Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

5.2 3-Jet Production• quarks are electrically chargedk, µ, app ′ = − ieQ q γ µ (and couple in the standard model (cf. (176) and (178)) to the Z 0 and W ± as well.Problem 20. Draw all Feynman diagrams without closed loops, that follow from the Feynmanrules of QED und QCD for the electroproction of 3 jetse + + e − → q + ¯q + g(at higher energies, there are additional contributions from Z 0 bosons).Write down the corresponding algebraic expression for the T-matrix.Problem 21. Verify the Ward identity for the gluonT 1∣∣ɛ∗ µ (k)=k µ+ T 2∣∣ɛ∗ µ (k)=k µ= 0 , (152)i. e. the fact that no gluons with unphysical polarization ɛ ∗ µ(k) = k µ are produced. Hint: usethe Dirac equation for the external quarks, e. g.1−/p 1 − /k − m + iɛ (/k + /p 1 + m) v(p 1 ) = −v(p 1 ) (für k 0) (153)• down the road, some additional shorthands will be usefulT n = e 2 Qg [¯v(p 2 )γ µ u(p 1 )] 1 s Jµ n(q 1 , q 2 , k, ɛ) (154)with the current matrix elementJ µ (q 1 , q 2 , k, ɛ) = J µ 1 (q 1, q 2 , k, ɛ) + J µ 2 (q 1, q 2 , k, ɛ)J µ 1 (q 1, q 2 , k, ɛ) = ū(q 1)T a /ɛ ∗ (k)(/q 1 + /k + m)γ µ v(q 2 )(q 1 + k) 2 − m 2 + iɛJ µ 2 (q 1, q 2 , k, ɛ) = ū(q 1)γ µ (−/q 2 − /k + m)T a /ɛ ∗ (k)v(q 2 )(q 2 + k) 2 − m 2 + iɛ(155a)(155b)(155c)• this allows to write∑|T 1 + T 2 | 2 = e4 g 2 Q 2spins, pol.s 2 L µν (p 1 , p 2 , 0)H µν (q 1 , q 2 , k) (156)32