# Research offers a mathematical tool to help understand the fractal structure of quark-gluon plasma

Quark-Gluon Plasma (QGP) is a state of matter existing at extreme temperatures and densities, such as those that occur in collisions of hadrons (protons, neutrons and mesons). Under so-called “normal” conditions, quarks and gluons are still confined within the structures that make up hadrons, but when hadrons are accelerated to relativistic velocities and made to collide with each other, as they are in experiments carried out at the Large Hadron Collider (LHC) operated by the European Organization for Nuclear Research (CERN), confinement is broken and quarks and gluons disperse, forming a plasma. The phenomenon lasts only a tiny fraction of a second, but its observation has produced important discoveries about the nature of material reality.

One of the findings, for which evidence is steadily accumulating, is that quark-gluon plasma has a fractal structure. As it decays into a stream of particles propagating in various directions, the behavior of particles in the jets is similar to that of quarks and gluons in plasma. Moreover, it decays in a cascade of reactions with a pattern of self-similarity on many scales typical of fractals.

A new study, published in The European Physical Journal Plus, describes a mathematical tool to better understand the phenomenon. The authors focus on a technical aspect of the solution of the Klein-Gordon equation for the dynamics of bosons, spin-zero relativistic particles sharing the same quantum states and therefore indistinguishable. Moreover, in a Bose-Einstein Condensate (BEC), the particles behave collectively as if they were a single particle. BEC research has produced new atomic and optical physics. Potential applications include more accurate atomic clocks and improved techniques for fabricating integrated circuits.

“Fractal theory explains the formation of the BEC,” said Airton Deppman, a professor at the Institute of Physics at the University of São Paulo (IF-USP) in Brazil, and principal investigator of the study, which was supported by FAPESP via three projects (21/12954 -5, 19/10889-1 and 16/17612-7).

“The study was part of a larger research program that had already resulted in the 2020 paper ‘Fractals, nonextensive statistics, and QCD’ published in Physical Review D, demonstrating that Yang-Mills fields have structures fractals and explaining some phenomena seen in high-energy collisions where a quark-gluon plasma forms,” Deppman added.

Formulated in the 1950s by Chinese physicist Chen-Ning Yang (co-winner of the 1957 Nobel Prize in Physics) and American physicist Robert Mills, Yang-Mills theory is very important for the Standard Model of particle physics because it describes three of the four fundamental forces in the Universe: electromagnetic, weak and strong forces (the fourth is gravitational interaction).

“In high-energy collisions, the main result is the particle momentum distribution, which follows Tsallis statistics instead of traditional Boltzmann statistics. We show that fractal structure is responsible for this. This leads to statistics of Tsallis rather than Boltzmann,” continued Deppman. Constantino Tsallis was born in Greece in 1943 and naturalized Brazilian in 1984. He is a theoretical physicist primarily interested in statistical mechanics. Ludwig Boltzmann (1844-1906) was a Austrian physicist and mathematician who made important advances in statistical mechanics, electromagnetism and thermodynamics.

“With this fractal approach, we were able to determine Tsallis’ entropy index q, which is calculated using a simple formula relating it to key Yang-Mills parameters,” Deppman said. “In the case of quantum chromodynamics [QCD, the theory of the strong interaction between quarks mediated by gluons], these parameters are the number of colors and flavors of the particles. With these parameters, we found q = 8/7, consistent with experimental results where q = 1.14,” he said.

Colors in QCD do not refer to the usual concept but to color charges, related to strong interactions between quarks. There are three possibilities, symbolized by red, green and blue. Quarks also have electric charges, which are related to electromagnetic interactions, but color charges are a different phenomenon. Flavors describe the six types of quarks: up, down, charm, strange, up, and down. This quaint nomenclature reflects the sense of humor of Murray Gell-Mann (1929-2019), an American physicist who won the Nobel Prize in Physics in 1969 for his work on the theory of elementary particles, and later scientists who also contributed to the QCD.

“An interesting aspect of the evolution of our knowledge is that before high-energy collisions were experimentally achieved in large particle colliders, and even before the existence of quarks was proposed, Rolf Hagedorn, a physicist German who worked at CERN, set out to predict the production of particles in these collisions,” Deppman said. “On the basis of cosmic ray research alone, he formulated the concept of fireballs to explain the cascade of particles created during high-energy collisions. With this hypothesis, he predicted the threshold temperature corresponding to the phase transition between confined and deconfined regimes. The key element of his theory is the self-similarity of fireballs Hagedorn didn’t use the term “fractal” because the concept didn’t yet exist, but after the term was coined by Mandelbrot, we saw that fireballs were fractals. oît Mandelbrot (1924-2010) was a Franco-American mathematician of Polish origin.

According to Deppman, Hagedorn’s theory can be generalized by including Tsallis’ statistics. Indeed, Deppman did so in an article published in Physica A in 2012.

“With this generalization, we obtain a self-consistent thermodynamic theory that predicts the critical temperature for the transition to quark-gluon plasma, and also provides a formula for the mass spectrum of hadrons, from lightest to heaviest,” said he declared. “There is strong evidence for conceptual continuity in the description of hadronic systems from quark-gluon plasma to hadrons, and for the validity of the fractal structure of QCD in both regimes.”

Deppman wonders if fractal structures could also be present in electromagnetism. This would explain why so many natural phenomena, from lightning to snowflakes, have fractal structures, as they are all governed by electromagnetic forces. This could also explain why Tsallis statistics are present in so many phenomena. “Tsallis statistics have been used to describe scale transformation invariance, a key ingredient of fractals,” he said.

Could the fractal theory be extended to gravitational phenomena? “Gravitation is outside the scope of our approach, as it does not fit into Yang-Mills theory, but nothing prevents us from speculating whether fractals express an underlying pattern in all material reality,” he said. he declares.