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 gφ ¯γ5 then - gγ5 ) add igγ5 to vertices; - φ ) the propagator is scalar; - ¯ ) 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 ∗ ∗ ∗ ∗ jMj = 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 ∗ ∗ ∗ ∗ ) jMj = 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 3.