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156

Supersymmetry Demystified

7.9 The Coleman-Mandula No-Go Theorem

A book on SUSY would be incomplete if it did not mention one of the most famous

no-go theorems of particle physics. Now that we have worked a bit with charges

and their algebra, the theorem will be easier to understand.

Since the discovery of relativity, it has been clear that particle states should belong

to representations of the Poincar´e group. On the other hand, it is natural to also group

particles into multiplets of some additional “internal” symmetry group, and this

viewpoint, of course, has been very successful in the construction of gauge theories.

An obvious question to ask is whether such a symmetry group actually could “mix”

nontrivially with the Poincar´e group. To make this rather vague statement more

precise, the question is whether the generators of the internal symmetry group

could have nonzero commutators with the generators of the Poincar´e group.

Let us write the charges (or generators) of the internal symmetry group as T

and their algebra as

, T

] = if

abc

(7.61)

The question is whether it is possible to write down some physically meaningful

theory where the charges T

do not commute with the generators of the Poincar´e

group M

μν

and P

(another way to put it is to say that the T

should carry some

Lorentz index). This is an interesting question because such a theory then would

mix spacetime transformations with “internal” transformations and would lead to

a highly nontrivial extension of gauge theories.

What Coleman and Mandula proved in 1967

(a derivation also can be found

) was that for an algebra of the form of Eq. (7.61) and under a few very reasonable

conditions, e.g., that elastic scattering amplitudes must be analytic functions of the

Mandelstam variables and that plane waves must in general scatter, the answer is a

resounding no! In other words, the generators T

cannot carry spacetime indices.

Exercise 7.7 illustrates the theorem in the simple case of a symmetric second-rank

tensor charge S

μν

= S

νμ

EXERCISE 7.7

Consider a symmetric tensor S

μν

that is conserved. Then its eigenvalue will be

conserved in a collision. Consider the scattering process a + b → a + b. Denote

the four-momenta of the particles p

, p

, and p

. Show that the conservation

of the eigenvalue of S

μν

together with the conservation of total momentum implies

that either there is no scattering at all, p

= p

and p

= p

, or that the particles

exchange their momenta, p

= p

and p

= p

. No other direction of scattering

is possible! This is clearly unphysical.

CHAPTER 7 Applications of the SUSY Algebra

157

Hint: Write the most general eigenvalue for this operator for a one-particle state.

Then use that the corresponding eigenvalue for a two-particle state is simply the

sum of the one-particle state eigenvalues.

This seemed to be the kiss of death for the idea of building theories mixing

spacetime transformations and internal symmetries nontrivially. One might think

that physicists would have been eager to ﬁnd some way to circumvent this no-go

theorem as soon as it came out, but this was not how SUSY was ﬁrst discovered.

We won’t get into the history of SUSY here; interested readers are invited to read

the ﬁrst chapter of Ref. 47 for a very informative historical summary.

Of course, SUSY does circumvent the theorem in a very simple way: The su-

percharges do not obey a set of commutation relations as in Eq. (7.61) but instead

obey anticommutation relations. Another way to carry the same information is that

supercharges transform as a spinor representation of the Lorentz group instead of

transforming as vector (or more generally, tensor) representations.

The theorem is still useful because it tells us that no other symmetry with charges

obeying commutation rules could mix nontrivially with spacetime transformations.

But are the supercharge anticommutators we wrote down the most general possible

ones? The next section addresses this question.

7.10 The Haag-Lopuszanski-Sohnius

Theorem and Extended SUSY

The discovery of SUSY raised another obvious question: Is the algebra we wrote

down the most general algebra that evades Coleman and Mandula’s theorem? The

answer was provided by Haag, Lopuszanski, and Sohnius in 1975,

and the answer

again was a resounding no. Not only did they show that other possibilities exist,

but they also worked out the most general algebra possible.

To write it down, we ﬁrst introduce a set of N copies of the supercharges, i.e.,

, Q

†I

, Q

†I

where I is an extra label ranging from 1 to N . The SUSY we have considered

so far in this book corresponds to N = 1 and therefore is referred to as N = 1

supersymmetry. The most general algebra satisﬁed by these charges is



, Q



= (−iσ

)



, Q

J †



= (σ

)



I †

, Q

J †



= (−iσ

)

∗

(7.62)

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Supersymmetry Demystified

where the Z

are called the central charges of the algebra (a central charge refers

to a quantity which commutes with all the generators of an algebra) and always can

be made antisymmetric, Z

=−Z

. In the case of N = 1 supersymmetry, the

unique central charge Z

is therefore necessarily zero, and the algebra Eq. (7.62)

reduces to what we obtained earlier in this chapter.

Extended SUSY is beyond the scope of this book, but one important point to make

is that from a purely phenomenological point of view, only N = 1 is attractive, at

least in theories describing physics at the low energies that will be accessible to the

Large Hadron Collider (LHC). The reason is that starting with N = 2, supersym-

metric theories are necessarily nonchiral; i.e., they must treat left- and right-chiral

states on equal footing. For N = 2, the chiral supermultiplet contains both a right-

and a left-chiral fermion, with h = 1/2 and h =−1/2 (as well as two states with

h = 0). But the members of a supermultiplet all transform the same way under a

gauge symmetry, so for N = 2 (and any larger value of N as well), the two chiral

states must be treated on an equal footing by any gauge force. But this conﬂicts

with what we know about weak force of the standard model, which treats differently

left- and right-chiral states; hence only N = 1 SUSY is relevant to phenomenology.

This is true if we think of the MSSM as some effective ﬁeld theory to be used with

energies on the order of a few TeV. A more fundamental theory could have a very

different structure, and who knows how symmetry may be broken down to what

we observe at small energies relative to the Planck scale. But even if we set aside

the issue of chirality, other problems arise, if we increase too much the value of N

because it does not seem possible to write down consistent quantum ﬁeld theories

containing ﬁelds with spin larger than 2.

More details on extended SUSY may be found in Refs. 8 and 47.

7.11 Quiz

1. What is the effect of the supercharge Q

on the four-momentum and helicity

of a state?

2. For N = 1, how many states are required to form a SUSY multiplet?

3. What is extended SUSY?

4. What can we say about the vacuum expectation value of the hamiltonian in a

supersymmetric theory?

5. How does SUSY evade the Coleman-Mandula no-go theorem?

CHAPTER 8

Adding Interactions:

The Wess-Zumino Model

So far we have considered a rather uninteresting theory: All the ﬁelds are massless,

and even more boring, there are no interactions at all. What we would like to do now

is to see if we can include masses and interactions that will preserve supersymmetry

(SUSY). The answer is obviously yes because SUSY would never have become

such a hot topic if interactions could not be included. However, as you surely expect,

this will prove to be a highly nontrivial undertaking.

In writing down possible interaction terms, we will follow certain rules:

•

We will not consider terms of dimension above 4. This requires some expla-

nation. Of course, any term in a lagrangian density is of dimension 4. What

we mean here is that if we ignore all the constants, the dimension must not

be more than 4.

Consider for example the four-fermion interaction χ ·χ ¯χ · ¯χ. Since the

dimension of fermion ﬁelds is 3/2, this term has dimension 6 and must be

160

Supersymmetry Demystified

rejected. Note that if we insisted on including this term in our lagrangian

density, we would have to divide it by a constant having the dimension of

energy squared:

int

χ ·χ ¯χ · ¯χ (8.1)

where μ would be a free parameter with dimension of energy. Therefore,

another way of saying that we reject terms of dimension more than 4 is that

we will not consider any term in the lagrangian that contains a dimensionful

constant in the denominator.

The reason for this condition is that any interaction of dimension 5 or higher

leads to a nonrenormalizable theory. There is nothing intrinsically wrong with

nonrenormalizable theories if they are treated as effective ﬁeld theories, but

we won’t consider these in this book.

•

All the terms we add must, of course, be Lorentz-invariant. This obviously

implies that any upper (contravariant) Lorentz index must be contracted with

a corresponding lower (covariant) Lorentz index. But this is not all. We also

must ensure that spinors are combined into Lorentz-invariant combinations.

This is where all our work of Chapter 2 will come in handy.

•

The lagrangian must be real, L

†

= L, in order to have CPT invariance.

•

And ﬁnally, of course, we will insist that the lagrangian is invariant (up to total

derivatives) under the SUSY transformations of the ﬁelds, which we repeat

here for convenience:

δφ = ζ · χ = ζ

(−iσ

)χ

δφ

†

= ¯χ ·

ζ = χ

†

iσ

∗

δχ =−iσ

(∂

φ)i σ

∗

+ Fζ

δF =−iζ

†

¯σ

∂

χ (8.2)

8.1 A Supersymmetric Lagrangian With

Masses and Interactions

Let’s begin. We want to write new terms built out of the ﬁelds φ, χ, F, and their

hermitian conjugates that will respect our four criteria. Consider ﬁrst interactions

that depend only on φ and its dagger φ

†

. To keep things general, let’s just write this

CHAPTER 8 The Wess-Zumino Model

161

interaction as

= G(φ, φ

†

) (8.3)

This is automatically Lorentz-invariant because φ is a scalar ﬁeld. We use a calli-

graphic G instead of F for this function in order to avoid possible confusion with

the auxiliary ﬁeld F. To satisfy the criterion of renormalizability, we may only

consider functions of at most four scalar ﬁelds. Without loss of generality, we may

take G to be real, G

†

= G (if it weren’t, we would simply add to it its hermitian

conjugate and rename the sum G).

What else can we write down? Well, we can consider some arbitrary function of

φ times the auxiliary ﬁeld F. We will write this term as

(φ, φ

†

)F (8.4)

The notation W

is a standard one. The reason for the subscript 1 is that we are

dealing with a single set of ﬁelds φ,χ, and F. Later on we will extend our discussion

to consider several sets of ﬁelds, and the 1 will be replaced by an index i.

The lagrangian Eq. (8.4) is automatically Lorentz-invariant because both φ and

F are scalar ﬁelds. For this to be of dimension 5 or less, W must contain the product

of at most two φ or φ

†

ﬁelds. However, Eq. (8.4) is not real. To get a real lagrangian,

we must add to Eq. (8.4) its hermitian conjugate, so we will consider instead

= W

F + W

†

(8.5)

Note that there is no reality condition imposed on W

Next, we also can consider a general function of the scalar ﬁelds multiplying a

Lorentz-invariant combination of the spinor ﬁeld χ. We have seen that a possible

Lorentz invariant made of χ is χ ·χ , so we write this new contribution to the

lagrangian as

−

(φ, φ

†

)χ ·χ (8.6)

Once again, we are using a standard notation for the name of the function of the

scalar ﬁeld, and the factor of −1/2 is purely conventional and chosen to simplify

a later result. There is no a priori relation between W

and W

, although we will

later ﬁnd that they must indeed be closely related in order for the theory to be

supersymmetric.

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Supersymmetry Demystified

Again, Eq. (8.6) is not real, so we need to consider instead the real combination

[using Eq. (A.44)]:

=−

χ ·χ −

†

(χ ·χ)

†

=−

χ ·χ −

†

¯χ · ¯χ (8.7)

There is no reality condition imposed on W

Next, we might think about considering terms containing a function of the aux-

iliary ﬁeld F times χ ·χ, but this is necessarily of dimension 5 or higher. For the

same reason, it is impossible to write any term that contains φ (or φ

†

) and two

auxiliary ﬁelds or to write terms containing all three types of ﬁelds (these start at

dimension 6).

Denoting by L

the free supersymmetric lagrangian obtained in Chapter 6, the

most general lagrangian that is real, renormalizable, and Lorentz-invariant is there-

fore

total

= L

+ L

= ∂

φ∂

†

+ χ

†

i ¯σ

∂

χ + FF

†

+ G + W

F + W

†

−

χ ·χ −

†

¯χ · ¯χ (8.8)

where one must keep in mind that G, W

, and W

are functions of φ and φ

†

. We did

not include any coefﬁcients in front of the new terms because they can be absorbed

into a redeﬁnition of G, W

, and W

Our goal now is to see if we can ﬁnd expressions for those three functions of the

scalar ﬁelds such that the total lagrangian is supersymmetric. We already know that

is invariant under SUSY so we only need to impose that L

+ L

must

be supersymmetric. From now on we will call the sum of these three terms L

int

because they contain interactions between the different ﬁelds, i.e.,

+ L

≡ L

int

(8.9)

Let us write the variation of L

int

under SUSY explicitly:

δL

int

= δG + δ(W

F) + δ(W

†

)

−

δ(W

χ ·χ) −

δ(W

†

¯χ · ¯χ) (8.10)

CHAPTER 8 The Wess-Zumino Model

163

At ﬁrst sight, it may seem like we will need to write down explicit expressions

for W

, W

, and G before we can calculate the supersymmetric variation of our

lagrangian, but this won’t be necessary. For example, we can simply write the total

variation of G(φ, φ

†

) under a SUSY transformation as

δG =

∂G

∂φ

δφ +

∂G

∂φ

†

δφ

†

(8.11)

using the rule familiar from calculus. We therefore get, using Eq. (8.2),

δG =

∂G

∂φ

ζ · χ +

∂G

∂φ

†

¯χ ·

ζ (8.12)

We can, of course, use the same trick to write down the variation of W

and W

For example, we have

δ(W

F) = (δW

) F + W

δF



∂W

∂φ

δφ +

∂W

∂φ

†

δφ

†



F + W

δF

∂W

∂φ

ζ · χ F +

∂W

∂φ

†

ζ · ¯χ F − iW

†

¯σ

∂

χ (8.13)

The variation of W

†

is obtained simply by taking the hermitian conjugate of this

expression.

On the other hand, we ﬁnd that

−

δ(W

χ ·χ) =−

δW

χ ·χ − W

χ ·δχ

=−



∂W

∂φ

δφ +

∂W

∂φ

†

δφ

†



χ ·χ − W

χ ·δχ

=−

∂W

∂φ

ζ · χχ·χ −

∂W

∂φ

†

ζ · ¯χχ· χ

−iW

iσ

(∂

φ) σ

iσ

∗

− W

F χ · ζ (8.14)

and again, the variation of −W

†

¯χ · ¯χ/2 is simply the hermitian conjugate of the

preceding.

Before writing out Eq. (8.10) in gory detail, let us make a key observation: The

ﬁrst two terms (containing G) cannot be canceled by anything arising from the last

164

Supersymmetry Demystified

four terms (containing W

and W

). This is clear because the variation of the terms

in W

and W

contains either a derivative, an auxiliary ﬁeld F, or three spinors χ.

But none of these appear in the terms in G. Therefore, if we want the theory to be

supersymmetric, G must be a constant, which always can be chosen to be zero, i.e.,

G = 0 (8.15)

This leaves the more manageable form [using Eqs. (8.13) and (8.14)]

δL

int

= δ(W

F) −

δ(W

χ ·χ) + h.c.

∂W

∂φ

ζ · χ F



 

∂W

∂φ

†

ζ · ¯χ F



 

−iW

†

¯σ

∂

  

III

−

∂W

∂φ

ζ · χχ·χ



 

−

∂W

∂φ

†

ζ · ¯χχ· χ



 

−iW

iσ

(∂

φ) σ

iσ

∗

  

− W

Fχ · ζ



 

VII

+h.c.

where we have labeled the terms with roman numerals in order to make it easier

to refer to them later. We did not write down the hermitian conjugate expressions

explicitly because all we really need is to make the terms shown here cancel each

other; if they cancel out, their hermitian conjugate will cancel out as well. Thus our

task is now to see if there is any choice of the potentials W

and W

such that the

terms shown cancel each other out (modulo total derivatives, as always).

First, let us concentrate on the two terms that contain four spinors. Term IV is

the only term that contains ζ · χχ· χ . However, this expression is identically zero

because it necessarily contains the square of a Grassmann number, so we get no

constraint on ∂W

/∂φ.

On the other hand, the quantity

ζ · ¯χχ· χ , which appears in term V, is not zero,

and since there are no other terms that could possibly cancel this one, we must

impose

∂W

∂φ

†

= 0 (8.16)

This is a result that deserves to be emphasized: The potential W

depends only on

φ, not on φ

†

. A common way to express this is to say that W

is a holomorphic

function of φ. This expression comes from complex analysis, where a complex

CHAPTER 8 The Wess-Zumino Model

165

function that depends only on the complex variable z and not on

z is said to be

holomorphic.

Next, let us turn our attention to II. There are no other terms of this form, so it

must be identically zero, which implies that W

is also holomorphic in φ:

∂W

∂φ

†

= 0 (8.17)

This leaves us with four terms (plus their hermitian conjugate). Two of these terms

contain derivatives, so they must cancel each other (or vanish separately). These

two terms are III and VI:

III+VI=−iW

†

¯σ

∂

χ −iW

iσ

(∂

φ) σ

iσ

∗

In the second term, we can move the ∂

φ in front and then use Eq. (A.5) to simplify

III+VI=−iW

†

¯σ

∂

χ +i (∂

φ) W

¯σ

μT

∗

which, using Eq. (A.36), may be written as

III+VI=−iW

†

¯σ

∂

χ −i (∂

φ) W

†

¯σ

In order to have the same spinor product in both terms, let us do an integration

by parts on the ﬁrst term (keep in mind that W

is a function of φ,soitisa

spacetime-dependent quantity):

III+VI= i (∂

)ζ

†

¯σ

χ −i (∂

φ) W

†

¯σ

Now we set this equal to zero and obtain the condition

∂

= (∂

φ) W

(8.18)

You see now why a factor of −1/2 has been included in Eq. (8.6); it was chosen in

order to make the relation between W

and W

as simple as possible.

Equation (8.18) provides a relation between W

and W

that can be made more

explicit by noting that, through an application of the chain rule,

∂

∂W

∂φ

∂

φ (8.19)