Gauge fields renormalization groups and thermofractals

The perturbative approach to QCD has shown to be limited, and the difficulties to obtain accurate calculations in the low-energy region seems to be insurmountable. A recent approach uses the fractal structures of Yang-Mills Field Theory to circumvent those difficulties, allowing for the determination of an analytic expression for the running coupling. The results obtained are in agreement with several experimental findings, and explain many of the observed phenomena at high-energy collisions. In this work, we address some of the conceptual aspects of the fractal approach, which are expressed in terms of the renormalization group equation and the self-energy corrections to the parton mass. We associate these concepts with the origins of the fractal structure in the quantum field theory.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil)Project INCT-FNA Proc. No. 464 898/2014-5FAPESP under grant 2016/17612-7Project PID2020-114767GB-I00 Financed by MCIN/AEI/10.13039/501100011033FEDER/Junta de Andalucía-Consejería de Economía y Conocimiento 2014-2020 Operational Programme under Grant A-FQM-178-UGR18Junta de Andalucía under Grant FQM-225Consejería de Conocimiento, Investigación y Universidad of the Junta de Andalucía and European Regional Development Fund (ERDF) under Grant SOMM17/6105/UGRRamón y Cajal Program of the Spanish MCIN under Grant RYC-2016-2067

Fractal Structure and Non-Extensive Statistics

The role played by non-extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics have been discussed in detail in many works and triggered an interesting discussion on the most deep meaning of entropy and its role in complex systems. Some possible mechanisms that could give rise to non-extensive statistics have been formulated over the last several years, in particular a fractal structure in thermodynamic functions was recently proposed as a possible origin for non-extensive statistics in physical systems. In the present work, we investigate the properties of such fractal thermodynamical system and propose a diagrammatic method for calculations of relevant quantities related to such a system. It is shown that a system with the fractal structure described here presents temperature fluctuation following an Euler Gamma Function, in accordance with previous works that provided evidence of the connections between those fluctuations and Tsallis statistics. Finally, the scale invariance of the fractal thermodynamical system is discussed in terms of the Callan–Symanzik equation.Conselho Nacional de Desenvolvimento Científico e Tecnológico: 464898/2014-5, Spanish MINEICO: FPA2015-64041-C2-1-P, SpanishMINEICO: FIS2017-85053-C2-1-P, Junta de Andalucía: Grant FQM-225, and Consolider Ingenio 2010 Programme CPAN: CSD2007-00042