1 4-dimensional Weyl spinor

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1 4-dimensional Weyl spinor Left moving ψ α, α =, Right moving ψ α, α =, They are related by the complex conjugation. The indices are raised or lowerd by the ϵ tensor as ψ α := ϵ αβ ψ β, α := ϵ α β β. (.) Then ψ α = ϵ αβ ψ β, α = ϵ α β ψ β, (.) where and similar for one with dotted indices. Inner product ϵ αβ = ϵ βα, ϵ αβ = ϵ βα, ϵ = +, ϵ =, (.3) ψ χ := ψ α χ α, χ := α χ α. (.4) Sigma matrices σ µ α β σ 0 = 0, σ = 0, σ = 0 i, σ 3 = 0. (.5) 0 0 i 0 0 We also define σ µ αα := ϵ α β ϵ αβ σ µ β β, or σ 0 = σ 0, σ i = σ i, (i =,, 3). Then they satisfy σ µ σ ν + σ ν σ µ = η µν, (.6) σ µ σ ν + σ ν σ µ = η µν. (.7) Lorentz generators σ µν = 4 (σ µ σ ν σ ν σ ν ), σ µν = 4 ( σ µ σ ν σ ν σ ν ) (.8) Spinor bilinear ψσ µ χ := ψ α σ µ α β χ β, σ µ χ := α σ µ αβ χ β. (.9) Hermitian conjugations (ψ χ ) = ψ χ, ( χ ) = ψ χ, (ψσ µ χ ) = χσ µ ψ = ψ σ µ χ, ( σ µ χ ) = χ σ µ ψ = ψσ µ χ, (.0)

2 Some useful formulae ψ χ = χψ, χ = χ ψ, ψσ µ χ = χ σ µ ψ, ψσ µν χ = χσ µν ψ, σ µν χ = χ σ µν ψ, (.) θ α θ β = ϵαβ θθ, θ α θ β = ϵ αβθθ, θ α θ β = ϵ α β θ θ, θ α θ β = ϵ α β θ θ, (.) (θ χ )(θψ ) = (θθ )(χψ ), ( θ χ )( θ ψ ) = ( θ θ )( χ ψ ), θσ µ θθσ ν θ = θθ θ θη µν, (.3) χσ µν θψσ ρσ θ = θθ χσ µν σ ρσ ψ, ϵ αβ (σ µν θ ) α (σ µν θ ) β = 4 θθ (ηµρ η νσ η µσ η νρ + iϵ µνρσ ), (.4) σ µ σ ν = η µν + σ µν, σ µ σ ν = η µν + σ µν, Tr[σ µ σ ν ] = η µν, (.5) Tr[σ µν σ ρσ ] = (ηµρ η νσ η µσ η νρ ) i ϵ µνρσ, (ϵ 03 = +). (.6) Superspace Superspace coordinates (x µ, θ α, θ α ). A superfiled F (x, θ, θ ) Differential operators Anti-commutation relations Q α := θ α iσ µ α α θ α µ, D α := θ α + iσ µ α α θ α µ, Q α := θ α + iθα σ µ α α µ, (.) D α := θ α iθα σ µ α α µ. (.) {Q α, Q α } = iσ µ α α µ, {D α, D α } = iσ µ α α µ, (.3) {Q α, Q β } = {D α, D β } = {Q α, Q β } = {D α, D β } = 0, (.4) SUSY transformation with parameters ξ α, ξ α {Q α, D β } = {Q α, D β } = {Q α, D β } = {Q α, D β } = 0. (.5) δ ξ F (x, θ, θ ) := (ξ Q + ξ Q)F (x, θ, θ ). (.6)

3 3 Chiral superfield A chiral superfield Φ(x, θ, θ ) satisfies D α Φ = 0. (3.) Let us introduce y µ := x µ + iθσ µ θ. By the superspace coordinates (y, θ, θ ), the differential operators are written as Q α = θ α, Q α = θ α + iθα σ µ α α y µ = D α + iθ α σ µ α α y µ, (3.) It is expanded as D α = θ α + iσ µ α α α θ y µ, D α =. (3.3) θ α Φ(x, θ, θ ) = ϕ(y) + θψ (y) + θθf (y) (3.4) = ϕ(x) + θψ (x) + θθf (x) + iθσ µ θ µ ϕ(x) + i θθ θ σ µ µ ψ (x) + 4 θθ θ θ ϕ(x). (3.5) The complex conjugate of a chiral superfield satisfies D α Φ = 0 and it is expanded as Φ(x, θ, θ ) = ϕ(x) + θ ψ (x) + θ θf (x) iθσ µ θ µ ϕ(x) + i θ θθσ µ µ ψ (x) + 4 θθ θ θ ϕ(x). (3.6) SUSY transformation δ ξ ϕ = ξψ, δ ξ ψ α = ξ α F + i(σ µ ξ ) α µ ϕ, δ ξ F = i ξ σ µ µ ψ, δ ξ ϕ = ξ ψ, (3.7) δ ξ ψ α = ξ α F + i( σ µ ξ ) α µ ϕ, (3.8) δ ξ F = iξ σ µ µ ψ. (3.9) 3. Super potential term Let us consider n chiral superfields Φ i, i =,..., n and their complex conjugates Φī, ī =,..., n. A possible SUSY invariant term is a super potential term. L W = W (Φ) θθ + (c.c.), (3.0) where W (ϕ) is a holomorphic function of ϕ,..., ϕ n, called "super potential." This super potential term is expanded as L W = F i i W i j Wψ i ψ j + (c.c.). (3.) 3

4 3. Kähler potential term Another possible SUSY invariant term is the Kähler potential term. L K = K (Φ, Φ) θθ θ θ, (3.) where K (ϕ, ϕ) is a real function, called "Kähler potential." It is expanded as L K =д i j ( µ ϕ i µ ϕ j i ψ i σ µ µ ψ j i ψ j σ µ µ ψ i + F i F j) i K ij k ψ k σ µ ψ i µ ϕ j i K ī jkψ k σ µ ψī µ ϕ j where K i...i p j... j q := K ij k ψ i ψ j F k K ī jk ψīψ j F k + 4 K ij k l ψ i ψ j ψ k ψ l, (3.3) ϕ i... ϕ i p... ϕ j ϕ j K (ϕ, ϕ), д q i j := K i j. (3.4) 4 Vector superfield A super field V (x, θ, θ ) is called vector super field if it is real V = V. A vector super field is used to describe a gauge theory. For a U () gauge theory the gauge parameter is a chiral super field Λ(x, θ, θ ), D α Λ = 0, and transformation law is V V = V + Λ + Λ. (4.) The following field strength is gauge invariant. W α := 4 DDD αv. (4.) This W α is a chiral super field, i.e. D α W α = 0. It is also checked that it satisfied the reality condition DW = DW. It is convenient to choose Wess-Zumino gauge", in which V is expanded as V (x, θ, θ ) = θσ µ θv µ (x) + iθθ θ λ(x) i θ θθλ(x) + θθ θ θd(x). (4.3) It is convenient to rewrite it as V = θσ µ θv µ (y) + iθθ θ λ(y) i θ θθλ(y) + θθ θ θ (D(y) i µ v µ (y)). (4.4) W α is calculated as W α = iλ α (y) + θ α D(y) i(σ µν θ ) α v µν (y) + θθ (σ µ µ λ(y)) α, (4.5) 4

5 and the bilinear of W α 4 WW = 4 λ(y)λ(y) i θλ(y)d(y) λσ µν θv µν (y) + θθ ( i λσ µ µ λ + 4 D 8 v µνv µν i8 ) v µνṽ µν, where ṽ µν := ϵ µνρσ v ρσ. A gauge invariant Lagrangian can be written, with a complex constant τ 0 = τ 0 + iτ 0 (4.6) L V = i 8π τ 0WW θθ + (c.c.) (4.7) = τ 0 π ( 4 v µνv µν i λσ µ µ λ i λ σ µ λ + D ) τ 0 8π v µνṽ µν. (4.8) 5 N = Lagrangian An N = vector multiplet contains an N = vector multiplet and an N = chiral multiplet. W α = iλ α (y) + θ α D(y) i(σ µν θ ) α v µν (y) + θθ (σ µ µ λ(y)) α, (5.) A = a(y) + θψ (y) + θθf (y). (5.) Since we do not have potential term for a, we do not have superpotential for A. Thus the generic Lagrangian which preserves N = SUSY can be written as L = K (A, A) θθ θ θ + i 8π τ (A)WW θθ + (c.c.), (5.3) where τ (a) = τ (a) + iτ (a) is a holomorphic function of a. Let us see the kinetic terms of the fermions L = д aā ( i ψσ µ µ i ψ ) σ µ µ ψ + τ π ( i λσ µ µ λ i λ ) σ µ λ In order to have N = SUSY, the kinetic term for ψ and λ must be the same. Thus In other words, there is a holomorphic function F (a) which satisfies +. (5.4) д aā = τ π = (τ (a) τ (a)). (5.5) 4πi F (a) = τ (a), K (a, ā) = 4πi (ā F (a) a F (a)). (5.6) If we obtain this holomorphic function F (a), we can completely determine the action. 5