What are the differences between the Jetphox, Pythia and Herwig event generators? I know Jetphox is a parton-level event NLO generator program. But I want to know more about other generator programs such as Pythia and Herwig. What are the differences?
I am undergraduate student so my knowledge is limited.
 A: Old question, but very briefly the difference is that JETPHOX and other "matrix element" (ME) partonic event generators simulate the hard process only, resulting in a partonic event with only a few final-state particles, while Herwig and Pythia turn such low-multiplicity systems into "realistic" events.
And in more detail...
To generate such high particle multiplicities, which would be intractable as analytic QCD calculations, these "shower+hadronisation generators" (SHGs) use a factorised approximation called "parton showering" (PS) to iterate QCD evolution of the matrix element final-state legs, until the low-energy limit where non-perturbative physics becomes dominant. At this point, SHGs apply a phenomenological model of the non-perturbative hadronization process. They also typically simulate the effects of secondary partonic interactions and their interleaving with the hard-scattering evolution. Finally they simulate hadron (and tau) decays.
The two approaches are complementary in modern particle physics. Many effects require going beyond the lowest orders of matrix element calculation, at which point dedicated ME generators tend to be needed. But experimenters and many phenomenologists also need to have more realistic pictures including QCD effects beyond fixed order -- in particular, large numbers of parton emissions as simulated by the shower are important in some regions of phase space. The business of how to smoothly join together higher-order ME event generation and parton showers has dominated MC generator development for at least the last decade.
The downside of parton showers is that they are approximations, which ignore quantum interference between amplitudes. It's reasonable to expect that after a large number of branchings the event system "decoheres" and this factorisation is safe, but for LHC physics the lowest order calculations are in most cases insufficient. Increased sophistication of the matrix element also helps to reduce calculation uncertainties, which are intrinsically larger for parton showers than for complete QFT calculations... as long as the merging process is well-defined!
I should say that historically (and still) Herwig and Pythia were leading-order generators, while those like JETPHOX and MCFM operate at next-to-leading order and higher. But these days the merging of ME and PS (and of different MEs) is their dominant role and many of the built-in LO processes in SHGs are just for simple preliminary studies and tests. Sherpa is the other major SHG, but is a rather different beast since it also includes significant elements of higher-order ME generation in the same box.
