Using some of the world’s most powerful supercomputers, a group of theorists has produced a major advance in nuclear physics, a calculation of the “diffusion coefficient of heavy quarks.” This number describes how quickly a molten soup of quarks and gluons, the building blocks of protons and neutrons, which are set free in collisions of nuclei in powerful particle accelerators, transfers its momentum to the heavy quarks.

The answer, it turns out, is very fast. As described in an article just published in *Physical Review Letters*, the transfer of momentum from the “liberated” quarks and gluons to the heavier quarks occurs at the limit of what quantum mechanics will allow. These quarks and gluons have so many short-range strong interactions with the heavier quarks that they drag the “boulder”-like particles along with their flow.

The work was led by Peter Petreczky and Swagato Mukherjee of the nuclear theory group at the US Department of Energy’s Brookhaven National Laboratory and included theorists from the universities of Bielefeld, Regensburg and Darmstadt in Germany and the University of Stavanger in Norway .

The calculation will help explain experimental results showing heavy quarks being captured in the flux of matter generated in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) in Brookhaven and at the Large Hadron Collider (LHC) at CERN’s European laboratory . The new analysis also adds corroborating evidence that this matter, known as “quark-gluon plasma” (QGP), is a near-perfect liquid, with a viscosity so low it even approaches the quantum limit.

“To see the heavy quarks flowing with the QGP at the RHIC and LHC was very surprising initially,” Petreczky said. “It would be like seeing a heavy stone dragged with water in a stream. Usually the water flows but the stone remains”.

The new calculation reveals why that startling image makes sense when you think about QGP’s extremely low viscosity.

## Frictionless flow

The low viscosity of matter generated in RHIC’s gold ion collisions, first reported in 2005, was a major reason for the new calculation, Petreczky said. When these collisions melt away the boundaries of individual protons and neutrons to free internal quarks and gluons, the fact that the resulting QGP flows with virtually no resistance is evidence that there are many strong interactions between the quarks and gluons in the hot quark soup.

“The low viscosity implies that the ‘mean free path’ between the ‘molten’ quarks and gluons in the hot dense QGP is extremely small,” said Mukherjee, explaining that the mean free path is the distance a particle can travel before interact with another particle

“If you think about trying to walk in a crowd, that’s the typical distance you can walk before you bump into someone or have to change course,” he said.

With a short mean free path, quarks and gluons interact frequently and strongly. The collisions dissipate and distribute the energy of the fast-moving particles, and the strongly interacting QGP exhibits collective behavior including nearly frictionless flow.

“It’s much more difficult to change the momentum of a heavy quark because it’s like a train to stop,” noted Mukherjee. “It would have to go through a lot of collisions to be pulled along with the plasma.”

But if the QGP is indeed a perfect fluid, the mean free path for the heavy quark interactions should be short enough to make this possible. Calculating the diffusion coefficient of heavy quarks, which is proportional to how strongly heavy quarks interact with plasma, was one way to test this understanding.

## Crunching the numbers

The calculations needed to solve the equations of quantum chromodynamics (QCD), the theory that describes the interactions between quarks and gluons, are mathematically complex. Several advances in theory and powerful supercomputers helped pave the way for the new calculus.

“In 2010/11 we started using a mathematical shortcut, which assumed that plasma consisted only of gluons, no quarks,” said Olaf Kaczmarek of the University of Bielefeld, who led the German side of this effort. Thinking only of gluons helped the team come up with their own method using lattice QCD. In this method, scientists run simulations of particle interactions on a discretized four-dimensional space-time lattice.

In essence, they “place” particles at discrete locations on an imaginary 3D grid to model their interactions with nearby particles and see how those interactions change over time (the 4th dimension). They use many different starting arrangements and include varying distances between particles.

After working out the method with just gluons, they figured out how to add the complexity of the quarks.

The scientists loaded a large number of sample configurations of quarks and gluons onto the 4D lattice and used Monte Carlo methods, repeated random sampling, to try to find the most likely distribution of quarks and gluons within the lattice.

“By averaging over these configurations, you get a correlation function related to the diffusion coefficient of the heavy quarks,” said Luis Altenkort, a graduate student at the University of Bielefeld who also worked on this research at the Brookhaven Lab.

As an analogy, think of estimating the air pressure in a room by sampling the positions and motion of molecules. “You try to use the most likely distributions of molecules based on another variable, like temperature, and rule out unlikely configurations like all the air molecules clustered in one corner of the room,” Altenkort said.

In the case of QGP, the scientists were trying to simulate a thermalized system in which even on a tiny time scale of split-second collisions of heavy ion particles, the quarks and gluons reach a certain equilibrium temperature.

They simulated the QGP at a range of fixed temperatures and calculated the heavy quark diffusion coefficient for each temperature to map the temperature dependence of the heavy quark interaction strength (and the mean free path of those interactions).

“These challenging calculations were only possible using some of the world’s most powerful supercomputers,” said Kaczmarek.

Computing resources included Perlmutter at the National Energy Research for Scientific Computing Center (NERSC), a DOE Office of Science user facility located at the Lawrence Berkeley National Laboratory; Juwels Booster at the Juelich Research Center in Germany; Marconi at CINECA in Italy; and dedicated lattice QCD GPU clusters at the Thomas Jefferson National Accelerator Facility (Jefferson Lab) and at the University of Bielefeld.

As Mukherjee noted, “These powerful machines don’t just do the work for us as we sit back and relax; it took years of hard work to develop the codes that can squeeze the most efficient performance out of these supercomputers to do our complex calculations.” .”

## Fast thermalization, short-range interactions

Calculations show that the heavy quark’s diffusion coefficient is highest at the very temperature at which QGP forms, and therefore decreases with increasing temperature. This result implies that the QGP reaches an equilibrium very quickly.

“You start with two nuclei, essentially having no temperature, then you crash them together and in less than a quadrillionth of a second you have a thermal system,” Petreczky said. Even heavy quarks are thermalized.

For this to happen, the heavy quarks must undergo a lot of scattering with other particles very quickly, implying that the mean free path of these interactions must be very small. Indeed, calculations show that, at the transition to QGP, the mean free path of the heavy quark interactions is very close to the shortest allowed distance. That so-called quantum limit is set by the inherent uncertainty of knowing both the position and momentum of a particle simultaneously.

This independent ‘measurement’ provides corroborating evidence of QGP’s low viscosity, confirming the picture of its perfect fluidity, the scientists say.

“The shorter the mean free path, the lower the viscosity and the faster the thermalization,” Petreczky said.

## Real collision simulation

Now that scientists know how heavy quark interactions with the QGP vary with temperature, they can use this information to improve their understanding of how actual heavy ion collision systems evolve.

“My colleagues are trying to develop more accurate simulations of how QGP interactions affect the motion of heavy quarks,” Petreczky said. “To do this, they have to account for the dynamic effects of how the QGP expands and cools throughout the complicated stages of collisions.”

“Now that we know how the diffusion coefficient of heavy quarks changes with temperature, they can take this parameter and plug it into their simulations of this complicated process and see what else needs to change to make those simulations compatible with the experimental data from RHIC and the LHC.”

This effort is the subject of a major collaboration known as Heavy-Flavor Theory (HEFTY) for the QCD Matter Topical Theory Collaboration.

“We will be able to better model the motion of heavy quarks in the QGP, and thus have a better theory for comparing the data,” Petreczky said.

**More information:**

Luis Altenkort et al, Heavy quark diffusion from 2+1 flavor lattice QCD with 320 MeV pion mass, *Physical Review Letters* (2023). DOI: 10.1103/PhysRevLett.130.231902

**About the magazine:**

Physical Review Letters

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