Concepts in Particle Physics

The following sections are included:

• Dedication
• Preface
• Contents

The Standard Model of particle physics is arguably the most important intellectual achievement of the human race. It describes a number of fundamental particles (fundamental to the best of our knowledge, to date) and their interactions. It is a conceptual and logical structure which is expected to have a very wide range of applicability essentially being the kernel for building up most of physics. And while many details of the phenomena which follow from it are still being worked out, there is good reason to believe that most of the observed results within its expected domain of applicability can be obtained from it. Admittedly, there is evidence for phenomena such as dark matter, which may be beyond the purview of the Standard Model. But as for observed results within its expected domain, we do not yet have any evidence for deviations from the Standard Model. This is both good and not so good. The range of observations we have carried out to date stretch from a microscopic distance scale of about 10⁻¹⁸ meters to cosmological scales of the order of 10²⁶ meters. The Standard Model, including classical Einstein gravity (with a cosmological constant), can account for most of these observations, at least in principle. This is a remarkably good situation, when we recall that at the time of Newton, a little over three hundred years ago, all physics was essentially confined to the terrestrial scales. In fact, one of Newton’s great achievements was to realize that the same physics which applies to terrestrial phenomena could also be applied to explain planetary motion and other phenomena at the scale of the solar system. However, the success of the Standard Model is far from complete because there are still many puzzles left, many features of the Standard Model are awkward and it is hard to believe that it is the final theory of the particle interactions, even apart from deeper issues such as quantum gravity and so on. For further progress, we need to push the limits of validity of the Standard Model, until we find deviations from it, so we can improve it and take it to the next level of understanding and explanation. We will talk about the inadequacies of the Standard Model towards the end of this course, but let us begin by accentuating the positive. Here is an immensely successful theory, arguably the most successful theory ever. What does it look like?

The theoretical underpinnings for much of what we want to discuss are what are often referred to as the twin pillars of twentieth century physics: the theory of relativity and quantum mechanics. It is not too difficult to see why these are important for particle physics. Most often we deal with particles which have speeds comparable to the speed of light in vacuum so that Newtonian (nonrelativistic) physics is totally inadequate. (To dramatize this by an extreme example, recall that the protons circulating in the rings of the Large Hadron Collider (LHC) travel at 0:999999991 times the speed of light in vacuum.) Further, particle collisions can lead to the annihilation of particles and/or the creation of new particles, both of which rely on the interconversion of mass and energy via the famous Einstein relation E = mc². So relativity is central to particle physics. In addition, the scales involved in particle interactions are such that quantum effects are not negligible. Let us note that the very concept of a point particle requires the possibility of localizing it to a very small region, otherwise how do we know that it has no extent, and this can be problematic because of the uncertainty relation Δx Δp ≥ ℏ/2. Thus the picture of a point particle as a little rigid ball of extremely small radius is meaningless, we have to define the concept of a particle via the observables and their algebra of commutation rules in quantum mechanics. Also, quantum fluctuations of observables (which are the hallmark of quantum mechanics) can affect the intermediate states in particle interactions and this too must be taken account of. So quantum mechanics is central to particle physics as well.

Let us now turn to the other great pillar of twentieth century physics, namely, quantum mechanics. We want to extract the essence of the subject in a few ideas which we can take to the realm of particle interactions. The key concept will be the propagator, which will denote the probability amplitude for a particle introduced at a certain point in spacetime to be detected at another spacetime point. If the particle is subject to interactions, either with other particles or with an externally applied field, all that should be reflected in how this probability amplitude is modified. So the propagator provides a simple and easy language to talk about particle interactions. Also being an idea posed directly in terms of spacetime points, it can be made consistent with relativity in a fairly straightforward way.

We shall now embark on a more detailed exploration of Feynman diagrams. These were invented by Richard Feynman in the late 1940s, based on certain intuitive ideas, as a method for calculating the probability amplitudes for various processes involving particle interactions. Subsequently, they have been derived more rigorously from quantum field theory. The great success of this approach is due to the fact that the diagrams themselves are fairly easy to construct based on some simple intuitive rules, yet each diagram is a mnemonic for the mathematical expression of the probability amplitude corresponding to the process that the diagram represents. So there is an easy translation from the diagrams to the amplitudes, bypassing much of the elaborate machinery of quantum field theory. We will now go over how this picture-to-mathematics dictionary is set up and how we can use it.

We are now ready to start discussing a more realistic theory, namely, of photons and the electromagnetic interaction, and on quantum electrodynamics (QED), which is truly a marvelous sub-theory embedded in the Standard Model. Photons are the quanta of the electromagnetic field. The Maxwell equations show that the electric and magnetic fields are dynamical, evolving in time as given by the equations or, equivalently, in a way controlled by a Hamiltonian. Thus they can, and must, be quantized just as any dynamical system should be. The eigenstates of the Hamiltonian, the energy eigenstates, for the free Maxwell theory (without currents or charge densities) are the photons. These particles are therefore central to electromagnetic interactions. While all this can be obtained by applying the principles of quantization to the fields themselves, here we will continue with our strategy of focusing on propagators and vertices and constructing Feynman diagrams.

In the last lecture we talked about the photon propagator and how we can couple charged particles to the electromagnetic field (and the photon). The principle we have enunciated seems very general and one can ask if it is truly universal. Is the coupling of the photon to any charged particle given by this principle? The answer is that it works for a charged point particle. If we consider a composite particle, then we can apply the same principle to the charged point particles which make up the composite particle of interest. The coupling of the photon to a field which represents the whole composite particle has to be obtained from the coupling of the fundamental constituents. This may show different or additional vertices at the level of the composite particles. We will comment again on this matter later. But for now, we continue with the construction of diagrams and processes with photons.

We have discussed the notion of the cross section. It is a central notion in particle physics, in many ways the bread and butter of particle physicists. As emphasized, it is a measure of the strength of the interaction which leads to the scattering, and by simple multiplications with the flux and running time, it gives the number of events. The expression for the cross section will, of course, depend on the momenta of the particles involved and on the nature of the interaction. But, amazingly, the fact that the cross section has the dimensions of an area is actually powerful enough to make some general statements. For this, we first note that the action is a quantity with no dimensions at all. It has the dimension of Energy × Time, which is dimensionless in natural units. Another way to think about this is to note that in the usual units, S had the dimensions of ħ, but since we use units in which ħ = 1, S has no dimension. Thus dim[S] = 0. Equivalently, we may say that we are really using S/ħ when we speak of the action.

In the last lecture, we started our discussion of the Dirac theory of spin-$\frac{1}{2}$ fermions. It provides a relativistically invariant description of such particles. Dirac himself arrived at this by trying to factorize the wave operator □ into a product of first order diffierential operators. He realized he could do it if he allowed for a matrix equation. He introduced the γ^μ matrices with their algebra to ensure that the square of the first order operator, namely, (γ^μ ∂_μ)², would indeed give the wave operator, little knowing if it had anything deeper to do with Lorentz symmetry. It was only later it was realized that the γ-matrices led to a representation of the Lorentz transformations. Almost magically, in attempting something very unusual, yet very down-to-earth and mundane as trying to factorize □, Dirac obtained one of the deepest results in physics.

We have considered the electromagnetic interactions of charged particles or Quantum Electrodynamics (QED), and in particular, the theory of electrons (and positrons) and photons. This theory is one of the most successful theories in the whole history of physics, making precise predictions which have also been experimentally verified to very high accuracy. After centuries of wondering about it, we could finally claim we have some understanding of the nature of light. But somewhat ironically, QED as a theory in its own right may not exist. There is a way to formulate field theories from a rigorous mathematical point of view, known as constructive field theory. What is done is to set up the theory on a lattice with a finite lattice spacing and a finite number of lattice points, making everything mathematically well-defined and then taking the limit of the number of lattice points becoming infinite and the lattice spacing becoming zero, thus recovering a continuum formulation. There are indications that QED formulated in this way does not exist as an interacting field theory, meaning that the theory becomes a free theory in the continuum limit. Nevertheless, it is sensible to work with QED viewing it as part of a larger theory such as the Standard Model (with other particles and other interactions) and, as we mentioned, it has been amazingly successful.

We will now begin our discussion of the gauge principle, which is the foundational principle, one might even say the soul, of the Standard Model. To define what we are trying to formulate, let us start by recalling that at the end of the last lecture, we were talking about how to improve the V - A theory of weak interactions. We mentioned how we can mimic the electromagnetic interaction to get a dimensionless coupling constant so that the problematic rise of cross sections with energy can be avoided. Thus, for example, we can try postulating a vertex of the form

Einstein’s theory of gravity is the quintessential example of a theory built on invariance under local symmetry transformations. It provides the basic paradigm for the gauge principle which we seek to generalize. Let us start by recalling that the gauge symmetry behind gravity was the symmetry under local coordinate transformations. For example, a vector A_μ transforms under coordinate transformations as

In the last lecture, we started the generalization of the idea of gauge symmetry to transformations using matrices rather than just a complex number. The key ideas were the following. We can consider a number of fields together, say, as a column vector with N entries. Let us designate this as Φ.

In the last lecture we formulated the matrix generalization of the idea of gauge symmetry, concluding with the definition of the field strength and the Yang-Mills action. It is very useful to see how this all works out for some simple examples.

We are now in a position to return to the discussion about particles and their interactions. Once again the fundamental particles, as we know them to date, are as listed in Table 1.1, which is reproduced here, as Table 14.1, for convenience. We should now consider the basic forces, all of which will be described by a gauge theory. In other words, we need to specify the type of symmetry transformations and what particles respond to them in what way. The forces of interest, putting aside gravity, are the electromagnetic forces, the strong nuclear force and the weak force. Of these, we have discussed the electromagnetic force to some extent and the weak force using the V - A theory, although we know from the problem of rising cross sections that the V - A theory has to be superseded by some more fundamental theory. The latter will turn out to have the Yang-Mills structure. The strong nuclear force, which will also be of the Yang-Mills type, is the simplest to formulate (although the hardest to analyze), so we will start with it.

We have talked about QCD, the theory of strong nuclear forces and its property of asymptotic freedom with the effective coupling decreasing at high energies or short separations, in contrast to quantum electrodynamics. We also discussed the notion of quark confinement which is crucial in understanding the bound states of quarks such as mesons and baryons. It also explains why we do not see free quarks, despite long and careful searches carried out in the 1960s and 1970s. The theory is described by an SU(3) symmetry. Thus there are 8 types of “gluons” interacting with the quarks, each species (or flavor) of quark having 3 colors or internal states. The action for the quarks is obtained from the Dirac action by the gauge principle, specifically using SU(3) symmetry. For example, for the up-quark, the action is

In the last lecture, we talked about how we might be able to use symmetry for understanding the bound states of quarks. The key ingredients are the notion of color (or quark) confinement (which tells us that physical states must be invariant under the color transformations), the idea that there is a flavor SU(3)_f symmetry and the fact that quarks are fermions, so we must take account of the Pauli exclusion principle when we have identical fermions. With these concepts, we can get quite a bit of useful information on the bound states.

In this lecture, we will start considering the ingredients for a theory of weak interactions. The key point we have made so far is that the theory at low energies is described by the V - A theory with an interaction term of the form

We have talked about spontaneous symmetry breaking and the Higgs mechanism. We also mentioned that much of the inspiration for this came from the theory of superconductivity. We will talk about this in some more detail here, since it provides a very concrete realization of the ideas we want to explore.

We have talked about the fermions and how they couple to the gauge fields which are responsible for the electroweak interactions. We also showed that one can recover the older V – A results in the low energy limit, if we use massive propagators for the W’s and the Z’s. However, this was done in something of an ad hoc fashion, because we did not really discuss how the mass arises. Let us talk about this question now. As mentioned earlier, the masses have to come from a Higgs mechanism. In the case of superconductivity, we had a field corresponding to the Cooper pairs, which are themselves bound states of two electrons. However, for the weak interactions, since we are talking about the real vacuum state of the world, we do not consider such a paired bound state, instead we simply postulate a fundamental field with the required properties. Such a field must give a mass to the W’s, so it must break the SU(2) symmetry. This means that it should transform under the SU(2) transformations, so that assigning a nonzero value to this field in the vacuum leads to spontaneous breaking of the symmetry. Further, we do not want the photon to be massive, so we must ensure that whichever field has a nonzero value in the vacuum does not break the electromagnetic symmetry of phase transformations. Since the phases for the latter are of the form e^iqθ where q is the electric charge, this can be done by making sure that the postulated Higgs field has a component with zero charge, and that it is this component which gets a nonzero expectation value in the vacuum. (The Higgs field obviously has to have more than one component since it transforms nontrivially under the SU(2) transformations.) The simplest choice consistent with these requirements is a doublet of the form

We have gone over some of the key features of the Standard Model. This is by no means a comprehensive study of the Standard Model, but we are now in a position to talk about some of the not-so-well-understood features. This also borders on the many puzzles which are still there.

We have talked about particle physics and the Standard Model. This theory achieves the unification of the weak and electromagnetic interactions and also provides the proper description of the strong nuclear forces. Further, the Standard Model, when combined with classical gravity, explains phenomena over a wide range of scales, from 10⁻¹⁸ m to 10²⁵ m or so, about 43 orders of magnitude in length scale. This is a truly amazing achievement. Remember that at the time of Newton, a little over three hundred years ago, we were still at the level of terrestrial phenomena occurring on a scale of a few meters, a typical human scale. In some ways, Newton's achievement was to realize that the phenomena on Earth and in the heavens should have the same set of physical laws and hence what was observed on Earth, such as gravity, could be used to understand planetary motion. So it was an enhancement of scale from the human dimensions to the scale of the solar system. Simultaneously, we started a journey to the smaller scales as well, with the work of Leeuwenhoek and Hooke. In about 300 years, we can now cover roughly 43 orders of magnitude. This does not mean we understand all physics and can give an explanation for all experimental results. No, many details are yet to be worked out, many collective phenomena may require new concepts. But there is good reason to believe that while there are so many details at so many levels to be worked out and understood, no new physical principles will be needed for the majority, with significant exceptions, of the phenomena we have observed in the range of length scales indicated. And yet, many big questions remain, the exceptions mentioned above, some of which may require new laws of physics. Let us talk about some of these remaining big questions, as we know of them, or understand them, at this point in history. We should keep in mind that our perspective on this may change over time as well.

The following sections are included:

• References
• Index