211 Appendix E: Standard pQCD Results The inverse transformation is then The Minkowski dot product in lightcone coordinates is: The length of a vector using lightcone coordinates is then: p 0 = 1 2 ( p + + p $^{−}$) (E.5) p z = 1 2 ( p + − p $^{−}$) (E.6) p = p . (E.7) p · k = p $^{0}$k0 − p $^{z}$kz − p · k = 1 2 ( p $^{+}$k − + p $^{−}$k $^{+}$) − p · k . (E.8) p · p = p $^{+}$p − − p · p (E.9) . Note that the eikonal approximation requires collinear emission, which in turn requires k + ≫ k − (for emitted boson with 4-momentum k ). If this were otherwise, the k z emission would be in the opposite direction of p $^{z}$: p z = 1 $_{2}$( p + − p $^{−}$) ≃ 1 $_{2}$p + = 1 $_{2}$(1 − x ) E $^{+}$; on the other hand k z = 1 $_{2}$( k + − k $^{−}$) must be approximately 1 $_{2}$k + or else the sign is wrong. E.1.2 Eikonality in Light Cone Coordinates