Supersymmetry. Theory, Experiment, and Cosmology

The chiral superfield 439

Indeed, the Lagrangian (C.46) provides the supersymmetric generalization of a generic nonlinear sigma model. We now show that supersymmetry provides a very nice geometrical description. Let us first rewrite the complete component form of (C.46):

All these seemingly complicated terms have a beautiful geometric interpretation which we now present. Indeed, one may note that the transformation

¯ †

where F (Φ) is an analytic function of the Φ fields (F (Φ ) is the complex conjugate in order that the total expression remains real), is an invariance of the action (C.46):

An invariance such as (C.52) is known as a K¨ahler invariance and K(Φ, Φ†) is called a K¨ahler potential. Let us note that it is a transformation on functions of fields and it may not be realized as a transformation on the fields themselves2.

i ≡ ¯¯ ≡

The space parametrized by the complex scalar fields φ φi, φ φj is a complex

2although it does in some well-known cases (see Exercise 9).

440 Superfields

is interpreted as a metric on the complex manifold. It is said that the metric derives from the K¨ahler potential K and a complex manifold with such a metric is called a K¨ahler manifold3.

K¨ahler manifolds have very specific properties; for example, the only nonzero components of the Christo el symbol (see Section D.1 of Appendix D) are

where gi¯ is the inverse metric (gi¯g¯k = δki ) and ∂i ≡ ∂/∂Φi, ∂¯ ≡ ∂/∂Φ†j . We have used

which expresses the fact that the fermion fields transform as contravariant vectors under a reparametrization of the K¨ahler manifold (see Exercise 8); the Riemann tensor

3For a mathematical introduction to K¨ahler manifolds, see Section 5.3 of the book by [196].

442 Superfields

Note therefore that the transformation

corresponds to a gauge transformation with parameter α(x) ≡ −g(a(x) + a (x)) for the vector component (cf. (A.31) of Appendix Appendix A). The complete gauge transformation reads

C(x) → C(x) + i[a(x) − a (x)]

√

χ(x) → χ(x) + 2ψ(x)

M (x) + iN (x) → M (x) + iN (x) + 2F (x)

1

Aµ(x) → Aµ(x) − g ∂µα(x)

λ(x) → λ(x)

D(x) → D(x).

We see that, through a choice of the parameters i[a(x) − a (x)], ψα(x) and F (x), we may set the components C(x), χα(x), M (x) and N (x) to zero. This is the socalled Wess–Zumino gauge, which is obviously not consistent with supersymmetry transformations. The gauge invariant degrees of freedom on the other hand are: the gauge field strength Fµν = ∂µAν − ∂ν Aµ, the gaugino field λ(x), and the auxiliary field D(x). They turn out to be the component fields of the chiral superfield:

where we used the fact that Λ is chiral and Λ† antichiral. We note that Wα satisfies the condition

In fact, W α can be defined as the chiral superfield which satisfies this constraint. It is reminiscent of a similar property in nonsupersymmetric gauge theories: Fµν can be defined from the gauge potential as ∂µAν − ∂ν Aµ or as a generic antisymmetric tensor field which satisfies the Bianchi identity µνρσ∂ν Fρσ = 0 (see Exercise 6).

The vector superfield 443

Note that one may avoid altogether to work in the Wess–Zumino gauge and take

A supersymmetric action must involve a term F µν Fµν and therefore be quadratic in W α. An obvious candidate is

Thus one recovers the Lagrangian described in Chapter 3.

One may construct the supersymmetry current associated with this system (as we saw in Section 3.1.5 of Chapter 3, this should not be confused with the Noether current associated with the transformation (C.70)):

If we want to couple this to matter described by a chiral supermultiplet φ, we must implement a gauge transformation at the superfield level. Since Λ in (C.65) is a chiral superfield and its scalar component gives the gauge parameter (α(x) = −2gRe a(x)), an obvious candidate is

444 Superfields

which reads in terms of component fields

where Dµφ ≡ ∂µφ − igqAµφ and Dµψ ≡ ∂µψ − igqAµψ are the covariant derivatives. Returning to R-symmetry, we see that V being real does not transform. Thus since

and W αWα has R-charge + 2: the action (C.71) is invariant under R-symmetry.

At the component field level, let us note that the gaugino λα being the lowest

The action (C.74) is obviously R-symmetric at the superfield level. One checks that as well, at the component field level, a term such as λψϕ has R-charge 1+(r −1)−r = 0.

For the case of nonabelian gauge symmetry, we refer the reader to the references below (see for example [362]) and only sketch the results. The vector superfield has a matrix structure V = V ata, where ta are the generators of the gauge group. The gauge transformation (C.66) takes the form

5Note that in the limit of abelian symmetry, Wα = −gWα. We have restored the gauge coupling in the nonabelian case because of the self-couplings of the gauge sector.

g being the gauge coupling.

Finally, the coupling of the chiral field to the vector field is given as in (C.74) by

We conclude this appendix by introducing another example of a superfield, which describes the supermultiplet associated with an antisymmetric tensor field bµν (known as a Kalb-Ramond field [244]). Such a field is present in string theory where it plays a central rˆole in cancelling anomalies (see Section 10.4.2 of Chapter 10).

The linear superfield is a real superfield L which satisfies the covariant constraints:

We note that none of the fields of the linear multiplet are auxiliary fields.

6The Cabc are the structure functions: ta, tb = iCabctc.

446 Superfields

We show in equation (10.120) of Chapter 10 that an antisymmetric tensor field is Hodge dual to a pseudoscalar field. This equivalence extends to the full supermultiplets and a linear superfield is dual to a chiral supermultiplet. Indeed, consider the following Lagrangian:

where X is a real superfield, S is chiral (and thus S† antichiral).

We may vary the action with respect to S, or better since S is a constrained (chiral)

¯

respect to Σ. Similarly for S†. This gives D2X = D2X = 0 which shows that X is a linear superfield L. Thus the action (C.90) coincides with (C.89).

Alternatively, one may minimize with respect to X, which gives the superfield

Thus (C.89) and (C.92) describe two theories equivalent on-shell (we have used equations of motion).

Further reading

•J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton series in physics.

•J.-P. Derendinger, Lecture notes on globally supersymmetric theories in four and two dimensions, preprint ETH-YH/90-21.

Exercises

Exercise 1 Show that

Exercise 2 Derive from (C.29) and (C.39) the supersymmetry transformations and Lagrangian for a chiral superfield in four-component notation, i.e. (3.10) and (3.19) in

7One may show that any chiral superfield may be written in this way. The unconstrained superfield Σ is known as a prepotential.

(a)Using (C.69), show that (C.71) is written in terms of the component fields as (C.72).

(b)Prove similarly (C.75).

(c)Show that the four-component spinor versions of (C.72), (C.70), and (C.75) are given by the equations (3.38), (3.40), and (3.43) of Chapter 3.

Exercise 5 Prove, using (C.19), that the components of the full vector supermultiplet transform under supersymmetry as:

δS C = i(ηχ − η¯χ¯)

δS χα = ηα(M + iN ) − i (σµη¯)α (Aµ − i∂µC),

Exercise 6 Express the constraint (C.68) in terms of the component fields using the decomposition (C.69).

Exercise 7

2¯ 2 †

(a)Express D D K(Φi, Φj ) in terms of derivatives of K with respect to Φi and/or

448 Superfields

Exercise 8 We discuss in this exercise the reparametrization invariance of the K¨ahler manifold associated with a supersymmetric nonlinear sigma model.

¯ ¯ ¯

Let us redefine the scalar fields according to φ = φ(φ ), φ = φ(φ ). The kinetic term (C.54) is invariant if the K¨ahler metric transforms as:

(a)How does the inverse metric gi¯ transform? Show that the Christo el symbol (C.56) transforms as:

(b)Show that in order that the kinetic term for the fermion fields in (C.58) be invariant, one must impose

(c)Using (a) show explicitly that, if ψi transforms as (C.96), then Dµψi, as defined in (C.59), transforms as a contravariant vector.

(d)Show that:

with ad − bc = 1 amounts to a K¨ahler transformation (C.52). Such a chiral superfield appears in string theory: its scalar component is the string dilaton whose vacuum expectation value provides the gauge coupling g ( S = 1/g2). Thus, the ‘S-duality’ transformation (C.98) with the choice a = d = 0 and b = −c = 1 corresponds to a strong/weak coupling duality (see Chapter 10).

Hints: F (S) = ln (icS + d).