PART I: Supersymmetry

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M2R PSA 10 f´evrier 2011, dur´ee 3h00 Physics beyond the standard model

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PART I: Supersymmetry

Formulae:

Space-time metric:

• η^µν = diag[1,−1,−1,−1]

In the Weyl representation:

γ^µ =

0 (σ^µ)_α_α^. (¯σ^µ)^αα^. 0

, γ⁵ = δ^α_β 0 0 δ^α^.^.

β

! , 1

2Σ^µν = i(σ^µν)_α^β 0 0 i(¯σ^µν)^α^.^.

β

!

with

(σ^µ)_α_α^. = (I₂, ~σ), (¯σ^µ)^αα^. = (I₂,−~σ) where ~σ = (σ¹, σ², σ³) are the Pauil matrices. Furthermore,

(σ^µν)_α^β = 1

4(σ^µσ¯^ν−σ^νσ¯^µ), (¯σ^µν)^α^.^.

β = 1

4(¯σ^µσ^ν −σ¯^νσ^µ).

Some spinor relations:

• χψ :=χ^αψ_α =ψχ, ¯χψ¯:= ¯χ_α^.ψ¯^α^. = ¯ψχ, (χψ)¯ ^† =ψ^†χ^†= ¯ψχ¯

• χσ^µψ¯:=χ^ασ^µ_α_α^.ψ¯^α^., ¯χ¯σ^µψ := ¯χ_α^.σ¯^µ^αα^. ψ_α

• (χσ^µψ)¯ ^†=ψσ^µχ¯

• χσ^µψ¯=−ψ¯σ¯^µχ

• χσ^µνψ =−ψσ^µνχ, ¯χ¯σ^µνψ¯=−ψ¯σ¯^µνχ¯

• (χσ^µνψ)^† = ¯χ¯σ^µνψ¯

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Problem 1: Variations on the SUSY algebra

a) Write down the (N = 1) SUSY algebra in terms of the 10 bosonic generators of the Poincar´e group (P^µ, J^µν) and the 4 fermionic generators (Q_α, and ¯Q_α^.). Here Q_α (α= 1,2), and ¯Q_α^. (α^. =1,^. 2) are 2-component Weyl spinors transforming according to^. the (1/2,0) and (0,1/2)-representations of the Lorentz group, respectively.

b) In 4-component notation we define the Majorana spinor Q_M,a (a= 1,2,1,^. 2):^. QM =

Q_α Q¯^α^.

.

Show that the Dirac-adjoint spinor is given by ¯Q_M = (Q^β,Q¯_β^.).

c) Show that

[P^µ, QM] = 0, [J^µν, Q_M] = −1

2 Σ^µν Q_M, {QM,Q¯M} = 2γ^µPµ

with γ^µ and Σ^µν := ₄ⁱ[γ^µ, γ^ν] in the Weyl representation.

Hint: Consider {Q_M,a,Q¯_M,b} for fixed a, b= 1,2,1,^. 2.^.

d) The SUSY algebra can be expressed entirely in terms of commutators with the help of anti-commuting parameters ξ^α, ¯ξ_α^. and η^α, ¯η_α^.:

{ξ^α, ξ^β}={ξ^α, Q_β}=. . .= [P_µ, ξ^α] = 0. Evaluate the following commutators using the algebra in a):

(i) [P_µ, ξQ] = [P_µ,ξ¯Q]¯ (ii) [J^µν, ξQ], [J^µν,ξ¯Q]¯ (iii) [ξQ, ηQ] = [ ¯ξQ,¯ η¯Q]¯ (iv) [ξQ,η¯Q]¯

where we use the summation convention ξQ :=ξ^αQ_α, ¯ξQ¯ := ¯ξ_α^.Q¯^α^..

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Problem 2: A supersymmetric Lagrangian

a) Write down the Wess-Zumino Lagrangian for massless (m = 0) and non-interacting (g = 0) fields A and ψ

b) In this case (m= 0, g = 0) the SUSY transformations are given by δ_ξA = √

2 ξψ , δ_ξψ = −i√

2 σ^µξ ∂¯ _µA

Here, we use again the summation convention and free indices αhave been suppressed.

Write down the transformation rules for the conjugate fields which can be obtained by hermitian conjugation:

(i) δ_ξA^∗ = (δ_ξA)^†=. . . (ii) δ_ξψ¯= (δ_ξψ)^†=. . . c) Show that

√1 2δ_ξ

(∂_µA^∗)(∂^µA)

=−ξ¯ψ¯A−ξψ A^∗+∂_µK₁^µ with :=∂_µ∂^µ. Specify K₁^µ.

d) Show that

√1 2δ_ξ

iψ¯σ¯^µ∂_µψ

= ¯ξψ¯A+ξψ A^∗+∂_µK₂^µ Specify K₂^µ.

Hint: σ^µσ¯^ν =η^µν+ 2σ^µν , ¯σ^µσ^ν =η^µν + 2¯σ^µν and σ^µν =−σ^νµ, ¯σ^µν =−¯σ^νµ. e) Conclude that the action is invariant under this symmetry: δ_ξS = 0.