How to Compute the Propagator in 2D Euclidean Space?

[constantinou1]

Homework Statement

Consider the following scalar theory formulated in two-dimensional Euclidean space-time;

S=∫d²x ½(∂_μφ∂^μφ + m²φ²) ,​

a) Determine the equations of motion for the field φ.

b) Compute the propagator;

G(x,y) = ∫d²k/(2π)² e^ik(x-y)G(k).​

Homework Equations

Euler-Lagrange equations for the field φ is;

∂L/∂φ - ∂_μ(∂L/∂(∂_μφ) = 0 .​

2. The attempt at a solution
I can solve (a) fine, but its the integral that confuses me.

1.(a)
Well Euclidean space-time has the particular metric g_μν = diag{1,1,1,1}, and the integrand in equation S yields the Lagrangian density L, whereby;

L=½(∂_μφ∂^μφ + m²φ²) .​

By then using the Euler-Lagrange equations for the field φ, we find;

∂_μ∂^μφ - m²φ = 0,
∴ (∂_μ∂^μ - m²) ⋅ φ = 0 .​

2.(a)
Since we are dealing with Euclidean space-time, then;

p_μp^μ = |p⁰|² + |pⁱ|² = m² + |p|²​

The general idea is then to use G(k) = 1/(m² + |k|²), so that;

G(x,y) = ∫d²k/(2π)² e^ik(x-y)/(m² + |k|²),​

and then compute the integral. But how can I possibly solve this integral?

Also, when its written d²k for a Fourier transform, it means dk_x.dk_y right, not integrate twice over dk.dk.

Any pointers or assistance on how to solve this problem is greatly appreciated.

 

[fzero]

The normal way to approach this type of integral is to do the $p^0$ integration by continuing to the complex plane and using the residue theorem. For instance, in Lorentzian 4d, there is an outline at eq. (2.95) of http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf. In your case, you are in Euclidean signature, so you need to look closely at the numerator to decide how you have to close the contour to see which pole should contribute the reside. It is possible that this use of the residue theorem is explained more clearly in your own text or notes, so you should consider checking those sources as well.