[Paper Review] Logarithmic enhancements in conformal perturbation theory and their real time interpretation

David Berenstein, Alexandra Miller|arXiv (Cornell University)|Jul 7, 2016

Mathematical functions and polynomials4 citations

TL;DR

This paper investigates logarithmic divergences in conformal perturbation theory using dimensional regularization, showing they arise from resonant behavior in time-dependent perturbation theory on the cylinder. It establishes a real-time interpretation of these divergences by mapping them to secular (resonant) transitions induced by oscillatory perturbations, linking UV/IR divergences in the plane to physical resonances in time evolution.

ABSTRACT

We study various corrections of correlation functions to leading order in conformal perturbation theory, both on the cylinder and on the plane. Many problems on the cylinder are mathematically equivalent to those in the plane if we give the perturbations a position dependent scaling profile. The integrals to be done are then similar to the study of correlation functions with one additional insertion at the center of the profile. We will be primarily interested in the divergence structure of these corrections when computed in dimensional regularization. In particular, we show that the logarithmic divergences (enhancements) that show up in the plane under these circumstances can be understood in terms of resonant behavior in time dependent perturbation theory, for a transition between states that is induced by an oscillatory perturbation on the cylinder.

Motivation & Objective

• To understand the origin of logarithmic divergences in conformal perturbation theory beyond standard UV/IR regularization.
• To connect divergences in correlation functions on the plane to resonant transitions in time-dependent perturbation theory on the cylinder.
• To provide a real-time physical interpretation of logarithmic enhancements using dimensional regularization and Schwinger parameter methods.
• To generalize the use of dimensional regularization in CFT by fixing operator dimensions while varying spacetime dimension d.
• To establish a correspondence between divergences in the plane and resonant behavior in time evolution via Weyl rescaling and position-dependent deformations.

Proposed method

• Uses dimensional regularization with fixed operator dimensions h_D and varying spacetime dimension d to regulate divergences in correlation functions.
• Applies Weyl rescaling to map correlation functions from the plane to the cylinder, introducing position-dependent scaling factors that act as infrared regulators.
• Employs Schwinger parameter representations to evaluate integrals involving three-point functions with oscillatory perturbations.
• Transforms integrals into forms involving modified Bessel functions K_ν via change of variables and gamma function identities.
• Introduces a fourth Schwinger parameter to handle the sum of inverse variables in the denominator, enabling extraction of divergence structure.
• Analyzes the exponential factor in the integral to isolate frequency-dependent terms that signal resonant behavior in time evolution.

Experimental results

Research questions

• RQ1 What causes logarithmic divergences in conformal perturbation theory when using dimensional regularization with fixed operator dimensions?
• RQ2 How do divergences in the plane relate to physical resonant behavior in time-dependent perturbation theory on the cylinder?
• RQ3 Can the divergent structure of correlation functions be interpreted as secular (resonant) growth in time due to oscillatory perturbations?
• RQ4 How does a spacetime-dependent deformation f(x) = |x|^{h_D - d} regulate infrared divergences and mimic the cylinder geometry?
• RQ5 What is the role of the modified Bessel function K_ν in encoding the frequency-dependent divergence structure of the integrals?

Key findings

• Logarithmic divergences in conformal perturbation theory arise from poles in gamma functions when operator dimensions and spacetime dimension d reach special values.
• The divergences in the plane are equivalent to resonant transitions in time-dependent perturbation theory on the cylinder, where oscillatory perturbations induce secular growth.
• The position-dependent deformation f(x) = |x|^{h_D - d} provides an infrared regulator and mimics the Weyl rescaling that maps the plane to the cylinder.
• The integral structure reduces to a form involving modified Bessel functions K_ν, with the frequency dependence encoded in the argument and order of K_ν.
• The divergence structure is fully captured by the exponential factor involving frequency differences ω_ij, which signals resonant behavior when frequencies align.
• The final expression for the integral exhibits explicit dependence on frequency differences through Ω_j^2 = (ω_k - ω_l)^2, confirming the resonant origin of the logarithmic enhancements.