[Paper Review] Analysis of the anomalous-dimension matrix of n-jet operators at 4 loops

Leonardo Vernazza|arXiv (Cornell University)|Dec 14, 2011

Black Holes and Theoretical Physics8 references3 citations

TL;DR

This paper extends the diagrammatic analysis of the anomalous-dimension matrix for n-jet operators in Soft-Collinear Effective Field Theory (SCET) to four loops, confirming that only two-parton correlations contribute to the cusp anomalous dimension. It finds that potential new four-parton color structures with momentum dependence on conformal cross ratios—though compatible with collinear and soft-collinear constraints—do not affect Casimir scaling of the cusp anomalous dimension, reinforcing the all-order conjecture for infrared divergences in massless gauge theories.

ABSTRACT

Recently, an all-order conjecture for the anomalous-dimension matrix of n-jet operators in SCET was proposed, which allows one to predict the structure of the infrared divergences of dimensionally regularized, massless gauge-theory scattering amplitudes with an arbitrary number of legs and loops. The conjecture is severely constrained by soft-collinear factorization, non-abelian exponentiation, and the behavior of amplitudes in collinear limits. Using these constraints, a diagrammatic analysis has shown that the anomalous dimension involves only two-parton correlators up to three loop order. The only exception is given by a single color structure multiplying a function of conformal cross ratios depending on the momenta of four external partons, which would have to vanish in all two-particle collinear limits. We extend this analysis by completing the diagrammatic analysis at four loop, and we find that additional functions which vanish in all two-particle collinear limits may arise.

Motivation & Objective

• To test the all-order conjecture for the anomalous-dimension matrix of n-jet operators in SCET at four-loop order.
• To determine whether new color structures beyond two-parton correlations can arise, which might break Casimir scaling of the cusp anomalous dimension.
• To assess consistency of potential four-loop contributions with soft-collinear factorization, non-abelian exponentiation, and collinear limits.
• To examine whether functions of conformal cross ratios—dependent on four-parton momenta—can contribute without violating known constraints.
• To rule out or constrain new momentum-dependent functions using high-energy Regge limit behavior.

Proposed method

• Perform a diagrammatic analysis of four-loop gluon webs in SCET, focusing on connected soft and collinear contributions to the anomalous dimension matrix.
• Use the color-space formalism to express the anomalous dimension as a sum over color structures involving products of T_i operators and functions of momentum invariants.
• Apply constraints from soft-collinear factorization, which require the anomalous dimension to factorize into single-connected webs on Wilson lines.
• Enforce the two-parton collinear limit condition: the anomalous dimension must reduce to a splitting function depending only on the two collinear partons.
• Use the high-energy Regge limit to constrain the behavior of functions of conformal cross ratios, ruling out terms with leading logarithmic behavior in s/t.
• Analyze color structures involving four and five partons, identifying allowed forms that respect all symmetries and constraints.

Experimental results

Research questions

• RQ1 Can new four-parton color structures with non-trivial momentum dependence on conformal cross ratios appear in the four-loop anomalous dimension matrix?
• RQ2 Do such structures violate Casimir scaling of the cusp anomalous dimension for quarks and gluons?
• RQ3 Are functions of the logarithm of conformal cross ratios compatible with the two-parton collinear limit?
• RQ4 Can the high-energy Regge limit rule out specific functional forms of momentum-dependent coefficients?
• RQ5 Are there any new contributions at four loops that would break the all-order conjecture for the anomalous dimension matrix?

Key findings

• At four loops, no new color structures beyond two-parton correlations are allowed that would affect Casimir scaling of the cusp anomalous dimension.
• New terms involving four-parton color structures with momentum dependence on conformal cross ratios are possible only if the corresponding function G₁(x,y) vanishes in all two-parton collinear limits.
• An explicit candidate function G₁(x,y) = x²(x² − y²)² is ruled out by the Regge limit, as it would generate unphysical leading logarithmic terms in the gluon Regge trajectory.
• More complex functions G₁(x,y) that cancel leading and next-to-leading logarithms may still be allowed, but they do not affect the cusp anomalous dimension's Casimir scaling.
• For five-parton structures, terms proportional to T_ijklm with functions G₂ of multiple conformal cross ratios are allowed only if G₂ vanishes in all collinear limits, and such terms do not affect Casimir scaling.
• The analysis confirms that the all-order conjecture for the anomalous-dimension matrix remains consistent at four loops, with no violation of Casimir scaling.