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Jan Plefka Exercise 1.6 – MHV 4-gluon amplitude i) Show that the four gluon color-ordered amplitude A is given by the compact formula i A(1−,2−,3+,4

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Exercise Sheet 3: Scattering amplitudes in gauge theories Discussion on Wednesday 15.05, NEW 15 2102, Prof. Dr. Jan Plefka

Exercise 1.6 – MHV 4-gluon amplitude

i) Show that the four gluon color-ordered amplitude A(1−,2+,3+,4+) is given by the compact formula

i A(1−,2−,3+,4+) = h12i4 h12i h23i h34i h41i by employing the color-ordered Feynman rules.

ii) Check that the helicities of the four legs are correct upon acting with the helicity operator

hi = 12(−λαi ∂

∂λαi + ˜λαi˙ ∂

∂λ˜αi˙) on the above result.

iii) Use the U(1) decoupling theorem (or the results of excercise 1.5) to show that i A(1−,2+,3−,4+) = h13i4

h12i h23i h34i h41i.

Exercise 1.7 – MHV and anti-MHV 4-gluon amplitude

Show that the 4-gluon MHV amplitude ist identical to the 4-gluon MHV amplitude as it should be by parity, i.e. show that

h12i4

h12ih23ih34ih41i = [34]4

[12][23][34][41], with

4

X

i=1

pi = 0

holds.

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