First-Order Perturbation Theory

First-order calculations introduce a single power of \(S_{\mathrm{int}}\), adding interaction vertices to the contraction.

Scalar Cubic Vertex

Consider \(S_{\mathrm{int}} = \int g\,\phi(z)\,\psi(z)\,\mathrm{d}z\) and the two-point function \(\langle\phi(x)\,\phi(y)\rangle_S\):

from sft_wick import Field, Vertex, Action, compute_moment, reset_uid_counter

reset_uid_counter()
phi = Field('phi', 'physical')
psi = Field('psi', 'response')

v = Vertex(fields=[phi, psi], coupling='g')
action = Action(vertices=[v])

obs = [phi('x'), phi('y')]
result = compute_moment(obs, action, order=1)

print("Order 0:")
print(result.order(0).to_latex())
print()
print("Order 1:")
print(result.order(1).to_latex())

Order 0:
C(x, y)

Order 1:
<first-order correction involving R and C propagators with integration>

The first-order correction involves an integration over the internal spatial variable \(y_0\) and products of R and C propagators.

Multi-Component Cubic Vertex

For \(S_{\mathrm{int}} = \int F_{ijk}\,\phi_i(z)\,\phi_j(z)\,\psi_k(z)\,\mathrm{d}z\):

reset_uid_counter()
phi = Field('phi', 'physical', n_components=3)
psi = Field('psi', 'response', n_components=3)

v = Vertex(fields=[phi, phi, psi], coupling='F')
action = Action(vertices=[v])

obs = [psi('a', 'x'), phi('b', 'x'), phi('c', 'x'), phi('d', 'x')]
result = compute_moment(obs, action, order=1)

print("Order 0:")
print(result.order(0).to_latex())
print()
print("Order 1:")
print(result.order(1).to_latex())

The first-order terms now include summations over internal component indices \(i_0, i_1, i_2\) and the coupling tensor \(F_{i_0 i_1 i_2}\).

Inspecting Diagrams

Each non-vanishing contraction at first order produces a diagram:

for d_info in result.diagrams_by_order[1]:
    fd = d_info.to_feynman_diagram()
    print(fd.summary())
    print(f"  Loops: {fd.n_loops}, Connected: {fd.is_connected}")

# Visualise
result.draw_diagrams(order=1)

First-order diagrams have one interaction vertex (square) connected to the external points (circles) by propagator lines.