The long journey

With the discovery of the Higgs boson, the Standard Model has emerged as the standard theory to describe nature.

The recently announced discovery of what in all probability is the Higgs boson is the cornerstone of the Standard Model of High Energy Physics. This theory is the culmination of 100 years of fundamental physics, starting with discoveries such as radioactivity and the electron. The Standard Model is now known to be the basis of almost all of known physics except gravity. In the context of the discovery, it is worthwhile to give a historical perspective of the evolution of our understanding of the constituents of the universe and the forces of interaction between them.

The earlier part of the 20th century was marked by two revolutions that rocked the foundations of physics: quantum mechanics and relativity. Quantum mechanics became the basis for understanding atoms, and coupled with Special Relativity, it provided the framework for understanding the atomic nucleus and what lies inside.

At the beginning of the 20 century, the quest to understand the atom topped the agenda of fundamental physics. This led to the unravelling of the atomic nucleus and then to the nucleon (the proton or the neutron). Now, we know that the nucleon itself is made of three quarks. The depth (or the distance scale) probed thus far is 10-17 cm, or 100 millionth of a billionth of a centimetre (Table 1).

The unravelling of the mysteries of matter and the forces that hold it together at deeper and ever deeper levels culminated at the end of the 20th century in the theory of fundamental forces on the basis of what is known in the language of physics as non-abelian gauge fields, which is also known by its rather prosaic name: the Standard Model.

In this theory, the strong forces operating within the nuclei and within the nucleons as well as the weak forces that were revealed through the discovery of radioactivity are understood to be generalisations of electrodynamics, whose foundations were laid by Michael Faraday and James Clerk Maxwell in the 19th century.

Electrodynamics was formulated around the year 1875 and its applications came in the 20th century. We owe a lot to the Faraday-Maxwell electrodynamics for the applications of electrodynamic technology that have become a part of modern life. People take out a small gadget from their pockets and speak to their friends living hundreds or thousands of kilometres away; somebody in a laboratory on the earth turns a knob of an instrument and controls a spacecraft that is hurtling across millions of kilometres away to a distant planet. All this has been possible only because of electromagnetic waves.

The dynamics of strong and weak forces was formulated around 1975, almost exactly 100 years after electrodynamics was formulated. We may expect that equally profound applications will follow, once the technologies of the strong and weak forces are mastered. That may be the technology of the 21st century.

Forces of Nature

The four fundamental forces of nature are the strong nuclear force, electromagnetism, the weak nuclear force and gravitation. Strong forces are responsible for binding nucleons into the nucleus (and for binding quarks into the nucleons). (Their relative strengths and ranges are given in Table 2.) Electromagnetic forces bind nuclei and electrons to form atoms and molecules and bind atoms or molecules to form solid matter. Weak interactions cause the beta decay of nuclei and also are responsible for the fusion reactions that power the sun and stars. Gravitational force binds the planets to the solar system, stars to galaxies, and so on. Although gravity is the weakest force, it becomes the dominant force for the universe at large because of its infinite range and because of its being attractive only (unlike electromagnetism where attraction can be cancelled by repulsion).

In quantum theory, the range of a force is inversely proportional to the mass of the quantum (the force carrier) that is exchanged. Since the photon mass is zero, electromagnetic force mediated by the exchange of photons is of infinite range. Since the strong interaction between nucleons has a finite range, it has to be mediated by a quantum (or particle) of finite mass. Since gravity has infinite range, quantum theory of gravity (if it is constructed) will have its quantum, called graviton with zero mass.

However, with developments in physics in the 20th century, the textbook classification of the four fundamental forces of nature has broken down. We now know that the weak force and electromagnetism are two facets of one entity called the electroweak force. Can one go further and unify the strong force with the electroweak force? It is possible. It is called the grand unification, but that is a speculative step which could be confirmed only in the future. The grander unification will be the unification with gravitation, which we may call Total Unification, which is what Einstein dreamed of. Perhaps that will be realised by string theory. For the present, we have the Standard Model, which is a theory of the electroweak and strong interactions and is based on a generalisation of electrodynamics.

Laws of Electrodynamics

The laws of electrodynamics (Fig. 1) were formulated by Maxwell on the basis of earlier experimental discoveries by Hans Christian Oersted, Andre-Marie Ampere, Faraday, and many others. Actually, from his observations and deep experimental studies of the electromagnetic phenomena, Faraday actually built up an intuitive physical picture of the electromagnetic field and Maxwell made this picture precise by his mathematical formulation.

Once Maxwell wrote down the complete and consistent system of laws, very important consequences followed. He could show that his equations admitted the existence of waves that travelled with a velocity that he could calculate purely from electrical measurements to be 31010 cm per second. Since the velocity of light was known to be this number, Maxwell proposed that light was an electromagnetic wave. This was a great discovery since until that time nobody knew what light was. Subsequently, Heinrich Hertz experimentally demonstrated the existence of the electromagnetic waves predicted by Maxwell.

Maxwells laws have stood the test of time for a century. Even the two revolutions of relativity and quantum mechanics have not invalidated them. In fact, Einstein resolved the confrontation between Newtons laws of particle dynamics and Maxwells laws of field dynamics in favour of the latter. He had to modify Newtons laws to be in conformity with the space-time picture of Maxwells laws; this is how the Special Theory of Relativity was born. Even quantum mechanics left the form of Maxwells equations unchanged. However, there was a profound reinterpretation of the continuous Faraday-Maxwell field with its continuous energy distribution in space-time. It was replaced by field quanta or discrete packets of energy. This was called field quantisation and this was the birth of the Quantum Field Theory.

Let us briefly compare the Faraday-Maxwell picture with the Quantum Field Theory. In the former, a charged particle, say, a proton, is surrounded by an electromagnetic field existing at every point in space-time. If another charged particle, an electron, is placed in this field, the field will interact with the electron, and that is how the electromagnetic interaction between proton and electron is to be understood in the classical electromagnetic theory. In the Quantum Field Theory, the proton emits an electromagnetic quantum, which is called the photon, and the electron absorbs it. This is how the interaction between the proton and the electron is to be understood (Fig.2). Exchange of the field quanta is responsible for the interaction. The Quantum Field Theory is the basic language in which the Standard Model is written.

The Standard Model consists of two parts: electroweak dynamics, which unifies electromagnetic and weak interactions, and chromodynamics, which governs strong interactions.

In electrodynamics we have an electromagnetic field described by the pair of electric and magnetic fields (E, B). The corresponding force mediator is the photon. Analogously, in electroweak dynamics we have four types of generalised electromagnetic fields, one of them being the Faraday-Maxwell electromagnetic field. Correspondingly, there exist four mediators of the electroweak force, or electroweak quanta, also called electroweak gauge bosons. One of them is the photon () mediating electromagnetic interaction and the other three W+, W and Z mediating the weak interaction. Fig. 3 shows, as an example of weak interaction, the decay of the neutron into a proton, an electron and an anti-neutrino.

The laws of electroweak dynamics (EWD) can be written analogous to Maxwells equations (Fig.4). In quantum chromodynamics (QCD), which governs strong interactions, we have eight types of generalised electromagnetic fields. The corresponding force carriers or quanta are called gluons. It is their exchange between quarks that bind or glue the quarks together to form the proton or neutron (Fig.5). Like EWD, laws of QCD can be written for the eight gluon fields. The analogy of the laws of EWD and QCD to the original laws of electrodynamics is obvious.

If these generalisations of Maxwells equations are as simple as they are made out to be, why did the Standard Model take another 100 years to be constructed? The answer lies in the dots in the equations (Fig.4) expressing the laws of EWD and QCD. Let us go back to electrodynamics, in which every electrically charged particle interacts with the electromagnetic field or (in the quantised version) emits or absorbs a photon. But the photon itself does not have charge and hence does not interact with itself.

In the generalisation described above, there are 12 generalised charges, four in EWD and eight in QCD, corresponding to a similar number of generalised electromagnetic fields. In contrast to electric charge, which is just a number (positive, negative or zero), these generalised charges are matrices which do not commute with each other that is, the order in which they are multiplied matters and hence are called non-abelian charges. (In mathematics, algebras with commuting and non-commuting objects are called abelian and non-abelian algebras respectively.) Electrodynamics, which is based on the abelian charge, is called the abelian gauge theory and the generalisation based on the non-abelian charges is called the non-abelian gauge theory. In contrast to the photon, which is the abelian force carrier, and does not carry the abelian electric charge, the non-abelian force carriers themselves carry the non-abelian charges and hence are self-interacting (Fig.6).

The non-linear terms expressing these couplings are hidden behind the dots in the equations of EWD and QCD, and it is these that make the theory of non-abelian gauge fields much more complex than the simple Maxwell theory. Non-abelian gauge fields were introduced by Chen Ning Yang and Robert Mills in 1954, but it took many more important steps in the next two decades before this theory could be used to construct the correct Standard Model.

The Field and Particle Sectors

The constituents of the universe, according to the Standard Model, come in two categories the field sector and the particle sector. In the field sector, we have the 12 gauge fields: the four electroweak fields that include the photon (, W+, W, Z ) and the eight gluon fields (G1, G2,... G8). Their quanta are all particles with one unit of the quantum attribute called spin (spin 1 particles), exactly like the first and most familiar one among them, the photon. All such particles having integral spin belong to the family of particles called bosons (and these obey what is called Bose-Einstein statistics).

The particle sector consists of spin particles belonging to the other great family of fermions (particles that obey Fermi-Dirac statistics). Among these, we have already encountered the two quarks ( u, d) and the two leptons (, e). The quarks make up the nucleon and the nucleons make nuclei. Nuclei and electrons make atoms, molecules and all known matter. The weak radioactive decays involve the neutrino. Thus the quartet of particles consisting of a quark doublet and a lepton doublet seems to be sufficient to make up the whole universe. However, nature has chosen to repeat this quartet two more times so that there actually exist three generations of particle quartets, each consisting of a quark doublet and a lepton doublet (Fig.7). The existence of three generations seems to be related to matter-antimatter asymmetry, which can solve the cosmological puzzle how did the universe, which started as a fireball with equal proportion of matter and antimatter, evolve into a state which has only matter?

There is an incompleteness in our description of the QCD sector. Both quarks and gluons are not seen directly in any experiment. They are supposed to be permanently confined inside the proton and the neutron. But this hypothesis of confinement, which is supposed to be a property of QCD, has not been proved. This important theoretical challenge remains a loophole. This problem is so intractable that it has been announced as one of the millennium problems of mathematics.

Symmetry breaking and Higgs

Remember the vast disparity between electromagnetism and the weak force as regards their ranges; one is of infinite range and the other is short-ranged. How does electroweak unification cope with this breakdown of the electroweak symmetry that is intrinsic to unification? This is achieved by a spontaneous breakdown of symmetry (SBS) engineered by the celebrated Higgs mechanism, which keeps photon massless while raising the masses of W and Z to finite values. Thus, weak interaction gets a finite range. The experimental discovery of W and Z with the masses predicted by the electroweak theory in the 1980s was a great triumph for the theory.

The idea of SBS in high-energy physics originates from Yoichiro Nambu although he applied it in a different context. But the stumbling block was the Goldstone theorem, which predicted the existence of a massless spin zero boson as the consequence of the SBS and prevented the application of SBS to construct any physically relevant theory, since such a massless particle is not seen. Thus, apparently, one had to choose between the devil (massless W boson) and the deep sea (massless spin zero boson).

It was Higgs who, in 1964, showed that this is not correct. He showed that there is no Goldstone theorem if the symmetry that is broken is gauge symmetry. The devil drinks up the deep sea and comes out as a regular massive spin one gauge boson. No massless spin zero boson is left. This is called the Higgs mechanism. Many other authors also contributed to this idea. By combining the electroweak non-abelian gauge theory of Sheldon Glashow with the Higgs mechanism, Steven Weinberg and Abdus Salam independently constructed the electroweak part of the Standard Model in 1967.

There is a bonus. The Higgs mechanism postulates the existence of an all-pervading field called the Higgs field and this field, which gives mass to W and Z, also gives mass to all the fermions of the particle sector, except to the neutrinos. Thus, in particular, the masses of the quarks and the electron come from the Higgs field.

But there is an important byproduct of the Higgs mechanism: a massive spin zero boson, called the Higgs boson, must exist as a relic of the original Higgs field. High-energy physicists searching for it in all the earlier particle accelerators had failed to find it. So the announcement on July 4, 2012, that the Higgs boson has been sighted finally at a mass of 125 GeV at the Large Hadron Collider at CERN (the European Organisation for Nuclear Research), Geneva, has been welcomed by physicists. More tests have to be performed to establish that the particle seen is indeed the Higgs boson.

In the past four decades, experimenters have succeeded in confirming every component of the full Standard Model with three generations of fermions. The Higgs boson was the only missing piece. With its discovery, the Standard Model has emerged as the standard theory to describe nature. This is a great scientific achievement. The Standard Model now deserves a better name.

In the Standard Model, neutrinos are massless. The Higgs mechanism does not give mass to neutrinos. About 15 years ago, experimenters discovered that neutrinos do have tiny masses. This may show us how to go beyond the Standard Model. Neutrinos may be the portal to go beyond the Standard Model, and that is the importance of the India-based Neutrino Observatory (INO), which is to come up in Tamil Nadu.

The biggest loophole in the Standard Model is that gravity has been left out. The most successful attempt to construct quantum gravity is the string theory, but it is still an incomplete theory. So, quantum gravity is the next frontier, and the journey continues.

G. Rajasekaran is an Emeritus Professor at the Institute of Mathematical Sciences and Adjunct Professor at the Chennai Mathematical Institute, both in Chennai.