19.1. Form factors of a free theory Consider the theory of a free bosonic field (x) associated...

19.1. Form factors of a free theory Consider the theory of a free bosonic field φ(x) associated to a particle A of mass m.

a. Compute the form factors of φ(x) and prove that 0|φ(0)|A = 1/sqrt2. Show that the Euclidean correlation function is given by

φ(x)φ(0) = 1

π

K0(mr).

b. Show that the arbitrariness of the one-particle formfactor of the trace of the stressenergy tensor F

1

= 0|(0)|A ≡ −

√

2π m2Q corresponds to the possibility of redefining the stress-energy tensor by adding a total divergence

(x) = 2π m2φ2 + Q √

π

φ.

c. Use the c-theorem and the form factors of (x) to show that the central charge in the ultraviolet region is given by c = 1+12Q2.