The study proposes a mathematical tool to help understand the fractal structure of quark-gluon plasmas

The study proposes a mathematical tool to help understand the fractal structure of quark-gluon plasmas

A chain of events caused by the collision of lead ions at the CMS detector of the Large Hadron Collider, recorded in November 2018. Credit: CMS/CERN

Quark-gluon plasma (QGP) is a state of matter that exists at extreme temperatures and densities, such as those that occur in the collision of hadrons (protons, neutrons, and mesons). Under so-called “normal” conditions, quarks and gluons are always confined to the structures that make up hadrons, but when hadrons are accelerated to relativistic velocities and collide with each other, as in experiments at the Large Hadron Collider (LHC) operated by the organization European Nuclear Research Institute (CERN), the sequestration and scattering of quarks and gluons are interrupted, forming plasma. This phenomenon only lasts a fraction of a second, but observing it has produced important discoveries about the nature of physical reality.

One of the discoveries, for which evidence is steadily accumulating, is that Quark-gluon plasma he have fractal structure. When it disintegrates into a stream of particles propagating in different directions, the particles’ behavior in jets is similar to that of quarks and gluons in plasma. Moreover, it decomposes in a series of interactions with a pattern of self-similarity on many scales typical of fractals.

A new study published in European Physical Journal PlusMathematical tool with which to understand more about this phenomenon. The authors focus on the technical aspect of solving the Klein-Gordon equation for the dynamics of bosons, which are relativistic particles with zero rotation that share the same quantum states and are therefore indistinguishable. in Bose-Einstein condensate (BEC); Moreover, particles that collectively behave as if they were a single particle. BEC’s research has resulted in new atomic and optical physics. Potential applications include more accurate atomic clocks and improved technologies for making integrated circuits.

“The fractal theory explains the formation of the BEC,” said Arton Dibmann, a professor at the University of São Paulo’s Institute of Physics (IF-USP) in Brazil and the study’s lead author.

‘The study was part of a broader research program that already in 2020 resulted in the article’Fractals, Non-Extended Statistics, and QCD‘ Posted in physical review dwhich shows that the Yang-Mills fields have fractal structures and explains some of the phenomena seen in high-energy collisions where quark-gluon plasmas are formed.”

The Yang-Mills theory, formulated by Chinese physicist Chen Ning Yang (winner of the 1957 Nobel Prize in Physics) and American physicist Robert Mills, in the 1950s, is extremely important to the Standard Model of particle physics because it describes three of the four. The basic forces in the universe: the electromagnetic, weak and strong forces (the fourth force is the interaction of gravity).

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

“With this fractal approach, we were able to determine the cascade index of entropy q, which is calculated using a simple formula that relates it to the main Yang-Mills parameters,” said Dibman. 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 in agreement with the experimental results where q = 1.14″.

The colors in QCD refer not to the usual concept but to the color charges related to the strong interactions between quarks. There are three possibilities denoted by red, green and blue. Quarks also contain electric charges related to electromagnetic interactions, but chromatic charges are a different phenomenon. Flavors describe the six types of quarks: up, down, charm, strange, up, and down quark. This brilliant designation reflects the humorous sense of Murray Gell-Mann (1929-2019), the American physicist who won the 1969 Nobel Prize in Physics for his work on elementary particle theory, and later scientists who also contributed to the QCD.

“One intriguing aspect of the development of our knowledge is that before high-energy collisions were performed experimentally in large particle collisions, and even before the existence of quarks was suggested, Rolf Heddorn, a German physicist who worked at CERN, began predicting the production of particles in these collisions,” said Dibmann. “Only based on research in cosmic raysHe coined the concept of fireballs to explain the chain of particles produced by high-energy collisions. By this hypothesis, he predicted the threshold temperature corresponding to the phase transition between the confined and unconstrained systems. A key element in his theory is the self-similarity of fireballs. Hagedorn didn’t use the term ‘fractal’ because the concept didn’t exist yet, but after Mandelbrot coined the term, we saw that fireballs were fractals. Benoît Mandelbrot (1924-2010) was a French-born American mathematician.

According to Dibmann, Hagedorn’s theory can be generalized by including the statistics of Tasalis. In fact, Dibman did so in an article published in Physica In 2012.

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

Dibman wonders if fractal structures can also be present in electromagnetism. This would explain why many natural phenomena, from lightning to snowflakes, have fractal structures, all of which are subject to electromagnetic forces. It may also explain why tessellation statistics exist in many phenomena. “Tasalis statistics have been used to describe scale-shift invariance, a key component of fractals,” he said.

Can fractal theory extend to gravitational phenomena? “Gravity is outside the scope of our approach, as it does not enter into Yang-Mills theory, but there is nothing to prevent us from speculating whether fractals express a fundamental pattern in all physical reality,” he said.


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more information:
Megias et al., The nonlinear Klein-Gordon equation and Bose-Einstein condensation, European Physical Journal Plus (2022). DOI: 10.1140 / epjp / s13360-022-02511-2

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