Labelle P. Supersymmetry DeMYSTiFied

Подождите немного. Документ загружается.

This page intentionally left blank

CHAPTER 13

Supersymmetric Gauge

Field Theories in the

Superfield Approach

Now, we would like to see how to construct gauge-invariant theories using super-

ﬁelds. As a warm-up, we will start with abelian gauge theories.

13.1 Abelian Gauge Invariance

in the Superfield Formalism

First, let’s consider a left-chiral superﬁeld  that carries a U (1) charge q. What

do we mean by this? Well, by direct analogy with the usual gauge transforma-

tion of charged particles, we will take this as meaning that the superﬁeld acquires

298

Supersymmetry Demystified

a phase

 → e

2iq



under a gauge transformation, with q being the charge of the ﬁeld and  the gauge

parameter. This is of the same form as a standard U (1) transformation except for the

factor of 2, which is, of course, purely conventional and has been chosen because

it will simplify a later result.

To build a gauge theory, we proceed as usual: We make the gauge parameter

spacetime-dependent, i.e.,  = (x), so it is now a ﬁeld. But we are working in

superspace, so it actually must be taken to be a superﬁeld! However, it cannot be

an arbitrary superﬁeld because we obviously want the left-chiral superﬁeld  to

remain a left-chiral superﬁeld after a gauge transformation. This means that we

must take the superﬁeld  to be a chiral superﬁeld itself, so it has the expansion

[see Eq. (12.8)]:

 = φ



−

θσ

θ∂



−

θ ·θ

θ ·

φ



+θ ·χ



−

θσ

θθ· ∂



θ ·θ

Even though this has the same expansion as the left-chiral superﬁelds we have

met so far, the dimensions of the ﬁelds are different. In order for the superﬁeld  to

appear in an exponential, it must be dimensionless. If we recall that each Grassmann

variable has dimension −1/2, then the dimensions of the component ﬁelds are

[φ



] = 0[χ



] =



] = 1

Now that we have deﬁned how left-chiral superﬁelds transform, we can build

supersymmetric and gauge-invariant interactions. Consider a superpotential W that

depends on a set of left-chiral superﬁelds 

,

,.... To have gauge invariance,

we may only consider products of superﬁelds whose charges add up to zero. Note

that if the superpotential is invariant when the gauge parameter is a constant, it will

remain invariant even after we make the gauge invariance local since it does not

contain derivatives acting on the superﬁelds.

Now consider the “kinetic” term for a left-chiral superﬁeld 

†

. When we

perform a gauge transformation of this, we get



†

 → e

2iq(−

†

)



†

 (13.1)

CHAPTER 13 Gauge Field Theories

299

where we have used the fact that left-chiral superﬁelds commute to move an expo-

nential around. We also have taken the charge q to be a real number. We see that this

term is not gauge-invariant! It might seem that we could make it gauge-invariant

by requiring 

†

= ,but is a left-chiral superﬁeld, which cannot be real (the

hermitian conjugate of a left-chiral superﬁeld does not have the form of a left-chiral

superﬁeld because it will contain

θ). So how are we to make 

†

 gauge-invariant?

We can’t simply drop it because it contains the kinetic terms of the component

ﬁelds. The way out is the same as in conventional ﬁeld theories: We must introduce

a gauge superﬁeld V whose transformation will compensate the variation appearing

in Eq. (13.1). We therefore replace 

†

 by



†

2qV

 (13.2)

with the gauge superﬁeld deﬁned to have a “super gauge transformation” given by

V → V − i( − 

†

) (13.3)

It is conventional to write the exponential in the middle, as in Eq. (13.2), but there

is no special reason for this here because the superﬁelds commute anyway. When

we consider nonabelian gauge theories, the superﬁelds actually will be matrices,

and then the order will be important, obviously.

The reason for including a factor i in the gauge transformation in Eq. (13.3) is

to make the combination i ( − 

†

) real. This then implies that we may take the

gauge superﬁeld V to be a real superﬁeld. Recall that the most general superﬁeld

contains nine terms, as we saw in Exercise 11.2, which we may write as

V(x) = A(x) +θ ·α(x) +

θ ·

β(x) + θ ·θ B(x) +

θ ·

θ H (x) + θσ

θ V

(x)

+θ ·θ

θ · ¯γ(x) +

θ ·

θθ· η(x) + θ · θ

θ ·

θ P(x ) (13.4)

where α, β, γ , and η are Weyl spinors, V

is a complex vector ﬁeld, and A, B, H ,

and P are all complex scalar ﬁelds. Let’s now impose

†

= V

It turns out that this is consistent with supersymmetry (SUSY) transformation;

i.e., this reality condition is preserved by SUSY transformations. Imposing this

condition on Eq. (13.4) leads to

†

= AV

†

= V

†

= P

†

= H α = βγ= η

300

Supersymmetry Demystified

which reduces the ﬁeld content to one real vector ﬁeld, two Weyl spinors, two real

and one complex scalar ﬁelds. This is still much more than the ﬁeld content we

needed in Chapter 10 to build the vector supermultiplet!

For reasons that will become clear very soon, a different representation is usually

chosen for the real superﬁeld V:

V = C(x) +

√

θ ·ρ(x) −

√

θ · ¯ρ(x) +

θ ·θ(M(x) + iN(x))

−

θ ·

θ(M

†

(x) − iN

†

(x)) +

θσ

θ A

(x)

√

θ ·θ



θ ·

λ +

θ ¯σ

∂



√

θ ·



θ ·λ −

θσ

∂

¯ρ



−

θ ·θ

θ ·



D(x) +

C(x)



(13.5)

where M, N, C, and D are all real scalar ﬁelds, A

is a real vector ﬁeld, and λ and

ρ are Weyl spinors.

This gauge superﬁeld is often called a vector superﬁeld because one of its com-

ponents is the vector ﬁeld A

. However, keep in mind that the vector superﬁeld V

itself is a scalar under Lorentz transformations, like the left-chiral superﬁelds 

The vector superﬁeld is dimensionless, which can be seen by looking at the

term containing A

. A vector ﬁeld has a dimension equal to 1, but θ and

θ have

dimension −1/2, which makes the term containing A

, and consequently the whole

superﬁeld, dimensionless.

Now let’s see how these ﬁelds change under a gauge transformation [Eq. (13.3)].

We have (we will not write explicitly the x dependence of the component ﬁelds

from now on)

V → V − i ( − 

†

)

→ V − i





− φ

∗





−i



θ ·χ



−

θ · ¯χ





−

θ ·θ F



θ ·

θ F

∗



−

θσ

θ∂





+ φ

∗





−



θ ·θ

θ ¯σ

∂



−

θ ·

θθσ

∂

¯χ





θ ·θ

θ ·

(φ



− φ

∗



) (13.6)

where we have used Eqs. (12.47) and (A.50). We use a complex conjugation on



and F



instead of a dagger because these are complex functions, not quantum

CHAPTER 13 Gauge Field Theories

301

ﬁelds. Using Eq. (13.5), we can easily pick up how each of the component ﬁelds of

V transform. Focus ﬁrst on the terms with no Grassmann variables, as well as on

those proportional to θ ·θ

θ ·

θ. These terms combined transform as

C −

θ ·θ

θ ·



D +

C



→ C − i





− φ

∗





−

θ ·θ

θ ·



D +

C



θ ·θ

θ ·







− φ

∗





(13.7)

from which we can read off the gauge transformations of the two ﬁelds to be

C → C − i





− φ

∗





D → D (13.8)

Now it should be clear why the last term of V was chosen to contain the appar-

ently strange combination D −

C. This choice allowed us to put all the gauge

dependence into the ﬁeld C, leaving a gauge-independent ﬁeld D.

Working out the gauge transformations of the other component ﬁelds, one ﬁnds

ρ → ρ −

√

2 χ



M +iN → M + iN − 2 F



→ A

− ∂





+ φ

∗





λ → λ (13.9)

Again, the reason for the peculiar combinations of the ﬁelds ρ and λ in Eq.

(13.5) was to have all the gauge dependence in one ﬁeld, leaving the other one

gauge-invariant.

We see that we have a lot of “gauge freedom” in Eqs. (13.8) and (13.9). In

particular, by an appropriate choice of χ



and F



, we can use a gauge transformation

to set the ﬁelds ρ, M, and N to zero. We also can choose the imaginary part of φ



so that the ﬁeld C also will vanish. This gauge, in which all these ﬁelds are set to

zero, is called the Wess-Zumino gauge. In this gauge, the vector superﬁeld is simply

V =

θσ

θ A

√

θ ·θ

θ ·

λ +

√

θ ·

θθ· λ −

θ ·θ

θ ·

θ D

(13.10)

where the ﬁelds λ and D are gauge-invariant. From now on, we will only work with

the gauge superﬁeld in the Wess-Zumino gauge.

302

Supersymmetry Demystified

The gauge is not completely ﬁxed, however. We still have one last gauge param-

eter, the real part of φ



. Under this residual gauge transformation, the vector ﬁeld

changes by

→ A

− 2 ∂



Re(φ



)



which is again the standard form for the U (1) gauge transformation of the gauge

ﬁeld A

apart from the factor of 2.

To summarize, we can write down gauge-invariant combinations of superﬁelds

at two conditions:

•

Recall that the superpotential must be a holomorphic function of the left-chiral

superﬁelds; i.e., it may not contain their hermitian conjugates 

†

. Therefore, in

any product of left-chiral superﬁelds appearing in the superpotential W(

the charges of the superﬁelds must add up to zero in order to have gauge

invariance.

•

The kinetic term of left-chiral superﬁelds 

†



must be replaced by 

†



(where the label i indicates the different superﬁelds).

To obtain SUSY-invariant lagrangians (up to total derivatives, as always) from

these terms, all we have to do is to extract the F term of the superpotential and the

D term of the gauge-invariant kinetic term:

L =

W(

)



+ h.c.





†





(13.11)

Note that if there is only one charged left-chiral superﬁeld, the superpotential is

automatically zero because it is impossible to write down a holomorphic function

of  that is gauge-invariant.

There is actually a third gauge-invariant and supersymmetric term that can be

added to Eq. (13.11). This is a sneaky one. The trick is to notice that even though

V is not gauge-invariant, its D term is! Indeed, the D term of V is simply [see

Eq. (13.10)]



=−

D (13.12)

which is gauge-invariant [see Eq. (13.8)]. We therefore can add to Eq. (13.11)

the term

−2 ξV



= ξ D

CHAPTER 13 Gauge Field Theories

303

where the parameter ξ has the dimension of a mass squared. This is, of course, the

Fayet-Illiopoulos term that we introduced in Section 10.2.

Incidentally, we now see why extracting the coefﬁcient of the combination θ ·

θ ·

θ/4 is referred to as ﬁnding the D term: When this is done to a gauge superﬁeld,

it indeed pulls out the D ﬁeld.

13.2 Explicit Interactions Between

a Left-Chiral Multiplet and the

Abelian Gauge Multiplet

Let us now work out explicitly the second term of Eq. (13.11), i.e.,



†

2qV





in terms of component ﬁelds (we have already worked out the expansion of the

superpotential, so we won’t repeat that here).

Let us start by Taylor expanding e

2qV

.Weget



†

2qV

 = 

†

(1 + 2 q V + 2 q

) (13.13)

The series terminates there because V

necessarily contains a Grassmann variable

squared. We get a large number of terms, but of course, we need only retain those

that have two θ and two

θ because we are interested only in the D term:



†

2qV





= 

†





+ 2 q 

†

V



+ 2 q



†

V



(13.14)

We have already worked out the D term of 

†

; it’s simply the free Wess-Zumino

lagrangian.

The last term of Eq. (13.14) is actually very simple because the only nonvanishing

term in V

θσ

θθσ

θ A

θ ·θ

θ ·

θ A

(13.15)

where we have used Eq. (A.56) for the last step. The only contribution to the D

term of 

†

V

therefore comes from the term with no Grassmann variable in 

†

,

304

Supersymmetry Demystified

which is φ

†

φ. Thus

2 q



†

V

= q

†

φ A

(13.16)

Now let us consider the D term of 

†

V. Since V contains terms with at least

one θ and one

θ, we need only consider in the product 

†

 terms with at most one

θ and one

θ. Keeping this in mind, we get



†

 = φ

†

φ +φ

†

θ ·χ + φ

θ · ¯χ −

†

θσ

θ∂

φ +

φθσ

θ∂

†

+θ ·χ

θ · ¯χ +···

where we have only written the terms that will contribute to the D term of 

†

 V.

Writing again only the relevant terms, we have



†

 V =−

θ ·θ

θ ·

θφ

†

φ D +



√

θ ·

θθ· χθ · λφ

†

+ h.c.



−



θσ

θθσ

θφ

†

∂

φ +h.c.



θσ

θθ· χ

θ · ¯χ A

+··· (13.17)

We need to rewrite all the terms so that they are in the form

Product of ﬁelds × θ · θ

θ ·

so that we can extract the D term. We can achieve this using identities we already

have. For the second term of Eq. (13.17), we use Eq. (A.54); for the third term, we

use Eq. (A.56), whereas for the last term, we use Eq. (12.47). This ﬁnally gives



†

 V =

θ ·θ

θ ·



−

†

φ D +

χσ

¯χ A

−



√

χ ·λφ

†

∂

φ +h.c.



+···



CHAPTER 13 Gauge Field Theories

305

from which we can read off the D term directly. Recall that what we are really

interested in is the quantity

2 q

†

V



=−q φ

†

φ D +q χσ

¯χ A

−



√

2 q χ · λφ

†

+iqA

†

∂

φ +h.c.



(13.18)

Finally, the D term of 

†

2qV

 is given by the sum of the free Wess-Zumino

lagrangians, of Eqs. (13.16) and (13.18):



†

2qV





= ∂

†

∂

φ +i ¯χ ¯σ

∂

χ + FF

†

φ A

−q φ

†

φ D

−



√

2 q χ · λφ

†

+iqA

†

∂

φ +h.c.



+q χσ

¯χ A

(13.19)

This must be compared with the supersymmetric lagrangian for the interaction of a

left-chiral Weyl spinor with an abelian gauge ﬁeld that we worked out in Chapter 10

[see Eq. (10.54)]. To make the relation more explicit, consider the following terms

appearing in Eq. (13.19):

∂

†

∂

φ + q

†

φ A

−



iqA

†

∂

φ +h.c.



= ∂

†

∂

φ +q

†

φ A

−iqA

†

∂

φ +iqA

φ∂

†

(13.20)

These terms can be combined into

φ)

†

φ)

if we deﬁne

≡ ∂

+iqA

(13.21)

the usual covariant derivative. It would be nice to also write the kinetic energy of

the spinor in terms of a covariant derivative by combining the following two terms:

i ¯χ ¯σ

∂

χ +q χσ

¯χ A

(13.22)

but we see that we can’t because the σ matrices in the two terms are of different

types. Luckily, we are saved again by an identity, this time Eq. (A.50), which allows