Supersymmetry. Theory, Experiment, and Cosmology

The minimal supersymmetric SU(5) model 239

ones have vanishing terms on the diagonal. Thus, the mixed terms (9.56) do not generate a mass term for the Hi superfields.

We still have not shown that we can generate a mass term for the triplets of Higgs while keeping the doublets light. But the preceding discussion shows that, if we succeed to do that at tree level, then we can count on supersymmetry to ensure that radiative corrections will follow the same pattern. Let us show how this may be realized in practice.

If we consider the following superpotential for H1 and H2:

where the minimization of the extra terms is supposed to fix Σ to its value (9.49).

with σ = M/Λ. Thus, the mass of the doublet is m2 = µ − 3λσ whereas the mass of the triplets is MT = µ + 2λσ. If we impose that

which is of the order of the grand unification scale, then one may realize a doublet– triplet splitting. This condition may seem at this point ad hoc but, as long as supersymmetry is unbroken, it is natural in the technical sense3.

It remains to see which dynamics might be responsible for the condition (9.59). In the sliding singlet approach [292,372], one introduces a chiral superfield S which

is a gauge singlet and one modifies the superpotential (9.57) as follows:

3Using the fact that the terms in the superpotential (9.57) are not renormalized and defining the wave function renormalization constant Z as Hi = Z1/2HiR , we have

and thus µR 3λR σR follows from (9.59).

240 Supersymmetric grand unification

The potential is now of the form (9.58) with µ replaced by κS. Its minimization leads to the conditions (κs − 3λσ)vi = 0, i = 1, 2 where s ≡ S . Hence, once SU (2) × U (1) is broken (v1 or v2 = 0), we have automatically κs − 3λσ = 0, which ensures massless doublets at this order. In other words, the vev of the gauge singlet slides in order to minimize the ground state energy, and correspondingly the mass of the doublets. Obviously, supersymmetry breaking will modify the analysis: it is thus important to make sure that the scale of supersymmetry breaking in this sector remains low.

Another line of attack when the scale of supersymmetry breaking is not small is to introduce a Higgs representation which contains triplets of SU (3) (i.e. (3, 1) under SU (3) × SU (2)) but no doublet of SU (2) (i.e. (1, 2) under SU (3) × SU (2)). Coupling this representation to H1 and H2, one thus gives a mass to the triplets but not to the doublets. This is the missing doublet mechanism [178, 208, 284].

An example is provided by the representation 50 of SU (5). Its decomposition under

50 = (8, 2) + (6, 3) + (6, 1) + (3, 2) + (3, 1) + (1, 1).

One chooses to break SU (5) down to SU (3) × SU (2) × U (1) with a 75 of SU (5)

9.3.3Fermion masses

Fermion masses arise from Yukawa interactions which are derived directly from the superpotential. Denoting, as above, by χmn and ηm the superfields in representations

¯

10 and 5, the superpotential reads:

of SU (5) and is the completely antisymmetric tensor.

One easily deduces the Yukawa couplings and, setting the Higgs fields at their

246 Supersymmetric grand unification

Fig. 9.6 Dimension-5 operators generated by color triplet exchange.

p → K+ν¯τ (right) through dimension-5 operators.

(9.95) The negative search performed by the Super-Kamiokande Cherenkov detector sets a limit of 6.7 × 1032 years (90% confidence level) for the partial lifetime of the proton in the decay channel K+ν¯ [138]. This puts MT in a mass range incompatible with the data on the gauge coupling unification (9.53). It thus rules out the minimal SU (5) model [203].

This, however, does not exclude a more general SU (5) unification. One may for example include more superheavy multiplets in order to push gauge unification to higher scales through threshold corrections. Alternatively, one may try to suppress the dimension-5 operators, as in the flipped SU (5) model. Finally, one may enlarge the grand unification symmetry, as we will now see.

9.4The SO(10) model

The SU (5) model does not truly unify the matter fields since one has to advocate

¯

two representations: 5 and 10. It turns out that, if we view SU (5) as a subgroup of

¯

SO(10), these two representations make a single one of SO(10): 16 = 5 + 10 + 1. The SU (5) singlet that is required to make the sixteenth field is singlet under SU (3) × SU (2) × U (1): it is interpreted as a right-handed neutrino. This provides a natural explanation for the cancellation of gauge anomalies: as explained in Section 9.1.2, anomalies naturally cancel for representations of SO(10).

The SO(10) model 247

Since SO(10) has rank 5, we expect to be able to define an extra gauge quantum number. In fact, the global B − L symmetry of SU (5) is promoted to the status of

a gauge symmetry in the context of SO(10). Noting that Tr(B − L)|¯5 = −3 and Tr(B − L)|10 = 2, we see that the presence of the right-handed neutrino gives the missing contribution to ensure a vanishing total contribution for Tr(B − L)|16, as is fit for a gauge generator. Moreover, SO(10) incorporates a natural left-right symmetry and thus a spontaneous breaking of parity may be implemented. We will see below that the two issues are somewhat linked.

All these properties, compared with the mixed success of the SU (5) model make SO(10) unification a very interesting candidate for a grand unified theory.

9.4.1Symmetry breaking

Among the subgroups of SO(10), we first focus on SU (5) × U (1). One can easily see why, on general grounds, SU (n) × U (1) is a subgroup of SO(2n). A transformation of SU (n) leaves invariant the scalar product of two complex n-component vectors U and V :

U † · V = Re U · Re V + Im U · Im V + i Re U · Im V − i Im U · Re V. (9.96)

Using instead the real 2n-component fields U = (Re U, Im U ) and V = (Re V, Im V ), one sees that the following scalar product is conserved:

Hence, a transformation of SU (n) is also a transformation of SO(2n). Moreover, under the U (1) transformation U → eiφU , which commutes with SU (n), U = (Re U, Im U ) transforms into (Re U cos φ − Im U sin φ, Re U sin φ +Im U cos φ). This transformation preserves the scalar product (9.97). One concludes that SU (n) × U (1) is indeed a subgroup of SO(2n).

It is thus possible that, at some scale larger than the scale obtained earlier for SU (5) unification, SO(10) is broken into SU (5) × U (1)χ. Quarks, leptons and their supersymmetric partners form NF chiral supermultiplets in 16 of SO(10), which are decomposed under SU (5) × U (1)χ as:

(Dc, E, N ) (D, U, U c, Ec) (N c)

where we have indicated as subscript for each SU (5) representation its charge yχ under U (1)χ. Similarly, gauge supermultiplets transform as 45 with the decomposition:

where we recognize in the first places the gauge bosons of SU (5) and U (1)χ.

The first breaking (SO(10) to SU (5)) may be realized through a 16 of Higgs, whereas the second breaking (SU (5) to the Standard Model) uses a 45, since 45 includes the necessary 24 of SU (5). There is no constraint on the scale of U (1)χ breaking and the corresponding gauge boson Zχ could thus be a low energy field.

248 Supersymmetric grand unification

Alternatively, since the group SO(m + n) contains SO(m) × SO(n), and SO(6) SU (4) whereas SO(4) SU (2) × SU (2), we may consider the breaking of SO(10) to SU (4) × SU (2) × SU (2). One of the SU (2) symmetries may be interpreted as the SU (2)L symmetry of the Standard Model and the other one is identified as its parity counterpart SU (2)R. Moreover, SU (4) naturally incorporates the color SU (3) and the abelian group U (1)B−L associated with the B − L quantum number.

The breaking of SO(10) is realized through a 54 = (6, 2, 2)+(20 , 1, 1)+(1, 3, 3)+ (1, 1, 1) and the final breaking to the Standard Model uses 16 + 16 or 126 + 126.

We note the following relation:

which shows that B − L is a generator of SO(10) (since the others are). This provides also a nice interpretation of hypercharge, which had a somewhat mysterious origin in the context of the Standard Model:

9.4.2Fermion masses

Since Yukawa couplings are trilinear, one expects to find the Higgs representations (or rather their conjugates) in the product 16 × 16 = (10 + 126)s + 120a (the subscripts refer, respectively, to the symmetric and the antisymmetric combination). The Higgs are thus searched for in 10, 120, or 126 (the first two are self-conjugates).

[We note that 16 is the spinor representation of SO(10) constructed in Section B.2.1 of Appendix B. We thus see that matter supermultiplets are in spinor representations of SO(10) whereas Higgs and gauge supermultiplets are in tensor representations (they appear in even products of spinor representations). This allows us to define a matter parity: −1 for spinors and +1 for tensors.]

If we take the Higgs in the representation 10H which decomposes under SU (5) ×

¯

U (1)χ as 10 = 52 + 5−2, the trilinear coupling yields the following decomposition under SU (5):