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High-energy strong interactions: from `hard' to `soft'

M. G. Ryskin, A. D. Martin, V. A. Khoze

TL;DR

This paper develops a unified framework that merges perturbative QCD with Reggeon Field Theory to describe both soft and semi-hard hadronic interactions at high energy. By incorporating BFKL-like diffusion in transverse momentum and absorptive multi-Pomeron corrections within a two-channel eikonal, the model explains the energy-dependent infrared cutoff and the smooth transition from hard to soft dynamics. It successfully describes soft cross sections and makes predictions for gluon PDFs, diffractive PDFs, and rapidity-gap survival probabilities, offering guidance for LHC and ultra-high-energy cosmic ray phenomenology. The approach contrasts with purely pole-based Regge or single-pole models and provides a physically motivated path toward a MC implementable description of high-energy hadron scattering.

Abstract

We discuss the qualitative features of the recent data on multiparticle production observed at the LHC. The tolerable agreement with Monte Carlos based on LO DGLAP evolution indicates that there is no qualitative difference between `hard' and `soft' interactions; and that a perturbative QCD approach may be extended into the soft domain. However, in order to describe the data, these Monte Carlos need an additional infrared cutoff k_min with a value k_min ~ 2-3 GeV which is not small, and which increases with collider energy. Here we explain the physical origin of the large k_min. Using an alternative model which matches the `soft' high-energy hadron interactions smoothly on to perturbative QCD at small x, we demonstrate that this effective cutoff k_min is actually due to the strong absorption of low k_t partons. The model embodies the main features of the BFKL approach, including the diffusion in transverse momenta, lnk_t, and an intercept consistent with resummed next-to-leading log corrections. Moreover, the model uses a two-channel eikonal framework, and includes the contributions from the multi-Pomeron exchange diagrams, both non-enhanced and enhanced. The values of a small number of physically-motivated parameters are chosen to reproduce the available total, elastic and proton dissociation cross section (pre-LHC) data. Predictions are made for the LHC, and the relevance to ultra-high-energy cosmic rays is briefly discussed. The low x inclusive integrated gluon PDF, and the diffractive gluon PDF, are calculated in this framework, using the parameters which describe the high-energy pp and p\bar{p} `soft' data. Comparison with the PDFs obtained from the global parton analyses of deep inelastic and related `hard' scattering data, and from diffractive deep inelastic data looks encouraging.

High-energy strong interactions: from `hard' to `soft'

TL;DR

This paper develops a unified framework that merges perturbative QCD with Reggeon Field Theory to describe both soft and semi-hard hadronic interactions at high energy. By incorporating BFKL-like diffusion in transverse momentum and absorptive multi-Pomeron corrections within a two-channel eikonal, the model explains the energy-dependent infrared cutoff and the smooth transition from hard to soft dynamics. It successfully describes soft cross sections and makes predictions for gluon PDFs, diffractive PDFs, and rapidity-gap survival probabilities, offering guidance for LHC and ultra-high-energy cosmic ray phenomenology. The approach contrasts with purely pole-based Regge or single-pole models and provides a physically motivated path toward a MC implementable description of high-energy hadron scattering.

Abstract

We discuss the qualitative features of the recent data on multiparticle production observed at the LHC. The tolerable agreement with Monte Carlos based on LO DGLAP evolution indicates that there is no qualitative difference between `hard' and `soft' interactions; and that a perturbative QCD approach may be extended into the soft domain. However, in order to describe the data, these Monte Carlos need an additional infrared cutoff k_min with a value k_min ~ 2-3 GeV which is not small, and which increases with collider energy. Here we explain the physical origin of the large k_min. Using an alternative model which matches the `soft' high-energy hadron interactions smoothly on to perturbative QCD at small x, we demonstrate that this effective cutoff k_min is actually due to the strong absorption of low k_t partons. The model embodies the main features of the BFKL approach, including the diffusion in transverse momenta, lnk_t, and an intercept consistent with resummed next-to-leading log corrections. Moreover, the model uses a two-channel eikonal framework, and includes the contributions from the multi-Pomeron exchange diagrams, both non-enhanced and enhanced. The values of a small number of physically-motivated parameters are chosen to reproduce the available total, elastic and proton dissociation cross section (pre-LHC) data. Predictions are made for the LHC, and the relevance to ultra-high-energy cosmic rays is briefly discussed. The low x inclusive integrated gluon PDF, and the diffractive gluon PDF, are calculated in this framework, using the parameters which describe the high-energy pp and p\bar{p} `soft' data. Comparison with the PDFs obtained from the global parton analyses of deep inelastic and related `hard' scattering data, and from diffractive deep inelastic data looks encouraging.

Paper Structure

This paper contains 27 sections, 58 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. LHC data versus Monte Carlo predictions
  3. The 'old' Regge description of 'soft' data
  4. Implications of the LHC data
  5. Diffusion in $\ln k_t$
  6. Low $p_t$ suppression and enhanced multi-Pomeron diagrams
  7. Matching the 'soft' and 'hard' contributions
  8. Smooth transition between the hard and soft regimes
  9. Outline of the procedure to link the hard and soft regimes
  10. The parameters of the model
  11. Simultaneous description of soft and semi-hard data
  12. Choice between hypotheses (a) and (b) for multi-Pomeron couplings, $g^n_m$
  13. The gluon PDFs: $g(x,\mu^2)$ and $g^D(x_P,z,\mu^2)$
  14. Results of the fit to soft data and predictions for semi-hard phenomena
  15. The parameters and their values
  16. ...and 12 more sections

Figures (12)

  • Figure 1: The theoretical formalisms appropriate to the various domains are indicated. As long as $x$ is not too small, we have a well justified theory based on the perturbative QCD and DGLAP evolution. For smaller $x$ we need to resum the terms enhanced by the large value of $\ln 1/x$ inside DGLAP evolution, and to account for the semi-enhanced absorptive corrections generated by the Balitsky-Kovchegov (BK) equation BK. Above the 'BK' curve more complicated multi-Pomeron diagrams enter and we cannot justify the result obtained by summation of 'fan' diagrams only, that is, the results based on the BK-equation. However, here we extend the partonic ladder structure of the Pomeron (generated by BFKL-like evolution in rapidity) to allow for a full set of multi-Pomeron exchange diagrams, and obtain a model which is applicable in the low $k_t$ region bounded by the dashed curve. Moreover, in principle, it is possible to use a more precise evolution kernel, which accounts for both BFKL and DGLAP logarithms, to cover the whole $k_t$ region in our approach.
  • Figure 2: Eikonal and enhanced multi-Pomeron contributions
  • Figure 3: The triple-Pomeron diagram, together with its ladder structure
  • Figure 4: The ladder structure of the bare Pomeron, $F(y,\boldsymbol{k}_t,\boldsymbol{b})$, together with a symbolic sketch of the hadronisation process, cf (\ref{['eq:D']}). This figure shows the symbols for the various transverse momenta used in the text.
  • Figure 5: A typical enhanced multi-Pomeron exchange diagram
  • ...and 7 more figures