Supersymmetry. Theory, Experiment, and Cosmology

The flavor problem 341

Let us illustrate here this line of reasoning on an example [44]. We consider an abelian gauge symmetry U (1)X which forbids any renormalizable Yukawa coupling except the top quark coupling. Hence the Yukawa coupling matrix Λu in (12.3) has the form:

where 1 in the last entry means a matrix element of order one. The presence of such a nonzero entry means that the charges under U (1)X obey the relation:

whereas similar combinations for the other field are nonvanishing and prevent the presence of a nonzero entry elsewhere in the matrix Λu.

We assume that this symmetry is spontaneously broken through the vacuum expectation value of a field θ of charge xθ normalized to −1: θ = 0. The presence of nonrenormalizable terms of the form QiUjcH2(θ/M )nij induces in the e ective theory below the scale of U (1)X breaking an e ective Yukawa matrix of the form:

where λ = θ /M and

(n33 = 0). Such nonrenormalizable interactions may arise through integrating out heavy fermions of mass M as in the Froggatt–Nielsen model or appear if the underlying theory incorporates gravity, e.g. in string theories, in which case the scale M is the Planck scale mP .

We note however that the form (12.18) is valid only if all nij are positive. If one nij is strictly negative, the corresponding entry vanishes: barring nonperturbative e ects, fields appear only in the superpotential with positive exponents and holomorphicity prevents us from using θ in the superpotential (and thus from writing a term of the form QiUjcH2(θ /M )−nij ). Such zero entries are thus called holomorphic zeros3.

charges ±1 [233]. They acquire equal vacuum expectation values along a D-flat direction and we may

≡ ¯ |n | write λ θ /M = θ /M . In this case, there are no holomorphic zeros and (Λu)ij λ ij .

342 The challenges of supersymmetry

Let us denote for example the X-charges of the standard supermultiplets as given in Table 12.2 (we assume that 3a8 + b8 > a3 + b3 > 0 and 3a8 + b8 > a3 + b3 > 0 and similarly for b3,8 replaced by c3,8).

Then the CKM matrix reads [43]

One has in particular VusVcb = Vub. As for the mass ratios, one finds (use the result proven in Exercise 1)

which is a classical relation [175].

It might seem on this example that, by choosing the charges of the di erent fields, one may accommodate any observed pattern of masses. There are however constraints on the symmetry: in particular those coming from the cancellation of anomalies. This is indeed one of the reasons to choose a local gauge symmetry. We will see that this gives very interesting constraints on the model.

Table 12.2 U (1)X charges for the standard supermultiplets (e.g. the charge of Q3 is a0−2a8).

The flavor problem 343

Anomalies

In order to make sense of the horizontal symmetry, one must make sure that the anomalies are cancelled: not only the anomaly CX corresponding to the triangle diagram with three U (1)X gauge bosons, but also the mixed anomalies Ci corresponding with triangle diagrams with one U (1)X gauge boson and two gauge bosons of the Standard Model gauge symmetry: U (1)Y , SU (2) and SU (3) for i {1, 2, 3}, respectively.

Since they are linear in the U (1)X charges the coe cients Ci depend only on the family independent part of the quantum numbers, respectively a0, b0, c0, d0 and e0 for Qi, Uic, Dic, Li and Eic. They read explicitly

where h1 and h2 are the x-charges of H1 and H2.

On the other hand, mass ratios are also given by the X-charges. In the example

Anomaly cancellation would require C1 = C2 = C3 = · · · = 0 which gives, after a redefinition of the x charge (see Exercise 2), a0 + b0 = a0 + c0 = 0 and 3(d0 + e0) = −(h1 + h2). Then comparing (12.24) and (12.25) with the data (12.14) yields h1 = 2 and h2 = 4. The last equation (12.26) is then incompatible with the same data. This shows that the observed pattern of masses is incompatible with a nonanomalous family symmetry: the U (1)X symmetry must be anomalous [33, 44, 233]. Is this the end of the story?

Before we address this question, let us derive from the mass spectrum (12.14) a relation among the anomaly coe cients. We have

Assuming the presence of a mu-term µH2 · H1 imposes that h1 + h2 = 0. Since the data (12.15) imposes this ratio of masses to be of order one, one then finds:

344 The challenges of supersymmetry

We have encountered in Section 10.4.5 of Chapter 10 a seemingly anomalous symmetry: in superstring models, there is a U (1) symmetry whose anomaly is compensated by the four-dimensional version [116] of the Green–Schwarz mechanism [205]. This is possible through the couplings of the gauge fields to a dilaton–axion–dilatino supermultiplet: the anomalous terms that arise when we perform a gauge transformation are cancelled by a Peccei–Quinn transformation of the axion. The necessary condition for the cancellation of anomalies a` la Green–Schwarz is

where the ki are the Kaˇc–Moody levels. Combined with the gauge unification condition k1g12(M ) = k2g22(M ) = k3g32(M ) = kX gX2 (M ), this gives

The relation (12.28) can now be discussed in this context. For example, in the standard case where all the nonabelian symmetries appear at the same Kac–Moody level, k2 = k3 and thus C2 = C3, we find:

Thus, if the horizontal abelian symmetry is precisely the pseudo-anomalous U (1)X , observed hierarchies of fermion masses are compatible with the standard value of sin2 θW at gauge coupling unification (see (9.35)). And this result is obtained without ever making reference to a grand unified gauge group (which is rarely present in superstring models).

There is another advantage of working in the context of string models. Indeed, in this case, the properties of the anomalous U (1)X are constrained. For example in the case of the weakly coupled heterotic string model, the absolute normalization is fixed [13] and

Hence, one naturally obtains the small parameter (the Cabibbo angle) that was necessary for the whole picture to make sense.

Let us stress, however, the main drawback of this approach. In all the preceding formulas we have neglected factors of order one and discussed only the orders of magnitude as powers of the small parameter λ sin θc 1/5. But since this is not such a small parameter, the actual value of the factor of order one introduces some uncertainties: for example λn/2 3λn+1.

Now, returning to our original motivation: does this help to align sfermion and fermion mass eigenstates? Obviously, sfermion masses are also constrained by the symmetry U (1)X ( [272, 273, 297]).

The flavor problem 345

Let us consider the LL squark mass matrix. If xQi > xQj , the following mass term

(the use of hermitian conjugates of fields is allowed since this is a supersymmetrybreaking contribution).

Thus, after U (1)X breaking, this yields m˜ 2q˜iq˜j λ|xQi −xQj | + h.c., where λ = θ /M =θ† /M and the scalar squared mass matrix reads:

Thus the squark mass matrices are approximately diagonal: the family symmetry induces a partial alignment [297].

One finds for the parameters introduced in (12.6)

where the matrix elements on the right-hand side are to be renormalized down to low energies where they tend to decrease.

This still gives some severe constraints on the models that do not seem to be satisfied without the presence of holomorphic zeros in the down quark mass matrix [296].

Neutrino masses

The neutrino sector is specific since both Dirac and Majorana mass terms are allowed for neutral leptons (see Appendix Appendix A, Section A.3). Introducing a righthanded neutrino NR , we write the Dirac mass term as

where mD arises through SU (2) × U (1) breaking and is of the order of the electroweak scale. On the other hand, a Majorana mass term involves a single chirality. Standard

346 The challenges of supersymmetry

Model gauge symmetry forbids such a term for the left-handed neutrino but not for the right-handed neutrino:

where M is not constrained by electroweak symmetry. The seesaw model [177, 383], which represents the prototype of neutrino mass models in all theories which involve large scales such as grand unified or superstring theories, includes both Dirac and Majorana mass terms:

In the case where M mD , the eigenvalues are respectively:

and, due to the presence of a zero in the matrix, the mixing angle is given in terms of

We have discussed the case of one family but the discussion easily generalizes to the three-family case with a mass matrix of the form:

where the Dirac and Majorana mass matrices, respectively MD and MM , are 3 × 3 matrices. Then the light neutrino mass matrix reads:

Of course, given the freedom we have on each of the specific entries in MD and MM , seesaw models really form a class of models and one has to go to specifics in order to discuss their phenomenology. This is precisely what a family symmetry provides us with.

In order to discuss neutrino masses in this context, we introduce one right-handed neutrino for each family: Nic, i {1, 2, 3}. The neutrino Dirac mass term is generated from the nonrenormalizable couplings:

The flavor problem 347

where Li is the left-handed lepton doublet and pij = xLi + xNjc + xH2 is assumed to be positive (otherwise, this coupling is absent, which leads to a supersymmetric zero in the mass matrix). This yields a Dirac mass matrix:

The entries of the Majorana matrix MM are generated in the same way, with nonrenormalizable interactions of the form:

Mν = −MD M−M1MTD are therefore fixed by the U (1)X symmetry.

Let us suppose that all entries of MD and MM are nonzero (i.e. pij , qij ≥ 0); one

which leads to the following light neutrino masses and lepton mixing matrix:

One therefore finds that the neutrino spectrum is hierarchical. Moreover, the structure of the lepton mixing matrix is very similar to the CKM matrix (12.1) with generically small mixing angles: Uij2 mνi /mνj for mνi < mνj . This is in disagreement with the experimental evidence for large angles both for µ − τ mixing (atmospheric neutrinos) and µ − e mixing (solar neutrinos).

It would be wrong, however, to consider that a hierarchical spectrum is a generic feature of this type of models. Indeed, one can work out models which allow for degeneracies in the light neutrino spectrum [42]. It is easy to see that such degeneracies are associated in these models with large mixings.

One may note that the situation in the neutrino sector might be very di erent from the quark and charged lepton sector where one Yukawa coupling (the top) dominates over all the rest, thus providing a clear starting point for the U (1)X symmetric situation, summarized in (12.16). In the case of neutrinos, the U (1)X symmetry, even

348 The challenges of supersymmetry

though it is abelian, may induce some degeneracies. Consider for example the following matrix:

where a is a number of order one. This pattern corresponds to the conservation of a combination of lepton number a` la Zeldovich-Konopinsky–Mahmoud ( [260, 386]): indeed Le and Lµ + Lτ are separately conserved. It has two degenerate eigenvalues and the corresponding diagonalizing matrix Rν has one large mixing angle:

where Dν = RνT Mν Rν .

Of course, as soon as the U (1)X symmetry is broken, the vanishing entries in (12.52) are filled by powers of the small parameter λ. This lifts the degeneracy at a level which is fixed by the charges under the U (1)X symmetry.

The large mixing angles observed in the neutrino sector may also point towards a nonabelian nature of the family symmetry, for example SU (3). This goes beyond the scope of this book and we refer the reader to reviews on the subject such as [325].

12.1.5Split supersymmetry

If one is ready to raise the mass of the supersymmetric particles to alleviate the flavor and CP problem, one may wonder how far one can go. Obviously, the fine tuning problem becomes more acute. However, we will see in the next section that supersymmetric theories have to deal with an even more severe problem of fine tuning associated with the vacuum energy. This has recently led several groups to propose to set aside the naturalness problem that was one of the bases of low energy supersymmetry: if the supersymmetric spectrum is heavy enough, one may expect to ease problems with flavor and CP violation, fast proton decay through dimension-5 operators, tight Higgs mass limits etc.

The next issue is obviously the status of gauge coupling unification. Arkani-Hamed and Dimopoulos [9] have shown that unification can still be achieved in a supersymmetric model, now referred to as split supersymmetry [192], where all scalars but one Higgs doublet are much heavier than the electroweak scale. Next comes the question of a dark matter candidate: if the presence of such a “light” state is imposed on the theory, this leaves hope to find some direct or indirect signal of this type of models at high energy colliders.