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Supersymmetry. Theory, Experiment, and Cosmology

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Appendix C

Superﬁelds

In this appendix, we construct representations of the supersymmetry algebra by using a trick that allows us to put under the rug the di culty that arises from having anticommutation relations. Supersymmetry transformations are understood as translations in a generalized space, known as superspace, where one adds to the standard spacetime coordinates anticommuting (Grassmann) variables. Fields in this superspace, known as superﬁelds, then describe general supermultiplets. Speciﬁc covariant constraints on these superﬁelds yield irreducible representations of the supersymmetry algebra.

Since this appendix lies on the theorist’s track, we use throughout two-component spinor notation, as introduced in appendix B.

C.1 Superspace and superﬁelds

Let us ﬁrst consider translations. A translation of vector yµ on a scalar ﬁeld φ(x) reads:

φ(x) → eiyµPµ φ(x)e−iyµPµ ,

(C.1)

where Pµ is the generator of translations. We can write φ(x) as the value of φ translated from the origin:

φ(x) → eixµPµ φ(0)e−ixµPµ .

(C.2)

Thus, under a translation,

φ(x) → ei(xµ+yµ)Pµ φ(0)e−i(xµ+yµ)Pµ .

If the translation is inﬁnitesimal, then according to (C.2) and (C.3), we have

δφ = −iyµ[φ, Pµ] = yµ ∂x∂µ φ(x),

which we can write

δφ = −iyµPµφ,

with

∂

Pµ = i ∂xµ .

(C.3)

(C.4)

(C.5)

(C.6)

430 Superﬁelds

In other words, Pµ is the representation of the translation generator Pµ as a di erential operator. For the simplicity of notation, we will denote it simply by Pµ in what follows.

To obtain a similar representation of supersymmetry transformations in terms of di erential operators, we have to introduce Grassmann variables. Let us recall that a Grassmann variable θ is an anticommuting variable

{θ, θ} = 0 θ2 = 0.

A general function F (θ) can be expanded in series

∞

F (θ) = an θn = a0 + a1 θ.

n=0

The derivative is simply

dθ = a1.

One may introduce two Grassmann variables θα, α = 1, 2:

F (θα) = a0 + θ1 a1 + θ2 a2 + θ2θ1 a3 = a0 + θα aα + 12 θθ a3.

where we used the notation introduced in Appendix B (equation (B.6)) since α is to

be interpreted as a spinor index. In fact, we will need two sets θ

, α = 1, 2, and θα˙ ,

α˙ = 1, 2, of Grassmann variables:

{θ

, θ

} =

{θα˙

, θβ˙ } = {θ

, θα˙ } = 0,

¯α˙

. One may then multiply the

corresponding to the supersymmetry charges Qα and Q

supersymmetry algebra

Pµ,

}

(C.7)

{Qα, Qα˙ } = 2σαα˙

{Qα, Qβ } = 0 = {Qα˙

, Qβ˙

by θ

α ¯α˙

and sum over α , α˙ to obtain

[θQ, θQ¯ ¯] = 2

θσµθ¯ Pµ,

[θQ, θ Q] = 0 = [θQ,¯ ¯ θ¯ Q¯]

(C.8)

thus recovering commutation relations (because the θ’s anticommute with the supersymmetry charges).

Using the standard procedure described above, we now realize the algebra (C.8) through differential operators acting on a “superspace” described by the spacetime

µ	and the Grassmann variables θ	α	¯	. An object in this superspace is a
variable x	and the Grassmann variables θ		, θα˙	. An object in this superspace is a

F ¯ superﬁeld (x, θ, θ).

Superspace and superﬁelds 431

We consider the “superspace translation” generalizing (C.1):

F ¯ → µ F ¯ −1 µ

(x, θ, θ) S (y , η, η¯) (x, θ, θ)S (y , η, η¯) ,

with

S (yµ, η, η¯) = ei(y	µ	¯
S (yµ, η, η¯) = ei(y		Pµ+ηQ+η¯Q).

Using the Hausdor formula

eA eB = eA+B+ 12 [A,B]

(C.9)

(C.10)

if [A, B] commutes with A and B, we obtain from the supersymmetry algebra (C.8):

S (yµ, η, η¯) S xµ, θ, θ¯ = S xµ + yµ + iησµθ¯ − iθσµη,¯ θ + η, θ¯ + η¯ .

(C.11)

Since any point can be obtained from the origin by translation, one may always write

F ¯ ¯ F −1 ¯

(x, θ, θ) = S(x, θ, θ) (0, 0, 0)S (x, θ, θ).

Thus using (C.11), a superspace translation of parameters yµ, η, and η¯ acts on the superﬁeld:

F (xµ, θ, θ¯) → F xµ + yµ + iησµθ¯ − iθσµη,¯ θ + η, θ¯ + η¯ .

Inﬁnitesimally,

∂

δS F

, θ, θ =

, θ, θ , −i y

Pµ + ηQ + η¯Q

= η

θ¯ + iη¯σ¯ θ

F .

∂θα

+ η¯α˙

∂θ¯α˙

+ y

+ iησ

∂xµ

(C.12)

as follows

(C.13)

(C.14)

This explicitly shows the action of the di erent generators on the superﬁeld F :

with	δS F xµ, θ, θ¯ = −iyµPµF − iηQF − iη¯Q¯F						(C.15)
	Pµ = i ∂µ,
	Qα = i		∂		+ i σααµ ˙ θ¯α˙ ∂µ ,

			∂θα
	Q¯α˙	= −i		∂		+ i θασααµ ˙ ∂µ ,	(C.16)

				∂θ¯α˙
	Q¯α˙	= i	∂		+ i σ¯µαα˙ θα ∂µ .

			∂θ¯α˙

(C.16) gives a representation of the supersymmetry algebra in terms of di erential operators. One may check directly that

¯	µ	∂µ
{Qα, Qα˙	} = 2i σαα˙	∂µ
	¯	¯	}.	(C.17)
{Qα, Qβ } = 0 = {Qα˙ , Qβ˙			}.	(C.17)

432 Superﬁelds

One may expand the superﬁeld F (x

, θ, θ) in terms of the Grassmann variables. The

series stops at the term

One has

θ2 θ1 θ¯2˙ θ¯1˙

(θα θα) θ¯α˙

θ¯α˙ .

¯2

n(x)

x, θ, θ

f (x) + θ ρ(x) + θ χ¯(x) + θ

m(x) + θ

θσµθ¯ vµ(x) + θ2 θλ¯¯(x) + θ¯2 θψ(x) + θ2θ¯2d(x)

(C.18)

α˙

¯α˙

, and ψα spinor ﬁelds and vµ complex

with f , m, n, d complex scalar ﬁelds, ρα, χ¯

, λ

vector ﬁeld. These ﬁelds form a representation of the supersymmetry algebra since one can write the explicit transformations using (C.15):

¯	−iη	α	¯α˙	¯
δS F (x, θ, θ) =	−iη		Qα − iη¯α˙ Q	F (x, θ, θ)
	δ			¯
	δ			+ θ δS χ¯(x) + · · ·	(C.19)
= S f (x) + θ δS ρ(x)				+ θ δS χ¯(x) + · · ·	(C.19)

But this representation is obviously reducible: there is more to it than simply a chiral supermultiplet, or a vector supermultiplet. The problem will be to impose constraints that reduce the representation and make it irreducible.

For that matter, we will need to introduce covariant spinor derivatives on super-

ﬁelds, that is spinor derivatives such that
Dα[δS F ] = δS [Dα F ].	(C.20)

In other words, covariant derivatives are spinor derivatives which commute with supersymmetry transformations.

It is easy to show that

Dα =

∂

¯α˙

(C.21)

∂θα

− i σαα˙

∂µ

∂

− i θ

Dα˙

σαα˙

∂µ

∂θ¯α˙

¯ α˙

= −

∂

+ iσ¯

µαα˙

(C.22)

θα ∂µ

∂θ¯α˙

satisfy

∂µ

}

{Dα, Dα˙

} = −2i σαα˙

{Dα, Dβ } = 0 = {Dα˙

, Dβ˙

} =

} = 0.

(C.23)

{Dα, Qβ } = {Dα, Qβ˙

{Dα˙ , Qβ } = {Dα˙

, Qβ˙

C.2 The chiral superﬁeld

C.2.1 Deﬁnition

We want to reduce the ﬁeld content of a general superﬁeld to the content of a chiral supermultiplet (φ, ψα, F ). Let us take this opportunity to note that we are dealing

The chiral superﬁeld 433

with the o -shell formulation here because the supersymmetry transformations are obviously linear. Hence some of the ﬁelds in the general decomposition (C.18) of the superﬁeld F are auxiliary ﬁelds.

There is unfortunately no straightforward procedure for, given a speciﬁc supermultiplet (of the type constructed in Chapter 4), identifying the relevant constraint. One often adopts a trial and error process which sometimes leaves the nonexpert bewildered.

One possibility is to use the covariant derivative which has just been introduced

and impose the constraint	xµ, θ, θ¯ = 0.
D¯ α˙ Φ	xµ, θ, θ¯ = 0.	(C.24)

Such a superﬁeld Φ is called chiral and, as we will show below, describes the ﬁelds of a chiral supermultiplet. The constraint (C.24) is chosen because it is invariant by supersymmetry. Indeed, using (C.20),

¯ α˙	¯	α˙	Φ = 0.
D	(δS Φ) = δS D		Φ = 0.

It is easy to see how the constraint (C.24) reduces the representation of the supersymmetry algebra. One notes that

¯ α˙

θα

= 0

µ ¯

α˙

i θσ

¯ α˙

θσ¯

−

= D

∂

+ i

ναα˙

µββ˙

= −

+ i σ¯

θα ∂ν

+ i θ¯β˙

σ¯

θβ

∂θ¯α˙

= i σ¯µαα˙ θα − i σ¯µαβ˙ θβ = 0.

Hence any function of θα and yµ = xµ

−

iθσµ ¯

constraint1.

θ automatically satisﬁes the

This can also be seen trivially by expressing everything in terms of θ, θ and y

∂

¯α˙

∂

Dα =

∂θα

− 2i σαα˙

∂yµ

¯ α˙

−

∂

(C.25)

∂θ¯α˙

α˙

Φ = 0 is Φ = Φ(y

, θ). A chiral superﬁeld is thus

and therefore the solution of D

written as:

Φ(y, θ) = φ(y) + √

θα ψα(y) + θ2 F (y).

(C.26)

We are left with three independent ﬁelds. Counting dimensions may suggest that F is an auxiliary ﬁeld: taking φ as a scalar ﬁeld of canonical dimension 1, we ﬁnd a dimension −1/2 for θα (so that ψα be of dimension 3/2) and a dimension +2 for F ; the latter dimension suggests that F gives a total divergence under a supersymmetry transformation and thus, as explained in Chapter 3, that it is an auxiliary ﬁeld.

1Note that, strictly speaking, the variable yµ is complex (use (B.64)). This is one of the many illustrations that supersymmetry “complexiﬁes” spacetime.

434 Superﬁelds
¯	µ	, using (B.15) and
One may rewrite (C.26) in terms of the variables θ, θ, and x		, using (B.15) and
(B.61),

µ ¯

Φ x , θ, θ

µ ¯

¯2

= φ(x) − i θσ θ ∂µφ(x) −

φ(x)

√

µ ¯

2 θ

ψα(x) +

√

∂µψ(x)σ θ + θ

F (x)

(C.27)

which clearly shows how the condition (C.24) constrains the component ﬁelds of a general superﬁeld (C.18).

Note that we can deﬁne the component ﬁelds of the chiral superﬁeld as:

φ = Φ|0 ,

χα =

√

DαΦ 0

F = −

Φ 0 ,

(C.28)

where the index 0 means that we take θ = θ¯ =

¯α˙

The supersymmetry transformations are easily obtained by expressing Qα and Q

in terms of θ, θ and y

Qα

= i

∂

∂θα

= i

∂

Q¯α˙

+ 2i σ¯µαα˙

θα

∂θ¯α˙

∂yµ

and using (C.19). Indeed

−iηα Qα Φ = √

η ψ(y) + 2 ηθ F (y)

α˙

Φ = 2i η¯σ¯

√

θ) θ

∂µψβ

−iηα˙ Q

θ ∂µφ + 2i 2 (¯ησ¯

= −2i θσ

√

α˙

η¯ ∂µφ − 2i 2 θ

σαα˙ θ

∂µψβ η¯

= −2i θσ

√

∂µψσ

η¯

η¯ ∂µφ + i 2 θ

where we have used θαθβ = − 21 εαβ θ2 (see (B.61) of Appendix B). Hence

δS φ = √

η ψ

√

α˙

∂µφ

δS ψα = 2 ηα F − i 2 σαα˙ η¯

δS ψ

η¯

F − i

σ¯

ηα ∂µφ

¯α˙

√

α˙

√

µαα˙

δS F = i√

∂µψσµη¯ = −i√

η¯σ¯µ∂µψ

(C.29)

which are the familiar supersymmetry transformations of a chiral supermultiplet. We check that F is indeed the auxiliary ﬁeld: it transforms into a total derivative.

C.2.2 Actions

Just as in Minkowski spacetime where we obtain actions invariant under translations by integrating over all spacetime, we will obtain supersymmetric actions by integrating over all superspace. In order to fulﬁll this program, we ﬁrst have to discuss the integration of a Grassmann variable.

The chiral superﬁeld 435

We start with one variable θ. We want to deﬁne an integration law for a general function f (θ) = a0 + a1θ such that one has the standard property of linearity and one

recovers the rule		df
			dθ = 0.

Clearly this requires		dθ
	1 · dθ = 0 and, if we want the integral to be nonvanishing,

θdθ = 0. Choosing the overall normalization, we thus set

		dθ = 0				θ dθ = 1.			(C.30)
Hence		df			=
		df
				= a1		f (θ) dθ.			(C.31)
		dθ		= a1		f (θ) dθ.			(C.31)
For two variables θα, α = 1, 2, we deﬁne
	d2θ ≡ −	1	εαβ			1		dθ1 dθ2.
					dθα	dθβ =
		4			dθα	dθβ =	2

Hence

d2θ · 1 = 0

d2θ θα = 0

d2θ θθ =		dθ1	dθ2 2θ2θ1 = 1.	(C.32)
	2

It is then straightforward to write a supersymmetric action using a chiral superﬁeld Φ:

S =

d4y

d2θ Φ =

= − 4

d4yF =

d4y Φ θ2

d4y D2Φ 0

(C.33)

where Φ

θ2

is the θ

component of Φ and D Φ 0

is the scalar component

of D Φ

αβ

αβ ∂

|∂

(we have D

DαΦ|0

= ε

Dβ DαΦ|0 = ε

Φ|0 = −4F using the results of

∂θβ

∂θα

Exercise 1).

Indeed since δS F is a total derivative

δS S =

d4y δS F = 0.

Hence any F component of a chiral superﬁeld provides a good supersymmetric action. Let us note, as an aside, that, as in (C.31), an integration over a Grassmann variable is equivalent to a di erentiation. In supersymmetric covariant notation, d2θ − 14 D2. We may now use the previous remark in order to derive supersymmetric

Lagrangians. Let us consider a set of n chiral superﬁelds Φi, i = 1, . . . , n. Any product of these (and thus any analytic function W (Φi)) is also a chiral superﬁeld. For example,

√

Φi = φi + 2 θ ψi + θ2 Fi

√

Φi Φj = φi φj + 2 θ (ψi φj + ψj φi) + θ2 (φi Fj + φj Fi − ψiψj )

√

Φi Φj Φk = φi φj φk + 2 θ (ψi φj φk + perm.) + θ2 (Fi φj φk − ψiψj φk + perm.)

436 Superﬁelds

and more generally for a general analytic function W (Φi)

W (Φi)\|θ2 = Σi	∂W	(φ) Fi −	1			∂2W	(φ) ψiψj .	(C.34)

	∂Φi			2	ij	∂Φi∂Φj

We thus recover the terms of a supersymmetric Lagrangian which arise from the superpotential. Let us note that the analyticity of the superpotential is intrinsically connected with the requirement of supersymmetry: a product of Φi and Φ†i ﬁelds is not a chiral superﬁeld; thus in this construction we must restrict ourselves to sums of products of Φi’s and thus to analytic functions.

This, however, does not provide us with kinetic terms for the component ﬁelds. Let us therefore consider more closely Φ†: Φ† is not chiral since it satisﬁes

Dα Φ† = 0.

(C.35)

This is known as an antichiral superﬁeld. In any case, Φ† is not a function of y and θ

only (it is in fact a function of y and θ) and it is better to revert to the x variable.

Using (C.27) (and (B.64) of Appendix B)

Φ†(x) = φ (x) + iθσµθ¯ ∂µφ (x) −

θ2 θ¯2 φ (x)

√

¯α˙

¯2

2 θα˙

(x) −

√

θσ

∂µψ(x) + θ

F (x) .

(C.36)

We may now consider the real superﬁeld F = Φ†Φ. The highest component of such a real superﬁeld (known as a vector superﬁeld) transforms under supersymmetry into a total derivative. This may be guessed since this component has the highest mass dimension and must therefore transform into derivatives of the other ﬁelds (of lower

dimensions). Thus F |θ2θ¯2

provides us with a good supersymmetric action

S =

d4x F

θ2θ¯2

d4x d2θd2θ¯ F =

d4x D2D¯ 2F 0

(C.37)

θ¯ (

= 1). Indeed

with obvious notation for

θ¯ θ¯

δS S = −i d4x d2θ d2θ¯

ηQ + η¯Q¯

F = 0

α˙

∂

since Qα and Q

involve only derivatives

. Again, integrating over all

∂x

∂θ

∂θα˙

superspace provides an action invariant under superspace translations, i.e. supersym-

metry transformations.

But using (C.27) and (C.36)

−

φ +

∂µφ ∂

φ + F F

F |θ

θ¯

−

∂µψσ

µ ¯

ψσ

∂µψ

= ∂µφ ∂µφ + iψσµ∂µψ¯ + F F +

total derivatives.

(C.38)

The chiral superﬁeld 437

To recapitulate, we may write the following supersymmetric action in the case of a single chiral superﬁeld Φ

S =

d4x d2θ d2θ¯ Φ†(x, θ, θ¯) Φ(x, θ, θ¯) +

d4y d2θ W (Φ(y, θ)) + h.c.

= d4x $∂µφ ∂µφ + i ψσµ∂µψ¯ + F F +

∂W

(φ)F +

∂W

(φ)

∂Φ

1 ∂2W

∂2W

−

(φ)ψψ −

(φ)

ψψ¯ ¯ .

(C.39)

∂Φ2

This is by construction invariant under the supersymmetry transformations (C.29). The ﬁeld F has no kinetic term: we may solve for this auxiliary ﬁeld:

F = −

∂W

(φ)

∂Φ

to obtain

(∂µφ ∂µφ + iψσµ∂µψ¯ −

∂W

∂2W

) .

S =

d4x

∂Φ

(φ)

−

∂Φ2

(φ) ψψ + h.c.

(C.40)

C.2.3 R-symmetry

We saw in Section 4.1 of Chapter 4 that N = 1 supersymmetry allows for a global U (1) symmetry which does not commute with supersymmetry: the R-symmetry whose generator satisﬁes

[Qα, R] = Qα
¯	¯	(C.41)
[Qα˙	, R] = −Qα˙ .	(C.41)

Let us illustrate on the example of a single chiral superﬁeld how this R-symmetry can be realized. Because the R-symmetry does not commute with supersymmetry, it must act di erently on the di erent components of the superﬁeld and thus cannot leave the Grassmann variable θ invariant. The R-symmetry thus acts generically on the chiral superﬁeld Φ(y, θ) as

R Φ(y, θ) = eirα Φ y, e−iαθ
†		¯	−		†		,
									θ
R Φ	(y , θ) = e			Φ		y		e

where r is the R-charge of the supermultiplet. In terms of the component ﬁelds, this reads

R φ(x) = eirα φ(x)

R ψ(x) = ei(r−1)α ψ(x)

R F (x) = ei(r−2)α F (x).

(C.42)

(C.43)

438 Superﬁelds

Let us note that, because of the Grassmann nature of the θ variable, if θ = e−iαθ,


d2θ	= e2iαd2θ (cf. (C.32)). Obviously, the kinetic term				d2θ d2		¯	Φ is R-invariant;
the	superpotential term d2θ W (Φ) is invariant if						θ Φ†
			2iα	W Φ(y, e−	iα	θ)		(C.44)
in which case		R W (Φ(y, θ)) = e		W Φ(y, e−		θ)		(C.44)
		d2θR W (Φ(y, θ)) =	d2	θ e2iα W Φ(y, e−iαθ)
		=	d2	θ W (Φ(y, θ )) .				(C.45)

This is the case if W (Φ) is a monomial in Φ and the R-charges add up to 2: for example W (Φ) = λΦ3 with Φ of R-charge r = 23 . This is not the case if W (Φ) is a general polynomial in Φ.

C.2.4 Supersymmetric nonlinear sigma models and K¨ahler invariance

If one considers several chiral superﬁelds Φi, the kinetic term for each of them is provided by Φ†i Φi. But one may try to be more general and consider a generic real function K of the Φi and Φ†i [389]:

4	2	2	¯	†	(C.46)
S = d	x d	θ d	θ K(Φi, Φi ).		(C.46)

Obviously, this leads to nonnormalized kinetic terms for the scalar ﬁelds. Indeed, following the same steps as before, one obtains

S =	d4x	∂2K	(φ, φ ) ∂µφi∂µφj + · · ·	(C.47)
		∂Φi∂Φj†

A ﬁeld redeﬁnition φi → φi(φj ) may lead in some cases to a normalized kinetic term

∂µ	0	0	but this is not	general. This is reminiscent of the situation encountered in
	φi ∂µφi		but this is not	0

the nonlinear sigma model. Let us recall that this model is obtained as a limit of the linear sigma model whose Lagrangian is described in terms of a triplet of ‘pion’ ﬁelds πi, i = 1, 2, 3, and a σ ﬁeld by:

L = 21 ∂µσ∂µσ + ∂µπi∂µπi − 41 λ σ2 + πiπi − v2 2 .

(C.48)

In the limit λ → +∞, we must impose the constraint σ2 +πiπi = v2 to keep the energy of the conﬁguration ﬁnite. Di erentiation of this constraint yields σ∂µσ = −πi∂µπi and (C.48) can be rewritten as

L =	1	δij +	πiπj	∂µπi∂µπj ,	(C.49)
	2		v2 − πkπk

which is the Lagrangian of the nonlinear sigma model: the ﬁelds πi cannot be redeﬁned to give a normalized kinetic term.

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