Supersymmetry. Theory, Experiment, and Cosmology

where LV is given in (3.48); the rest of the first three lines corresponds to (3.19) where we have just replaced the ordinary derivatives by covariant derivatives:

The last line of (3.52) should be compared with (3.43) where the abelian symmetry allows a much lighter (and thus transparent) notation.

One can solve for the auxiliary fields Fi or F i and Da to obtain the scalar potential. One obtains

In this last equation, we have been slightly more general than previously:

•We have included the possibility of having a gauge group G which is a product of simple groups: for all generators ta corresponding to the same simple group

factor, the coupling ga is identical. For example, in the case of the Standard Model where G|nonabel. = SU (3) × SU (2), all the generators of SU(3) have coupling gs and all generators of SU (2) have coupling g.

•We have included in the last line the possibility of having a certain number of

abelian U (1) gauge groups, labelled by m (qim is the charge of the field φi under U (1)m), with possible Fayet–Iliopoulos terms (labelled ξm for U (1)m). Let us note that a Fayet–Iliopoulos term (3.41) is not allowed for a nonabelian symmetry because it is not gauge invariant12.

12Compare the gauge transformations of the auxiliary fields in (3.39) and (3.50).

50 Basic supermultiplets

The general scalar potential (3.54) with its F -terms and D-terms will be the central object of study when we discuss gauge or supersymmetry breaking.

Exercises

Exercise 1

(a) Show that, under the supersymmetry transformation (3.3), (3.4), we have:

[δ1, δ2] A = 2i ε¯1γµε2 ∂µA [δ1, δ2] B = 2i ε¯1γµε2 ∂µB

i.e. supersymmetry transformations on scalar fields close o -shell.

(b) Prove (3.7) using Fierz transformations (equation (B.29) of Appendix B).

Hints:

(a)Use (B.40) of Appendix B. The fact that supersymmetry transformations close o -shell on scalar fields indicates that there is no need to introduce auxiliary fields in the supersymmetry transformations of scalar fields.

(b)Use equation (B.29) of Appendix B to prove:

ε2ε¯1 − ε1ε¯2 − (γ5ε2) (¯ε1γ5) + (γ5ε1) (¯ε2γ5) = −γµ (¯ε1γµ 2) .

Exercise 2 Compute the spectrum for the O’Raifeartaigh model of Section 3.1.4 in the case A = 0 but for any value of X along the flat direction. Check that STr M 2 = 0.

Hint: This time, the A and Y supermultiplets mix in the mass matrices. The eigenval-

√

ues are for the system (ΨA, ΨX ): m2 + λ2X2 ± λX, for the system (Re A, Re X):

"

m2 + λ2(2X2 − M 2) ± λ λ2(2X2 − M 2)2 + 4m2X2

and for the system (Re A, Re X):

"

m2 + λ2(2X2 + M 2) ± λ λ2(2X2 + M 2)2 + 4m2X2.

Exercise 3 We generalize here the O’Raifeartaigh model to make explicit the structure that leads to supersymmetry breaking.

Let us consider a set of N supermultiplets (Ai, ΨAi ) and P supermultiplets (Xm, ΨXm ) whose interactions are described by the superpotential:

P

m=1

where fm are P analytic functions of the variables Ai.

(a)Write the scalar potential.

(b)Show that, when N < P , supersymmetry is spontaneously broken in the generic case. The O’Raifeartaigh model (3.22) corresponds to N = 1 and P = 2.

[(c) Give an R-symmetry that would make natural the choice (3.55).]

this is not possible in the general case: it would require specific values of the parameters.

(c) R(Ai) = 0, R(Xm) = 2.

Exercise 4 Prove (3.27) from (3.26) using the formulas (B.36) of Appendix B.

c ¯ ¯ c c

Hint: To obtain δS ΨR , introduce a spinor Ψ1 and use Ψ1δS ΨL = Ψ1δS ΨR .

Exercise 5 We write in this exercise the supersymmetric version of quantum electrodynamics as first introduced by [363].

If we want to couple a chiral supermultiplet with a Majorana spinor to an abelian gauge supermultiplet, we face the problem that a gauge (phase) transformation is incompatible with the reality imposed by the Majorana condition. We must thus introduce two Majorana√spinors Ψ1 and Ψ2 in order to form a complex Majorana spinor Ψ± = (Ψ1 ±iΨ2)/ 2 which undergoes a phase transformation under the abelian symmetry.

We thus start from the Lagrangian (3.43) with two chiral supermultiplets (φ±, Ψ±, F±) of charge ±q:

− √ ¯ √ ¯ c L gq Dφ−φ− + 2 λΨ−L φ− + 2 λΨ−R φ− + V ,

where LV is given in (3.38). This Lagrangian is invariant under (3.40) and the following

4

The supersymmetry algebra and its representations

Now that we have acquired some familiarity with supersymmetry and the techniques involved, we may discuss more thoroughly the supersymmetry algebra and its representations. As an example of application, we will then discuss in more details the construction of the representations of N = 2 supersymmetry. Of great importance is the appearance of short representations of supersymmetry, associated with the concept of BPS states in soliton physics. Their study allows us to introduce such notions as moduli, moduli space, etc. important when we discuss supersymmetry breaking or string and brane theory. This chapter is intended for the more theoretically oriented reader1.

4.1Supersymmetry algebra

We use two-component notation (see Appendix B). The basic commutator of N = 1 supersymmetry algebra

1Since this chapter is on the Theorist track we will assume in Section 4.3 familiarity with the superspace techniques developed in Appendix C. We also use the Van der Waerden notation for spinors (see Appendix B).

Supermultiplet of currents 55

invariant mass since [Qiα, P µPµ] = 0). Also, once we have proven that Xij = 0, it follows from (4.5) that Yij = 0. Hence

{Qiα, Qjβ } = εαβ Zij

$ %

The complex constants Zij are called central charges because they are shown to commute with all generators (see Exercise 3). Note that the central charges have the dimension of a mass.

The supersymmetry charges usually also transform under an internal compact symmetry group G which is a product of a semisimple group and an abelian factor. Denoting by Br the hermitian generators of this group, which satisfy

yields (br )j i = (br)i j . Hence the matrices Br are hermitian and the largest possible internal symmetry group which acts on the supersymmetry generators is U (N ).

In the case of N = 1 supersymmetry, one recovers (4.1) since Z, being antisymmetric, vanishes (Z = −Z): there are no nonvanishing central charges. Also, by rescaling the generators Br by (br)11, one obtains from (4.9)

[Qα, Br] = Qα

¯ − ¯

Qα˙ , Br = Qα˙ .

There is only one independent combination R of the generators Br which acts nontrivially on the supersymmetry charges

Thus N = 1 supersymmetry possesses an internal global U (1) symmetry known as

¯ −

the R-symmetry: Qα has R-charge +1 and Qα˙ has R-charge 1. We will see later the important rˆole played by this symmetry (see Section C.2.3 of Appendix C and Chapter 8).

4.2Supermultiplet of currents

The supersymmetry current plays an important rˆole when we discuss the issue of supersymmetry breaking. A key property is that it belongs itself to a supermultiplet,

Representations of the supersymmetry algebra 57

five degrees of freedom. Similarly, Jrµ accounts for 16 − 4 − 4 = 8 and JµR for 4 − 1 = 3. The full supersymmetry transformations read

δJµR = −i¯rγrs5 Jsµ,

This, together with (2.10) of Chapter 2 and (A.230) of Appendix Appendix A, gives the algebra of conformal supersymmetry.

4.3Representations of the supersymmetry algebra

Remember that, in the case of the Poincar´e algebra, representations are classified

according to the mass operator

P 2 = P µ Pµ

and the spin operator which is built out of the Pauli–Ljubanski operator

W 2 has eigenvalues −m2j(j+1) for massive states of spin j = 0, 12 , 1, . . . and Wµ = λPµ for massless states of helicity λ.

Of the two “Casimir” operators, when we go supersymmetric, P 2 remains unchanged since it commutes with supersymmetry, whereas W 2 includes some extra contributions.

4We do not include the commutators of R with the generators of the algebra of the conformal group (Pµ, Mµν , D and Kµ) since they all vanish.