SLIDE 7 J . C . C
l i n s , D . E . S
e r / B a c k
a c k j e t s
407 shown in fig. 4.1. A typical diagram contributing to fig. 4.1 is the one shown in fig. 5.7, where now we imagine that the gluon with momentum q" is coupled to the initial quark with a square vertex of fig. 4.2. Each such graph can be decomposed according to the garden formula (5.18). It is useful to rewrite eq. (5.18) by grouping together all gardens that have the same largest tulip. Thus G = GR q-~, S(T) { 1 +
T
Y
i n e q u i v a l e n t g a r d e n s
Tn=T;N>I
(-1) N iS(T1)''" S(Tn-1)]G.
(6.1) Consider now a particular term in the sum over largest tulips T. The tulip T can be considered to be a Feynman graph in its own right, and we can define a subtracted version of T by TR = T+
S ( T n
) T .
(6.2)
i n e q u i v a l e n t g a r d e n s
T,,=T;N>I
Let us denote by t~ the graph that is left over from the complete graph G if its subgraph T and all of the gluons leaving T are erased. The T term of eq. (6.1) is a particular insertion of TR into (~ in the soft approximation. We now sum over all "soft" insertions of Ta into G. From the discussion of subsect. 5.3 it is evident /"
I
k
m
J
- Fig. 6.1. The fate of fig. 5.7. After making the soft approximation on the gluons leaving each tulip in
- fig. 5.7 and summing over attachments of the four gluons leaving the larger tulip, one obtains this
diagram, as well as several others.
Approach 2: Pinch surfaces
June 19, 2014 Matthew Schwartz
.L
/ Back-to-back jets
401
- Fig. 5.6. Eikonal rotation matrix next to a collinear three-gluon vertex. The "'soft" q~' and q~' are
neglected in the expression for the vertex. (The 8 here signifies that the octet SU(3) representation matrices appear on the eikonal line.) Thus we can adopt the convention that k"+q~+q~ is replaced by k" at the three-gluon vertex in the "soft" approximation. In coordinate space with the vertex at x ", this amounts to saying that the O/Ox" at the vertex does not differentiate the eikonal rotation matrix U(x ; A) that appears at the end of the collinear gluon line. 5.5. A BOTANICAL CONSTRUCTION Consider a cut Feynman diagram G contributing to (OldJ(k')a+(P)a(P)×
T{A(q)d~(k)}[O), as illustrated in fig. 5.7. When the gluon A(q) is coupled to the
21 I 2
k I ~ ~ ~ k ~ I x ./ q
I
- Fig. 5.7. A two-tulip garden.
404 J.C. Collins, D.E. Soper / Back-to-back jets
S
\ A_.~" ,¢~.;_B
A A
d
- Fig. 5.10. Adding tulip B to the garden produces a cancellation.
the finger of TA enters S. There the boundary of TB follows the boundary of S. Of course it is supposed that the finger of which we have been speaking has no subfingers interior to it; if it does, we apply the argument to the smallest subfinger. If the garden contains several tulip fingers extending from S into J, there is an independent cancellation for each finger. There are also (independent) cancellations when the exterior of a tulip has a finger that extends from J into S. The cancellations for a three tulip garden of reasonably complicated geometry are illustrated in fig. 5.11. This argument shows that GR-- 0 when soft and collinear integration regions are taken into account. We now extend this argument to cover ultraviolet momenta, proceeding in two stages. In the first stage, imagine that ultraviolet loop momenta in vertex and self-energy subgraphs are allowed provided that no soft gluon from the upper soft subgraph S in fig. 4.3a ends on an ultraviolet line in J. Upon rereading the original argument for GR--0, one discovers that no change in the argument is required to cover this case.
I
2 :5
- Fig. 5.11. Cancellations for a complicated garden. The shaded area is the soft subgraph. The solid lines
are tulip boundaries. Addition of tulips with new boundary portions along one or more of the dashed
- r dotted lines produces cancellations.
Collins & Soper, 1981
Collins, Soper, Sterman: pinch surfaces factorize
