Supersymmetry. Theory, Experiment, and Cosmology

R-parity breaking 189

Table 7.1 Best limits on some of the couplings λijk . The bounds are obtained from experimental results on the following processes: neutrinoless double beta decay [ββ0ν], charge current universality in the quark sector [Vud], semileptonic decays of mesons [B− → Xq τ − ν¯], atomic parity violation [QW (Cs)].

We refer the reader to more extensive reviews on the subject (see for example [27]) for complete lists of constraints. It should be stressed that, whereas the bounds on the λijk couplings scale like me˜kR , other bounds may have a more complicated dependence on the masses of the supersymmetric particles, as can be seen for example from Table 7.1

in the case of λijk.

The main phenomenological consequences of allowing for R-parity violations are as follows.

First, supersymmetric particles need not be produced in pairs, which lowers the threshold for supersymmetric particle production. This is certainly a welcome feature for experimental searches. On the other hand, the fact that, at the time of writing, no supersymmetric particle has been found thus finds a more natural explanation in the context of R-parity conservation.

Table 7.2 gives, for each class of high energy colliders, the resonant production mechanisms allowed by the di erent types of R-parity breaking couplings.

Second, the LSP is not stable and can decay in the detector. We note that, since the LSP is not stable, cosmological arguments about it being charge and color neutral drop. It could for example be a squark, a slepton, a chargino... We give in Table 7.3 the direct decays of supersymmetric fermions through R-parity violating couplings. Charginos and neutralinos decay into a fermion and a virtual sfermion which subsequently decays into standard fermions through R-parity violating couplings: this yields a three-fermion final state.

For example, in the case of a nonvanishing λijk coupling, the sneutrino R-parity violating partial width reads18

The corresponding mean decay length L reads

where γ is the Lorentz boost factor.

18If the dominant coupling is λijk , replace in the following formulas λ2ijk with 3λijk2 , where 3 accounts for color summation.

190 Phenomenology of supersymmetric models: supersymmetry at the quantum level

Table 7.2 s-channel resonant production of sfermions at colliders in the case of R-parity violation.

Table 7.3 Direct decays of supersymmetric particles via R-parity violating couplings.

Similarly, the partial width for a pure photino neutralino decaying with λijk is [99]

The issue of phases 191

Table 7.4 Charges of the fields and of the parameters under the symmetries U (1) and U (1)R of the minimal supergravity model.

Thus, for couplings of order 10−5, one expects displaced vertices in the detector. For much smaller couplings, the neutralino decays outside the detector whereas for larger couplings, the primary and displaced vertices cannot be separated.

7.7The issue of phases

In our discussions above, we have not paid su cient attention to the fact that the parameters of a supersymmetric theory are complex. The presence of nontrivial complex phases leads to possible deviations from the standard phenomenology that has been presented until now.

These phases appear already at the level of the minimal supergravity model. We may take as parameters of this model19 M1/2, m0, A0 , Bµ and µ. One can redefine the gaugino and Higgs fields in order to make M1/2 and Bµ real. One is then left with two independent complex phases for A0 and µ, which one defines traditionally as

One may alternatively note [110] that, in the limit where both µ and the soft parameters Bµ, M1/2 and A0 vanish, the model acquires new abelian symmetries U (1) and U (1)R: the charges of the di erent superfields are given in Table 7.4. We note that the second symmetry is of a type known as R-symmetry, encountered in Chapter 4 or in Section 7.4.2: it does not commute with supersymmetry. Thus it is a symmetry of the scalar potential but not of the superpotential; similarly, gauginos transform whereas vector gauge fields are invariant. We will study in more details such symmetries in the next chapter.

If one allows the parameters of the model to transform as (spurion) fields, with charges given in Table 7.4, we may turn U (1) and U (1)R as symmetries of the full model. We note that we have the following invariant combinations:

M1/2µBµ, A0µBµ, A0M1/2.

Two of their phases are independent: they are ϕA and ϕB .

19This list is somewhat di erent from the usual one given for example in Section 6.8 of Chapter 6. We have replaced tan β by Bµ. Moreover, in Chapter 6, we assumed implicitly Bµ to be real and |µ| was fixed by electroweak radiative breaking: we were thus left only with an ambiguity with the sign of µ. Here it is the phase of µ which remains to be fixed.

192 Phenomenology of supersymmetric models: supersymmetry at the quantum level

These phases are constrained by experimental data, such as the electric dipole moment of the neutron. As an example, the down quark electric dipole moment receives a one-loop gluino contribution which is computed to be [56, 309]:

where m˜ is a mass scale of the order of the gluino mass mg˜ or a squark mass mq˜, and A is the low energy value of the A-term (the experimental constraint is found in [144]). This shows that the ϕA,B have to be small. This need not be accidental. For example, if A0 and M1/2 arise from a single source of supersymmetry breaking, their phases might be identical, in which case ϕA = 0.

In the case of nonminimal models, other phases appear in the squark and slepton mass matrices. We will return to this question in Chapter 12 and see that a generic supersymmetric extension of the Standard Model has 44 CP violating complex phases!

Further reading

•G.F. Giudice and R. Rattazzi, Theories with gauge-mediated supersymmetry breaking, Physics Reports 322 (1999) 419.

•R. Barbier et al., R-parity violating supersymmetry, Physics Reports 420 (2005) 1.

Exercises

Exercise 1 We prove the cancellation of the one loop contributions to the magnetic operator in the supersymmetric QED model of Wess and Wess and Zumino [363], studied in Exercise 5 of Chapter 3.

Using Lorentz covariance, charge conservation and hermiticity, one may write the matrix element of the electromagnetic current in the Dirac theory in terms of

where qµ = pµ2 − pµ1 and p21 = p22 = m2.

(a)Show that parity conservation implies under F3 = 0. We will assume that parity is conserved from now on (hence we do not take into account weak interactions).

(b)Since (p/ − m)u(p) = 0, one may use the projectors (p/ ± m)/2m and write the decomposition above as

(p/2 + m)M µ(p/1 + m) = (p/2 + m) &γµF1(q2) − σµν qmν F2(q2)' (p/1 + m). (7.115)

(7.116)

where P ≡ p1 + p2.

Exercises 193

(c)We now consider the supersymmetric QED model studied in Exercise 5 of Chapter 3. The Lagrangian is given by its equation (3.58) in terms of the two Majorana spinors Ψ1 and Ψ2 and their supersymmetric partners (we add mass terms to give these fields a common mass m). Show that the magnetic operator is necessary of

2) on the fermion and scalar lines of the Feynman graphs of Fig. 7.2.

(d)Compute the respective contributions of diagrams (a), (b), and (c) of Fig. 7.2 to the Pauli form factor (7.116) and show that they cancel.

Hints:

(a)Under parity ui(˜pi) = γ0ui(pi) up to a phase (see equation (A.107) of Appendix A).

(b)Use (7.116) to compute Tr [γµ(p/2 + m)M µ(p/1 + m)] and Tr [Pµ(p/2 + m)M µ(p/1 + m)].

¯

(c) Because of the Majorana nature (see (B.40) of Appendix B), Ψiσµν Ψi = 0.

(d)

Exercise 2 Show that, in the case where the hidden sector gauge symmetry is broken at an intermediate scale MI , the expression (7.66) for the gaugino condensates is

where one will express the scale M in terms of MU , MI , and the one-loop beta function coe cient b0 (resp. b0) of the corresponding gauge group above (resp. below) the scale MI .

Hints: Write

to obtain M/MU = (MI /MU )1−(b0/b0).

Exercise 3 Using Appendix C, write in components the Lagrangian

√

where S = S + 2θψS + θ2FS is a chiral superfield.

Hints: Use for example Exercise 4, question (a) of Appendix C.

8

Dynamical breaking. Duality

A motivation for supersymmetry is the stability of the electroweak breaking scale under quantum corrections arising from fundamental physics at a much higher scale (naturalness problem). One may however turn the argument around: in a theory with a very large fundamental scale, how does one generate a scale as low as the supersymmetry breaking scale (hierarchy problem)? Again, the supersymmetry breaking mechanism should be devised in order to provide a rationale for such a behavior.

We will argue in what follows that nonperturbative dynamics may be the answer: there are several examples where nonperturbative phenomena account for the generation of very large ratios of scales. This is referred to under the generic name of dynamical breaking. As we will see, supersymmetry provides new tools to control quantum fluctuations in the nonperturbative regime. One will uncover some duality relations between two regimes of a given theory, or two regimes of two di erent theories. This duality is often of a strong/weak coupling nature and is thus very promising to study strongly interacting theories.

The theories that we will consider have a dynamical scale below which some of the symmetries are broken. In the low energy regime, below this scale, the fields are usually composite bound states of the fundamental fields. This leaves the possibility that the fields of the supersymmetric theory at low energy (such as the MSSM studied in Chapter 5) are not fundamental. Supersymmetry may thus be realized at a preonic level.

8.1Dynamical supersymmetry breaking: an overview

8.1.1Introduction

The expression “dynamical breaking of a symmetry” refers to the spontaneous breaking of this symmetry when it is generated by a nonperturbative dynamics.

The standard example is quantum chromodynamics: keeping only the three light quark flavors u, d and s, QCD is described by the action

where Fµν is the gluon field strength and Dµqf the SU (3) covariant derivative of the quark field qf . In the limit of vanishing quark masses mqf → 0, this action is invariant under a global symmetry SU (3)L × SU (3)R, known as the chiral symmetry (independent SU (3) rotations respectively on the left and the right chirality quark fields). Such

Dynamical supersymmetry breaking: an overview 197

The justification may be found in string theory where, as we will see in Chapter 10, there is a single fundamental scale, the string scale MS and all dimensionless parameters may be expressed in terms of vacuum expectation values of scalar fields. For example, in the weakly coupled heterotic string theory, the gauge coupling g and the Planck scale MP are expressed in terms of the vacuum expectation value of the string dilaton S:

More generally, our requirement of generalized holomorphy for the superpotential is simply a translation of the fact that, in a supersymmetric Feynman diagram which involves the interactions described by the superpotential, there is no possibility to find a coupling λ or a mass m .

Once couplings are interpreted on the same footing as fields, one may attribute them quantum numbers. For example, the superpotential (8.3) is invariant under the abelian symmetries U (1) × U (1)R, with quantum numbers given in Table 8.1.

Let us pause for a moment to discuss the di erence of status of the two abelian symmetries. The U (1) symmetry is of a standard nature and commutes with supersymmetry: if (φi, ΨiL ) are the bosonic and fermionic components of a chiral multiplet of charge qi, they both transform under U (1) as:

The superpotential W must be invariant under such a symmetry. On the other hand, U (1)R is known as a R-symmetry, a concept which we have encountered several times: it does not commute with supersymmetry and, as such, plays a central rˆole in all discussions of supersymmetry and supersymmetry breaking. One can show [see Section 4.1 of Chapter 4] that, in the case of N = 1 supersymmetry, the only symmetries which may not commute with supersymmetry are abelian symmetries. Under such a symmetry, fermion and boson components of a given multiplet (φi, ΨiL ) of charge ri transform di erently:

and the superpotential W has nonzero charge r = 2.

Table 8.1