24.2. Exponential behaviour of the fermionic zero mode The poles present in the Fourier transform of (0)

24.2. Exponential behaviour of the fermionic zero mode The poles present in the Fourier transform of ψ(0)

ab (x) are determined by the exponential behaviour at x→−∞of this function. This behaviour depends on the interaction term V(ϕ(x)), which we assume can be expanded nearby the vacuum values ϕ(0)

a as Vab(ϕ(x)) = va +V

ab(ϕ(x)−ϕ(0)

a )+ 1 2

V

ab(ϕ(x)−ϕ(0)

a )2+· · · .

a. Substitute in Vab(x) the asymptotic expression of the kink configuration ϕ(x) and show that around x→−∞it can be generally written as Vab(ϕ(x)) = va +

∞

n=1 dnenωax, x→−∞, where ωa is the curvature of the bosonic potential U(ϕ) at ϕ = ϕ(0)

a , while dn are coefficients determined by the various derivatives Vn ab and the expansion coefficients μ(a)

n in eqn. (23.2.5).

b. Show that, for x→−∞, it holds

 x x0 Vab(t) dt = −vax+

n=1

ˆd nenωax, x→−∞, where all terms but the first one are exponentially small.

c. Use the results above to show that for x→−∞the fermionic zero mode has the asymptotic expansion ψ(0)
ab (x) = A ˆψ exp−  x x0 Vab(t) dt = A ˆψ exp−vax+
n=1 ˆd nenωax
= A ˆψ evax 1+
n=1 tn enωax, where tn are the final coefficients resulting from the series expansion of the exponential function and the coefficients ˆd n.