Supersymmetry. Theory, Experiment, and Cosmology

6

Supergravity

There are two directions one might take to generalize supersymmetry and its models as we have presented them until this point: (a) to try to incorporate gravity as the fourth fundamental interaction, or (b) by using our experience with gauge theories, to try to make supersymmetry local. It turns out that these two attempts converge: local supersymmetry makes it necessary to introduce a graviton field (and its supersymmetric partner the gravitino) and therefore gravity. Thus the theory of local supersymmetry is called supergravity.

Let us note immediately that this is not yet a full quantum theory of gravity: just as gravity, supergravity is nonrenormalizable; its coupling constant κ is still the dimensionful Newton’s constant κ G1N/2 and interactions scale like powers of κ.

It can thus be understood as an e ective theory of a full-fledged quantum theory

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of gravity whose fundamental scale is typically the Planck scale MP ≡ hc/G¯ N 1.22 ×1019 GeV. We will return to these considerations at the beginning of Chapter 10, when we discuss string theory, which represents, to date, the most elaborate attempt to write a quantum theory of gravity coupled with matter.

For the time being, we will content ourselves with this nonrenormalizable theory of supergravity, which still represents major progress since it allows us to consider the theory up to scales a few orders of magnitude smaller than the Planck scale. We recall that it was precisely one of the motivations for supersymmetry to push the appearance of new physics to superheavy scales, the hope being that the theory becomes simpler. In this chapter, we will thus renormalize the low energy supersymmetric models up to scales close to the Planck scale. This leads to a new way of envisaging model building: a few assumptions on masses and couplings at this high energy scale, to be discussed later, allow us to restrict the parameter space of the MSSM. This gives in particular the so-called minimal supergravity model which has provided a benchmark for discussing the phenomenology of supersymmetry at high energy colliders, especially hadron colliders.

6.1Local supersymmetry is supergravity

Let us start from global supersymmetry and make the supersymmetry transformations local. In other words, the parameter ε of the transformation is made spacetime dependent: ε(x). Then following (3.5) of Chapter 3, we have

Local supersymmetry is supergravity 121

Hence the commutator of two local supersymmetry transformations is a spacetime dependent translation, what is known as a spacetime general coordinate transformation. It is precisely by making the relativity theory invariant under local coordinate transformations that Einstein has derived the theory of general relativity. Thus one expects, as immediately stressed by Wess and Zumino [364] in the first article on fourdimensional supersymmetry, that local supersymmetry will lead to a theory of gravity. This theory is known as supergravity.

We expect to find in the particle spectrum a graviton field (of spin 2) and its supersymmetric partner, a field of spin 3/2, the gravitino. The gravitino plays a fundamental rˆole in supergravity, as we will now see.

Let us first recall how one makes a gauge symmetry local in order to follow the same steps with supersymmetry (see Section A.1.3. of Appendix Appendix A for more details). The simple action

4 ¯ µ

S0 = d x Ψ(x) iγ ∂µ Ψ(x)

is not invariant under the local transformation: Ψ(x) → eiθ(x) Ψ(x). Infinitesimally

be invariant under local gauge transformations. Finally one includes a dynamics for the gauge field by adding a kinetic term

Let us now consider supersymmetry transformations. Since the parameter of the transformation carries a spinor index r (εr(x)), the equivalent of (6.3) necessarily reads

There is therefore a need for a field carrying a spin 1 and a spin 1/2 index, which amounts to a spin 3/2 (and spin 1/2) field: the gravitino field. As above, we follow the Noether procedure to couple this field to matter and then give this field a dynamics.

122 Supergravity

Let us start, as an example, with the kinetic terms of the Wess–Zumino model:

¯ µ

S0 = Ψγ ∂µΨ (6.7)

√

where φ = (A +iB)/ 2 and Ψ is a Majorana spinor. We have seen that S0 is invariant under the global transformation (3.3), (3.4) of Chapter 3:

the canonical dimension of the Majorana spinor εr being −1/2. If we make εr spacetime dependent, S0 is no longer invariant:

We thus introduce a new field ψrµ(x) with vector and spinor indices and a new term in the action

Since ψµr is a spinor field, it has canonical dimension 32 , just as Ψ; and the coupling κ has dimension −1. In other words, the theory has a dimensionful coupling and is obviously nonrenormalizable. The supersymmetry transformation of ψµr is chosen in order to cancel (6.9):

Note that dimensions match.

However, contrary to the previous case, the story does not end here since S0 +S1 is not invariant under local supersymmetry. The transformation of S1 yields for example

where one notices between the brackets the energy–momentum tensor Θµν for the scalar field. Remembering that Θµν is obtained by varying the action with respect to

The corresponding spin 2 field – the graviton – is obviously the supersymmetric partner of the gravitino, a spin 3/2 field, and the constant κ is nothing but the gravitational

manifold and the K¨ahler metric is the metric on this manifold. One speaks of a flat K¨ahler metric if gi¯ = δij . If the metric is nonflat, the nonnormalized kinetic term is obviously not renormalizable. Nonrenormalizable interaction terms are also generated besides the kinetic term, for example a four-fermion interaction: they are proportional to derivatives of the K¨ahler metric. [A more technical introduction to K¨ahler potential is given in Appendix C.]

The complete supergravity Lagrangian is invariant under the K¨ahler trans-

where F φi is an analytic function of the scalar fields φi. For example, the K¨ahler metric is invariant under (6.21). The specific choice κ2F φi = ln W φi shows that the supergravity Lagrangian only depends on the combination [91]:

We refrain from using this function in what follows because the K¨ahler invariance of the Lagrangian is no longer explicit.

Finally, we note that the K¨ahler transformation acts, in a way similar to a R-symmetry, as a chiral U (1) rotation on the individual fermion fields. More precisely, one has:

where the chiral weight is w = 1 for the gravitino and gaugino fields, w = −1 for a fermion in a chiral supermultiplet 2.

•A kinetic function f (φi) for the gauge fields which determines the gauge field kinetic term:

2We have assumed here that all scalar fields have vanishing chiral weights. [More precisely, using the superfield language of Appendix C, one should note that the θ variables have chiral weight w = 1, i.e. transform as θα → e−iImF/2θα. Thus, if a scalar fields has chiral weight w0, its fermion partner has weight w0 − 1.]

There are several comments to be made on this (locally) supersymmetric term in the action.

First, the term quadratic in the gravitino field looks very much like a mass term:

where we have used the gauge condition (see next Section) γµψµ = 0. But although this resembles a gravitino mass term, this is not su cient to identify it as such: if there is only the term (6.39) besides (6.18), supersymmetry is conserved and the gravitino is massless!

Second, the cosmological constant term in (6.39) corresponds to a negative vacuum

energy: E0 = −3m2 /κ2. The corresponding spacetime is no longer Minkowski but

3/2

it is known as an anti-de Sitter space4. [This is the first time, and not the last one, that we encounter this type of space. It su ces to say that (anti)de Sitter space can be described geometrically as the surface of equation

with upper sign for de Sitter, lower sign for anti-de Sitter space.

The spacetime symmetries of such anti-de Sitter spaces are not described by the Poincar´e group O(1, 3) of translations and Lorentz transformations (as in Minkowski spacetime), but rather by the group O(2, 3).5]

In anti-de Sitter space, the definition of mass from quadratic terms in the action is subtle. Indeed, if one redefines the covariant derivatives and Riemann tensor in the field equations to account for the symmetries specific of such spacetime, the two terms coming from the action (6.39) are absorbed [348]. This solves the puzzle of the seemingly spurious mass term discussed above.

However, if we combine (6.38) with (6.39) and assume cancellation of the vacuum

energy:

3m2

4The de Sitter space familiar in the inflation scenario would correspond to a positive vacuum energy.

5The group O(m, n) is the group of linear transformations that leave invariant a metric with m positive and n negative diagonal components. Clearly, m = 2 and n = 3 for the metric (6.41).

We have thus traded the vacuum energy for the gravitino mass: the criterion for the spontaneous breaking of local supersymmetry is a nonvanishing gravitino mass. The physical meaning is obvious: since the graviton stays massless, a massive gravitino is a clear sign of supersymmetry breaking in the spectrum.

This is quite similar to the situation in spontaneous gauge symmetry breaking: the criterion is a non zero mass for the gauge vector field. One can in fact push the analogy further and speak of a super-Higgs mechanism. Indeed, the longitudinal degree of freedom of the massive vector field is provided by the Goldstone particle associated with the spontaneous breaking of the continuous gauge symmetry. Similarly, the missing degrees of freedom for the massive gravitino are provided by the massless Goldstone fermion or Goldstino that was necessarily present in the spectrum when we discussed the spontaneous breaking of global supersymmetry. The two degrees of freedom of a massless gravitino and the two degrees of freedom of the Goldstino make up the four degrees of freedom of the massive gravitino. Just as in the gauge case, the Goldstino thus does not appear in the physical spectrum of the theory when supersymmetry is local.

The previous analysis was obviously sketchy since it did not include the dynamics of supersymmetry breaking. One may take a more realistic approach by using the explicit coupling of matter to supergravity discussed in the preceding Section. This is relevant if supersymmetry is broken by the F -component of some chiral supermultiplet.

Besides the scalar potential (6.36), the complete Lagrangian includes a term quadratic in the gravitino field:

If eκ2K/2W (φ) = 0 in the ground state, then supersymmetry is spontaneously broken:

where we have used (6.30).

Assuming a vanishing vacuum energy V = 0, we have the following relation from

i.e. the vacuum value of an auxiliary field. As discussed in Section 3.1.2 of Chapter 3,