Labelle P. Supersymmetry DeMYSTiFied

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206

Supersymmetry Demystified

10.1 Free Supersymmetric Abelian

Gauge Theory

We consider a U (1) free gauge theory where the superpartner of the photon ﬁeld

will be a left-chiral Weyl spinor that we will denote by λ. The spin 1/2 super-

partners of gauge bosons are generically referred to as gauginos. The gaugino for

the supersymmetric version of quantum electrodynamics (QED) is also referred to

as a photino.

Our starting point for the lagrangian is therefore a sum of kinetic terms for the

photon and the spinor

L =−

μν

+i λ

†

¯σ

∂

λ (10.1)

But this is incomplete because, as in the case of the Wess-Zumino model, we will

need to introduce an auxiliary scalar ﬁeld in order for the SUSY algebra to close

off-shell. We will set aside this issue for now and come back to it in the next section.

The members of a SUSY multiplet all must transform the same way under any

gauge group acting on them. Since the photon ﬁeld is neutral, we take the photino

to be neutral as well so we don’t need to replace the derivative ∂

acting on it by

a covariant derivative, and the lagrangian (10.1) is gauge-invariant as it is. Note

that when we say that the photon is neutral, this is a statement about the gauge

invariance of the ﬁeld strength F

μν

, not of A

(which, of course, is not gauge-

invariant). In nonabelian gauge theories, the ﬁeld strength is not invariant, which is

why the gauge bosons are “charged,” i.e., interact directly (at tree level) with one

another. In supersymmetric extensions of nonabelian gauge theories, the gauginos

also will be charged under the gauge group.

Because there is no interaction between the spinor and the vector ﬁeld, this is a

free theory. We will include interactions later.

Now consider the SUSY transformations of the ﬁelds. The gauge ﬁeld A

is real

(in the sense A

μ†

= A

), so its transformation must be real as well. Let us try

δ A

= ζ

†

¯σ

λ + λ

†

¯σ

ζ (10.2)

where we had to add the hermitian conjugate to make the transformation real. Since

the dimension of A

is equal to 1, and since λ has a dimension of 3/2, we get that

the SUSY parameter ζ is of dimension −1/2 as before.

EXERCISE 10.1

Show that Eq. (10.2) is real.

CHAPTER 10 Supersymmetric Gauge Theories

207

Now consider the transformation of the photino. There is a new constraint that

we did not have previously: The transformation must be gauge-invariant. Thus we

can’t use δλ  A

. Instead, we start with the gauge-invariant combination

δλ  Cζ F

μν

where C is an arbitrary constant. Since F

μν

has dimension 2, the dimensions work

out. However, the Lorentz properties are not all right, obviously. We need to contract

the two Lorentz indices of the ﬁeld strength without changing the dimensions

of the right-hand side. We should use some combination of σ

and ¯σ

but we

must be careful; we also must make sure that we end up with something that will

transform like a left-chiral spinor. Recall that P

¯σ

χ transforms like a right-chiral

spinor, whereas P

η transforms like a left-chiral spinor. Therefore P

¯σ

transforms like a left-chiral spinor. So we see that the quantity

δλ = CF

μν

¯σ

ζ (10.3)

transforms like a left-chiral spinor, as desired (note that ¯σ

must be applied on the

spinor ﬁrst).

Taking the hermitian conjugate of Eq. (10.3), we get

δλ

†

= C

∗

μν

†

¯σ

Let’s now ﬁnd out if there is some value of C that will make the lagrangian

invariant. Consider ﬁrst the variation of the Maxwell lagrangian:



−

μν



=−

μν

(∂

δ A

− ∂

δ A

)

=−F

μν

∂

δ A

=−F

μν

∂

(ζ

†

¯σ

λ) − F

μν

∂

(λ

†

¯σ

ζ)

=−F

μν

†

¯σ

∂

  

−F

μν

(∂

†

) ¯σ

  

(10.4)

where in the second step we used the antisymmetry of F

μν

Now consider the variation of the spinor term in Eq. (10.1):



iλ

†

¯σ

∂



= iC

∗

μν

†

¯σ

∂

  

III

+iCλ

†

¯σ

ζ∂

μν

  

208

Supersymmetry Demystified

Terms I and III must cancel each other (modulo total derivatives) because they are

the only ones containing ζ

†

. Likewise, the second and fourth terms must cancel out

because they contain ζ .

Consider III ﬁrst. We shall use identity (A.11), which we will repeat here for

convenience:

¯σ

= η

μν

¯σ

− η

νρ

¯σ

+ η

μρ

¯σ

−i 

νμρδ

¯σ

We use this to write term III as

III = iC

∗

μν

†

(η

μν

¯σ

− η

νρ

¯σ

+ η

μρ

¯σ

−i 

νμρδ

¯σ

)∂

The ﬁrst piece is identically zero because η

μν

is symmetric in the two Lorentz

indices, whereas F

μν

is antisymmetric. The term with the Levi-Civita symbol also

vanishes, although this is not as obvious. To prove it, we do an integration by parts

and rewrite



νμρδ

(∂

λ)F

μν

=−

νμρδ

λ∂

μν

=−

νμρδ

λ∂

∂

+ 

νμρδ

λ∂

∂

Now we see that this is identically zero because both terms contain a quantity

symmetric under the exchange of two indices (the two derivatives) times the anti-

symmetric Levi-Civita symbol.

We are left with

III =−iC

∗

μν

†

¯σ

∂

λ +iC

∗

μν

†

¯σ

∂

If we relabel μ ↔ ν in the ﬁrst term and use the antisymmetry of the ﬁeld strength,

we may combine the two terms to get the simple expression

III = 2iC

∗

μν

†

¯σ

∂

λ (10.5)

This will be canceled by the ﬁrst term, namely,

I =−F

μν

†

¯σ

∂

at the condition of choosing C = i/2.

CHAPTER 10 Supersymmetric Gauge Theories

209

Let’s now look at IV. Using the same tricks we used for the third term, we ﬁnd

that it may be written as

IV = 2iC(∂

†

) ¯σ

ζ F

μν

=−2iC(∂

†

) ¯σ

ζ F

μν

which cancels out II at the condition that C = i/2 once again.

The lagrangian (10.1) is therefore supersymmetric under the transformations

δ A

= ζ

†

¯σ

λ + λ

†

¯σ

δλ =

μν

¯σ

δλ

†

=−

μν

†

¯σ

However, as noted at the beginning of this chapter, this is incomplete because

the algebra does not close off-shell. Let’s discuss this now.

10.2 Introduction of the Auxiliary Field

The next step would be to compute the commutator of two SUSY transformations

on all the ﬁelds, as we did for the left-chiral multiplet of the Wess-Zumino model.

We won’t do the calculation here but only quote the ﬁnal result that, again, the

algebra does not close off-shell.

This should not be surprising if we count the fermionic and bosonic degrees of

freedom in our theory. On-shell, A

has two degrees of freedom (the two transverse

polarization states), and so does a left-chiral Weyl fermion, as we have mentioned

several times. Off-shell, a vector ﬁeld A

has three degrees of freedom, whereas a

left-chiral Weyl fermion has four. Therefore, we need to introduce one extra bosonic

degree of freedom (as opposed to the two needed for the chiral multiplet) in the

form of a real scalar ﬁeld that is conventionally denoted D. As with the photino

and the ﬁeld strength, we take D to be gauge-invariant.

We therefore add the following term to the lagrangian:

aux

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Supersymmetry Demystified

Like the auxiliary ﬁeld F, D has dimension 2. For its variation, we may take the

same expression we had for the variation of F, at the condition of adding the

hermitian conjugate of that variation, because D is a real ﬁeld. This gives

δ D =−iζ

†

¯σ

∂

λ +i(∂

λ)

†

¯σ

Note that the transformation is a total derivative. Interestingly, the transformation

of the auxiliary ﬁeld F is also a total derivative, (which is not a coincidence, as will

be explained in Section 12.5). Since D is real, gauge invariant and transforms with

a total derivative under SUSY, we can include a term linear in D in the lagrangian.

We therefore include a term

= ξ D (10.6)

where the arbitrary constant ξ has the dimension of a mass squared because D is of

dimension 2. This term is known as the Fayet-Illiopoulos term (hence the choice of

subscript) and will play an important role in our discussion of SUSY spontaneous

breaking in Chapter 14. Note that we could not have added a term linear in F to

the Wess-Zumino lagrangian because F is not a real ﬁeld.

Now consider the variation of D

/2, i.e.,





=−iDζ

†

¯σ

∂

λ +iD(∂

λ)

†

¯σ

As before, to cancel these terms, we introduce an extra piece to the variation of the

spinor ﬁeld λ:

δλ = previous term + Dζ

(10.7)

δλ

†

= previous term + Dζ

†

where we have used the fact that D is real.

One can check that the lagrangian is still invariant under SUSY (modulo total

derivatives, as always) after including those modiﬁcations.

The complete supersymmetric lagrangian of a free abelian gauge multiplet is

therefore

L =−

μν

+i λ

†

¯σ

∂

λ +

+ ξ D (10.8)

CHAPTER 10 Supersymmetric Gauge Theories

211

with the SUSY transformation of the ﬁelds given by

δ A

= ζ

†

¯σ

λ + λ

†

¯σ

δλ =

μν

¯σ

ζ + Dζ

δ D =−iζ

†

¯σ

∂

λ +i(∂

λ)

†

¯σ

ζ (10.9)

Let’s now consider nonabelian gauge theories.

10.3 Review of Nonabelian Gauge Theories

Before discussing supersymmetric nonabelian gauge theories, let’s do a quick re-

view of some basic facts about group theory. We will restrict our attention to

semisimple groups, i.e., groups with no U(1) invariant subgroup. The generators

of a group obey an algebra of the form

, T

] = if

abc

(10.10)

where the constants f

abc

are called the structure constants of the group, and the

indices run from 1 to the order of the group. A basis always may be chosen such

that the structure constants are completely antisymmetric.

It is easy to check that for any generators A, B, and C, the following Jacobi

identity holds:



[A, B], C





[B, C], A





[C, A], B



= 0

This leads to the following identity for the structure constants:

abe

ecd

+ f

bce

ead

+ f

cae

ebd

= 0 (10.11)

The generators are traceless, but the trace of the product of two generators is in

general nonzero. It is always possible to choose a basis such that

Tr(T

) = C(R)δ

where the coefﬁcient C(R) depends on the representation used for the generators

and on their normalization. However, once the value of C(R) for one particu-

lar representation is ﬁxed, the values of C(R) for all the other representations is

automatically determined.

212

Supersymmetry Demystified

We will now focus on SU(N ) because these are the only groups [beside the

trivial U (1)] that will be of interest to us in this book. For SU(N ), the number of

generators is equal to N

− 1, so the indices in Eqs. (10.10) and (10.11) run from

1toN

− 1.

We will be concerned with only two speciﬁc representations: the fundamental

and adjoint representations. The fundamental representation is of dimension N

(the generators are N × N matrices) and is the smallest irreducible representa-

tion. There is also an “antifundamental” representation related to the fundamental

representation by complex conjugation. For N = 2, the fundamental and antifun-

damental representations are actually equivalent; i.e., they are related by a unitary

transformation. On the other hand, for N > 2, the fundamental and antifundamen-

tal representations are not equivalent. We will see an explicit illustration of the

difference between the two representations in the next section, where we discuss

quantum chromodynamics (QCD).

It is customary to normalize the generators of the fundamental representation to





(10.12)

As mentioned earlier, this then sets the normalization of all other representations.

For SU(2), the structure constants are given by the Levi-Civita symbol f

ijk



ijk

. We won’t make any distinction between upper and lower indices for the SU(N )

indices. The generators of the fundamental representation are taken to be half the

Pauli matrices:

for SU(2) (10.13)

where the σ

are just the usual Pauli matrices. These obey the normalization con-

dition (10.12).

The structure constants for SU(3) can be found in all books on particle physics

(often with typos, unfortunately), and we won’t repeat them here (to avoid writing

them with typos!). The generators of the fundamental representation are half the

Gell-Mann matrices:

for SU(3)

Do not confuse the Gell-Mann matrices with the components of the photino λ

introduced earlier in this chapter!

CHAPTER 10 Supersymmetric Gauge Theories

213

In the standard model, all the fermions, as well as the Higgs ﬁelds, transform in

the fundamental and antifundamental representations of the gauge groups.

The adjoint representation is also of particular interest because it is the repre-

sentation in which the gauge ﬁeld strengths transform. The adjoint representation

can be obtained from the structure constants by deﬁning





≡−if

abc

(10.14)

On the left-hand side, the index i labels the generator with which we are dealing,

whereas the jk denote the speciﬁc entry of that generator. Clearly, the dimension of

these matrices is N

− 1, the same as the number of generators. Using the identity

(10.11), one can show that these matrices indeed obey the commutation relations

(10.10), which proves that Eq. (10.14) is indeed a valid representation. With the

normalization (10.12) for the fundamental representation, the adjoint representation

satisﬁes





= N δ

(10.15)

Consider now a set of N ﬁelds ψ

(which we take to be fermions but which could

be bosons as well, such as the Higgs doublet of the standard model) that transform

in the fundamental representation of SU(N ). They can be assembled into an N -

component column vector ψ. This means that under a gauge transformation, they

transform as

ψ → ψ



= exp



igT



(x)



≡ U ψ (10.16)

where g is the gauge coupling constant, and the 

(x) are a set of N

− 1 spacetime-

dependent gauge parameters. In this chapter, the symbol U will represent the matrix

implementing the gauge transformation; this should not be confused with the ma-

trices U used in Chapter 6 for Lorentz and SUSY transformations.

We will suppress the x dependence of the gauge parameters in the following to

alleviate the notation. It is also convenient to deﬁne



≡ 

and to refer to  as the gauge parameter, even though it is really an N by N matrix

containing a set of N

− 1 parameters.

214

Supersymmetry Demystified

We deﬁne the gauge covariant derivative on this multiplet to be

ψ ≡



∂

+ig A



ψ (10.17)

When we consider the standard model, we will use the notation A

≡ W

for the

SU(2)

gauge ﬁelds and A

= G

for the gluons.

We see that there is the same number of gauge ﬁelds as there are generators of

the group, i.e., N

− 1. To simplify the notation, we also deﬁne

≡ A

(10.18)

which we shall simply call the gauge ﬁeld. The transformation of this gauge ﬁeld

is taken to be

→ A



= UA

†

(∂

U )U

†

(10.19)

With this deﬁnition, the covariant derivative D

ψ transforms the same way as ψ

itself:

ψ → D



= UD

ψ (10.20)

For a set of Dirac spinors 

transforming in the fundamental representation,

a gauge-invariant lagrangian is obtained simply by replacing ∂

by the gauge-

covariant derivative in the Dirac lagrangian:

L =





iγ

− m





where, again,  is to be understood as a column vector containing the N Dirac

spinors transforming in the fundamental representation.

We have a similar expression for Weyl spinors. Consider a set of N left-

chiral Weyl spinors χ

transforming in the fundamental representation. The gauge-

invariant kinetic energy then is given by

= χ

†

i ¯σ

As mentioned earlier, for SU(N ) with N > 2, there are two nonequivalent fun-

damental representations. One involves the generators T

, and the second involves

their complex conjugate T

a∗

(we then say that the group admits complex repre-

sentations). It turns out that when we express the QCD lagrangian in terms of the

left-chiral quark and antiquark Weyl spinors χ

and χ

, we ﬁnd that they transform

CHAPTER 10 Supersymmetric Gauge Theories

215

under the fundamental and antifundamental representations of SU(3) (this will be

discussed in more detail in the next section). This is noteworthy because when we

will write down the minimal supersymmetric standard model (MSSM) in Chapter

15, it will be entirely in terms of left-chiral spinors, and it will be important to

be able to check that the standard model lagrangian will be contained within the

MSSM. We will get back to this point in the next section.

Be warned that there is much freedom in the choice of signs in Eqs. (10.19) and

(10.17), and in the exponential of (10.16), the only constraint being the transfor-

mation property (10.20). Many different permutations of the three signs are found

in the literature.

The ﬁeld strength corresponding to a gauge ﬁeld is

μν

= ∂

− ∂

−ig[A

, A

] (10.21)

where it is understood that each A is meant to be the matrix deﬁned in Eq. (10.18).

The commutator is, of course, zero in the case of an abelian gauge theory, and we

then recover the familiar ﬁeld strength of electromagnetism. Since the ﬁeld strength

is also a matrix in the Lie algebra, it may be expanded over the generators as

μν

≡ F

μν

Written in terms of components, Eq. (10.21) corresponds to

μν

= ∂

− ∂

−ig A



, T



Using Eq. (10.10), this leads to

μν

= ∂

− ∂

+ gf

abc

Using Eq. (10.19), one can show that the ﬁeld strength transforms as

μν

→ F



μν

= UF

μν

†

≈ F

μν

+i [, F

μν

]

= F

μν

+i 

μν



, T



= F

μν

− f

abc



μν

(10.22)

from which we can read off the transformation of F

μν



μν





= F

μν

− f

abc



μν