Wednesday, April 10, 2013

About Hadrons

This post recaps a few basic facts about Standard Model particle physics that is neither particularly controversial nor at the cutting edge of experimental discoveries to provide context when discoveries are made in the future. This analysis has consulted standard reference sources, my notes from a number of QCD journal articles, and a spreadsheet created for comparison purposes.

Quarks and other fundamental particles

There are six kinds of quarks (up, down, strange, charm, bottom (formerly also know as "beauty") and top), and all of them except the top quark are always observed confined to two quark (meson) or three quark (baryon) composite structures bound by gluons.  The top quark, which is the most massive of all of the fundamental particles with a mass of about 173.3 +/- 1.4 GeV based on the most recent LHC measurements, decays via the nuclear weak force too quickly to form observable hadrons, although in principle a vanishingly small share of top quarks produced might last long enough to allow hadronization since particle lives are probabilities of decay per time period and not certainties.    (The central value of the Tevatron measurement of the top quark mass was 173.2 +/- 0.8  GeV, with a combined estimate of a bit closer to 173.2 GeV than 173.3 GeV with a two sigma confidence interval of about 172.8 GeV to 174 GeV which probably overestimates the true error since independent measurements are so much closer to each other than we would expect if the error bars were as great as they are stated to be).

There are eight basic kinds of gluons, which are defined by the combinations of color charges that they carry, since they are otherwise identical. Gluons have a zero rest mass, but can acquire mass dynamically as they interact with each other and quarks.

Leptons (electrons, muons, tauons, and neutrinos) can interact with particles made of quarks via the exchange of five particles that are associated with the elecromagnetic and weak forces, including the Higgs boson (although they don't interact via the strong nuclear force) but don't form composite particles with them that binds quarks together. The heaviest quark has a rest mass of about 4.2 GeV. Up and down quarks are believed to have rest masses in the single digit MeV (about a thousand times lighter).

Each kind of quark comes in three color charges, has an electric charge of either +/- 1/3 (for up, charm and top quarks and their antiquarks) or +/- 2/3 (for down, strange and bottom quarks and their antiquarks), can have a left or right parity (sometimes called even or odd), and can come in matter or antimatter varieties. A particular quark has a particular rest mass associated with it which is the same for both the particle and its antiparticle (which also have opposite electric charges). Apart from these properties, quarks are entirely identical expect for their current momentum and location (both of which can't be determined at the same time beyoond a certain level of precision as a fundamental principle of physics) and their history of entanglement with other quantum particles.

Quark "color" which is neutral for every confined hadron, like the five lighest quark masses,  is something that is never directly observed. We know what observable results would flow from a different number of color charges, and those predictions are inconsistent with what we see in experiments, but there is no device that exists to directly tell you if a particular quark has a red, green or blue QCD charge.

Most importantly, a three color charge system (with three corresponding anticolors) constrains all hadrons to have integer electromagnetic charges and to have particular combinations of matter and antimatter in baryons and mesons, while forbidding all other combinations.


Only A Finite Number Of Hadrons Are Theoretically Possible

There are roughly one hundred theoretically possible kinds of hadrons and their quantum numbers (charge, spin, etc.), which can be set down from first principles by any graduate student in physics in an afternoon from the basic rules of quantum chromodynamics, although a handful of observed states which are combinations of different electromagnetically neutral hadron states, or are excited states, are not obvious from a mere rudimentary understanding of the laws of quantum chromodynamics.  With the more sophisticated nuances like excited states and a high but not utterly unlimited bound on energy levels, maybe you can get to twice that number.

In many practicle applications, an approximation of reality that ignores the masses of the lightest three kinds of quarks, and the existence of some or all of the heaviest three kinds of quarks, is adequate to provide results that are as accurate as can be calculated because the ephemeral particles made of heavier quarks are hard or impossible to form due to matter-energy conservation, and often have only a minor impact on lower energy physical systems. These models exclude the vast majority of these exotic hadrons.

Several other kinds of composite quark and gluon particles are not obviously forbidden by the Standard Model, but have not been observed and definitively identified. These include quarkless "glueballs" made entirely of gluons, tetraquarks and pentaquarks.

Potential tetraquark resonnances seen to date have turned out to be, in fact, "molecules" of discrete mesons rather than single coherent four quark composite particles. Numerous theoretical papers have described the properties that glueballs ought to have, but in the absence of experimental evidence (which is hard to amass since glueballs would be similar in many ways to hadrons with neutral electrical charge), we can't be certain that some law of nature not currently known to us forbids their formation or makes them so absurdly rare that we will never see one.

Mean Hadron and Fundamental Particle Lifetimes

The only hadrons that are stable are the proton and the bound neutron. The proton is stable with a mean lifetime at least as long as the age of the universe and a neutron which is not stable when not confined within an atom of a stable isotype, has a mean lifetime of about 886 seconds (about fourteen minutes and 46 seconds). The runner up, the charged pion, has a mean lifetime of about 2.6*10^-8 seconds, followed closely by the neutral kaon with a mean lifetime of about 1.2*10^-8 seconds, followed by others with mean lifetimes hundreds to trillions of times shorter. Protons, neutrons and pions are comprised only of up quarks, down quarks and their antiparticles. Kaons also incorporate strange quarks. The longest lived hadrons that contain charm or bottom quaraks have mean lifetimes on the order of 10^-12 seconds (one 1000 billionth of a second).

By way of comparison, electrons are stable, second generation heavy electrons which are called muons have a mean lifetime on the order of 10^-6 seconds, and third generation even heavier electrons called tauons have a mean lifetime on the order of 10^-13 seconds. The massive bosons of the electroweak force (the W+, W-, and Z bosons and the Higgs boson) ae likewise ephemeral, as are solitary top quarks which essentially always decay before they can form hadrons.

Hadron Volume

While quarks and other fundamental particles in the Standard Model are conceived of as being point-like, hadrons have a radius (that can be defined in several ways) on the order of 0.8.5*10^-15 meters (one femtometer), the experimentally measured size of a proton, which is small, but is trillions of times longer than the hypothetical minimum Planck length favored in many quantum gravity theories.

This scale is set largely by the form of the equations of the strong nuclear force.  At very small distances relative to this distance it is repulsive.  At longer distances it grows incredibly strong.  In between, the quarks bound by it are "asymptotically free".

While exotic hadron volume is rarely directly measured, it can be expected to vary to a similar extent to the strong force energy field binding energy of hadrons, which is pretty much all within an order of magnitude.

Electron orbits around atomic nuclei are much more tight than the gravitationally bound orbits of objects in our solar system which are often used as an analogy to them.  But, like our solar system, the vast majority of atoms and matter made of atoms and molecules is empty space.

Hadron Masses

The lighest of the hadrons (the "neutral pion") has a mass of about 0.1349766 (6) GeV. Both the proton and neutron have masses that are almost, but not quite identical, of about 0.93827013 (23) GeV for the proton and 0.939565346 (23) Gev for the neutron (about one part in four million). Approximately thirty meson masses and forty-two baryon masses have been measured to date. Several dozen more possible combinations are theroretically possible but belong to the mountain of experimental data for which some basic properties and approximate mass resonnances have been observed, but which have not been succeptible to a definitive identification with a particular predicted composite particle of QCD.

The heaviest observed three quark particle (i.e. baryon) whose mass has been precisely measured called a "bottom omega", made of two strange quarks and a bottom quark bound by gluons has a mass of about 6.165 (23) GeV (and is the least precisely measured mass of the lot at an accuracy of about a third of a percent precision). The heaviest observed two quark particle (i.e. meson) whose mass has been precisely measured is called an upsilon, is made of a bottom quark and an anti-bottom quark bound by gluons has a mass of about 9.46030 (26) GeV. The heaviest theoretically possible meson or baryon (that does not have a top quark as a component), which has not yet been observed, called the triple bottom omega baryon should have a mass of about 15 GeV.

The heaviest theoretically possible hadron is about 100 times as heavy as the lighest possible hadron, a much narrower range of masses than the range of masses for the fundamental particles of the Standard Model (which range over about 21 orders of magnitude from the top quark to the lighest neutrino), or even the quarks themselves (which have a range of masses of about 100,000 to 1). The range of hadron masses is bounded in part because the heaviest quark, the top quark, does not form hadrons. The range of rest masses of the five quarks that form hadrons is about 3,000 to one.

Equally or more importantly, this is because the color charge interactions of any kinds of quarks in two and three quark particles respectively, are almost (but not exactly) the same. The amount of strong nuclear force field energy necessary to bind an exotic spin-3/2 baryon made of the heaviest quarks is only about 30% greater than the amount of energy necessary to bind an ordinary proton or neutron and is basically the same for all spin-3/2 baryons (the ones with the most binding energy have only about 3% more binding energy than the ones with the least binding energy about ten to one hundred times more than the uncertainty in the experimental measurements themselves).

There is more variation in the amount of strong nuclear force field energy that binds together spin-1/2 baryons, but none have a binding energy that is more than about 40% greater than that of an ordinary proton or neutron (which have the least).

Moreover, this range is greatly inflated by a handful of the heaviest and most rare varieties of hadrons.

The stability of the amount of hadron mass attributable to this binding energy matters a great deal because in a proton or neutron, the sum of the three fundamental up and down quark rest masses is equal to roughly 1% of the total mass of nucleon. In contrast, in a bottom omega baryon, sum of the rest masses of the constituent quarks is about 71% of the whole particle's mass, and in the heaviest experimentally measured hadron, the upsilon, the sum of the masses of the constituent quarks is about 89% of the whole particle's mass.  In the heaviest theoretically possible hadrons, the ratio of fundamental particle mass to hadron mass would be even greater.

Of course, hadrons in turn bind themselves into one of about 120 different kinds of atoms, in a wide variety of isotypes, i.e. numbers of neutrons in an atom of N protons (only a small portion of which are stable), whose nuclei are made entirely of protons and neutrons.

Hadron Density

Atomic nuclei, in general, have approximately the same density of neutron stars, which are the most dense known objects in the universe outside of black holes.  Indeed, large black holes have less mass per volume defined by their event horizons than neutron stars do.  Black holes have a declining mass per event horizon volume has they acquire more mass.  Atomic nuclei are significantly below the density needed to form a black hole at their scale according to the General Relativity, although it isn't obvious that General Relativity applies at such small scales without modification since it is a clasical rather than a quantum theory that is applied to a quantum scale in this context.

It is possible that higher generation quarks like strange quarks or hadrons made of them may be stable in extreme circumstances like extremely dense neutron stars (or perhaps utterly beyond our observation inside black holes), but there is no solid evidence that such quark stars actually exist.

Theory Lags Behind Experiment In Hadron Physics

First principles theoretical calculations of proton and neutron masses are accurate to about 1% in absolute terms, although more precise theoretical predictions can be made for heavier exotic hadrons and it is also possible to calculate from first principles to an order of magnitude accuracy the much smaller mass difference between the proton and neutron massses, even though this is only about 0.1% of the mass of the proton. Even the more precise theoretical determination of the difference of the two masses is about 4,000 times less precise than experimental measurements of this mass difference.

Quantum chromodynamics is the virtually unchallenged contender for this part of the Standard Model mostly because none of its theoretical predictions have been contradicted and because no one else has come up with any really credible alternatives that make more precise predictions. The only particles it predicts that we haven't seen are particles that are hard to observe and classify which we may actually have seen already. Every particle that we have been able to observe carefully enough to classify has been succeptible to being fit into QCD's built in taxonomy with a modicum of ingenuity.

An important reason for this difficulty in making accurate calculations is that it is quite difficult to determine the quark masses precisely from the roughly seventy-two available hadron data points, and some less direct data points (like measurements of the strong force coupling constant which are accurate to at least about four significant digits), and the known values of the Standard Model coupling constants. The ligher the quark, the less precisely its mass is known, because mass attributable to gluon interactions so profoundly overwhelms the fundamental quark masses. Turning those data points into theoretical constant values involves great computational difficulties, so mostly physicists resort to methods only moderately more sophisticated than a basic spreadsheet comparing the seventy-two data points with the known composition of the particles in question (determined based upon their other properties like charge and spin).

These percentages are a bit slippery because fundamental particle masses "run" with the energy level of the circumstances where they are observed so any single mass value for them necessarily includes contextual assumptions about the measurement that may be inconsistent with the context in which the hadron that contains it is observed. Often the 2 GeV mass level of about two protons at rest, is used as a standard. Likewise, the conventional view that the additional mass associated with quarks within hadrons is localized in the dynamically generated gluon masses within the confined quark system is to a great extent a model dependent feature and one could imagine a coherent model in which that additional mass was apportioned to the constituent quarks which is hard (although not necessarily impossible) to distinguish experimentally.

In a calculation like the calculation of the proton-neutron mass difference, uncertainty regarding the values of the fundamental constants gives rise to something on the order of two-thirds of the uncertainty in the theoretical prediction, while about a third of the theoreticallly predicted value or so is due to truncation of the infinitely long series of equation terms that the QCD equations tell us give the exact value which can be approximately solved numerically but not calculated precisely in all but the most simple cases.

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