Supersymmetry. Theory, Experiment, and Cosmology

82 The supersymmetry algebra and its representations

Exercise 5 Write equation (III) of Section 2.2 of Chapter 2 in two-component notation:

indicated in Section 4.3.2.

Hint: Use (B.23) and (B.24) of Appendix B, as well as the explicit forms of σµ and σ¯µ.

Exercise 6 Prove that massless representations are only compatible with vanishing central charges. For this purpose, choose the frame of Section 4.3.1 where Pµ = (E, 0, 0, E) and show that Q2ar annihilates any state |Φ .

Hint: See the footnote 6 of Section 4.3.1; if Q2ar annihilates any state |Φ , then zr = 0 from (4.25).

Exercise 7 Show that the N = 2 supersymmetry transformations (4.37) include the N = 1 supersymmetry transformations of the vector and chiral supermultiplets of parameters ηα1 , η¯1α˙ (given in equation (C.99) of Appendix C).

which corresponds to the transformations (4.37) of parameter η1, once one uses the four-component symplectic Majorana notation for spinors and the form (4.35) for the auxiliary fields.

Exercise 8 Consider in electromagnetism the energy density 12 (E2 + B2) and Poynting vector 41π E B. By expressing them in terms of E + iB, show that they are invariant under the duality transformation (4.66).

Hint: |E + iB|2 = (E2 + B2), (E + iB) (E + iB) = 2i E B.

Exercises 83

Problem 1 We study in this problem the two-dimensional supersymmetric model of Section 4.5.1. Two-dimensional supersymmetry plays also an important rˆole in string models because the string, being a one-dimensional object, follows a two-dimensional world-sheet in its motion: string theory can be understood as a two-dimensional quantum theory on the world-sheet (see Section 10.1 of Chapter 10).

1.We first discuss spinors in d = 2 dimensions. Since the spacetime dimension falls in the class d = 2 mod 8, we can impose at the same time the Majorana and Weyl conditions. Moreover, the Lorentz group is simply SO(2) U (1) (or rather SO(1, 1) in the Lorentz case that we consider here). Thus its representations are characterized by a single quantum number. The spinor representation has dimension 2; it can be decomposed into spinors of chirality ±1: a Majorana–Weyl spinor is thus a one-dimensional object and chirality is the additive quantum number just mentioned.

More precisely, we consider the two-dimensional gamma matrices:

(the metric being (+, −)). We define also the two-dimensional analog of the γ5 matrix:

(a) Show that C = −ρ0 is the matrix that intertwines ρµ with −ρTµ :

¯ † 0 c ¯T

One defines as usual ψ = ψ ρ and ψ = Cψ . Show that the Majorana condition ψc = ψ is simply a reality condition with the gamma matrix basis chosen here.

(b) Show that, for Majorana spinors ψ and χ,

which are the analogs of the four-dimensional formulas (B.40) of Appendix B.

(c)Compute Z for the kink state when V (φ) = W 2(φ)/2 = λ(φ2 − φ20)2/2.

(d)Expressing Q+ ± Q− for a static solution (∂0φ = 0), show that one of these combinations vanishes for the kink (antikink), i.e. for a solution of (4.52).

Hints: 1(b) Remember that spinors anticommute: hence transposition of a quadratic combination of spinor fields introduces a minus sign.

¯ µ

2(a,c) δψ = iερ¯ ∂µφ + F ε¯.

In order to compute the Noether current, it is advisable to introduce a spacetime dependent ε(x) spinor parameter and to make use of the relation

δS = d2x∂µεj¯ µ.

This allows us to take full account of the anticommuting nature of ε. One finds the following variation for the Lagrangian L:

+∂µε¯[∂ν φρν ρµψ + iW ρµψ] .

(b)O -shell, two scalar (φ, F ) and two spinor (ψ±) degrees of freedom. On-shell, one must take into account the special nature of 1+1 dimensional spacetimes. Indeed, the free field equations of motion ∂µ∂µφ = 0 and iρµ∂µψ = 0 are

solved by φ = φ+(x + t) + φ−(x − t), ψ+ = ψ+(x + t), ψ− = ψ−(x − t). Hence one scalar and one spinor left-moving degrees of freedom. And the same for right-movers.

To compute [ψ±(x), P0], use the equations of motion:

5

The minimal supersymmetric model

As a part of the general e ort to implement supersymmetry at the level of fundamental interactions, the minimal supersymmetric extension of the Standard Model is motivated and described here at tree level. The presentation of its phenomenology requires a discussion of radiative corrections which play here a central rˆole ; it is therefore postponed to Chapter 7. Of great interest for astrophysics is the lightest supersymmetric particle (LSP) if it is stable: as a weakly interacting massive particle (WIMP), it provides a candidate for dark matter. The last section of this chapter is devoted to these considerations.

5.1Why double the number of fundamental fields?

5.1.1First attempts

If supersymmetry rules the microscopic world of elementary particles, then one should be able to classify the known particles in supersymmetric multiplets or supermultiplets. As is customary (cf. baryons in the SU (3) classification), some elements of the puzzle, i.e. some components of the supermultiplets might be missing and experimental e ort should be looking for them.

In the years 1976–1977, P. Fayet undertook such an e ort, to come to the conclusion that half of the jigsaw puzzle is presently missing. This is such a sweeping statement (although not such an implausible one: after all, the prediction of antiparticles which also led to an almost doubling of the spectrum of known particles, was later confirmed by experiment) that it is worth following P. Fayet’s early e orts to put known particles in supermultiplets.

Obviously, the mass spectrum is not a reliable tool since supersymmetry is (at best) spontaneously broken and thus not visible in the particle spectrum. However, photon and neutrino are a boson–fermion pair of massless particles and it is tempting to associate their charged electroweak partners W± and e±. Indeed a model could be built along these lines [146].

This does not, however, solve the problem at hand: what to do with the other two neutrinos? how to relate the colored particles: the bosons (gluons) are in a di erent representation of SU (3) than the fermions (quarks)? etc.

88 The minimal supersymmetric model

It will prove to be useful in what follows to remember this supersymmetric limit in which mH0 → MZ , mH± → MW and mh0 → mA0 .

5.2Model building

5.2.1Chiral supermultiplets and their supersymmetric couplings

The Standard Model is chiral, in the sense that left-handed and right-handed components transform di erently under the gauge interactions. When discussing supersymmetry, it is however more convenient to describe all fermions by spinors of the same chirality, say left-handed. If necessary, we will use charge conjugation in order to do so; for example we will trade uR for ucL . One thus introduces, for each generation of quarks and leptons:

(SU (3), SU (2), Y )

where the capital letter indicates the superfield. For example U contains the lefthanded quark uL and the squark field uL ; whereas U c contains the left-handed antiquark ucL and the antisquark uR (respectively, charge conjugates of the right-handed quark uR and of the squark uR ). Note that uL and uR are completely independent scalar fields and that the index refers to the chirality of their supersymmetric partners.

Moreover we have just seen that we must introduce two Higgs doublets:

Since by definition, the Minimal SuperSymmetric Model (MSSM) has a minimal field content, the former fields form the minimal set that we are looking for.

Their supersymmetric couplings are described by the superpotential, which is an analytic function of the (super)fields. We may form quadratic terms using the Higgs fields only