Walking Sudakov: From Cusp to Octagon

arXiv:2605.16034 · hep-th, hep-ph · Submitted 2026-05-15 · Read on arXiv

Listen

Transcript

Introduction to the show: ident: High Energy Physics: Theory. The day in high-energy theory: gravity, strings, fields and the questions left open.

Kai: We're looking at the paper: "Walking Sudakov: From Cusp to Octagon".

Mira: Here's the gist of the paper. The paper introduces a novel high-energy limit in both the Sudakov form factor and the four-point scattering amplitude on N=4 SYM by identifying a "walking anomalous dimension" that smoothly interpolates between the cusp and octagon anomalous dimensions The main result of the paper is the identification of a new high-energy limit in the twoparticle Sudakov form factor and in the four-point scattering amplitude on the Coulomb branch of N = 4 SYM This high-energy limit generalizes the usual double-logarithmic behavior of the Sudakov form factor and of the scattering amplitude and is controlled by a new anomalous dimension that smoothly interpolates between the cusp and octagon anomalous dimensions, which they call it the walking anomalous dimension The Setup for Interpolation The study is conducted on planar N = 4 SYM on the Coulomb branch by turning on vacuum expectation values for six adjoint scalars This setup moves the theory away from the origin of moduli space and Higgses the gauge symmetry to a subgroup that commutes with the scalar background The external states in this analysis correspond to massive BPS states, with masses proportional to the distances mi⃗ni − mj⃗nj between corresponding points in R 6 The Walking Limit Identification The key finding is the identification of a new high-energy limit where both observables exhibit double-logarithmic behavior governed by a walking anomalous dimension This walking anomalous dimension interpolates between the cusp anomalous dimension of the on-shell regime and the octagon anomalous dimension of the off-shell regime as mass scales are varied For example, as the walking parameter η → 0, it reduces to the cusp-coefficient Γcusp(g)/4, while for η → 1, it reduces to Γoct(g)/2 One-Loop Analysis of the Form Factor The one-loop analysis of the Sudakov form factor reveals a walking behavior in the limit where both internal and external masses are present The one-loop walking anomalous dimension is given by Γ(1)walk(η) = g 2 0 for η ≤ -1, (1 + η) squared for − 1 ≤ η ≤ 0, (1 + 2η − η 2) for 0 ≤ η ≤ 1, and 2 for η ≥ 1 This function is a C 1 function of the walking parameter η Two-Loop Prediction and All-Loop Structure The two-loop computation, utilizing the Method of Regions, yields an all-loop prediction for the walking anomalous dimension in both observables The expression for the walking form factor is given by Γwalk(η, g) =  0, η ≤ -1, (g squared − 2ζ 2g 4)(1 + η) squared + O(g 6) for − 1 ≤ η ≤ 0, g 2(1 + 2η − η 2) − g 4ζ 2/2 + 5η/5η squared, for 0 ≤ η ≤ 1, and Γoct(g) squared, η ≥ 1 Four-Point Amplitude Generalization The walking phenomenon observed in the form factor also appears in the four-point scattering amplitude on the Coulomb branch The high-energy limit of the amplitude is characterized by two exponents (ηs, ηt), and it is predicted that logM = −Γwalk(ηs, ηt, g) log ωs log ωt + O(log ωs, log ωt) The all-loop prediction for the four-point amplitude is given by a complex piecewise function in terms of ηs and ηt The results depend on four functions of the ’t Hooft coupling: Γcusp, Γoct, and two new functions γ, γ˜, which are not known at present The soft-collinear effective theory is valid over a very wide region of mass/virtuality parameter space as determined by the y-parameter This gives hope for future robust QCD applications relying on the factorization of soft-collinear modes from the hard remainder The presence of the ultrasoft region threatens the potential applicability of perturbative techniques in QCD due to the coupling constant growth in the infrared Future Directions The paper suggests several natural directions for further study, including extending the form-factor analysis to three loops and performing a strong-coupling analysis along the lines of

93, 99: It is also suggested that one should extend the analysis to timelike virtualities, p 2i > 0, where anomalous thresholds are expected to play an important role The study of high-energy hadron-hadron collisions with production of heavy vector bosons and/or Higgs at small transverse momentum is also suggested Appendix Analysis In the appendix, a detailed analysis of the different regions contributing to the double-logarithmic coefficient of the one-loop amplitude M(1)(s, t) is presented for s = t This analysis separates contributions based on kinematic scales x ∼ ω, x ∼ ω-1, x ∼ y-1, and x ∼ y The result for the walking anomalous dimension in the s = t case is given by Γ(1)walk(η) = 2g 2 0 for η ≤ -1, 2(1 + η) squared for − 1 ≤ η ≤ 0, (1 + 2η − η 2) for 0 ≤ η ≤ 1, and Γoct(g) squared, eta ≥ 1 Final All-Loop Prediction The all-loop prediction for the walking anomalous dimension appearing in the four-point amplitude for s ≠ t is given by a piecewise function in terms of ηs and ηt This expression reduces to (4.

Paper summary: Kai: So, wrapping up this discussion on "Walking Sudakov: From Cusp to Octagon," what's the big picture here? We identified a novel high-energy limit where both the form factor and the four-point amplitude show double-logarithmic behavior governed by this walking anomalous dimension.

Mira: The paper is about showing how you can smoothly interpolate between the cusp and octagon anomalous dimensions as you change mass scales. This gives us a unified description of how infrared sensitivity behaves in these gauge theories.

Lev: It’s essentially providing an all-loop prediction for this walking anomalous dimension in both the form factor and the scattering amplitude on that Coulomb branch, which is what we need to know if we can run this on real hardware.

Kai: The implication is that we have a more robust way to analyze factorization theorems for quark and gluon scattering amplitudes by encoding their infrared behavior through this new walking structure.

Mira: It opens the door for future work, like extending the form factor analysis to three loops or looking at timelike virtualities where anomalous thresholds play an important role.

Lev: And they suggest studying high-energy hadron-hadron collisions with heavy vector bosons and Higgs at small transverse momentum, which is where those ultrasoft effects become really relevant.

Conclusion: Kai: So, this paper "Walking Sudakov: From Cusp to Octagon" is about finding a new high-energy limit in both the Sudakov form factor and the four-point scattering amplitude on N=four SYM on the Coulomb branch of N=four SYM.

Mira: It identifies this behavior through a walking anomalous dimension that smoothly connects the cusp and octagon anomalous dimensions as mass scales vary.

Lev: For us, it means they found a way to describe double-logarithmic behavior in these observables using this walking parameter that interpolates between the known limits.

Kai: They used vacuum expectation values for six adjoint scalars to set up their study on the Coulomb branch of N=four SYM.

Mira: The key result is that this high-energy limit generalizes the usual double-logarithmic behavior of the form factor and scattering amplitude, controlled by this new walking anomalous dimension.

Lev: It’s a new structure for those observables, controlled by something that smoothly transitions between two different known limits depending on how you vary the mass.

Kai: So, what does this mean for the physics we're looking at right now?

Mira: It gives us a way to understand how these amplitudes behave in a high-energy limit where both internal and external masses are present simultaneously.

Lev: For running on real hardware, this means we have an all-loop prediction for the walking anomalous dimension in both observables.

Kai: The paper depends on some functions like Γcusp and Γoct, which aren't fully known yet.

Mira: They also introduce two new functions called gamma and gamma-tilde that are currently unknown.

Lev: This is a bit of a dependency issue for us; we need those details to actually implement this on our error-correction codes.

Kai: But the soft-collinear effective theory they use is valid over a wide range of mass and virtuality, which is good for QCD applications.

Mira: That's promising because it suggests factorization of soft-collinear modes from the hard remainder might work better in those regions.

Lev: It’s a hopeful sign for future QCD modeling, but we still have to deal with the ultrasoft region where coupling constant growth becomes an issue in the infrared.

Kai: So, this paper points toward a new framework for analyzing these high-energy scattering processes.

Mira: It sets up some clear directions for future work, like extending the form factor analysis to three loops or looking at timelike virtualities where anomalous thresholds are important.

Lev: And they suggest studying high-energy hadron-hadron collisions involving heavy vector bosons and Higgs at small transverse momentum, which is where those ultrasoft effects really show up.

Mathematical Institute, University of Oxford · CERN Department of Theoretical Physics, CERN · Fields and Strings Laboratory, Institute of Physics, Ecole Polytechnique Férale de Lausanne (EPFL) · Department of Physics, Arizona State University · Institute for Theoretical Physics, University of Amsterdam

hep-th, hep-ph

Submitted: 2026-05-15

Updated: 2026-10-29

Comments: 43 pages, 11 figures, calculational mistake fixed in v2

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 91/100

The gist: The paper introduces a novel high-energy limit in both the Sudakov form factor and the four-point scattering amplitude on N=4 SYM by identifying a "walking anomalous dimension" that smoothly

Key concepts

Walking Anomalous Dimension: This is a new mathematical function that describes how a certain physical quantity changes with energy scales. It smoothly transitions between the cusp anomalous dimension (relevant at low energies) and the octagon anomalous dimension (relevant at high energies), characterizing the theory's behavior across different regimes.
Sudakov Form Factor: This is a mathematical tool used to calculate high-energy scattering processes in quantum field theories. In this paper, it is analyzed to find a new high-energy limit that exhibits double-logarithmic behavior controlled by the walking anomalous dimension.
Coulomb Branch: This refers to a specific configuration of N=4 SYM where the gauge symmetry is reduced by turning on vacuum expectation values for adjoint scalars. This setup allows researchers to study massive BPS states and their interactions in a way that moves away from simpler theories.
Cusp/Octagon Anomalous Dimensions: These are two established anomalous dimensions that describe the behavior of the theory's coupling constant at different energy scales. The walking anomalous dimension is introduced as an interpolation between these two, showing how the theory behaves when transitioning between low-energy and high-energy limits.

Terminology

Regularization — 10× in this paper · explained in 3 episodes
Accuracy — 5× in this paper · explained in 1 episode
Momentum transfer — 3× in this paper · explained in 2 episodes
European Research Council — 2× in this paper
Gauge Invariance — 2× in this paper · explained in 2 episodes
Generalization — 2× in this paper · explained in 14 episodes
Integrability — 2× in this paper · explained in 2 episodes
Pair Production — 2× in this paper · explained in 2 episodes
Dispersion Relation — 1× in this paper · explained in 2 episodes
Generator — 1× in this paper · explained in 2 episodes
Symmetry Breaking — 1× in this paper · explained in 4 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

The paper introduces a novel high-energy limit in both the Sudakov form factor and the four-point scattering amplitude on N=4 SYM by identifying a walking anomalous dimension that smoothly interpolates between the cusp and octagon anomalous dimensions The main result of the paper is the identification of a new high-energy limit in the twoparticle Sudakov form factor and in the four-point scattering amplitude on the Coulomb branch of N = 4 SYM This high-energy limit generalizes the usual double-logarithmic behavior of the Sudakov form factor and of the scattering amplitude and is controlled by a new anomalous dimension that smoothly interpolates between the cusp and octagon anomalous dimensions, which they call it the walking anomalous dimension

The Setup for Interpolation

The study is conducted on planar N = 4 SYM on the Coulomb branch by turning on vacuum expectation values for six adjoint scalars This setup moves the theory away from the origin of moduli space and Higgses the gauge symmetry to a subgroup that commutes with the scalar background The external states in this analysis correspond to massive BPS states, with masses proportional to the distances mi⃗ni − mj⃗nj between corresponding points in R 6

The Walking Limit Identification

The key finding is the identification of a new high-energy limit where both observables exhibit double-logarithmic behavior governed by a walking anomalous dimension This walking anomalous dimension interpolates between the cusp anomalous dimension of the on-shell regime and the octagon anomalous dimension of the off-shell regime as mass scales are varied For example, as the walking parameter η → 0, it reduces to the cusp-coefficient Γcusp(g)/4, while for η → 1, it reduces to Γoct(g)/2

One-Loop Analysis of the Form Factor

The one-loop analysis of the Sudakov form factor reveals a walking behavior in the limit where both internal and external masses are present The one-loop walking anomalous dimension is given by Γ(1)walk(η) = g 2 0 for η ≤ -1, (1 + η) squared for − 1 ≤ η ≤ 0, (1 + 2η − η 2) for 0 ≤ η ≤ 1, and 2 for η ≥ 1 This function is a C 1 function of the walking parameter η

Two-Loop Prediction and All-Loop Structure

The two-loop computation, utilizing the Method of Regions, yields an all-loop prediction for the walking anomalous dimension in both observables The expression for the walking form factor is given by Γwalk(η, g) =  0, η ≤ -1, (g squared − 2ζ 2g 4)(1 + η) squared + O(g 6) for − 1 ≤ η ≤ 0, g 2(1 + 2η − η 2) − g 4ζ 2/2 + 5η/5η squared, for 0 ≤ η ≤ 1, and Γoct(g) squared, η ≥ 1

Four-Point Amplitude Generalization

The walking phenomenon observed in the form factor also appears in the four-point scattering amplitude on the Coulomb branch The high-energy limit of the amplitude is characterized by two exponents (ηs, ηt), and it is predicted that logM = −Γwalk(ηs, ηt, g) log ωs log ωt + O(log ωs, log ωt) The all-loop prediction for the four-point amplitude is given by a complex piecewise function in terms of ηs and ηt

Conclusions on Physical Relevance

The results depend on four functions of the ’t Hooft coupling: Γcusp, Γoct, and two new functions γ, γ˜, which are not known at present The soft-collinear effective theory is valid over a very wide region of mass/virtuality parameter space as determined by the y-parameter This gives hope for future robust QCD applications relying on the factorization of soft-collinear modes from the hard remainder The presence of the ultrasoft region threatens the potential applicability of perturbative techniques in QCD due to the coupling constant growth in the infrared

Future Directions

The paper suggests several natural directions for further study, including extending the form-factor analysis to three loops and performing a strong-coupling analysis along the lines of [93, 99] It is also suggested that one should extend the analysis to timelike virtualities, p 2i > 0, where anomalous thresholds are expected to play an important role The study of high-energy hadron-hadron collisions with production of heavy vector bosons and/or Higgs at small transverse momentum is also suggested

Appendix Analysis

In the appendix, a detailed analysis of the different regions contributing to the double-logarithmic coefficient of the one-loop amplitude M(1)(s, t) is presented for s = t This analysis separates contributions based on kinematic scales x ∼ ω, x ∼ ω-1, x ∼ y-1, and x ∼ y The result for the walking anomalous dimension in the s = t case is given by Γ(1)walk(η) = 2g 2 0 for η ≤ -1, 2(1 + η) squared for − 1 ≤ η ≤ 0, (1 + 2η − η 2) for 0 ≤ η ≤ 1, and Γoct(g) squared, eta ≥ 1

Final All-Loop Prediction

The all-loop prediction for the walking anomalous dimension appearing in the four-point amplitude for s ≠ t is given by a piecewise function in terms of ηs and ηt This expression reduces to (4.

Improvements for AI systems

Precision prediction of high-energy observables can be improved by implementing an all-loop prediction for anomalous dimensions that interpolates between cusp and octagon limits, as described in equation (3.66). This allows AI systems to make more accurate predictions for form factors and four-point scattering amplitudes across different kinematic regimes.
AI systems can be enhanced with the ability to calculate the two-loop correction to the walking anomalous dimension, given by equation (3.57) for the form factor or equation (3.60) for the amplitude in certain limits. This provides higher precision than one-loop calculations, which is crucial for refining theoretical predictions in high-energy physics.
The AI can be equipped to handle complex kinematic dependencies and mass scaling by utilizing the walking parameter eta, which smoothly interpolates between regimes, as seen in equation (1.1). This enables the system to accurately model observables across both on-shell and off-shell kinematic conditions of gauge theories like N = 4 SYM.
The AI can perform a detailed analysis of the regions contributing to the double-logarithmic coefficient in one-loop amplitudes, as presented in appendix A, specifically distinguishing between regions like y omega minus 1. This capability allows for better understanding of how different kinematic constraints affect observable behavior.
The system can be trained on the complex analytic expressions for the one-loop walking anomalous dimension across all parameter ranges, such as equation (3.28). This enables the AI to accurately predict the behavior of observables in both regimes defined by y 1, including transitions like the octagon shoulder.
The AI can be extended to handle non-equal Mandelstam variables s and t by utilizing equation (4.40) for the one-loop walking anomalous dimension in the s not equal to t case. This allows for more general predictions of scattering amplitudes beyond the simplified s = t limit, which is essential for realistic collider simulations.
The AI can be designed to understand the physical implications of different infrared power counting schemes, differentiating between contributions from soft-collinear modes and ultrasoft-collinear modes as discussed in section 5. This understanding helps in identifying regimes where perturbative techniques are valid or where nonperturbative physics might become relevant.

Abstract

We study the Sudakov form factor and the four-point scattering amplitude on the Coulomb branch of planar N=4 SYM as functions of the Coulomb-branch parameters and kinematic invariants. This setup provides a controlled probe of the interpolation between on- and off-shell regimes of infrared-sensitive quantities in gauge theories. We identify a novel scaling limit in which both observables exhibit double-logarithmic behavior governed by a walking anomalous dimension. As the mass scales are varied, this walking anomalous dimension interpolates between the cusp anomalous dimension of the on-shell regime and the octagon anomalous dimension of the off-shell regime. Based on the explicit two-loop result and the expected all-order structure, we propose an all-loop form for the walking anomalous dimension both for the form factor and for the four-point scattering amplitude. These all-loop expressions depend on new, presently unknown functions of the 't Hooft coupling.