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 ). Performing the Takagi-diagonalization yields two degenerate two- component fermions of mass m. However, the corresponding mass-eigenstates are not eigenstates of charge.
This is the analog of a free field theory of a complex scalar boson Φ with a mass term m2 Φ 2. Writing Φ=(φ + iφ )/√2, we can write Lagrangian | | 1 2 in terms of φ1 and φ2 with a diagonal mass term. But, φ1 and φ2 do not correspond to states of definite charge.
Together, χ and η† constitute a single (four-component) Dirac fermion. More generally, consider a collection charged Dirac fermions represented by i pairs of two-component interaction eigenstate fields χˆαi(x), ηˆα(x), with
L = iχˆ†iσµ∂ χˆ + iηˆ†σµ∂ ηˆi M i χˆ ηˆj M jχˆ†iηˆ† , µ i i µ − j i − i j where M is a complex matrix with matrix elements M i , and M j (M i )∗. j i ≡ j i Introduce the mass eigenstate fields χi and η and the unitary matrices L k i i k and R, such that χˆi = Li χk and ηˆ = R kη and
T L MR = m = diag(m1, m2,...), where the mi are non-negative real numbers. This is the singular value decomposition of a complex matrix. Noting that R†(M †M)R = m2 , the diagonal elements of m are the non-negative square roots of the corresponding eigenvalues of M †M. In terms of the mass eigenstate fields,
L = iχ†iσµ∂ χ + iη†σµ∂ ηi m (χ ηi + χ†iη†) . µ i i µ − i i i Fermion–scalar interactions
The most general set of interactions with the scalars of the theory φˆI are then given by:
L 1 ˆ Ijk ˆ ˆ ˆ 1 ˆ ˆI ˆ† j ˆ† k int = −2Y φIψjψk − 2YIjkφ ψ ψ ,
Ijk ∗ I ∗ where YˆIjk = (Yˆ ) and φˆ = (φˆI) . The flavor index I runs over a collection of real j ∗ scalar fields ϕˆi and pairs of complex scalar fields Φˆ j and Φˆ ≡ (Φˆ j) [where a complex field and its conjugate are counted separately]. The Yukawa couplings Yˆ Ijk are symmetric under interchange of j and k.
The mass-eigenstate basis ψ is related to the interaction-eigenstate basis ψˆ by a unitary transformations: ξˆ Ω 0 0 ξ ψˆ ≡ χˆ = Uψ ≡ 0 L 0 χ , ηˆ 0 0 R η where Ω, L, and R are constructed as described previously. Likewise a unitary transformation yields the scalar mass-eigenstates via φˆ = Vφ. Thus, in terms of mass-eigenstate fields:
L 1 Ijk 1 I † j † k int = −2Y φIψjψk − 2YIjkφ ψ ψ ,
Ijk I j k Jmn where Y = VJ Um Un Yˆ . Fermion–gauge boson interactions
In the gauge-interaction basis for the two-component fermions the corresponding interaction Lagrangian is given by L µ ˆ† i a j ˆ int = −gaAaψ σµ(T )i ψj ,
µ where the index a labels the (real or complex) vector bosons Aa and is summed over. If the gauge symmetry is unbroken, then the index a runs over the adjoint representation of a j † the gauge group, and the (T )i are hermitian representation matrices of the gauge group acting on the fermions. There is a separate coupling ga for each simple group or U(1) factor of the gauge group G.
In the case of spontaneously broken gauge theories, one must diagonalize the vector boson a squared mass matrix. The above form still applies where Aµ are gauge boson fields of a definite mass, although in this case for a fixed value of a, gaT is some linear combination a a of the original gaT of the unbroken theory. Henceforth, we assume that that the Aµ are the gauge boson mass-eigenstate fields.
†For a U(1) gauge group, the T a are replaced by real numbers corresponding to the U(1) charges of the 1 (2, 0) fermions. In terms of mass-eigenstate fermion fields,
L µ † i a j int = −Aaψ σµ(G )i ψj ,
a † a where G = gaU T U (no sum over a).
The case of gauge interactions of charged Dirac fermions can be treated as follows. Consider 1 i pairs of (2, 0) interaction-eigenstate fermions χˆi and ηˆ that transform as conjugate representations of the gauge group (hence the difference in the flavor index heights). The Lagrangian for the gauge interactions of Dirac fermions can be written in the form:
L µ † i a j µ † a i j int = −gaAaχˆ σµ(T )i χˆj + gaAaηˆi σµ(T )j ηˆ ,
a where the Aµ are gauge boson mass-eigenstate fields. Here we have used the fact a j i that if (T )i are the representation matrices for the χˆi, then the ηˆ transform in the complex conjugate representation with generator matrices −(T a)∗ = −(T a)T . In terms of mass-eigenstate fermion fields,
L µ † i a j † a i j int = −Aa χ σµ(GL)i χj − η i σµ(GR)j η , h i a † a a † a where GL = gaL T L and GR = gaR T R (no sum over a). Four-component spinor notation
The correspondence between the two-component and four-component spinor language is most easily exhibited in the basis in which γ5 is diagonal (this is called the chiral representation). In 2×2 blocks, the gamma matrices are given by:
0 σµ β µ αβ˙ 0 1 2 3 −δα 0 γ = , γ5 ≡ iγ γ γ γ = . µαβ˙ α˙ σ 0 0 δ β˙ ! 1 1 The chiral projections operators are: PL ≡ 2(1 − γ5) and PR ≡ 2(1 + γ5).
1 1 In addition, we identify the generators of the Lorentz group in the (2, 0) ⊕ (0, 2) representation:‡ µν β i σ α 0 1Σµν ≡ [γµ,γν] = , 2 µνα˙ 4 0 σ β˙ !
µν µν 1 µνρτ where Σ satisfies the duality relation, γ5Σ = 2iǫ Σρτ . ‡In most textbooks, Σµν is called σµν. Here, we use the former symbol so that there is no confusion with the two-component definition of σµν. A four component Dirac spinor field, Ψ(x), is made up of two mass-degenerate two-component spinor fields, χα(x) and ηα(x) as follows:
χα(x) Ψ(x) ≡ . η† α˙ (x) Note that PL and PR project out the upper and lower components, respectively. The Dirac conjugate field Ψ and the charge conjugate field Ψc are defined by
† α † Ψ(x) ≡ Ψ A = (η (x),χα˙ ) ,
T ηα(x) Ψc(x) ≡ CΨ (x) = , χ† α˙ (x) where the Dirac conjugation matrix A and the charge conjugation matrix C satisfy
AγµA−1 = 㵆 , C−1γµC = −γµT .
† It is conventional to impose two additional conditions: (i) Ψ = A−1Ψ [which guarantees that ΨΨ is hermitian] and (ii) (Ψc)c = Ψ. It follows that
A† = A , CT = −C, (AC)−1 = (AC)∗ . In the chiral representation, A and C are explicitly given by
α˙ 0 δ ˙ ǫαβ 0 A = β , C = . β α˙ β˙ δα 0 ! 0 ǫ !
Note the numerical equalities, A = γ0 and C = iγ0γ2, although these identifications do not respect the structure of the undotted and dotted indices specified above.
One can relate bilinear covariants in two-component and four-component notation. † † Ψ1Ψ2 = η1ξ2 + ξ1η2
† † Ψ1γ5Ψ2 = −η1ξ2 + ξ1η2
µ µ † µ Ψ1γ Ψ2 = ξ1σ ξ2 − η2σ η1
µ † µ † µ Ψ1γ γ5Ψ2 = −ξ1σ ξ2 − η2σ η1
µν µν † µν † Ψ1Σ Ψ2 = 2(η1σ ξ2 + ξ1σ η2)
µν µν † µν † Ψ1Σ γ5Ψ2 = −2(η1σ ξ2 − ξ1σ η2) . Relating bilinear covariants in two-component and four-component notation
ξ1(x) ξ2(x) Ψ (x) ≡ , Ψ (x) ≡ . 1 † 2 † η1(x) η2(x)
c c Ψ1PLΨ2 = η1ξ2 Ψ1PLΨ2 = ξ1η2
† † c c † † Ψ1PRΨ2 = ξ1η2 Ψ1PRΨ2 = η1ξ2
c c Ψ1PLΨ2 = ξ1ξ2 Ψ1PLΨ2 = η1η2
c † † c † † Ψ1PRΨ2 = ξ1ξ2 Ψ1PRΨ2 = η1η2
µ † µ c µ c † µ Ψ1γ PLΨ2 = ξ1σ ξ2 Ψ1γ PLΨ2 = η1σ η2
c µ c µ † µ µ † Ψ1γ PRΨ2 = ξ1σ ξ2 Ψ1γ PRΨ2 = η1σ η2
µν µν c µν c µν Ψ1Σ PLΨ2 = 2 η1σ ξ2 Ψ1Σ PLΨ2 = 2 ξ1σ η2
µν † µν † c µν c † µν † Ψ1Σ PRΨ2 = 2 ξ1σ η2 Ψ1Σ PRΨ2 = 2 η1σ ξ2
µν i µ ν µ † µ Σ ≡ 2[γ ,γ ]. Note that we may also write: Ψ1γ PRΨ2 = −η2σ η1, etc. c T For Majorana fermions defined by ΨM = ΨM = CΨM , the following additional conditions are satisfied:
ΨM1ΨM2 = ΨM2ΨM1 ,
ΨM1γ5ΨM2 = ΨM2γ5ΨM1 ,
µ µ ΨM1γ ΨM2 = −ΨM2γ ΨM1 ,
µ µ ΨM1γ γ5ΨM2 = ΨM2γ γ5ΨM1 ,
µν µν ΨM1Σ ΨM2 = −ΨM2Σ ΨM1 ,
µν µν ΨM1Σ γ5ΨM2 = −ΨM2Σ γ5ΨM1 .
In particular, if ΨM1 = ΨM2 ≡ ΨM , then
µ µν µν ΨM γ ΨM = ΨM Σ ΨM = ΨM Σ γ5ΨM = 0 .
One additional useful result is:
µ µ ΨM1γ PLΨM2 = −ΨM2γ PRΨM1 .