The heart of matter

Print edition : November 05, 2004

THE 2004 Nobel Prize in Physics has been jointly awarded to three United States physicists David J. Gross of the University of California at Santa Barbara, H. David Politzer of California Institute of Technology (Caltech) and Frank Wilczek of Massachusetts Institute of Technology (MIT) for the "discovery of asymptotic freedom in the theory of strong interaction" that they made in 1973. This discovery has enabled a consistent picture of matter in terms of fundamental constituents called quarks and the force of interaction among them. However, quarks have never been seen in isolation and the discovery has helped to understand this paradox. To appreciate this, in particular the italicised phrase in the citation, means, one must have an idea of how physicists today describe matter and the fundamental forces of nature. Matter at the sub-atomic scale is made up of positively charged protons, electrically neutral neutrons and negatively charged electrons, These were detected as individual particles during the late 19th century and early part of the 20th century. Protons and neutrons make up the positively charged nucleus. Around the nucleus are electrons that are bound to the nuclei by the force of electromagnetism forming electrically neutral atoms. Aggregates of atoms form matter in the macro-scale that we see around us.

David Politzer.-AP

It is also known that there are four fundamental forces of nature that govern the behaviour of matter. Two of these manifest themselves in the macro-scale, which we experience in our daily life: gravity - that causes matter in the large to attract one another, keeps us rooted to the earth and governs the motion of planets, stars and galaxies; electromagnetism - which makes charged bodies exert force on one another and produces electricity. They manifest themselves in the large because these are mediated over an infinite range and are hence called long-range interactions. Both these forces vary inversely as the square of the distance between objects.

Gravity may appear to be strong considering, for example, the energy it takes to launch rockets into space countering earth's attraction. But it is actually a very weak force. Between two electrons, for instance, the gravitational force is about 1040 times weaker than the electromagnetic force. The other two forces are short-range interactions and, therefore, manifest themselves only in the micro-scale. These are the `weak' nuclear force; which causes radioactivity and the Sun to shine, and the `strong' nuclear force, which binds neutrons and protons (collectively called nucleons) into nuclei.

The discovery of many "elementary" particles beyond the neutron, the proton and the electron during the last century called for a radical thinking about the nature of matter. To make sense of the proliferation of particles, in terms of a fewer number of fundamental constituents, it was postulated that the strongly interacting nuclear particles such as neutron and proton were made up of what were called `quarks'. A consistent picture became possible if one assumed that the particles were made up of three kinds of quarks and their anti-matter counterparts, the anti-quarks. Further, in order that the value of the electrical charge of the composite particle like the proton or the `pion' was an integer, quarks had to be fractionally charged, like +2/3 and -1/3.


But, despite intense searches in accelerator experiments, where particles have been bombarded with increasing energies, a proton or a neutron or any of the other scores of particles that these machines produced, was never found to break up to reveal the quarks. Experimental data from various accelerators around the world in the 1970s were, however, only increasingly pointing to the consistency of the emerging picture in terms of quarks. The fact that these quarks could not be observed led to a paradoxical situation that physicists found hard to explain. Thus arose the concept of `quark confinement' - that is, quarks were somehow permanently bound in some `bag'-like structure (of about 10-13 cm) to yield strongly interacting composite particles. At this stage, it was no more than an unconvincing hypothesis. Matter is thus understood to be essentially composed of two kinds of particles: hadrons (that included the strongly interacting particles like the nucleons) and leptons (like the electron and the muon and their chargeless and massless partners, the neutrinos). All charged particles have electromagnetic interaction. In addition, while hadrons interact both via the weak and strong nuclear forces, leptons interact only through the weak nuclear force. In the quark picture, the hadrons are composite and made of the more fundamental quarks. However, quarks cannot be seen but experiments provide indirect evidence to this internal sub-structure of hadrons. Leptons, on the other hand, do not seem to have any further intrinsic structure and can be regarded as fundamental point particles.

A quantum mechanical description of particle interactions requires, in addition, the existence of force carriers or particles that are exchanged to give rise to a given force between interacting particles. The electromagnetic force was well described in terms of the exchange of light quanta, the massless and chargeless photons. Masslessness of photons immediately explained its long-range nature as well. It is important to keep in mind that since a photon is chargeless, it cannot interact with itself. The analogous massless force carrier for gravity (called graviton) is yet to be discovered, perhaps because of the very weak nature of the force.

Through the 1940s and 1950s, a self-consistent framework to describe electromagnetic interactions at the micro-level, consistent with special relativity and quantum mechanics was developed. It was called Quantum Electro Dynamics (QED). In the quantum mechanical description, the photon, besides its zero charge and mass, has an intrinsic angular momentum, or `spin', of 1 unit. That is, photon is a spin-1 particle (a `vector boson') and QED is a theory of a non-self-interacting spin-1 particle. In technical parlance, it is an `abelian gauge theory' and the word `gauge' arises from a certain intrinsic mathematical symmetry in the equations of QED. The equations remain invariant under a prescribed change in the `measure' or `gauge' of the photon amplitude.

QED is one of the most successful theories of physics. It agrees with results of experiments to a precision of nearly one part in a billion. One of the chief reasons why QED is so successful is the smallness of the electromagnetic coupling constant, a measure of the strength of interaction. Its value 1/137 is much smaller than 1. This enables calculation of all electromagnetic effects to be done as a series of successive approximations, with successive terms being proportional to increasing powers of the coupling constant. Since the constant is less than one, successive terms are smaller and smaller and one arrives at the correct value fairly quickly in the series. This method is called `perturbation calculation', which will fail if the constant were of the order of 1 or larger.

Following QED's remarkable success, self-consistent theoretical description of the other disparate forces of nature in a similar framework was being attempted. But there was an essential difference. Since both strong and weak nuclear forces were short-range forces, the carriers had to be massive. Furthermore, since the coupling constant for the strong nuclear force was of the order of 1 as compared to the electromagnetic force (1/137) and the weak force (10-14), a framework in which perturbation calculations for strong interactions could be meaningfully performed seemed remote.

As regards the weak nuclear force, though the coupling constant was small, it was not clear how a self-consistent theory with massive spin-1 particles (required by the nature of the force) could be built. In QED, the inherent gauge symmetry of the interaction equations, which arose from the masslessness of the photon, had been exploited to yield a self-consistent formalism. For a theory of massive spin-1 particles - the weak interaction carriers - this symmetry principle of `gauge invariance', was absent.


If one were to construct a theory of the weak force with gauge symmetry, one was faced with twin problems. One, it had to be a non-abelian gauge theory; that is, unlike the photon, the force carriers would interact with themselves. Two, to give mass to these particles that the short range required. The prototype of such a gauge theory is the Yang-Mills theory, named after C.N. Yang and R.L. Mills who developed it in 1954. It is basically a theory of matter with self-interacting massless spin-1 particles as force carriers. Even in this massless version, the non-abelian or the self-interacting nature of the force carriers made the proof of self-consistency complex.

The 1970s witnessed a double breakthrough to the problem. It essentially hinged on a concept (borrowed from solid state physics) of `spontaneous break down of the gauge symmetry'. That is, while there is an inherent symmetry in the equations of interaction, this symmetry is hidden in the observed world where the symmetry appears broken. This approach culminated in the Nobel Prize-winning work of Abdus Salam, Steven Weinberg and Sheldon Glashow, which achieved the unification of electromagnetism and weak force in a single mathematical framework. The hidden gauge symmetry could still be mathematically exploited to yield a self-consistent theory, analogous to QED. The proof of self-consistency was the Nobel Prize-winning work of Martinus Veltman and Gerardus 'tHooft.

In this unified electro-weak theory, there are six quarks (and their anti-quarks) classified into three families and six leptons (and anti-leptons) classified into three families. The theory also has four force carriers, one of which is the massless photon. The breaking of symmetry can be suitably designed to give appropriate masses to both matter and force carriers. The disparate nature of electromagnetic and weak forces, which otherwise have a common origin, is the manifestation of the symmetry breakdown in the observable world.

The massive carriers of the weak nuclear force, the charged W+- and the neutral Z particles, were subsequently discovered at the European Centre for Particle Physics, CERN, in Geneva. However, in this theory, mass is purchased at the cost of postulating a particle (called Higgs) to mediate the symmetry breakdown. The Higgs particle is yet to be discovered and physicists hope that it will show up in the next generation accelerator experiments.

Given the success of the electro-weak unification as a non-abelian theory, physicists hoped that if a consistent non-abelian theory of strong interactions could be constructed, a `grand unification' scheme, which will unite the strong force as well, could be achieved. The quark picture seemed to immediately lend itself to such a non-abelian structure.

In building hadrons out of quarks, consistency with quantum mechanics forces one to ascribe a new quantum attribute (like the electrical charge) to the quarks. Physicists call this attribute `colour' (though this is just whimsical nomenclature and has nothing to do with actual colour). Quarks can be in one of the three coloured states: red, blue or green. For every quark, there is also an anti-quark with opposite (fractional) electric charge and opposite `colour charge'. Aggregates of quarks (hadrons), which are observed, are, however, colour neutral. The three quarks in proton, for example, will have different colour charges so that the total colour charge is neutral.

The force between quarks (the colour force) is mediated by eight massless spin-1 particles, analogous to the photon, called gluons. However, unlike the photon, which is electrically neutral, gluons carry colour charge. This implies that gluons must interact among themselves; that is, it has to be a non-abelian theory.

IN the late 1960s and early 1970s, a series of experiments called `deep inelastic scattering' were carried out the Stanford Linear Accelerator Centre (SLAC) and CERN. These involved scattering electrons and neutrinos off protons by imparting a large energy kick to the proton. High energy is equivalent to probing at very small distances and the energies achieved in these experiments were probing distances less than 10-13 cm, which meant one was probing inside the proton. The results revealed an unexpected and surprising phenomenon known as `scaling'.

Contrary to expectations, where the results of electron-proton scattering would depend on the energy transferred to the proton, here they seemed to depend on a certain ratio of two energy variables.

Richard Feynman, the great theoretical physicist and one of the architects of QED, demonstrated that one could explain this result if one pictured proton to consist of a sea of non-interacting particles. That is, the constituents of proton (quarks or whatever) seemed to be free inside. How does one reconcile this observation with the basic strong nature of the interaction?

Further, by providing a handle on momentum distribution of the constituents of the proton, these scattering experiments also provided an indirect evidence for gluons, which seemed to carry nearly half of proton's momentum. So, scaling - equivalently, existence of free constituent particles inside the proton - was sought to be explained in the quark-gluon framework.

In QED, quantum theory tells us that the electromagnetic coupling constant varies with energy; it increases with energy (short distances). That is, the closer one is to the electron, the higher will be the strength of interaction. In experiments with current accelerators, at 100 giga electron volt (GeV) energy, its value has been measured to be 1/128 rather than 1/137. As a function of energy, therefore, the curve for the electromagnetic `running coupling constant' slopes upwards. In physicists' parlance "beta function for QED is positive". This is also intuitively clear from the quantum picture of an electron as a bare electron "dressed" with a cloud of virtual electron-positron pairs. These virtual particles "screen" the bare charge so that the effective charge is smaller at large distances (or low energies). At higher energies one is probing closer to the bare electron through the cloud so that the screening effect is less and the effective charge higher.

If the beta function were positive in a theory of strong interactions as well, the impossibility of doing meaningful calculations at low energies would only worsen at higher energies. The late German theoretical physicist Kurt Symanzik had realised that the correct strong interaction theory should have a negative beta function. This would not only allow calculations to be performed but also would immediately explain scaling and why quarks appeared as free particles in deep inelastic scattering experiments. But a negative beta function is counter-intuitive, as it would correspond to an anti-screening effect.

Illuminated Large Hadron Collider of the European Centre for Particle Physics, CERN, in Geneva.-AP

ENTER Gross, Wilczek and Politzer, this year's Nobel Prize winners essentially proved, after heroic calculations, that this counter-intuitive picture is indeed true for the strong colour force between quarks and gluons. That is, with increasing energy, strong interaction's `running coupling constant' decreases in value. In other words, at very small distances, the strong colour force between quarks, mediated by spin-1 gluons, decreases. (In retrospect, the anti-screening effect is akin to paramagnetism - what one obtains in normal magnetic materials like iron - and has been shown to arise essentially due to the spin-1 nature of the gluons.)

Thus, in the asymptotic limit of infinite energy, the force between quarks is zero. That is, quarks are essentially free when they are very close to each other and we have "asymptotic freedom". As quarks move apart, the force increases rapidly much like a stretched rubber band and to liberate a quark from a nucleus one would need infinite energy. When a particle beam imparts very high energy to the quark within the proton, in a bid to tear it away, the quarks within the proton separate for a while but this eventually results in the energy being converted into the creation of a quark-anti-quark pair. These would combine with the separating quarks resulting in two quark pairs (two hadrons) instead of any free quark. (This is similar to trying to saw a bar magnet to isolate magnetic poles. Since isolated magnetic poles cannot exist, this results only in creating two new bar magnets from one.) That is why quarks (and gluons) remain confined within hadrons and we do not see free quarks.

Thus a non-abelian quark-gluon gauge theory achieves both confinement and scaling, the two critical observations that a theory of strong interaction had to explain. Two papers, one by Gross and Wilczek and the other by Politzer, were published back to back in the now famous June 1973 issue of Physical Review Letters.

Both Wilczek and Politzer were graduate students then, the former with Gross at Princeton University and the latter with Sidney Coleman at Harvard University. And both the groups were aware that they were attacking the same problem. The idea seems to have been realised earlier elsewhere. In his book In Search of the Ultimate Building Blocks, `tHooft says that he had this result (of negative beta function) in 1972 and at a small conference in Marseille, which Symanzik too had attended, he had stated it. However, he did not follow Symanzik's advice and publish it.

The discovery of asymptotic freedom laid the foundation of a theory of colour force called Quantum ChromoDynamics (QCD), which allowed the calculation of small distance interaction between quarks and gluons in quantitative detail to be done for the first time. The theory showed remarkable agreement with experiments. Integration of QCD with the electro-weak unification in a single theoretical framework is what is referred to as the Standard Model (SM) description of fundamental particles and forces of nature.

It must be emphasised here that what we have is a theory of colour force, the force between quarks and gluons. This is not the force of interaction between hadrons, like proton and neutron, the bound states of quarks. This force would be analogous to what produces interaction between electrically neutral molecules - the van der Waal force - to form chemical bonds and form macro-scale matter. The force between protons and neutrons inside the nucleus occurs through the colour forces that "leak" from their constituent quarks and gluons.

An important proof of QCD and Standard Model is provided by collisions between electrons and their antiparticles positrons in high energy accelerators. Electrons are not made of quarks, but the energy released in electron-positron annihilations can give rise to quarks with mass and kinetic energy in accordance with E = mc2. These quarks are created very close to each other. As they move away from each other at high velocities, QCD comes into play and, as explained earlier, will give rise to creation of hadrons made of the newly created quark-anti-quark pairs. These new particles move in the direction of the originally separating quarks in the form of oppositely directed jets of particles. Using asymptotic freedom, it is possible to calculate these processes very precisely and such experiments have found remarkable agreement with QCD's predictions.

As indicated earlier, the most appealing consequence of QCD is that opens up the possibility of a unified description of the forces. If one studies the energy dependence of the strong coupling constant, the electromagnetic and the weak, they almost meet at one point - but not quite - and have the same value at some high energy (around 1015 GeV). If they actually met at one point, grand unification would have been achieved.

However, Standard Model would have to be modified if this is to be realised. One possible scheme is introduction of new symmetries (supersymmetry) and corresponding new (supersymmetric) particles. The evidence for these will be looked for in the Large Hadron Collider (LHC), the upcoming accelerator at CERN. Unification with gravity takes us to the realm the emerging String Theory.

But there is still a major unsolved problem with QCD itself, which is a theory of massless quarks and massless gluons. And yet - that is, without an intrinsic mass and length parameter in the theory - it is supposed to give rise to the real world of massive bound states like the proton and the neutron and the short range defined by the typical size of, say pion, the smallest hadron.

How does this "Problem of Mass Gap in QCD", as it is called, come about. No one has an answer to this. And this is one of the Millennium Problems posed by the Clay Mathematical Institute, United States, with attractive prize money. No doubt that would fetch a Nobel Prize as well.

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