The latter will be discussed in the lecture on 03.07

Exercise Sheet 6: Scattering amplitudes in gauge theories

Discussion on Wednesday 10.07, NEW 15 2102, Prof. Dr. Jan Plefka

Schedule for the coming two weeks: Lectures: 03.07 and 11.07, Excercises: 10.07, no lecture on 04.07.

Exercise 6.1 – On-shell loop-momenta for cut triangles and bubbles Consider the scalar triangle and bubble integrals in four dimensions

I₃ =

Z d⁴l (2π)⁴

1 d0d1d2

I₂ =

Z d⁴l (2π)⁴

1 d0d1

with d₀ =l² and d_i = (l+q_i)² (i= 1,2). Show that the maximal cuts of these integrals, i.e. putting all propagators on shell d_a = 0 with a = 0,1,2 respectively a = 0,1, lead to the two independent one-parameter family solutions for the triangle cuts

¯l^µ_±(α) = −¹₂(q₁²v₁^µ+q₂²v₂^µ)± ¹₂ q

−(q₁²v^µ₁ +q₂²v₂^µ)² (cos(α)n^µ₃ + sin(α)n^µ₄) , α ∈R, and to the two independent two-parameter families of solutions for the bubble cuts

¯l±(α1,α2) =−¹₂q₁²v₁^µ± q

−α²₁ −α²₂− ¹₄(q₁²)²v₁²n^µ₂ +α1n^µ₃ +α2n^µ₄ , αi ∈R.

Exercise 6.2 – Box coefficients of A^1-loop₄ (1⁻,2⁺,3⁻,4⁺)

Determine the box coeffiecient of the four gluon amplitude A^1-loop₄ (1⁻,2⁺,3⁻,4⁺) in pu- re Yang-Mills theory using either unitarity (two-particle cuts) or generalized unitarity (maximal cuts). The latter will be discussed in the lecture on 03.07.

Exercise 6.3 – The remaining two box coefficients of A^1-loop₅ (1⁻,2⁻,3⁺,4⁺,5⁺) Complete the discussion of the lecture on 03.07 on generalized unitarity and the 5-gluon example and derive the remaining coefficients d23 and d34 using the same methods.

1