Results

7	Results

7.1

Examples

process Diagrams O'Mega

# vertices #prop. vertices

e⁺e^-® e⁺n_e^- du^- 20 80 14 45

e⁺e^-® e⁺n_e^- du^- g 146 730 36 157

e⁺e^-® e⁺n_e^- du^- gg 1256 7536 80 462

e⁺e^-® e⁺n_e^- du^- ggg 12420 86940 168 1343

e⁺e^-® e⁺n_e^- du^- gggg 138816 1110528 344 3933

Table 1: Radiative corrections to four fermion production: comparison of the computational complexity of scattering amplitudes obtained from Feynman diagrams and from O'Mega. (The counts correspond to the full Standard Model---sans light fermion Yukawa couplings---in unitarity gauge with quartic couplings emulated by cubic couplings of non-propagating auxiliary fields.)

process Diagrams O'Mega

# vertices #prop. vertices

e⁺e^-® e⁺n_e^- du^- bb^- 472 2832 49 232

e⁺e^-® e⁺n_e^- du^- bb^- g 4956 34692 108 722

e⁺e^-® e⁺n_e^- du^- bb^- gg 58340 466720 226 2212

Table 2: Radiative corrections to six fermion production: comparison of the computational complexity of scattering amplitudes obtained from Feynman diagrams and from O'Mega. (The counts correspond to the full Standard Model---sans light fermion Yukawa couplings---in unitarity gauge with quartic couplings emulated by cubic couplings of non-propagating auxiliary fields.)

Tables 1 and 2 show the reduction in computational complexity for some important processes at a e⁺e^--linear collider including radiative corrections. Using the asymmetric keystones can reduce the number of vertices by some 10 to 20 percent relativ to the quoted numbers for symmetric keystones.

7.2

Comparisons

HELAC's [5] diagnostics report more vertices than O'Mega for identical amplitudes. This ranges from comparable numbers for Standard Model processes with many different flavors to an increase by 50 percent for processes with many identical flavors. Empirically, O'Mega's straight line code is twice as fast as HELAC's DO-loops for identical optimizing Fortran95 compilers (not counting HELAC's initialization phase). Together this results in an improved performance by a factor of two to three.

The numerical efficiency of O'Mega's Fortran95 runtime library is empirically identical to HELAS [7]. Therefore, O'Mega's performance can directly be compared to MADGRAPH's [8] by comparing the number of vertices. For 2®5-processes in the Standard Model, O'Mega's advantage in performance is about a factor of two and grows from there.

The results have been compared with MADGRAPH [8] for many Standard Model processes and numerical agreement at the level of 10^-11 has been found with double precision floating point arithmetic.

7.3

Applications

O'Mega generated amplitudes are used in the omnipurpose event generator generator WHIZARD [15]. The first complete experimental study of vector boson scattering in six fermion production for linear collider physics [16] has been facilitated by O'Mega and WHIZARD.

process	Diagrams		O'Mega
	#	vertices	#prop.	vertices
e⁺e^-® e⁺n_e^- du^-	20	80	14	45
e⁺e^-® e⁺n_e^- du^- g	146	730	36	157
e⁺e^-® e⁺n_e^- du^- gg	1256	7536	80	462
e⁺e^-® e⁺n_e^- du^- ggg	12420	86940	168	1343
e⁺e^-® e⁺n_e^- du^- gggg	138816	1110528	344	3933