Supersymmetry. Theory, Experiment, and Cosmology

Dynamical supersymmetry breaking: an overview 199

As a first example consider a U (1) supersymmetric gauge theory with two supermultiplets of charge ±1. We denote their scalar components by φ±. In the case of a

characterizes the flat direction. This flat direction consists of all the degenerate vacua; borrowing the terminology to solitons, in particular monopoles (see Chapter 4), one often refers to this set of all degenerate vacua as the classical moduli space.

For any nonvanishing value of a, the U (1) gauge symmetry is spontaneously broken and the vector field becomes massive. In fact, because supersymmetry is not broken (the D-term vanishes), a whole vector supermultiplet becomes massive. As we have seen several times earlier (for example, in Section 5.1.2 of Chapter 5), such a supermultiplet has the same number of degrees of freedom as a massless vector supermultiplet plus a chiral supermultiplet (three vector, four spinor and one scalar): this may be interpreted as the supersymmetric version of the Higgs mechanism. Since we introduced in the theory one vector and two chiral supermultiplets, we are left with a single chiral superfield to describe the light degrees of freedom of the e ective theory much below the scale of gauge symmetry breaking. The light scalar degree of freedom is obviously X = φ+φ−, since it must be gauge invariant. In the vacuum which is labelled by a, one has simply X = a2. The field X is called a modulus: its vacuum expectation value labels the classical moduli space. The dynamical field X(xµ) corresponds to a continuous variation through the space of vacua as one moves in spacetime.

There is obviously no classical potential for X and no strong interaction present to generate it dynamically: W (X) = 0.

The K¨ahler potential, which fixes the normalization of the kinetic term, is obtained from the K¨ahler potential in the original theory. Assuming normalized kinetic terms

We see that the theory is singular as X → 0. This generally means that one is missing degrees of freedom in the e ective theory. Indeed, as a → 0, the symmetry remains unbroken and the low energy theory involves the two chiral supermultiplets.

The rest of this chapter lies in the “Theoretical Introduction” track.

200 Dynamical breaking. Duality

8.2Perturbative nonrenormalization theorems

8.2.1Wilson e ective action

Requiring holomorphy with respect to some parameters of a supersymmetric theory seems to be in contradiction with the renormalization program. Indeed, in the context of supersymmetry, all renormalization e ects appear through the wave function renormalization factors Z. Since these are intrinsically nonholomorphic (recall that scalar kinetic terms are given by a real function, the K¨ahler potential), it seems that the relation between renormalized and tree-level couplings introduces nonholomorphicity which endangers any of the arguments presented above. The solution to this puzzle has been explained by [337, 338].

Let us illustrate this on the example of massive supersymmetric QED [338] (see Section C.3 of Appendix C or Exercise 5 of Chapter 3):

The low energy limit of this theory (i.e. the theory at a scale smaller to the mass scale m0) is a theory of free massless gauge boson and gaugino. The associated gauge coupling is simply given by the exact formula [337]

where ΛU V is the ultraviolet cut-o . Since the superpotential m0Φ+Φ− is not renormalized, one obtains for the low energy physical mass m = Zm0, where Z is the common field renormalization factor. One can thus express the low energy gauge coupling in terms of the physical mass as3:

The factor Z is by essence nonholomorphic: it can be computed through the D-term renormalization and is a real function of ln |ΛU V /m|. Its presence accounts for the multiloop contribution.

It turns out that this nonholomorphic contribution is due to the presence of massless fields. The approach to e ective theories that we have implicitly followed is based on the e ective action formalism described in Section A.5.3 of Appendix Appendix A. The e ective action Γ, which generates proper Green’s functions, is expressed as a series in , which corresponds to an expansion in the number of loops. Because loop

which gives the standard super-QED relation β(α) = α2 (1 − γ(α)) /π with β(α) = ∂α/∂ ln µ and γ(α) = ∂ ln Z/∂ ln µ (see, for a similar relation in the Wess–Zumino model, (8.39)).

Perturbative nonrenormalization theorems 201

integration is performed down to zero momentum, infrared e ects due to the presence of massless fields are naturally included into the e ective action.

An alternative approach has been pursued by [367] and [310]. If one is interested in studying the e ects of a theory at a scale µ, one decomposes the quantum fields into a high frequency part (E > µ) and a low frequency part (E < µ). The Wilson e ective action is then obtained by integrating over the high frequency modes. At a scale µ m, it simply reads

which does not contain higher powers of g i.e. it is renormalized only at one loop. Thus, the relation between the gauge coupling (8.19) appearing in the e ective action Γ and the one appearing in the Wilson e ective action SW is

The last term may be interpreted in terms of the [259]. The parameters of the Wilson e ective action are free from infrared contributions, and one-loop contributions exhaust all renormalization e ects (see (8.21)), which saves holomorphicity. This is not so for the e ective action Γ parameters. Infrared contributions lead to ln p2 terms at the loop level, which results in nonholomorphicity of the nonlocal action.

One may now turn to nonabelian gauge theories. In the case of a supersymmetric SU (N ) gauge theory, direct instanton calculations give the following result for the

where C is a constant and the prefactor 1/g02 accounts for zero modes (infrared e ects).

Since we are discussing holomorphy, one may allow a θFµν ˜µν term which has a

F

physical e ect in an instanton background. Introducing thus the complex parameter

the gauge action simply reads

However, if we replace all factors 1/g02 in (8.23) by −iτ /4π, we find a dependence which is not periodic in the vacuum angle θ. Hence, the expression (8.23) does not seem to allow for a complex gauge coupling, i.e. to be consistent with holomorphy.

202 Dynamical breaking. Duality

Again, the way out of this dilemma is to go to the Wilson e ective action, whose coupling is given in terms of the coupling in the e ective action as:

We see that, as θ changes continuously from 0 to 2π, the N distinct vacua (with a phase dependence e2iπk/N , k = 0, . . . , N −1) are covered. Finally, just as in the abelian case, the evolution of the Wilson e ective action gauge coupling is purely one-loop:

which allows us to write the θ-independent part of the condensate (8.27) as Cµ3 exp

parameters can be viewed as the vevs of chiral fields, one may consider the cut-o s as field-dependent. In order to have a holomorphic description in the e ective theory, one may need to introduce di erent field-dependent regulators for the di erent sectors of the theory.

8.2.2Flat directions

To discuss the rˆole played by flat directions of the scalar potential, we have relied mostly on the tree level potential. A key property is that flat directions remain flat to all orders of perturbation theory [371]. We will prove here this statement in the context of global supersymmetry since this is the relevant framework when one discusses flat directions (see, for example, Section 6.12 of Chapter 6).

Flat directions are associated with the vanishing of auxiliary fields and are thus referred to as F -flat or D-flat directions. They are lifted if one of the auxiliary fields acquires a vacuum expectation value. The tree level potential is quadratic in these fields. In the context of renormalizable theories, a nontrivial vacuum appears if a

discussed in Chapter 1 precisely forbid the generation of such terms at higher orders. If a nonvanishing D-term is to lead to spontaneous supersymmetry breaking, it must be associated with a massless fermion (the Goldstino). This is only possible if the corresponding gauge symmetry is not broken. If the gauge symmetry group is semisimple (no U (1) factor), a term linear in D always appears multiplied with fields which transform nontrivially under the gauge symmetry. Since the symmetry must remain unbroken, these fields have a vanishing vacuum expectation value, which

206 Dynamical breaking. Duality

8.3.1Phases of gauge theories

The simplicity of gauge theories is somewhat misleading. They can represent very diverse regimes with strikingly di erent properties (mass gap, confinement, etc.) as one probes deeper into the infrared, i.e. larger distances. Let us consider two test charges separated by a large distance r.

If we are in a regime where there are unconfined massless gauge fields, the interaction potential V (r) is typically of the form g2(r)/r, where the gauge coupling is evaluated at the distance r. The gauge theory in this regime is not asymptotically free and the gauge coupling diverges at some ultraviolet scale Λ: g2(r) 1/ log (rΛ). Thus V (r) 1/ (r log(rΛ)). This is referred to as the free electric phase. But, it is possible that g2 is constant (free abelian gauge theory or renormalization group fixed point of a nonabelian gauge theory): this is the Coulomb phase where V (r) 1/r. The third possibility is the presence of massless magnetic monopoles. It is now 4π/g2 which is renormalized and V (r) log(rΛ)/r. This is known as the free magnetic phase.

There is, however, the further possibility that the gauge symmetry be spontaneously broken, in which case the gauge bosons become massive through the Higgs mechanism. The vacuum condensate provides a constant contribution to the potential, whereas the Higgs field exchange yields a Yukawa interaction, which is exponentially decreasing at large distances. Thus V (r) is constant in the Higgs phase.

Finally, in the case of asymptotically free nonabelian gauge theories, the interaction potential is confining at large distance: V (r) r.

8.3.2Renormalization group fixed points

In the renormalization group evolution, fixed points correspond to values of the coupling for which the beta function vanishes. The presence of nontrivial fixed points may enhance the symmetries of the theory. Indeed, at a fixed point, a nonsupersymmetric theory has conformal invariance. One may use this to prove general results [279]. For example, the scaling dimension d of a scalar field satisfies d ≥ 1, with equality only for a free field.

Supersymmetry may provide useful constraints on possible fixed points. As a first example, we consider the Wess–Zumino model and show that the nonrenormalization theorems discussed above forbid the existence of nontrivial fixed points [158]. We have seen in Section 8.1.2 that the superpotential (8.3) is not renormalized. This leaves us only with wave function renormalization. Adding the appropriate counterterm to the scalar field kinetic term in the renormalized Lagrangian yields

where Z is the wave function renormalization constant: the bare field is φ0 = Z1/2φ. Then, expressing the superpotential (8.3) in terms of φ0,

gives the renormalized coupling λ in terms of the bare coupling λ0: λ = Z3/2λ0. We deduce a relation between the beta function β(λ) and the anomalous dimension γ(λ) of the field φ:

208 Dynamical breaking. Duality

In the case of supersymmetric theories, the fixed point regime has superconformal invariance, which has some nontrivial consequences. For example, if we consider the supermultiplet of currents (Tµν , Jrµ, JµR) which consists of the energy–momentum tensor, the supersymmetric current and the R-symmetry current (see Section 4.2 of Chapter 4), superconformal invariance implies:

Hence, at the fixed point, the R-symmetry is conserved. Moreover scaling dimensions of superfields satisfy

where R is the R-charge, with equality for chiral or antichiral superfields. We will make use of these results below.

8.3.3Symmetries and anomalies. The ’t Hooft consistency condition

In a confining regime (such as in QCD), it is important to identify correctly the light (massless) degrees of freedom. This is not a problem for QCD since we observe them: they are for example the nucleons. When we consider other theories, identifying them might be a challenge.

’t Hooft [349] has devised a general consistency condition which has proved to be a very powerful tool to identify massless fermionic bound states. It rests on a clever use of the cancellation of anomalies.

Following ’t Hooft, we consider a Yang–Mills theory with gauge symmetry group G, coupled to chiral fermions in various representations of G. These massless fermions form multiplets of a global symmetry group GF . If we now consider the low energy theory, one encounters massless composite bound states. If we now consider three currents associated with the global symmetry GF , the corresponding triangle anomalies may be computed using the elementary fermions of the short-distance description or using the composite massless bound states. Consistency requires that the two computations coincide.

The proof is rather straightforward. Let us gauge GF by introducing weaklycoupled “spectator” gauge bosons. In order to cancel the anomalies of the short distance theory, one must introduce massless “spectator” fermions. At low energy (large distance), one finds the gauge symmetry GF with the massless bound states and the “spectator” fermions: anomalies must cancel i.e. the contribution of spectators must cancel that of bound states. Hence the latter coincides with the anomalies computed at short distance with the elementary (nonspectator) fermions.

8.3.4E ective theories and decoupling

Another way of extracting information is based on the decoupling of heavy flavors [8]. If we make one of the fundamental fermions massive, then, in the limit of large mass, this flavor decouples: in the low energy theory, all bound states containing this fermion disappear. In the context of supersymmetric theories, this means that if we start with a theory with Nf + 1 massless flavors with superpotential W (Φ1, . . . , Φn+1) and make the (n+1)th flavor heavy by adding a mass term mΦ2n+1, one may solve the F -flatness condition for Φn+1 and thus eliminate this field from the low energy action, recovering an e ective theory with Nf fundamental flavors.