Supersymmetry. Theory, Experiment, and Cosmology

Exercises 221

SU (2) indices a, b and c). What are their quantum numbers under SU (2) ×

U (1)?

(c)Deduce a basis B32 of monomials invariant under SU (3) × SU (2).

(d)Finally deduce a basis B321 invariant under SU (3) × SU (2) × U (1).

Hints:

(a)dS = 49; 12 D-term constraints and 12 phases which may be gauge fixed leave dD = 37 complex dimensions.

(b)see [184].

Exercise 2 One considers an extension of the MSSM with two fields singlet under SU (3) × SU (2) × U (1):

•a right-handed neutrino N c;

•a Froggatt–Nielsen scalar θ (see Section 12.1.4 of Chapter 12).

These fields have respective charges xN and xθ under an abelian family symmetry U (1)X . Because this symmetry is pseudo-anomalous, its D-term includes a Fayet– Iliopoulos term ξ:

The scalar potential should be such that θ = 0 (to generate family hierarchies) andN c = 0.

(a)How should one choose the signs of xθ and xN ? What is then a dangerous flat direction?

(b)Which terms should be present in the superpotential in order to prevent this flat direction?

Hints:

(a)xθ > 0 and xN < 0; since xθxN < 0, one can form a holomorphic invariant using both θ and N c: θ and N c could be both nonvanishing;

(b)N cθn would forbid θ = 0; hence (N c)p θn, p ≥ 2 and n = 0 mod p (see [41]).

Exercise 3 In the case of SU (Nc) with Nf < Nc flavors (Section 8.4.1), we probe formula (8.53) for the dynamical superpotential in various regimes of the theory.

1.We first assume that QαNf = vNf δαNf (see end of Section 8.2.3) and study the e ective theory at a scale much smaller than vNf . At scale vNf , SU (Nc) is broken to SU (Nc − 1). Thus the e ective theory is SU (Nc − 1) with Nf − 1 flavors.

− ˜

(a) The SU (Nc 1) gauge coupling diverges at a scale Λ. Neglecting threshold

222Dynamical breaking. Duality

(a)What is the U (1)R charge of m? Deduce that the following combination is dimensionless and has vanishing U (1)R charge:

(b)Write the most general superpotential compatible with the symmetries. By considering the limit of small mass m and small gauge coupling, deduce that:

(c) The e ective theory at scales much smaller than m is SU (Nc) with Nf − 1

ˆ

flavors. Express the corresponding dynamical scale Λ in terms of Λ, m, Nc and Nf .

(d)Starting from (8.53), obtain the e ective low energy superpotential and derive thus a relation between CNc,Nf −1 and CNc,Nf (depending on Nc and Nf ).

3.Use the preceding results to show that

CNc,Nf = (Nc − Nf ) C1/(Nc−Nf ),

where C is a constant.

Hints:

1.(a) Define b(Nc,Nf ) = 3Nc − Nf . By matching the couplings of the full theory and of the e ective theory at the scale µ = vNf , one obtains

or

Λ3Nc−Nf

vN2 f

= Λ3(Nc−1)−(Nf −1). (8.93)

Note that the exact scale where the matching occurs is renormalization scheme dependent. It is precisely vNf in the DR scheme (see Section E.8 of Appendix E).

(b) Following (8.53),

Use (8.93) to show that this corresponds to the formula for the (Nc −1, Nf −1) theory.

Exercises 223

2.(a) One obtains the U (1)R charge of m from the tree level superpotential: 2Nc/Nf .

(b)We have

3Nc−Nf 1/(Nc−Nf )Λ

Consider the limit m → 0, Λ → 0 but X fixed, to obtain F (X) = CNc,Nf +X.

ˆ

(c) Λ3Nc−(Nf −1) = m Λ3Nc−Nf .

(d)Solve for MNf Nf by using its equation of motion derived from (8.92): dWdyn/ dMNf Nf = 0. One obtains:

Exercise 4 Check the anomalies (8.61) using first the fundamental degrees of freedom, then the composite degrees of freedom of the low energy theory.

that there are Nc2 − 1 gaugino fields. Thus the U (1)3R anomaly computed with these

9

Supersymmetric grand unification

With all its successes, the Standard Model does not really achieve complete unification. Indeed, it trades the two couplings αe.m. and GF describing respectively electromagnetic and weak interactions for the two gauge couplings g and g . Moreover, even if one disregards gravitational interactions which are so much weaker, one should at some point take into account the remaining fundamental gauge interaction, namely quantum chromodynamics (QCD), which describes strong interactions. Indeed, as one goes to higher energies, the QCD running coupling becomes weaker, and thus closer in magnitude to the electroweak couplings. Earlier work on quark–lepton unification [302] helped to bridge the gap towards a real unification of all known gauge interactions and in 1973 Georgi and Glashow proposed the first such grand unified model based on the gauge group SU (5) [181].

The most spectacular consequence of this unification is the presence of new gauge interactions which would mediate proton decay: the corresponding gauge bosons have to be very heavy in order to make the proton almost stable. On the more technical side, grand unified models provide a clue to one of the most intriguing aspects of the Standard Model: the cancellation of gauge anomalies. Quarks and leptons are arranged in larger representations of the grand unified symmetry which are anomaly-free [179] (as we will see, this is actually only partially true in minimal SU (5) but is completely realized in the case of SO(10) or larger groups).

We noted above that strong interactions become weaker as one increases the energy. Georgi, Quinn and Weinberg [183] showed that indeed strong and electroweak interactions reach the same coupling regime at a mass scale MU of order 1015 GeV. The three couplings g3 for color SU (3), g and g have approximately the same value: they are unified in the single coupling of the grand unified theory and the corresponding symmetry is e ective above 1015 GeV. This value proved to be a blessing: it ensured that the gauge bosons which mediated proton decay were superheavy, which put (at that time) proton decay in a safe range. Also, the fundamental scale of the grand unified theory was only a few orders of magnitude away from the fundamental scale of quantum gravity, i.e. the Planck scale MP : this led to possible hopes of an ultimate unification of all known fundamental unifications; it will be the subject of the next chapter.

The next step was to go supersymmetric [108] and the supersymmetric version of the minimal SU (5) model was proposed by Dimopoulos and Georgi [107]. We stress that the move to supersymmetry was necessary because the fundamental scale of

An overview of grand unification 225

grand unified theories was found to be so large. In a nonsupersymmetric theory, the superheavy scale would destabilize the electroweak scale. This is the hierarchy or naturalness problem discussed in Chapter 1.

Moreover, the unification of couplings turned out to be in better agreement with the values measured for the low energy couplings [108]. Some 20 years later, with an impressive increase in the precision of measurements, this remains true and contributes significantly to the success of the grand unification scenario. In what follows, after a presentation of grand unified theories, we will thus start by discussing gauge coupling unification. We will then study some aspects specific to supersymmetric grand unification.

9.1An overview of grand unification

9.1.1The minimal SU(5) model

The minimal simple gauge group that incorporates SU (3) × SU (2) × U (1) is SU (5). This is why particular attention has been paid to grand unified models based on SU (5).

The gauge fields transform under the adjoint of SU (5) which has the same dimension 24 as the group. Under SU (3) × SU (2) × U (1) this 24 transforms as follows:

We recognize on the right-hand side: the gluon fields (8, 1, 0), the triplet Aaµ of SU (2) gauge bosons (1, 3, 0), the hypercharge gauge boson Bµ (1, 1, 0). We note the remaining gauge fields in (3, 2, 5/3) as Xαµ, α = 1, 2, 3 color index, for the t3 = 1/2 component of charge 4/3 and Yαµ for the t3 = −1/2 component of charge 1/3 (their

¯ ¯ ¯ −

antiparticles Xαµ and Yαµ are found in (3, 2, 5/3)). They will play an important rˆole in what follows since their exchange is a source of proton decay.

The Xαµ and Yαµ gauge bosons acquire a mass of the order of the scale of breaking of the grand unified theory. This spontaneous breaking is realized minimally by introducing a set of scalar fields H transforming as a 24 of SU (5): this breaks SU (5) into SU (3) × SU (2) × U (1).

In order to ensure the electroweak breaking, we must also incorporate the Standard Model Higgs into this framework. This is done minimally by introducing a set of scalar

We recognize in (1, 2), which we note Φ, the Standard Model SU (2) × U (1) doublet:Φ 250 GeV. The color triplet (3, 1), noted ΦT , must have a vanishing vacuum expectation value in order not to break color: ΦT = 0. However a severe problem arises in this arrangement: the mass scales of the two fields Φ and ΦT must be vastly di erent. Indeed, the doublet mass is constrained to be of the order of the electroweak scale, whereas the triplet must be superheavy because it mediates proton decay. This di culty is known as the “doublet–triplet splitting problem”. In some sense, it is another disguise of the hierarchy problem discussed in Chapter 1.