Supersymmetry. Theory, Experiment, and Cosmology

No-scale models 149

6.12No-scale models

In the context of electroweak symmetry radiative breaking, we have seen that the electroweak symmetry breaking scale comes out naturally small compared to the Planck scale. This however does not determine the gravitino mass which fixes the masses of the supersymmetric partners. Indeed, only dimensionless quantities such as mass ratios (e.g. gaugino mass over m3/2 ) or A-terms play a rˆole in the determination of the electroweak scale. However, if we do not want to destabilize the electroweak vacuum through radiative corrections, the overall supersymmetric mass should be a low energy scale. It is thus desirable to look for a determination of the scale m3/2 through low energy physics.

Obviously, in this case, the gravitino mass should remain undetermined by the hidden sector: in other words, it should correspond to a flat direction of the scalar potential. Let us get some orientation with a hidden sector reduced to a single field z [92]. For the time being, we ignore low energy observable fields. The scalar potential may be written from (6.37)

One notes that the corresponding curvature, defined in (6.51), is proportional to the K¨ahler metric, i.e. Rzz¯ = − 23 Gzz¯. This property defines an Einstein–K¨ahler manifold.

One may redefine the coordinates on the K¨ahler manifold by performing the replacement f (z) → z. Then,

In order to discuss the symmetries associated with such a Lagrangian, we define z = (y + 1)/(y − 1) such that L = 3 ∂µy∂µy/¯ 1 − |y|2 2, which is invariant under:

12A representation of this group is provided by the matrices ¯ which, acting on vectors , b a¯ y2

leave invariant the quadratic form |y1|2 − |y2|2 (to be compared with |y1|2 + |y2|2 for the compact group SO(2)).

150 Supergravity

This includes:

˜ →

(i) imaginary translations, forming a noncompact U (1)a group: z z + iα;

(ii)dilatations: z → β2z;

(iii)conformal transformations: z → (z + i tan θ)/(i tan θ + 1).

The complete supergravity Lagrangian is invariant under this symmetry, except

˜

This is invariant under Ua(1) but not under dilatations since it has dimension d = 3 (see Section A.5.1 of Appendix Appendix A), nor on conformal transformations (under which the gravitino and the Goldstino transform di erently). This term thus breaks

˜

SU (1, 1) down to the noncompact Ua(1).

Because, by construction, the tree level scalar potential is flat, we must rely on

quantum fluctuations to determine m2 = eG : since supersymmetry is broken, these

3/2

radiative corrections should lift the flat direction. If only light fields (with a mass of order MW or µEW as defined in (6.85)) contribute, then one expects m3/2 to be of the same order. On the other hand, if superheavy supermultiplets of mass of order MU (with mass splittings of order m3/2 ) contribute, then one expects a vacuum energy

towards MU .

We thus need to find models where superheavy matter supermultiplets do not feel supersymmetry breaking. We construct in what follows such models [130]. Let us

where h is a real function of n chiral supermultiplets (i = 1, . . . , n). We note φ , I = 0, . . . , n the scalar fields φ0 ≡ z and φI=i ≡ φi. The scalar potential then reads, from (6.37), setting κ = 1,

From the explicit form of the K¨ahler potential, we deduce that Gi = G0hi, G¯ı = G0h¯ı and

Hence the scalar potential (6.141) is reduced to its D-terms and is flat along the D-flat directions.

Exercises 153

Exercise 5 In the model described by the superpotential (6.128) and the K¨ahler potential (6.129), show that, in the limit κ → 0, the e ective scalar interactions are described by the Lagrangian (6.132).

Hints: Start from the supergravity scalar potential (6.36). Writing the full K¨ahler

where gpq¯ = gp(0)q¯ +κgp(1)q¯ +κ2gp(2)q¯ (gp(iq¯) is the metric associated with the K¨ahler potential K(i)). The inverse metric then reads:

where gqp¯ is the inverse metric of gpq¯:

gqp¯ = g(0)¯qp − κg(0)¯qrgr(1)s¯ g(0)¯sp − κ2 g(0)¯qrgr(2)s¯ g(0)¯sp − g(0)¯qrgr(1)s¯ g(0)¯stgt(1)u¯ g(0)¯up .

Hints: (b) Use (6.143) to show that the first term in (6.149) does not contribute to

ST rM 2.

7

Phenomenology of supersymmetric models: supersymmetry at the quantum level

Since the Standard Model has been tested to a level where quantum corrections play an important rˆole, it is expected that any theory beyond it will have to be tested at the same level. From this point of view, the odds are in favor of supersymmetric theories since supersymmetry has been constructed from the start to control quantum corrections. Since this chapter deals with the confrontation of supersymmetry with the real world, we first discuss in some detail this important issue of stability under quantum corrections.

Obviously, supersymmetry is broken. The way supersymmetry is broken should have some decisive impact on the search of supersymmetric particles. We thus review the main scenarios of supersymmetry breaking, mostly from a phenomenological point of view. More theoretically oriented issues are postponed till Chapter 8.

We conclude this chapter by a concise review of the search for supersymmetric particles. Since this is constantly evolving with the accumulation of new data and the progress on collider energies, I have taken the conservative approach to give the limits obtained at the LEP collider, except otherwise stated. The reader is referred to the proceedings of recent summer conferences for updated values.

7.1Why does the MSSM survive the electroweak precision tests?

In view of the many successes of the Standard Model under precision tests (see Section A.4 of Appendix Appendix A), it might come as a surprise to realize that the MSSM is also surviving these tests. Indeed, most of the theories beyond the Standard Model fail to do so because the extra heavy fields that one introduces induce undesirable radiative corrections. Supersymmetry has been introduced in order to better control these radiative corrections. It is thus not completely surprising that it fares better in this respect. It actually does so much better than other theories that it is often stated that the more standard the theory of fundamental interactions seems to be, the more likely it is to be supersymmetric (spontaneously broken). Although we will see in Chapter 12 that such a statement has its limits, this is the reason why

Why does the MSSM survive the electroweak precision tests? 155

supersymmetry is considered as the most serious candidate for a theory beyond the Standard Model.

One of the predictions of the Standard Model that is most di cult to reproduce is the fact that the ρ parameter defined as

M 2

ρ = W , (7.1) MZ2 cos2 θW

has a value very close to 1. Indeed, this value of 1 is predicted at tree level (see equation (A.136) of Appendix Appendix A) and receives only corrections of a few per cent from radiative corrections (see Section A.4 of Appendix Appendix A). To discuss departures

from the tree level value ρ = 1 it proves to be useful to use a global symmetry known as the custodial SU (2) symmetry. This concept was developed [341] in the context of technicolor theories for which this test proved to be very damaging. It can be used to discuss radiative corrections to the ρ parameter in the framework of the Standard Model as well as of any theory beyond the Standard Model.

This custodial SU (2) symmetry is a global symmetry under which the gauge fields Aaµ, a = 1, 2, 3 of the SU (2) symmetry of the Standard Model transform as a triplet: as we will see in the next Section, the presence of such a symmetry ensures that the parameter ρ is 1. Deviations from 1 are associated with the breaking of this symmetry: for example, (t, b) is a doublet of custodial SU (2), and the fact that the top mass is much larger than the bottom mass is a major source of breaking. The precision tests from the LEP collider have taught us that the breaking of custodial SU (2) in the Standard Model is in good agreement with the one observed in the data. We develop these considerations in the next two sections.

7.1.1Custodial symmetry

We first prove the following statement:

In any theory of electroweak interactions which conserves charge and a global SU (2) symmetry under which the gauge fields Aaµ, a = 1, 2, 3 of the SU (2) symmetry of the Standard Model transform as a triplet, one has ρ = 1.

A mass term for the gauge bosons Aaµ necessarily transforms under this custodial SU (2) as1 (3 × 3)s = 1 + 5. In order not to break the global SU (2) the mass term must be a SU (2) singlet and the corresponding mass matrix is simply of the form

M 2δab.

We then add the U (1)Y gauge field Bµ. Since charge is conserved, there is no mixed term of the form AaµBµ, a = 1, 2, in the mass matrix (cf. (A.114) of Appendix Appendix A). Thus the complete mass matrix for the electroweak gauge bosons in the basis A1µ, A2µ, A3µ, Bµ is of the form:

156 Phenomenology of supersymmetric models: supersymmetry at the quantum level

charge is assumed to be conserved, we expect one massless neutral field, the photon,

This proves the statement above.

This has many interesting applications:

(i)Strong interactions conserve electric charge and strong isospin. We may thus choose strong isospin as our global SU (2). For it to be a symmetry in the presence of the gauge fields Aaµ, we must transform these fields under this global symmetry. We conclude that the relation ρ = 1 remains true to all orders of strong interaction. On the other hand, weak interactions violate strong isospin and we expect violations of order g2 GF MW2 .

(ii)Let us consider the Higgs potential ((A.127) of Appendix Appendix A):

which shows that V is invariant under O(4) SU (2) × SU (2). This symmetry

√

is spontaneously broken into O(3) SU (2) by φa = δa0 v/ 2. Thus ρ = 1 to all orders of scalar field interactions. This remains true for any number of Higgs doublets (since one can redefine them in such a way that a single one acquires a vacuum expectation value). On the other hand, in presence of a triplet of Higgs, O(3) is broken down to O(2), which is insu cient as a custodial symmetry: ρ = 1.