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ELEMENTARY PARTICLES IN PHYSICS 1
Elementary Particles in Physics
S. Gasiorowicz and P. Langacker
Elementary-particle physics deals with the fundamental constituents of mat- ter and their interactions. In the past several decades an enormous amount of experimental information has been accumulated, and many patterns and sys- tematic features have been observed. Highly successful mathematical theories of the electromagnetic, weak, and strong interactions have been devised and tested. These theories, which are collectively known as the standard model, are almost certainly the correct description of Nature, to first approximation, down to a distance scale 1/1000th the size of the atomic nucleus. There are also spec- ulative but encouraging developments in the attempt to unify these interactions into a simple underlying framework, and even to incorporate quantum gravity in a parameter-free “theory of everything.” In this article we shall attempt to highlight the ways in which information has been organized, and to sketch the outlines of the standard model and its possible extensions.
Classification of Particles
The particles that have been identified in high-energy experiments fall into dis- tinct classes. There are the leptons (see Electron, Leptons, Neutrino, Muonium), all of which have spin 12. They may be charged or neutral. The charged lep- tons have electromagnetic as well as weak interactions; the neutral ones only interact weakly. There are three well-defined lepton pairs, the electron (e−) and the electron neutrino (νe), the muon (μ−) and the muon neutrino (νμ), and the (much heavier) charged lepton, the tau (τ ), and its tau neutrino (ντ ). These particles all have antiparticles, in accordance with the predictions of relativistic quantum mechanics (see CPT Theorem). There appear to exist approximate “lepton-type” conservation laws: the number of e−^ plus the number of νe mi- nus the number of the corresponding antiparticles e+^ and ¯νe is conserved in weak reactions, and similarly for the muon and tau-type leptons. These conser- vation laws would follow automatically in the standard model if the neutrinos are massless. Recently, however, evidence for tiny nonzero neutrino masses and subtle violation of these conservations laws has been observed. There is no un- derstanding of the hierarchy of masses in Table 1 or why the observed neutrinos are so light. In addition to the leptons there exist hadrons (see Hadrons, Baryons, Hy- perons, Mesons, Nucleon), which have strong interactions as well as the elec- tromagnetic and weak. These particles have a variety of spins, both integral and half-integral, and their masses range from the value of 135 MeV/c^2 for the neutral pion π^0 to 11 020 MeV/c^2 for one of the upsilon (heavy quark) states. The particles with half-integral spin are called baryons, and there is clear ev- idence for baryon conservation: The number of baryons minus the number of antibaryons is constant in any interaction. The best evidence for this is the stability of the lightest baryon, the proton (if the proton decays, it does so with a lifetime in excess of 10^33 yr). In contrast to charge conservation, there is no
Table 1: The leptons. Charges are in units of the positron (e+) charge e =
- 602 × 10 −^19 coulomb. In addition to the upper limits, two of the neutrinos have masses larger than 0.05 eV/c^2 and 0.005 eV/c^2 , respectively. The νe, νμ, and ντ are mixtures of the states of definite mass. Particle Q Mass e−^ − 1 0 .51 MeV/c^2 μ−^ − 1 105 .7 MeV/c^2 τ −^ − 1 1777 MeV/c^2 νe 0 < 0 .15 eV/c^2 νμ 0 < 0 .15 eV/c^2 ντ 0 < 0 .15 eV/c^2
Table 2: The quarks (spin- 12 constituents of hadrons). Each quark carries baryon number B = 13 , while the antiquarks have B = − 13. Particle Q Mass
u (up) 23 1. 5 − 5 MeV/c^2
d (down) − 13 5 − 9 MeV/c^2
s (strange) − 13 80 − 155 MeV/c^2
c (charm) 23 1 − 1 .4 GeV/c^2
b (bottom) − 13 4 − 4 .5 GeV/c^2
t (top) 23 175 − 180 GeV/c^2
deep principle that makes baryon conservation compelling, and it may turn out that baryon conservation is only approximate. The particles with integer spin are called mesons, and they have baryon number B = 0. There are hundreds of different kinds of hadrons, some almost stable and some (known as resonances) extremely short-lived. The degree of stability depends mainly on the mass of the hadron. If its mass lies above the threshold for an allowed decay channel, it will decay rapidly; if it does not, the decay will proceed through a channel that may have a strongly suppressed rate, e. g., because it can only be driven by the weak or electromagnetic interactions. The large number of hadrons has led to the universal acceptance of the notion that the hadrons, in contrast to the leptons, are composite. In particular, experiments involving lepton–hadron scattering or e+e−^ annihilation into hadrons have established that hadrons are bound states of point-like spin- 12 particles of fractional charge, known as quarks. Six types of quarks have been identified (Table 2). As with the leptons, there is no understanding of the extreme hierarchy of quark masses. For each type of quark there is a corresponding antiquark. Baryons are bound states of three quarks (e. g., proton = uud; neutron = udd), while mesons consist of a quark and an antiquark. Matter and decay processes under normal terrestrial con- ditions involve only the e−, νe, u, and d. However, from Tables 2 and 3 we
The equations of motion show that the current is conserved,
∂ ∂xα
jα(x) = 0 , (6)
so that the charge
Q =
d^3 r j 0 (r, t) (7)
is a constant of the motion. The form of the interaction is obtained by making the replacement ∂ ∂xα^
∂xα^
− ieAα(x) (8)
in the Lagrangian for a free lepton. This minimal coupling follows from a deep principle, local gauge invariance. The requirement that ψ(x) can have its phase changed locally without affecting the physics of the lepton, that is, invariance under ψ(x) → e−iθ(x)ψ(x) , (9)
can only be implemented through the introduction of a vector field Aα(x), cou- pled as in (8), and transforming according to
Aα(x) → Aα(x) −
e
∂θ(x) ∂xα^
This dictates that the free-photon Lagrangian density contains only the gauge- invariant combination (2), and that terms of the form M 2 A^2 α(x) be absent. Thus local gauge invariance is a very powerful requirement; it implies the existence of a massless vector particle (the photon, γ), which mediates a long-range force [Fig. 1(a)]. It also fixes the form of the coupling and leads to charge conservation, and implies masslessness of the photon. The resulting theory (see Quantum Electrodynamics, Compton Effect, Feynman Diagrams, Muonium, Positron) is in extremely good agreement with experiment, as Table 3 shows. In working out the consequences of the equations of motion that follow from (3), infinities appear, and the theory seems not to make sense. The work of S. Tomonaga, J. Schwinger, R. P. Feynman, and F. J. Dyson in the late 1940s clarified the nature of the problem and showed a way of eliminating the difficulties. In creating renormalization theory these authors pointed out that the parameters e and m that appear in (3) can be identified as the charge and the mass of the lepton only in lowest order. When the charge and mass are calculated in higher order, infinite integrals appear. After a rescaling of the lepton fields, it turns out that these are the only infinite integrals in the theory. Thus by absorbing them into the definitions of new quantities, the renormalized (i. e., physically measured) charge and mass, all infinities are removed, and the rest of the theoretically calculated quantities are finite. Gauge invariance ensures that in the renormalized theory the current is still conserved, and the photon remains massless (the experimental upper limit on the photon mass is 6 × 10 −^17 eV/c^2 ).
ELEMENTARY PARTICLES IN PHYSICS 5
Fig. 1: (a) Long-range force between electron and proton mediated by a photon. (b) Four-fermi (zero-range) description of beta decay (n → pe−^ ν¯e). (c) Beta decay mediated by a W −. (d) A neutral current process mediated by the Z.
Table 3: Extraction of the (inverse) fine structure constant α−^1 from various experiments, adapted from T. Kinoshita, J. Phys. G 29, 9 (2003). The con- sistency of the various determinations tests QED. The numbers in parentheses (square brackets) represent the uncertainty in the last digits (the fractional uncertainty). The last column is the difference from the (most precise) value α−^1 (ae) in the first row. A precise measurement of the muon gyromagnetic ratio aμ is ∼ 2. 4 σ above the theoretical prediction, but that quantity is more sensitive to new (TeV-scale) physics. Experiment Value of α−^1 Difference from α−^1 (ae) Deviation from gyromagnetic 137 .035 999 58 (52) [3. 8 × 10 −^9 ] – ratio, ae = (g − 2)/2 for e− ac Josephson effect 137 .035 988 0 (51) [3. 7 × 10 −^8 ] (0. 116 ± 0 .051) × 10 −^4 h/mn (mn is the neutron mass) 137 .036 011 9 (51) [3. 7 × 10 −^8 ] (− 0. 123 ± 0 .051) × 10 −^4 from n beam Hyperfine structure in 137 .035 993 2 (83) [6. 0 × 10 −^8 ] (0. 064 ± 0 .083) × 10 −^4 muonium, μ+^ e− Cesium D 1 line 137 .035 992 4 (41) [3. 0 × 10 −^8 ] (0. 072 ± 0 .041) × 10 −^4
ELEMENTARY PARTICLES IN PHYSICS 7
particle into another, or the corresponding creation of an e−^ ν¯e pair. Similarly, J α† describes a charge-raising transition such as n → p. Equation (14) describes a zero-range four-fermi interaction [Fig. 1(b)], in contrast to electrodynamics, in which the force is transmitted by the exchange of a photon. An additional class of “neutral-current” terms was discovered in 1973 (see Weak Neutral Currents, Currents in Particle Theory). These will be discussed in the next section. Jα(x) consists of leptonic and hadronic parts:
Jα(x) = J leptα (x) + J hadα (x). (15)
Thus, it describes purely leptonic interactions, such as
μ−^ → e−^ + ¯νe + νμ , νμ + e−^ → νe + μ−^ ,
through terms quadratic in Jlept; semileptonic interactions, most exhaustively studied in decay processes such as
n → p + e−^ + ¯νe (beta decay) , π+^ → μ+^ + νμ , Λ^0 → p + e−^ + ¯νe ,
and more recently in neutrino-scattering reactions such as
νμ + n → μ−^ + p (or μ−^ + hadrons) , ν ¯μ + p → μ+^ + n (or μ+^ + hadrons) ;
and, through terms quadratic in J hadα , purely nonleptonic interactions, such as
Λ^0 → p + π−^ , K+^ → π+^ + π+^ + π−^ ,
in which only hadrons appear. The coupling is weak in that the natural di- mensionless coupling, with the proton mass as standard, is Gm^2 p = 1. 01 × 10 −^5 , where G is the Fermi constant. The leptonic current consists of the terms
J leptα (x) = ¯eγα(1 − γ 5 )νe + ¯μγα(1 − γ 5 )νμ + ¯τ γα(1 − γ 5 )ντ. (16)
Both polar and axial vector terms appear (γ 5 = iγ^0 γ^1 γ^2 γ^3 is a pseudoscalar matrix), so that in the quadratic form (14) there will be vector–axial-vector interference terms, indicating parity nonconservation. The discovery of this phenomenon, following the suggestion of T. D. Lee and C. N. Yang in 1956 that reflection invariance in the weak interactions could not be taken for granted but had to be tested, played an important role in the determination of the phe- nomenological Lagrangian (14). The experiments suggested by Lee and Yang all involved looking for a pseudoscalar observable in a weak interaction experi- ment (see Parity), and the first of many experiments (C. S. Wu, E. Ambler, R.
W. Hayward, D. D. Hoppes, and R. F. Hudson) measuring the beta decay of polarized nuclei (^60 Co) showed an angular distribution of the form
W (θ) = A + Bpe · 〈J〉 , (17)
where pe is the electron momentum and 〈J〉 the polarization of the nucleus. The distribution W (θ) is not invariant under mirror inversion (P ) which changes J → J and pe → −pe, so the experimental form (17) directly showed parity nonconservation. Experiments showed that both the hadronic and the leptonic currents had vector and axial-vector parts, and that although invariance under particle–antiparticle (charge) conjugation C is also violated, the form (14) main- tains invariance under the joint symmetry CP (see Conservation Laws) when restricted to the light hadrons (those consisting of u, d, c, and s). There is evi- dence that CP itself is violated at a much weaker level, of the order of 10−^5 of the weak interactions. As will be discussed later, this is consistent with second- order weak effects involving the heavy (b, t) quarks, though it is possible that an otherwise undetected superweak interaction also plays a role. The part of J hadα relevant to beta decay is ∼ uγ¯α(1−γ 5 )d. The detailed form of the hadronic current will be discussed after the description of the strong interactions. Even at the leptonic level the theory described by (14) is not renormalizable. This manifests itself in the result that the cross section for neutrino absorption grows with energy:
σν = (const)G^2 mpEν. (18)
While this behavior is in accord with observations up to the highest energies studied so far, it signals a breakdown of the theory at higher energies, so that (14) cannot be fundamental. A number of people suggested over the years that the effective Lagrangian is but a phenomenological description of a theory in which the weak current Jα(x) is coupled to a charged intermediate vector boson W (^) α− (x), in analogy with quantum electrodynamics. The form (14) emerges from the exchange of a vector meson between the currents (see Feynman Diagrams) when the W mass is much larger than the momentum transfer in the process [Fig. 1(c)]. The intermediate vector boson theory leads to a better behaved σν at high energies. However, massive vector theories are still not renormalizable, and the cross section for e+e−^ → W +W −^ (with longitudinally polarized W s) grows with energy. Until 1967 there was no theory of the weak interactions in which higher-order corrections, though extraordinarily small because of the weak coupling, could be calculated.
Unified Theories of the Weak and Electromagnetic Inter-
actions
In spite of the large differences between the electromagnetic and weak interac- tions (massless photon versus massive W , strength of coupling, behavior under P and C ), the vectorial form of the interaction hints at a possible common origin. The renormalization barrier seems insurmountable: A theory involving
theories are renormalizable because the form of the interactions in (21) and (23) leads to cancellations between different contributions to high-energy am- plitudes. However, gauge invariance does not allow mass terms for the vector bosons, and it is this feature that was responsible for the general neglect of the Yang–Mills theory for many years. S. Weinberg (1967) and independently A. Salam (1968) proposed an ex- tremely ingenious theory unifying the weak and electromagnetic interactions by taking advantage of a theoretical development (see Symmetry Breaking, Spon- taneous) according to which vector mesons in Yang–Mills theories could acquire a mass without its appearing explicitly in the Lagrangian (the theory without the symmetry breaking mechanism had been proposed earlier by S. Glashow). The basic idea is that even though a theory possesses a symmetry, the solutions need not. A familiar example is a ferromagnet: the equations are rotationally invariant, but the spins in a physical ferromagnet point in a definite direction. A loss of symmetry in the solutions manifests itself in the fact that the ground state, the vacuum, is no longer invariant under the transformations of the sym- metry group, e. g., because it is a Bose condensate of scalar fields rather than empty space. According to a theorem first proved by J. Goldstone, this implies the existence of massless spin-0 particles; states consisting of these Goldstone bosons are related to the original vacuum state by the (spontaneously broken) symmetry generators. If, however, there are gauge bosons in the theory, then as shown by P. Higgs, F. Englert, and R. Brout, and by G. Guralnik, C. Hagen, and T. Kibble, the massless Goldstone bosons can be eliminated by a gauge trans- formation. They reemerge as the longitudinal (helicity-zero) components of the vector mesons, which have acquired an effective mass by their interaction with the groundstate condensate (the Higgs mechanism). Renormalizability depends on the symmetries of the Lagrangian, which is not affected by the symmetry- violating solutions, as was elucidated through the work of B. W. Lee and K. Symanzik and first applied to the gauge theories by G. ’t Hooft. The simplest theory must contain a W +^ and a W −; since their generators do not commute there must also be at least one neutral vector boson W 0. A scalar (Higgs) particle associated with the breaking of the symmetry of the solution is also required. The simplest realistic theory also contains a photon- like object with its own coupling constant [hence the description as SU (2) × U (1)]. The resulting theory incorporates the Fermi theory of charged-current weak interactions and quantum electrodynamics. In particular, the vector boson extension of the Fermi theory in (19) is reproduced with gw = g/ 2
2, where g is the SU (2) coupling, and G ≈
2 g^2 / 8 M (^) W^2. There are two neutral bosons, the W 0 of SU (2) and B associated with the U (1) group. One combination,
A = cos θW B + sin θW W 0 , (25)
is just the photon of electrodynamics, with e = g sin θW. The weak (or Wein- berg) angle θW which describes the mixing is defined by θW ≡ tan−^1 (g′/g), where g′^ is the U (1) gauge coupling. In addition, the theory makes the dramatic
ELEMENTARY PARTICLES IN PHYSICS 11
prediction of the existence of a second (massive) neutral boson orthogonal to A:
Z = − sin θW B + cos θW W 0 , (26)
which couples to the neutral current
JαZ =
i
T 3 (i) ψγ¯α(1 − γ 5 )ψi − 2 sin^2 θW jαγ , (27)
where jαγ is the electromagnetic current in (13) and T 3 (i) [+ 12 for u, ν; − 12 for e−, d] is the eigenvalue of the third generator of SU (2). The Z mediates a new class of weak interactions (see Weak Neutral Currents),
(ν/ν¯) + p, n → (ν/ν¯) + hadrons , (ν/¯ν) + nucleon → (ν/ν¯) + nucleon , νμ + e−^ → νμ + e−^ ,
characterized by a strength comparable to the charged-current interactions [Fig. 1(d)]. Another prediction is that of the existence, in electromagnetic interactions such as e−^ + p → e−^ + hadrons ,
of tiny parity-nonconservation effects that arise from the exchange of the Z between the electron and the hadronic system. Neutral current-induced neu- trino processes were observed in 1973, and since then all of the reactions have been studied in detail. In addition, parity violation (and other axial current effects) due to the weak neutral current has been observed in polarized M¨oller (e−e−) scattering and in asymmetries in the scattering of polarized electrons from deuterons, in the induced mixing between S and P states in heavy atoms (atomic parity violation), and in asymmetries in electron–positron annihilation into μ+μ−, τ +τ −, and heavy quark pairs. All of the observations are in excel- lent agreement with the predictions of the standard SU (2) × U (1) model and yield values of sin^2 θW consistent with each other. Another prediction is the existence of massive W ±^ and Z bosons (the photon remains massless because the condensate is neutral), with masses
M W^2 =
A^2
sin^2 θW
, M Z^2 =
M W^2
cos^2 θW
where A ∼ πα/
2 G ∼ (37 GeV)^2. (In practice, a significant, 7%, higher-order correction must be included.) Using sin^2 θW obtained from neutral current processes, one predicted MW = 80. 2 ± 1 .1 GeV/c^2 and MZ = 91. 6 ± 0 .9 GeV/c^2 ’. In 1983 the W and Z were discovered at the new ¯pp collider at CERN. The current values of their masses, MW = 80. 425 ± 0 .038 GeV/c^2 , MZ = 91. 1876 ± 0 .0021 GeV/c^2 , dramatically confirm the standard model (SM) predictions. The Z factories LEP and SLC, located respectively at CERN (Switzerland) and SLAC (USA), allowed tests of the standard model at a precision of ∼ 10 −^3 ,
ELEMENTARY PARTICLES IN PHYSICS 13
MW (as did experiments at the Fermilab Tevatron ¯pp collider (USA)), measured the four-fermion cross sections e+e−^ → f f¯, and tested the (gauge invariance) predictions of the SM for the gauge boson self-interactions. The Z-pole, neutral current, and boson mass data together establish that the standard (Weinberg–Salam) electroweak model is correct to first approximation down to a distance scale of 10−^16 cm (1/1000th the size of the nucleus). In particular, this confirms the concepts of renormalizable field theory and gauge invariance, as well as the SM group and representations. The results yield the precise world average sin^2 θW = 0. 23149 ± 0 .00015. (It is hoped that the value of this one arbitrary parameter may emerge from a future unification of the strong and electromagnetic interactions.) The data were precise enough to allow a successful prediction of the top quark mass (which affected higher order corrections) before the t was observed directly, and to strongly constrain the possibilities for new physics that could supersede the SM at shorter distance scales. The major outstanding ingredient is the Higgs boson, which is hard to produce and detect. The precision experiments place an upper limit of around 250 GeV/c^2 on the Higgs mass (which is not predicted by the SM), while direct searches at LEP II imply a lower limit of 114.4 GeV/c^2. Some physicists suspect that the elementary Higgs field may be replaced by a dynamical or bound-state symmetry-breaking mechanism, but the possibilities are strongly constrained by the precision data. Unified theories, such as superstring theories, generally imply an elementary Higgs. It is hoped that the situation will be clarified by the next generation of high energy colliders.
The Strong Interactions
The strength of the coupling that manifests itself in nuclear forces and in the interaction of pions with nucleons is such that perturbation theory, so useful in the electromagnetic interaction, cannot be applied to any field theory of the strong interactions in which the mesons and baryons are the fundamental fields. The large number of hadronic states strongly suggests a composite structure that cannot be viewed as a perturbation about noninteracting systems. In fact, it is now generally believed that the strong interactions are described by a gauge theory, quantum chromodynamics (QCD), in which the basic entities are quarks rather than hadrons. Nevertheless, prior and parallel to the development of the quark theory a wealth of experimental information concerning the hadrons and their interactions was accumulated. In spite of the absence of guidance from field theory, and in spite of the fact that each jump in available accelerator energy brought a shift in the focus of attention, certain simple patterns were identified.
Internal Symmetries
The first hint of a new symmetry can be seen in the remarkable resemblance between neutron and proton. They differ in electromagnetic properties, and, other than that, by effects that are very small; for example, they differ in mass
by 1 part in 700. W. Heisenberg conjectured that the neutron and proton are two states of a single entity, the nucleon (see Nucleon), just as an electron with spin up and an electron with spin down are two states of a single entity, even though in an external magnetic field they have slightly different energies. Pursuing this analogy, Heisenberg and E. U. Condon proposed that the strong interactions are invariant under transformations in an internal space, in which the nucleon is a spinor (see Isospin). Thus, the nucleon is an isospin doublet, with Iz (p) = 12 and Iz (n) = − 12 , and isospin (in analogy with angular momen- tum) is conserved. In the language of group theory, the assertion is that the strong interactions are invariant under the transformations of the group SU (2), and that particles transform as irreducible representations. The electromagnetic and weak interactions violate this invariance. The expression for the charge of the nucleons and antinucleons,
Q = Iz + B/ 2 , (29)
shows that the charge picks out a preferred direction in the internal space. (It is now believed that the strong interactions themselves have a small piece which breaks isospin symmetry, in addition to electroweak interactions.) With the discovery of the three pions (π+, π^0 , π−) with mass remarkably close to that predicted by H. Yukawa (1935) in his seminal work explaining nuclear forces in terms of an exchange of massive quanta of a mesonic field, the notion of isospin acquired a new significance. It was natural, in view of the small π±^ −π^0 mass difference, to assign the pion to the I = 1 representation of SU (2). The invariance of the pion–nucleon interaction under isospin transformations led to a number of predictions, all of which were confirmed. In particular, states initiated in pion-nucleon collisions could only have isospin 12 or 32. Early work on pion–nucleon scattering led to the discovery of a resonance with rest mass 1236 MeV/c^2 , width 115 MeV/c^2 , and angular momentum and parity JP^ = (^32)
. This resonance occurred in π+p scattering, so that it had to have I = 32 , and its effects seen in π−p → π−p and π−p → π^0 n should be the same as those in π+p → π+p. This prediction was borne out by experiment. Formally, SU (2) invariance is described by defining generators Ii; (i = 1, 2 , 3) obeying the Lie algebra [Ii, Ij ] = ieijk Ik , (30)
where eijk is totally antisymmetric in the indices and e 123 = 1. The statement that a pion is an I = 1 state then means that the pion field Πa transforms according to [Ii, Πa] = −(Ii)abΠb , a = 1, 2 , 3 , (31)
where the Ii are 3×3 matrices satisfying (30). In relativistic quantum mechanics conservation laws must be local, so the conservation law
dIi dt
really follows from the local conservation law
∂ ∂xμ^
Iiμ (x) = 0 (33)
Table 4: Table of low-lying mesons and baryons, grouped according to SU (3) multiplets. There may be considerable mixing between the SU (3) singlets η′, ϕ, and f ′^ and the corresponding octet states η, ω, f. Mass Quark Particle B Q Y I JP^ (GeV/c^2 ) content π 0 1, 0, − 1 0 1 0 −^ 0.14 u d, u¯ ¯u − d d, d¯ ¯u K 0 1, 0 1 12 0 −^ 0.49 u¯s, d¯s K^ ¯ 0 0, − 1 − 1 1 2 0
− (^) 0.49 s d, s¯ u¯
η 0 0 0 0 0 −^ 0.55 uu¯ + d d¯ − 2 s¯s
η′^0 0 0 0 0 −^ 0.96 uu¯ + d d¯ + s¯s
ρ 0 1, 0, − 1 0 1 1 −^ 0.77 u d, u¯ ¯u − d d, d¯ ¯u K∗^0 1, 0 1 12 1 −^ 0.89 u¯s, d¯s K^ ¯∗^0 0, − 1 − 1 1 2 1
− (^) 0.89 s d, s¯ u¯
ω 0 0 0 0 1 −^ 0.78 uu¯ + d d¯
φ 0 0 0 0 1 −^ 1.02 s¯s
A 2 0 1, 0, − 1 0 1 2 +^ 1.32 u d, u¯ ¯u − d d d¯ ¯u K∗(1430) 0 1, 0 1 12 2 +^ 1.43 u¯s, d¯s K^ ¯∗(1430) 0 0, − 1 − 1 1 2 2
f 0 0 0 0 2 +^ 1.28 uu¯ + d d¯
f ′^0 0 0 0 2 +^ 1.53 s¯s
N 1 1, 0 1
0.94 uud, udd
Λ (^1 0 0 0 )
1.12 uds − dus Σ 1 1, 0, − (^1 0 1 )
1.19 uus, uds + dus, dds
Ξ 1 0, − 1 − (^1 )
1.32 uss, dss
1.23 uuu, uud, udd, ddd
Σ(1385) 1 1, 0, − (^1 0 1 )
1.39 uus, uds, dds
Ξ∗(1530) 1 0, − 1 − (^1 )
1.53 uss, dss
Ω−^1 − 1 − (^2 0 )
1.67 sss
ELEMENTARY PARTICLES IN PHYSICS 17
also contains (I = 1, Y = 0) and (I = 12 , Y = −1) states and an isosinglet Y = −2 particle. The symmetry-breaking pattern that explained the mass splittings among the isospin multiplets in the octet predicted equal mass splittings. Thus, when the I = 1 Σ(1385) was discovered, predictions could be made about the I = 12 Ξ∗, found at mass 1530 MeV/c^2 , and the Ω−, predicted at 1675 MeV/c^2. The latter mass is too low to permit a strangeness-conserving decay to Ξ^0 K−, so the Ω−^ had to be long-lived, only decaying by a chain of ∆S = 1 weak interactions with a very clear signature. The dramatic discovery in 1964 of the Ω−^ with all the right properties convinced all doubters. [see SU(3) and Higher Symmetries, Hyperons, Hypernuclear Physics and Hypernuclear Interactions].
S-Matrix Theory
The construction of higher-energy accelerators, the invention of the bubble chamber by D. Glaser, and the combination of large hydrogen bubble chambers, rapid scanning facilities, and high-speed computers into a massive data produc- tion and analysis technology, pioneered by L. Alvarez and collaborators, led to the discovery of many new resonances during the 1950s and 1960s. The basic procedure was to measure charged tracks in bubble-chamber pictures, taken in strong magnetic fields, and to calculate the invariant masses (
Ei)^2 −(
pic)^2 for various particle combinations. Resonances manifest themselves as peaks in mass distributions, and the events in the resonance region may be further ana- lyzed to find out the spin and parity of the resonance. Baryonic resonances were also discovered in phase-shift analyses of angular distributions in pion–nucleon and K–nucleon scattering reactions. The patterns of masses and quantum num- bers of the resonances showed that all the mesonic resonances came in SU (3) octets and singlets, and the baryonic ones in SU (3) decuplets, octets, and sin- glets. There was good evidence that there was no fundamental distinction between the stable particles and the highly unstable resonances: The ∆ and the Ω−, discussed above, are good examples, and theoretically it was found that both stable (bound) states and resonant ones appeared in scattering amplitudes as pole singularities, differing only in their location. Furthermore, the role assigned by Yukawa to the pion as the nuclear “glue” – it was the particle whose exchange was largely responsible for the nuclear forces – had to be shared with other par- ticles: Various vector and scalar mesons were seen to contribute to the nuclear forces, and G. F. Chew and F. E. Low explained much of low-energy pion physics in terms of nucleon exchange. Chew, in collaboration with S. Mandelstam and S. Frautschi, proposed to do away with the notion of any particles being “fun- damental.” They hypothesized that the collection of all scattering amplitudes, the scattering matrix, be determined by a set of self-consistency conditions, the bootstrap conditions (see S-Matrix Theory), according to which, crudely stated, the exchange of all possible particles should yield a “potential” whose bound states and resonances should be identical with the particles inserted into the exchange term. Much effort was devoted to bootstrap and S-matrix theory during the 1960s
ELEMENTARY PARTICLES IN PHYSICS 19
Table 5: The u, d, and s quarks. B Q Y I Iz
u (^1323131212)
d 13 − 13 13 12 − (^12)
s 13 − 13 − 23 0 0
(quark + antiquark) if they have baryon number B = 0 and of (qqq) (three quarks) if they have baryon number B = 1. They proposed that there exist three different kinds of quarks, labeled u, d, and s. These were assumed to have spin 12 and the internal quantum numbers listed in Table 5. The quark contents of the low-lying hadrons are given in Table 4. The vector meson octet (ρ, K∗, ω) differs from the pseudoscalars (π, K, η) in that the total quark spin is 1 in the former case and zero in the latter. The (A 2 , K∗(1490), f ) octet are interpreted as an orbital excitation (^3 P 2 ). All of the known particles and resonances can be interpreted in terms of quark states, including radial and orbital excitations and spin. The first question was answered automatically, since products of the simplest representations decompose according to the rules
3 × 3 ∗^ = 1 + 8 ,
3 × 3 × 3 = 1 + 8 + 8 + 10.
A problem immediately arose in that the decuplet to which Σ(1236) belongs, being the lowest-energy decuplet, should have its three quarks in relative S states. Thus the ∆++, whose composition is uuu, could not exist, since the spin- statistics connection requires that the wave function be totally antisymmetric, which it manifestly is not when the ∆++^ is in a Jz = 32 state, with all spins up, for example. The solution to this problem, proposed by O. W. Greenberg, M. Han, and Y. Nambu and further developed by W. A. Bardeen, H. Fritzsch, and M. Gell-Mann, was the suggestion that in addition to having an SU (3) label such as (u, d, s) – named flavor by Gell-Mann – and a spin label (up, down), quarks should have an additional three-valued label, named color. Thus according to this proposal there are really nine light quarks:
uR uB uY dR dB dY sR sB sY
Hence, the low-lying (qqq) state could be symmetric in the flavor and spin labels, provided it were totally antisymmetric in the color (red, blue, yellow) labels. More colors could be imagined but at least three are needed. Transformations among the color labels lead to another symmetry, SU (3)color. The totally anti- symmetric state is a color singlet. The mesons can also be constructed as color
singlets, for example,
π+^ =
(uR d¯R + uB d¯B + uY d¯Y ).
The existing hadronic spectrum shows no evidence for states that could be color octets, for example, so the present attitude is that either color nonsinglet states are very massive compared with the low-lying hadrons or that it is an intrinsic part of hadron dynamics that only color singlet states are observable. The first evidence that there are three (and not more) colors came from the study of π^0 → 2 γ decay. Using general properties of currents, S. Adler and W. A. Bardeen were able to prove that the π^0 decay rate was uniquely determined by the process in which the π^0 first decays into a u¯u or a d d¯ pair, which then annihilates with the emission of two photons. The matrix element depends on the charges of the quarks, and a calculation of the width yields 0.81 eV. With n colors, this is multiplied by n^2 , and the observed width of 7. 8 ± 0 .6 eV supports the choice of n = 3. Subsequent evidence for three colors was provided by the total cross section for e+e−^ annihilation into hadrons (see below), and by the elevation of the SU (3)color symmetry to a gauge theory of the strong interactions. The quark model has been extremely successful in the classification of ob- served resonances, and even predictions of decay widths work very well, with much data being correlated in terms of a few parameters. The ingredients that go into the calculation are (a) that quarks are light, with the (u, d) doublet al- most degenerate, with mass in the 300-MeV/c^2 range (one-third of a nucleon), (b) that the s quark is about 150 MeV/c^2 more massive – this explains the pat- tern of SU (3) symmetry breaking – and (c) that the low-lying hadrons have the simple q q¯ or qqq content, without additional q q¯ pairs. However, nobody has ever observed an isolated quark (free quarks should be easy to identify because of their fractional charge). It is now generally believed that quarks are con- fined, i. e., that it is impossible, even in principle, for them to exist as isolated states. However, in the 1960s this led most physicists to doubt the existence of quarks as real particles. That view was shattered by the deep inelastic electron scattering experiments in the late 1960s.
Deep Inelastic Reactions and Asymptotic Freedom
In 1968 the first results of the inelastic electron-scattering experiments (Fig. 3),
e + p → e′^ + hadrons
measured at the Stanford Linear Accelerator Center (SLAC), were announced. The experiments were done in a kinematic region that was new. Both the momentum transfer squared (that is, the negative mass squared of the virtual photon exchanged) and the “mass” of the hadronic state produced were large. The cross section could be written as
d^2 σ dE′dΩ
dσ dΩ
point
W 2 (x, Q^2 ) + 2W 1 (x, Q^2 ) tan^2
θ 2
