Labelle P. Supersymmetry DeMYSTiFied

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Supersymmetry Demystified

there is actually no problem because Dirac conjugation can only be applied to four-

component spinors, not to Weyl spinors. In this book we will be working almost

exclusively with Weyl spinors, which will always be denoted by lowercase Greek

letters. The few four-component spinors we will need will always be denoted by an

uppercase . So whenever a bar appears above a Weyl spinor, you will know that

it is in the sense of indicating dotted indices, as in Eq. (3.5).

So far we have only introduced three types of indexed quantities: left-chiral

spinors components with lower undotted indices χ

and right-chiral spinors with

upper dotted and upper undotted indices ¯η

˙a

and η

. Of our six Lorentz invariants

listed in Eq. (2.42), the only one we can recover with our new notation is η

†

χ,

which corresponds to

†

χ = ( ¯η

˙a

)

†

= η

(3.6)

after using Eq. (3.4).

Note that the combination ¯η

˙a

is not a Lorentz invariant [if it were, we would

have η

χ listed in Eq. (2.42)]. This is actually a nice feature that will turn out to

be general: in order to obtain a Lorentz invariant, an upper index be matched with

a lower index of the same type, i.e., both indices must be dotted or both must be

undotted.

We clearly need to extend the notation in order to have expressions corresponding

to all six invariants of Eq. (2.42). Consider the third invariant in Eq. (2.42):

†

η = (χ

)

†

¯η

˙a

(3.7)

Remember that what we would like is a notation where an upper index must be

matched with a lower index of the same type to yield a Lorentz invariant, so it is

natural to deﬁne (keeping the tradition that spinors with dotted components have a

bar)

†

η ≡ ¯χ

˙a

¯η

˙a

(3.8)

Comparing Eqs. (3.7) and (3.8), we see that we must deﬁne

¯χ

˙a

≡ (χ

)

†

(3.9)

This is a pleasing result because it has the same basic structure as Eq. (3.5): Taking

the hermitian conjugate changes a dotted index, into an undotted index, and vice

versa, without changing the position of the index.

CHAPTER 3 New Notation for Weyl Spinors

To be explicit, Eq. (3.9) corresponds to

¯χ

= χ

†

¯χ

= χ

†

(3.10)

We now have deﬁned left-chiral spinors with both dotted and undotted lower

indices as well as right-chiral spinors with the two types of upper indices. What

about left-chiral spinors with upper indices or right-chiral spinors with lower in-

dices? To deﬁne those, we could continue with the other invariants in Eq. (2.42),

but it is more instructive instead to go back to the Lorentz transformations given in

Eq. (2.41).

Of course, we want to deﬁne quantities with the same type of indices to have

the same Lorentz transformations, so we should deﬁne, e.g., the components ¯χ

˙a

to transform the same way as the ¯η

˙a

. Looking at Eq. (2.41), we see that it is the

quantity i σ

†T

that transforms the same way as a right-chiral spinor, i.e., with the

matrix A. We therefore deﬁne

¯χ

˙a

≡ components of iσ

†T

In order for the indices to match, we must write

¯χ

˙a

= (iσ

)

˙a

†

(3.11)

or, using Eq. (3.9),

¯χ

˙a

= (iσ

)

˙a

¯χ

(3.12)

where the indices of the iσ

were chosen in order to follow the rules that indices

must be of the same type to be summed over. Note that in Eq. (3.11) we used a

dotted

b index on the Pauli matrix even if the spinor had an undotted index because

of the presence of the hermitian conjugation, which we know changes an undotted

index into a dotted one. However, some authors write that equation using (iσ

)

˙ab

so there is no universally accepted convention in this case.

Don’t be alarmed by the freedom in the type of indices assigned to iσ

;in

contradistinction with spinor indices, the indices on σ

are mostly decorative. We

will come back to this point shortly, but for now, just know that no matter what

indices we are assigning to it, σ

always represents the usual Pauli matrix.

To be explicit, Eq. (3.12) gives

¯χ

= ¯χ

¯χ

=−¯χ

(3.13)

Supersymmetry Demystified

Using Eqs. (3.10) and (3.13), we may express the components with superscripted

dotted indices in terms of the components with subscripted undotted indices:

¯χ

= χ

†

¯χ

=−χ

†

The result [Eq. (3.12)] shows that the matrix iσ

may be used to raise dotted

indices. We have only shown this for left-chiral spinors, but we will soon prove that

this is also valid for right-chiral spinors.

Now that we have deﬁned the components of left-chiral spinors with upper

dotted indices, the obvious next step is to deﬁne the components χ

. From our

earlier discussion, it seems natural to use the deﬁnition

= ( ¯χ

˙a

)

†

Using Eq. (3.12), we ﬁnd



(iσ

)

˙a

¯χ



†

= [(iσ

)

˙a

]

∗

( ¯χ

)

†

where we used the fact that the (iσ

)

˙a

are just numbers, so applying a dagger on

them amounts to a simple complex conjugation. Now we use Eq. (3.9) to get

= [(iσ

)

˙a

]

∗

(χ

†

)

†

= (iσ

)

(3.14)

Since the components of iσ

are real, the complex conjugation did not do anything,

except that we changed the dotted indices into undotted indices in order, once again,

to ensure proper “index etiquette.” As promised earlier, we will come back to the

vexing question of the mutating indices of iσ

shortly.

Note that Eq. (3.14) implies that the matrix i σ

may be used to raise undotted

indices too.

If the derivation of Eq. (3.14) does not satisfy you, we may repeat it in “matrix

language” without explicitly showing the indices. We start with

χ = (iσ

¯χ)

†T

where now we have to specify that the χ on the left has an upper index and the

¯χ has a lower index (we don’t have to specify whether the indices are dotted or

CHAPTER 3 New Notation for Weyl Spinors

undotted because the presence or absence of a bar tells us that). We then get

χ = (iσ

¯χ)

†T

= ( ¯χ

†

(iσ

)

†

)

= (χ

(−iσ

))

= (−iσ

)

= i σ

It must be understood that the χ on the right has a lower index. This agrees with

Eq. (3.14) if we reinstate the indices. The result [Eq. (3.14)] is important enough

to deserve its own little box:

= (iσ

)

(3.15)

To be explicit,

= χ

=−χ

(3.16)

Now that we know how to ﬁnd the components of left-chiral spinors with upper

indices in terms of the components with lower indices using Eqs. (3.12) and (3.15),

we may invert those two relations to express components with lower indices in

terms of components with upper indices. All we need to use is the fact that the

inverse of the matrix iσ

is −iσ

, which is obvious if we recall that (σ

)

= 1.

Imposing the indices in the equations to match properly ﬁxes uniquely the indices

of −iσ

, giving

¯χ

˙a

= (−iσ

)

˙a

¯χ

and

= (−iσ

)

Therefore, the matrix −iσ

is used to lower both dotted and undotted indices.

We have only shown this for left-chiral spinors, but we will soon show that it is also

valid for right-chiral spinors.

So far we have only deﬁned right-chiral spinors with upper indices. If we impose

that the η

transform like the χ

and look at Eq. (2.41), we see that the quantity that

Supersymmetry Demystified

transforms like χ is −iσ

†T

. Thus we deﬁne

≡ (−iσ

)

(η

)

†

= (−iσ

)

As promised, the matrix −iσ

is seen to also lower undotted indices on right-chiral

spinors. Deﬁning ¯η

˙a

≡ η

†

and following the same steps as in Eq. (3.14), we also

can show that

¯η

˙a

= (−iσ

)

˙a

¯η

(3.17)

which shows that −iσ

may be used to lower dotted indices of right-chiral spinors

as well.

Let us pause to comment on the “chameleon indices” of iσ

that change to

whatever the equations require. This is something that may be quite confusing

at ﬁrst. Note that when we write spinors with different types of indices, e.g., η

and η

˙a

, we really do mean different quantities. However, it is important to keep

in mind that in all our equations, the components of the σ

matrix are always the

usual components, no matter if the indices are dotted or undotted, no matter if they

are upstairs or downstairs. For example, the following components are equal:

(iσ

)

= (iσ

)



−10



= 1

whereas

(−iσ

)

= (−iσ

)



0 −1



=−1

It may be confusing to have the matrices with upper and lower indices differing

only by a minus sign. If we see the components (iσ

)

in an equation, what prevents

us from rewriting this as −(−iσ

)

? The key point is that we will never deal with

matrices in isolation; they will always be part of equations containing spinors

and other quantities with well-deﬁned Lorentz transformations, and consistency

of the equations under those transformations will always clearly dictate what the

correct indices are.

Let us summarize what we have accomplished so far. We have deﬁned four types

of components for both left- and right-chiral spinors. These are all related to one

another following three simple rules:

CHAPTER 3 New Notation for Weyl Spinors

1. Taking a hermitian conjugate switches dotted and undotted indices without

changing their position (and of course adds or removes a bar since a bar always

accompanies dotted indices).

2. The matrix iσ

may be used to raise indices (both dotted and undotted).

3. The matrix −iσ

may be used to lower indices (both dotted and undotted).

One last comment before moving on to the topic of Lorentz invariants: It is

important to keep in mind that all four sets of components always may be written

in terms of a single set and its hermitian conjugate. For example, all types of

components of a left-chiral spinor may be written in terms of χ

and χ

and their

hermitian conjugates χ

†

and χ

†

, as we have seen explicitly in this section. So

despite the fact that we have deﬁned eight different components associated to a

single left-chiral spinor, only four of them are independent.

3.1 Building Lorentz Invariants

Now let us use our new notation to build Lorentz invariants. As hinted before, the

rule to obtain invariants turns out to simply be that an upper index must be matched

with a lower index of the same type. We will show that this is the correct rule by

recovering all the invariants of Eq. (2.42) this way and nothing else.

Following this rule, there are only eight combinations made out of one left-chiral

and one right-chiral spinor that can be written:

¯η

˙a

¯χ

˙a

¯χ

˙a

¯η

˙a

¯χ

˙a

¯χ

˙a

¯η

˙a

¯η

˙a

(3.18)

Of course, reversing the order of any of the spinors does not give a new invariant.

For example, χ

is simply equal to −η

. In our list (3.18) we chose to list the

quantities with undotted indices going diagonally down and dotted indices going

diagonally up for a reason that will become clear later on. This does not mean

that the quantities with the opposite order are not invariants, just that they are not

independent.

Still, we seem to get eight invariants, whereas we had found only six in Chapter

2, so our eight expressions must not all be independent. This becomes evident if

we recall that we may always write components with upper indices in terms of

Supersymmetry Demystified

components with lower indices (without taking a hermitian conjugate) so that, for

example, η

and χ

must somehow be related. Indeed, we have



(iσ

)



=−(iσ

)

= (iσ

)

= χ

(3.19)

where the minus sign in the second line came from moving the η

to the right of χ

In the third line, we have used the antisymmetry of iσ

to write (iσ

)

=−(iσ

)

and in the last line, we have used the iσ

to raise the index of χ. To summarize, we

have shown that

= χ

(3.20)

EXERCISE 3.1

If the proof in Eq. (3.19) does not convince you, express explicitly η

in terms

of the components with lower indices only (χ

,χ

,η

, and η

). Do the same with

, and show that the two expressions are equal.

Likewise, one can show that

¯η

˙a

¯χ

˙a

= ¯χ

˙a

¯η

˙a

(3.21)

Therefore, we only have six independent invariants in the list in Eq. (3.18), which

we will take to be

¯χ

˙a

¯η

˙a

¯χ

˙a

¯χ

˙a

¯η

˙a

¯η

˙a

(3.22)

It is easy to show that these six invariants correspond indeed to the invariants of

Eq. (2.42). The only tricky point is that we must remember that in all the formulas

of Chapter 2 the left-chiral spinor χ is always written in terms of its components

with lower undotted indices χ

, whereas the right-chiral spinor η is always written

in terms of its components with upper dotted indices η

˙a

. The claim then is that if we

express all the spinors in Eq. (3.22) in terms of these components, we will recover

the six expressions of Eq. (2.42).

CHAPTER 3 New Notation for Weyl Spinors

For example, consider

(3.23)

We need to express χ

in terms of lower indices in order to compare it with

Eq. (2.42). We know that the matrix iσ

raises indices, so we write



(iσ

)



= χ

(iσ

)

(3.24)

where we have used the freedom to move the (iσ

)

around because they are

simply numbers. Now, in order to compare with Eq. (2.42), we need to write Eq.

(3.24) in matrix form, with the σ

matrix sandwiched between the two spinors. To

do this, we have to switch the order of the indices of iσ

, which brings in a minus

sign because of the antisymmetry of iσ

. Thus we get

= χ

(−iσ

)

= χ

(−iσ

)χ

where in the last line it is understood that the indices on the spinors are all lower

undotted indices. Sure enough, this is the ﬁrst Lorentz invariant in Eq. (2.42). The

other invariants can be obtained in a similar manner.

EXERCISE 3.2

Show that ¯η

˙a

¯η

˙a

corresponds to one of the invariants of Eq. (2.42).

EXERCISE 3.3

At some point we will need the relations equivalent to Eqs. (2.54) and (2.55) but

written in terms of upper indices. Prove the two identities

=−

(iσ

)

χ ·χ

¯χ

˙a

¯χ

(iσ

)

˙a

¯χ · ¯χ (3.25)

3.2 Index-Free Notation

In order to alleviate the notation, it is useful to be able to write expressions without

explicit indices by writing, e.g., ηχ instead of η

or ¯χ ¯η instead of ¯χ

˙a

¯η

˙a

. This

is what most SUSY references do, but in this book we will instead use the “spinor

Supersymmetry Demystified

dot product notation” introduced in Chapter 2; i.e., we will write η · χ and ¯χ · ¯η.

This makes complex expressions easier to read by showing clearly which spinors

are contracted together (e.g., it’s easier to see which spinors are contracted together

in χ ·ηλ · χ than in χηλχ).

However, not showing explicit indices obviously leads to an ambiguity. We can

distinguish spinors with dotted indices from those with undotted indices because

the former have a bar over them, but once we drop indices, we can’t tell if they are

supposed to be in the upper or in the lower position. Consider, for example, η ·χ.

Does this represent η

or η

? These two are not equivalent! They differ by a

minus sign, as is easily shown:

=−χ

=−η

where in the ﬁrst step we have used the Grassmann nature of spin components and

in the second step we made use of Eq. (3.20).

The same ambiguity arises if we drop the indices of any of the six invariants of

Eq. (3.22). We therefore see that if we want to be able to not write explicit indices,

we need a new rule to tell us when indices are in the upper position and when they

are in the lower position.

The convention that has been chosen is that a dot product between unbarred

spinors corresponds to the undotted indices written diagonally downward; i.e.,

η · χ always stands for η

. This convention deserves to be emphasized:

The dot product between two unbarred spinors corresponds to the product of

the components with the undotted indices going diagonally downward

What do we do if we want to write η

as a dot product? We simply need to

change the order ﬁrst:

=−χ

=−χ ·η

As for barred spinors, the convention is the reverse: A dot product between

barred spinors corresponds to the dotted indices written diagonally upward, i.e.,

CHAPTER 3 New Notation for Weyl Spinors

¯η · ¯χ always stands for ¯η

˙a

¯χ

˙a

. Again, let us emphasize this important convention:

The dot product between two barred spinors corresponds to the product

of the components with the dotted indices going diagonally upward

It may seem confusing (and annoying) at ﬁrst that the convention for the barred

and unbarred spinors are reversed, but one way to see that it is natural is to remember

that we have deﬁned the original components of the left-chiral spinor χ as being

and the original components of the right-chiral η as being ¯η

˙a

. Therefore, in both

conventions stated above, the original component of the spinors appears on the

right of the spinor dot product. In that sense, the two conventions are consistent

with each other.

It is easy to check that the dot products we just introduced in terms of components

with indices are in fact exactly equal to the dot products introduced in Eq. (2.50).

For example, consider η · χ . Using the notation of upper and lower indices, we

have

η · χ = η

= η

+ η

On the other hand, in the notation of Chapter 2,

η · χ = η

(−iσ

)χ = η

− η

Using Eq. (3.16), we see that these two expressions are indeed equal.

Now recall the identities (3.21) and (3.20). Translated into dot-product notation,

these become

η · χ = χ ·η

¯η · ¯χ = ¯χ · ¯η (3.26)

which we already mentioned in Eq. (2.60) and which you proved in Exercise 2.6.

The fact that the order does not matter in a dot product is familiar from relativity

or even from three-dimensional vector algebra, where



A ·



B =



B ·



A. But keep

in mind that here the identities (3.26) are more subtle. There is one minus sign

introduced by the exchange of the two spinor components, which are Grassmann

quantities, and a second minus sign owing to the switch from upper to lower indices

on both spinors (or, if you prefer, owing to the antisymmetry of the matrix iσ

The end result is that we can switch the order of the spinors in those dot products

without introducing a minus sign, a result that will come in handy many times.