# How do quarks determine which quarks to pair with?

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1. Sep 29, 2015

### Ryan Reed

Quarks join up with other quarks to form composite particles like protons and neutrons, but in the center of something like a nucleus, how do they know which quarks are in THEIR proton or neutron? When all the quarks are together and it becomes a "soup" of quarks, why doesn't it form things like delta particles as well as neutrons and protons?

2. Sep 29, 2015

### mathman

I don't know the details, but essentially it is an energy question.

3. Oct 3, 2015

### vanhees71

Indeed, it's an energy question.The quarks and gluons form complicated bound states, the hadrons. The underlying fundamental theory of the strong interaction is quantum chromodynamics, which has the remarkable feature of confinement, i.e., you cannot observe the fundamental quanta, quarks and gluons, in any way as free "particles" or fields like electrons and photons, which are described by QED (or more precisely electroweak theory). It is also very difficult to understand the hadrons as bound states of quarks and gluons, because this is a problem you cannot treat with perturbation theory, and there is not much else than pertubation theory you can use to analytically solve problems of relativistic quantum-field theory. Fortunately there is an alternative numerical approach, "lattice QCD", where you put QCD on the computer in considering it on a space-time grid ("lattice") (technically to be precise, it's not spacetime in the usual sense, but in socalled Euclidean field theory, where you formally make the time coordinate imaginary in the sense of a Wick rotation). Then you can evaluate quantities with help of the socalled Feynman-integral technique in big computer Monte-Carlo calculations. The result is remarkable since with this technique the lattice-QCD physicists were able to reproduce the masses of the known light hadrons, which makes us confident that QCD is the right theory also in the realm, where we cannot use perturbation theory.

Another remarkable feature of QCD is "asymptotic freedom", which is a phenomenon you can address by perturbative methods, i.e., you start with non-interacting quarks and gluons and treat the interaction among them as small perturbations. This can obviously not work at low energies, where we never see free quarks and gluons but only very strongly bound hadronic states. As it turns out, however, when you scatter, e.g., electrons on hadrons (usually protons) at very high energies, you find that they behave as if scattered from pointlike quasi-free constituents, the partons. This indicates not only the compositeness of hadrons, but the scattering cross sections also show scaling laws, which indicate that at high energies the quarks and gluons behave as free particles.

The explanation for this came only with the discovery of QCD: In quantum field theory, when you do the above mentioned perturbation theory (which you can do for QCD when assuming that the coupling constant is very small), you always stumble over integrals that diverge, and this was a great obstacle in making sense of relativistic quantum-field theories of interacting particles, but then physicists like Feynman, Schwinger, and Tomonaga (Nobel prize 1965) and also Dyson, developed a mathematical tool, called perturbation theory. The idea is that of course non-interacting quarks and gluons don't make sense, because then they would not have color charges and no color field around them. So these non-interacting particles you start with to do perturbation theory are fictitious (and by the way couldn't be observed, because they wouldn't interact with anything, and particularly not with any detector to observe them). So you have at each order of perturbation theory (which is a formal expansion in powers of the coupling constant) to "renormalize" some fundamental properties of the theory. Then you can lump the infinities of the perturbative corrections to the mass and charge of the particles (as well as the field normalization) into the unobservable "bare quantities", the non-interacting particles would have and works with the perfectly finite observable quantities of the real particles that are interacting.

The important point is that in this renormalization procedure one necessarily has to introduce some energy-momentum scale, the renormalization scale, where you fix the parameters of the theory. Of course the physics cannot depend on this arbitrary choice, and this leads to a "running" of the parameters of the theory with the choice of the renormalization scale. The theory also tells you, how these quantities change via the socalled renormalization-group equation. Of course, all this only works in practice, in a realm, where the coupling constant is small since only than the perturbation theory is a good approximation. To the bewilderment of Wilczek and Gross as well as Politzer in the early 1970ies, when analyzing the "running" of the coupling in QCD, using this renormalization theory, they found that the coupling constant goes down hat high renormalization scales. First they thought they made a sign mistake in their calculations, because up to then in more simple theories (like $\phi^4$ theory, which is a great toy model to learn perturbative methods in quantum field theory, or quantum electrodynamics) the coupling behaved in the opposite way, i.e., it became larger at higher renormalization scales. Nevertheless, after carefully checking, Gross and Wilczek and independently also Politzer found that it was not a bug but a feature of their calculations: This explains why the strong interaction becomes weak at large renormalization scales, i.e., when probing hadrons at high collision energies with other hadrons or electrons. This is called asymptotic freedom, and the renormalization group methods where also successful in explaining small violations of scaling laws in such deep inelastic scatterings. This is the success of perturbative QCD also adding to the confidence that it is the right way to describe the strong interaction and it earned Gross, Wilczek, and Politzer the Nobel prize in 2004.

Now we can come back to your original question and explain the answer that this is indeed a question of energy. The idea is that you could create an environment, where the strong interaction becomes weak, because the hadrons are scattering with each other at high energies, because then according to asymptotic freedom the interactions should become weak. The only thinkable way is to create a state of hadrons, where they are together at high tempertures and densities, so that they rattle around and collide so violently that the hadrons melt to a soup of quarks and gluons which within this soup (the socalled quark gluon plasma, QGP). Effective theories of hadrons, based on some symmetry principles of QCD (chiral symmetry) indicate that the temperatures to be needed to reach such a state are equivalent to energies of $k_B T \simeq 150-200 \; \mathrm{MeV}$, which means a temperature of some trillion Kelvin ($10^{12} \; \text{K}$)! According to the big-bang model of the evolution of the universe such conditions were present in nature some microseconds after the big bang, but there is not much left to observe some signals from the transition, where the QGP formed hadrons (big-bang nucleosynthesis).

Fortunately, there is a way out of this dilemma! You can scatter big atomic nuclei, with the electrons of the corresponding atoms stripped off (which is why one talks of relativistic heavy ions in this context). As has been found out over more than 40 years of heavy-ion research, indeed one is able to create for a very short moment a hot and dense soup of quarks and gluons. Nowadays this is mostly done at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Lab (BNL) on Long Island and in the heavy-ion program at the Large Hadron Collide (LHC) at CERN in Geneve. The problem here is, however, that the dense and hot "droplets" of QGP matter (if formed at all) rapidly fly appart, becoming diluter and thus cooling down. This means that you can only observe the then again formed hadrons at the very end of this "fireball expansion", the socalled freeze-out state.

So you need theory to describe this process and follow the evolution of the fireball to figure out which observables can indicate that the hadrons measured in the detectors came from a hot and dense QGP state. First of all there is again Lattice QCD, which can also be used to evaluate the properties of hot and dense strongly interacting matter, i.e., doing quantum field theory at finite temperature. Among other things, Lattice QCD can evaluate the equation of state of the QGP, i.e., energy density and pressure as a function of temperature and also figure out the temperature, where the transition from a QCD state of matter and hadronic matter really occurs. The now valid value is a temperature of around 150-160 MeV.

Now one of the most simple observables is just to count how many hadrons of each species occur, i.e., the "chemistry" of the produced hadrons. What comes out is that these particle abundances can be quite well explained by the socalled thermal model, i.e., one assumes that the hadron chemistry is fixed at the socalled chemical freezeout, i.e., when the fireball has expanded and cooled down so much that no more particles are destroyed and recreated, i.e., the socalled inelastic processes do not happen anymore. That's called "chemical freezeout". If now the medium is thermalized, i.e., in equilibrium, you can predict the abundances of the various hadrons. You only need to know temperature (and chemical potentials), and indeed one can fit the particle abundances quite well, and at the highest collision energies the chemical freezeout temperature happens to come out around the confinement-deconfinement transition temperature of around 160 MeV.

Of course, the detectors at RHIC and LHC can measure much more details. Another important measurement are the momentum spectra of each of the different hadrons. An analysis of this momentum spectra leads to the conclusion that the particles come from a collectively expanding hot fireball! I.e., they do not look like the spectra of particles in single collisions (like in pp collisions also done at RHIC and LHC) but like from a hydrodynamically evolving droplet of strongly interacting particles. One observable are the transverse-momentum spectra (i.e., the momentum components perpendicular to the beam direction), which can be explained by a model, where one assumes that the particles come from a locally thermalized medium expanding radially (i.e., perpendicular to the beam axis). This leads to an effective slope of the $p_T$ spectra that is determined by the temperature of the medium and its radial collective flow. Indeed one can simulate this with relativistic hydrodynamics, and it turns out that the spectra are quite well understood by such hydrodynamical models. The corresponding temperature is around 100 MeV, indicating the socalled thermal freezeout, i.e., the moment, where the fireball is so cold and dilute that also the elastic collisions become very rare, and the hadrons are more or less freely streaming towards the detectors after that.

Another indication is the socalled "elliptic flow". The idea behind this observable is that in most of the heavy-ion collisions that nuclei do not collide head on but with a finite impact parameter. The reaction zone is thus not spherical but more like an almond shaped region. If now the initial fireball is shaped like such an ellipsoid and if it really behaves like a collectively flowing fluid, the pressure gradient must be larger along the short axis than along the long axis of this ellipsoid. According to hydrodynamics a larger pressure gradient means that there develops more flow in the corresponding direction than in the other direction with lower pressure gradient. This means that the momentum distribution in the transverse plane cannot be axially symmetric: There should be more flow in the reaction plane than perpendicular to it. What one does is a Fourier analysis of the angular dependence of the $p_T$ spectrum, and indeed one finds that the 2nd Fourier coefficient, called $v_2$ is indeed different from 0 and follows again quite well hydrodynamical simulations of the fireball evolution.

Finally one can also observe jets. These are showers of hadrons at very high energy. One finds that in heavy-ion collisions there are jets that are not counterbalanced in the opposite direction, as in pp collisions. This should be so because of energy-momentum conservation. But what happens in heavy-ion collisions is that a jet that must go through a lot of the hot and dense medium, gets more or less stuck there, i.e., it becomes part of the hot and dense collective fireball rather than coming out as a high-energy particle jet. This socalled jet suppression allows for an estimate of the density (and energy density) of the medium, and this comes out to be pretty large, much larger than the critical densities for the transition between a QGP and hadronic matter, as known again from the thermal Lattice-QCD calculations.

Of course, there are much more observables and underlying models used to figure out this fascinating very exotic state of matter (strangeness production, $J/\psi$ suppression, electromagnetic probes (dileptons and photons),...)

There are some good Wikipedia articles on this fascinating physics:

https://en.wikipedia.org/wiki/Relativistic_Heavy_Ion_Collider
https://en.wikipedia.org/wiki/Quark–gluon_plasma and the links to other Wikipedia articles therein.

4. Oct 3, 2015