Elementary Particle Physics in a Nutshell

Index

Preface

The 21st century is a time of great change in particle physics. A new energy frontier recently opened up at the Large Hadron Collider (LHC) at CERN. It's a time of great excitement with the anticipation of unexpected outcomes. At the same time, the most widely used university-level texts on high-energy physics date back to the time leading up to the W and Z boson discoveries. Since then, the Standard Model of particle physics has been thoroughly explored at the Large Electron Positron (LEP) collider at CERN, the Tevatron at Fermilab, HERA at DESY and at two B-factories, KEKB and PEP-II. A decade of neutrino physics has brought an exciting new view on these elementary and light, but massive, particles. This text is an attempt to capture the modern understanding of particle physics in a snapshot of time leading up to the start-up of the LHC. I believe that the pause in the development of texts has been due in part to the anticipated discovery of the Higgs boson and the implications that the observed Higgs field properties will have in defining the high-energy unification of the fundamental interactions. However, it is difficult for a new generation of high-energy physics to prepare for the challenge of the LHC without having the perspective needed to look beyond the limitations of the current Standard Model. In this text, I attempt to introduce a complete working knowledge of the SU(3)C × SU(2)L × U(1)Y Standard Model as early as possible and then focus on the many experimental confirmations in the context of the full theory. Ultimately, this will lead us to the next generation of high-energy experiments with a focus on what we hope to learn.

The final editing of this book was completed while on sabbatical at the Institute for Advanced Study with support from the IBM Einstein Fellowship Fund.

Chris Tully

Princeton, 2010

1

Particle Physics: A Brief Overview

Particle physics is as much a science about today's universe as it is of the early universe. By discovering the basic building blocks of matter and their interactions, we are able to construct a language in which to frame questions about the early universe. What were the first forms of matter created in the early universe? What interactions were present in the early universe and how are they related to what we measure now? While we cannot return space-time to the initial configuration of the early universe, we can effectively turn back the clock when it comes to elementary particles by probing the interactions of matter at high energy. What we learn from studying high-energy interactions is that the universe is much simpler than what is observed at room temperature and that the interactions are a reflection of fundamental symmetries in Nature. An overview of the modern understanding of particle physics is described below with a more quantitative approach given in subsequent chapters and finally with a review of measurements, discoveries, and anticipated discoveries, that provide or will provide the experimental facts to support these theories.

We begin with the notion of a fundamental form of matter, an elementary particle. An elementary particle is treated as a pointlike object whose propagation through space is governed by a relativistically invariant equation of motion. The equation of motion takes on a particular form according to the intrinsic spin of the particle and whether the particle has a nonzero rest mass. In this introduction, we begin by assuming that elementary particles are massless and investigate the possible quantum numbers and degrees of freedom of elementary particle states.

1.1 Handedness in the Equation of Motion

A massless particle with nonzero intrinsic spin travels at the speed of light and has a definite handedness as defined by the sign of the dot product of the momentum and spin. The handedness of a massless particle, of which there are two possible values, is invariant and effectively decouples the elementary particles into two types, left-handed and right-handed. However, the association of handedness to a degree of freedom has to be extended to all solutions of the relativistic equation of motion. The time evolution of a solution to a wave equation introduces a time-dependent complex phase, where for a plane-wave solution of ordinary matter, we have

Relativistic invariance and, in particular, causality introduces solutions that propagate with both positive and negative frequency. Relative to the sign of the frequency for matter solutions, a new set of solutions, the antimatter solutions, have the opposite sign of frequency, exp(iωt), so as to completely cancel contributions of the relativistic wave function outside the light cone. Therefore, in the relativistic equation of motion, there is always an antiparticle solution that is inseparable from the particle solution. In terms of handedness, an antiparticle solution has the sign of the dot product of momentum and spin reversed relative to the corresponding particle solution. We define a new quantity, called the chirality, that changes sign for antiparticles relative to particles. Therefore, a particle solution with left-handed chirality is relativistically linked through the equation of motion to an antiparticle solution that also has left-handed chirality. We can now separate in a relativistically invariant way two types of massless particles, left-handed and right-handed, according to their chirality.

1.2 Chiral Interactions

The existence of an interaction is reflected in the quantum numbers of the elementary particles. We introduce here a particular type of chiral interaction, one in which left-handed particle states can be transformed into one another in a manner similar to a rotation. However, unlike a spatial rotation, the chiral interaction acts on an internal space termed isospin, in analogy to a rotation of intrinsic spin. The smallest nontrivial representation of the isospin interaction is a two-component isospin doublet with three generators of isospin rotations. Left-handed particles interact under the chiral interaction, and, therefore, the symmetry associated with this interaction imposes a doubling of the number of left-handed elementary particles. There is an up and down type in each left-handed isospin doublet of elementary particles. If we further tailor our chiral interaction, we can begin to construct the table of known elementary particles. Namely, we do not introduce a right-handed chiral interaction. Furthermore, elementary particles that have right-handed chirality are not charged under the left-handed chiral interaction and are therefore singlets of the left-handed chiral symmetry group.

The evidence for the left-handed chiral interaction was initially observed from parity violation in the radionuclear decay of unstable isotopes emitting a polarized electron and an undetected electron antineutrino in the final state. While we have not introduced mass or an interaction for electric charge as would be expected for the electron, we can ignore these properties for now and construct a left-handed doublet from the elementary particles consisting of the electron (down-type) and the electron neutrino (up-type). The electron and neutrino are part of a general group of elementary particles known as the leptons.

1.3 Fundamental Strong Interaction

We now consider the force that leads to the formation of protons and neutrons, and is ultimately responsible for nuclear forces. This force is the fundamental strong interaction and, similar to the chiral interaction, is an interaction that acts on an internal space. In this case, the internal space is larger and has a smallest nontrivial representation given by a triplet with a set of eight generators of rotation. The triplet is referred to as a triplet of color, with components denoted red, green, and blue. As with the electron and electron neutrino, a left-handed triplet of color is also a doublet of the chiral interaction. The lightest down-type color triplet is called the down-quark. Correspondingly, the lightest up-type color triplet is called the up-quark. In contrast to the quarks, leptons are charge neutral with respect to the strong interaction.

1.4 Table of Elementary Particles

The chiral and fundamental strong interactions are a sufficient starting point to introduce the table of elementary particles, shown in and Z), respectively.

The properties of the elementary particles not explained by the chirality and color interactions are the electric charges and masses, and so too the unexplained presence of the photon. In order to explain the properties of mass and charge, here we look to a predicted and yet still elusive element in the particle table, the particle shown in the center of figure 1.1, the Higgs boson.

1.5 Mass and Electric Charge

While mass and electric charge are second nature from classical physics, they are highly nonobvious quantities in the elementary particles. In other words, their origin is believed to be linked to the properties of the physical vacuum rather than an inherent quantity that one would assign based on first principles, as explained below.

The poor assumption in the above discussions on elementary particles is the requirement of indistinguishability of the components of the particle doublets that represent the internal space of the chiral interaction. For the strong interaction, the components of the triplet of color are indistinguishable and hence are not explicitly labeled in and Z bosons in the way that they do. Moreover, for the chiral interaction to be based on an exact symmetry, the masses of elementary particles would have to be identically zero; otherwise, particles of left-handed chirality and right-handed chirality could be transformed into each other through a relativisitic transformation, changing the quantum numbers.

) are on the right. The central particle, the Higgs boson (spin-0), has yet to be observed (Credit: Fermilab).

1.6 Hypercharge Interaction of the Standard Model

The Standard Model is a theory that solves the paradox of the hidden symmetry of the chiral interaction. To restore the indistinguishability of components of a chiral doublet, the Standard Model eliminates electromagnetism as an elementary interaction of massless fermions. The road to reintroducing electomagnetism as an interaction of massive fermions begins by postulating an alternative elementary interaction for massless fermions, called the hypercharge interaction. Hypercharge has every similarity to electric charge with the exception of the charge assignments to the elementary particles. A left-handed electron and electron neutrino are assigned the same hypercharge, and similarly for left-handed up- and down-quarks. Hypercharge assignments preserve the indistinguishabilityof the components of isospin doublets. The right-handed chiral particles are singlets in the chiral interaction and hence a right-handed up-quark can be assigned a different hypercharge than the left-handed up-quark or the right-handed down-quark without breaking the chiral symmetry. Thus, the critical step in constructing the Standard Model is to throw out mass and electric charge as a starting point for building a table of elementary particles. However, of what purpose is the hypercharge interaction in explaining the physically observed masses and charges of elementary particles and from where does the photon of electromagnetism originate? This brings us to the heart of the Standard Model theory, the electroweak symmetry breaking.

1.7 Higgs Mechanism

The central concept of the Standard Model is that the properties of the physical vacuum do not have the same symmetries as the fundamental interactions. This notion seems absurd at first, but physical examples of such systems, such as low-temperature superconductivity, clearly demonstrate such behavior in nonvacuous environments. Indeed, in a superconductor, the vacuum symmetry of zero electric charge is no longer present and photons do not propagate as massless particles in the space of a superconductor. The space of a low-temperature superconductor is filled with charge 2e electron pairs, known as Cooper pairs, that behave as a condensate of bosons. The photon is unable to propagate freely in a superconductor as it encounters nonzero electric charge at every point in space. The same type of mechanism can be postulated in the physical vacuum if there exists a condensate with nonzero hypercharge everywhere in space. Furthermore, the lack of an observed chiral symmetry in Nature would imply that a condensate of nonzero isospin is present in the vacuum ground state.

If there is a condensate in the ground state of the physical vacuum, what is it? One possible mechanism of electroweak symmetry breaking is known as the Higgs mechanism. The Higgs mechanism predicts the existence of a new type of elementary matter, called the Higgs bosons. The Higgs bosons are a set of spin-0 particle states with nonzero isospin and nonzero hypercharge quantum numbers. Similar to the Cooper pairs of superconductivity, the Higgs bosons interact with each other and do so in such a way so as to prefer a nonzero expectation value in the ground state. In other words, the Higgs condensate is present everywhere in space. Such an occurrence completely redefines our concept of the elementary particles and interactions. The table of elementary particles are those mass eigenstates that arise from particles interacting with the Higgs condensate. Similarly, the particles mediating the elementary interactions are hindered from propagating freely, resulting in a transformation of the elementary particle interactions. The photon of electromagnetism is the zero-mass eigenstate that propagates in the electrically neutral physical vacuum. It is in this way that the physical vacuum imposes the definition of electric charge and mass eigenstates in the elementary particles—these are not properties that come directly from the fundamental hypercharge and chiral interactions.

1.8 Program of Study

The Higgs mechanism of the Standard Model is a great leap beyond the notion of an empty vacuum and symmetry-preserving interactions. Indeed, the timeline of when the universe developed a vacuum filled with a symmetry-breaking condensate is not clear. Perhaps the most trying part of studying elementary particle physics is the lack of direct evidence to prove the existence of the Higgs mechanism or alternative electroweak symmetry-breaking mechanisms. Nevertheless, the experimental verification of the Standard Model is extensive with no apparent deviations with respect to all known predictions. Many internal consistencies of the Standard Model overwhelmingly support the concept of a symmetry-breaking physical vacuum, whether the fundamental source is the Higgs mechanism or something else. It is this predicament that has brought elementary particle physics into its most challenging and potentially the most revolutionary stage in its development. The energy scale that will confirm or refute the Higgs mechanism will be fully explored by the Large Hadron Collider (LHC) at the CERN laboratory in Geneva, Switzerland. The LHC experiments will be able to detect evidence for the Higgs bosons and probe possible extensions to the known physical symmetries in the elementary particles that would explain what stabilizes the electroweak scale.

The purpose of this text is to bring students a full understanding of the Standard Model, from relativistic kinematics and the Dirac equation through the concept of gauge interactions, and then to review in the context of the full theory the many areas of experimental investigation that have tested and subsequently confirmed the validity of the Standard Model predictions. Each chapter is concluded with a section that lists references that provide detailed background on the topics discussed. These references contain a wealth of interesting perspectives and lessons that were invaluable in the development of the Standard Model. In contrast, this book teaches the Standard Model as an established theory and applies the predictions to directly explain the mountain of experimental evidence that in retrospect was intended to challenge its validity. By presenting a fresh look at the Standard Model outside of the historically important questions that led to its creation, the intention is to prepare the ground for the next generation of exploration of elementary particle physics at the LHC.

1.9 Exercises

1. Fundamental interactions.

(a) What are the interactions described by the Standard Model in a symmetry-preserving vacuum? Ignore the Higgs interactions.

(b) What Standard Model interactions in part (a) are unaffected by the reduced symmetries of the physical vacuum?

2. Elementary fermions.

(a) The right-handed up-type leptons are unusual fermion components in the Standard Model. These particles, known as right-handed neutrinos, are color singlets and isospin singlets and have zero hypercharge, and, hence, zero electric charge. In a symmetry-preserving vacuum, do right-handed neutrinos interact? What property of the left-handed neutrinos would imply that the right-handed neutrinos interact with the Higgs condensate?

(b) If one counts the distinguishable and indistinguishable components of the strong and chiral multiplets, how many fermions make up the table of elementary particles, including the particles described in part (a)?

1.10 References and Further Reading

A selection of general introductions to the early universe, particle physics, experimental particle physics, and historical accounts can be found in the following reference texts: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11].

A selection of the original articles on the theory of the Standard Model of particle physics and the Higgs mechanism can be found here: [12, 13].

[1]  Robert Cahn and Gerson Goldhaber. The Experimental Foundations of Particle Physics. Cambridge University Press, 2009. ISBN 978-0-521-52147-5.

[2]  Stephen Hawking and Leonard Mlodinow. A Briefer History of Time. Bantam Dell Publishing, 2005. ISBN 978-0-553-38546-5.

[3]  Don Lincoln. Understanding the Universe. World Scientific Publishing, 2004. ISBN 981-238-705-6.

[4]  Martinus Veltman. Facts and Mysteries in Elementary Particle Physics. World Scientific Publishing, 2003. ISBN 981-238-149-X.

[5]  Gerard ‘t Hooft. In Search of the Ultimate Building Blocks. Cambridge University Press, 1997. ISBN 0-521-57883-3.

[6]  Harvey B. Newman and Thomas Ypsilantis , editors. History of Original Ideas and Basic Discoveries in Particle Physics. Plenum Press, 1996. ISBN 0-306-45217-0.

[7]  Val L. Fitch and Jonathan L. Rosner. Elementary Particle Physics in the Second Half of the Twentieth Century. IOP Publishing, 1995. ISBN 1-56396-048-6.

[8]  Gordon Kane. The Particle Garden. Perseus Publishing, 1995. ISBN 0-201-40826-0.

[9]  Leon Lederman. The God Particle. Dell Publishing, 1993.ISBN 0-385-31211-3.

[10] Steven Weinberg. The First Three Minutes. BasicBooks, Harper Collins Publishers, 1993. ISBN 0-465-02437-8.

[11] Sheldon L. Glashow. The Charm of Physics. Simon and Schuster, 1991. ISBN 0-671-74013-X.

[12] S. L. Glashow, Nucl. Phys. 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, Weak and Electromagnetic Interactions, in Elementary Particle Theory, Ed. N. Svartholm. Almquist and Wiksell, 1968, p. 367, LCCN 68055064; H. D. Politzer, Phys. Rev. Lett. 30 (1973) 1346; D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343.

[13] P. W. Higgs, Phys. Lett. 12 (1964) 132, Phys. Rev. Lett. 13 (1964) 508, and Phys. Rev. 145 (1966) 1156; F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321.

2

Dirac Equation and Quantum Electrodynamics

particle, for example, an electron at rest. The rotations of the electron spin are described by the SU(2) group and the states of the system are labeled by the values of the total spin, or specifically S², and the projection of the spin onto an arbitrary axis of quantization, for example, Sz, the component along the z-direction. Now take the electron and Lorentz boost into a frame moving to the right (along the x-axis). How do we construct a wave function of a relativistic electron?

The answer to this question will lead us to one of the most profound equations in elementary particle physics, the Dirac equation. The Dirac equation introduces the first of several symmetry extensions to the structure of matter, namely, the antiparticles.

2.1 Natural Units and Conversions

By choosing natural units, where

the quantities of mass, inverse length, and inverse time can be described by a single dimensional unit. The choice in this text is to measure all quantities in units of GeV. The conversion to units of meters and seconds is handled, for the most part, by inserting values of c where needed:

The charge of the electron is denoted e with e < 0. In general, the conventions of the Peskin and Schroeder [1] text are followed for all calculations.

2.2 Relativistic Invariance

A wave function in nonrelativistic quantum mechanics describes the probability of finding a particle at time t in a volume element d³x centered at position x The normalization of the wave function is defined to be unity when integrated over all space, or a periodic box of volume V for plane-wave solutions,

indicating that in a fixed snapshot of time the particle must be somewhere. The generators of infinitesimal translations in time and space are the familiar energy and momentum operators

respectively, giving rise to the commutation relation and hence the Heisenberg uncertainty principle,

respectively.

Dirac wanted to describe the dynamics, infinitesimal time translations, of a particle with an equation that is first-order in time and Lorentz-invariant. To do this, he started with the general form:

The question then became, "What are the conditions on α and β?" The relativistic energy-momentum relation for a free particle requires that

This implies the anticommutation { } relations

and that the squares are unity

As pure numbers α and β would not satisfy the above conditions. Dirac proposed that α and β are matrices and that the wave function Ψ a multicomponent column vector, known as a Dirac spinor.

2.3 Pauli-Dirac Representation and Connection with Nonrelativistic QM

We can find 4 × 4 matrix representations for α and β that satisfy the Dirac conditions, equations (2.8) and (2.9). However, a four-component Dirac spinor, Ψ, has twice the number of degrees of freedom needed to describe a nonrelativistic spin-1/2 particle. We can therefore appeal to the low (kinetic) energy limit of the Dirac equation (2.6) to find the nonrelativistic correspondence of solutions to the Dirac equation and begin to associate physical meaning to the components of the spinor. The specific form for α and β is not unique, and therefore there is the freedom to choose a particular representation. The Pauli-Dirac representation assigns

where σi are the 2 × 2 Pauli spin matrices and I2 the unit 2 × 2 matrix. Recall the Pauli representation for σi,

We will see from equation (2.6) that the choice of β diagonal favors a nonrelativistic decoupling of the two 2-component solutions.

For an electron at rest, the Dirac equation (2.6) reduces to

and has the following four solutions:

Two of these solutions are positive-frequency solutions (Ψ¹,² ) and two are negative (Ψ³,⁴ ), corresponding to the sign on the right-hand side of equation (2.12). In the absence of an interaction, we do not know how to interpret the different free-particle solutions. We will therefore exercise the Dirac equation and introduce the electromagnetic interaction from an external four-potential

via the minimal coupling substitution

where e < 0 is the electron charge. The origin of the minimal coupling substitution is described later in the section on local gauge invariance.

The Dirac equation with the electromagnetic minimal coupling substitution becomes

which we will use to study the interactions of a point charge with an applied electromagnetic field.

Initially, we have considered the matrices α and β as introducing a static rearrangement of the components of Ψ. In the Heisenberg interpretation (H ), these matrices become operators whose time evolution is governed by the Heisenberg equation of motion:

The correspondence with the Schrödinger (S) time-independent operators is described as follows:

, we get

which by relativistic extension of the Ehrenfest classical correspondence principle indicates that α is we find

for the electric E and magnetic B fields

reproducing the motion of a point charge e with α as the velocity operator.

, where the L and S refer to the large and small components, respectively. The relative magnitude of the two 2-component spinors results from specifying positive-frequency solutions, which we know from the free-particle rest-frame solutions, equation (2.13), come dominantly from the top two components of the four-component Dirac spinor. Therefore, writing

and substituting into equation (2.16) yields

If we now assume that the rest energy m is the largest energy in the system, the dominant time dependence of the positive-frequency solution can be factorized out,

This approximation results in the following simplification:

The time derivative of Ψ will estimate the energy of the electron relative to its rest mass, as is the convention for energies in nonrelativistic mechanics. In the nonrelativistic limit where m the approximate solution for the lower two components of equation (2.25) is

which when substituted back into the upper two components of equation (2.25) yields

This is further reduced by the identity for Pauli spin matrices

which holds even if a and b are operators. Therefore, evaluating the operator cross-product term yields

will act on everything to the right,

is acting only on A. With this simplification, equation (2.27) becomes

where Ψin the—μ . B interaction term and writing the magnetic moment μ as

reveals the landmark result from the Dirac equation of g = 2 for the gyromagnetic ratio g of the electron.

2.3.1 Constants of Motion

The Heisenberg equation (2.17) of motion can be used to determine whether a given observable is a constant of the motion. If we apply this to the angular momentum operator

then we find for a free-particle Dirac Hamiltonian (2.6)

This result is contrary to that in nonrelativistic mechanics. For a free Dirac particle, L is not a constant of the