + d d + s s)

charmed heavy mesons:
D-meson (uc u c, dcd c, scs c)
J/ψ-meson (cc¯c \bar c)
bottom heavy mesons:
B-meson (qbq b)
ϒ-meson (bb¯b \bar b)
baryonsnucleons:
proton (uud)(u u d)
neutron (udd)(u d d)

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

sfermions:

dark matter candidates

Exotica

auxiliary fields

Contents

Idea

In quantum field theory the term gravitino refers to the superpartner of the graviton, a Rarita-Schwinger field of spin 3/23/2 that appears in supergravity.

In supergravity a field history is a connection on super spacetime locally given by a super Lie algebra-valued differential form

(E,Ω,Ψ):TX𝔦𝔰𝔬( 1,10|32) (E, \Omega, \Psi) \,\colon\, T X \longrightarrow \mathfrak{iso}\big(\mathbb{R}^{1,10\vert \mathbf{32}}\big)

on spacetime with values in the super Poincaré Lie algebra. Its components Ψ\Psi in the spin representation 32𝔦𝔰𝔬( 1,10|32)\mathbf{32} \subset \mathfrak{iso}\big(\mathbb{R}^{1,10\vert \mathbf{32}}\big) is the gravitino field.

The name derives from the fact that the other two components are identified in gravity with the graviton field.

Examples

Gravitino in 11d Supergravity

The Rarita-Scwinger-like equation of motion for the gravitino in D=11 N=1 supergravity is (on any chart)

(1)Γ ab 1b 2ρ b 1b 2=0 \Gamma^{a \, b_1 b_2} \, \rho_{b_1 b_2} \;=\; 0

(due to Cremmer, Julia & Scherk 1978, p. 411, cf. Castellani, D’Auria & Fré 1991, §III.8, p. 910),

where

Proposition

(implications of 11d gravitino equation)

We have the following implications of the gravitino equation Γ ab 1b 2ρ b 1b 2=0\Gamma^{a b_1 b_2} \rho_{b_1 b_2} \;=\; 0 (1) in D=11 supergravity:

(2)Γ b 1b 2ρ b 1b 2=0 \Gamma^{b_1 b_2} \, \rho_{b_1 b_2} \;=\; 0
(3)Γ b 1ρ b 1b 2=0 \Gamma^{b_1} \, \rho_{b_1 b_2} \;=\; 0
(4)Γ aaρ ab=ρ a b \Gamma^{a a'} \, \rho_{a' b} \;=\; - \rho^a{}_b
(5)Γ c 1c 2 b 1b 2ρ b 1b 2=2ρ c 1c 2. \Gamma_{\!c_1 c_2}{}^{ b_1 b_2 } \rho_{b_1 b_2} \;=\; - 2\rho_{c_1 c_2} \,.

Proof

Equation (2) follows immediately by Clifford contraction:

Γ aΓ ab 1b 2ρ b 1b 2=9Γ b 1b 2ρ b 1b 2 \Gamma_{\!a} \Gamma^{a b_1 b_2} \,\rho_{b_1 b_2} \;=\; 9 \, \Gamma^{b_1 b_2} \,\rho_{b_1 b_2}

Equation (3) follows by the contraction

Γ caΓ ab 1b 2ρ b 1b 2=18Γ bρ cb+8Γ cb 1b 2ρ b 1b 2 \begin{array}{l} \Gamma_{\!c a} \Gamma^{a b_1 b_2} \, \rho_{b_1 b_2} \;=\; 18 \, \Gamma^{b} \rho_{ c b } \;+\; 8 \, \Gamma^{c b_1 b_2} \rho_{b_1 b_2} \end{array}

and using that the second summand vanishes by assumption (1).

For equation (4) we compute as follows:

Γ aaρ ab =12(Γ aΓ aρ abΓ aΓ aρ ab) =12Γ aΓ aρ ab =12Γ aΓ aρ abη aaρ ab =ρ a b, \begin{array}{l} \Gamma^{a a'}\rho_{a' b} \\ \;=\; \tfrac{1}{2} \big( \Gamma^a \Gamma^{a'} \rho_{a' b} - \Gamma^{a'} \Gamma^a \rho_{a' b} \big) \\ \;=\; - \tfrac{1}{2} \Gamma^{a'} \Gamma^a \rho_{a' b} \\ \;=\; \tfrac{1}{2} \Gamma^a \Gamma^{a'} \rho_{a' b} - \eta^{a a'} \rho_{a' b} \\ \;=\; -\rho^a{}_b \,, \end{array}

where in the second and fourth step we used (3).

For (5) we consider this contraction:

Γ c 1c 2aΓ ab 2b 3ρ b 2b 3 =16Γ c 1 bρ c 2b16Γ c 2 bρ c 1b18ρ c 1c 2+7Γ c 1c 2 b 1b 2ρ b 1b 2 =16ρ c 1c 216ρ c 2c 118ρ c 1c 2+7Γ c 1c 2 b 1b 2ρ b 1b 2 =14ρ c 1c 2+7Γ c 1c 2 b 1b 2ρ b 1b 2, \begin{array}{l} \Gamma_{c_1 c_2 a} \Gamma^{a b_2 b_3} \rho_{b_2 b_3} \\ \;=\; 16 \Gamma_{c_1}{}^{b} \rho_{c_2 b} - 16 \Gamma_{c_2}{}^{b} \rho_{c_1 b} - 18 \rho_{c_1 c_2} + 7 \Gamma_{c_1 c_2}{}^{ b_1 b_2 } \rho_{b_1 b_2} \\ \;=\; 16 \rho_{c_1 c_2} - 16 \rho_{c_2 c_1} - 18 \rho_{c_1 c_2} + 7 \Gamma_{c_1 c_2}{}^{ b_1 b_2 } \rho_{b_1 b_2} \\ \;=\; 14 \rho_{c_1 c_2} + 7 \Gamma_{c_1 c_2}{}^{ b_1 b_2 } \rho_{b_1 b_2} \,, \end{array}

where in the second step we used (4).

References

General

See also

Classification of long-range forces

Classification of possible long-range forces, hence of scattering processes of massless fields, by classification of suitably factorizing and decaying Poincaré-invariant S-matrices depending on particle spin, leading to uniqueness statements about Maxwell/photon-, Yang-Mills/gluon-, gravity/graviton- and supergravity/gravitino-interactions:

Review:

As a dark matter candidate

Discussion of the gravitino as a dark matter candidate:

A proposal for super-heavy gravitinos as dark matter, by embedding D=4 N=8 supergravity into E10-U-duality-invariant M-theory:

following the proposal towards the end of

  • Murray Gell-Mann, introductory talk at Shelter Island II, 1983 (pdf)

    in: Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics. MIT Press. pp. 301–343. ISBN 0-262-10031-2.

Further discussion:

Last revised on October 27, 2024 at 10:30:35. See the history of this page for a list of all contributions to it.