i) Using the color ordered Feynman rules show that the the color ordered tree-level amplitude

Exercise Sheet 2: Scattering amplitudes in gauge theories Discussion on Wednesday 02.05, NEW 15 2102, Prof. Dr. Jan Plefka

Exercise 1.3 – Spinor technology i) Prove that [1|p|2i = h2|p|1].

ii) Determine the value of the constant c in the Fierz rearrangement formula [i| γ ^µ |j i [k| γ _µ |li = c [i k] hl ji .

Exercise 1.4 – Tree level qqgg ¯ scattering

i) Using the color ordered Feynman rules show that the the color ordered tree-level amplitude

A ^tree ₄ (1 ⁻ _{q ¯} ,2 ⁺ _q ,3 ⁺ ,4 ⁺ ) = 0

vanishes using the spinor helicity formalism and a suitable choice of reference mo- menta for the gluon polarizations. Here ¯ q is a helicity −1/2 anti-quark state, q a helicity +1/2 quark state and the n ^± states refer to gluons of helicity ±1.

ii) Go on to show by using the color ordered Feynman rules and a suitable choice for the gluon-polarization reference null-vector q ₃ ^{α α ˙} = µ ^α ₃ µ ˜ ^α ₃ ^˙ that the first non-trivial

¯

qqgg scattering amplitude is given by

A ^tree _{¯ qqg}

(1 ⁻ _{q ¯} , 2 ⁺ _q , 3 ⁻ ,4 ⁺ ) = i h13i ³ h23i h12i h23i h34i h41i .

Also convince yourself that the result for this amplitude has the correct helicity assignments.

Exercise 1.5 – Independent four and five gluon amplitudes

Using the general properties for partial amplitudes discussed in class (cyclicity, parity, reflection, photon decpoupling) determine the independent set of partial amplitudes for 4 and 5 gluon scattering at tree-level.

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