PART I: Supersymmetry

M2R PSA 03 f´evrier 2010, dur´ee 3h00 Physics beyond the standard model

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PART I: Supersymmetry

Hints:

• η^µν = diag[1,−1,−1,−1] is the (flat) space-time metric.

• σ^µ= (I₂, ~σ) , ¯σ^ν = (I₂,−~σ) withI₂ = diag[1,1] and the Pauli matrices

σ¹ =

0 1

1 0

, σ² =

0 -i

i 0

, σ³ =

1 0

0 -1

• σⁱ =σ^i† (Hermitian)

• Trσⁱ = 0 (Traceless)

• {σⁱ, σ^j}= 2 δ^ij I₂ (Clifford algebra)

• [σⁱ, σ^j] = 2 i^ijkσ^k (Lie algebra)

• σⁱσ^j = ¹₂ {σⁱ, σ^j}+ ¹₂ [σⁱ, σ^j] =δ^ijI₂+i^ijkσ^k

Problem 1: The supersymmetric ground state

a) Show that Tr(σ^µσ¯^ν)≡σ^µ

α

.

βσ¯^ν

.

βα = 2η^µν.

b) Show that {Q_α,Q¯_β^.}σ¯^ν

.

βα = 4P^ν.

c) Show that the operator H :=P⁰ has real and non-negative eigenvalues E ≥0.

d) If |0iis the ground state (vacuum state) show that

h0|H|0i= 0 ⇔ Q_α|0i= ¯Q_α^. |0i= 0 (α,α^. = 1,2).

Conclusion: A ground state with positive energy breaks supersymmetry spontaneously:

[H, Q_α] = 0 (SUSY algebra) but Q_α |0i 6= 0.

Problem 2: Number of bosonic and fermionic degrees of freedom in SUSY multiplets

Recall the Casimir operators of the Poincar´e algebra, P² and W² where Wµ is the Pauli- Lubanski vector. Note that: [P², Q_α] = [P²,Q¯_α^.] = 0 but [W², Q_α]6= 0, [W²,Q¯_α^.]6= 0. Thus, irreducible (and therefore also reducible) representations of the supersymmetry algebra will contain states with different spins. Schematically, we can write

Q_α |Bi= |Fi , Q_α |Fi= |Bi , where |Bi is a bosonic and |Fi a fermionic state.

Definition: (−1)^NF is an operator defined such that

(−1)^NF |Bi= + |Bi , (−1)^NF |Fi=− |Fi . a) Show that Q_α(−1)^NF =−(−1)^NFQ_α.

b) Show that Tr[(−1)^NF] = 0 (for fixed non-zero P_µ) where the trace takes all states of the representation/multiplet into account. (Hint: Evaluate Tr[(−1)^NF{Q_α,Q¯_β^.}] direct- ly and by using the right side of the corresponding supersymmetry algebra relation.)

c) Conclude that every representation of the supersymmetry algebra contains an equal number of bosonic and fermionic states.