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Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit

Vittorio Del Duca, Alberto Frizzo, Fabio Maltoni

TL;DR

The paper develops a comprehensive framework to factorize tree-level QCD amplitudes in the high-energy and collinear limits, systematically extracting universal, process-independent building blocks for NNLO and NNNLO jet production. It constructs forward clusters of three and four partons, delivering the tree parts of NNLO and NNNLO impact factors and deriving the NNLO Lipatov vertex, all while validating against known splitting functions in triple and quadruple collinear limits. A novel color ladder decomposition for purely gluonic amplitudes clarifies color flows and connects amplitudes directly to BFKL-building blocks, enabling a modular approach to multiparton dynamics. These results substantively advance NNLL BFKL resummation prospects and higher-loop evolution kernels, and suggest scalable amplitude decompositions beyond Parke–Taylor-type bases.

Abstract

In the high-energy limit, we compute the gauge-invariant three-parton forward clusters, which in the BFKL theory constitute the tree parts of the NNLO impact factors. In the triple collinear limit, we obtain the polarized double-splitting functions. For the unpolarized and the spin-correlated double-splitting functions, our results agree with the ones obtained by Campbell-Glover and Catani-Grazzini, respectively. In addition, we compute the four-gluon forward cluster, which in the BFKL theory forms the tree part of the NNNLO gluonic impact factor. In the quadruple collinear limit we obtain the unpolarized triple-splitting functions, while in the limit of a three-parton central cluster we derive the Lipatov vertex for the production of three gluons, relevant for the calculation of a BFKL ladder at NNLL accuracy. Finally, motivated by the reorganization of the color in the high-energy limit, we introduce a color decomposition of the purely gluonic tree amplitudes in terms of the linearly independent subamplitudes only.

Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit

TL;DR

The paper develops a comprehensive framework to factorize tree-level QCD amplitudes in the high-energy and collinear limits, systematically extracting universal, process-independent building blocks for NNLO and NNNLO jet production. It constructs forward clusters of three and four partons, delivering the tree parts of NNLO and NNNLO impact factors and deriving the NNLO Lipatov vertex, all while validating against known splitting functions in triple and quadruple collinear limits. A novel color ladder decomposition for purely gluonic amplitudes clarifies color flows and connects amplitudes directly to BFKL-building blocks, enabling a modular approach to multiparton dynamics. These results substantively advance NNLL BFKL resummation prospects and higher-loop evolution kernels, and suggest scalable amplitude decompositions beyond Parke–Taylor-type bases.

Abstract

In the high-energy limit, we compute the gauge-invariant three-parton forward clusters, which in the BFKL theory constitute the tree parts of the NNLO impact factors. In the triple collinear limit, we obtain the polarized double-splitting functions. For the unpolarized and the spin-correlated double-splitting functions, our results agree with the ones obtained by Campbell-Glover and Catani-Grazzini, respectively. In addition, we compute the four-gluon forward cluster, which in the BFKL theory forms the tree part of the NNNLO gluonic impact factor. In the quadruple collinear limit we obtain the unpolarized triple-splitting functions, while in the limit of a three-parton central cluster we derive the Lipatov vertex for the production of three gluons, relevant for the calculation of a BFKL ladder at NNLL accuracy. Finally, motivated by the reorganization of the color in the high-energy limit, we introduce a color decomposition of the purely gluonic tree amplitudes in terms of the linearly independent subamplitudes only.

Paper Structure

This paper contains 29 sections, 173 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Tree Amplitudes
  3. Gluon amplitudes
  4. Quark-gluon amplitudes
  5. The Leading Impact Factors
  6. The Next-to-leading Impact Factors
  7. The NLO impact factor $g\, g^* \rightarrow g\, g$
  8. The NLO impact factor $g\, g^* \rightarrow \bar{q} q$
  9. The NLO impact factor $q\, g^* \rightarrow q\, g$
  10. NLO impact factors in the collinear limit
  11. The Next-to-next-to-leading Impact Factors
  12. The NNLO impact factor $g\, g^* \rightarrow g\, g\, g$
  13. The NNLO impact factor $g\, g^* \rightarrow g\, g\, g$
  14. The NNLO impact factor $g\, g^* \rightarrow g \,\bar{q} \, q$
  15. The NNLO impact factor $q\, g^* \rightarrow q\, g\, g$
  16. ...and 14 more sections

Figures (8)

  • Figure 1: $(a)$ Amplitude for $g\, g \to g\, g$ scattering and $(b),\,(c)$ for $q\, g \to q\, g$ scattering. We label the external lines with momentum, color and helicity, and the internal lines with momentum and color.
  • Figure 2: Amplitudes for the production of three partons, with partons $k_1$ and $k_2$ in the forward-rapidity region of parton $p_a$.
  • Figure 3: Amplitudes for the production of four partons, with partons $k_1$, $k_2$ and $k_3$ in the forward-rapidity region of parton $p_a$.
  • Figure 4: Limits of the amplitude for the production of three gluons in the forward-rapidity region of gluon $p_a$, for $y_1 \gg y_2 \simeq y_3$ (a) and $y_1 \simeq y_2 \gg y_3$ (b).
  • Figure 5: Same as Fig. \ref{['fig:ggglim']} for the production of a quark-antiquark pair and a gluon in the forward-rapidity region of gluon $p_a$.
  • ...and 3 more figures