Stable branches of a solution for a fermion on domain wall
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ELEMENTARY PARTICLES AND FIELDS Theory
Stable Branches of a Solution for a Fermion on Domain Wall* V. A. Gani1), 2) , V. G. Ksenzov2) , and A. E. Kudryavtsev2) Received December 7, 2010
Abstract—The case when a fermion occupies an excited nonzero frequency level in the field of domain wall is discussed. It is demonstrated that a solution exists for the coupling constant in the limited interval 1 < g < gmax ≈ 1.65. It is shown that indeed there are different branches of stable solution for g in this √ interval. The first one corresponds to √ a fermion located on the domain wall (1 < g < 4 2π). The second branch, which belongs to the interval 4 2π ≤ g ≤ gmax , describes a polarized fermion off the domain wall. The third branch with 1 < g < gmax describes an excited antifermion in the field of the domain wall. DOI: 10.1134/S1063778811050085
1. INTRODUCTION In our previous paper [1] we studied the problem “domain wall + excited fermion”. Initially the problem of the spectrum of a fermion coupled to the field of a static kink was discussed in [2–6]. These papers were devoted mainly to a zero-frequency fermion bound by the domain wall. Fermionic bound states in the field of external kink were studied in [7] for the case of the λφ4 model and in [8] for the case of the sine-Gordon model. However, the authors of these publications have considered kink as given external field. As we shall demonstrate in this our work, the presence of the fermion changes drastically the kink profile. So the excitation spectrum for the problem “fermion coupled to kink” looks quite different from that calculated in the external field approximation. We studied the system of the interacting scalar (φ) and fermion (Ψ) fields in two-dimensional space– time (1 + 1). In terms of dimensionless fields, coupling constant g and space–time variables (x, t), the Lagrangian density was taken in the form 2 1 2 1 ˆ − gΨΨφ. ¯ ∂Ψ ¯ φ − 1 + Ψi (1) L = (∂μ φ)2 − 2 2 The equation of motion for scalar field φ(x, t) in the presence of a fermionic field Ψ reads: ¯ (2) ∂μ ∂ μ φ − 2φ + 2φ3 = −gΨΨ, ¯ = Ψ† β, β is the Pauli matrix, see below (5). where Ψ ∗
The text was submitted by the authors in English. Department of Mathematics, National Research Nuclear University MEPhI, Moscow, Russia. 2) Institute for Theoretical and Experimental Physics, Moscow, Russia. 1)
If the coupling of the scalar field to fermions is switched off, g = 0, the equation of motion (2) has a static solution called “kink”, (3) φK (x) = tanh x. In three space dimensions this solution corresponds to a domain wall that separates two space regions with different vacua φ± = ±1, see, e.g., [9, 10] for more details. Let us discuss the fermionic sector of the theory. After the substitution Ψ(x, t) = e−iεt ψε (x) the Dirac equation for the massless case reads: ∂ − gβφ(x) ψε (x) = 0, (4) ε + iαx ∂x where αx and β are the Pauli matrices, ⎛ ⎞ ⎛ ⎞ 0 −i 0 1 ⎠, β = ⎝ ⎠. αx = ⎝ i 0 1 0 In Eq. (4),
⎞
⎛ ψε (x) = ⎝
(5)
uε (x)
⎠
vε (x) is the two-component spinor wave function. In terms of functions uε (x) and vε (x), E
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