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the Presentations in article, pre-discovery Questions my of in “Unanswered the- construction given electroweak theory the electroweak complements the the presentation condensed- in The together and ory. came theories, physics, weak-interaction field matter from gauge mainly drawn phenomenology, concepts, of range hc icse n erdcsmn lsi papers. classic Goldhaber, many & reproduces Cahn by and book discusses of the which creation is the model to standard for led the source that superb observations A suc- experimental explain. a the to what need establish would weak theory to cessful (charged-current) order in the developed, of interactions phenomenology the how ftewlsrnso oenpyis nasottime, short radioac Rutherford a of by In sorts classified several tivity, physics. distinguish to modern learned of physicists wellsprings the of h rtga fti ril st kthhwabroad a how sketch to is article this of goal first The eqee’ discovery Becquerel’s of evocation compressed a to devoted is section This ∗ I XEIETLROSO THE OF ROOTS EXPERIMENTAL II. edsoey ilas uvywhat survey also will I discovery. he ies da aetgte nthe in together came ideas diverse sbsnrmi ob established, be to remain boson gs LCRWA THEORY ELECTROWEAK gestwr nesadn the understanding toward ogress A β Z ea,osre as observed decay, → 30 A Z+1 + 1) + (Z frdociiyi 86i one is 1896 in radioactivity of FERMILAB–PUB–15/058–T 31 codn otechar- the to according β − , (1) 26 29 y - 2 where β− is Rutherford’s name for what was soon iden- Detecting a particle that interacts as feebly as the neu- tified as the electron and AZ stands for the nucleus with trino requires a massive target and a copious source of charge Z and mass number A (in modern language, Z neutrinos. In 1953, Clyde Cowan and Fred Reines39 used protons and A Z neutrons). Examples are tritium β the intense flux of antineutrinos from a fission reactor and 3 3− 3 decay, H1 He2 + β−, neutron β decay, n p + β−, a heavy target (10.7 ft of liquid scintillator) containing → 214 214 → 28 and β decay of Lead-214, Pb Bi + β−. about 10 protons to detect the inverse neutron-β-decay 82 → 83 For two-body decays, as indicated by the detected reaction ν¯+p e++n. Initial runs at the Hanford Engi- → products, the Principle of Conservation of & Mo- neering Works were suggestive but inconclusive. Moving mentum says that the β particle should have a definite their apparatus to the stronger fission neutrino source at energy. What was observed, as experiments matured, the Savannah River nuclear plant, Cowan and Reines and was very different: in 1914, James Chadwick32 (later to their team made the definitive observation of inverse β discover the neutron) showed conclusively that in the de- decay in 1956.40 cay of Radium B and C (214Pb and 214Bi), the β energy Through the 1950s, a series of experimental puzzles follows a continuous spectrum. led to the suggestion that the weak interactions did not 41 The β-decay energy crisis tormented physicists for respect reflection symmetry, or parity. In 1956, C. S. years. On December 4, 1930, Wolfgang Pauli addressed Wu and collaborators detected a correlation between the 60 an open letter33 to a meeting on radioactivity in Tübin- spin vector J~ of a polarized Co nucleus and the direc- 42 gen. In his letter, Pauli advanced the outlandish idea tion pˆe of the outgoing β particle. Now, parity inver- of a new, very penetrating, neutral particle of vanish- sion leaves spin, an axial vector, unchanged ( : J~ J~) P → ingly small mass. Because Pauli’s new particle interacted while reversing the electron direction ( :p ˆe pˆe), P → − very feebly with matter, it would escape undetected from so the correlation J~ pˆe should be an “unobservable” null any known apparatus, taking with it some energy, which quantity if parity is· a good symmetry. The observed would seemingly be lost. The balance of energy and mo- correlation is parity violating. Detailed analysis of the mentum would be restored by the particle we now know 60Co result and others that came out in quick succession as the electron’s antineutrino. Accordingly, the proper established that the charged-current weak interactions scheme for beta decay is are left-handed. By the same argument, the parity op- eration links a left-handed neutrino with a right-handed A A Z (Z+1)+ β− +¯ν . (2) neutrino. Therefore a theory that contains only νL would → be manifestly parity-violating. What Pauli called his “desperate remedy” was, in its way, Could the neutrino indeed be left-handed? M. Gold- very conservative, for it preserved the principle of energy haber and collaborators inferred the electron neutrino’s and momentum conservation and with it the notion that helicity43 from the longitudinal polarization of the recoil the laws of physics are invariant under translations in nucleus in the electron-capture reaction space and time. 152 m 152 e− + Eu (J = 0) Sm∗(J =1)+ νe After Chadwick’s discovery of the neutron in 1932 in → (3) highly penetrating radiation emitted by beryllium irra- | γ +152 Sm . diated by α particles,34 Fermi named Pauli’s hypothet- → ical particle the neutrino, to distinguish it from Chad- A compendious knowledge of the properties of nuclear wick’s strongly interacting neutron, and constructed his levels, together with meticulous technique, enabled this four-fermion theory (what we would today call a low- classic experiment. energy effective theory) of β decay, which was the first Following the observation of maximal parity violation step toward the modern theory of the charged-current in the late 1950s, a serviceable effective Lagrangian for .35 In retrospect, nuclear β decay was the weak interactions of and neutrinos could be the first hint for flavor, the existence of particle families written as the product of charged leptonic currents, containing distinct species. That hint was made mani- fest by the discovery of the neutron, nearly degenerate GF µ V A = − νγ¯ µ(1 γ5)e eγ¯ (1 γ5)ν + h.c., (4) in mass with the proton, which suggested that neutron L − √2 − − and proton might be two states of a nucleon, with the n - p mass difference attributed to electromagnetic ef- where Fermi’s coupling constant is GF =1.1663787(6) 5 2 × fects. The inference that neutron and proton were part- 10− GeV− . This Lagrangianhas a V A (vector minus ners was strengthened by the observation that nuclear axial vector) Lorentz structure,44–47 whereas− Fermi’s ef- forces are charge-independent, up to electromagnetic cor- fective Lagrangian for β decay was a (parity-conserving) rections.36 The accumulating evidence inspired Heisen- vector interaction. A straightforward lepton-current– berg37 and Wigner38 to make an analogy between the times–nucleon-current generalization of Eq. (4) that proton and neutron on the one hand and the up and down takes account of the fact that nucleons are not simple spin states of an electron. Isospin symmetry, based on Dirac particles leads to an effective Lagrangian for β de- the spin-symmetry group SU(2), is the first example of a cay and associated processes. Many applications and ex- flavor symmetry. perimental tests are detailed in.48–50 3

The direct phenomenological consequences of parity vi- and cannot reliably be computed beyond leading order. olation in the weak interactions, which shattered the re- A classic application55 of partial-wave unitarity (proba- ceived wisdom of the era, were themselves dramatic, lead- bility conservation) to inverse muon decay, νµe µνe, ing for example to a factor-of-three difference between the leads to the conclusion that the four-fermion effective→ La- total cross sections for νe and νe¯ scattering. Parity viola- grangian Eq. (7) can only make sense for c.m. tion is also a harbinger of a particular challenge to be met √s 617 GeV. That comfortably encompasses most lab- by a true theory of the weak interactions. In quantum oratory≤ experiments, but as a matter of principle gives electrodynamics, it is perfectly respectable (and correct!) a clear lesson: new physics must intervene below about to write a Lagrangian that includes a term for electron 600 GeV c.m. energy. mass, Although Fermi took his inspiration from the theory of electromagnetism, he did not posit a force carrier anal- =e ¯(iγµ m)e =e ¯(iγµ∂ m)e qA eγ¯ µe, (5) L Dµ − µ − − µ ogous to the photon. This is a perfectly reasonable first step, given that electromagnetism acts over an infinite where Aµ is the four-vector potential of electromag- netism. The left-handed and right-handed components range, whereas the influence of the β-decay interaction 15 of the electron have the same charge, and so appear sym- extends only over about 10− cm. One may hope to ob- metrically. If fermions are chiral, which is to say that the tain a more satisfactory theory by taking the next step, left-handed and right-handed components behave differ- supposing that the weak interaction, like quantum elec- ently, a mass term conflicts with symmetries. This will trodynamics, is mediated by vector-boson exchange (of be made precise in §IIIB. nonzero range) to soften the high-energy growth of am- A second charged lepton, the muon, was discovered plitudes. The weak intermediate boson must carry charge 1, because the familiar manifestations of the weak inter- and identified as lacking strong interactions in the decade ± beginning in 1937.51–53 In common with the electron, actions (such as β-decay) are charge-changing; be rather 1 massive ( 100 GeV), to reproduce the short range of the muon is a spin- 2 Dirac particle, structureless at our ≈ present limits of resolution. It is unstable, with a mean the weak force; and accommodate parity violation. In- lifetime of approximately 2.2 µs and a mass 207 that troducing a weak boson W ± in this ad hoc manner indeed of the electron. It might be tempting, therefore, to× con- mitigates the unitarity problem for inverse muon decay, sider the muon as an excited electron, but the transi- but introduces incurable unitarity problems for reactions + + + tions µ eγ, µ ee+e , and µ eγγ has never been such as e e− W W − or νν¯ W W −, as detailed − 9 → → seen. The→ limits→ on these decays→ are so stringent [e.g., in §6.2 of. 13 the branching fraction for µ eγ is < 5 10− at 90% It is also worth mentioning the discovery of strange confidence level24] that we regard→ the muon× as a distinct particles in the early 1950s, because it was essential to lepton species. establishing that the leptonic and hadronic weak interac- If the muon is distinct from the electron, what is the tions have the same strength and stimulated the inven- nature of the neutrino produced in association with the tion of quarks.56–58 Semileptonic decays of hyperons59 muon in pion decay, π+ µ+ν? In 1962, Lederman, were an essential testing ground for Cabibbo’s formula- → Schwartz, Steinberger, and collaborators carried out a tion of the universality of the charged-current weak in- two-neutrino experiment using neutrinos created in the teractions60 which was the forerunner of today’s 3 3 decay of high-energy pions from the new Alternating Gra- quark-mixing matrix.61 × dient Synchrotron at Brookhaven.54 They observed nu- merous examples of the reaction νN µ+X, but found no evidence for the production of electrons.→ Their study established that the muon produced in pion decay is a distinct particle, ν , that is different from either ν or III. THE DEVELOPMENT OF THE µ e ELECTROWEAK THEORY ν¯e. This observation suggests that the leptonic charged- current weak interactions exhibit a two-doublet family structure, This section is a brief historical survey of the ideas that came together in the notion of a gauge theory for ν ν e µ . (6) the weak and electromagnetic interactions. What follows e µ  − L  − L is neither a complete intellectual history (which would occupy a book) nor an abbreviated course, but a tour We are led to generalize the effective Lagrangian (4) to of key themes, including Yang–Mills theory, the insight include the terms from superconductivity that spontaneous breaking of a

(eµ) GF µ gauge symmetry endows gauge bosons with mass, and V A = − ν¯µγµ(1 γ5)µ eγ¯ (1 γ5)νe + h.c., (7) the development of the electroweak theory as we know L − √2 − − it. The aim here is to stress the interplay of ideas from in the familiar current-current form. diverse sources and to show how the electroweak theory Because the weak interaction acts at a point, the effec- responds to the established phenomenology of the weak tive Lagrangians hold only over a finite range of energies, interactions. 4

µ A. Symmetries and Interactions A Aµ. Thus has local gauge invariance led to the exis- tence of a massless photon. Notions of symmetry lie at the heart of much of sci- The construction of Quantum Electrodynamics as the 66 ence, and a confidence in the importance of symmetry gauge theory based on U(1)em phase symmetry pro- is a guiding principle for scientists in many disciplines. vides a template for building other interactions derived Werner Heisenberg’s quasi-Biblical pronouncement, »Am from symmetries. In 1954, as isospin emerged as a re- Anfang war die Symmetrie« [“In the beginning was Sym- liable classification symmetry for nuclear levels and as 67 metry”],62 resonates in much theoretical work from the a tool for understanding nuclear forces, Yang & Mills 68 early twentieth century to the present. An essential in- [see also Shaw ] asked whether isospin, promoted to a sight of our modern conception of nature is that symme- local symmetry, could lead to a theory of nuclear forces. tries dictate interactions. It is a lovely idea: derive the strong interactions among While Heisenberg’s assertion can be challenged as mere nucleons by requiring that the theory be invariant un- opinion, physicists have learned over the past century der independent choices at every point of the convention how to connect symmetries with conservation laws, and defining proton and neutron. symmetries with interactions. The 1918 work of Emmy The construction begins with the free-nucleon La- Noether63,64 which took the form of two mathematical grangian theorems, showed that to every continuous global sym- = ψ¯(iγµ∂ m)ψ, (9) metry of the laws of nature there corresponds a conser- L0 µ − p vation law. Thus, translation invariance in space—the written in terms of the composite fermion fields ψ ( n ). statement that the laws are the same everywhere—is The Lagrangian (9) is invariant under global isospin≡ ro- connected with conservation of momentum. Invariance tations ψ exp(iτ α/2)ψ, where τ is a Pauli isospin under translations in time is correlated with the conser- → · Jµ ¯ µ τ matrix, and the isospin current = ψγ 2 ψ is con- vation of energy. Invariance under rotations implies the served. Now require invariance under a local gauge trans- conservation of angular momentum. Noether’s theorem formation, ψ(x) ψ′(x) = G(x)ψ(x), with G(x) shows how conservation laws could arise, and indeed how exp(iτ α(x)/2).→ The construction is similar to the one≡ they could be exact statements, not merely summaries of made for· QED, but is more involved because of the non- empirical evidence. Abelian nature of the SU(2) isospin gauge group. In this The derivation of interactions from symmetries was case, we find an isovector of gauge fields, corresponding to initiated by Weyl65 in a series of papers from 1918 the adjoint representation of SU(2). The gauge fields sat- b b b to 1929, spanning the invention of quantum mechan- isfy the transformation law µ′ = µ α µ (1/g)∂µα ics. In the version that became a prototype for mod- l l − j×k − l or, in component form, bµ′ = bµ εjklα b (1/g)∂µα : ern gauge theories, Weyl showed that by requiring that the translation familiar from QED− plus an− isospin rota- the laws of nature be invariant under local changes of tion. Here g is the coupling constant of the theory. The the phase convention for the quantum-mechanical wave field-strength tensor is F l = ∂ bl ∂ bl + gε bj bk. iα(x) µν ν µ µ ν jkl µ ν function, ψ(x) ψ′(x) = e ψ(x), one could derive 1 −l l → It is convenient to define Fµν = 2 Fµν τ . Then we may the laws of electrodynamics. Invariance under global write the Yang–Mills Lagrangian as (coordinate-independent) U(1)em phase rotations implies the conservation of electric charge; invariance under local ¯ µ g b ¯ µ 1 µν YM = ψ(iγ ∂µ m)ψ µ ψγ τ ψ 2 tr(Fµν F ) (coordinate-dependent) U(1)em phase rotations implies L − − 2 · − g µ 1 µν the existence of a massless vector field—the photon—that = 0 bµ ψγ¯ τ ψ tr(Fµν F ) , (10) couples minimally to the conserved current of the theory. L − 2 · − 2 A straightforward derivation leads to the Lagrangian, namely a free Dirac Lagrangian plus an interaction term that couples the isovector gauge fields to the conserved = J µA 1 F F µν (8) LQED Lfree − µ − 4 µν isospin current, plus a kinetic term that now describes ¯ µ ¯ µ 1 µν = ψ(iγ ∂µ m)ψ qAµψγ ψ 4 Fµν F , both the propagation and the self-interactions of the − − − gauge fields. As in the case of electromagnetism, a mass where ψ is the electron field, ∂µ + iqAµ(x) µ is the term quadratic in the gauge fields is incompatible with gauge-covariant derivative, J µ = qψγ¯ µψ is the≡ conservedD local gauge invariance, as in electromagnetism, but noth- electromagnetic current, and the field-strength tensor is ing forbids a common nonzero mass for the nucleons. The µν νµ ν µ µ ν µν F = F = ∂ A ∂ A . The Fµν F term, which quadratic term in the gauge fields present in the field- accounts− for photon propagation,− is called the kinetic strength tensor gives rises to self-interactions among the term. Under a local phase rotation, the photon field gauge bosons that are not present in Abelian theories transforms as Aµ(x) Aµ(x) ∂µα(x), the familiar form such as QED. of a gauge transformation→ in− (even classical) electrody- The discovery that interactions may be derived from namics. The electron mass term ( mψψ¯ ) respects the lo- isospin symmetry, and from a general gauge group,69 pro- cal gauge symmetry. A photon mass− term would have the vides theorists with an important strategy for deriving 1 2 µ form γ = 2 m A Aµ, which conflicts with local gauge potentially well-behaved theories of the fundamental in- invarianceL because AµA (Aµ ∂µα)(A ∂ α) = teractions. Nuclear forces are not mediated by massless µ → − µ − µ 6 5 spin-1 particles, so the Yang–Mills theory does not suc- where the kinetic term for the gauge fields is ceed in the goal that motivated it. Nevertheless, the ap- 1 l lµν 1 µν proach underlies two new laws of nature: quantum chro- gauge = F F fµν f (13) L − 4 µν − 4 modynamics and the electroweak theory. and the field-strength tensors are F l = ∂ bl ∂ bl + µν ν µ − µ ν gε bj bk for the SU(2) gauge fields and f = ∂ jkl µ ν L µν ν Aµ − B. SU(2)L ⊗ U(1)Y ∂ for the U(1) gauge field. The matter term is µAν Y

The Yang–Mills experience shows that there is no guar- ig′ = R¯ iγµ ∂ + Y R (14) antee that a gauge theory built on a particular symmetry Lleptons e  µ 2 Aµ  e will faithfully describe some aspect of matter. A great L¯ µ ig′ ig L deal of art and, to be sure, trial and error, goes into + eiγ ∂µ + µY + τ bµ e.  2 A 2 ·  the selection of the right gauge symmetry. In the late 1950s and early 1960s, several authors advanced propos- The coupling of the weak-isospin group SU(2)L is called als for a gauge theory of the weak interactions, or of g, as in the Yang-Mills theory, and the coupling constant a “unified” theory of the weak and electromagnetic in- for the weak-hypercharge group U(1)Y is denoted as g′/2. teractions, reading clues from experiment as best they 1 the factor 2 being chosen to simplify later expressions. could. Even after what would become the standard Similar structures appear for the hadronic weak interac- SU(2) U(1) electroweak theory had emerged and was L ⊗ Y tions, now expressed in terms of quarks. The universal elaborated, imaginative theorists put forward alternative strength of charged-current interactions follows from the ideas, guided either by experimental hints or by aesthet- fact that both the left-handed quarks and the left-handed ics. We do not (yet) have a way of deducing the correct leptons reside in weak-isospin doublets. gauge symmetry from higher principles. The theory of weak and electromagnetic interactions What turned out to be the correct choice was elabo- 1 described by the Lagrangian Eq. (13) is not a satisfac- rated by Glashow in 1961. Let us review the essential tory one, for two immediately obvious reasons. It con- structure to recall why a new idea was needed to arrive tains four massless gauge bosons (b1,b2,b3, ), whereas at a successful theory, even after the correct symmetry Nature has but one, the photon. In addition,A the ex- had been chosen. The leptonic elements of the theory pression (15) represents a massless electron; it lacks the will suffice to exhibit the motivation and the principal meee¯ term of the QED LagrangianEq. (8), and for good features. reason.− A fermion mass term links left-handed and right- We begin by designating the spectrum of fundamental 1 1 handed components: ee¯ = e¯(1 γ5)e + e¯(1 + γ5)e = fermions of the theory. It suffices for the moment to 2 − 2 e¯ReL +¯eLeR. The left-handed and right-handed compo- include only the electron and its neutrino, which form nents of the electron transform differently under SU(2)L a left-handed “weak-isospin” doublet (cf. Eq. (6)), Le ν 1 ≡ and U(1)Y , so an explicit fermion mass term would break ( ) , where the left-handed states are νL = (1 γ5)ν e L 2 the SU(2)L U(1) gauge invariance of the theory: such 1 − ⊗ Y and eL = 2 (1 γ5)e. For the reasons we reviewed in §II, a mass term is forbidden. it is convenient− to assume that the right-handed state 1 νR = 2 (1 + γ5)ν does not exist. Thus we designate only R 1 one right-handed lepton, e = eR = 2 (1 + γ5)e, which is C. Insights from Superconductivity a weak-isospin singlet. This completes a specification of the charged weak currents. How gauge bosons can acquire mass is a conundrum To incorporate electromagnetism, Glashow defines a for both the Yang–Mills theory as a description of nuclear “weak hypercharge,” Y . Requiring that the Gell-Mann– forces and for the SU(2)L U(1) theory as a description 1 ⊗ Y Nishijima relation for the electric charge, Q = I3 + 2 Y , be of the weak and electromagnetic interactions. An impor- satisfied leads to the assignments Y = 1 and Y = 2. L − R − tant general insight is that the symmetries of the laws of By construction, the weak-isospin projection I3 and the nature need not be manifest in the outcome of those laws. weak hypercharge Y are commuting observables. Hidden (or secret) symmetries are all around us in the We now take the (product) group of transformations everyday world—for example, in the ordered structures generated by I and Y to be the gauge group SU(2) L ⊗ of crystals and snowflakes or the spontaneous magneti- U(1)Y of the theory. To construct the theory, we intro- zation of a ferromagnetic substance, configurations that duce the gauge fields belie the O(3) rotation symmetry of electromagnetism. [See70 for an interesting tour of spontaneous symmetry b1 ,b2 ,b3 for SU(2) , µ µ µ L (11) breaking in many physical contexts.] The common fea- for U(1) . Aµ Y ture of these phenomena is that the symmetry exhibited Evidently the Lagrangian for the theory may be written by the state of lowest energy, the vacuum, is not the as full symmetry of the theory. In addition, the vacuum is degenerate, characterized by many states of the same = + (12) energy, and the choice of any one is aleatory. L Lgauge Lleptons 6

As it happens, superconductivity, a rich and fascinat- which describes a pair of scalar particles with common ing phenomenon from condensed-matter physics, points mass µ. Thus the introduction of a symmetric interaction the way to understanding how gauge bosons can ac- preserves the spectrum of the free theory with λ =0. quire mass. In 1911, shortly after he succeeded in liq- For the choice µ2 < 0, a line of minima| lie| along uefying helium, and therefore could conduct experiments φ2 = µ2/ λ v2, a continuum of distinct vac- 0 − | | ≡ at unprecedented low temperatures, Heike Kamerlingh uum states, degenerate in energy. The degeneracy follows Onnes71 observed the sudden vanishing of electrical re- from the SO(2) symmetry of the potential (15). Desig- sistance in a sample of mercury cooled to 4.2 K. This first nating one state as the vacuum selects a preferred direc- miracle of superconductivity is of immense technological tion in (φ1, φ2) internal symmetry space, and amounts to importance, not least in magnets that are essential com- a spontaneous breakdown of the SO(2) symmetry. Let ponents of the Large Hadron Collider. us select as the physical vacuum state the configuration v The second miracle, which for me marks supercon- φ 0 = ( 0 ), as we may always do with a suitable defini- ductivity as truly extraordinary, was discovered in 1933 tionh i of coordinates. Expanding about the vacuum con- 72 η by Meissner & Ochsenfeld [for an English translation, figuration by defining φ′ φ φ , we obtain ≡ − h i0 ≡ ζ see73]: magnetic flux is excluded from the superconduct- the Lagrangian for small oscillations  ing medium. A typical penetration depth74 is on the = 1 [(∂ η)(∂µη)+2µ2η2]+ 1 [(∂ ζ)(∂µζ)], (18) order of 10 µm. This means that, within the supercon- Lso 2 µ 2 µ ductor, the photon has acquired a mass. Here is the germ plus an irrelevant constant. There are still two particles of the idea that leads to understanding how the force in the spectrum. The η-particle, associated with radial particles in gauge theories could be massive: QED is a oscillations, has (mass)2 = 2µ2 > 0. The ζ-particle, gauge theory, and under the special circumstances of a however, is massless. The mass− of the η-particle may be superconductor, the normally massless photon becomes viewed as a consequence of the restoring force of the po- massive, while electric charge remains a conserved quan- tential against radial oscillations. In contrast, the mass- tity. lessness of the ζ-particle is a consequence of the SO(2) Two decades would pass before the idea would be fully invariance of the Lagrangian, which means that there is formed and ready for application to theories of the fun- no restoring force against angular oscillations. It is ironic damental interactions. The necessary developments in- that the η-particle, which here seems so unremarkable, is cluded the elaboration of relativistic quantum field the- precisely what emerges as the “Higgs boson” when the ory and the full realization of QED, a focus on the con- hidden symmetry is a gauge symmetry. sequences of spontaneous symmetry breaking, the emer- The splitting of the spectrum and the appearance of gence of informative theories of superconductivity, and the massless particle are known as the Goldstone phe- attention to the special features of gauge theories. nomenon. Such massless particles, zero-energy excita- tions that connect possible vacua, are called Nambu– Goldstone bosons. Many occurrences are known in par- D. Spontaneous Symmetry Breaking ticle, nuclear, and condensed-matter physics.76 In any field theory that obeys the “usual axioms,” including lo- A key insight into hidden symmetry in field theory was cality, Lorentz invariance, and positive-definite norm on achieved by Jeffrey Goldstone,75 who considered the La- the Hilbert space, if an exact continuous symmetry of grangian for two scalar fields φ1 and φ2, the Lagrangian is not a symmetry of the physical vac- uum, then the theory must contain a massless spin-zero = 1 [(∂ φ )(∂µφ )+(∂ φ )(∂µφ )] V (φ2 +φ2). (15) L 2 µ 1 1 µ 2 2 − 1 2 particle (or particles) whose quantum numbers are those of the broken group generator (or generators).77 The Lagrangian is invariant under the group SO(2) of This strong statement seemed a powerful impediment rotations in the φ1-φ2 plane. It is informative to consider to the use of spontaneous symmetry breaking in realis- the effective potential tic theories of the fundamental interactions, as the dis- ease of unobserved massless spin-0 particles was added V (φ2)= 1 µ2φ2 + 1 λ (φ2)2, (16) 2 4 | | to the disease of massless gauge bosons. Motivated by analogy with the plasmon theory of the free-electron gas φ φ1 φ2 2 2 78 where = φ2 and = φ1 + φ2, and distinguish two Anderson put forward a prescient conjecture that one cases.   zero-mass ill might cancel the other and make possible a A positive value of the parameter µ2 > 0 corresponds realistic Yang–Mills theory of the strong interactions. to the ordinary case of unbroken symmetry. The unique The decisive contributions came at a time of intense minimum, corresponding to the vacuum state, occurs at interest in superconductivity—in the intricacies of the 0 79 φ 0 = ( 0 ), and so for small oscillations the Lagrangian Bardeen–Cooper–Schrieffer (BCS) theory and in un- takesh i the form derstanding the role of symmetry breaking in the Meiss- ner effect. From the remove of a half century, it seems 1 µ 2 2 1 µ 2 2 so = 2 [(∂µφ1)(∂ φ1) µ φ1]+ 2 [(∂µφ2)(∂ φ2) µ φ2], to me that preoccupation with the microscopic BCS the- L − − (17) ory might have complicated the search for a cure for the 7 massless gauge bosons. An easier path is to analyze the gauge symmetry to complete the program set out by Shel- phenomenological Ginzburg–Landau80,81 description of don Glashow1 that we recalled in §IIIB. The construc- the superconducting phase transition in the framework tion of the spontaneously broken SU(2)L U(1)Y theory of QED. It is then easy to see how the photon acquires of the weak and electromagnetic interactions⊗ is detailed mass in a superconducting medium [see, e.g., Problem in many places, including §2 of “Unanswered Questions,” 5.7 of Ref. 9, §21.6 of Ref. 82 and the “Abelian Higgs Ref. 26, so we may focus here on a few important con- Model,”83]. But that is hindsight and speculation! ceptual matters. Searching for a solution to the problem of mass- If SU(2) U(1) proves to be the apt choice of 18 L Y less gauge bosons in field theory, Englert & Brout, gauge symmetry⊗ for a theory of weak and electromag- 19,20 21 Higgs, and Guralnik, Hagen, & Kibble showed that netic ineractions, then that symmetry must be hidden, gauge theories are different. They do not satisfy the or broken down to the U(1)em symmetry we observe assumptions on which Goldstone theorem is based, al- manifestly. The simple choice made by Weinberg and though they are respectable field theories. Recall that Salam, which now has significant empirical support, is to quantize electrodynamics, an exemplary gauge theory, to introduce a complex weak-isospin doublet of auxil- one must choose between the covariant Gupta–Bleuler iary scalar fields, and to contrive their self-interactions formalism with its unphysical indefinite-metric states or to create a degenerate vacuum that does not exhibit quantization in a physical gauge for which manifest co- the full SU(2)L U(1)Y symmetry. Before spontaneous variance is lost. Through their work, we understand that symmetry breaking,⊗ we count eight degrees of freedom the would-be Goldstone bosons that correspond to bro- among the four massless gauge bosons and four degrees ken generators of a gauge symmetry become the longi- of freedom for the scalar fields. After spontaneous break- tudinal components of the corresponding gauge bosons. ing of SU(2)L U(1)Y U(1)em, following the path What remains as scalar degrees of freedom is an incom- we have just reviewed⊗ in→ §III D, the scalar field ob- plete multiplet—defined by the unbroken generators of 0 tains a vacuum-expectation value, φ = , where the gauge symmetry—of massive particles that we call 0 v/√2 1 h i   Higgs bosons. v = (G √2)− 2 246 GeV to reproduce the low-energy F ≈ Their collective insight did not, as many had hoped, phenomenology. make a proper description of the strong nuclear force out The charged gauge bosons, W ±, which mediate the of Yang–Mills theory. It did, however, set the stage for V A charged-current interaction, acquire mass gv/2. the development of the electroweak theory and for plau- The− neutral gauge bosons of Eq. (11) mix to yield a 0 sible, if still speculative, unified theories of the strong, massive (Mz = MW / cos θW) neutral gauge boson, Z , weak, and electromagnetic interactions. that mediates a hitherto unknown weak-neutral-current It is inaccurate to say that the work of these theorists interaction and a massless photon, γ. [The weak mix- 3 solved a problem in the standard model—the standard ing angle θW, which parametrizes the mixing of bµ and model did not yet exist! Indeed, they were not con- µ, is determined from experiment.] The photon has cerned with the weak interactions, and the implications Apure vector couplings, as required, whereas Z0 has a mix for fermion mass shifts are mentioned only in passing. of vector and axial-vector couplings that depend on the [Recall that for nonchiral theories such as QED and the quantum numbers of the fermion in question. Eleven of Yang-Mills theory, the origin of fermion masses does not the twelve bosonic degrees of freedom now reside in the arise, in the sense that fermion mass is consistent with vector bosons: 3 3 massive bosons + 1 2 massless the gauge symmetry.] Their work can be said to have photon. The last× degree of freedom corresponds× to the triggered the conception of the electroweak theory, which Higgs boson: it is a massive scalar, but the Weinberg– is a very considerable achievement. Salam theory does not fix its mass. Because the Higgs bo- Following the discovery of the Higgs boson of the elec- son and the longitudinal components of the gauge bosons troweak theory, François Englert84 and Peter Higgs85 share a common origin, the Higgs boson plays an essen- shared the 2014 Nobel Prize for Physics. Gerald Gu- tial role in ensuring a sensible high-energy behavior of ralnik and Richard Hagen86 have published a memoir of the electroweak theory.93 their work. In addition, several of the leading actors in All this is fixed by the construction of the theory: once the discovery of spontaneous gauge symmetry breaking the representation of the auxiliary scalar fields is chosen as an origin of particle mass have described their personal and the weak mixing parameter determined, all the cou- involvement: Anderson,87 Englert,88 Guralnik,89,90 and 91,92 plings of gauge bosons to fermions and couplings among Higgs. Their words carry a special fascination. gauge and Higgs bosons are set. The new neutral-current interactions among the leptons are flavor-diagonal. Also, to this point, we have solved only one of the outstand- E. The Electroweak Theory and the ing problems of the unbroken SU(2)L U(1)Y theory: Standard-Model Higgs Boson the masslessness of all the gauge bosons.⊗ What of the fermions? Weinberg and Salam saw the possibility to In the late 1960s, Steven Weinberg2 and Abdus Salam3 generate fermion masses in the spontaneously broken the- used the new insights about spontaneous breaking of ory by adding to the Lagrangian a gauge-invariant inter- 8 action, = ζ [R¯ (φ†L ) + (L¯ φ)R ], where the ery of charm (hidden first, then open), which was re- LYukawa − e e e e e Yukawa coupling, ζe, is a phenomenological parameter. quired in the framework of the electroweak theory, once When the gauge symmetry is hidden, the Yukawa term neutral currents had been observed. The discovery of becomes the W and Z was the second great triumph of the elec- troweak theory. Experiment also brought new evidence ζev ζeH Yukawa = ee¯ ee,¯ (19) of richness, including the the discovery the τ lepton and L − √2 − √2 evidence for a distinct τ neutrino and the discovery of the b quark. The top quark completed a third quark where H is the Higgs boson. The electron has acquired a generation; the top-quark mass became an essential in- mass m = ζ v/√2, and the Hee¯ coupling is im /v. It e e e put to quantum corrections to predictions for precisely is pleasing that the electron mass arises spontaneously,− measured observatbles. Moreover, finding a third quark but frustrating that the parameter ζ must be put in by e generation opened the way to understanding, at least at hand, and does not emerge from the theory. The same an operational level, the systematics of CP violation.61 strategy carries over for all the quarks and charged lep- Highly detailed studies at many laboratories confirmed tons, and may also be seen as the origin for the parame- the predictions of the electroweak theory to an extraor- ters of the quark-mixing matrix. dinary degree. If the Higgs field is the source of the quark and charged- As the electroweak theory emerged as a new law of lepton masses, that does not mean that the “Higgs boson nature, the question of how the electroweak symme- is the source of all mass in the Universe,” as is frequently try was hidden came to center stage. While the de- stated—even by physicists. The overwhelming major- fault option—the one emphasized in textbooks—was an ity of the visible mass in the Universe is in the form elementary scalar Higgs boson, electroweak symmetry of atoms, and most of that is made up of nucleon mass, breaking through some sort of new strong dynamics, or as which arises as confinement energy in Quantum Chromo- a message from extra spatial dimensions, or as an emer- dynamics.94 Electroweak symmetry breaking is decidedly gent phenomenon arising from strong interactions among a minor player. the weak bosons all received attention from both theory Only three quark flavors (u,d,s) were known when the and experiment. electroweak theory was formulated. The weak-isospin u doublet ( d cos θC+s sin θC )L, where θC is the Cabibbo angle, captured the known structure of the hadronic charged- current interaction and expressed the universal strength IV. AFTER THE DISCOVERY—OUTLOOK of quark and lepton interactions. Within the Weinberg– Salam framework, however, this single quark doublet The most succinct summary we can give is that the gives rise to flavor-changing s d neutral-current in- data from the ATLAS and CMS experiments are devel- teractions that are not observed↔ in nature. Glashow, Il- oping as if electroweak symmetry is broken spontaneously iopoulos, and Maiani95 noted that the unwanted interac- through the work of elementary scalars, and that the em- tions could be cancelled by introducing a second quark blem of that mechanism is the standard-model Higgs bo- c doublet, ( s cos θC d sin θC )L, involving a new “charmed” son. I refer to Refs. 22–25 for details and to Ref. 17 for quark and the orthogonal− combination of d and s. The perspective. absence of flavor-changing neutral currents generalizes to The bare facts are these: the LHC experiments have more (complete) quark doublets, and is a striking feature found a new unstable particle H, with a mass in the + of the experimental data. neighborhood of 125 GeV. It decays into γγ, W W −, To save writing, I have outlined here a theory of a sin- and Z0Z0 in approximately the proportions expected for gle generation of leptons; the other lepton families are a standard-model Higgs boson. The new particle is nar- included as simple copies. However, a theory of leptons row, for its mass, with the current bounds measured alone would be inconsistent. In our left-handed world, in tens of MeV. The dominant production mechanism each doublet of leptons must be accompanied by a color- has characteristics compatible with gluon fusion through triplet weak-isospin doublet of quarks, in order that the a heavy-quark loop, as foreseen. Topological selections theory be anomaly free, i.e., that quantum corrections re- have identified a subsidiary mechanism compatible with spect the symmetries on which the theory is grounded.96 vector-boson fusion. Some evidence has been presented + Since its invention, the electroweak theory has been for the decays H b¯b and τ τ −. No decays that entail supported again and again by new observations, in many lepton-flavor violation→ have been observed. The new par- cases arising from experiments conceived or reoriented ticle does not have spin 1, studies of decay angular distri- explicitly to test the electroweak theory. I treated this butions and correlations among decay products strongly + question in some detail in §3 of “Unanswered Ques- favor spin-parity 0 over 0−, and while spin 2 has not tions”,26 to which I refer the reader for specific refer- been excluded in the most general case, that assignment ences. It will suffice here to mention some of the major is implausible. supporting elements. The first great triumph of the elec- As one measure of the progress the discovery of the troweak theory was the discovery of weak neutral cur- Higgs boson represents, let us consider some of the ques- rents. This was followed in short order by the discov- tions I posed before the LHC experiments in Ref. 26. 9

Future Issues (from “Unanswered Questions . . . ”) 2. Do the HZZ and HW W couplings indicate that H(125) is solely responsible for electroweak sym- 1. What is the agent that hides the electroweak symme- metry breaking, or is it only part of the story? try? Specifically, is there a Higgs boson? Might 3. Does the Higgs field give mass to fermions beyond To the best of our knowledge, there be several? the third generation? Does H(125) account quan- displays the characteristics of a standard- H(125) titatively for the quark and lepton masses? What model Higgs boson, an elementary scalar. Searches sets the masses and mixings of the quarks and lep- will continue for other particles that may play a role tons? in electroweak symmetry breaking. 4. What stabilizes the Higgs-boson mass below 2. Is the “Higgs boson” elementary or composite? How 1 TeV? does the Higgs boson interact with itself? What triggers electroweak symmetry breaking? We have 5. Does the Higgs boson decay to new particles, or via not yet seen any evidence that H(125) is other new forces? than an elementary scalar. Searches for a com- posite component will continue. The Higgs-boson 6. What will be the next symmetry recognized in self-interaction is almost certainly out of the reach Nature? Is Nature supersymmetric? Is the elec- of the LHC; it is a very challenging target for fu- troweak theory part of some larger edifice? ture, very-high-energy, accelerators. We don’t yet 7. Are all the production mechanisms as expected? know what triggers electroweak symmetry breaking. 8. Is there any role for strong dynamics? Is elec- 3. Does the Higgs boson give mass to fermions, or only troweak symmetry breaking related to to the weak bosons? What sets the masses and through extra spacetime dimensions? mixings of the quarks and leptons? The experi- mental evidence suggests that H(125) couples to tt¯, 9. What lessons does electroweak symmetry breaking + b¯b, and τ τ −, so the answer is probably yes. All hold for unified theories of the strong, weak, and these are third-generation fermions, so even if the electromagnetic interactions? evidence for these couplings becomes increasingly robust, we will want to see evidence that H cou- 10. What implications does the value of the H(125) ples to lighter fermions. The most likely candidate, mass have for speculations that go beyond the stan- perhaps in High-Luminosity LHC running, is for dard model? . . . for the range of applicability of the the Hµµ coupling, which would already show that electroweak theory? the third generation is not unique in its relation to In the realms of refined measurements, searches, and the- H. Ultimately, to show that spontaneous symmetry breaking accounts for electron mass, and thus en- oretical analysis and imagination, great opportunities lie ables compact atoms, we will want to establish the before us! Hee¯ coupling. That is extraordinarily challenging because of the minute branching fraction. Summary Points 10. What lessons does electroweak symmetry break- ing hold for unified theories of the strong, weak, 1. The ATLAS and CMS Collaborations, working at and electromagnetic interactions? Establishing that CERN’s Large Hadron Collider, have discovered a scalar fields drive electroweak symmetry breaking new particle, H(125 GeV), that matches the profile will encourage the already standard practice of us- of the Higgs boson of the electroweak theory. ing auxiliary scalars to hide the symmetries that 2. Observation of decays into the weak bosons, H underlie unified theories. + 0 0 → W W − and H Z Z , establishes a role for the Higgs boson in→ hiding the electroweak symmetry To close, I offer a revised list of questions to build on and endowing the weak bosons with mass. what our first look at the Higgs boson has taught us. + 3. Evidence for the decays H b¯b and H τ τ −, together with characteristics→ of H production→ that implicate gluon fusion through a top-quark loop, Issues Sharpened by the Discovery of H (125) suggest that the Higgs boson also plays a role in giving mass to the fermions. 1. How closely does H(125) hew to the expectations for a standard-model Higgs boson? Does H have 4. It will be important to show that H couples to any partners that contribute appreciably to elec- quarks and leptons of the first two generations and troweak symmetry breaking? to test its role in generating their masses. 10

5. If the electron mass, in particular, does arise from mediated by virtual heavy particles are replaced the vacuum expectation value of the Higgs field, by local interactions that yield the same low-energy we will have a new understanding of why compact limit. The effective theory can only be a valid de- atoms exist, why valence bonding is possible, why scription of physics at energies below the masses of liquids and solids can form (cf. §4.4.2 of Ref. 26). the heavy particles, and must be superseded by a more complete (but perhaps still effective theory) 6. The spin-parity of H, which is strongly indicated on that energy scale. as 0+, favors the interpretation as an elementary scalar. Flavor-changing neutral current: A transition that changes quark or lepton flavor, without chang- 7. Even after its apparent completion by the obser- ing electric charge; strongly inhibited by the GIM vation of a light Higgs boson, the electroweak the- mechanism in the standard electroweak theory. ory raises puzzles. An outstanding question is why the electroweak scale is so much smaller than other GIM mechanism: Observation by Glashow, Iliopoulos, plausible physical scales, such as the unification and Maiani95 that flavor-changing neutral-current scale and the Planck scale. interactions vanish at tree level, and are strongly in- hibited at higher orders, provided that quarks (and 8. It is possible that the Higgs boson experiences new leptons) occur in SU(2) doublets. Argued for the forces or decays into hitherto unknown particles. L necessity of the charm quark.

Goldstone Phenomenon: The appearance of massless DISCLOSURE STATEMENT modes whenever a global continuous symmetry of the Lagrangian is broken, in the sense that the vac- The author is not aware of any affiliations, member- uum state does not display the full symmetry of the ships, funding, or financial holdings that might be per- Lagrangian. One massless scalar or pseudoscalar ceived as affecting the objectivity of this review. appears for each broken generator of the full sym- metry.

ACKNOWLEDGMENTS Higgs boson: Elementary scalar particle that is the avatar of electroweak symmetry breaking in the Fermilab is operated by Fermi Research Alliance, standard electroweak theory, an excitation of the LLC under Contract No. DE-AC02-07CH11359 with the auxiliary scalar fields introduce to contrive a vac- United States Department of Energy. I am grateful uum that does not respect the full SU(2)L U(1) ⊗ Y to Professor Heinrich Meier and the Carl Friedrich von symmetry on which the electroweak theory is built. Siemens Foundation for gracious hospitality during the An unstable particle with mass 125 GeV that final stages of writing. closely fits the profile of the Higgs boson was dis- I want also to pay my respects to the designers, covered in experiments at CERN in 2012. builders, and operators of the Large Hadron Collider and of the ATLAS and CMS experiments; to the experimen- Lepton: An elementary (at the present limits of resolu- 1 tal teams for their outstanding achievement; and to the tion) spin- 2 particle that does not experience the generations of scientists whose work has brought us to . The current roster is composed the point of savoring the discovery of the Higgs boson. of three charged particles, e−,µ−, τ −, and three neutrinos, νe,νµ,ντ .

Appendix A: Terms / Definitions Neutral current: The weak interaction mediated by 0 the Z -boson, first observed in the reactions νµe ν e and ν N ν + anything. → Anomaly: The violation by quantum corrections of a µ µ → µ symmetry of the Lagrangian. If anomalies violate Quark: An elementary (at the present limits of resolu- gauge symmetry, the theory becomes inconsistent, 1 tion) spin- 2 particle that experiences the strong in- so the freedom from anomalies becomes a powerful teraction. The current roster is composed of six condition on candidate theories. species: up, down, charm, strange, top, bottom, Charged current: The weak interaction mediated by grouped in three weak-isospin doublets. the W -boson, first observed in nuclear β decay. ± Superconductivity: A phenomenon that occurs in Effective field theory: A description valid over a par- many materials when they are cooled to low tem- ticular range of energies or distance scales, based on peratures or subjected to high pressure, supercon- the degrees of freedom most relevant to the phe- ductivity entails zero electrical resistance and the nomena that occur there. Nonlocal interactions expulsion of magnetic fields. 11

Appendix B: Abbreviations and Acronyms CMS: The Compact Muon Solenoid, one of two general- purpose experiments for the Large Hadron Collider. ATLAS: One of two general-purpose experiments for It is located in Cessy, France. cms.cern.ch. the Large Hadron Collider, located adjacent to CERN’s main campus. atlas.ch. LHC: The Large Hadron Collider at CERN is a two- bore proton synchrotron 27 km. in circumference. CERN: The European Laboratory for Particle Physics It is designed to provide proton-proton collisions straddles the French-Swiss border near Geneva. up to 14-TeV c.m. energy and luminosity exceed- Its principal research instrument is now the Large ing 1034 cm 2 s 1, as well as Pb-Pb and proton-Pb Hadron Collider. One of Europe’s first common − − collisions. lhc.web.cern.ch. undertakings at its founding in 1954, CERN now includes twenty-one Member States. cern.ch

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