Labelle P. Supersymmetry DeMYSTiFied

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Supersymmetry Demystified

The matrices σ

must be sandwiched between an η

†

and an η

to yield a four-vector,

and ¯σ

must be sandwiched between χ

†

and χ

EXERCISE 2.3

Prove that η

†

transforms as a four-vector under a Lorentz transformation. Hint:

Consider ﬁrst η

†

and show that it transforms like the energy E in Eq. (2.33).

Then show that η

†

transforms like the momentum three-vector.

2.5 A First Notational Hurdle

Here we should pause to discuss a ﬁrst (of many!) possible source of confusion

owing to notation. This ﬁrst hurdle concerns the use of the hermitian conjugate

(or “dagger”) symbol, †. In linear algebra, taking the hermitian conjugate means

taking the complex conjugate and transpose of a matrix or vector. However, in

quantum ﬁeld theory, the symbol † has in addition a second role: When applied to

a quantum ﬁeld, it consists of taking a complex conjugate of any complex number

as well as replacing annihilation operators by creation operators (and vice versa).

For example, writing a free complex scalar ﬁeld φ(x) as

φ(x) =



2E(2π)

[a(k)e

−ik·x

+ b

†

(k)e

ik·x

]

the hermitian conjugate of the ﬁeld is given by

†

(x) =



2E(2π)

†

(k)e

ik·x

+ b(k)e

−ik·x

] (2.34)

Clearly, if the dagger is applied to a scalar ﬁeld, it only represents taking the

hermitian conjugate in the “quantum ﬁeld operator sense” shown in Eq. (2.34).

Consider now quantum ﬁelds that are themselves components of column or row

vectors, e.g., the spinor χ . Both components of χ—let’s call them χ

and χ

—

are quantum ﬁelds, and it makes sense to talk about χ

†

and χ

†

. This would be

nonsensical for components which are simple complex numbers because we can’t

take the hermitian conjugate of a complex number. In the case of quantum spinor

ﬁelds, taking the dagger of a component has a clear meaning: It stands for the

hermitian conjugation in the quantum ﬁeld operator sense (complex conjugation

plus exchange of creation and annihilation operators).

The problem arises when we consider something like χ

†

. What is meant by this

is not completely clear, a priori. Does this represent the column vector with the

CHAPTER 2 A Crash Course on Weyl Spinors

hermitian conjugation in the quantum ﬁeld operator sense applied to the compo-

nents, namely,

†



†



(2.35)

or is the hermitian conjugate applied in both the operator and the linear algebra

sense, as in

†



†

,χ

†



Most authors simply use χ

†

to represent both expressions, letting the context

dictate which meaning is implied. With some practice, it is not difﬁcult to get used

to this double role of the dagger, especially when expressions are fairly short, but it

is distracting to have to worry about making sense of an ambiguous notation when

learning a new subject. For this reason, we will use the following convention in this

book: When a dagger will be applied to a spinor ﬁeld, it will represent the hermitian

conjugation of each component of the spinor ﬁeld in the quantum ﬁeld operator

sense as well as a transpose. Therefore, the quantity χ

†

will always represent the

row vector with components χ

†

, i.e.,

†



†

,χ

†



Note that this is the convention we used in the preceding section.

The question is, then, what do we do if we want to represent the column vector

(2.35) unambiguously? We will simply have to “undo” the transpose included in

the linear algebra dagger, so we will write

†T

≡



†



(2.36)

Of course, this notation would never be used if we were dealing with vectors made

of complex numbers because we would simply write χ

∗

instead of χ

†T

. Here, the

quantum operator dagger is applied to each component of the spinor, which is why

we cannot think of χ

†T

as χ

∗

This notation is admittedly a bit awkward, but it is a small price to pay for the

gain in clarity we get by explicitly distinguishing the column and row spinors χ

†T

and χ

†

Note that the possible confusion arises only when the dagger is applied to spinor

quantum ﬁelds. When applied to matrices such as Dirac matrices or Pauli matrices,

Supersymmetry Demystified

the dagger obviously stands for the usual hermitian conjugation of linear algebra,

and when applied to a scalar ﬁeld, it clearly represents the hermitian conjugation

in the quantum ﬁeld operator sense.

2.6 Building More Lorentz Invariants

Out of Weyl Spinors

The simplest supersymmetric theory, which we will construct in Chapter 5, involves

a single Weyl spinor (chosen by convention to be left-chiral) coupled to scalar ﬁelds.

Later we will work with theories containing several left-chiral spinors. It is therefore

important to ﬁgure out all the possible ways to construct Lorentz-invariant terms

out of left-chiral spinors only. For the sake of completeness, we also will write down

Lorentz-invariant expressions containing only right-chiral spinors or mixing left-

and right-chiral spinors, although these won’t be needed in the rest of this book.

Consider, then, a single left-chiral spinor χ

. We already know one Lorentz-

invariant expression that contains only χ

: the kinetic term appearing in Eq. (2.28).

On the other hand, the mass term that we obtained from the Dirac lagrangian mixes

a left- and right-chiral spinor. So an interesting question arises: Is it possible to

build a mass term out of a left-chiral spinor only? It turns out that it is possible, but

not completely trivial.

We know that χ

†

is invariant, so what we need is to somehow build out of the

left-chiral ﬁeld χ

something that transforms like a right-chiral ﬁeld η

! If we then

multiply χ

†

by this new quantity, we will have obtained a Lorentz-invariant mass

term for the left-chiral ﬁeld.

As a ﬁrst guess to what this quantity could be, let’s consider χ

†T

. Recall from the

preceding section that this is a column vector whose components are the dagger of

the quantum ﬁelds [see Eq. (2.36)]. Keeping in mind that the hermitian conjugate is

applied only to the quantum ﬁelds and that only a complex conjugation is applied

to everything else, and using Eq. (2.32), we ﬁnd that χ

†T

transforms as

†T

→ (χ

†T

)



= (χ



)

†T



1 +

 ·σ +



β ·σ



∗

†T



1 −

 ·σ

∗



β ·σ

∗



†T

Unfortunately, this does not transform like a right-chiral spinor [see ﬁrst line of Eq.

(2.32)]. There are two problems: The two signs are wrong, and the Pauli matrices

are complexed conjugated. But it turns out that both problems can be solved in one

CHAPTER 2 A Crash Course on Weyl Spinors

shot if we recall Eq. (2.12):

σ

∗

=−σσ

which is exactly what the doctor ordered! To see why this helps, consider the

Lorentz transformation of the quantity iσ

(the inclusion of a factor of i is

purely conventional but convenient when we relate particle and antiparticle states,

as we will see in Chapter 4):

iσ

†T

→



iσ

†T





= (iσ

)

∗



1 +

 ·σ +



β ·σ



∗

†T

= i σ



1 −

 ·σ

∗



β ·σ

∗



†T



1 +

 ·σ −



β ·σ



iσ

†T

which is the transformation of a right-chiral spinor! We have therefore built some-

thing that transforms like a right-chiral spinor out of a left-chiral spinor. This is a

key result, so let us emphasize it:

iσ

†T

transforms like a right-chiral spinor

(2.37)

The matrix iσ

will appear in so many of our calculations that it is useful to

summarize some of the properties that will often be used:

(iσ

)

=−iσ

(iσ

)

∗

= iσ

(iσ

)

†

=−iσ

(iσ

)

=−1

Now we are ready to build Lorentz invariants that contain only left-chiral spinors.

We consider only invariants with no Lorentz indices for now. Since the expression

†

is invariant under Lorentz transformations and iσ

†T

transforms like η

,we

know that the following expression is also invariant:



iσ

†T



†

= χ

(iσ

)

†

= χ

(−iσ

)χ

(2.38)

This is what we were seeking: a Lorentz invariant mass term containing only left-

chiral spinors!

Supersymmetry Demystified

Since we know that χ

†

is invariant, we obtain a second invariant containing

only left-chiral spinors:

†

iσ

†T

(2.39)

The two invariant combinations (2.38) and (2.39) play such an important role in

the construction of supersymmetric theories that they are assigned special notations

that we will discuss shortly. Before doing that, let us build other invariants made of

right-chiral spinors alone or mixing left- and right-chiral spinors. First, we need to

build something out of the right-chiral spinor η

that will transform like a left-chiral

spinor. It is easy to check that

−iσ

†T

transforms like a left-chiral spinor

(2.40)

Again, the reason why we choose to include a factor of −i will become clear when

we discuss antiparticles in Chapter 4.

To build other invariants, we could proceed again by relying on the invariance of

the terms appearing in the Dirac lagrangian. However it is also instructive to write

down explicitly the Lorentz transformation properties of the various quantities we

are dealing with.

Let’s deﬁne the matrix A as

A ≡ 1 + i

 ·σ

−



β ·σ

With this deﬁnition, right-chiral spinors have for Lorentz transformation η

→

A η

Using the fact that the Pauli matrices are hermitian, we can easily ﬁnd the

hermitian conjugate of the matrix A:

†

= 1 −i

 ·σ

−



β ·σ

The inverse matrix is obtained simply by changing the sign of  and η because these

are inﬁnitesimal quantities. Thus we ﬁnd

−1

= 1 −i

 ·σ



β ·σ

CHAPTER 2 A Crash Course on Weyl Spinors

In turn, this implies

−1

)

†

= (A

†

)

−1

= 1 +i

 ·σ



β ·σ

where we stressed that the order of the dagger and inverse operations does not

matter. This is the matrix that appears in the transformation of left-chiral spinors.

Thus the quantities we have at our disposal, together with their behavior under

Lorentz transformations, are

→ Aη

iσ

†T

→ A



iσ

†T



→ (A

†

)

−1

−iσ

†T

→ (A

†

)

−1



−i σ

†T



(2.41)

It is then clear how to build invariants. If we call type I the expressions that transform

with the matrix A and type II those that transform with the matrix (A

†

)

−1

, then the

possible invariants are either of the form

(type I)

†

(type II)

or of the form

(type II)

†

(type I)

This gives eight combinations (counting as different combinations related by her-

mitian conjugation), but it turns out that only six of those are independent, which

may be taken to be

− χ

iσ

†

iσ

†T

†

− η

†

iσ

†T

iσ

(2.42)

The ﬁrst two expressions involve only the left-chiral spinors that we obtained earlier.

The next two expressions mix the two types of spinors and are the ones appearing

in a Dirac mass term. The last two expressions contain only right-chiral spinors.

Supersymmetry Demystified

EXERCISE 2.4

Show that the two remaining combinations, involving both iσ

†T

and −i σ

†T

are equivalent to two of the expressions already listed.

2.7 Invariants Containing Lorentz Indices

Let us now consider terms that contain derivatives of spinor ﬁelds. We have seen,

by looking at the Dirac equation written in terms of Weyl spinors [Eq. (2.26)], that

(after using P

= i∂

)

i∂

transforms like a left-chiral spinor (2.43)

whereas

¯σ

i∂

transforms like a right-chiral spinor (2.44)

Note how the matrix σ

“belongs” with right-chiral spinors and ¯σ

“belongs”

with left-chiral spinors, in the sense that these are the combinations required to build

quantities that have well-deﬁned Lorentz transformation properties. Applying a σ

to a left-chiral spinor does not give anything with well-deﬁned Lorentz properties,

so it’s not a useful thing to do. However, applying a ¯σ

to a left-chiral spinor and

then applying a σ

does produce something interesting, as will be illustrated in

Exercise 2.5.

To build Lorentz invariants containing derivatives, we simply have to use the

expressions listed in Eq. (2.42) with the χ

appearing in those equations replaced

by Eq. (2.43) or the η

replaced by Eq. (2.44).

As examples, consider building invariants containing only left-chiral spinors and

only one derivative. These are obtained by taking the terms in Eq. (2.42), which

contain both χ

and η

, and replacing η

by Eq. (2.44). If we do this for the term

†

of Eq. (2.42), we get

†

¯σ

i∂

(2.45)

which we recognize as the kinetic term of the left-chiral spinor that appears in the

Dirac lagrangian [Eq. (2.28)]. A second possibility is the term η

†

, which, after

replacing η

by Eq. (2.44), gives



¯σ

i∂



†

=−i



∂

†



¯σ

(2.46)

CHAPTER 2 A Crash Course on Weyl Spinors

where we have used the fact that ( ¯σ

)

†

= ¯σ

because the Pauli matrices are hermi-

tian. As part of a lagrangian density, however, an integration by parts can be used

to show that the term in Eq. (2.46) is equivalent to Eq. (2.45) after discarding a

surface term.

We will frequently use integrations by parts on expressions appearing in la-

grangian (lagrangian densities, to be exact). We will always assume that the surface

terms vanish so that we may write



xA∂

B =−



x (∂

A)B

where A and B are arbitrary expressions containing quantum ﬁelds. This then shows

clearly that the right-hand side of Eq. (2.46) is equal to Eq. (2.45).

We also can construct invariants containing two derivatives of a left-chiral ﬁeld

by replacing both η

in the last two terms of Eq. (2.42) by Eq. (2.44). We will

never have to deal with these terms, however, because they are not renormalizable

(an issue that we will address in Chapter 8). But there are plenty of other things

of interest that we can build with spinors: Lorentz scalars, vectors, and tensors!

Exercise 2.5 offers two examples.

EXERCISE 2.5

Build a contravariant vector (i.e., with one upper Lorentz index) and a second rank

contravariant tensor using two left-chiral spinors and no derivatives.

In principle, this is all we need to start studying SUSY! Unfortunately, in practice,

we need to introduce much more notation. However, as mentioned in the Introduc-

tion, the choice has been made to keep the notation as simple as possible until

Chapter 11 in order to keep it from getting in the way of understanding the basic

concepts of SUSY. Nevertheless, we will cheat a little and introduce a tiny bit of ex-

tra notation that will make many equations far less cumbersome. Before doing this,

however, let’s pause to mention a simple identity that will help us countless times.

2.8 A Useful Identity

One of the identities that will be used the most often is a simple modiﬁcation of a

relation familiar from elementary linear algebra.

Consider two plain column vectors v and w whose components are ordinary

complex numbers (“ordinary” in the sense that they are commuting quantities) and

a matrix A whose elements are also complex numbers. Then the following equality

Supersymmetry Demystified

holds:

Aw = w

v (2.47)

This can be checked easily by writing the indices explicitly. This identity follows

from the fact that

Aw = (v

Aw)

which is true because the quantity v

Aw is just a number, so it is trivially equal to

its own transpose. Applying the the transpose to the parenthesis on the right-hand

side, we recover Eq. (2.47).

Of course, if complex conjugates are taken on any of the quantities, they will

appear on both sides of the equation. For example,

†

∗

= w

†

∗

We will not be dealing with vectors with complex numbers as components but

with spinors, whose components are Grassmann quantities, which anticommute. In

that case, we must introduce a minus sign when we switch the order of the spinors

so that Eq. (2.47) is replaced by

A β =−β

α (2.48)

where α and β are arbitrary spinors. Of course, if a hermitian conjugate is applied to

either spinor or to the matrix, we need to take this into consideration. For example,

A β

†T

=−β

†

We will use Eq. (2.48) or its equivalent form with hermitian conjugates applied

to some of the terms repeatedly throughout this book.

2.9 Introducing a New Notation

We have achieved our goal for this chapter, which was to build Lorentz invariants

out of Weyl spinors.

In principle, before moving on, we should discuss the shorthand notation of

dotted and undotted indices, also called the van der Waerden notation, used by

“superphysicists” when they do calculations in SUSY. This notation is very useful

CHAPTER 2 A Crash Course on Weyl Spinors

in that it makes it much easier to identify Lorentz invariant combinations of Weyl

spinors, without the need to remember where to put the iσ

matrices or where

to place hermitian conjugates. Unfortunately, it is also a real pain for beginners

because each spinor now comes in four different versions.

Of course, we could avoid altogether this shorthand notation because it is not

absolutely necessary to do SUSY calculations. But there would be two major draw-

backs to this. One is that it would be more difﬁcult for you to consult other publica-

tions on SUSY if you haven’t been exposed to this notation, which is used widely.

The second problem is that it does make calculations much simpler when we have to

deal with complicated expressions (and expressions do get fairly messy in SUSY!).

However, for the ﬁrst several chapters, our calculations will be simple enough that

using the shorthand notation would not provide a noticeable advantage and would

make the topic harder to learn. On the other hand, starting in Chapter 11, where

we begin working in superspace, calculations would become awkward if we would

not take advantage of the shorthand notation of dotted and undotted indices.

So we have opted for a compromise. The van der Waerden notation is introduced

in Chapter 3 because it is a logical extension of the material covered in Chapter 2,

but it will not be used until Chapter 11 (aside from part of an exercise in Chapter 6).

By the time we reach Chapter 11, the basic concepts and tools of SUSY hopefully

will be sufﬁciently familiar that it should not be too much of a problem to add a

new layer of notation. We would therefore invite the reader to skip Chapter 3 on a

ﬁrst reading and to get back to it before tackling superspace, in Chapter 11.

In the rest of this chapter we will only be introduced to the bare minimum of

extra notation needed until Chapter 11.

In practice, SUSY theorists work almost exclusively with left-chiral spinors,

even when considering supersymmetric extensions of the standard model (using

some tricks that will be explained later), so we really only need to focus on the two

invariants built out of a left-chiral spinor:

†

iσ

†T

(−iσ

)χ

(2.49)

From now on we will drop the subscript L; it will be implicit that χ always

stands for a left-chiral spinor.

We deﬁne two new types of spinor dot products between left-chiral Weyl spinors

as a shorthand notation for the two Lorentz invariants of Eq. (2.49):

χ ·χ ≡ χ

(−iσ

)χ

¯χ · ¯χ ≡ χ

†

iσ

†T

(2.50)