# Parton model: impulse approximation vs rough factorization criterion in Drell-Yan process

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Consider Drell-Yan process

$$P + P' \to l + l^{\dagger}, \qquad (1)$$

where $P, P'$ are partons inside colliding nucleons $N, N'$, and $l$ is lepton. The process $(1)$ describes the general process $N+N' \to l+l^{\dagger}+X$ in the case
$$t_{\text{parton}}\gg t_{\text{int}}, \qquad (2)$$

where $t_{\text{parton}}$ is a lifetime of the parton virtual state inside the nucleon, and $t_{\text{int}}$ is characteristic interaction time of the process $(1)$. The assumption $(2)$ is called impulse approximation and historically appeared in Drell and Yan papers on parton model.

In https://arxiv.org/pdf/1409.0051.pdf is written that on the modern QCD language $(2)$ is replaced by the  (roughly speaking) factorization condition

$$Q^{2}\gg \Lambda_{\text{QCD}}^{2}, \qquad (3)$$

see Eq. (33), where $Q^{2}$ is the invariant pair of partons pair. The condition $(3)$ is simply interpreted as perturbativity condition in QCD. However, I don't clearly understand its relation to $(2)$, since the latter should obviously depend on the type of the interaction (as the interaction rate of the process, i.e. the right hand-side of $(2)$, is given by the transition probability per unit time).

Could anyone please explain why people use condition $(3)$ as the criterion of applicability of the Drell-Yan process for describing lepton pair production?

edited Jul 15, 2018

it is a coincidence that references 33 and 34 are cited near formulas 33 and 34, mainly the "Factorization of Hard Processes in QCD"

@igael : regarding my question (relation between the impulse approximation and perturbativity condition) in the reference you cited is written only the following: "...The interactions which produce the distributions of each such parton occur on a scale which is again much longer than the time scale of the annihilation and, in addition, ﬁnal-state interactions between the remaining partons take place too late to aﬀect the annihilation..." But this doesn't provide the desired explanation.

@NAME_XXX: for your precise question, he states in 1.1 "We now assume that non perturbative long-distance effects in the complete theory factorize in the same way as do perturbative long-distance effects. Once this assumption is made, we can interpret our perturbative calculation of Hμν a as a prediction of the theory.". Then he shows it ( 1.1 1.2 1.3 etc ).

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