Do $x$ and $Q^2$ associate with particular directions in the infinite momentum frame?

Question

In deep inelastic scattering, you describe a collision using the variables $Q^2 = -q^2$ (probe virtuality) and $x = Q^2/2p\cdot q$ (Bjorken x, parton momentum fraction). Now, I seem to remember reading somewhere that in the infinite momentum frame, one of these variables primarily depends on momentum in the transverse direction and the other primarily depends on momentum in the longitudinal direction, but I can't remember which, and checking my most likely sources hasn't turned up anything. Is this actually the case, or is my memory playing tricks on me? And if it is true, which is which?

Apologies for asking a vague question, but I'm really drawing a blank on what it is I might have read, and asking here seemed like the quickest way to figure it out ;)

Answer 1

David, your memory is right. You want to use the Drell-Yan-West frame with $q^+=0$, using the light-cone coordinates. In those coordinates, $Q^2=-q^2$ is clearly $q_\perp^2$ because the $q^+ q^-$ term vanishes and only depends on the transverse components of $q$.

In the Drell-Yan limit, the fraction $x$ becomes $$x_1 = p_1^- / P_1^-$$ and only depends on the longitudinal components as in equation (26) of

http://arxiv.org/PS_cache/arxiv/pdf/0909/0909.4159v2.pdf

where I send you to refresh some memory - as well as provide you with new keywords and references you may search through. I won't try to make my answer perfect, sorry.