Supersymmetry. Theory, Experiment, and Cosmology

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Appendix B

Spinors

B.1 Spinors in four dimensions

In this appendix, we ﬁrst discuss spinors in four dimensions and introduce two-component spinors, a notation that we heavily use in the superspace formulation of supersymmetry. We then review spinors in higher dimensional spacetime, which are in spinor representations of a general SO(1, D − 1) Lorentz group. This is relevant when we discuss higher dimensions, for example in the context of string theories (chapter 10).

B.1.1 Van der Waerden notation

The metric used is ηµν = (1, −1, −1, −1). Let us consider the spinor representation of the Lorentz group SO(1, 3) which decomposes into irreducible representations of given chiralities: 4 = 2L + 2R. The corresponding spinor ﬁelds are called Weyl spinors. We will denote the representation 2L ≡ (ξα, α = 1, 2) by (1/2, 0). Under a proper Lorentz transformation1

ξα = M αβ ξβ , det M = 1.

(B.1)

One can then check that, introducing a second such spinor ηα, ξ2η1 − ξ1η2 is invariant under M .

Let us note

ξα = εαβ ξβ

(B.2)

where εαβ is the antisymmetric tensor (εαβ = −εβα, we choose ε21 = 1), invariant under a Lorentz transformation:

εαβ M αγ M β δ = εγδ det M = εγδ .	(B.3)
One may write also
ξα = εαβ ξβ	(B.4)

where εαβ is the invariant antisymmetric tensor which is the inverse of εαβ :

εαβ εβγ = δγα

(ε12 = 1).

1Since the dimension of space time is di erent from 2 mod 8, ξα being of deﬁnite chirality cannot be chosen real (otherwise it would be a Majorana–Weyl spinor) and M SL(2, C).

420 Spinors

Then using η1 = η2 and η2 = −η1, one may write, using the Einstein convention

ξ2η1 − ξ1η2 = ξ1η1 + ξ2η2 ≡ ξαηα.

Let us note that the spinor ξα introduced in (B.2) transforms as:

ξ = εαβ ξβ			= εαβ M β δ ξδ = εγδ (M −1)γ					α	ξδ
α								α
= M		1T	γ	ξ					(B.5)
where we have used (B.3).	−		α		γ
One may then check that ξαηα is invariant under a Lorentz transformation:
ξ αηα = M αβ ξβ M −1T α						γ ηγ = M −1M γ β ξβ ηγ = ξβ ηβ .
We will use from now on the following notation:
		ξαηα ≡ ξη,					ξξ ≡ ξ2.		(B.6)

Our convention is that spinors always anticommute. It follows that

ξη = ηξ.

(B.7)

The complex conjugate (ξα) transforms with the matrix M . To make the di erence explicit, one introduces a di erent notation

ξ¯α˙	= (M )α˙		˙	(B.8)
ξ¯α˙	= (M )α˙	˙	ξ¯β .	(B.8)
		β

¯α˙ ¯α˙

The ξ form an inequivalent spinor representation: ξ , α˙ = 1, 2 = 2R = (0, 1/2).

We use the antisymmetric tensors ε ˙ and εα˙ β to raise and lower dotted indices

α˙ β

˙ ˙

(ε˙ ˙ = 1 = ε12), and we note:

¯	α˙	¯	¯¯	¯2	.	(B.9)
ξα˙	η¯	≡ ξ η,¯	ξξ	≡ ξ	.	(B.9)

The convention is the following: when indices are not written, undotted indices are descending and dotted indices are ascending.

We will also take as a convention to reverse the order of spinors when performing complex conjugation:

(ξη)† = (ξαηα)† = η¯α˙ ξ¯α˙ = η¯ξ¯ = ξη¯¯.

(B.10)

B.1.2 Relation of spinors with vectors

As we have just seen, one may form a spin-zero object ξη out of two spinors. One may also form a spin-one object. In order to do so, let us introduce the Pauli matrices σµ,

µ = 0, . . . , 3,

σ0	=	1 0	σ1	=	0 1
		0 1			1 0

		σ2 =			0	−i		σ3 =	1	0	(B.11)
					i	0			0	−1
and form the vector V	µ	≡ ξ	α	µ	¯α˙		.
and form the vector V		≡ ξ		σαα˙	ξ		.

Spinors in four dimensions 421

The matrices σµ, being hermitian, satisfy:

(σµ ˙ ) = (σµ )αβ˙ = (σµ†)βα˙ = (σµ)βα˙ .
αβ
One introduces also the matrices σ¯µ deﬁned by:
σ¯µ αα˙		˙
σ¯µ αα˙		= εα˙ β εαβ σµ ˙
					ββ
(¯σ0 = σ0, σ¯i = −σi). They have the property
σαµα˙ σ¯ναβ˙ + σαν α˙ σ¯µαβ˙				= 2ηµν δαβ
σ¯µαα˙ σν ˙ + σ¯ναα˙ σµ			˙	= 2ηµν δα˙˙		,
αβ		αβ			β
from which one deduces
Tr (σµσ¯ν ) = 2 ηµν .
One may then show using (B.61) and (B.15) that
V	µ	Vµ = 2 ξ	2	¯2	.
V		Vµ = 2 ξ		ξ	.

(B.12)

(B.13)

(B.14)

(B.15)

Hence V µVµ is invariant under the transformations (B.1) and (B.8) and V µ → V µ =

α ¯ α˙

ξ σαα˙ ξ is a Lorentz transformation. The vector representation may thus be referred to as (1/2, 1/2).

B.1.3 Dirac spinors
A four-component Dirac spinor describes two Weyl spinors:
χα
Ψ = ξ¯α˙ .	(B.16)

One may introduce the following basis for gamma matrices, called the Weyl basis:

γµ =	0	σαβµ ˙		, γ5 = γ5 = iγ0γ1γ2γ3 =	−	i	µνρσγµγν γργσ =	−I 0	(B.17)
γµ =		σαβµ ˙		, γ5 = γ5 = iγ0γ1γ2γ3 =			µνρσγµγν γργσ =	−I 0	(B.17)
	σ¯µαβ˙ 0					4!		0 I
where µνρσ is the fully antisymmetric tensor with the convention:
				0123 = 1 = − 0123.					(B.18)
Indeed one checks, using (B.14), that
				{γµ, γν } = 2ηµν .					(B.19)
If we deﬁne
γ[µγν γρ] =		1	(γµγν γρ + γργµγν + γν γργµ − γν γµγρ − γµγργν − γργν γµ) ,						(B.20)

		3!

antisymmetric in µνρ, it must necessarily be of the form µνρσAσ where Aσ can be determined by taking speciﬁc values of µ, ν, ρ:

γ[µγν γρ] = −i µνρσγ5γσ .

(B.21)

422 Spinors

Similarly, with obvious notation,

γ[µγν γργσ] = −i µνρσγ5.

We will also use

	1			σµν β		0
σµν =		[γµ, γν ] =			0α	σ¯µνα˙ β˙	,
	4
where
σµν αβ	=		1	σαµα˙ σ¯ναβ˙	− σαν α˙ σ¯µαβ˙

			4
σ¯µνα˙ β˙	=		1	σ¯µαα˙ σαβν ˙	− σ¯ναα˙ σαβµ ˙ .

			4

We have

σµν = − 4i µνρσγ5γργσ .

(B.22)

(B.23)

(B.24)

(B.25)

The chirality eigenstates are:

1 − γ5

Ψ =

χα

and Ψ

1 + γ5

Ψ =

0 .

ξ¯α˙

Since we have Ψ¯ = Ψ†A where2 A =

0 δα˙˙

δαβ

δα˙˙

Ψ = (χ¯α˙

ξ )

δαβ

= (ξ

χ¯

˙ ).

(B.27)

One then easily proves that, for two Dirac spinors Ψ1 and Ψ2:

Ψ2 = ξ1 χ2

Ψ1

+ χ¯1 ξ2

Ψ1

γ5 Ψ2 = −ξ1 χ2 + χ¯1 ξ2

Ψ2 = ξ1

χ¯1

Ψ1

ξ2 − χ2 σ

Ψ2 = ξ1σ

µ ¯

+ χ2 σ

χ¯1.

(B.28)

Ψ1

ξ2

The completeness of the 16 Dirac matrices can be used to derive the Fierz rearrangement formula for two anticommuting spinors


Ψ2Ψ¯ 1 = −	1	1 Ψ¯	1Ψ2 −	1	γµ	Ψ¯ 1	γµΨ2 −			1	σµν		Ψ¯ 1	σµν Ψ2
	4			4						8
+	1	γµγ5	Ψ¯ 1γµγ5Ψ2			−	1	γ5	Ψ¯ 1γ5Ψ2			.		(B.29)

	4						4

2A is the matrix that intertwines the representation γµ with the equivalent representation γµ† :

AγµA−1 = γµ†

(B.26)

(C plays a similar rˆole between γµ and −γµT , as seen from (B.31)). As is well known, A coincides

numerically with γ0. We refrain from using the same notation because the two-component indices do not match as can be seen by comparing the explicit form of A with (B.17).

Spinors in four dimensions 423

The charge conjugation matrix C reads, in this notation,

C =

C−1 =

εαβ 0

0αβ

εα˙ β˙

0 εα˙ β˙

and satisﬁes

C−1

C γT

− µ

The charge conjugate spinor Ψc is thus simply:

ξα

Ψc = CΨ¯ T

= χ¯α˙ .

Thus a Majorana mass term reads:

¯ c

¯α˙

= Ψ

ΨR .

Ψ = χ

χα + ξα˙ ξ

R ΨL + Ψ L

The following properties may be checked from (B.28) and (B.64):

¯ c		c	¯	Ψ1,
Ψ1		Ψ2	= Ψ2
¯ c	γ5	c	¯	γ5Ψ1,
Ψ1		Ψ2	= Ψ2
¯ c		c	¯		,
Ψ1γµ		Ψ2	= −Ψ2γµΨ1
¯ c	γ5	c	¯	γµγ5Ψ1,
Ψ1γµ		Ψ2	= Ψ2

and more generally

¯ c			c	n ¯	· · · γµ1 Ψ1,
Ψ1	γµ1 · · · γµn Ψ2			= (−) Ψ2γµn	· · · γµ1 Ψ1,
¯ c			c	¯
Ψ1γµ1		· · · γµn γ5Ψ2		= Ψ2γµn · · · γµ1 γ5Ψ1,
and
¯		· · · γµn Ψ2		n ¯ c	c
Ψ1γµ1		· · · γµn Ψ2		= (−) Ψ1γµ1 · · · γµn Ψ2,
Ψ¯ 1γµ1 · · · γµn γ5Ψ2			= (−)n+1Ψ¯ 1c γµ1 · · · γµn γ5Ψ2c .
The Majorana condition

¯ T Ψ = CΨ

reads simply

(B.30)

(B.31)

(B.32)

(B.33)

(B.34)

(B.35)

(B.36)

(B.37)

χα = ξα.

(B.38)

Thus a Majorana spinor ΨM is expressed in terms of a single two-dimensional spinor as

χα
ΨM = χ¯α˙ .	(B.39)

Majorana spinors have the following properties, as can be checked from (B.35) and (B.36):

	¯				n ¯
Ψ1γµ1 · · · γµn Ψ2					= (−) Ψ2γµn · · · γµ1 Ψ1,
¯			· · · γµn γ5Ψ2		¯	· · · γµ1	γ5Ψ1,		(B.40)
Ψ1γµ1			· · · γµn γ5Ψ2		= Ψ2γµn	· · · γµ1	γ5Ψ1,		(B.40)
and
¯	γµ1			· · γµn Ψ2	n ¯
Ψ1	γµ1		·	· · γµn Ψ2	= ( ) Ψ1γµ1 · · · γµn Ψ2,				(B.41)
Ψ¯ 1γµ1			·	γµn γ5Ψ2	= (−)n+1	Ψ¯ 1γµ1	· · ·	γµn γ5Ψ2.	(B.41)
		· · ·			−		· · ·

424 Spinors

B.2 Spinors in higher dimensions

We ﬁrst review the representations of the group SO(2n) before deﬁning spinors in D dimensions.

B.2.1 Representations of SO(2n)

SO(2n) is deﬁned as the group of 2n-dimensional orthogonal matrices Λ of determinant unity, i.e.

ΛT Λ = 1, Det Λ = 1.	(B.42)
Turning to the algebra, Λ = eJ 1 + J + · · · , these two conditions become
JT = −J.	(B.43)

There are 2n(2n − 1)/2 independent 2n-dimensional matrices satisfying (B.43). Hence the dimension of the SO(2n) algebra is n(2n − 1). A basis of generators is

(1 ≤ K, L ≤ 2n)				·		·


				1		1 · · ·
			· · · · · · · · ·			1 · · ·	K
				·		·	L
	JKL =			·		·		.	(B.44)

		· · ·		·	· · · · · · · · ·
				·		·
				K		L

From this one obtains the commutation relations

[JIJ , JKL] = δJK JIL − δJLJIK + δILJJK − δIK JJL ,

(B.45)

which deﬁne the SO(2n) algebra.

A maximal set of commuting generators (known as the Cartan subalgebra) is given by the n generators3 Nk ≡ J2k−1,2k, k = 1, . . . , n. The number of such generators is the rank of the algebra, hence the rank of SO(2n) is n.

The representation (B.44) is the vector representation of dimension 2n. The dimension of the algebra gives the dimension of the adjoint representation: n(2n − 1). In the following, we build the spinor representations of dimension 2n−1.

The standard procedure is to consider a set of n fermion creation and annihilation

operators ak† , ak, k = 1, . . . , n:
&ak, al†' = δkl, [ak, al] = 0 = &ak† , al†' .	(B.46)

One then constructs a Fock space by deﬁning a vacuum |0 and acting on it with creation operators:

a†

a† a†

, . . . ,

a†

· · ·

a† 0

(B.47)

k l

form a set of 2n independent states.

3Each of them is a block diagonal matrix with only nonzero block

0 1

in the (2k − 1, 2k)

−1 0

entry.

Spinors in higher dimensions 425

The following 2n matrices

ΓK ≡ aK + a†K , K = 1, . . . , n

ΓK ≡	aK−n − aK†	−n	, K = n + 1, . . . , 2n	(B.48)
	ı
act on this 2n-dimensional Fock space. They satisfy the Cli ord algebra
	{ΓK , ΓL} = 2δKL ,			(B.49)

and are therefore generalizations of the gamma matrices to dimensions higher than four. The point of interest to us is that the Σ matrices built out of them have precisely the commutation relations (B.45) of SO(2n):

ΣKL ≡	1		[ΓK , ΓL]
		4
[ΣIJ , ΣKL] =	δJK ΣIL − δJLΣIK + δILΣJK − δIK ΣJL .			(B.50)

Since the ΣKL matrices act on the 2n-dimensional Fock space, we have constructed a (2n)-dimensional representation known as the spinor representation.

This representation is not irreducible, however. Indeed, deﬁne

Γ(2n+1) ≡ ın(2n−1)Γ1 · · · Γ2n, Γ(2n+1) 2 = 1 . (B.51)

Let us choose a Fock vacuum of deﬁnite chirality, i.e. Γ(2n+1)\|0 = +\|0 . Then, since
$Γ2n+1, ak† %	= 0,
	Γ(2n+1)ak†	1 · · · ak†p \|0 = (−1)pak†	1 · · · ak†p \|0 ,	(B.52)

and we can divide our Fock space (B.47) into states of opposite chirality. We have therefore built two (2n−1)-dimensional irreducible representations, the spinor representations of SO(2n) of opposite chirality.

B.2.2 Spinors in D-dimensional spacetimes

Consider a D-dimensional Riemannian4 manifold. We need to introduce the tangent space Tp, the set of tangent vectors at point p which is generated by the inﬁnitesimal displacements dxM , M = 1, . . . , D, at this point. The discussion will rest on the notion of tangent space group, the set of orthogonal transformations on Tp. Writing such a transformation as dxM = ΛM N dxN , the property of orthogonality is expressed as usual by (gMN is the metric of the manifold)

gM N dxM dxN = gMN ΛM P ΛN S dxP dxS = gP S dxP dxS
that is
gM N ΛM P ΛN S = gP S .	(B.53)

4A Riemannian manifold is a smooth manifold with a symmetric metric tensor gM N such that the bilinear form gM N V M W N (V, W Tp) is nondegenerate (gM N V M W N = 0 for all V i W = 0).

426 Spinors

In the case of Minkowski spacetime M4, with gµν = (+1, −1, −1, −1), this tangent space group is the proper Lorentz group SO(1, 3) (if we restrict to det Λ = +1). For a D-dimensional manifold, with metric gM N = (+1, −1, −1, . . . , −1), the tangent space group is SO(1, D − 1).

The corresponding Cli ord algebra reads
{ΓM , ΓN } = 2gM N ,	(B.54)

where the gamma matrices have dimension 2[D/2], with [D/2] the integral part of D/2. As in the previous section, the generators of SO(1, 9) are the ΣM N = 1/4[ΓM , ΓN ].

Just as before, one may introduce for even spacetime dimension D, the matrix Γ(D+1) which anticommutes with all gamma matrices5:

Γ(D+1) ≡ ı(D−2)/2Γ0Γ1 · · · ΓD−1 .

(B.55)

We note that γ5 deﬁned in (B.17) is precisely Γ(5) for D = 4. One then deﬁnes the chirality eigenstates by the condition:

Γ(D+1)ΨL,R = ΨL,R .	(B.56)
Thus, a Dirac spinor has 2D/2 degrees of freedom whereas a Weyl6	fermion has
2(D−2)/2.

Can we always deﬁne a Majorana spinor? In order to answer this question, let us deﬁne the matrix B which intertwines the representation ΓM with the equivalent

representation −ΓM :
BΓM B−1 = −ΓM .	(B.57)

We have BB = ηI with η = ±1. Clearly, B = CΓ0 where C is deﬁned as in (B.31) by

C−1ΓM C = −ΓM T .

(B.58)

If a spinor ﬁeld Ψ of charge q satisﬁes the Dirac equation coupled with the electromagnetic ﬁeld, one easily checks that B−1Ψ satisﬁes it with charge −q: it is associated with the antiparticle. One deﬁnes a Majorana spinor by the condition that particle and antiparticle are identical:

Ψ = BΨ .

(B.59)

This Majorana condition is easily shown (see Exercise 6) to be equivalent to the condition (B.32) given above. It obviously implies η = 1.

It turns out that η can be computed as a function of the dimension D (see [331]

5The di erence in the overall factor between (B.51) and (B.55) is due to the change of metric between Euclidean and Minkowski. In both cases, it is chosen in such a way that Γ(D+1) 2 = 1.

6In the case of odd dimension D , there are no Weyl spinors. If we write D = D + 1 where D is even, then Γ0, Γ1, . . . , ΓD−1 and iΓ(D+1) given in (B.55) yield a set of gamma matrices. In what follows, we only consider the case of even D.

Exercises 427

Thus η = 1 for D = 2, 4 mod 8. One can then show that there exists a representation of the gamma matrices which is purely imaginary, in which case B = 1 and C = Γ0: the Majorana condition reads Ψ = Ψ. A Weyl condition (B.56) is then only possible if Γ(D+1) in (B.42) is real i.e. for D/2 odd. Hence one can deﬁne a Majorana–Weyl spinor only in D = 2 mod. 8 dimensions.

					1
θσµθ¯ θσν θ¯ =							ηµν θ2θ¯2.
θσµθ¯ θσν θ¯ =						2	ηµν θ2θ¯2.
Exercise 2 Prove that
χ σ	µ		¯		¯			µ	χ,
χ σ			ψ = −ψ σ¯						χ,
χ σµ ψ¯ = ψ σµ χ¯.
Exercise 3 Use
σµ		σ		˙	= 2εαβ ε				˙
αα˙			µββ						α˙ β
to show the following Fierz reordering formula:
(ψ1ψ2) (χ¯1χ¯2) =				1	(ψ1σµ		χ¯1) (ψ2σµχ¯2) .
(ψ1ψ2) (χ¯1χ¯2) =					(ψ1σµ		χ¯1) (ψ2σµχ¯2) .
			2

(σµν σρ)αβ˙	=	1	gνρgµλ − gµρgνλ − i µνρλ σλαβ˙	,
		2
(σµσ¯νρ)αβ˙	=	1	gµν gρλ − gµρgνλ − i µνρλ σλαβ˙	,

		2
(¯σµν σ¯ρ)αβ˙	=	1	gνρgµλ − gµρgνλ + i µνρλ σ¯λαβ˙	,

		2
(¯σµσνρ)αβ˙	=	1	gµν gρλ − gµρgνλ + i µνρλ σ¯λαβ˙	.	(B.68)

		2

ξα ξβ		=	1	εαβ ξ2,
ξα ξβ		=	2	εαβ ξ2,
			2
¯	¯	= −		1	¯2
ξα˙	ξβ˙	= −		2	εα˙ β˙ ξ	.

(σµν )		β =	i	µνρσ (σρσ)		β	,
	α					α
		2

(¯σµν )α˙		β˙ = −		i	µνρσ (¯σρσ)α˙		β˙ ,

				2