Supersymmetry. Theory, Experiment, and Cosmology

Gauge coupling unification 231

We find, for the evolution of SU (3) and SU (2) couplings above MZ , with NF

Indeed, we have defined (see Section A.3.2 of Appendix Appendix A) the gauge coupling g through the minimal coupling of a field Φ of hypercharge y:

In the case of an abelian group, it is not possible to use the commutation relations to normalize the generators. One may thus replace y by ky if one substitutes g /k for g . To make this explicit, let us write y ≡ 2Ct0:

where αe.m. ≡ e2/(4π).

If there is an underlying theory where U (1)Y is imbedded in a nonabelian group G, then C is fixed by the normalization imposed by the commutation relations. One can obtain a simple expression for the normalization constant C under such an assumption.

Indeed, if t3 and t0 are the generators of a simple nonabelian group G in a represen-

232 Supersymmetric grand unification

For example, if we assume that all known fermions form one or more full represen-

where we have counted the contributions of the leptons e, µ, τ (and their antiparticles) and of the three colors of quarks u, c, t and d, s, b (and their antiparticles). Also

TrR t3 2 = 12 (3 + 3 × 3) = 6

to account for all the fermion doublets (T (R) = 1/2): three leptons and three quarks in three colors. Hence,

This can be checked directly on the example of SU (5) by identifying the explicit form of the hypercharge generator t0 (see Exercise 1).

We then obtain the one-loop coe cient for the g1 beta function:

where −1/10 is the contribution from the Higgs, and we recall that NF is the number of families.

9.2.2The nonsupersymmetric case

We suppose the existence of an underlying gauge symmetry described by a simple gauge group G, which is broken at a superheavy scale MU much larger than the TeV scale. This gauge invariance is only explicit above the scale MU . This theory contains superheavy fields, such as gauge fields, with mass of order MU . We note two important properties of the renormalization group equations for the gauge couplings αi (i = 1, 2, 3) [183]:

(i)any representation R which contains only superheavy fields decouples from these equations;

(ii)any representation R which contains only light fields contributes equally to b1, b2 and b3.

For example, the generators of SU (3) t(3)i are generators of G in the representation R: δij T (3)(R) ≡ Tr t(3)it(3)j = δij T (R). Similarly, the generators of SU (2) t(2)a are generators of G in R and δabT (2)(R) ≡ Tr t(2)at(2)b = δabT (R). Hence, they contribute equally to b2 and b3. This is why the coe cients of NF in (9.20), (9.21), and (9.31)

Gauge coupling unification 233

are identical: a family of quarks and leptons yields complete representations of the minimal SU (5) group.

Let us thus suppose that the only multiplet which contains both light and super-

and we see, using (9.14), that the combination of couplings α1−1(µ)−3α2−1(µ)+2α3−1(µ) is not renormalized at one loop, i.e. is independent of the renormalization scale µ at this order.

If we suppose that these couplings are equal at the scale MU , as expected since they unify into the single coupling g corresponding to the simple group G, then

Thus, since C has a computable value for any given theory, (9.35) yields a testable relation between quantities measurable at µ = MZ . We note the value of sin2 θW at unification: (1 + C2)−1 that is 3/8 for C2 = 5/3.

One can also determine the scale of unification MU and the common value αU of the gauge couplings at unification. Indeed, since

Let us take the example of NF = 3 and C2 = 5/3 as in SU (5). We have, using (9.10) and (9.12),

It is a remarkable coincidence that one obtains such a superheavy scale for grand unification. Indeed, any lower scale would be catastrophic for the theory since it would

2Note that this agrees with (9.20), (9.21), and (9.31), if we disregard the Higgs contribution. Indeed, the Higgs representation contains both light and heavy fields (cf. the doublet–triplet splitting problem discussed above for SU (5)).

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induce rapid proton decay, the gauge bosons Xµ and Y µ being too light. We will see that even such a large scale puts in jeopardy the minimal SU (5) model.

As for the relation (9.35), it reads

The order of magnitude comes out right and this is a second success of the grand unification picture. This seems to indicate that there is at least an approximate unification of couplings at a scale much larger than we might have expected. Indeed, this does not leave much room for physics in the range between MZ and MU : this has become known as the “grand desert hypothesis”.

There is, however, still a discrepancy of some 10 standard deviations in (9.40) between the experimental result (first number) and the prediction (second number). Where could such a disagreement come from?

•Higher orders in the evolution of the renormalization group equations. One may include the two-loop contribution to the beta functions. This gives an extra contribution δh.o. 0.0029 on the right-hand side of (9.40).

•Threshold e ects. One needs to be more quantitative to account for the decoupling of heavy particles in the evolution of the gauge couplings. One might also include possible intermediate thresholds. We call δth the corresponding contribution.

•Mixed representations. It is also possible to have representations which mix light and superheavy fields. We have seen for example that the representation for the Higgs doublet falls in such a category in the case of SU (5) since the corresponding triplet must be superheavy to prevent too rapid proton decay. We call δm such a contribution.

The precision measurements in (9.10)–(9.12) provide useful indications on the magnitude of the last terms. In any case, they show that the minimal SU (5) model is not compatible with gauge unification. In other words, given their allowed range of values, the gauge couplings of the minimal SU (5) model do not intersect at a single point.

9.2.3The supersymmetric case

We now turn to the supersymmetric case. We will describe supersymmetric models of grand unification in the next section. For the time being, it su ces to stress that, for scales µ larger than the supersymmetric thresholds, one must take into account the supersymmetric partners in the evolution of the running gauge couplings. There is no need for further computations, since equation (9.19) summarizes it all.

Indeed, for a gauge vector supermultiplet (adjoint representation), gauge bosons

For a chiral supermultiplet in a representation R of the group G (complex) scalars contribute 13 T (R) and the Weyl fermion 23 T (R), adding up to T (R).

Gauge coupling unification 235

Thus, (9.19) reads, in the supersymmetric case,

which gives for NF families and two Higgs doublets:

b1 = −2NF − 35

where −3/5 and −1 are the contributions of the two doublets of Higgs supermultiplets. If we make the hypothesis that the only multiplets which contain light and superheavy fields are the gauge vector supermultiplets, then one recovers in the supersym-

metric case, the relation (9.32) and thus (9.41) follows.

What is new is that in δm, the contribution of mixed light–superheavy representations, there are now two complete chiral supermultiplets corresponding to the representations of H1 and H2, instead of a single Higgs doublet in the nonsupersymmetric case.

A more complete analysis is performed by assuming that all supersymmetric particles have a common mass MSUSY . Since squarks and sleptons form complete representations of SU (5), they modify in the same way the evolution of all couplings. Also, the contribution of gauginos does not change the relation (9.33). In the case of Higgs, in the simplest approach, one is taken with a mass MZ whereas the other has mass

MSUSY . Hence the determination of δm may be turned into a determination of MSUSY . It is a remarkable feature that for a value of MSUSY in the TeV range, a region which was favored earlier by the naturalness criterion, there is unification of the three gauge

couplings at a scale of order 1016 GeV (see Fig. 9.2).

Fig. 9.2 Unification of gauge couplings in the supersymmetric case for MSUSY 1 TeV.

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9.3The minimal supersymmetric SU(5) model

Following the procedure developed in Chapter 5 which is to introduce supersymmetric partners for every field of the nonsupersymmetric model, we introduce:

• a vector supermultiplet in the representation 24 of SU (5);

• ¯

NF chiral supermultiplets in the representations 5 and 10 to accomodate the NF families of quarks and leptons; as above, we denote them, respectively, ηi and χij ;

•a chiral supermultiplet Σ in the representation 24 to break the grand unified symmetry SU (5);

We note at this stage why we cannot identify the 5 of Higgs with one of the 5 of the

leptons (and hence consider the Higgs doublet as the supersymmetric partner of one

¯

of the lepton doublets): the fermionic component of the corresponding 3 would be the light dc whereas the prevention of rapid proton decay imposes it to be superheavy (see Section 9.3.4 below).

the unusual form being due to the implementation of the constraint Tr Σ = 0 (see Exercise 2).

The minimum corresponds to