Exercise 5.1 – Tricks for delta-function maniplulations Prove the following relations for the helicity spinors λ and µ

Exercise Sheet 5: Scattering amplitudes in gauge theories Discussion on Wednesday 12.06, NEW 15 2102, Prof. Dr. Jan Plefka

Exercise 5.1 – Tricks for delta-function maniplulations Prove the following relations for the helicity spinors λ and µ

δ ⁽²⁾ (λ ^α a + µ ^α b) =







δ(a) δ(b)

|hλµi| for a,b Grassmann even δ(a) δ(b) hλµi for a,b Grassmann odd Use this to show that

δ ⁽⁸⁾

3

X

i=1

λ ˜ ^α _i ^˙ η ¯ _{i A}

!

= [12] ⁴ δ ⁽⁴⁾

¯

η _1,A − [23]

[12] η ¯ _{3 A}

δ ⁽⁴⁾

¯

η _2,A − [31]

[12] η ¯ _{3 A}

which we used in the derivation of the A ^MHV 3 amplitude in class.

Exercise 5.2 – The MHV ₄ superamplitude and component field results

Use the super-BCFW recursion to prove the MHV super-amplitude formula at n-points A ^MHV n = δ ⁴ (p) δ ⁸ (q)

h12ih23ih34i . . . hn1i .

Use this to establish the four point gluino-quark component field amplitudes A 4 (1 ⁻ _{¯ ˜ g} , 2 ⁺ _{g ˜} , 3 ⁻ ,4 ⁺ ) = δ ⁽⁴⁾ (p) h31i ³ h23i

h12i . . . hn1i , A 4 (1 ⁻ _{g ¯ ˜} , 2 ⁺ _{g ˜} , 3 ⁻ _{¯ ˜ g} , 4 ⁺ _{˜ g} ) = −δ ⁽⁴⁾ (p) h31i ³ h24i

h12i . . . hn1i .

Exercise 5.3 – The NMHV five-point superamplitude

Derive the five-point next-to-MHV superamplitude A 5 from the BCFW super recursion introduced in class

A ^NMHV n =

Z d ⁴ P P ²

Z

d ⁴ η _{P ˆ} A ^MHV 3 (z _P ) A ^NMHV n−1 (z _P ) +

n−1

X

i=4

Z d ⁴ P _i P _i ²

Z

d ⁴ η _P

A ^MHV i (z _P

) A ^MHV n−i+2 (z _P

) . That is show that

A ^NMHV 5 = δ ⁴ (p) δ ⁸ (q)

h12ih23ih34ih45ih51i R _5;2,4 with the “R-invariant” defined in eq. (2.129) in the script.

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