23.2. Short and long kinks Consider a simplified version of a potential with three vacua configurations, realized by the potential shown in the Figure below V(ϕ) = m2 2

⎧⎨

⎩

(ϕ +2b)2, ϕ ≤−b;

ϕ2, −b ≤ ϕ ≤ a;

(ϕ −2a)2, ϕ >a.

Shape of V (ϕ) with a = b (left hand side) and with a > b (right hand side).

a. Prove that the explicit configurations of the long and short kink of this potential are given by

ϕL(x) = aemx, x ≤ 0, 2a−ae−mx, x ≥ 0, ϕ¯S (x) = −bemx, x ≤ 0,

−2b+be−mx, x ≥ 0

b. Show that their classical masses are ML = ma2, MS = mb2.

c. Compute the form factors involving the long and the short kinks and show that they are given by Fϕ

L¯L

(θ) = i ML(iπ −θ)

 1

θ −iπ(1−ξL)

− 1

θ −iπ(1+ξL)

, Fϕ

¯ SS

(θ) = − i MS(iπ −θ)

 1

θ −iπ(1−ξS)

− 1

θ −iπ(1+ξS)

.

Determine ξL and ξS.

d. Show that there are apparently two towers of particles, of different masses, above the vacuum | 0, expressed by the formulae mL = 2ma2 sin π
2a2 , mS = 2mb2 sin π
2b2 .

e. Argue that one of these two spectra has to be spurious.