Feynman Rules and Scattering

Yichen Shi Michaelmas 2013

Note that we use the metric convention (− + ++).

1 Feynman Rules

1. Draw a Feynman diagram of the process and put momenta on each line consistent with momentum conser- vation.

2. Associate with each internal propagator −i (scalar propagator); p2 + m2 − i

−i(−p/ + m) (fermion propagator); p2 + m2 − i −iη ab (photon propagator). p2 − i

3. Associate vertices with coupling constants obtained from the interaction term in the Lagrangian, and impose momentum conservation 4 (4) X X  (2π) δ pin − pout .

4. Integrate over momenta associated with loops, with the measure

d4k , (2π)4 and multiplied by -1 for fermions and 1 for bosons.

5. Add in wavefunction terms for external particles of momentum p and spin s:

u(p, s) incoming fermions;u ¯(p, s) outgoing fermions;

v¯(p, s) incoming antifermions; v(p, s) outgoing antifermions; ∗ a incoming photons; a outgoing photons.

6. Take into account symmetry factors, which originate from various combinatorial factors counting the different δ ways in which the δJ ’s can hit the J’s in the path-integral formulation.

Note that we look at the Lagrangian for information. For example, if it contains the term gφψγ¯ 5ψ then - gγ5 ⇒ add igγ5 to vertices; - φ ⇒ the propagator is scalar; - ψψ¯ ⇒ particles are fermionic so we need to subtract one diagram from the other when calculating A.

1 2 Examples

• Fermion propagating through n vertices labelled a1 to an, emitting a photon at each vertex −i(−p/ + m) −i(−p/ + m) n 0 0 an n an−1 n−1 an−2 a1 (−ie) u¯(p , s )γ 2 2 γ 2 2 γ ··· γ u(p, s). pn + m − i pn−1 + m − i

• Anti-fermion propagating through n vertices labelled an to a1, emitting a photon at each vertex −i(−p/ + m) −i(−p/ + m) n an n an−1 n−1 an−2 a1 0 0 (−ie) v¯(p, s)γ 2 2 γ 2 2 γ ··· γ v(p , s ). pn + m − i pn−1 + m − i

• Fermion loop with n vertices emitting photons

−i(−p/ + m) −i(−p/ + m) −i(−p/ + m) n an n an−1 n−1 an−2 a1 1 (−ie) γ 2 2 γ 2 2 γ ··· γ 2 2 . pn + m − i pn−1 + m − i p1 + m − i

• e−µ− → e−µ− (photon internal line) −iη (−ie)2u¯ (p0, s0)γau (p, s) ab u¯ (q0, r0)γbu (q, r). e e (p − p0)2 − i µ µ

• e−e− → e−e− (photon internal line) −iη A = (−ie)2u¯ (p0, s0)γau (p, s) ab u¯ (q0, r0)γbu (q, r); 1 e e (p − p0)2 − i e e −iη A = (−ie)2u¯ (p0, s0)γau (q, r) ab u¯ (p0, s0)γbu (q, r). 2 e e (p − p0)2 − i µ µ So due to Fermi-Dirac statistics, A = A1 − A2. And scattering cross-section 2 ∗ ∗ ∗ ∗ |M| = A1A1 + A2A2 − (A1A2 + A2A1).

• e−e+ → µ−µ+ (photon internal line) −iη (−ie)2u¯(p0)γav(q0) ab v¯(q)γbu(p) (p + q)2 − i

• Compton scattering e−γ → e−γ (fermion internal line)

−i(−(p/ + k/) + m) A = (−ie)2u¯(p0, s0)γb∗ γa u(p, s); 1 b (p + k)2 + m2 − i a

0 −i(−(p/ − k/ ) + m) A = (−ie)2u¯(p0, s0)γb∗ γa u(p, s). 2 b (p − k0)2 + m2 − i a

⇒ A = A1 + A2. 2 ∗ ∗ ∗ ∗ ⇒ |M| = A1A1 + A2A2 + (A1A2 + A2A1).

2 • Two scalars to two scalars (scalar internal line)

−i A = (2π)4δ(p + q − p0 − q0)(−iλ)2 ; 1 (p + q)2 + m2 − i

−i A = (2π)4δ(p + q − p0 − q0)(−iλ)2 ; 2 (p − p0)2 + m2 − i −i A = (2π)4δ(p + q − p0 − q0)(−iλ)2 . 3 (p − q0)2 + m2 − i

We use Mandelstam variables

s = (p + q)2, t = −(p − p0)2 u = −(p − q0)2.

Our definitions of A and M have been ambiguous as to whether i(2π)4δ(··· ) is included. I guess A does but M doesn’t. Also we defined them to include the factor of i. Some people don’t. Anyway, here

 −i −i −i  A = A + A + A := (2π)4δ(p + q − p0 − q0)(−iλ)2 + + . 1 2 3 s + m2 − i t + m2 − i u + m2 − i

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