Supersymmetry. Theory, Experiment, and Cosmology

Example of supersymmetric SU(Nc) with Nf flavors. The rˆole of R-symmetries 209

We take this opportunity to note that, in order to integrate out a heavy degree of freedom in a supersymmetric way, one has to work at the level of the superpotential, and not at the level of the Lagrangian (potential).

8.4Example of supersymmetric SU (Nc) with Nf flavors. The rˆole of R-symmetries

We use the methods introduced in the preceding sections to study the case of supersymmetric QCD, or more generally of a SU (Nc) gauge theory with Nf flavors of quarks and antiquarks. As we will see, the discussion depends on whether the number of flavors Nc is smaller or larger than the number of colors.

8.4.1Nf < Nc

If at some large scale M0, the gauge coupling has a perturbative value g, then the running coupling g(µ) evolves, at one loop, as

where b = 3Nc − Nf is the coe cient of the one-loop beta function (see Chapter 9). Due to asymptotic freedom, or rather asymptotic slavery (b > 0), it explodes at a scale

Above the scale Λ, we have a theory of elementary excitations, the quarks and antiquarks, whereas, below, the e ective theory is a theory of mesons. We wish to study the symmetries of the original theory in order to identify the dynamical interactions of the meson fields. Under independent global rotations of left and right chiralities, SU (Nf )L × SU (Nf )R, the quark superfields Q transform as (Nf , 1) (since they

¯

include the left-handed quark field), whereas the antiquark superfields Q transform

¯

as (1, Nf ) (they include the right-handed chirality). Similarly, baryon number con-

¯

servation is associated with a global U (1)B symmetry: Q (resp. Q) has charge +1 (resp. −1).

We now identify a R-symmetry which is nonanomalous. We have seen in (8.7) that the fermionic component of a superfield of R-charge r (the charge of its scalar

Also gaugino fields transform with charge +1 (see equation (C.77) of Appendix C):

210 Dynamical breaking. Duality

One computes the mixed U (1) − SU (Nc) − SU (Nc) triangle anomalies. The goal is to choose r in order to cancel these mixed anomalies. Writing T a (resp. ta) the generators of SU (Nc) in the adjoint (resp. fundamental) representation, we have:

Then the condition of anomaly cancellation reads:

Nc + 12 (r − 1)Nf + 12 (r − 1)Nf = 0.

Hence

The meson superfields Mij of the e ective theory transform under the nonanomalous U (1)R symmetry as:

We now use the full symmetries in order to extract the dynamical interactions of these e ective mesonic degrees of freedom. They are described by a superpotential Wdyn(Mij ). Since this superpotential cannot have a matrix structure – indeed it must be invariant under SU (Nf )L × SU (Nf )R – it must depend on det(M ). We summarize in Table 8.2 the transformation properties of the di erent, fundamental or e ective, fields.

Since Wdyn(M ) must have R-charge +2, we conclude that it must be proportional to (det M )1/(Nf −Nc). Using dimensional analysis ([W ] = 3 and [det M ] = 2Nf ) and the fact that the only dynamical scale available is the scale Λ, we conclude that the dynamical e ective potential is

where CNc,Nf is a dimensionless constant. This result may be tested in several di erent regimes and the constant computed to depend on Nc and Nf as CNc,Nf = (Nc −

1/(Nc−Nf ), with C constant (see Exercise 3).

Table 8.2

Example of supersymmetric SU(Nc) with Nf flavors. The rˆole of R-symmetries 211

In the special case Nf = Nc − 1, we may then write (8.53) as

where we have used (8.46). We recognize in the last factor the standard one-instanton contribution. Indeed, a direct instanton calculation provides the same answer and allows us to compute the remaining constant: C = 1 in the DR scheme [163].

Finally, taking Nf = 0 yields Wdyn(M ) = Λ3. This is obviously related to gaugino condensation: the dynamical potential being a function of λλ alone, dimensional

8.4.2 Nf = Nc or Nc + 1

Let us start with Nf = Nc. Then, according to Table 8.2, the quark superfields Q

¯

and Q have vanishing R-charge and it is not possible to use them only to write a superpotential of R-charge 2. However, there are now enough flavors to be able to form baryonic states:

[335] has, however, argued that this relation is only valid at the classical level. At the quantum level, he proposes to include a nonperturbative contribution:

where Λ is the dynamical scale.

One may first check that the new term Λ2Nc has the right dimension (2Nc) and R-charge (0). But the main argument in favour of this addition is that it yields the right theory when one flavor decouples. Let us see this in detail. As discussed above in Section 8.3.4, we make the Nf th flavor decouple by introducing the following tree level term in the superpotential:

212 Dynamical breaking. Duality

We expect that the theory at scales much below m is supersymmetric SU (Nc) with Nc − 1 flavors. Thus the low energy superpotential should simply read, according to (8.53),

where b = 2Nc is the one-loop beta function coe cient of the theory with Nc fla-

To extract the low energy theory, we must place ourselves in the F -flat direction cor-

in agreement with (8.58).

Another nontrivial check is provided by the ’t Hooft anomaly condition (see Section 8.3.3 above). The complete global symmetry SU (Nf = Nc)L × SU (Nf = Nc)R × U (1)B × U (1)R is partially broken by the vacua which satisfy the quantum constraint (8.56). The ’t Hooft consistency condition is all the more useful that the residual

λ) as well as composite fermion fields (supersymmetric partners of the meson scalars

We note that among the e ective degrees of freedom, the mesons form an adjoint representation of dimension Nf2 − 1 of SU (Nf = Nc)V : one component of the matrix Mij is not independent because of the quantum constraint (8.56) which fixes det M .

N = 2 supersymmetry and the Seiberg–Witten model 217

once the gauge symmetry is spontaneously broken. Defining A(µ±) ≡ A1µ iA2µ /√2, the mass term simply reads

We conclude that the massive states can be organized in two multiplets of re-

√

spective charge ±1 but same mass a 2. Each multiplet has eight degrees of freedom. But since supersymmetry is not broken in the vacuum (8.72), these multiplets must be N = 2 supermultiplets. This seems in contradiction with what we have seen in Section 4.3.2 of Chapter 4: massive representations of N supersymmetry theories have a multiplicity of 22N (2j + 1), which gives for N = 2 a minimum of 16 (for j = 0), unless there are central charges.

This necessarily means that the theory has a central charge and that the supermultiplets that we consider are the short supermultiplets discussed in Section 4.3.3 of Chapter 4. The mass of these supermultiplets is directly related to the central charge of the theory: m = z/2. Let us recall that the central charge is an operator which commutes with all the generators of the supersymmetry algebra. We deduce an expression

Let us note that this expression might be incomplete since it has been obtained in the ultraviolet regime of the theory where the degrees of freedom have only electric charge. As expected from the considerations of Chapter 4, and as seen below, we expect that, in other regimes of the theory, appear solitons which carry magnetic charge.

We now try to identify the e ective low energy action. In the regime where we have easily access to it, it corresponds to a N = 2 supersymmetric U (1) theory, which we may describe in terms of a N = 1 abelian gauge supermultiplet Wα and a N = 1

where the holomorphic function F (Φ) is a N = 2 prepotential which generalizes the discussion of Section 4.4.1 of Chapter 4: it determines simultaneously the holomorphic gauge kinetic function f (Φ) iF (Φ) and the K¨ahler potential K(Φ, Φ†) = ImF (Φ) which fixes the normalization of the scalar kinetic term.

We have already noted that the zeroes of the K¨ahler potential represent singular points in the scalar field parameter space where the parametrization is inadequate,

218 Dynamical breaking. Duality

usually because the light degrees of freedom have not been identified properly. Indeed, because F (Φ) is holomorphic, K(Φ, Φ†) = ImF (Φ) is harmonic and has thus no minimum in the complex plane. If it is not constant, it should reach zero at some point.

We will see however that the e ective action (8.74) has alternate descriptions which may thus describe other regimes of the theory. We start by performing what is known as a duality transformation.

We define

Hence the second term of (8.74) allows us to identify FD as the prepotential in the dual formulation. Before checking this with the first term, let us note that

Hence the duality transformation yields an alternative description where the parameter τ introduced in (8.66) transforms as

as in (8.72). Moreover (8.67) tells us that this is a duality relation of the weak coupling/strong coupling type, i.e. g2 ↔ 1/g2, that we have encountered when discussing electric–magnetic duality.

Before dwelling more on this, let us show that the gauge part of the action (8.74) is invariant under the duality transformation. For this we recall that the gauge superfield Wα can be defined as a generic chiral superfield which satisfies the constraint (C.68) of