Techniques of evaluation of QCD low-energy physical quantities with running coupling with infrared fixed point

2013/09/30 by Gorazd Cvetič
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Fixed point #Gravitational singularity #Infrared fixed point #Mathematical analysis #Mathematical physics #Mathematics #Particle physics #Particle physics theoretical and experimental studies #Physics #Power series #Quantum Chromodynamics and Particle Interactions #Quantum chromodynamics #Quantum electrodynamics #Quantum mechanics #Renormalization #Renormalization group #Renormalon #Resummation #Series (stratigraphy) #hep-ph

paper · pdf · doi:10.1103/physrevd.89.036003

v3: 24 pages, 10 figures; improved presentation; additional references [33,40,42,43,45,49,55,63-66,68,70]; the version to appear in Phys.Rev.D

arxiv created 2014/01/07 · openalex publication_date 2014/02/14 · arxiv updated 2015/06/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

Perturbative QCD (pQCD) running coupling a(Q²) (\ensuremath≡\ensuremathα_s(Q²)/\ensuremathπ) is expected to get modified at low spacelike momenta 0<Q²\ensuremath\lesssim1 GeV² so that, instead of having unphysical (Landau) singularities, it remains smooth and finite there, due to the infrared (IR) fixed point. This behavior is suggested by the Gribov-Zwanziger approach, Dyson-Schwinger equations and other functional methods, lattice calculations, light-front holographic mapping AdS/CFT modified by a dilaton background, and most of the analytic (holomorphic) QCD models. All such couplings A(Q²) differ from the pQCD couplings a(Q²) at |Q|\ensuremath\gtrsim1 GeV by nonperturbative (NP) terms, typically by some power-suppressed terms \ensuremath∼1/Q^2N. Evaluations of low-energy physical QCD quantities in terms of such A(Q²) couplings (with an IR fixed point) at a level beyond one loop are usually performed with a (truncated) power series in A(Q²). We argue that such an evaluation is not correct, because the NP terms in general get out of control as the number of terms in the power series increases. The series consequently become increasingly unstable under the variation of the renormalization scale and have a fast asymptotic divergent behavior compounded by the renormalon problem. We argue that an alternative series in terms of logarithmic derivatives of A(Q²) should be used. Furthermore, a Pad'e-related resummation based on this series gives results which are renormalization scale independent and show very good convergence. Timelike low-energy observables can be evaluated analogously, by using the integral transformation which relates the timelike observable with the corresponding spacelike observable.

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