[Paper Review] No-$π$ Theorem for Euclidean Massless Correlators

P.A. Baikov, K.G. Chetyrkin|arXiv (Cornell University)|Aug 1, 2018

Advanced Algebra and Geometry4 references3 citations

TL;DR

This paper establishes a No-π theorem for Euclidean massless correlators in QCD, proving that π-dependent terms in 6-loop correlators and 7-loop renormalization group functions arise only through specific zeta-function combinations. Using a 'hatted' representation of p-integrals, the authors derive exact algebraic relations for π-dependent contributions, showing they are fully determined by lower-loop anomalous dimensions and β-functions, with explicit formulas up to 7 loops.

ABSTRACT

We provide the reader with a (very) short review of recent advances in our understanding of the $π$-dependent terms in massless (Euclidean) 2-point functions as well as in generic anomalous dimensions and $β$-functions. We extend the considerations of [1] by one more loop, that is for the case of 6-loop correlators and 7-loop renormalization group (RG) functions.

Motivation & Objective

• To understand the origin and structure of π-dependent terms in massless 2-point functions and renormalization group functions in QCD.
• To extend previous results on π-suppression in correlators and anomalous dimensions to 6-loop (correlators) and 7-loop (RG functions) order.
• To provide a generic proof of the absence of certain π-powers (e.g., π², π⁴, π⁶) in scale-invariant correlators up to 5-loop order.
• To derive exact algebraic constraints relating π-dependent contributions in anomalous dimensions and β-functions at 7 loops.
• To demonstrate that a specific scheme redefinition can eliminate all π-dependence in the 5-loop correlator and β-function, confirming a nontrivial identity.

Proposed method

• Introduces a 'hatted' representation of transcendental numbers (ζ-values) in p-integrals, enabling systematic classification of π-dependent terms.
• Uses the formalism of renormalization group equations and the Adler-Bell-Jackiw anomaly to relate anomalous dimensions and β-functions to correlator structures.
• Derives recursive relations between π-dependent coefficients in anomalous dimensions and β-functions using the structure of the renormalization group equation.
• Applies the hatted representation to compute explicit expressions for π-dependent terms in γ₇ and β₇ at 7 loops, expressed in terms of lower-loop coefficients.
• Validates results against known 7-loop results in the O(n) φ⁴ model and QCD with n_f flavors, confirming consistency across multiple independent calculations.
• Demonstrates that a scheme redefinition involving c₁, c₂, c₃ and a 4-loop correction can fully remove π-dependence in the 5-loop correlator and β-function.

Experimental results

Research questions

• RQ1 Why do certain π-powers (e.g., π², π⁴, π⁶) vanish in scale-invariant 5-loop correlators despite individual diagrams containing ζ(3), ζ(4), etc.?
• RQ2 What algebraic structure governs the appearance of π-dependent terms in 6-loop correlators and 7-loop RG functions?
• RQ3 Can the absence of π² and π⁶ in 5-loop scale-invariant correlators be explained by a deeper algebraic mechanism?
• RQ4 Is there a scheme in which the 5-loop β-function and correlator become π-independent, and if so, what are the conditions on the scheme parameters?
• RQ5 How are π-dependent contributions in γ₇ and β₇ at 7 loops related to lower-loop coefficients through exact algebraic identities?

Key findings

• The 5-loop scale-invariant correlator F⁰⁵ is free of π² and π⁶ terms, but contains π⁴ terms, confirming a nontrivial pattern in π-suppression.
• The QCD β-function first acquires π-dependent terms at 5 loops, exclusively through ζ(4) = π⁴/90, and no higher π-powers appear at this order.
• A 7-loop identity is derived: β₇^ζ₄ = (3/8)β₄^ζ₃ β₁ + (9/10)β₂β₅^ζ₃ - (1/2)β₃^ζ₃ β₄^(1) + (5/4)β₁β₆^ζ₃, showing exact algebraic dependence.
• The 7-loop anomalous dimension γ₇ has nonvanishing π-dependent terms only for ζ₃ζ₇, ζ₄ζ₃², ζ₄ζ₆, ζ₄ζ₅, ζ₃ζ₆, ζ₄ζ₇, and ζ₅ζ₆, with specific coefficients derived from lower-loop data.
• The 7-loop β-function β₇ has π-dependent terms only for ζ₄, ζ₆, ζ₃ζ₄, ζ₈, ζ₃ζ₆, ζ₄ζ₅, ζ₁₀, ζ₄ζ₃², and ζ₄ζ₇, with explicit formulas in terms of lower-loop coefficients.
• A scheme redefinition a = a̅(1 + c₁a̅ + c₂a̅² + c₃a̅³ + (1/3)(β₅/β₁)a̅⁴) can eliminate all π-dependence in the 5-loop correlator and β-function, confirming a nontrivial identity from [3].