THE DIRAC FIELD

X

λ1,λ2

dσ

dcosθ_L. (2.148)

This can be evaluated using (2.121) for both theλ₁ andλ₂sums,¹⁵ with k₁^µ=k1(1,0,0,1), k^µ₂ =k2(1,sinθ_L,0,cosθ_L)

k^µ_1r=k₁(1,0,0,−1), k^µ_2r=k₂(1,−sinθ_L,0,−cosθ_L), (2.149) yielding

1 2

X

λ1λ2

|₁·^∗₂|²= 1

2(1 + cos²θ_L). (2.150) In this case, however, it is simpler to evaluate the sum using the explicit forms

1(1) = (0,1,0,0), 2(1) = (0,cosθL,0,−sinθL)

1(2) =2(2) = (0,0,1,0). (2.151)

for the transverse polarization vectors.

2.7 THE DIRAC FIELD

The Dirac field ψ_α(x), where α = 1· · ·4 is the spinor index, describes a four-component spin-¹₂ particle, i.e.,ψannihilates the two possible spin states for a particle and creates the two spin states for the antiparticle. In the absence of interactions, the Lagrangian density is

L⁰= ¯ψ(x)α(i6∂−m)_αβψ(x)β= ¯ψ(x) (i6∂−m)ψ(x). (2.152) The sum over α and β is written in matrix notation in the second form, in whichψ is a four-component column vector; theDirac adjoint

ψ(x)¯ ≡ψ^†(x)γ⁰ (2.153)

is a four-component row vector; and

6∂≡γ^µ ∂

∂x^µ =γ^µ∂_µ, (2.154)

whereγ^µ, µ= 0· · ·3, are the 4×4 Diracmatrices, defined by

{γ^µ, γ^ν}= 2g^µν. (2.155)

(There is an implicit 4×4 identity matrix on the right side of (2.155) and aftermin (2.152).) Theγ^µ must also satisfy

(γ^µ)^†=γ⁰γ^µγ⁰=γ_µ, (2.156)

15Since we have already evaluatedMf iin a specific gauge, wecannotdrop the second term on the right of (2.121).

so thatL⁰=L^†0. It is useful to define

γ⁵=γ5≡iγ⁰γ¹γ²γ³, σ^µν ≡ i

2[γ^µ, γ^ν]. (2.157) γ⁵enters for spin and chirality projections, for coupling fermions to pseudoscalars, and for axial vector currents in the weak interactions. From (2.155),γ⁵=γ^5†, γ⁵γ^µ =−γ^µγ⁵, and (γ⁵)² =I.σ^µν is useful, e.g., in the description of electric and magnetic dipole moments.

An arbitrary 4×4 matrix can be written as a linear combination of I, γ⁵, γ^µ, γ^µγ⁵, and σ^µν. For example,σ^µνγ⁵ is given in Problem 2.9.

2.7.1 The Free Dirac Field

From (2.152), one obtains the free-field (Dirac) equation

(i6∂−m)ψ(x) = 0. (2.158) The solution to (2.158) is

ψ(x) =

Z d³~p (2π)³2Ep

2

X

s=1

u(~p, s)a(~p, s)e^−ip·x+v(~p, s)b^†(~p, s)e^+ip·x

, (2.159)

wherea^†(~p, s) andb^†(~p, s) are the creation operators fore⁻ ande⁺ states, respectively,¹⁶ a^†(~p, s)|0i=|e⁻(~p, s)i, b^†(~p, s)|0i=|e⁺(~p, s)i. (2.160)

~

prefers to the physical momentum for both|e^∓(~p, s)i, andsruns over the two independent spin states. aand b and their adjoints satisfy the anticommutation rules in (2.10)–(2.12), e.g.,{a(~p, s), a^†(~p⁰, s⁰)}= (2π)³2Epδ³(~p−~p⁰)δss⁰.

In (2.159),u(~p, s) andv(~p, s) are four-componentDirac spinors(complex column vec-tors). From (2.158) they are the solutions to the momentum space Dirac equation, i.e.,

(6p−m)u(~p, s)≡(pµγ^µ−m)u(~p, s) = 0

(6p+m)v(~p, s) = 0. (2.161)

Taking the adjoint of (2.161) and using (2.156),

[(6p−m)u]^†=u^†(pµγ^µ†−m) =u^†(pµγ⁰γ^µγ⁰−(γ⁰)²m)

= ¯u(6p−m)γ⁰= 0, (2.162)

where ¯u≡u^†γ⁰ and ¯v≡v^†γ⁰ are the Dirac adjoints. Thus,

¯

u(~p, s)(6p−m) = ¯v(~p, s)(6p+m) = 0. (2.163) Before proceeding to the electrodynamics of fermions, let us consider some properties of the Dirac matrices and spinors.

16By convention, we are takingψ≡ψ_e− to be thee⁻field. Thee⁺ fieldψ^c≡ψ_e+is of the same form as (2.159) exceptaandbare reversed. Charge conjugation and space reflection will be discussed in more detail inSection 2.10.

2.7.2 Dirac Matrices and Spinors Explicit Forms for the Dirac Matrices

The Dirac matrices are defined by (2.155) and (2.156). For most calculations one does not need their explicit form, but it is occasionally useful to have one. For example, the Pauli-Diracrepresentation is useful for studying the non-relativistic limit of an interaction, while thechiralrepresentation is useful at high energy and forWeylorMajorana fields, encounted in neutrino physics and supersymmetry. In the Pauli-Dirac representation

γ⁰=

where I is the 2×2 identity matrix and σⁱ are the Pauli matrices in (1.17). Similarly, in the chiral representation

It is sometimes convenient to rewrite the chiral representation matrices as γ^µ= Traces and Products of Dirac Matrices

Most calculations can be carried out without using the specific forms of the Dirac matrices by using varioustrace identities, where the trace is TrA=P

αAααfor any square matrix

for an arbitrary four-vector a_µ. Then, from (2.155)

6a6b=− 6b6a+ 2a·bI ⇒ 6a6a=a²I. (2.170) One has immediately that

Tr (6a6b) = 4a·b. (2.171) Other useful trace identities include (see, e.g., Bjorken and Drell, 1964),

Tr (6a6b6c6d) = 4(a·b c·d+a·d b·c−a·c b·d) Tr (γ⁵6a6b) = 0

Tr (γ⁵6a6b6c6d) = 4i^µνρσ a_µb_νc_ρd_σ,

(2.172)

where^µνρσ is the totally antisymmetric tensor with₀₁₂₃= +1 and⁰¹²³=−1. Also, Tr (6a₁· · · 6a_n) = Tr (γ⁵6a₁· · · 6a_n) = 0 (fornodd)

Tr (6a1· · · 6an) =a1·a2Tr (6a3· · · 6an)−a1·a3Tr (6a26a4· · · 6an)

· · ·+a1·a_nTr (6a2· · · 6a_n−1) (forneven) Tr (6a₁6a₂· · · 6a_n) = Tr (6a_n· · · 6a₂6a₁).

(2.173)

Related useful identities are

γ_µγ^µ= 4I, γ_µ6a γ^µ=−26a

γ_µ6a6b γ^µ= 4a·b I, γ_µ 6a6b6c γ^µ=−26c6b6a. (2.174) Finally, we record the identities

Tr

γ_µ6a γ_ν 6b 1 +λγ⁵ Tr

γ^µ6c γ^ν 6d 1±λγ⁵

=

64a·c b·dfor +

64a·d b·cfor− , (2.175)

where λ=±1. These results can be derived using the identities in (2.172) and Table 1.2, as will be shown in detail in Section 7.2.1. They are extremely useful for calculating four-fermion polarization effects, as well as for weak interaction decay and scattering processes.

Spinor Normalization and Projections

The Dirac spinors satisfy the normalization and orthogonality relations

¯

u(~p, s)u(~p, s⁰) =−¯v(~p, s)v(p, s⁰) = 2m δss⁰

u^†(~p, s)u(~p, s⁰) =v^†(~p, s)v(~p, s⁰) = 2Epδss⁰ (2.176)

¯

u(~p, s)v(~p, s⁰) =u^†(p, s~ )v(−~p, s⁰) = ¯v(~p, s)u(~p, s⁰) =v^†(~p, s)u(−~p, s⁰) = 0, which are just numbers (e.g., Bjorken and Drell, 1965). The projections

X

s

u(~p, s) ¯u(~p, s) =6p+m, X

s

v(~p, s) ¯v(~p, s) =6p−m, (2.177) which are useful in summing over spin orientations in physical rates, are 4×4 matrices. The form of (2.177) follows from the Dirac equation, while the normalization follows by taking the trace and using the first equation in (2.176). (The second equation (2.176) for s =s⁰ then follows by right-multiplying (2.177) byγ⁰ and then taking the trace.)

The projections

u(~p, s) ¯u(~p, s) = (6p+m)

1 +γ⁵6s 2

v(p, s~ ) ¯v(~p, s) = (6p−m)

1 +γ⁵6s 2

(2.178)

are useful when one does not want to sum over spins. Here, s^µ is the spin four-vector. In the particle rest frame,p= (m, ~0 ), it is just a unit vector

s= (0,s)ˆ (2.179)

in the spin direction, so thats²=−1 andp·s= 0. Boosting to an arbitrary frame in which

~

p=βE~ _p=γ ~βm,

s= (γ ~β·ˆs, γˆs_k+ ˆs⊥), (2.180) where ˆs_k = ˆs·βˆβˆ and ˆs⊥ = ˆs−sˆ_k are, respectively, the components of ˆs parallel and perpendicular to ˆβ. Thus,s= (0,s) for ˆˆ s·βˆ= 0, whiles≡s_±=±γ(β,β) for ˆˆ s·βˆ=±1. The latter are known as the positive and negative helicity states, respectively, or alternatively as right- and left-handed states. The projections in (2.178) simplify greatly for the helicity states in the relativistic limit:

(6p+m)

1 +γ⁵6s_± 2

−−−→_m→0 1±γ⁵

2 6p≡P_R,L6p (6p−m)

1 +γ⁵6s±

2

−−−→_m→0 1∓γ⁵

2 6p=PL,R6p,

(2.181)

where the chiral projection operatorsPR,L will be discussed below.

The Propagator

From (2.159) and the spinor sums (2.177), one obtains the Feynman propagator for the free Dirac field¹⁷

h0|T[ψ(x),ψ(x¯ ⁰)]|0i=i

Z d⁴k

(2π)⁴e^{−ik·(x−x0)}S_F(k), (2.182) where the momentum space propagatorSF(k), which is a 4×4 matrix, is

S_F(k) = 1

6k−m+i = 6k+m

k²−m²+i. (2.183)

The last equality follows from (2.170).

Explicit Spinor Forms

Explicit forms for the Dirac spinors are occasionally useful, e.g., for considering non-relativistic limits. In the Pauli-Dirac representation

u(~p, s) =p

E_p+m φ_s

~σ·~p E_p+mφs

!

, (2.184)

whereφ_s is a two-component Pauli spinor describing the spin orientation. Thus, φ₁=

1 0

, φ₂= 0

1

(2.185) describe spin orientations in the±zˆdirections, respectively.φ_±, defined by

hφ_±≡ 1

2~σ·p φˆ _±=±1

2φ_±, (2.186)

17The time-ordered product of two fermion fields is defined as in (2.27) except there is a minus sign before the second term.

describe positive (negative) helicity states. Similarly, thev spinors are represent spins in the±zˆdirections. The positive (negative) helicity spinors, which continue to mean that the physical spin is parallel (antiparallel) to the physical momentum, satisfy

1

2~σ·p χˆ ±=∓1

2χ±. (2.190)

The counter-intuitive results in (2.189) and (2.190) are considered in Problem 2.13and in the discussion of charge conjugation inSection 2.10. Explicit forms for the helicity spinors are given in Table 2.1, using a phase convention for which

iσ²φ^∗_±=∓φ_∓, iσ²χ^∗_±=∓χ_∓. (2.191)

TABLE 2.1 Explicit forms and properties of the helicity spinors^acorresponding to spherical angles (θ, ϕ) for ˆp. apply to the fixed spin axis basis.

The explicituandvspinors in the chiral representation are

where σ^µ and ¯σ^µ are defined in (2.167). The second form is easily verified using (1.19) on page 4. In the helicity basis, these are

u(±) =

The chiral representation is especially useful in the case of massless or relativistic fermions, for which the upper (lower) two components of the positive (negative) helicity u spinors vanish, and oppositely forv,

For a fermion fieldψ, one can define left (L)- and right (R)-chiral projections ψ_L =P_Lψ= 1−γ⁵

2 ψ, ψ_R=P_Rψ=1 +γ⁵

2 ψ, (2.196)

where P_L,R² =P_L,R, P_LP_R =P_RP_L = 0, P_L,R^† =P_L,R, andP_L+P_R =I. ψ_L and ψ_R can be viewed as independent degrees of freedom, withψ =ψ_L+ψ_R. For a massless fermion theL- andR-chiral components correspond to particles with negative and positive helicity, respectively, i.e.,ψ_L and ψ_R annihilate fermions with helicity h=∓¹2. For antifermions it is just the reverse,ψ_L andψ_R create antifermion states withh=±¹2. For massm6= 0 the chiral states of energyEassociated withψ_L,Rhave admixtures ofO(m/E) of the “wrong”

helicity (Problem 2.15). The free-field Lagrangian density in (2.152) can be rewritten in terms of the chiral projections as

L= ¯ψ_Li6∂ ψ_L+ ¯ψ_Ri6∂ ψ_R−m ψ¯_Lψ_R+ ¯ψ_Rψ_L

, (2.197)

where ¯ψL,R are defined¹⁸ by

ψ¯_L≡(ψ_L)^†γ⁰=ψ^†P_Lγ⁰= ¯ψP_R

ψ¯_R≡(ψR)^†γ⁰=ψ^†P_Rγ⁰= ¯ψP_L. (2.198)

18Some authors use the notationψLor (ψL) rather than ¯ψLto emphasize that the Dirac adjoint operation acts onψLrather than onψ.

The chiral projections ψ_L,R are also known as Weyl spinors or Weyl two-component fields. They can be described as above infour-component notation, i.e., as two-dimensional projections of four-component fieldsψ, but it is often convenient to discard the superfluous components and work in two-component notation. As the name “chiral” suggests, this is most conveniently displayed in the chiral representation, for which

P_L=

where ΨL,R are the Weyl two-component fields.

The Lagrangian density (2.152) for the free Dirac field can be written in terms of the Weyl fields as

L⁰= Ψ^†_Li¯σ^µ∂µΨL+ Ψ^†_Riσ^µ∂µΨR−m

Ψ^†_LΨR+ Ψ^†_RΨL

, (2.201)

where the four-vectorsσ^µ and ¯σ^µ are the 2×2 matrices defined in (2.167). The Dirac mass term couples the L and R components, while the kinetic energy terms are diagonal. The free-field Dirac equation becomes

i¯σ^µ∂µΨL−mΨR= 0, iσ^µ∂µΨR−mΨL= 0. (2.202) Above, we introduced the chiral fields as projections of a four-component Dirac field.

Alternatively, one can simply define chiral fields as those satisfying ψ_L =P_Lψ_L or ψ_R = P_Rψ_R, i.e., not necessarily as projections of another field ψ, and in fact this was done for theL-chiral neutrinos in the original formulation of the standard model. Equivalently, Weyl fields ΨLor ΨRcan be introduced independently of each other. For example, a single Weyl L field with L⁰ = Ψ^†_Li¯σ^µ∂_µΨL would describe a massless negative helicity particle and a positive helicity antiparticle.

One can also define the chiral projections of theuandv spinors,

u_L,R(~p, s)≡P_L,Ru(~p, s), v_L,R(~p, s)≡P_L,Rv(~p, s). (2.203)

thenu_L,Rsatisfy the Dirac equation

p·σ¯u_L= (EpI+~σ·~p)u_L=mu_R

p·σuR= (EpI−~σ·~p)uR=muL. (2.205) The chiral components of v = (vL v_R)^T satisfy similar equations, only with m→ −m. It follows easily that the solutions foruandv are given by (2.192).

For m → 0 the equations for u_L,R decouple. The u and v spinors of definite helicity given in (2.195) then coincide with the left- and right-chiral projections,

PRu(~p,+) =u(~p,+), PLu(~p,−) =u(~p,−)

P_Rv(~p,−) =v(~p,−), P_Lv(~p,+) =v(~p,+), (2.206)

showing the flip between chirality and helicity for the v spinors, consistent with (2.181).

The free-field expressions forψ_L,Ror ΨL,R are especially simple in this case. From (2.159) and (2.195)

Ψ(x)L,R=

Z d³~p (2π)³2Ep

p2Ep

φ_∓(ˆp)a(~p,∓)e^−ip·x±χ_±(ˆp)b^†(~p,±)e^+ip·x

, (2.207) which is similar to the free Dirac field except there is no sum on spins.

The Weyl two-component formalism is further developed inSection 2.11and inChapters 9 and10.

Bilinear Forms

Consider the bilinear form ¯w₂M w₁, where w_1,2 are any two Dirac u or v spinors. They may even correspond to particles with different masses, which is relevant, e.g., for weak interaction transitions. M is an arbitrary 4×4 matrix. Then,

( ¯w2M w1)^∗= ¯w1M w2, (2.208) whereM ≡γ⁰M^†γ⁰is theDirac adjointofM. One finds

M = M forM =I, γ^µ, γ^µγ⁵, σ^µν M =−M forM =γ⁵, σ^µνγ⁵

M₁M₂=M₂M₁ ⇒ 6a₁6a₂· · · 6a_n=6¯a_n· · · 6a¯₂6¯a₁.

(2.209)

There is an equivalent relation,

ψ¯2M ψ1†

= ¯ψ1M ψ2, (2.210)

for two Dirac fields, which may be the same or different. Then, for example, X

s1,s2

|u¯2M u1|²= X

s1,s2

¯

u2M u1u¯1M u2

=X

s2

¯ u_2α

M(6p₁+m₁)M

αβu_2β

= Tr

"

M(6p1+m1)MX

s₂

u2¯u2

#

= Tr

M(6p1+m1)M(6p2+m2) ,

(2.211)

which allow one to express a physical rate in terms of a trace. (This is sometimes referred to as the Casimir trick.)

For chiral spinors

w_L,R≡(wL,R)^†γ⁰=w^†P_L,Rγ⁰=w^†γ⁰P_R,L= ¯wP_R,L. (2.212) Therefore, for any two spinorsw₁ andw₂,

w1LΓw2L= 0 =w1RΓw2R (2.213)

for Γ =I, γ⁵, σ^µν, orσ^µνγ⁵, while

w_1LΓw_2R= 0 =w_1RΓw_2L (2.214)

for Γ = γ^µ or γ^µγ⁵. Equivalent relations for the chiral fieldsψ_L,R were used in (2.197).

Thus, scalar, pseudoscalar, and tensor transitions reverse the chirality between an initial and final fermion, while vector and axial vector transitions maintain it.

The Fierz Identities TheFierz identitiesare

ψ¯_1Lγ^µψ_2L ψ¯_3Lγ_µψ_4L

=−η_F ψ¯_1Lγ^µψ_4L ψ¯_3Lγ_µψ_2L ψ¯_1Rγ^µψ_2R ψ¯_3Rγ_µψ_4R

=−η_F ψ¯_1Rγ^µψ_4R ψ¯_3Rγ_µψ_2R ψ¯_1Rγ^µψ_2R ψ¯_3Lγ_µψ_4L

= 2ηF ψ¯_1Rψ_4L ψ¯_3Lψ_2R

(2.215) ψ¯_1Rψ_2L ψ¯_3Rψ_4L

= η_F

2 ψ¯_1Rψ_4L ψ¯_3Rψ_2L +η_F

8 ψ¯_1Rσ^µνψ_4L ψ¯_3Rσ_µνψ_2L , where ψ_iL and ψ_jR are anticommuting chiral fields and η_F = −1. There are analogous relations foruandv spinors, but with η_F = +1, e.g.,

( ¯w_1Lγ^µw_2L) ( ¯w_3Lγ_µw_4L) =−( ¯w_1Lγ^µw_4L) ( ¯w_3Lγ_µw_2L). (2.216) The Fierz identities are easily derived by expressing the 4×4 matrices such as ψ_2Lψ¯_3L in terms of the complete setI, γ⁵, γ^µ, γ^µγ⁵, andσ^µν. A related useful identity is

ψ¯1Rσ^µνψ2L ψ¯3Lσµνψ4R

= 0. (2.217)

The Fierz identities are frequently very useful in computations, and are often used in con-junction with those for charge conjugation (Section 2.10).

Im Dokument The Standard Model and Beyond (Seite 49-58)