Labelle P. Supersymmetry DeMYSTiFied

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Supersymmetry Demystified

incorrect; we can easily write a mass term for a single left-chiral ﬁeld. Actually, we

have already done it in Eq. (4.24)! So the lagrangian for a massive left-chiral Weyl

fermion is simply

= χ

†

¯σ

i∂

−

(χ

· χ

+ ¯χ

· ¯χ

) (4.35)

We see that Eq. (4.35) is exactly the same as the Majorana lagrangian expressed

in two-component notation given in Eq. (4.24)! There is no physical difference at all

between the theory of a massive Weyl fermion and the theory of a Majorana particle.

The only difference is a technical one: The Majorana theory is usually written in

terms of the four-component Majorana spinor, as in Eq. (4.21), whereas the massive

Weyl theory is written in terms of the two-component spinor, as in Eq. (4.35). In

other words, we could say that the Majorana theory is simply the four-component

version of a massive Weyl fermion (note that the degrees of freedom match; both

theories contain two observables states).

It’s because of this equivalence that sometimes people refer to Eq. (4.35) as the

lagrangian of a Majorana spinor, not of a massive Weyl spinor, keeping the term

Weyl spinor for massless spinors. But, as mentioned earlier, in this book we choose

to deﬁne Weyl spinors through their Lorentz transformations (2.32), which hold

regardless of the mass of the particles. We will reserve the expression Majorana

spinor for the four-component spinor satisfying the condition of Eq. (4.7).

Let us close this section with an important remark. In the last two sections we

chose to express everything in terms of the left-chiral state χ and its hermitian

conjugate χ

†

. It is, of course, possible to build Weyl spinors and Majorana spinors

out of η and η

†

instead, but it is conventional in SUSY to express all fermions in

terms of left-chiral states (a choice most likely inﬂuenced by the habit of working

with the standard model, in which there are no right-chiral neutrino states). This

is why we focused on left-chiral spinors and did not include the corresponding

expressions and identities for right-chiral spinors (which, however, can be worked

out easily with the information that has been provided).

4.11 Relation Between Weyl Spinors

and Majorana Spinors

Because of the equivalence between the Majorana and Weyl descriptions, we can

always write any expression containing a Weyl spinor as one containing a Majorana

spinor and vice versa. Indeed, all supersymmetry can be couched in either Weyl

or Majorana language, and many references use only one of the two. Which one

CHAPTER 4 The Physics of Spinors

is preferred is obviously a matter of taste, but it could be argued that Weyl spinors

are more fundamental because they are the simplest building blocks that can be

used to construct supersymmetric theories. The main reason that Majorana spinors

nevertheless are used widely is probably because calculations in that language

resemble more those made with the familiar Dirac spinors, especially when it

comes to working out scattering amplitudes. In this book the emphasis has been

put on Weyl spinors, but the main results also will be expressed in the Majorana

language, and some calculations of scattering amplitudes will be performed using

the Majorana language.

In this section we will establish some basic relations between Weyl and Majorana

spinors. Let’s start with lagrangian terms without derivatives. We have already seen

in Section 4.7 that



= χ ·χ + ¯χ · ¯χ (4.36)

(we may drop the label p without causing any ambiguity because there is no

distinction between the antiparticle and particle states for a Majorana fermion).

Keep in mind that the bar over a Majorana spinor denotes the usual Dirac adjoint,



= 

†

Obviously, we need a second relation if we want to be able to isolate χ ·χ and

¯χ · ¯χ in terms of the Majorana spinor. What we are looking for is provided by

sandwiching γ

between two Majorana spinors. It is easy to check that



=−χ ·χ + ¯χ · ¯χ (4.37)

Inverting Eqs. (4.36) and (4.37), we obtain

χ ·χ =



¯χ · ¯χ = 



(4.38)

where the projection operators are deﬁned in Eq. (2.18). Actually, it is straightfor-

ward to generalize these relations to products involving distinct spinors. Consider

a second left-chiral Weyl spinor λ. Let’s represent the Majorana spinor built out of

λ as 

. Then the generalizations of Eq. (4.38) are

χ ·λ =





¯χ ·

λ = 



Recall that the order in spinor dot products does not matter; i.e., χ · λ = λ · χ and

¯χ ·

λ =

λ · ¯χ. This implies that the order in the corresponding Majorana expressions

Supersymmetry Demystified

does not matter either, i.e.,





= 





= 



which can be checked easily.

The next obvious thing to try is to sandwich γ

and γ

between Majorana

spinors. One can show that





= χ

†

¯σ

λ − λ

†

¯σ





= χ

†

¯σ

λ + λ

†

¯σ

which imply

†

¯σ

λ = 



†

¯σ

χ =−



Note that no σ

matrices appear here. These would appear if we were expressing

all the spinors in terms of right-chiral states.

Of course, all these relations also can be veriﬁed explicitly by using Eq. (4.20)

for a Majorana spinor in terms of Weyl spinors.

4.12 Quiz

1. What is the CPT conjugate state of a right-chiral particle?

2. What is the difference between a Dirac mass term and a Majorana mass term

(after they are both expressed in terms of Weyl spinors)?

3. What is the difference between the theory of a Majorana spinor and the theory

of a Weyl spinor?

4. Is it possible to write a Dirac spinor in terms of left-chiral ﬁelds only? What

about in terms of right-chiral ﬁelds only?

5. Write down the mass term of a left-chiral Weyl spinor. Prove that it is real.

6. What symmetry, is a symmetry of a Dirac spinor but not a symmetry of a Majo-

rana spinor (in the sense that a Dirac spinor has a well-deﬁned transformation

under this symmetry, whereas a Majorana spinor doesn’t)?

CHAPTER 5

Building the Simplest

Supersymmetric

Lagrangian

We are ﬁnally ready to construct our ﬁrst supersymmetric theory! This is where the

fun really begins. It took a while to get here, but it was necessary to become familiar

with Weyl and Majorana spinors and the notation used to describe them before we

could attack supersymmetry (SUSY). This chapter will be short and sweet. The

lagrangian considered truly will be the simplest one we could possibly imagine:

There will be no mass terms and no interactions. Still, it will be sufﬁcient to learn

the essential concepts and to see SUSY in action (no pun intended).

Supersymmetry Demystified

5.1 Dimensional Analysis

Before building our ﬁrst supersymmetric lagrangian, it will be useful to review how

one works out the dimensions of the various quantities we will need. In natural units,

both the speed of light and Planck’s constant are set equal to 1. This implies that

dimensions of time and length are equal, i.e., [T ] = [L]. It also implies that mass

and energy have the same dimensions because E = mc

becomes E = m in natural

units. In addition, since Planck’s constant had dimensions of energy multiplied by

time, setting it equal to 1 tells us that [T ] = [L] = [E]

−1

. We now can express the

units of everything as powers of energy. Instead of writing [E]

for some exponent

n, it is simpler to give the exponent. We therefore have the following dimensions

for the fundamental scales of time, length, and mass:

[L] = [T ] =−1[M] = 1

Note that this implies that the dimension of a derivative ∂

is 1.

The action S =−i (



xL) is a dimensionless quantity which implies that

lagrangian densities (which we often simply call lagrangians) must have dimen-

sion 4. Using this, we can work out the dimensions of the ﬁelds we will deal with.

For example, the kinetic energy of a scalar ﬁeld is given by

∂

φ∂

φ (5.1)

which tells us that a scalar quantum ﬁeld has dimension 1 (because [∂

] = 1). The

quantum electrodynamic (QED) lagrangian density is given by

QED

(P

− m) −

μν

which tells us that a spinor quantum ﬁeld has dimension 3/2. We worked out the

dimension of a Dirac ﬁeld but of course Weyl and Majorana quantum ﬁelds have the

same dimension. We also see that the energy-momentum tensor F

μν

has dimension

2. Since it is deﬁned as

μν

= ∂

− ∂

we also ﬁnd that the gauge ﬁeld A

has dimension 1. The result is the same for

nonabelian gauge ﬁelds such as the gluon ﬁelds.

CHAPTER 5 Building the Langrangian

Let us summarize our results:

[∂

] = [φ] = [A

] = 1

[

] = [

] = [χ] = [η] =

(5.2)

5.2 The Transformation of the Fields

The simplest supersymmetric model we may consider is composed of two free

massless ﬁelds, with one bosonic and the other one fermionic. We therefore consider

a Weyl spinor and a complex scalar ﬁeld. By free, we mean that there is no interaction

between the two ﬁelds. We could take the Weyl spinor to be either right- or left-

chiral, but the convention is to use a left-chiral ﬁeld. Our starting point therefore is

the lagrangian

L = ∂

φ∂

†

+ χ

†

i ¯σ

∂

χ (5.3)

Now we are ﬁnally ready to introduce the SUSY transformations of the ﬁelds!

At this point, we could simply throw at you the transformations of the ﬁelds, and

we could check that they leave the lagrangian invariant (up to total derivatives).

However, the transformations are quite complicated, and it’s not obvious why any-

one would have chosen these particular expressions in the ﬁrst place. Since the

goal of this book is to demystify SUSY, we will devote a fair part of this chapter to

understanding the logic behind the choice of the supersymmetric transformations.

Consider ﬁrst the transformation of the scalar ﬁeld. We consider an inﬁnitesimal

transformation proportional to a small parameter ζ . Again, there is no standard

notation for this parameter, so always check ﬁrst the convention used when looking

at a new reference. It might seem that  would be a better choice of notation, but as

we saw in Section 3.4, this letter is also used in most SUSY references to represent

the matrices ±iσ

In the following calculations, always keep in mind that ζ is inﬁnitesimal, so we

will drop terms of order ζ

or higher in all our equations. Another important point

is that we take ζ to be spacetime-independent; i.e.,

∂

ζ = 0

a relation we will use repeatedly. We are therefore considering global SUSY trans-

formations, as opposed to local transformations, for which ζ would be spacetime-

dependent (another expression commonly used in the literature for global SUSY is

Supersymmetry Demystified

rigid supersymmetry). Making ζ spacetime-dependent—in other words, gauging

SUSY—forces one to introduce a gauge ﬁeld that turns out to have the properties

of a graviton (the force carrier of gravity), the same way that gauging the global

U (1) symmetry of the Dirac lagrangian leads to electromagnetism. For this rea-

son, theories with local SUSY invariance are called supergravity theories (SUGRA

for short). However, supergravity necessitates familiarity with general relativity, in

particular the tetrad formalism, and is beyond the scope of this book.

We postulate that the variation of the scalar ﬁeld is proportional to the Weyl

spinor χ (this is, after all, the whole idea of SUSY: Bosonic ﬁelds, are transformed

into fermionic ﬁelds, and vice versa). Let us then write, tentatively,

φ → φ



= φ + δφ (5.4)

with

δφ  ζχ

We used the symbol  because this relation is still qualitative. We need a bit more

work to make this precise.

When we write down transformations of ﬁelds, we must ensure two things: the

two sides of the equation must have the same dimension, and they must have the

same Lorentz transformations. Let’s consider the second requirement ﬁrst. Since φ

is a scalar ﬁeld, we must somehow build a Lorentz invariant out of ζ and χ. Since

χ is a Weyl spinor, it is clear that we have no choice other than taking ζ to be a

Weyl spinor too! This is a key result, so let us emphasize it:

The parameter of a global SUSY transformation is a (constant) Weyl spinor

What does this mean? It means that ζ is a two-component quantity, namely,

ζ =





that takes different values in different frames. We must now decide whether we

should take ζ to be a right-chiral or a left-chiral spinor. We do have the choice,

but again, the convention is to take it to be left-chiral. This tells us that if we do

a Lorentz transformation, ζ transform as χ

in Eq. (2.32). Note that ζ is not a

quantum ﬁeld, even though it is a spinor, so there is no particle associated to it.

Going back to the transformation of φ, we therefore must write down a Lorentz

invariant made out of the two left-chiral spinors ζ and χ . But we already know

how to do this; we simply take the spinor dot product ζ · χ ! Recall that the order

does not matter, so we could as well have used χ · ζ . We can ﬁnally write down the

CHAPTER 5 Building the Langrangian

precise expression for the transformation of φ:

δφ = ζ · χ (5.5)

Now let us turn to the question of the dimensions. Since φ has dimension 1 and

χ has dimension 3/2, we obtain that

The inﬁnitesimal SUSY parameter ζ has dimension −

Let’s now consider the transformation of χ . We want it to be linear in ζ and

φ or φ

†

(it would be extremely difﬁcult to build a consistent theory with ﬁelds

transforming nonlinearly). We will use φ

†

for a reason that will become clear in an

instant. So we take

δχ  C ζφ

†

(5.6)

where C is an arbitrary constant that we will have to determine. The reason we did

not include any such constant in the transformation of φ is that it could have been

reabsorbed into the deﬁnition of ζ . However, once the normalization of ζ is ﬁxed

by the transformation of φ, we have no freedom left to get rid of the constant in the

transformation of χ, and we will indeed see that C is not equal to 1.

Before going any further, we need to check whether the right-hand side of Eq.

(5.6) has the same Lorentz transformation properties as the ﬁeld χ (since the vari-

ation δχ must, of course, transform the same way as χ itself). In this respect,

everything is ﬁne: Since φ is a scalar, the right-hand side of Eq. (5.6) transforms

like a left-chiral spinor because ζ is left-chiral.

But there is a problem: The left side has dimension 3/2, whereas the right side

has dimension 1 −1/2 = 1/2. So we need to increase the dimension of the right

side by one. We do not want to put in another factor of φ because we want the

transformation to be linear in the ﬁelds. The only other quantity with dimension 1

at our disposal is ∂

,sowetry

δχ  C ζ∂

†

Now the dimensions are right, but the Lorentz properties do not match! We need

to contract the μ index with something, and it has to be something dimensionless.

The only possible terms ﬁtting the bill at our disposal are σ

and ¯σ

. Looking at

Eq. (2.44), we see that when applied to a left-chiral spinor, it is the matrices ¯σ

that have well-deﬁned Lorentz transformations. We therefore try

δχ  C ¯σ

ζ∂

†

Supersymmetry Demystified

but unfortunately, this still doesn’t work! To see the problem and how to ﬁx it, let’s

move ∂

†

in front (which we can do because it’s not a matrix),

δχ  C (∂

†

) ¯σ

ζ (5.7)

Recall that the quantity ¯σ

∂

transforms as a right-chiral spinor [see Eq. (2.44)].

The fact that the derivative is acting on a scalar ﬁeld instead of the spinor does not,

obviously, affect the behavior of the expression under a Lorentz transformation, so

(∂

†

) ¯σ

ζ also transforms as a right-chiral spinor!

Thus the right-hand side of Eq. (5.7) transforms as a right-chiral spinor, whereas

what we need is something that transforms as a left-chiral spinor. From Chapter 2,

however, we know how to do make something out of a right-chiral spinor that will

transform as a left-chiral spinor! The rule is to take the hermitian conjugate and

transpose of the right-chiral spinor and then multiply it by −iσ

[see Eq. (2.40)].

Applying this to the ﬁrst line of Eq. (5.7), we ﬁnally take the transformation of

χ to be:

δχ =−iσ



C (∂

†

) ¯σ



†T

This has the right dimension and also transforms as a left-chiral spinor.

Recall that the operation † applies a hermitian conjugate both in the quantum

ﬁeld operator sense and in the matrix sense (a transpose followed by complex

conjugation). Obviously, the transpose has no effect on the scalar ﬁeld, so

(φ

†

)

†T

= (φ

†

)

†

= φ

The only reason we chose to use φ

†

in our initial guess for the transformation of the

spinor was so that the ﬁnal expression would contain φ. Of course, this is purely a

matter of taste; there would have been nothing wrong with having φ

†

in the ﬁnal

result instead!

On the other hand, the effect of †T on anything that is not a quantum ﬁeld,

including the spinor ζ , is to simply apply a complex conjugation. We therefore get

δχ =−iσ

∗

(∂

φ) ¯σ

μ∗

∗

=−C

∗

(∂

φ)iσ

¯σ

μ∗

∗

where

∗



∗



CHAPTER 5 Building the Langrangian

Using the fact that ¯σ

μ∗

= ¯σ

μT

and (σ

)

= 1, we may write

−C

∗

(∂

φ)iσ

¯σ

μ∗

∗

=−C

∗

(∂

φ)iσ

¯σ

μT

∗

which, after using Eq. (A.8), gives our ﬁnal result for the transformation of the

spinor:

δχ =−C

∗

(∂

φ) σ

iσ

∗

(5.8)

5.3 Transformation of the Lagrangian

We haven’t even really started yet! All we have done is to write down transforma-

tions of the ﬁelds that have the correct dimensions and correct behavior under the

Lorentz group. What we want to do now is to see if there is a choice of constant C

such that the preceding transformations will leave the action invariant. Recall that

the action is the integral of the lagrangian density:

S =



The action is invariant if δS = 0, which implies



xδL = 0

This does not imply that δL itself must be zero. If we assume that the ﬁelds van-

ish sufﬁciently rapidly at spatial inﬁnity, the action will be invariant if either the

variation of the lagrangian density is zero or if it is a total derivative of the ﬁelds:

δL = ∂

F(φ,χ,...)

where F is some function of the ﬁelds, and the dots are included because later on

we will have other ﬁelds in our lagrangians.

Now let us build our ﬁrst supersymmetric theory! Again, the goal is to see if we

can ﬁnd a value of the constant C such that the variation of the lagrangian density

will be zero after dropping any total derivative. The variation of the lagrangian