Supersymmetry. Theory, Experiment, and Cosmology

Chiral supermultiplet 39

Note that the first two coincide with (3.3)–(3.4) when one makes use of the equation of motion (3.9). The novel feature is that one recovers (3.6) (and similar relations for φ and F ), without making use of any equation of motion: in this formulation the supersymmetry algebra closes o -shell.

It is not so surprising that we now have an o -shell formulation of supersymmetry: with the introduction of auxiliary fields, the number of o -shell bosonic degrees of freedom (four: A, B, F1, F2) equals the number of o -shell fermionic degrees of freedom (four: Ψr). We will see that this generalizes to all types of multiplets.

The supersymmetry transformations (3.10) call for several comments.

Let us first look for signs of spontaneous supersymmetry breaking. Since supersymmetry is global (i.e. the parameter ε is constant), we should look for a Goldstone field. As recalled in Section A.2.1 of Appendix Appendix A, in the case of a continuous symmetry which is spontaneously broken through the vacuum expectation value v = 0 of a scalar field, a constant term in the transformation law, i.e.

where α is the parameter of the transformation, characterizes φ as a Goldstone boson. In our case, supersymmetry is of a fermionic nature and, as discussed in Chapter 2, we expect the Goldstone field to be a fermion, the Goldstino, namely Ψ in our example. A look at (3.10) shows that a vacuum value for φ (i.e. A or B) does not generate a constant term in the transformation law of Ψ whereas a constant value for the auxiliary field does. For example, if F1 = 0 (this is obviously not the case in the free field theory that we consider here), then

Thus a non-vanishing ground state value for the auxiliary field is a signature of spontaneous supersymmetry breakdown.

If a continuous bosonic symmetry is local, the finite version of (3.11) shows that one can choose the parameter α(x) locally in such a way that it cancels the field φ(x): this field disappears from the spectrum. This is the Higgs mechanism. We will see in Chapter 6 that a similar phenomenon, the super-Higgs mechanism, occurs when supersymmetry is local: the Goldstone fermion disappears from the spectrum.

Finally, one should note that, since we are dealing with global supersymmetry, both δF1 and δF2 are total derivatives. This is a characteristic property of auxiliary fields, which will allow us to construct easily invariant action terms.

40 Basic supermultiplets

3.1.3Interacting theory

There is no di culty in adding interactions. As we have seen earlier, this is precisely what makes supersymmetry special. Indeed, the Lagrangian

which is analytic in the field φ and is called the superpotential. All interaction terms involve the superpotential and its derivatives

Let us write for the sake of completeness the Lagrangian describing n such supermultiplets (φi, ψi, i = 1, . . . , n) with a general superpotential W (φi), analytic in the fields φi:

Solving for Fi yields the scalar potential

Chiral supermultiplet 41

Let us stress that the analyticity of the superpotential W (it depends on the fields φi but not on φi ) is a crucial ingredient for supersymmetry. As for supersymmetry breaking, we see that supersymmetry is broken spontaneously if and only if ∂W/∂φi = 0, that is

It is this criterion – an auxiliary field acquires a non-zero vacuum expectation value (vev) – which will remain the basic one when we move to local supersymmetry. Indeed, as discussed above, it is associated with the presence of a Goldstone fermion, and is a necessary condition for the super-Higgs mechanism to take place.

3.1.4An example of F -term spontaneous supersymmetry breaking: O’Raifeartaigh mechanism

It is easy to see that, with a single field φ and a polynomial superpotential, it is not possible to break supersymmetry spontaneously: since φ is complex, dW/dφ = 0 always has solutions, which are the ground states of the theory and V = 0.

The simplest example of F-term supersymmetry breaking involves three scalar fields and was devised by [300]. Let us indeed consider the superpotential

where λ > 0, m and M are real parameters. The corresponding potential is given by (3.20). It reads

If M = 0, one cannot set the three terms separately to zero. Thus V = 0 and supersymmetry is spontaneously broken.

Let us see this in more detail. One may always choose Y and X in order to set FA = 0. Indeed, there is a direction in field space (mY + 2λAX = 0) where the potential is flat for a fixed value of A: this is the first example that we encounter of a flat direction of the scalar potential, a characteristic of supersymmetric models that will have important phenomenological and cosmological consequences.

Once this is done, one is left with a potential depending solely on A. One finds the following ground states:

One may check that the spectrum is no longer supersymmetric. Take the first caseA = 0; we will take the opportunity of the flat direction to compute the spectrum at X = 0. Note that X and ΨX are massless: ΨX is the Goldstone fermion associated with the spontaneous breaking ( FX = 0). On the other hand, the masses

where ε is the Majorana spinor of the transformation.

This supersymmetry transformation is similar to the one found for a Majorana spinor (3.10), except for the chirality projector L ≡ (1−γ5)/2 in the transformation law of the fermion field: this ensures that it remains left-handed under a supersymmetry transformation.

By taking the hermitian conjugate of (3.26), one may obtain, using formulas (B.36)

ginal free field Lagrangian, much in the way of (3.13) but with a special attention to chiralities:

One recognizes the first and second derivatives of the superpotential (3.17). The mass term is a Majorana mass term and (once one solves for F ) the interaction term is precisely the one which was given in (2.17) of Chapter 2 when we first discussed supersymmetric interactions as a way of evading Coleman–Mandula no-go theorem.

Now, if we look closely at (3.27), we realize that, just as (φ, ΨL , F ) transformed as (3.26) with a left-handed chirality projector, (φ , ΨcR , F ) transform with a righthanded chirality projector. By convention, the first transformation law (3.26) is the

44 Basic supermultiplets

ˇ ˇ c ˇ

Obviously then, (φ , ΨL , F ) transforms as a chiral multiplet. When we come to the Standard Model, we will use the relation just found to put all the fermion fields into chiral supermultiplets. For example, denoting the complex scalar which is the supersymmetric partner of eR by e˜R , we will describe their supersymmetric interactions in terms of e˜R and ecL , members of the associate chiral supermultiplet.

For future use, we may write the general interaction of a set of chiral supermultiplets with Weyl spinor fields (φi, ΨiL , Fi)3 (to be compared with the Majorana case (3.19)4):

Let us take this opportunity to introduce the general notion of an F -term. Since the kinetic terms of the free field theory are invariant under supersymmetry, the terms depending on W should be invariant on their own. Indeed we have, using (3.26) and (3.27) (ε = εc)

This transformation law is actually a consequence of the fact that this combination may be considered as the auxiliary field of a composite chiral supermultiplet5. We thus note, for any analytic function W of the fields φi,

We end this section by discussing some subtleties associated with the derivation of the supersymmetric current. We start with the Lagrangian (3.28) which we write, using the notation just introduced,

3In the following equation, we have adopted the convention to put lower index i for fields whose supersymmetry transformation follows the chiral supermultiplet transformation rule (3.26), such as φi, ΨiL , . . ., and upper index i to fields whose supersymmetry transformation follows the antichiral

supermultiplet transformation rule (3.29), such as φ i, ΨciR , . . ..

4Using (B.34) of Appendix B, one shows that for, for a Majorana spinor (Ψ = Ψc), ΨR iγµ∂µΨR = ΨL iγµ∂µΨL , up to a total derivative. Hence the di erent normalization of the fermion kinetic term in the two Lagrangians.

5[See Section C.2.2 of Appendix C for details.]

Vector supermultiplet and gauge interactions 45

The Noether current Jµ(N ) associated with the supersymmetry transformations (3.26) and (3.27) reads

This is, however, not the supersymmetry current Jµ because the supersymmetry transformations are not an invariance of the Lagrangian (3.33). They transform it into a total derivative

δS L = ∂µ (¯Kµ) ,

3.2Vector supermultiplet and gauge interactions

We will be much briefer in this case since the discussion is parallel to the previous one.

3.2.1Vector supermultiplet

The o -shell formulation of a vector supermultiplet uses a real vector field Aµ, a Majorana spinor λ and a real auxiliary pseudoscalar field D. This makes 3 + 1 bosonic degrees of freedom and 4 fermionic degrees of freedom in the o -shell formulation; on-shell, the auxiliary field is no longer independent and we have two bosonic and two fermionic degrees of freedom. The free field Lagrangian reads:

46 Basic supermultiplets

where Fµν = ∂µAν − ∂µAν . It is invariant under the abelian gauge transformation:

Once again, if D = 0, the supersymmetry transformation of the spinor field includes a constant term, which is the trademark of a Goldstone fermion or Goldstino. Then, the supersymmetry transformation of the auxiliary field D is a total derivative. This means that the following Lagrangian

leads by itself to a supersymmetric term in the action7. This is the so-called FayetIliopoulos [145] term. If the vector field is an abelian gauge field, it turns out that this term is gauge invariant and may be added to the Lagrangian (3.38).

Now that we have introduced auxiliary D fields, we may present the concept of a D-term (which somewhat parallels the F -term introduced in the previous section). It turns out8 that the kinetic terms of a chiral supermultiplet may be understood (up

to a total derivative) as the D auxiliary field of a composite supermultiplet which we note φ†φ:

This is indeed the reason why this provides a supersymmetric invariant action (as an auxiliary field, it transforms into a total derivative). The vector component (known as the K¨ahler connection) of this composite supermultiplet is Aµ = −iφ ∂µφ + iφ∂µφ −

¯ γµΨ . Such a construction can be generalized9 to any real function K(φ†, φ) (called

ΨL L

a K¨ahler potental). We thus see that full supersymmetric actions may easily be constructed as F components (cf. (3.32)) or D components (cf. (3.42)) of some composite supermultiplets.

6The γ5 in the transformation law of Aµ might seem surprising. However, as can be seen from (B.41) of Appendix B, it ensures the reality of δS Aµ. Following a standard convention [362], one

interactions.

7A similar term of the form aF + h.c. is in principle possible if the corresponding scalar field φ is a gauge singlet. However, as can be seen from (3.19), it corresponds to a linear term in the superpotential which can be absorbed through a redefinition of the field φ.

8[See Section 3.2.2 for details.]

9[See Section 3.2.4.]

Vector supermultiplet and gauge interactions 47

3.2.2Coupling of a chiral supermultiplet to an abelian gauge supermultiplet

We will consider here the case of a chiral supermultiplet with a Weyl spinor (see Exercise 5 for the case of Majorana spinors). The following Lagrangian may be shown to be invariant under supersymmetry transformations:

where Dµ = ∂µ − igqAµ is the covariant derivative, g is the gauge coupling and q the U (1) charge. More precisely, it is invariant under (3.40) and a variation of (3.10) where one replaces derivatives by covariant derivatives and one completes the auxiliary field transformation:

If we add LV given by (3.38) and possibly LFI in (3.41), we have a supersymmetric theory of a chiral supermultiplet coupled with an abelian gauge supermultiplet. The novel feature is that gauge interactions give a contribution to the scalar potential through the D auxiliary field. Indeed solving for D yields

If there are several such chiral supermultiplets with scalar fields φi of charge qi, the potential reads:

If we apply this to the abelian U (1)Y hypercharge symmetry of the Standard Model, then since the charges yi of the quarks and leptons (and of their scalar supersymmetric partners) have both signs, the minimum of the potential corresponds to zero energy and supersymmetry is not spontaneously broken. One is therefore tempted to introduce another abelian symmetry under which all charges qi have the sign opposite to the sign of ξ. Obviously, the minimum of the potential corresponds to all scalar fields vanishing and V = ξ2/2: supersymmetry is spontaneously broken. And, by expanding the potential (3.46), we check that the scalar squared masses are simply −gqiξ > 0: if ξ is large enough, they can be made much larger than typical quark and

48 Basic supermultiplets

lepton masses. However, such an abelian gauge symmetry has potential problems with quantum anomalies10.

We will see in Section 6.3.2 of Chapter 6 that the phenomenological problems that we have encountered both with F -type breaking and D-type breaking leads one to isolate the supersymmetry-breaking sector from the sector of quarks, leptons and their supersymmetric partners.

3.2.3Nonabelian gauge symmetries

The previous considerations are easily generalized to nonabelian gauge theories. Let us consider a gauge group G with coupling constant g (G is a simple Lie group) and structure constants Cabc (the index a runs over the adjoint representation of the group): the generators of the group satisfy

The gauge supermultiplet is simply (Aaµ, λa, Da), where one introduces a gaugino λa and a real auxiliary Da for each gauge vector field Aaµ. The straightforward generalization of (3.38) reads

are, respectively, the covariant field strength and the covariant derivative of the gaugino field11. This action is invariant under the supersymmetry transformation (3.40) where a superscript a should be added to all fields.

Likewise, the coupling of chiral supermultiplets to this gauge supermultiplet follows the same lines as before. Consider a set of chiral supermultiplets (φi, Ψi, Fi) transforming under the gauge group G in a representation with hermitian matrices (ta)i j :

10Remember that the triangle anomaly diagram with three abelian gauge bosons is proportional to i qi3.

11The corresponding infinitesimal gauge transformations of the gauge supermultiplet are: