The Standard Model of Particle Physics

Quantum Chromo Dynamics (QCD), the theory of strong interactions, forms a cornerstone of our understanding of the matter surrounding us. It is a theory in the general class of theories called Relativistic Quantum Field Theories with local gauge freedom. In this class of theories, one describes the physical systems as composed of matter (usually fermionic) degrees of freedom interacting via the medium of gauge fields which are bosonic. An important aspect of this theory is the gauge freedom because of which the simplest and explicitly local description of the theory involves the introduction of spurious degrees of freedom. In other words, the entire gauge field is not really an observable, but simplistic attempts at eliminating the unphysical degrees break the manifest locality of the formulation.

Gauge theories are usually specified by giving their `matter content' (i.e. the nature of all the fields other than the gauge fields) and specifying the `gauge group', a mathematical entity that condenses the interactions. (Such a description may not be unique: the same theory might be described in multiple ways. The important point is that a description like this specifies the theory almost completely.) The earliest such theory to be investigated at great depth was Quantum Electro Dynamics (QED), which describes the electromagnetic interaction between the familiar charged particles. In fact, it was the first great success of Relativistic Quantum Field Theories, theories that were born out of an attempt at a consistent merger of Einstein's Special Theory of Relativity and Quantum Mechanics. Even though QED alone can be shown to be almost surely incapable of defining a consistent and complete theory of interactions a property, called `perturbative renormalizability', of this theory guarantees that this incompleteness has little effect at moderate energies (which for the parameters describing QED encompasses all energies at which we expect such field theories to be meaningful), provided the coupling is weak. In fact, the weakness of the electromagnetic interactions, exemplfied by the binding energy of positronium being six orders of magnitude smaller than its mass, allows us to obtain predictions with an astonishingly high precision by the use of analytic techniques like semi-classical analysis and perturbation theory.

A second gauge theory, Quantum Flavour Dynamics (QFD), much more complicated than the first, explains the so-called weak interactions of nuclear matter. The intrinsic strength of these interactions is similar to those of the electromagnetic interactions at distances important in hadronic physics, and hence this theory is amenable to the same kinds of treatment which was so successful for QED. The reason this theory is quite irrelevant to every day physics, except to explain our very existence(!), is a peculiarity of the bosonic part of the matter sector that goes with this theory. This sector, the so-called Higgs' sector, is the least understood part of this theory, but manages to neutralize weak charges: so that the only time these interactions are really visible is when one probes distance scales smaller than the intrinsic scale at which this neutralization takes place: about 10-17 meters or 10-25 seconds. Because of the fuzziness of quantum mechanics, it does have weak observable effects even at long distances; thus, for example it is the ultimate cause of free neutrons decaying to protons. As opposed to QED, and barring our lack of knowledge of the Higgs' sector, QFD is believed to be mathematically consistent as a fundamental theory, though a proof is lacking. Nature, of course, shows us that it is not complete, other interactions do exist.

The third major component of the standard model of particle physics is called Quantum Chromo Dynamics (QCD). These `strong interactions' are an important part of hadronic physics, so much so that one can, to the first approximation, describe nuclear matter like neutrons and protons to be entirely described by QCD. Unfortunately, this interaction is extremely strong at typical nuclear distances, and QCD cannot be dealt, even at an intuitive level, in manners similar to QFD and QED. To provide just one example, the charged (talking, of course, of QCD charge and not electromagnetic charge) matter sector of this theory interacts so strongly that an isolated charged particle is believed to be a theoretical impossibility. Only at extremely high energies and small distances (because the interaction energy rises sufficiently slowly as the distance is decreased) can we manage to think of this theory in the same way as QED. Thus, below about 10-16 meters, we can think of quarks whitting past and radiating gluons, but these particles, if by abuse of terminology we call them so, never make it out past their little bags of sizes about 10-15 meters, the interaction with their neighbours hold them fast.

The last interaction, about which we know very little, is gravitation. Except at incredibly short distances (10-34 meters) or times (10-44 seconds), or correspondingly high energies, this force is extremely weak by comparison to the forces I described above, and is usually neglected in particle physics. This force becomes important at astronomical scales because the stronger electromagnetic (and strong) interactions forces positive and negative charges to come together and roughly neutralize each other. The `gravitational charges', or the quantity that measures the source of gravitational forces, are all of the same sign (barring some little understood quantum mechanical effects), so they cannot be neutralized and persist at long distances. At all scales relevant to current experiments (a few centimeters to the scale of the entire universe) gravity is well described by Einstein's General Theory of Relativity (to which Newtonian gravity is a very good approximation at laboratory scales), but, at short distances and, possibly, at cosmologically large times, the theory is very poorly understood at the moment. They are not gauge theories in the standard meaning of the term and the interaction energy rises so fast as we bring things closer, that by about 10-34 meters the gravitational forces could dominate even in the presence of uncancelled QCD charges. As those distance scales are very far beyond our experimental reach, they worry us only for consistency reasons.

The standard model of particle physics comprises the above components, and so far, nobody has found any inconsistency between what is calculable in the theory and what is experimentally measurable. Its field content is pretty clear: everything can be though of as composed of leptons (four possibly stable particles: electron and three kinds of neutrinos; muon and the tau), quarks (six flavours: up, down, charm, strage, top and bottom; each in three varieties commonly called `colors'), gauge bosons (photon which is stable, the unstable W+, W- and Z, and the eight `colors' of gluons), and the higgs boson in the simplest model. (To avoid confusion, I ought to mention that the concept of `particle' is not very well defined in quantum mechanics: `unstable particle' is already a difficult concept, and quarks and gluons are even less like the other `particles'. If we concentrate on possibly absolutely stable particles, in addition to the ones mentioned above, there are two more: the proton and its antiparticle; which can be thought of as being `composed of' the above.) Each of these `particles' (except the gauge bosons, whose antiparticles are also gauge bosons; and the Higgs boson, which may be its own antiparticle) has a distinct antiparticle.

However, the standard model is not a very neat theory: there are problems with consistency mentioned above, and, even apart from gravity there are a large number of unexplained `basic' parameters in the theory. QED has one parameter that gives the strength of the electromagnetic interactions (α), and all electromagnetic charges seem to come in multiples of a basic unit (e), a fact that is left unexplained by the theory. QFD brings in another parameter (θW, controlling the ratio of the QFD and QED interaction strengths), and the higgs sector is responsible for at least two more (in the simplest model of the Higgs sector. These are MW, the mass of the W gauge boson induced by the Higgs sector, and MH, the mass of the Higgs particle. Like QED, this simplest model may be inconsistent as a part of a fundamental theory). The interaction between the Higgs and the matter sector give rise to the masses of the particles: and are described by at least thirteen more parameters (the masses of the three massive leptons: electron, muon and tau and the six kinds of quarks: up, down, charm, strange, top and bottom; three parameters describing how the interactions can change one form of quark into another; and a fourth describing how time asymmetric the interactions are). QCD itself contributes at least one more parameter: αs describing its strength. If the three kinds of neutrinos have mass, as now seems likely, then the standard model needs parameters to describe these as well four more parameters to boot to describe the mixings and time asymmetry, just as in the quark sector. In addition, QCD (and QFD), in principle, have the oppotunity to be fundamentally time asymmetric: the standard model has no explanation why experiments seem to be consistent with absence of such asymmetry.


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Tanmoy Bhattacharya [Jan 16, 2000]

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