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Consider scattering process at partonic level occuring at low factorization scale $\mu^{2}\lesssim 1\text{ GeV}^{2}$ (for example, with the "hard" process $2\to 1$ with low mass produced particle, with the factorization scale chosen to be the squared mass of this particle). Using parton distribution functions generated by package LHAPDF, I obtain suppressed (in comparison with production at larger factorization scale) but non-zero result for the cross-section.

My question is whether the parton model (and therefore parton distributuon function) is well-defined at $\mu^{2}\lesssim 1\text{ GeV}^{2}$. Could you help me please?

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The interacting probe with momentum transfer $Q$ has resolving capabilities $\sim 1/Q$ of the hadronic target so, in particular, if $Q$ is of the order of $1 \,\text{GeV} \sim m_H$ then the resolution is at most of the order of the characteristic size of the hadron. We therefore do not really see the partonic structure of the proton, perhaps at best its valence content. At such energy resolutions one cannot justify the applicability of the parton model.

Typically the parton model is to be understood within a (collinear) factorisation in which hard scattering matrix elements $C_i$ are convoluted with the PDFs $F_i$ to ascertain observables such as structure functions, cross sections etc. One usually writes e.g $$\mathcal M \sim \sum_i C_i \otimes F_i + \mathcal O \left(\frac{\Lambda_{\text{QCD}}^2}{Q^2}\right)$$ with the higher twist terms neglected because of their suppression due to the relatively large scale $Q^2$. In your case, if $Q \sim \Lambda_{\text{QCD}}$ then these terms are $\mathcal O(1)$ and the factorisation into this nice sector split is spoiled.

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  • $\begingroup$ Thank you! Could you please recommend some overview article or book in which your answer is discussed in details? $\endgroup$ – Name YYY Jul 2 '18 at 17:53
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    $\begingroup$ No problem. Actually I read mostly papers on the subject because there seems to be generally very little treatment on factorisation in books, see e.g end of Schwartz or Ellis, Webber and Stirling's 'QCD and Collider Physics' colloquially named 'The Pink Book' :) $\endgroup$ – CAF Jul 2 '18 at 20:01
  • $\begingroup$ Thank you again! Reading Schwartz, I notice the statement "...the momentum sloshes around among proton constituents at time scales $\sim \Lambda_{\text{QCD}}^{-1} \sim m_{p}^{-1}$...", where by "momentum" is meaned the distribution of the proton's momentum among partons. Could you please comment this statement? I don't clearly understand why the time scale is of order of $\Lambda_{\text{QCD}}^{-1}$. $\endgroup$ – Name YYY Jul 2 '18 at 21:37
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    $\begingroup$ I would say this is just the time-energy form of the uncertainty principle. The characteristic time of the momentum ‘sloshes’ is proportional to the characteristic size of the proton which in turn is reciprocal to its mass which is then of the order of $\Lambda_{\text{QCD}}$. $\endgroup$ – CAF Jul 2 '18 at 22:10
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    $\begingroup$ Ah do you mean like in Drell-Yan? Actually I confess I never thought about that but see here arxiv.org/pdf/1409.0051.pdf .. under something called the ‘impulse approximation’ the authors show the validity of the Parton model to s channel annihilation over t channel scattering. Usually such calculations are done in the infinite momentum frame so perhaps this justifies its use if similar time dilation arguments can be used etc $\endgroup$ – CAF Jul 3 '18 at 8:19
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I'm a little confused by what you mean by a $2\rightarrow1$ process, because one cannot have a $2\rightarrow1$ process for which all particles are on-shell (the usual assumption when doing perturbative QCD calculations). But, assuming the formulae you've used don't necessarily break down at any particular scale (which is common), the parton model doesn't make sense when the scales are $\sim\Lambda_{QCD}$. In particular, you should expect large corrections from higher orders in $\alpha_s((\mu/\Lambda_{QCD})^2)$.

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  • $\begingroup$ Examples of $2\to 1$ hard process is the process $GG \to h$ where $h$ is hypothetical Higgs-like light scalar, or the process $q\bar{q} \to A'$ where $A'$ is hypothetical dark photon. Conservation laws simply require the invariant mass of partons pair be equal the squared mass of produced particle. $\endgroup$ – Name YYY Jul 1 '18 at 16:49
  • $\begingroup$ Thank you for the answer, it sounds reasonable. What is $\Lambda_{\text{QCD}}$ in your answer? Is it close to $1\text{ GeV}$? I don't have a clear definition of $\Lambda_{\text{QCD}}$: from one side, it is the scale defined by $\alpha_{s}(\Lambda_{\text{QCD}}) = 1$, while from the other side, it is defined as $\langle \bar{q}_{i}q_{j}\rangle \simeq \Lambda_{\text{QCD}}^{3} \delta_{ij}$? And what is the numerical value of this coupling? $\endgroup$ – Name YYY Jul 1 '18 at 16:53
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    $\begingroup$ Yes, can have a $2\rightarrow1$ process to an unstable particle that decays to two or more on-shell particles, that's true. I think of $\Lambda_{QCD}\sim160$ MeV from $\alpha_s(\Lambda_{QCD})\sim1$. But the point is that for scales $\mu\lesssim$ few GeV, higher order corrections will be at best large and at worst your expansion no longer makes any sense. $\endgroup$ – WAH Jul 1 '18 at 18:30

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