By Biggio, C

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The periodicity can be easily determined from the fact that ǫ(y) is periodic, whereas ǫ(y/2) and ǫ(y/2 + πR/2) are anti-periodic. There are three types of spectra: first, the ordinary KK tower n/R that includes a zero mode; second, an identical spectrum with the absence of the zero mode and, finally, the KK tower shifted by 1/2R. 1. 40). 8) are related. 38 CHAPTER 2. 4: Eigenfunctions of −∂y2 , for a real odd field ϕ(y), versus y/(πR). For each boundary condition, labelled by (V0 , Vπ , Vβ ), the eigenfunction corresponding to the lightest non-vanishing mode is displayed.

5). Obviously, in order to assign these generalized boundary conditions to fermions, Uγ must be a symmetry of the theory. 2 One Fermion Field To illustrate how these general boundary conditions determine the physics, we focus now on the case of a single 5D fermion. We start by deriving the lagrangian and 28 CHAPTER 2. GENERALIZED SCHERK-SCHWARZ MECHANISM the equation of motion in terms of 4D spinors and then we solve it with general boundary conditions. A 5D spinor is composed by two 4D Weyl spinors and can be represented with different notations: ψ1 Φ= (A) ψ2 .

GENERALIZED SCHERK-SCHWARZ MECHANISM the equation of motion in terms of 4D spinors and then we solve it with general boundary conditions. A 5D spinor is composed by two 4D Weyl spinors and can be represented with different notations: ψ1 Φ= (A) ψ2 . 7) ψ1 (B) Ψ= ψ2 With notations (A) we have Φ = (ψ2 ψ 1 ) while with notations (B) we have Ψ = (ψ 1 ψ 2 ). We remember that in this case ψ is not the usual Dirac Ψ = Ψ† γ 0 , but it is defined by ψ = ψ ∗ . Within formalism (A) we can write the 5D lagrangian in the usual way: L(Φ, ∂Φ) = iΦΓM ∂M Φ .

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### Symmetry Breaking in Extra Dimensions by Biggio, C

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