Supersymmetry. Theory, Experiment, and Cosmology

Further reading 331

where A(t) is the amplitude of oscillations at time t. The energy of oscillations is thus m2φ |A(t)|2: it can be viewed as a coherent state of particles of mass mφ and number

density m2φ |A(t)|2. The baryon number per particle is thus of order ϕ φ20/M 2. More precisely one finds

This shows that the A eck–Dine mechanism is very e cient to generate a baryon asymmetry.

Further reading

•T. Damour, Gravitation, experiment and cosmology, Proceedings of the 5th Hellenic School of Elementary Particle Physics (arXiv:gr-qc/9606079).

•D. Lyth and A. Riotto, Particle physics models of inflation and the cosmological density perturbation, Phys. Rep. 314 (1999) 1.

12

The challenges of supersymmetry

Although supersymmetry provides a satisfactory, and probably necessary, framework to address the questions left aside by the Standard Model, it is facing some di - cult challenges. We take the opportunity in this last chapter to discuss two of the most di cult questions: the flavor problem and the cosmological constant problem. The lack of a clear solution to either of these problems would be (is?) probably a sign that we are missing an important piece of the jigsaw puzzle and that, if there is supersymmetry, it is probably not realized in the exact way that we presently think it is.

12.1The flavor problem

One of the motivations for going beyond the Standard Model is to try to explain the variety of masses, i.e. the variety of Yukawa couplings. Any theory of mass, in fact any theory beyond the Standard Model, introduces new degrees of freedom. The quantum corrections associated with these new degrees of freedom tend to spoil the delicate balance achieved by the Standard Model in the flavor sector: mainly the strong suppression of Flavor Changing Neutral Currents (FCNC) and the presence of a single CP-violating parameter, the phase δCKM (besides the poorly understood θQCD parameter).

The latter property has been successfully verified in recent years. A useful tool to describe these experimental results is the unitary triangle. We refer the reader to Section A.3.4 of Appendix Appendix A for the definition of this triangle in the context of the Standard Model and we give here in Fig. 12.1 the experimental status in 2005. The parameters ρ and η appear in the Wolfenstein parametrization [379] of the Cabibbo–Kobayashi–Maskawa matrix:

The fact that all experimental data available are consistent with a small region with nonvanishing η shows that CP is violated and that its violation is consistent with a single origin, the phase of the CKM matrix.

The flavor problem 333

1.5

0.5

ρ

Fig. 12.1 Unitarity triangle and the experimental limits obtained in 2005 [76].

We note that the question of CP violation is intimately connected with the question of mass. For example, the condition for CP violation in the Standard Model may be summarized as (see (A.171) of Appendix Appendix A)

(m2t − m2c )(m2t − m2u)(m2c − m2u)(m2b − m2s)(m2b − m2d)(m2s − m2d)JCKM = 0,

(12.2)

which clearly involves the fermion mass spectrum besides the complex phases encapsulated in the quantity JCKM. CP violation has been a recurring theme in the last chapters. We know that all experimental results are in agreement with the CKM framework of the Standard Model. On the other hand, we expect other sources of CP violation to explain how baryon number was generated in the Universe. This is one of the basis for expecting physics beyond the Standard Model.

The natural procedure to account for the diversity of fermion masses is to introduce family symmetries. Such symmetries are not observed in the spectrum and thus must be broken at some large scale. A possible hint about this scale may be found in the seesaw mechanism for neutrino masses. In any case, fine tuning arguments require us to discuss such symmetries in a supersymmetric context.

But supersymmetry introduces many new fields, in particular sfermions, and thus many new sources of FCNC and CP violation. As discussed below [217], a generic extension has 44 CP-violating phases! This poses a severe problem and ways to control these sources have to be devised. There is thus a necessary connection between the discussion of supersymmetry breaking and the resolution of the (s)fermion mass problem.

334 The challenges of supersymmetry

In a supersymmetric framework, the Yukawa coupling matrices appear in the superpotential

generalizing the one-family superpotential considered in (5.2) of Chapter 5.

If we organize quark and lepton fields in columns associated with family and lines associated with Standard Model quantum numbers

we can distinguish

•vertical symmetries, such as the symmetries of the Standard Model or grand unification; such symmetries may be advocated to explain the intragenerational hierarchies of masses. Infrared fixed points or grand unified relations provide examples of how such vertical symmetries can be used.

•horizontal (or family) symmetries; such symmetries address the question of intergenerational hierarchies, in particular the question of why the first family is lighter than the second which is lighter than the third.

We will review the possible uses of both types of symmetries in what follows. Before doing this, we will recall where we stand experimentally to try to evaluate the seriousness of the mass problem in the context of supersymmetric extensions of the Standard Model.

We conclude this section by computing, as promised, the number of independent phases in a generic supersymmetric Standard Model. The three Yukawa 3 × 3 complex matrices in (12.3) involve 3 × 9 complex parameters. Similarly for the A-terms. The five 3 × 3 hermitian mass-squared sfermion masses involve 5 × 3 real and 5 × 3 complex parameters. Finally in the gauge sector, we count four real parameters (g1, g2, g3 and

θQCD ) and three complex (M1, M2, M3); in the Higgs sector, two real (m2H1 , m2H2 ) and two complex (µ, Bµ) parameters. In total, 95 real parameters and 74 imaginary

phases.

We have not yet taken into account the possible field redefinitions. We follow the method already used in Section 7.7 of Chapter 7: the various couplings are spurion fields that break the global symmetry U (3)5 × U (1) × U (1)R (the charges under the latter two symmetries are given in Table 7.4 of Chapter 7) into U (1)B ×U (1)L. We can thus remove 15 complex parameters and 15 phases. We are left with 80 real parameters and 44 imaginary phases.

12.1.1The supersymmetric CP problem

When we consider supersymmetric extensions of the Standard Model, new sources of CP violation appear both in flavor diagonal and flavor violating interactions.

We have encountered the first case when we discussed the neutron electric dipole moment in Section 7.7 of Chapter 7. Even in a universal model (with two residual

The flavor problem 335

CP violating phases ϕA,B ), one computes a neutron electric dipole moment of the order of

which misses the experimental bound [144] by two orders of magnitude for m˜ 100 GeV and ϕA,B 1.

As for flavor violating interactions, sfermion mixing leads to new sources of FCNC processes both CP conserving and CP violating. It has become customary to parametrize these flavor violations in the so-called mass insertion approximation [219]. The idea is to work in the basis for fermion and sfermion states where all couplings to neutral gauginos are diagonal: flavor violating e ects appear through the nondiagonality of mass matrices for sfermions of the same electric charge. Denoting by m2δ such nondiagonal terms (m is an average sfermion mass), one may parametrize the

336

More precise limits are given in Table 12.1. One sees that the experimental constraint

stringent limits on δLRd than on δLLd . In the case of , this is because of a cancellation between the contributions of the box and the penguin diagrams to δLLd . In the case of b → sγ, the helicity flip needed with a δLRd mass insertion is found in the internal gluino line, which leads to an enhancement factor of mg /mb over the amplitude with

a δLLd insertion.

We give also in Table 12.1 limits on the flavor-conserving CP-violating mass

in the limit mg = mq ≡ m˜ .

12.1.2Supersymmetry breaking versus flavor dynamics

Obviously the spectrum of masses and mixing angles in the supersymmetric sector plays a key rˆole in the solution to the supersymmetric FCNC and CP problem.

If we go to the basis of squark mass eigenstates, we have for example

where δmQ(D) is the mass di erence between the two left (right) down squarks and

K12dL(R) the gluino couplings to left (right) handed down quarks and squarks.

We have seen in Section 6.7 of Chapter 6 that there are three possible solutions to the supersymmetric flavor problem which requires small mass insertions:

•universality (e.g. δm2Q m2Q);

•e ective supersymmetry (e.g. m2Q MW2 );

•quark-squark alignment (e.g. K12d 1);

Regarding CP violations specifically, an approximate CP symmetry which would ensure all CP-violating phases to be small is no longer a valid possibility since the phase measured in B → ψKS is of order one [136, 137].

A general discussion of these issues relies on two scales: the scale ΛS of supersymmetry breaking and ΛF the scale of flavor dynamics. If ΛF ΛS , one expects a sparticle spectrum which is approximately flavor-free. Universality is thus favored. On the other hand, if ΛF ≤ ΛS , supersymmetry breaking and flavor dynamics are intimately connected: one has to resort to other solutions such as heavy supersymmetric partners or alignment.

We now discuss models illustrating the di erent possibilities.