Supersymmetry. Theory, Experiment, and Cosmology

Supersymmetry breaking 129

it is also this nonvanishing vacuum value which generates a field independent term in the Goldstino supersymmetry transformation.

6.3.2Hidden sector

When discussing the issue of supersymmetry breaking in Chapter 3, we have encountered di culties when there is a direct coupling between the supersymmetrybreaking fields and the quark and lepton fields. In the case of F -term breaking (O’Raifeartaigh model of Section 3.1.4), the spectrum obtained is unrealistic (STr M 2 = 0). In the case of D-term breaking (see Section 3.2.2), one faces quantum anomalies.

The idea is therefore to somewhat decouple the sector of supersymmetry breaking from the observable sector of quarks, leptons and gauge interactions. One refers to the former as a hidden sector. Obviously, one cannot totally “hide” this sector since the information of supersymmetry breaking must reach the observable sector. But, in the context of supergravity, gravity itself provides a necessary coupling between the two sectors. As we will see, the gravitational coupling between the two sectors tends to soften the divergences: whereas supersymmetry breaking e ects are of orderF MSB2 m3/2 mP l in the hidden sector, they are damped by the gravitational coupling κ m−P l1 in the observable sector and one expects there e ects of order m3/2 . We will see below that indeed soft supersymmetry breaking terms are of the order of the gravitino mass (or smaller if more powers of κ are involved). This means that, barring accidental cancellations, one expects STr M 2 ≡ TrM02 − 2TrM12/2 to be of

order m2 . For example, for a single (gauge singlet) chiral superfield z coupled to

3/2

supergravity, the supertrace, including terms up to spin 3/2, reads [209]

Appendix C]. Then, for a flat K¨ahler manifold (Gzz¯ = 1 hence Rzz¯ = 0), we have, for

√

z = (A + iB)/ 2 (the corresponding fermion field is the Goldstino; it disappears from the physical spectrum):

We have seen in Chapter 3, after equation (3.23) that STr M 2 measures approximately the di erence between an average supersymmetric particle mass and an average quark/ lepton mass. As such, it should be of the order of the (TeV)2. This explains why a gravitino mass of a few hundred GeV or a TeV is often quoted as a preferred value (although there exists viable models with a superlight or superheavy gravitino).

We now illustrate these ideas on a very simple model, known as the Polonyi [313] model, which has provided for a long time a guideline for this type of gravity-mediated supersymmetry breaking.

We recognize:

•in the first term the standard scalar potential of global supersymmetry;

•in the second term, scalar masses of order m3/2 ;

•in the third term, analytic terms in φi plus their hermitian conjugates. The con-

stant A has been introduced because for a cubic potential W (φ) = λφ3, these

terms simply read Am3/2 λ φ3 + φ 3 . This is a standard A-term of the type discussed in Section 5.2.3 of Chapter 5.

We see that we have generated through gravitational interactions soft supersymmetrybreaking terms in the observable sector precisely of the order of the gravitino mass m3/2 : scalar masses6 and A-terms. One may wonder about the last type of soft terms, gaugino masses. It turns out that the full supergravity Lagrangian includes a term:

where the gauge kinetic function f (z, φi) is defined in (6.25). Using (6.32), we findFz of order m3/2 κ−1. Thus, we obtain gaugino masses of order m3/2 as well if the gauge kinetic function depends on z.

6The other type of analytic mass term in equation (5.6) of Chapter 5 may be recovered from the last line in (6.63) if W (φ) includes a quadratic term ρφ2: then δm 2 = (A − 1)ρm3/2 .

in footnote 11 of Chapter 5. Hence u+ has helicity +1/2 and ε+µ u+ represents the component of helicity +3/2; similarly ε−µ u− is the component of helicity −3/2.

Let us recapitulate the counting of the number of on-shell degrees of freedom for a massless gravitino: it has the degrees of freedom of a massless vector field (2) times those of a Majorana spinor (2) but the gauge condition γρψρ = 0 kills those of the spin 1/2 part (2); we are left with 2 × 2 − 2 = 2 degrees of freedom.

This should be put in parallel with the counting of on-shell degrees of freedom of a graviton. The graviton field is obtained as a fluctuation of the metric gµν =

there is, as usual, a residual gauge invariance corresponding to ξµ satisfying ∂µξµ = 0 and ξµ = 0, which eliminates another three degrees of freedom (four massless scalars minus one condition). We are left with two on-shell degrees of freedom for the graviton.

It is thus perfectly consistent to have a gravity supermultiplet with only the graviton and the gravitino. As we have seen in Chapter 3 for other supermultiplets, the counting of o -shell degrees of freedom, on the other hand, imposes the presence of auxiliary fields, as described at the end of Section 6.1.

We now turn to the massive gravitino case. The equation of motion reads:

Then, contracting (6.74) with γµ and using the equation just obtained gives, again for m3/2 = 0,

Thus the gauge condition of the massless case is obtained as a consequence of the equations of motion (a similar situation is found for the case of a massive vector field, see the comment following equation (A.95) of Appendix Appendix A). One obtains:

which is the standard Dirac equation for a massive spinor field.

Whereas the MSSM was strictly speaking a low energy theory, the inclusion of gravity, even at the level of the nonrenormalizable theory that is supergravity, allows

134 Supergravity

us to consider supersymmetric models up to scales close to the Planck scale. The hope is that, as one renormalizes the model up to higher scales, it becomes simpler, reflecting a symmetry principle that becomes transparent at a high fundamental scale (we will study for example in later chapters grand unification theories or string theories which provide explicit examples of such paradigms). In the remaining sections, we will therefore use renormalization group techniques to bring the low energy supersymmetric models discussed in the previous chapter to higher scales, and before considering more elaborate high energy models, we will make some simplifying assumptions at a scale close to the Planck scale to introduce a toy model known as minimal supergravity which will represent our first attempt at writing a simple high energy paradigm.

6.5Radiative breaking of SU (2) × U (1)

We start by studying the breaking of the electroweak symmetry. We have seen in Chapter 5, equation (5.13) that, in the Higgs sector, the scalar potential

has a nontrivial minimum under the conditions

m21 + m22 > 2|Bµ|

m21m22 < |Bµ|2

which are not compatible with m21 = m22. Luckily enough, because H1 and H2 couple to di erent fields, radiative corrections give a di erent treatment to H1 and H2, even if supersymmetry breaking yields the same contribution of order m3/2. Let us see in more detail how this arises.

The main di erence between H1 and H2 is that H2 couples to the top quark. Restricting therefore our attention to the heavy quark loops, we must consider the diagrams of Fig. 6.1 which contribute to the renormalization of the soft scalar mass

cancelled in the sum, since supersymmetry is spontaneously broken– and a logarithmic divergence. The fourth diagram arises from an A-term proportional to the same term λtQ3 · H2T c in the superpotential: Atλtq3L · H2tR + h.c. and the last term arises

where t = ln(µ/µ0), µ being the renormalization scale (not to be confused with the supersymmetric parameter µ!) This should be solved with the boundary condition

m2H2 (µ = mP ) = m23/2.

Radiative breaking of SU(2) × U(1) 135

Fig. 6.1 One-loop diagrams contributing to the renormalization of m2H2 . Q, tR solid (resp. dashed) lines are quark (resp. squark) lines, solid W or H2 lines are gaugino or Higgsino lines.

We see that the term proportional to λt tends to decrease m2H2 as µ decreases. One should remember the relation (5.27) of Chapter 5 which we rewrite

This gives the condition that the soft supersymmetry breaking parameters m2H1 and m2H2 must fulfill at low energy in order to have the right breaking: the smaller m2H2 is (and possibly negative), the easier it is to fulfill.

Similar equations are obtained for the evolution of m2T and m2Q

If we can neglect the e ect of the gaugino masses M2, M3, we see that, as µ decreases, all three m2H2 , m2T and m2Q decrease but m2H2 faster than the other two: hence SU (2)× U (1) is broken before there is a danger that SU (3) gets broken (through m2T < 0 or m2Q < 0). If the e ect of M3 is nonnegligible, it will tend to slow down the evolution of m2T , m2Q, which in turn enhances the e ect on m2H2 .

To get a crude estimate of the scale µEW at which the electroweak symmetry is broken, we may neglect the running of all quantities besides mH2 . The leading contribution to the renormalization of mH2 is then, from (6.80),

Scalar masses 137

We have seen that the equation of evolutions for the scalar masses are rather more complicated. If we assume that scalar masses are smaller than gaugino masses at all scales, the gaugino loop contribution dominates in equations such as (6.82–6.83) and the renormalization group evolutions are easy to solve. One finds, to a good approximation,

6.7Scalar masses

There is however a severe problem connected with scalar masses arising from supersymmetry breaking: the problem of flavor changing neutral currents (FCNC). The origin of these FCNC is similar to the Standard Model but there is, in general, no equivalent to the GIM cancellation mechanism (see Section A.3.4 of Appendix Appendix A).

Let us recall that, in general, the nondiagonal nature of Yukawa couplings leads, after gauge symmetry breaking, to nondiagonal quark and lepton mass matrices. One thus needs to diagonalize them through some matrices V q, .

Similarly, we have seen earlier that supersymmetry breaking leads to nondiagonal squark and slepton mass matrices, which may be diagonalized through some matrices

V q, .

As long as the two issues – supersymmetry breaking and Yukawa couplings – are not connected, there is no reason to relate V q, with V q, , i.e. the rotations among quarks and leptons and the rotation of squarks and sleptons. This leads to undesirable

Fig. A.6 of Appendix Appendix A). As long as W = 1, there is a new contribution coming from the following box diagram involving gluinos and squarks as shown in Fig. 6.2.

where mi is the ui squark mass. For a generic W matrix, this is much too large.

138 Supergravity

We will return in detail to this problem in Chapter 12. Let us say here only that there are several ways of accounting for it:

(i) Universality. To first order, all squarks and sleptons are degenerate. One can then

q q

choose V = V , in which case W = 1 and the supersymmetric box diagram vanishes. Of course, one can only make this assumption at a given scale, of order mP l or MU for supergravity models, and renormalization to low energies will tend to generate nondiagonal terms in the mass matrices and thus FCNC at low energy.

(ii)E ective supersymmetry [79]. FCNC constraints are only crucial for the first two families, whereas the fine-tuning associated with the hierarchy problem concerns mostly the third family in the radiative breaking scenario (since the breaking of

SU (2) × U (1) and thus the scale MZ arises through the top Yukawa coupling). A possible solution is therefore to make the squarks of the first two families heavy (TeV scale), whereas the third family squarks remain relatively light in order to avoid an unwanted fine-tuning. In this case, because of the squark propagators, the function f (m2i /M32, m2j /M32) is very small.

(iii)Quark–squark alignment. If one “aligns” the squark flavors along the quark fla-

vors, that is if the same matrix may diagonalize quark and squark mass matrices,

indeed be obtained by imposing flavor symmetries known as horizontal symmetries (in contrast with the ordinary gauge symmetries of the Standard Model which are vertical, i.e. act within a given family of quarks and leptons).

6.8The minimal supergravity model

In the case of the minimal supergravity model, one therefore assumes, besides (6.90), that all scalar masses are equal at unification scale, as well as all A-terms: