17.9. S-matrix with resonances.

Consider an S-matrix for a neutral scalar particle.

a. Show that the unitarity and crossing invariance equations S(θ)S(−θ) = 1, S(θ) = S(iπ −θ), imply that that S(θ) is a periodic function along the imaginary axis of the rapidity variable, i.e.

S(θ) = S(θ +2πi).

b. Assume that S(θ) also presents a periodic behaviour along the real axis of the rapidity variable θ with a period T S(θ) = S(θ +T).

Show that in the domain 0 ≤ Reθ ≤ T and −iπ ≤ Imθ ≤ iπ the S-matrix must have necessarily at least two zeros and two poles unless it is a constant.

c. Argue that the presence of poles with a real part in the physical strip 0 ≤ Imθ ≤ iπ

would spoil the causality properties of the scattering theory.

d. Let

θn,m = iπa+2mπi +nT an infinite sequence of zeros in the physical strip, where 0 < a < 1 is a parameter linked to the coupling constant of the model. Argue that these zeros correspond to resonances and compute their mass and width.