Leading logarithms of the two-point function in massless O(N) and SU(N) models to any order from analyticity and unitarity

  • B. Ananthanarayan
  • Shayan GhoshEmail author
  • Alexey Vladimirov
  • Daniel Wyler
Special Article - New Tools and Techniques


Leading (large) logarithms in non-renormalizable theories have been investigated in the recent past. Besides some general considerations, explicit results for the expansion coefficients (in terms of leading logarithms) of partial wave amplitudes and of scalar and vector form factors have been given. Analyticity and unitarity constraints have been used to obtain the expansion coefficients of partial waves in massless theories, yielding form factors and the scalar two-point function to five-loop order in the \( O(4)/O(3)\) model. Later, all order solutions for the partial waves in any \( O(N+1)/O(N)\) model were found. Also, results up to four-loop order exist for massive theories. Here we extend the implications of analyticity and unitarity constraints on the leading logarithms to arbitrary loop order in massless theories. We explicitly obtain the scalar and vector form factors as well as the scalar two-point function in any O(N) and SU(N) type models. We present relations between the expansion coefficients of these quantities and those of the relevant partial waves. Our work offers a consistency check on the published results in the O(N) models for form factors, and new results for the scalar two-point function. For the SU(N) type models, we use the known expansion coefficients for partial waves to obtain those for scalar and vector form factors as well as for the scalar two-point function. Our results for the form factor offer a check for the known and future results for massive O(N) and SU(N) type models when the massless limit is taken. Mathematica notebooks which can be used to calculate the expansion coefficients are provided as supplementary material.

Supplementary material

10050_2018_12555_MOESM1_ESM.nb (100 kb)
Supplementary material
10050_2018_12555_MOESM2_ESM.nb (189 kb)
Supplementary material


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Copyright information

© SIF, Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • B. Ananthanarayan
    • 1
  • Shayan Ghosh
    • 1
    Email author
  • Alexey Vladimirov
    • 2
  • Daniel Wyler
    • 3
  1. 1.Centre for High Energy PhysicsIndian Institute of ScienceBangaloreIndia
  2. 2.Institut für Theoretische PhysikUniversität RegensburgRegensburgGermany
  3. 3.Physik-InstitutUniversität ZürichZurichSwitzerland

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