Supersymmetry. Theory, Experiment, and Cosmology

The SO(10) model 249

The first two terms were already considered in (9.64) and they yield the usual SU (5) mass relations6. The last term yields a Dirac mass for the neutrino, equal to the up quark mass at grand unification. In other words, the SU (5) prediction (9.66) is replaced by

where λ is the (16 × 16) × 10H Yukawa coupling. The last prediction is obviously in contradiction with the small neutrino mass.

It is precisely in this context that Gell-Mann, Ramond and Slansky [177] and Yanagida [383] proposed the seesaw mechanism. Indeed, if one breaks SO(10) down to

SU (5) with a 126 (which contains a SU (5) singlet φ(126) of charge yχ = 10 or B −L = −2, as seen from (9.106)), a Yukawa coupling of the form (16 × 16) × 126 includes a

Majorana mass for the right-handed neutrino of the order of the unification scale (i.e. fixed by the B − L breaking scale φ(126) ). This combined with the electroweak Dirac mass yields the mass matrix of the seesaw mechanism, as discussed in Section 1.1.1 of

In the case of theories where a 126 representation is not available, one may also use higher-dimensional operators of the type 16i × 16j × 16H × 16H /MP (i, j are family

indices). This yields a right-handed neutrino mass proportional to φ(16) 2/MP , where

φ(16) is the singlet of 16H with yχ = 5 i.e. B − L = −1. For φ(16) MU , this gives

a right-handed mass of order 1013 GeV. This time R-parity is broken since φ(16) has odd (B − L).

It may also be necessary to introduce terms of the type 16i ×16j ×16H ×16H /MP in order to generate CKM mixings.

9.4.3[Doublet–triplet splitting

The SO(10) theory allows for a nice implementation of the missing doublet mechanism (see Section 9.3.2) to generate the doublet–triplet splitting. Indeed, the 45 involves triplets (3, 1) but no doublet (1, 2) of SU (3) × SU (2) as can be readily checked from (9.99). In a model with two representations 10H and 10H , the coupling 10H · 45 · 10H leaves four doublets in the light spectrum. This is too many and one has to somewhat complicate the scheme to remain with only two doublets at low energy. A superpotential that achieves this [17] involves also 16H and 16H :

6Since both 120 and 126 include a 45 of SU (5), one may invoke the Georgi–Jarlskog mechanism to modify these. More precisely,

E6 251

As expected, the nonvanishing vacuum expectation value 45H in the (B − L) direction does not contribute to the doublet mass matrix. Introducing D1(10) and D2(10) the two doublets in 10H (with, respectively, yχ = −2 and yχ = 2), D(16) the doublet and

φ(16) the SU (5) singlet in 16H (with, respectively, yχ = 5 and yχ = −3), the doublet mass matrix simply reads:

with tan γ = λ φ(16) /M16. We have thus D1(10) = v1 cos γ and D(16) = v1 sin γ. We finally note that, in the case where nonrenormalizable terms involving 16H

and 16H are also introduced to generate right-handed Majorana masses and CKM

9.5E6

If one further enhances the gauge symmetry, the next simple gauge group encountered is E6, the exceptional gauge group of rank 6 with 78 generators. As we will see in more details in the next chapter, this group is naturally encountered in the context of the weakly heterotic string theory.

The fundamental representation of E6 is complex and 27-dimensional. It decomposes under the subgroup SO(10) × U (1)ψ as

where we have indicated as subscript the charge yψ associated with the abelian factor,

¯

noted U (1)ψ . This provides new fields: the 10 transforming under SU (5) as 5 + 5 yields new charge −1/3 (weak isosinglet) quarks and new (weak isodoublet) leptons; the 1 is a new neutral lepton.

The maximal subgroup SU (3) × SU (3) × SU (3) of E6 plays an important rˆole in discussions of superstring models. One identifies the first group with color SU (3)c, the second one with the SU (3)L group containing the electroweak symmetry SU (2)L and the third one with its parity counterpart SU (3)R.

For example, we will consider in Section 10.4.3 of Chapter 10 the breaking of E6 with an adjoint 78 representation which decomposes as:

Since SU (3)c remains unbroken, one must use (1, 8, 1) to break SU (3)L. But breaking SU (3) down to SU (2) with an octet always leaves an extra U (1) factor. We thus write

252 Supersymmetric grand unification

the residual symmetry SU (2)L × U (1)YL . Since the corresponding abelian charge yL cannot coincide with hypercharge (otherwise right-handed fields would have vanishing hypercharge), one must invoke a second conserved abelian charge qR, where U (1)QR is a subgroup of SU (3)R. Hypercharge is a linear combination of yL and qR. The orthogonal combination, noted yη , is also conserved. The low energy symmetry is SU (3)c × SU (2)L × U (1)Y × U (1)η .

For definiteness, let us consider the fundamental representation, which decom-

The first representation includes quark superfields, the second antiquarks and the third leptons. One may decompose SU (3)L,R into SU (2)L,R × U (1)YL,YR (both SU (2)L,R groups have been encountered above with SO(10)). Then Table 9.3 gives the full content of the 27 representation. One has

Obviously, the U (1)η charge is a combination of the other two: yη = (3yχ − 5yψ )/12.

Exercises

Exercise 1 We determine explicitly in this exercise the form of the generator of SU (5) associated with hypercharge in the fundamental representation 5 and check that its normalization agrees with the result obtained in (9.30).

Since t0, the generator of SU (5) which corresponds to hypercharge, must commute with the generators of SU (3) and SU (2), we write it, in the fundamental representation

(a)Determine α and β.

(b)Deduce the value of the normalization constant C defined in (9.24).

with the low energy definitions given in Chapter 5, equations (5.1) and (5.2).

10

An overview of string theory and string models

It is widely believed that string theory provides the consistent framework which lies behind low energy supersymmetry. The purpose of this chapter is to provide a nontechnical overview of the main ideas of string theory1, as well as to present some of the new concepts which arise and which might lead to new developments in the study of the low energy theory. Due to lack of space, there is no room for doing justice to the richness of the complete string picture and we refer the reader to other monographs for a more thorough treatment e.g. [312]. However, string models are to be considered very seriously because they provide a complete picture of fundamental physics, including gravity. As such, they should be considered in their entirety because they have an internal coherence. Thus using them for low energy phenomenology (or cosmology) requires us at least to be aware of the general consistency of the string picture.

10.1The general string picture

There is a well-known contradiction between the two fundamental theories which emerged in the first half of the twentieth century: quantum theory and general relativity. General relativity is a classical theory. When one tries to quantize it, one encounters divergences which prevent us from making predictions once the quantum regime of the theory is reached. As was discussed in Chapter 1, the fundamental scale of quantum gravity is obtained from Newton’s constant GN and Planck’s constant . This gives the Planck mass2

1Some of the more technical developments are placed in boxes in this chapter. The reader interested in a nontechnical overview may skip them.

2A word of caution: absolute lab measurements of Newton’s constant have only been conducted on distance scales in the 100 µ to m range (GN = 6.673 × 10−11 m3 kg−1. s−2) whereas Planck’s constant is by essence a microscopic scale. We are thus making the implicit assumption that gravity at the microscopic level is still described by the Newton’s constant which is measured macroscopically. We will return to this in Section 10.2.1.

256 An overview of string theory and string models

0

σ

π

Fig. 10.2 World-sheet for an open string parametrized by the spatial coordinate σ, 0 ≤ σ ≤ π, and time τ .

Let us note immediately that this type of approach does not treat spacetime in a truly quantum mechanical fashion. The string propagates in a background spacetime which is often taken to be flat and is in any case classical. In other words, the two-dimensional world-sheet is plunged into a classical background spacetime, often referred to as the target space. We will see momentarily in which sense one may still say that string theory is a quantum theory of gravity.

The “fundamental particles” appear as oscillation modes of the string. Indeed, these string vibrate and each oscillation mode is an eigenstate of the energy, and thus a particle. Just as we may reconstruct a violin string out of the sequence of its harmonics, one may reconstruct a fundamental string out of the particles which form its oscillation modes. In the low energy limit, one can only reach the fundamental mode. Similarly, our low energy world (much below MP ) has only access to the zero modes of the fundamental strings, i.e. the point particles that we observe.

Among these massless particles one encounters a spin 2 resonance which is interpreted as the graviton field: its long wavelength interactions are found to be in agreement with general relativity. Thus, string theory is a quantum theory of gravity (even though spacetime is treated as a classical background).

The closed string

Parametrizing the coordinate along the string as σ, 0 ≤ σ ≤ π, and time along the world-line of any point of the string as τ (in a way similar to the open string of Fig. 10.2), the world-sheet is described in D-dimensional spacetime by the coordinates XM (σ, τ ), M = 0, . . . , D − 1. Making use of the symmetries of the problem, one may write the string equation of motion as a simple wave

The standard solution is a superposition of left-moving and right-moving waves:

The general string picture 257

The general solution of (10.3) is obtained as a Fourier series:

where α , which has the dimension of a length squared, can be related to the string tension T = 1/(2πα ).

The usual canonical quantization yields commutation relations reminiscent of (an infinite set of) harmonic oscillators:

In order to obtain the mass spectrum, it is easiest to work in a light-cone formalisma which is free from ghosts although not manifestly covariant. The mass formula reads

One constructs the Fock space of quantum states just as in the case of the harmonic oscillator. Introducing a vacuum state |0 annihilated by the αnI , we see from (10.8) that this state corresponds to a tachyon of negative squared

bWe use the standard notation of normal ordering (: . . . :).