Supersymmetry. Theory, Experiment, and Cosmology

where m0 and A0 are typically of order m3/2.

The original motivation was that gravity is the messenger of supersymmetry breaking and that gravity is “flavor blind”. For example, the Polonyi model which illustrated in Section 6.3.2 the simplest example of gravitationally hidden sector yields universal soft terms. However, this turns out to be often an undue simplification: in many explicit models of supersymmetry breaking in a gravitationally hidden sector (such as those inspired by superstring theories), the soft supersymmetry breaking terms are not flavor blind. It remains however true that this property is recovered in some special limits: the minimal supergravity model thus addresses such cases.

As stressed earlier, such a universality of scalar masses is not stable under quantum fluctuations and renormalization down to low energies leads to nonuniversalities and thus FCNCs.

The attractiveness of the minimal supergravity model is its small number of parameters, besides those of the Standard Model:

m0, M1/2, tan β, A0, sign µ.

Let us note in particular that |µ| is determined through the condition (6.81) of SU (2)× U (1) breaking. The low energy parameters m21 and m22 may be expressed in terms of the high energy parameters m0, A0 and M1/2, which gives for example for tan β = 1.65

This already shows that, for small tan β, the minimal supergravity model requires some level of fine-tuning: as the experimental limits on squark, slepton and gaugino masses are raised, this pushes higher m0, M1/2 and thus (6.99) requires some finetuning among large parameters. We return to this issue in Section 6.10. We also see

The remaining sections, which lie in the “Theoretical Introduction” track, are somewhat technical, although they deal with important issues: dynamical determination of key parameters, and fine tuning. The phenomenology oriented readers may have interest to skip these sections and return to them at a later stage.

6.9Infrared fixed points, quasi-infrared fixed points and focus points

If the fundamental theory underlying the Standard Model lies at a scale which is close to the Planck scale, this means that, by studying today the Standard Model, we are looking at the deep infrared regime of the theory. The renormalization group flow might thus lead us to some specific corners of the parameter space. We will illustrate in this section this possibility with a few examples.

142 Supergravity

fixed point, probably to stress that, although the behavior has become universal, the couplings have not completely fallen yet into the Pendleton–Ross (quasi-)fixed point.

If the value of the top Yukawa coupling is large at some superheavy scale such as the grand unification scale, then we just saw that we can estimate the value of tan β through (6.100). The top mass which is present in this equation is the running mass mt(mt). This should be distinguished from the physical top mass Mt, which receives two important corrections at one-loop: QCD gluon corrections10 and stop/gluino corrections. Taking these corrections into account [69], one obtains a value of tan β of the order of 1.5. As we will see in Chapter 7, such a value is not favored by data because it gives too light a Higgs. For tan β ≥ 30, the bottom Yukawa coupling λb = λt(mb/mt) tan β cannot be neglected and the previous analysis must be changed: it turns out that, for values of tan β larger than 30, mt is decreasing with tan β.

6.9.2Focus points

Besides fixed points, there is the possibility of renormalization group trajectories focussing towards a given point: the evolution does not stop there and, as in the focussing of light, the trajectories diverge again beyond the focus point. It has been pointed out [153–155] that such focus points could be of interest if they correspond to the electroweak scale. Then, the low energy physics would be less dependent on the boundary values of parameters at high energy. This puts in a di erent perspective the discussion of naturalness which we will undertake in the next section. We will therefore discuss in some details the nature of such focus points.

The set of renormalization group equations that one has to solve at the one loop level is presented in Appendix E. They turn out to have a very specific structure which allows us to solve them, at least formally. We start by solving for the gauge couplings from which we can infer the evolution of the gaugino masses (see (6.88) above). This allows us to solve for the A-terms. In parallel, we may obtain the evolution of the Yukawa couplings following the methods developed in the previous section (we will apply it in Section 9.3.3 of Chapter 9 to obtain the evolution of λb).

Finally, there remains mainly to solve the evolution equations of the soft scalar squared masses, i.e. equations (E.17)–(E.23) of Appendix E. They turn out to involve only the following combinations

The evolution equations can thus be transformed into a set of equations for these variables, and by adding to each mass squared m2i (i = H1, H2, B, . . .) an appropriate combination of Xt, Xb and Xτ , we obtain a new variable m0i whose evolution depends only on gauge couplings and gaugino masses. We will illustrate this below.

10In the relevant renormalization scheme (DR), described in Section E.8 of Appendix E,

See Section 9.3.3 of Chapter 9 for a discussion of similar corrections in the case of the bottom quark.

The issue of fine tuning 145

It expresses the scale of electroweak symmetry breaking in terms of the parameters of the low energy theory (say at µ = MZ ). In the approach that we adopt in this chapter and the following ones, the relevant parameters are rather the ones at the high energy scale (µ0 = mP or MU or the string scale). One should therefore express m2H1,2 and |µ|2 in terms of the high energy parameters. This is what we did in (6.99) for the minimal supergravity model. More generally it takes the following form:

where mi represents a generic parameter of the softly broken supersymmetric Lagrangian (scalar mass, gaugino mass, A-term or |µ|). Obviously, the parameters corresponding to the largest coe cients Ci or Cij will have the most impact on the discussion of fine tuning: a small variation of these parameters induces large corrections which must be compensated by other terms in the sum (6.122). Moreover, the nonobservation of supersymmetric partners imposes lower limits on these parameters which are often much higher than MZ : the sum on the right-hand side thus equals MZ at the expense of delicately balancing large numbers one against another.

It turns out that, quite generically, it is the gluino mass M3 which appears in the sum with the largest coe cient. We are going to show this analytically by using the methods developed in the previous sections. We make the following simplifying assumptions: we neglect in the renormalization group equations terms proportional to g12, g22, λ2b and λ2τ ; we also set the A-terms to zero. Then from (E.5), (E.17), and (E.18) of Appendix E, m2H1 is not renormalized whereas we have

where Yt and Xt are defined in (6.104) and (6.112), and t = ln µ/µ0. The solution for Yt is written in (6.108). The equation of evolution for Xt is given in (6.113). We see that it involves a term in g32(t)M32(t) = g36(t)M32(0)/g34(0) (we have used (6.88)): it is this contribution that induces a term proportional to M32(0). Solving (6.113) for Xt (see Exercise 4), one obtains, in the limit λt(0) g3(0):

We see that the fine tuning gets larger for values of tan β close to one (the coe cients Ci behave as (tan2 β − 1)−1). On the other hand, the fine tuning becomes independent of tan β for values significantly larger than 1 (but not too large, otherwise the λb coupling may not be neglected).

Our approximate result may be compared with complete computations. For exam-

The large coe cient of the M32(0) term (and the nondiscovery so far of the gluino, which sets a lower limit for M3) imposes to envisage a cancellation among large numbers, in order to obtain the right scale MZ . We note on the other hand that neither M1 nor M2 play a significant rˆole in this discussion about fine tuning, except obviously if they are related to M3 as in the minimal supergravity model.

Several authors [23, 128] have tried to quantify the amount of fine tuning in the following way: writing MZ2 = MZ2 (m1 · · · mn) as above, they define

One obtains, at the tree level, ∆max > 100 for tan β = 1.65 or ∆max > 40 for tan β < 4. But including one-loop corrections [25] stabilizes the model and the fine-tuning is only ∆max > 8 for tan β < 4.

One may note that correlations between the di erent parameters arising from a given supersymmetry breaking mechanism may decrease the degree of fine tuning. Obviously, a correlation between M3 and µ would serve this purpose but might be di cult to realize. Otherwise, one may envisage correlations between M3 and mH2 or between the gaugino masses (which would then imply M3 < M1 or M2).

If we return to the Higgs mass, we may note an apparent paradox. As the experimental limits on the lightest Higgs mass are larger than MZ cos 2β, one must take into account loop corrections to the mass (all the more as tan β is closer to 1).

The µ problem 147

These corrections given in equation (5.36) of Chapter 5, depend logarithmically on the stop mass. They may be increased by exponentially increasing the stop mass, i.e. by increasing accordingly the values of the soft parameters, in particular M3(0) [246]. This is obviously at the expense of increasing the amount of fine tuning.

6.11The µ problem

We have seen in Chapter 5 that the only mass scale of the low energy supersymmetric theory is the µ parameter: all other mass scales are provided by supersymmetry breaking. If the energy scale of the underlying fundamental theory is superheavy (grand unification, string or Planck scale), it is di cult to understand why µ is not of this order. This has been called the µ problem [250].

Of course, a symmetry could ensure that µ vanishes at the large scale. But this has to be broken at low energy, otherwise the low energy theory has a U (1) Peccei– Quinn symmetry [303] which is broken by the vacuum expectation values of H1 and H2, leading to an unwanted axion. Moreover, in the simplest supergravity models, the soft parameter Bµ turns out to be proportional to µ and Bµ = 0 implies H10 = 0 or

H20 = 0 (see (5.25) of Chapter 5).

It is thus important to find at low energy a dynamical origin for the µ term. One possibility is the one discussed in Section 5.6 of Chapter 5: a cubic term λSH2 · H1 gives rise to the µ term when the singlet field S acquires a vacuum expectation value at low energy (µ = λ S ). The µ term may also be generated by radiative corrections or by nonrenormalizable interactions.

To date, the most elegant solution has been given by Giudice and Masiero [190] who have shown that, in a supersymmetric theory with no dimensional parameters (besides the fundamental scale), the µ parameter may be generated by supersymmetry breaking. Of course, the fundamental problem of how to generate a scale of supersymmetry breaking much smaller than the fundamental scale remains open. We will return to it in Chapter 8.

To illustrate the Giudice–Masiero mechanism, we consider a supergravity model along the lines of Section 6.3.2: the hidden sector consists of Nh chiral superfields with scalar component zp and the observable sector of No chiral superfields with scalar component φi (and their conjugates). We now make the assumption that there is no mass scale in the observable sector: observable fields have only renormalizable couplings among themselves and gravitational couplings (suppressed by powers of κ) with the hidden sector. Introducing the dimensionless hidden fields ξp ≡ κzp, we thus