By Pierre Binetruy
ISBN-10: 0198509545
ISBN-13: 9780198509547
This publication is an advent to the speculation of supersymmetry, that's a cornerstone to knowing the physics of common debris past the so-called "Standard Model". Supersymmetry is the 1st introductory booklet in this glossy and more and more renowned topic.
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Extra resources for Supersymmetry. Theory, Experiment and Cosmology
Sample text
59, Plenum Press, p. 135–157. Exercises Exercise 1 Majorana and Dirac masses. The purpose of this first exercise is to familiarize the reader with the notions of Majorana, Weyl, and Dirac spinor. 1, but will prove to be useful in the rest of the book when we deal with supersymmetric particles. Consider a spinor Ψ. 42) and its charge conjugate as T Ψc ≡ CΨ where C satisfies CγµT C −1 = −γµ . In the representation chosen, C T = C † = −C and CC † = C † C = 1. (a) Show that T (Ψc )L (x) = C(ΨR ) (x), T (Ψc )R (x) = C(ΨL ) (x).
However, we do not observe in nature a boson field with the same mass as the electron. We conclude that, if the Hamiltonian describing the fundamental interactions is supersymmetric, supersymmetry is not explicit in the spectrum: it must be spontaneously broken. Secondly, we may obtain from (I) an expression for the Hamiltonian of the system. Indeed, (I) reads 0 µ = 2γrs Pµ . {Qr , Qt }γts 0 , one obtains Contracting with γsr {Qr , Qt } γ 02 tr = 2 T r γ 0 γ µ Pµ . r,t BINE: “CHAP02” — 2006/10/5 — 06:39 — PAGE 24 — #5 Supersymmetry breaking 25 Using γ 02 = 1l and Tr(γ 0 γ µ ) = 4g 0µ , one obtains Q2r = 4P 0 = 4P0 r thus H= 1 4 Q2r .
One finds the following ground states: • if M 2 ≤ m2 /(2λ2 ), A =0 • if M 2 ≥ m2 /(2λ2 ), A2 = M 2 − m2 /(2λ2 ) and 2 V = λ2 M 4 = FX and V = m2 M 2 − m2 /(4λ2 ) . One may check that the spectrum is no longer supersymmetric. Take the first case A = 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 BINE: “CHAP03” — 2006/10/5 — 06:39 — PAGE 41 — #5 42 Basic supermultiplets of the (A, ΨA ) and (Y, ΨY )√supermultiplets are no longer supersymmetric: one finds √ m2 − 2λ2 M 2 for Re A, m2 + 2λ2 M 2 for Im A and m for Y .
Supersymmetry. Theory, Experiment and Cosmology by Pierre Binetruy
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