On-Shell Methods in Field Theory. David A. Kosower International School of Theoretical Physics, Parma, September 10-15, 2006 Lecture III. Factorization in Gauge Theory. Collinear limits with splitting amplitudes. Twistors and CSW. On-Shell Recursion Relation. =.
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On-Shell Methods in Field Theory
David A. Kosower
International School of Theoretical Physics, Parma, September 10-15, 2006
Lecture III
Collinear limits with splitting amplitudes
=
Let’s compute it with complex momenta chosen so that
that is,
but
compute
Choose common reference momentum q
so we have to compute
Not manifestly gauge invariant
but gauge invariant nonetheless,
and exactly the n=3 case of the general Parke–Taylor formula!
Pick a shift, giving one diagram
Three-point vertices with z-dependent momentum flow ~ z
Four-point vertices with z-dependent momentum flow ~ 1
Propagators with z-dependent momentum flow ~ 1/z
(one more vertex than propagators & two εs)
Compute the (1−,j−) amplitude: choose shift
Other diagrams vanish because
or
Risager, th/0508206
Consider NMHV amplitude: 3 negative helicities m1, m2, m3, any number of positive helicities
Choose shift
Momenta are still on shell, and
because of the Schouten identity
Bjerrum-Bohr, Dunbar, Ita, Perkins & Risager, th/0509016
unphysical singularity— cancels between terms
physical singularities
Grisaru, Pendleton & van Nieuwenhuizen (1977)