Introduction to Scattering Amplitudes
in Gauge Theories, WiSe 2016/17
Prof. Dr. Livia Ferro
Theresienstr. 37, Zi. 434
Mr. Federico Roccati
[email protected]
7. BCFW Recursion Relations II
Exercise 7.1: Six-point split helicity YM amplitude
Derive the 6-point color-ordered tree-level Yang-Mills amplitude
A6 (1− , 2− , 3− , 4+ , 5+ , 6+ )
using BCFW recursion relations under a [1, 2i-BCFW shift.1 Show explicitly that the BCFW
recursion under this shift does not get a contribution from a 23-channel diagram, i.e. there
will not be a diagram with a (p2 + p3 )2 propagator. Check also the little group scaling of the
resulting expression.
Exercise 7.2: Six-point alternating helicity YM amplitude
Show that the BCFW recursion relations based on the [2, 3i-shift give the following representation of the six-point “alternating helicity” gluon amplitude
A(1+ , 2− , 3+ , 4− , 5+ , 6− ) = {M2 } + {M4 } + {M6 },
where
hi, i + 2i4 [i + 3, i − 1]4
{Mi } = 2
P̃i hi|P̃i |i + 3]hi + 2|P̃i |i − 1]hi, i + 1i]hi + 1, i + 2i[i + 3, i − 2][i − 2, i − 1]
and P̃i = Pi,i+1,i+2 = (pi + pi+1 + pi+2 )2 .
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Hint: Keep in mind that a four-point amplitude can be either understood as MHV or MHV: picking the
right representation might drastically reduce the amount of work needed to simplify your results.
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