Supersymmetry. Theory, Experiment, and Cosmology

Model building 89

where H1 · H2 ≡ εij H1i H2j selects the combination which is invariant under SU (2)L (ε12 = −ε21 = 1). The parameter µ will turn out to be the only dimensionful supersymmetric parameter2 (cubic terms in the superpotential have dimensionless couplings). If the fundamental scale of the underlying theory is very large, unification scale MU of order 1016 GeV or Planck scale, one should explain why the scale µ is so much smaller (as we will see, a superheavy µ would destabilize the electroweak vacuum). This is known as the µ-problem.

Cubic terms in the superpotential yield the Yukawa couplings of the Standard Model

with notation similar to (5.1) and color indices suppressed. Indeed, plugging this superpotential into equation (3.30) of Chapter 3 gives (beware of signs which arise from the ij contractions denoted with a dot)

Let us stress that hypercharge conservation requires to use the Higgs doublet H2 in the up-type quark coupling because the “chiral” nature of the superpotential forbids to use H1 . We thus need H1 to give nonvanishing mass to down-type quarks and charged leptons, and H2 to give mass to up-type quarks3.

A very nice property of the Standard Model lies in the fact that its Yukawa couplings form the most general set of couplings compatible with gauge invariance and renormalizability. This is unfortunately not so in the case of its supersymmetric extensions, because there are more scalar fields (the squarks and sleptons) and more fermion fields (the Higgsinos). Indeed the following superpotential terms are allowed by gauge invariance and renormalizability (which allows terms up to dimension three in the superpotential):

They are potentially dangerous because they violate lepton or baryon number.

We will define the MSSM as the model with minimal field and coupling content. Whereas the couplings in (5.2) are necessary in order to give quarks and leptons a mass, the latter couplings are not. We therefore assume that they are absent in the MSSM. But, before going further, we will give a rationale for such an absence.

5.2.2R-parity

The presence of the couplings (5.4) in the theory leads through the exchange of a squark or a slepton to e ective four-fermion interactions in the low energy regime which violate lepton number (for the first two) or baryon number (for the third one). This was clearly undesirable, especially at a time when one thought that squarks and

2Beware that there is no general agreement on the convention for the sign of µ, i.e. on the sign in front of (5.1). This is quite unfortunate because this sign turns out to play an important rˆole in the physical consequences, as we will see later. It is therefore important when comparing results to check the authors’ convention on the sign of µ.

3This is why one often finds in the literature the notation: H1 ≡ Hd and H2 ≡ Hu.

90 The minimal supersymmetric model

Table 5.2

sleptons might be fairly light. In order to avoid this, P. Fayet [148, 149] postulated the existence of a discrete symmetry, known as R-parity. The fields of the Standard Model have R-parity +1 and their supersymmetric partners R-parity −1.

which shows the connection of R-parity with baryon (B) and lepton (L) numbers. Such a parity obviously forbids the single exchange of a squark or a slepton between ordinary fermions

At the level of the superpotential, Q, L, U c, Dc and Ec have R-parity −1 whereas H1 and H2 have R-parity +1 (in each case, the R-parity of the corresponding scalar field). Then obviously terms in (5.2) are invariant whereas terms in (5.4) are not.

This R-parity may find its origin in a R-symmetry, a continuous U (1) symmetry which treats di erently a particle and its supersymmetric partner, i.e. a continuous U (1) symmetry which does not commute with supersymmetry (see Chapter 4 for details). Gaugino fields are for example charged under such a symmetry. Supersymmetry breaking induces gaugino masses and thus breaks this continuous symmetry down to a discrete symmetry, R-parity. We will see later that R-symmetries often play a central rˆole in supersymmetric theories.

For the time being, we remain at a phenomenological level. The assumption of R-parity has some remarkable consequences:

(i)Supersymmetric partners are produced by pairs: since the initial state is formed of ordinary matter, it has R-parity +1. A final state such as slepton–antislepton or squark–antisquark pair has also R-parity (−1)2 = +1.

(ii)The lightest supersymmetric particle (LSP) is stable: since it has R = −1, it cannot decay into ordinary matter; since it is the lightest, it cannot decay into supersymmetric matter. We will see in Section 5.5 that this provides us in many cases with an excellent candidate for dark matter.

Even though R-parity might be well motivated, there is nothing sacred with it, all the more so with the existing limits on the supersymmetric spectrum. Indeed, a

single-sfermion exchange amplitude is typically of order λ2 /m2 where λR/ is a R-violating

R/ ˜ f

coupling and mf˜ the sfermion mass. It is thus possible to include R-parity violations under the condition that the R-violating coupling λR/ is not too large4. We will study in detail this possibility in Section 7.6 of Chapter 7. Let us note here only that R-parity

4Similarly, Higgs exchange between electrons has an amplitude of order λ2e /m2h m2e /(v2m2h) 1/v2 GF , where λe is the electron Yukawa coupling. It is therefore a minor correction to weak amplitudes.

The Minimal SuperSymmetric Model (MSSM) 91

violation is connected with B or L violations. As long as both B and L are not violated simultaneously, there is no problem with proton decay: proton decay channels (e.g. p → π0e+, p → K0µ+, p → K+ντ ) are forbidden by either B or L conservation.

5.2.3Soft supersymmetry breaking terms

We have until now discussed the supersymmetric couplings. Supersymmetry breaking will obviously generate new couplings. If we do not want to restrict ourselves to a given scenario of supersymmetry breaking, we may just require that this mechanism is chosen such that it does not generate quadratic divergences: otherwise the raison d’ˆetre of supersymmetry is lost. One says that supersymmetry is only broken softly. [The analysis at the end of Chapter 1 indicates which terms are allowed by this condition of soft supersymmetry breaking that we impose.] It turns out that the allowed terms have been classified by Girardello and Grisaru [188] and are remarkably simple. Soft supersymmetry breaking terms are of three kinds:

•Scalar mass terms. These terms are of two di erent forms: for a complex scalar field φ

multiplet equivalently whereas the other term (δm 2) introduces a gap between them.

also allowed in the superpotential. For reasons that will be more clear in the case of supergravity, it has become customary to write A ≡ Aλ where λ is the corresponding superpotential coupling. In other words,

5.3The Minimal SuperSymmetric Model (MSSM)

We study in this section the archetype of supersymmetric models, the Minimal SuperSymmetric Model (MSSM). This is the linearly realized supersymmetric version of the Standard Model with the minimal number of fields and the minimum number of couplings. From the discussion of the previous sections, this means that the field content is one of vector supermultiplets (gauge fields, gauginos) associated with the gauge symmetry SU (3) × SU (2) × U (1) and chiral supermultiplets describing quarks, leptons, the two Higgs doublets and their supersymmetric partners (squarks, sleptons, Higgsinos). As for the couplings, our minimality requirement leads us to impose

94The minimal supersymmetric model

(ii)Gauge symmetry breaking condition: H10 = H20 = 0 should not be a minimum.

We note immediately that m21 = m22 is not compatible with (5.16) and (5.18) simultaneously. This requires that the soft masses m2H1 and m2H2 di er. We will see in Section 6.5 of Chapter 6 that this may be induced by radiative corrections: top quark loops a ect the H2 propagator but not H1 to first order (cf. (5.2)).

Mass eigenstates are obtained by performing independent rotations in the pseudoscalar and the scalar sectors:

where the massless pseudoscalar field ZL0 provides the longitudinal degree of freedom of the massive Z0 gauge boson. The three other fields are physical scalar fields; by convention h0 is the lightest scalar: mh0 < mH0 . If we reintroduce the charged Higgs fields H1± and H2±, one finds that the same rotation as in (5.19) leads to the charged mass eigenstates:

Again, the massless eigenstates WL± are the longitudinal degrees of freedom of the massive W ± and H± is the physical charged Higgs.

One may replace the three parameters m21, m22 and Bµ by the more physical set: the vacuum expectation values

The Minimal SuperSymmetric Model (MSSM) 95

which can be chosen to be real through a SU (2) × U (1) rotation, and the mass of the lightest scalar mh0 or of the pseudoscalar mA0 . But, a combination of v1 and v2,

(v corresponds to the vev of the Higgs field in the Standard Model; cf. (A.131) of

and mh0 or mA0 . It turns out that the angle β thus defined is precisely the one that appears in the pseudoscalar (5.19) or charged (5.21) scalar rotation matrix. This angle β plays a central rˆole in supersymmetric models.

Obviously one may express the new set of parameters in terms of the previous one;

Following (5.25), we note that, had we forgotten the Bµ term in (5.12), we would have found sin 2β = 0, i.e. H10 = 0 or H20 = 0: up-type quarks or down-type quarks would be massless.

Another relation of importance for the future discussion on the issue of fine-

Of course, all physical Higgs masses and mixing angles can be expressed, at tree level, in terms of tan β and mh0 (or mA0 ). For instance, we have7

7Note the two interesting limits: when m2h0 → MZ2 cos2 2β, all other three masses become infinite;

when m2h0 → 0, then mH0 → MZ , mA0 → 0, mH± → MW and α → −β: one recovers the supersymmetry limit discussed at the end of Section 5.1.2.

96 The minimal supersymmetric model

The stability and gauge symmetry breaking conditions lead to the following bound:

As expected, the lightest scalar is found to be relatively light. Such a bound has led to the hope that the lightest Higgs of the MSSM could be detected or ruled out by the LEP collider at CERN. We will now see, however, that top/stop radiative corrections modify the bound (5.34).

We will have a general discussion of the radiative corrections to the Higgs potential in Chapter 7 but a simple argument [22] may be used to obtain the bulk of the one-loop contributions to the lightest Higgs mass. It rests on an analysis of the standard Higgs quartic coupling λ. We will make the simplifying hypothesis that the only relevant degrees of freedom are the Higgs scalar (assimilated to the lightest scalar h0), the top

quark and its supersymmetric partner, the stop (of mass mt˜).

For mass scales µ larger than mt˜, the model is supersymmetric and, as we have seen in (5.13) the quartic coupling λ(µ) is of the order of the gauge coupling squared

g2. For scales mt < µ < mt˜, the e ective theory is no longer supersymmetric (the stop decouples) and one typically recovers the Standard Model. The evolution of λ is governed by the Standard Model renormalization group equation, which is dominated

by the top Yukawa coupling λt (of order 1, and thus much larger than λ: λ(mt˜) g2):

This term is simply due to a diagram with four scalars attached to a top quark loop.

The fact that the top quark is heavy, i.e. that λt is of order one, thus leads to a

sharp increase in the magnitude of λ when one goes down in scale from mt˜ to mt. For scales MZ < µ < mt, the top has decoupled and the evolution of λ(µ) is milder.

The global e ect on λ is thus of order (3λ4/16π2) ln(m2/m2). This gives for the

t ˜ t t

Higgs mass (m2h = 2λv2 from (A.138) of Appendix Appendix A)

√

where we have used MW2 = 14 g2v2 and mt = −λtv/ 2 (cf. (A.141) of Appendix Appendix A).

We see that the e ect is negligible if mt is small with respect to MW (this is why these corrections were assumed to be small at an early period when one expected the

top to have a typical quark mass) as well as if mt mt˜ (approximate supersymmetry in the top sector). We will give in Chapter 7 a more detailed account of radiative corrections, but (5.36) represents the bulk of these corrections. If one allows a generous 185 GeV for the top mass and 1 TeV for the stop mass, this gives an upper limit of 130 GeV for the lightest Higgs of the MSSM.

The Minimal SuperSymmetric Model (MSSM) 97

Table 5.3 Couplings of fermions to the neutral scalars.

To conclude this section, we give here for future reference the couplings of the fermion fields to the neutral Higgs. They are obtained straightforwardly from the superpotential (5.2) using the decomposition (5.19)–(5.20):

=u,d,e

where the couplings are given in Table 5.3.

5.3.2The gaugino–Higgsino sector

The coupling of the Higgs fields to the gauge fields induces by supersymmetry a Higgsino –Higgs–gaugino coupling. This in turn generates, once the Higgs field is set to its vacuum expectation value, a mixed Higgsino –gaugino mass term. Mass eigenstates are therefore mixed Higgsino –gaugino states. Since color is not broken, charged (resp. neutral) fields mix among themselves: the mass eigenstates are called charginos (resp. neutralinos).

In this respect, the gluino g, supersymmetric partner of the gluon, stands alone: color conservation prevents it from mixing with Higgsino fields. Only soft supersym- metry-breaking terms generate a mass term: