[Paper Review] The gluon Reggeization in perturbative QCD at NLO

V. S. Fadin|ArXiv.org|Nov 9, 2005

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper proves the gluon Reggeization hypothesis in next-to-leading order (NLO) perturbative QCD by demonstrating that an infinite set of bootstrap relations—required for consistency with s-channel unitarity—is satisfied when specific conditions on the gluon trajectory and Reggeon vertices are fulfilled. The key result is the complete verification of all bootstrap conditions, including the previously unproven condition (22), thereby establishing the Reggeized form of QCD amplitudes at NLO.

ABSTRACT

Compatibility of the Reggeized form of QCD multi-particle amplitudes with the s-channel unitarity requires fulfilment of an infinite number of the "bootstrap" relations. On the other hand, it turns out that fulfillment of all these relations ensures the Reggeized form of energy dependent radiative corrections order by order in perturbation theory. It is extremely nontrivial, that all these relations are fulfilled if the Reggeon vertices and trajectory satisfy several bootstrap conditions. The full set of these conditions in the next-to-leading order was derived in the last year and the ultimate condition was shown to be satisfied recently. It means that the Reggeization hypothesis is proved now in the next-to-leading approximation.

Motivation & Objective

• To establish the gluon Reggeization hypothesis in next-to-leading order (NLO) QCD, which remains a critical assumption in high-energy scattering amplitudes.
• To resolve the long-standing challenge of proving that energy-dependent radiative corrections maintain the Reggeized form of amplitudes beyond leading logarithmic approximation.
• To verify the consistency of the Reggeized ansatz with s-channel unitarity by proving that all infinite bootstrap relations are satisfied.
• To complete the proof by confirming the final, previously unverified bootstrap condition (22) for Reggeon-gluon vertices.
• To provide a rigorous foundation for the BFKL approach in NLO by confirming the universality and consistency of the Reggeon vertices and trajectory.

Proposed method

• Derivation of the full set of bootstrap relations from the requirement of s-channel unitarity and analyticity in multi-Regge kinematics (MRK).
• Use of the Reggeized form of QCD amplitudes, factorized into Reggeon propagators, impact factors, and production vertices, to express scattering amplitudes in terms of the gluon trajectory ω(t) and effective vertices.
• Formulation of four key bootstrap conditions: (1) impact factor universality via eigenstates of the kernel K̂, (2) normalization condition (21), (3) and (4) vertex equations (22) for Reggeon-gluon interactions.
• Application of the BFKL kernel and its eigenstates |Rω(q)⟩ to describe the universal Reggeon state with eigenvalue ω(t).
• Use of light-cone kinematics and rapidity factorization to express amplitudes in terms of energy-dependent Regge factors s_i^{ω(q_i)}.
• Verification of all bootstrap conditions using known results from previous works and a recent proof of condition (22), completing the consistency check.

Experimental results

Research questions

• RQ1 Does the Reggeized form of QCD amplitudes remain consistent with s-channel unitarity at next-to-leading order?
• RQ2 Are the infinite set of bootstrap relations required for Reggeization satisfied in NLO QCD?
• RQ3 Can the final, previously unverified bootstrap condition (22) for Reggeon-gluon vertices be proven to hold?
• RQ4 Is the gluon Reggeization hypothesis fully justified in the next-to-leading approximation?
• RQ5 Do the Reggeon vertices and trajectory satisfy all necessary consistency conditions for the BFKL approach to be valid beyond leading logarithmic accuracy?

Key findings

• The gluon Reggeization hypothesis is rigorously proven in next-to-leading order (NLO) QCD through the full satisfaction of all bootstrap conditions.
• All infinite bootstrap relations are reduced to a finite set of four conditions: (19), (20), and (22), which are now fully verified.
• The final, previously unproven condition (22) for Reggeon-gluon vertices has been confirmed, completing the proof.
• The normalization condition (21) relating the trajectory ω(t) to the Reggeon state |Rω(q)⟩ is satisfied, ensuring consistency of the eigenvalue structure.
• The universal Reggeon state |Rω(q)⟩ is confirmed as an eigenstate of the BFKL kernel K̂ with eigenvalue ω(t), validating its role in the BFKL framework.
• The proof establishes that the Reggeized form of amplitudes, with energy dependence governed by s_i^{ω(q_i)}, is consistent with unitarity and analyticity at NLO.