Spin-1/2 fermions in quantum field theory µ First, recall that 4-vectors transform under Lorentz transformations, Λ ν, as p′ µ = Λµ pν, where Λ SO(3,1) satisfies Λµ g Λρ = g .∗ A Lorentz ν ∈ ν µρ λ νλ transformation corresponds to a rotation by θ about an axis nˆ [θ~ θnˆ] and ≡ a boost, ζ~ = vˆ tanh−1 ~v , where ~v is the corresponding velocity. Under the | | same Lorentz transformation, a generic field transforms as: ′ ′ Φ (x )= MR(Λ)Φ(x) , where M exp iθ~·J~ iζ~·K~ are N N representation matrices of R ≡ − − × 1 1 the Lorentz group. Defining J~+ (J~ + iK~ ) and J~− (J~ iK~ ), ≡ 2 ≡ 2 − i j ijk k i j [J± , J±]= iǫ J± , [J± , J∓] = 0 . Thus, the representations of the Lorentz algebra are characterized by (j1,j2), 1 1 where the ji are half-integers. (0, 0) is a scalar and (2, 2) is a four-vector. Of 1 1 interest to us here are the spinor representations (2, 0) and (0, 2). ∗ In our conventions, gµν = diag(1 , −1 , −1 , −1). (1, 0): M = exp i θ~·~σ 1ζ~·~σ , butalso (M −1)T = iσ2M(iσ2)−1 2 −2 − 2 (0, 1): [M −1]† = exp i θ~·~σ + 1ζ~·~σ , butalso M ∗ = iσ2[M −1]†(iσ2)−1 2 −2 2 since (iσ2)~σ(iσ2)−1 = ~σ∗ = ~σT − − Transformation laws of 2-component fields ′ β ξα = Mα ξβ , ′ α −1 T α β ξ = [(M ) ] β ξ , ′† α˙ −1 † α˙ † β˙ ξ = [(M ) ] β˙ ξ , ˙ ξ′† =[M ∗] βξ† . α˙ α˙ β˙ 2 0 1 αβ α˙ β˙ 2 −1 2 We use iσ = = ǫ = ǫ and (iσ ) = iσ = ǫ = ǫ ˙ to −1 0 − αβ α˙ β α αβ † † β˙ raise and lower spinor indices: ξ = ǫ ξβ ; ξα˙ = ǫα˙ β˙ ξ , etc. Dotted and undotted indices are related by hermitian conjugation: ξ† (ξ )†. α˙ ≡ α Finally, we introduce the σ-matrices: σµ =(I ; ~σ) , σµ αβ˙ =(I ; ~σ) , αβ˙ 2 2 − where I is the 2 2 identity matrix. The spinor index structure derives from 2 × the relations: † µ µ ν −1 µ −1 † µ ν M σ M =Λ νσ , M σ (M ) =Λ νσ . † α˙ µβγ˙ δ µ ν αδ˙ For example, (M ) β˙ σ Mγ =Λ νσ . Note that the matrix M and its inverse have the same spinor index structure. Some useful identities: ˙ ˙ σµ = ǫ ǫ σµ ββ , σµ αα˙ = ǫαβǫα˙ βσµ . αα˙ αβ α˙ β˙ ββ˙ The utility of σµ is that Lorentz 4-vectors can be built from spinor bilinears: ˙ ˙ χ ′ α(x′)σµ ξ′† β(x′)= χα(x)[M −1σµ(M −1)†] ξ† β(x) αβ˙ αβ˙ µ α ν † β˙ =Λ ν χ(x) σαβ˙ξ (x) . Spinor indices can be suppressed as long as one adopts a summation convention where we contract indices as follows: α α˙ α and α˙ . α For example, ξη ≡ ξ ηα, † † † † α˙ ξ η ≡ ξα˙ η , † µ † µαβ˙ ξ σ η ≡ ξα˙ σ ηβ, ˙ ξσµη† ≡ ξασµ η† β. αβ˙ In particular, for anticommuting spinors, ηξ ηαξ = ξ ηα =+ξαη = ξη. ≡ α − α α Note the behavior of spinor products under hermitian conjugation: † † † † † † † † † (ξΣη) = η Σrξ , (ξΣη ) = ηΣrξ , (ξ Ση) = η Σrξ, where in each case Σ stands for any sequence of alternating σ and σ matrices, and Σr is obtained from Σ by reversing the order of all of the σ and σ matrices. From the sigma matrices, one can construct the antisymmetrized products: i (σµν) β σµ σνγβ˙ σν σµγβ˙ , α ≡ 4 αγ˙ − αγ˙ µν α˙ i µαγ˙ ν ναγ˙ µ (σ ) ˙ σ σ ˙ σ σ ˙ . β ≡ 4 γβ − γβ 1 1 We may write the (2, 0) and (0, 2) transformation matrices, respectively, as: M = exp i θµνσ , −2 µν (M −1)† = exp i θµνσ , −2 µν where θµν is antisymmetric, with θij = ǫijkθk and θi0 = ζi. Consider a pure boost of an on-shell spinor from its rest frame to the frame where µ 2 2 1/2 ij p =(Ep , ~p), with Ep =( ~p + m ) . Setting θ = 0, | | p·σ (E + m)I − ~σ·~p 1~· p 2 M = exp −2ζ ~σ = = , m 2m(Ep + m) r p·σ (Ep+ m)I + ~σ·~p −1 † 1~· p 2 (M ) = exp +2ζ ~σ = = . m 2m(Ep + m) r p Useful identities and Fierz relations γδ γ δ δ γ γ˙ δ˙ γ˙ δ˙ δ˙ γ˙ ǫ ǫ = −δ δ + δ δ , ǫ ˙ ǫ = −δ δ + δ δ , αβ α β α β α˙ β α˙ β˙ α˙ β˙ µ ββ˙ β β˙ σαα˙ σµ = 2δαδα˙ , µ µαα˙ ββ˙ αβ α˙ β˙ σαα˙ σµββ˙ = 2ǫαβǫα˙ β˙ , σ σµ = 2ǫ ǫ , µ ν ν µ β µν β [σ σ + σ σ ]α = 2g δα , µ ν ν µ α˙ µν α˙ [σ σ + σ σ ] β˙ = 2g δβ˙ , µ ν ρ µν ρ µρ ν νρ µ µνρκ σ σ σ = g σ − g σ + g σ + iǫ σκ , µ ν ρ µν ρ µρ ν νρ µ µνρκ σ σ σ = g σ − g σ + g σ − iǫ σκ , 0123 where ǫ = −ǫ0123 = +1 in our conventions. Computations of cross sections and decay rates often require traces of alternating products of σ and σ matrices: Tr[σµσν] = Tr[σµσν] = 2gµν , Tr[σµσνσρσκ]=2(gµνgρκ − gµρgνκ + gµκgνρ + iǫµνρκ) , Tr[σµσνσρσκ]=2(gµνgρκ − gµρgνκ + gµκgνρ − iǫµνρκ) . Traces involving an odd number of σ and σ matrices cannot arise, since there is no way to connect the spinor indices consistently. We shall deal with both commuting and anticommuting spinors, which we shall denote generically by zi. Then, the following identities hold A z1z2 = −(−1) z2z1 † † A † † z1z2 = −(−1) z2z1 µ † A † µ z1σ z2 = (−1) z2σ z1 µ ν A ν µ z1σ σ z2 = −(−1) z2σ σ z1 † µ ν † A † ν µ † z1σ σ z2 = −(−1) z2σ σ z1 † µ ρ ν A ν ρ µ † z1σ σ σ z2 = (−1) z2σ σ σ z1 , where (−1)A = +1[−1] for commuting [anticommuting] spinors. Finally, the Fierz identities are given by: (z1z2)(z3z4)= −(z1z3)(z4z2) − (z1z4)(z2z3) , † † † † † † † † † † † † (z1z2)(z3z4)= −(z1z3)(z4z2) − (z1z4)(z2z3) , µ † † † † (z1σ z2)(z3σµz4)= −2(z1z4)(z2z3) , † µ † † † (z1σ z2)(z3σµz4) = 2(z1z3)(z4z2) , µ † † † † (z1σ z2)(z3σµz4) = 2(z1z3)(z4z2) . Free field theories involving fermions 1 The (2, 0) spinor field ξα(x) describes a neutral Majorana fermion. The free-field Lagrangian is: L = iξ†σµ∂ ξ 1m(ξξ + ξ†ξ†) , µ − 2 which is hermitian up to a total divergence since we can rewrite the above Lagrangian as ↔ 1 † µ 1 † † L = iξ σ ∂µ ξ m(ξξ + ξ ξ )+ total divergence , 2 − 2 ↔ † µ † µ † µ where ξ σ ∂µ ξ ξ σ (∂µξ) (∂µξ) σ ξ. ≡ − Generalizing to a multiplet of two-component fermion fields, ξαi(x), labeled by flavor index i. L = iξˆ† iσµ∂ ξˆ 1M ijξˆ ξˆ 1M ξˆ† iξˆ† j , µ i − 2 i j − 2 ij where hermiticity implies that M (M ij)∗ is a complex symmetric matrix. ij ≡ To identify the physical fermion fields, we express the so-called interaction −1 eigenstate fields, ξˆαi(x), in terms of mass-eigenstate fields ξ(x) = Ω ξˆ(x), where Ω is unitary and chosen such that T Ω M Ω= m = diag(m1, m2,...), where the mi are non-negative real numbers. In linear algebra, this is called the Takagi-diagonalization of a complex symmetric matrix M. To compute the values of the diagonal elements of m, one may simply note that ΩTMM †Ω∗ = m2. MM † is hermitian, and thus it can be diagonalized by a unitary matrix. Thus, the mi of the Takagi diagonalization are the non-negative square-roots of the eigenvalues of MM †. In terms of the mass eigenstate fields, L = iξ† iσµ∂ ξ 1m (ξ ξ + ξ† iξ† i) . µ i − 2 i i i Example: the see-saw mechanism The see-saw Lagrangian is given by: = i ψ1 σµ∂ ψ + ψ2 σµ∂ ψ M ijψ ψ M ψi ψj , L µ 1 µ 2 − i j − ij where 0 m M ij = D , mD M ! and (without loss of generality) mD and M are positive. The Takagi T diagonalization of this matrix is Ω MΩ= MD, where i cos θ sin θ m− 0 Ω= , MD = , i sin θ cos θ ! 0 m+ ! − with m = 1 M 2 + 4m2 M and sin2θ = 2m / M 2 + 4m2 . ± 2 D ± D D hp i p If M m , then the corresponding fermion masses are m m2 /M and ≫ D − ≃ D m M, while sin θ m /M. The mass eigenstates, χ are given by + ≃ ≃ D i j ψi = Ui χj; i.e. to leading order in md/M, m m iχ ψ D ψ , χ ψ + D ψ . 1 ≃ 1 − M 2 2 ≃ 2 M 1 Indeed, one can check that: m2 1m (ψ ψ + ψ ψ )+ 1Mψ ψ +h.c. 1 D χ χ + Mχ χ +h.c. , 2 D 1 2 2 1 2 2 2 ≃ 2 M 1 1 2 2 which corresponds to a theory of two Majorana fermions—one very light and one very heavy (the see-saw). In any theory containing a multiplet of fields, one can check for the existence 1 of global symmetries. The simplest case is a theory of two-component (2, 0) fermion fields χ and η, with the free-field Lagrangian, L = iχ†σµ∂ χ + iη†σµ∂ η m(χη + χ†η†) . µ µ − This Lagrangian possesses a U(1) global symmetry, χ eiθχ and η e−iθη. → → That is, χ and η are oppositely charged. The corresponding mass matrix 0 m is ( m 0 ).