Previous Up Next
5 Color
We will implement a variation of numeric color diagonalization [9].
Here's a sketch of the algorithm:
  1. expand the DAG D to a list L of trees
  2. numerically calculate the matrix C of color factors for the squared matrix element
  3. diagonalize C
  4. tag the wave functions in D by the list of their appearances in L
  5. for each wavefunction in D, calculate the coefficients of the eigenvectors corresponding to non-zero eigenvalues of C
  6. (like for Fermi statistics) keep only the factors that are not already in the daughter wave functions
This multiplies the complexity of the colorless amplitude by the number of eigenvectors with non-zero eigenvalues of C. Asymptotically, this will beat [8], but it is not obvious where the break even point is for many eigenvectors. Therefore more precise estimates will be useful ...
The same approach might be workable for spin and flavor sums. The gains are not obvious (they depend on the number of eigenamplitudes), but they could be huge.
For the sums over Feynman diagrams, color eigenamplitudes and wave functions, we introduce the following conventions:
i Î { 1, 2, ..., NFD}                     (7)
a Î { 1, 2, ..., Nev, ..., NFD}                     (8)
n Î { 1, 2, ..., NWF}                     (9)

A wavefunction is given by a sum over all Feynman diagrams
Wn =
 
å
i
wn,i = á0|f|nñ     (10)
where
wn,i = á0|f|nñdiagram #i     (11)
corresponds to the contribution of diagram i to the wavefunction Wn.
Aa =
 
å
i
cai ai     (12)

Wn,a =
 
å
i
cai wn,i     (13)
and
wn,i =
 
å
a
(c-1)ia Wn,a     (14)

Fusion coefficients
Fa,bc
=
 
å
i
cai(c-1)ib(c-1)ic
                    (15)
Fa,bcd
=
 
å
i
cai(c-1)ib(c-1)ic(c-1)id
                    (16)
can be calculated numerically, since cai can be extended to a non-singular square matrix, even if we need only small part of it.

Previous Up Next