Origins of Mass

Cent. Eur. J. Phys. • 10(5) • 2012 • 1021-1037 DOI: 10.2478/s11534-012-0121-0

Central European Journal of Physics

Origins of mass

Review Article

Frank Wilczek1∗

1 Center for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA

Received 01 July 2012; accepted 16 August 2012

Abstract: Newtonian mechanics posited mass as a primary quality of matter, incapable of further elucidation. We now see Newtonian mass as an emergent property. That mass-concept is tremendously useful in the approximate description of baryon-dominated matter at low energy – that is, the standard “matter” of everyday life, and of most of science and engineering – but it originates in a highly contingent and non-trivial way from more basic concepts. Most of the mass of standard matter, by far, arises dynamically, from back-reaction of the color gluon fields of quantum chromodynamics (QCD). Additional quantitatively small, though physically crucial, contributions come from the intrinsic masses of elementary quanta (electrons and quarks). The equations for massless particles support extra symmetries – specifically scale, chiral, and gauge symmetries. The consistency of the standard model relies on a high degree of underlying gauge and chiral symmetry, so the observed non-zero masses of many elementary particles (W and Z bosons, quarks, and leptons) requires spontaneous symmetry breaking. Superconductivity is a prototype for spontaneous symmetry breaking and for mass-generation, since photons acquire mass inside superconductors. A conceptually similar but more intricate form of all-pervasive (i.e. cosmic) superconductivity, in the context of the electroweak standard model, gives us a successful, economical account of W and Z boson masses. It also allows a phenomenologically successful, though profligate, accommodation of quark and lepton masses. The new cosmic superconductivity, when implemented in a straightforward, minimal way, suggests the existence of a remarkable new particle, the so-called Higgs particle. The mass of the Higgs particle itself is not explained in the theory, but appears as a free parameter. Earlier results suggested, and recent observations at the Large Hadron Collider (LHC) may indicate, the actual existence of the Higgs particle, with mass mH ≈ GeV. In addition to consolidating our understanding of the origin of mass, a Higgs particle with mH ≈ GeV could provide an important clue to the future, as it is consistent with expectations from supersymmetry.125 PACS (2008): 14.80.Bn, 11.30.Qc,125 12.10.Kt, 12.38.Gc, 11.30.Rd Keywords: mass • dimensional transmutation • Higgs particle • unification • supersymmetry © Versita sp. z o.o.

1. What is mass?

∗ 1.1. Newtonian mass E-mail: [email protected]

In classical physics, as epitomized in Newtonian mechanics, mass is a primary, conserved and irreducible property 1021 Origins of mass

quantity of matter of matter. Reflecting that significance, Newton spoke of highlight an elementary but profound point that is widely mass as [1]. Mass was so foundational overlooked. m v within Newton’s view of the world that he took its cen- Let us contrast the Newtonian and relativistic equations tral feature – what I call Newton’s zeroth law of motion – for momentum, in terms of mass and velocity pN mv ; for granted, without stating it explicitly. Newton’s zeroth law of motion, which underpins use of the others, is the pR mv ; = − v Newtonian inertia (1) conservation of mass. The Newtonian mass of a body is c 1 2 assumed to be a stable property of that body, unaffected = q 2 relativistic inertia (2) by its motion. In any collision or reaction, the sum of c 1 p the masses of the incoming bodies is equal to the sum of the masses of the products; mass can be redistributed, where is the speed of light. is a measure of the body’s but neither created nor destroyed.the Mass also occurs, of resistance to acceleration, or inertia. Both Newtonian and course, in Newton’s gravitational force law. Einsteinian mechanics posit that different observers, who Newtonian mass is, in fact, new primary quality of move at constant velocity relative to one another, construct matter introduced into the foundation of classical physics. equally valid descriptions of physics, using the same laws. It supplements size and shape, which the strict “mechan- To two such observers the body appearsm to move with dif- ical philosophy” of Descartes and of many of Newton’s ferent velocities, and also to have different momenta, but scientific contemporaries had claimed should be sufficient. both observers will infer the same . Similarly, for the energy we havem Relativity, and then quantum field theory, profoundly EN v ; changed the status of mass within physics. Both main 2 properties of Newton’s mass-concept got undermined. In ER = mc Newtonian kinetic: energy (3) 2 v special relativity we learn that energy is conserved, but 2 − c 1 2 mass is not. In general relativity we learn that gravity, in = q 2 relativistic energy (4) the form of space-time curvature, responds to energy, not 1 to mass. The word mass still appears in modern physics, v c

and the modern usage evolved from the earlier one, but it The relationship between Eqs. (1) and (2) is straightfor-ER denotes a radically different, more fluid concept. ward:v Forc , Eq. (2) goes over into Eq. (1). Not so Though these profound changes began in earnest more the relationship between Eqs. (3) and (4). Expanding than a hundred years ago, the old concept of mass re- for , we have approximatelym ER ≈ mc v mc EN : mains deeply embedded in common language and in the 2 2 2 folk physics of everyday life, not to mention in successful + = + (5) engineering practice. 2 E mc The demotion of mass from its position as a logical primi- The first term on the right-hand side2 of Eq. (5) is of course tive in the foundation of physics challenges us to rebuild it the famous mass-energy = associated with bodies on deeper foundations, and opens up the central question at rest. of this paper: What is the Origin of Mass? 1.2. Relativistic mass Now consider two slowly-movingi.e. bodies that interact with each other weakly, so that we can neglect potential energy. If only conservation ( , constancy) of the total m m energy isE assumed,≈ m c v m c v In modern physics energy and momentum are the primary R; 2 1 2 2 2 2 dynamical concepts, while mass is a parameter that ap- total 1 1 2 2 + + + (6) pears in the description of energy and momentum of iso- 2m ; m 2m m lated bodies [2]. The early literature of relativity em- 1 2 1 2 ployed some compromise definitions of mass – specifically, then wem v cannotm v deduce that , or even + is velocity-dependent mass, in both longitudinal and trans- rigorously1 2 constant,2 2 nor that the Newtonian kinetic en- verse varieties. Those notions have proved to be more con- ergy 2 + 2 is constant. Formal hocus-pocus can’t fusing than useful. They do not appear in modern texts or conjure up three independentm m conservation laws from just research work, but they persist in some popularizations, one! Rather, it would be natural to expect, from Eq. (6), 1 2 and of course in old books. To forestall confusion and that small changes in and could accompany the dy-

set the stage, I’ll now brieﬂy recall the pertinent facts namical evolution. Alternatively: Eq. (6) in itself doesE notR and deﬁnitions, using exclusively the now-standard rela- explain why the Newtonian kinetic energy, which after all tivistic mass concept. This will also be an opportunity to is the second, subdominant term in the expansion of , 1022 Frank Wilczek

should be separatelyv c conserved,v v even; v approximately.m m All time of creation. Modern quantum field theory opens the that we can legitimately inferv is/c that if the bodies move possibility of an alternative explanation. 1 2 1 2 slowly, with forc both = 2 2 , then + is ap- For our problem of the origin of mass, this general prin- proximately constant,2 to order . Indeed, if we divide ciple gives most welcome simplification and guidance [3]. Eq. (6) through by , we arrive at v Rather than having to address the mass of each object in ER; /c ≈ m m : c2 the universe separately, we can focus on the properties of 2 total 1 2 2 a few quantum fields, whose excitations (quanta) are the ( + ) 1 + order (7) building blocks of matter. Thus for instance if we understand the properties (including mass) of one electron we These difficulties of principle actually come into play understand the properties of all electrons. More gener- in describing nuclear reactions, where neither mass nor ally: If we understand the properties of the fields associ- Newtonian kinetic energy is separately conserved, even ated with the building blocks of matter, we should be able when all the bodies involved move slowly. to deduce the properties – including mass! – of matter In the most radical departure from the Newtonian frame- itself, and those deductions will be valid universally. work, we are allowed to considerpR bodies;ER with zero mass. 2. Emergent mass (And, as we’ll see,m → that possibilityv → provesc to be very fruit- ful.) The momentum andm energy can have sensible, 2.1. Mass for standard matter finite limits as 0, with appropriately. Thus isolated bodies with = 0 move at the speed of light, and for such bodies wepR have ER /c We can reach a deeply penetrating – though, as we shall

= pR ER (8) see, still incomplete and in part tentative – understanding of the origin of the (Newtonian) mass of ordinary matter but no other restriction on the values of or . following the strategy Newton called Analysis and Syn- These considerations sharpen the challenge of under- thesis, or essentially, in modern language, Reductionism. standing the emergence of Newtonian mass as a valid More specifically, we can build up our description of mat- 1.3.approximation Masses in theof quanta physical world. ter mathematically starting from rigorouslywhy tested properties of a few reproducible building blocks that obey ideally simplehow laws. I will return to further discuss this strategy works, but first let us consider, in meaningful detail, Relativistic quantum field theory introduces a powerful it works. constructive principle into the reductionisti.e program [3]. (By ordinary or standard matter I understand the mat- Since quantum fields create (and destroy) particles, ter encountered in everyday life, and studied in materials space-time uniformity of the fields – . their invariance science, chemistry, biology, geology, and stellar astro- under space and time translation – implies that their as- physics. Excluded for now are the highly unstable species sociated particles will have the same properties, indepen- produced at high-energy accelerators and the astronomi- dent of where and when they are observed. Thus all elec- cal dark matter and dark energy.) trons, for example, have the same properties, because they I will carry out the analysis in two stages: a semi- are all excitations of a single universal quantum field, the phenomenological stage, to identify the basic issues, and electron field. then an exploration of those issues. The first stage occu- Thoughtful atomists, notably including Newton and pies this section of the paper; the second stage most of Maxwell, were highly aware that the most elementary its remainder. facts of chemistry – that is, the exact reproducibility of Atoms, in the sense of modern physics and chemistry, are chemical reactions, including their intricate specific rules a convenient point of departure. An atom consists> : of its of combination – called for the building blocks of matter nucleus, wherein is concentrated all the positive electric to have this feature of accurate sameness, or universality, charge and overwhelmingly most of the mass ( 99 9%, in across space and time. The macroscopic bodies of every- all cases), surrounded by a cloud of much lighter electrons. day experience, of course, do not – they come in different So the first order of business is to understand the origin sizes, shapes, and composition, and can accrete or erode of the mass of atomic nuclei. i.e. over time. In our ordinary experience, only artfully man- The conventional model of nuclei is built on the idea that ufactured products can approximate to uniformity. Both they are made up from protons and neutrons ( , nu- Newton [4] and Maxwell [5] inferred that the basic build- cleons). This picture has both rigorous and approximate ing blocks of matter were manufactured by God at the aspects. The rigorous aspect is based on discrete, ad- 1023 Origins of mass

Ze ditive quantum numbersZ and very accurate conservationA Before leaving the subject of inter-nucleon forces, how- laws. Thus a nucleusP Z with electric chargeN A − Z(or alterna- ever, a few brief comments are in order. Within the con- < − tively, atomic number ) and baryon number uniquely text of nucleon-based models of nuclei, thev/c smallness∼ of corresponds to = protons and = ( ) neutrons. ∆ can be interpreted to imply the smallness of kinetic1 as The approximate aspect is the statement that in nuclei opposed to rest energies, and specifically 10 . In the protons and neutrons retain their identity, so that it this non-relativistic framework, it is natural to use ideas is appropriate to model the nucleus as a collection of in- like effective instantaneous interactions through potentials teracting protons and neutrons. In the context of quantum and many-body position space wave functions. chromodynamics – QCD – this is a highly non-trivial is- Within that circle of ideas, we can address the above- sue! For in QCD the protons and neutrons themselves are − < < − mentioned “foremost× issue”∼ r from∼ the× nuclear side. Why complex objects, built up from the more basic quarks and do nucleons, within a13 nucleus, stay separate?13 At long gluons. One might have anticipated that when nucleons range (that is, 1 10 2 10 cm) the domi- are brought together their quarks and gluons would un- nant inter-nucleon force can be ascribed to pion exchange, dergo complete reorganization, within which the original and is attractive. At nearer distances repulsive vector nucleon structure would become unrecognizable. (In the meson exchanges kick in, but once many channels con- language of the bag model, the issue is why the sepa- tribute, this exchange picture becomes complicated and rate nucleon membranes do not fuse, to produce one big < − then breaksr ∼ × down altogether. Phenomenologically, the bag [6,7].) Why such reorganization does not occur, is the inter-nucleon force14 features a so-called hard core repul- foremost issue regarding the relationship between funda- sion at 5 10 cm. That repulsion, together with the mental QCD and traditional nuclear physics. Before ad- effective repulsion associated with Fermi statistics, under- dressing that issue, however, it is appropriate to recall lies the self-consistency of the nucleon model of nuclei. its empirical grounding. The simplest and most important Though this explanation is convincing, it is both semi- success of the (nearly) independent nucleon model of nu- phenomenological and only semi-quantitative; a fuller, clei is directly related to our central concern, namely its moreZ;A microscopically based derivation of the foundations accountZ;A of nuclear masses. of nuclear physics would be important progress. For the ground state of nuclei with quantum numbers ∆( ) is well approximated using the famous “semi- ( ) we find M Z;A ZMp A − Z Mn Z;A ; empirical mass formula” of Bethe and von Weiszäcker, Z;A M Z;A ; which can be further refined to incorporate shell effects. ( ) = + ( ) + ∆( ) Altogether, then, we have a richly detailed, empirically ∆( ) ( ) (9) successful framework in which nuclear masses can be un- M Z;A ;Mp;Mn derstood, given the mass of protons and neutrons. whereinmass of defect course Z;A( ) denote theM nuclear,Z;A The mass of atoms is, as mentioned previously, overwhelm- proton, and neutron masses respectively and, importantly, ingly dominated by the mass of nuclei. The approximate the ∆( ) is much smaller than ( ). framework that is broadly successful for nuclei – that is, Thus, roughly speaking, the mass of a nucleus is simply slowly moving independent particles (here electrons, to- the sum of the masses of theZ;A protons and neutrons indi- gether with a central nucleus), with interactions described e ≡ α ≈ cated by its charge andEB baryonZ;A number.− Z;A Wec can easily by potentials – becomes extremelyπ precise and accurate Z 2 Zα make this quantitative. ∆( ) is negative, correspond-2 for atoms. (The measure of the electron’s typical1 velocity 4 137 ing to binding energy ( ) = ∆( ) . Nuclear is the fine structure constant . For inner electrons in atoms with high theelectron’s measure is , which binding energies perE Z;A nucleon/A < peak at B ∼ can approach unity. In that case the binding energy can be a significant fraction of the rest energy. A ( ) 10 MeV (10) relativistic treatment is then required, and has been car- < − | Z;A |/M Z;A ∼ : ried through.) The intrinsic mass of electrons is more than so that 2 a thousand times smaller than that of nucleons, and the ∆( ) ( ) 10 (11) atomic analogue of the nuclear mass defect ∆ is smaller still. Thus the change in mass due to interaction, while sig- Thus the problem of understanding origin of atomic masses nificant and of course eminently detectable, is relatively reduces, to a good approximation, to the problem of un- small. In this quantitative sense intra-nucleon forces are derstanding the origin of proton and neutron masses. To far more important than inter-nucleon forces for determin- achieve a superb approximation, we must also understand ing the mass of standard matter. the origin of electron mass. In the course of this reduc- 1024 Frank Wilczek

α tion we have appealed to the smallness of , and to some ally, this degeneracy is broken, but gives rise to some qualitative facts about the inter-nucleon forces; a full un- extremely low-energy degrees of freedom and excitations. derstanding of the origin of the mass of atoms should en- Indeed, magnetic resonance imaging exploits the sensitiv- 2.2.compass The those quantum facts as well. censor, gaps, and seques- ity of nuclear spin excitations to their chemical environ- tration ment. But precisely because so little energy is involved in nuclear spin interactions – in other words, because the spins interact so feebly – the associated transitions have only a minuscule effect on the total mass. The arguments of the preceding subsection concerned the Photons are another potential source of very low-energy ground states of isolated nuclei and individual atoms. Let excitations, immune to quantum censorship. The radia- us now consider how one builds from understanding of the tion fields in standard matter, however, must arise from origin of mass in those cases to understanding the origin transitions in the nuclear or atomic (electronic) degrees of freedom. However nuclei are inert for the reasons just of Newtonian mass for macroscopic bodies.quantum censor discussed, while electronic process put only tiny amounts Three closely related ideas, deeply rooted in quantum theof energy into play, compared to the rest-energy of the ory, make this transition possible.sequestered The nuclei, and so the impactdo of electromagnetic radiation on can effectively suppress the complexity of compositeenergy sys-gap the mass of matter is negligible. Weak decays and other tems. When such systems are , so that they forms of radioactivity put nuclear energies of order ∆ are subject only to perturbations below their , into play. Were these effects faster and more common, they appear as ideally simple particles, in a unique state they would vitiate the Newtonian mass-concept for accu- (their ground state). rate work. Let me illustrate these general ideas in the case that is The same logic of quantum censorship, gaps, and seques- most central to the mass problem for standard matter, that tration that applies to nuclei in standard matter also ap- is nuclear masses. For simplicity of exposition, I will tem- plies another level down, to nucleons within nuclei. Pro- porarily neglect the nuclear spin degree of freedom, and tons and neutrons, in our modern understanding, are com- then return to discuss its inclusion. Nuclei generally sup- plex composite objects, made from essentially massless port, in addition to their ground state, many excited states. quarks and gluons. Their excitations – baryonic reso- The excited states, however, are separated by gaps, typ- nances – are many tens of MeV higher in energy, however. ically of order a few MeV; that is, their energy exceeds Within nuclei, the hard core and fermi statistics sequester that of the ground state by at least a fewT ∼ MeV. Now an the protons and neutrons, as we discussed earlier, and MeV is, by everyday standards, an enormous energy:10 it so nucleons act as structureless particles,e.g with the same corresponds, for example, to temperature 10 K. Re- (conserved) mass as they have in isolation. Should quarks arrangements of the electrons surrounding a nucleus, such or gluons have internal structure, . if they are strings, as occur in chemical reactions or mechanical responses, then that structure can be, and empirically it must be, rig- do not bring anything approaching that energy into play. orously censored and sequestered through similar mech- Thus the nuclei remain in their ground states, and effec- anisms. tively behave as ideal, structureless particles. In particular: Although they are not isolated particles in the math- 3. “Mass without mass” for nucle- ematical sense, within standard matter nuclei behave es- ons sentially as if they were, and the potential complications we mentioned earlier in connection with mass (non-) conservation do not arise. Nuclear excited states and internal degreesi.e. × of freedom− 120 years ago Lorentz proposed his electromagnetic the- would come into play if nuclei could approach each13 other ory of the origin of the (then undiscovered!) electron’s within the range of the strong force, 2 10 cm. mass [8,9]. According to Lorentz’s idea, back-reaction of However the mutual repulsion of inner electron clouds, due electric and magnetic fields resists accelerated motion of to repulsive Coulomb forces and fermi statistics, effectively charges, thus giving rise dynamically to effects we call sequesters the nuclei. If that mechanism of sequestration “mass”, or inertia. lapses (as it does for itinerant protons) then Coulomb re- Lorentz’s idea, in its original form, no longer appears vi- pulsion between the nuclei themselves provides a further able, for several reasons. In modern quantum electrody- barrier. namics (QED) the electromagnetic correction to the elec- Isolated nuclei typically have a degenerate ground state, tron’s mass appears as a multiplicative factor. It can- due to their spin. Inside atoms, or inside matter gener- not, therefore, bootstrap a vanishing intrinsic mass to a 1025 Origins of mass

many

nonzero value. This no-go result arises because in the Note that distinct localized organizations of energy, limit of zero electron mass the equations for electrons in- or “particles”, with different masses, arise here as self- teracting with electromagnetic fields have enhanced sym- consistent solutions of the nonlinear back-reaction equa- metry (chiral symmetry), which forbids mass non-zero tions. mass, even when back reaction is taken into account. Fur- The Lorentzian perspective on the origin of mass in QCD, thermore, notoriously, the multiplicative correction factor for all its historical resonance, is perhaps less straight- is infinite, or rather ill-defined, and requires renormal- forward than an alternative, simpler yet no less beautiful ization.sui Aside generis from these very serious technical problems, perspective. Once we admit that the fields of QCD can there is the simple, overarching fact that the electron is no conspire to produce quasi-stable, localized concentrations longer . If we somehow succeeded in explain- of energy, then special relativity tells us that these enti- ing the electron’s mass as an electromagneticm effect,µ we’d ties will behave as particles with mass immediately be facedmτ with the challenge of accounting for m E/c ; the very different values1 of the muon’s mass , and the 2 tau lepton’s mass . = (12) E But something very close to Lorentz’s beautiful∼ idea is realized in quantum chromodynamics (QCD), the modern where is the total field energy of the configuration in theory of the strong nuclear force. Indeed most ( 95%) a frame where its center-of-energy is at rest or, more for- of the mass of protons and neutrons, and thus of ordinary mally, where its total momentum vanishes. matter, is due to the back-reaction of color gluon fields Figure 1 speaks eloquently for itself. In view of this tri- resisting accelerated motion of the quarks and gluons in- umph, it is appropriate to consider its conceptual roots side. Very detailed and impressive calculations, based on closely. A good entry into the discussion is to ask: What direct numerical solution of the equations and exploiting is the input, that underpins such impressive output? the full power of modern computers, stand behind those Decades of intense theoretical work have reinforced and words. indeed strengthened the intuition, originally based on perturbation theory, that 3+1 space-time dimensional quantum field theories, which are the only theories com- bining special relativity and quantum theory that are known to be consistent, are highly constrained. The source of this constraint is the possibility of ultraviolet divergences. In more detail: Special relativity appears to demand that the interactions be local; local interactions bring in field modes with arbitrarily large energy and momenta; couplinge.g. to quantum fluctuations in those field modes can integrate up to become so large as to disrupt the expected ( finite, relativistically invariant) behavior of physical process in the theory, rendering it incalculable and effectively nonexistent [3]. Nonabelian Figure 1. Spectrum of low-lying mesons and baryons, calculated gauge theories can avoid that fate, because asymptotic from first principles in quantum chromodynamics, compared to experiment [10]. π and K mesons masses are freedom weakens the couplings of the dangerous modes. used to fix the light quark and strange quark masses, re- That happy result only occurs, however, for the minimal spectively, and to fix the overall scale. “N” denotes nucleon. This result reveals the origin of the bulk of the mass gauge invariant couplings. One cannot tolerate, for exam- of standard matter.Ξ ple, anomalous color magnetic moments, for either quarks or gluons. SU For this reason QCD, a nonabelian gauge theory based on the gauge group (3) and triplet quarks, supports 1 very few parameters [11]. One can specify a mass for each These comments are rooted in the perturbative under- quark field, and2 one overall coupling strength; nothing standing of QED, which seems appropriate to its observed else is allowed . weak coupling. Nonperturbatively, both chiral symmetry θ breaking and nonlinear generation of a complex spectrum 2 of masses can occur – and in QCD, as we’ll presently dis- Here for once I’ll pass over in silence the saga of the cuss, they do! term. 1026 Frank Wilczek

The mass parameters associated with quark fields do not out mass” from QCD dynamics accounts for almost all of correspond directly to the mass of any physical particle, the mass of protons and neutrons - and therefore, accord- since quarks famously do not occur as individual particles, ing to the preceding analysis, almost all of the mass of but only confined within more complicated composites. We what we ordinarily consider matter. can, however, infer how energetic quarks propagate over 4. Masslessness and symmetry very short times, by reconstructing them from the jets they induce in high-energy collisions. That propagation is of course affected by the quark’sm mass,t and in principle allows its measurement. This technique is the only method The question of the origin of mass acquires a newmass- level mbyb; which mc m theb topmc quark mass can be measured, and ofless interest throughi.e. its deep connection with the symme- it also allows access to the bottom and charm masses try of physical law, for in some important cases . and can also be inferredb fromb thecc masses fields ( , fields whose quanta are massless parti- of bottomonium and charmonium resonances,m whichs; md; are mu cles) support forms of symmetry that are inconsistent with fairly well described as non-relativistic ¯ and ¯ bound non-zero mass. In such cases, the origin of mass gets tied states, respectively. The light quarkπ; masses K ; η 4.1.up with Scale the question symmetry of dynamical symmetry breaking. can be inferred from hadronms; m spectroscopy;d; mu the masses of the low-lying pseudoscalar mesons in particular depend sensitively upon u;. d; s From thec; b;perspective t of QCD dynamics, the quarks divide Thec simplest mass-forbidding symmetryh is scale invari- into two classes, the light quarks and the heavy ance.c In ah relativisticc field theory it is appropriate to use quarks . The reason for this division is that QCD dy- as the measure of velocity and ¯ as the measure of action namics is characterized by another scale with dimensions and , so ¯ = = 1. In such units mass has dimensions of inverse length: h of mass, to which quark masses can be compared. Thisg Q is M → : < qcd cL L the scaleQ ∼ Λ associated with the running, as a function of momentum transfer, of the dimensionless coupling ( ). ¯ 1 = x → λx (13) Forquark massesΛ the couplingmq is strong, and fluctuations in L → λL the gluon field dominate the dynamics. Hence the effects Symmetry under scale transformations , which in- of with Λ will,quarks ordinarily, bemq dominated by gluon fluctuations, and will have limited impact volves , is therefore incompatible with the existence on the underlying dynamics; while with Λ 4.2.of any invariant Chiral symmetry mass parameter. willqq ordinarily have limited impactmu; m ond the gluon mc; dynam- mb; mt < ψ ics, sincems ∼ the fluctuations are insufficient to produce heavy 1 ¯ pairs. Phenomenologically Λ , 2 while Λ. Relativistic spin- fields are said to be left-handed if they satisfy the equation γ Based on this dichotomymu; md it; m iss → amusing andmc; minstructiveb; mt → ∞ to R ψ ≡ ψ consider an idealization of QCD which I call QCD Lite, 5 1 + wherein one takes 0 and . Π = 0 (14) 2 In this idealization the heavy quarks effectively decouple, and we arrive at the theory of three flavors of massless and right-handed if they satisfy− γ quarks. In view of the preceding paragraph, one might an- Lψ ≡ ψ : ticipate that the masses of hadrons in QCD Lite closely re- 5 1 sembleπ ρ those of realistic QCD, especially for hadrons free Π = 0 (15) 2 L R of strange quarks (such as protons, neutrons, ∆ baryons, Lψ ψ R ψ ψ and mesons, and their rotational excitations). That Because the projection operators satisfy Π + Π = 1, we expectation can be checked against accurate numerical have Π = for a left-handed field and Π = for simulations, and it proves out. no a right-handed field. Fields with a definite handedness, In this way we are led to a most remarkable conclusion. left or right, are said to be chiral. Minimal chiral fields QCD Lite is a theory containing explicit mass parame- are spinor fields with two independent components. − ters; its only dimensional scale is set by the coupling evo- The quanta of a free left-handed field carry negative he-1 lution. Yet it supports a rich spectrum of massive excita- licity – that is, their spin in the direction of motion is 2 . tions, notably including protons and neutrons, that closely That condition can only be relativistically invariant for resembles the spectrum of QCD proper. Thus “mass with- particles that move at the speed of light – otherwise, one 1027 Origins of mass

can overtake the particle and reverse the helicity. So chi- with Λ an arbitrary space-time function. (Here for sim- ral ﬁelds describe massless quanta. Massive spinor ﬁelds plicity I suppress the intricacies of nonabelian gauge must contain both left- and right-handed components. In symmetry.) a Lagrangian formulation, the mass term for free quanta Whereas conventional symmetries imply relations among takes the form− Lm mψψ physical processes, gauge symmetry is an essentially

m R ψ Lψ Lψ R ψ mathematical statement, about theory formulation. Gauge = = ¯ symmetry asserts the absence of degrees of freedom that ≡ m ψR ψL ψLψR : = Π Π + Π Π change under its transformations, rather than, as for non- gauge symmetries, invariant relationships among physical + ψR ψR(16)→ iθ degrees of freedom. In quantum theory, it is implemented e ψR by constructing the physical Hilbert space through projec- Note that a phase transformation on the field, tion onto gauge-invariant states. The formalism of lattice , does not leave the right-hand side of Eq. (16) in- gauge theory is manifestly gauge invariant, implementing3 variant. Looking at the effect of this transformation, we projection by direct averaging over theA gauge group [12]. also see that it is natural to contemplate complex values µ Conventional perturbation theory works instead by using of− massL terms,µ∗ψ inψ the generalizedµψ ψ formµ ψψ i µ ψγ ψ : µ R L L R the gauge freedom to put the external field in a canon- 5 ical form – “gauge fixing” – while allowing off-shell fluc- = + = Re + Im tuations, not subject to projection, at intermediate stages (17) in its calculations. One must then work to show that the Candidate symmetries can act on left- or right-handed canonical form carries through to the final result, so that fields separately. Thus for example if we have two the desired projection is consistent. This can raise chal- spinorU × fieldsU with both left- and right-handed pieces, we L R lenging technical issues, especially with regard to reg- can contemplate, consistent with relativistic invariance,U L R ularization and renormalization. It can even go wrong, (2) (2) transformations among them; but for mas- + leading to so-called gauge anomalies. Failure to imple- sive fields the possible symmetry is reduced to (2) , ment a consistent projectionA is highly problematic,Ai specif- because the the left- and right- handed components must ically because the sign of covariant quantum-mechanical transform together. Symmetries that distinguish left- from 0 amplitudes involving and the spatial will be oppo- right-handed quanta, and thus force them to be massless, site, leading to assignment of negative probabilities. 4.3.are called Gauge chiral symmetry symmetries. Gauge symmetry is a central postulate of the standard model. It is appropriate to ask whether it is a completely independent postulate, or tied up with other fundamental Gauge symmetry applies to the quantum fields that de- principles. Special relativity and quantum mechanics in- scribe spin-1 bosons. It has a beautiful geometrical inter- evitably lead to local quantum field theory, but non-trivial pretation, but here I would like to emphasizem some more quantum field theories of this kind are notoriously difficult directly physical issues which relate closely to the ques- to construct, due to ultraviolet divergences. Perturbation tion of mass and the special status of =1 0. theory with massive vector particles involves the propaga- k k 2 g − µ ν By analogy with chiral symmetry for spin , just discussed, tor µν M one might seek to formulate theories of spin-1 fields whose k − M 2 quanta are only left-handed, or only right-handed; or in 2 2 other words, only definite circular polarizations. Those quanta would necessarily1 be massless, for the same reason we discussed in the spin 2 case. Consistency requirements related to the CPT theorem require that both helicities be near mass shell (so in the rest frame the time-likek → ∞ polar-

present,Aµ but apart from that subtlety gauge symmetry is ization is projected out). This has miserable ultraviolet precisely the implementation of this idea. A vector field behavior, being completely undamped as . For contains longitudinal polarizations, in addition to the massless vectors with gauge symmetry, on the other hand, desired circular polarizations. Gauge invariance insures we can work in Fermi-type gauges where the propagator that the longitudinal polarizations have no physical effect, 3 since it requires invariance of the theory under addition It is not known, however, how to formulate theories with of a longitudinally polarizedAµ → Aµ field:∂µ ; gauged chiral symmetries, including notably the full standard model (as opposed to QCD and QED), in that frame- + Λ (18) work [13]. 1028 Frank Wilczek

kµ kν gµν − α k assumes the form k 2 smaller scales, and that influence alters their prop- k 2 erties. That is the simple physical reason why scale symmetry, while valid in the classical formulation of with much better large- behavior. The renormalization the theory, is necessarily broken upon quantization. program delivers on this promise, by giving a procedure Thus this symmetry barrier to mass generation falls. for calculating physical amplitudes as a power series in Though it is literally false, scale symmetry in- the renormalized coupling, with finite coefficients. That volves important truths. First of all, it becomes is not good enough, however, since the series can fail to true asymptotically, and thereby governs the high- converge; we must investigate the non-perturbative be- energy behavior of many physical processes. Two havior. In principle the simplest, and in practice the most other profound consequences relate directly to our powerful, approach to constructing interacting quantum overarching theme, the origin of mass. Since the field theories is to introduce some sort of cutoff to reg- scalethe invariance bulk of of the classical mass of QCD matter Lite is would essentially make ulate the ultraviolet behavior, calculate within the reg- itquantum-mechanical impossible for nucleons in origin to have mass, we see ulated theory, and remove the regulator by a limiting that procedure, while holding some physical observable fixed. . Scale invariance is In general this approach will fail, because as one raises replaced by an emergent relationship between cou- the cutoff, one (re)introduces significant effects from ever pling strength and scale, whereby neither coupling higher-frequency fluctuations, and no definite limit can strength nor scale can be freely varied, but variation emerge. In asymptotically free theories, however, the in one can be compensated by variation in the other. high-frequency fluctuations decouple, and this construc- In place of scale invariance of the theory, we have tive strategy works. Since asymptotic freedom is an is- what Sidney Coleman called “dimensional transmu- sue concerning the ultimate weakness of couplings, it can tation”: Different values of the coupling strength do be investigated perturbatively. Famously, one finds that not yield a family of essentially different theories,the asymptotic freedom is a feature unique to (nonabelian) butbulk rather of the correspond mass of matter to a unique arises theory from a equipped unique, gauge theories. Thus gauge invariance – indeed, non- withparameter-free different units theory of length (or mass). Thus abelian gauge invariance – is at least helpful, and prob- ably necessary, for the harmonious marriage of specialm Gauge Symmetry . relativity with quantum mechanics. Thereby gauge invariance, and the consequent existence of fields with = 0 • : Gauge invariance states that the vector quanta, appears deeply inherent in the nature of physical Hilbert space contains only singlet states. things. 5. Mass and symmetry breaking in Since all physical states are gauge singlets, gauge invariance has no directfields implication for the phys- QCD ical spectrum, even though it constrains the mass parameter of gauge , which are used in constructing the theory, to vanish. Putting it more simply and physically, though more loosely: The nom- QCD features all three of the mass-forbidding symme- inally massless plane-wave gluons are confined, tries just discussed, yet manages to produce the mass of while the “cavity modes” associated with allowed, matter. This is especially clear and dramatic in realistic finite-extent configurations have a non-zero mini- QCD’s quasi-realistic cousin QCD Lite, whose formulation mum frequency. contains no quantity with units of mass or length. In this It seems appropriate to remark here that confine- subsection I’ll elaborate a bit on the symmetry-breaking ment of colornot gluons, and quarks, is a very nat- aspects of mass generation in QCD. Aside from their in- ural consequence of gauge symmetry, since those trinsic interest, these considerations shed interesting light quanta are gauge singlets. From this perspec- on the Higgs mechanism and the Higgs particle, as will Scale Symmetry tive it is the deconfined phases of QED and elec- appear below. troweak theory, which appear so natural in pertur- • : The classical scale invariance of bationChiral theory,Symmetry that appear paradoxical. QCD Lite, which characterizes its Lagrangian formulation, is violated by quantum fluctuations, lead- • : We might also invoke confine- ing to scale dependence of its dimensionless cou- ment to explain away massless quarks. Chiral sym- pling. Because the theory is nonlinear, larger- metry has profound implications for the spectrum, scale fluctuations are influenced by fluctuations on nevertheless. 1029 Origins of mass

φ ; φ SU V 0 1 To simplify the discussion,SU I willL ignore× SU theR strange with complexvU → scalarvUV fields . These formSU a dou-L quark, and consider a 2-flavor version of QCD Lite. blet under multiplication by (2) matrices act- Representations of chiral (2) (2) are of ingv ash φ i , orσ in other words under (2) . two kinds. We can have massless fermion multi- In this notationσ ≈ φ the−v vacuum expectation value is 0 plets, as for free fermions, or massive multiplets with = Re and the meson field, for small fluctu- 0 degenerate states of opposite parity. Neither alter- ations, is Re . This corresponds precisely, native is a feature of the observed hadron spectrum. as we’ll see, to the minimal Higgs particle. Fortunately, in QCD chiral symmetry is broken, dynamically (or as we say “spontaneously”), through 6. Summary: The origin of mass for formation of a quark-antiquark condensate in the k k standard matter ground state.G U WeqjR q canL G writeU vUj ;

( ) L;R( ) = (19) j; k v 6 U where the subscripts denote left- and right- By way of summary, some major statements concerning handed chirality, are flavor indices, = 0 is the mass of standard matter: theU magnitude of the condensate, and is a uni- tary matrix with determinant|G unity.U i Any fixed choice 1. In classical mechanics Mass is an irreducible, defin- of , independent of space-time coordinates, gives ing property of matter. In modern physics there is a possible ground state ( U) , and these states no equivalent fundamental concept. Mass in the are all orthogonal, but energetically degenerate. original, Newtonian sense appears as an approxi- Slow space-time variation of creates low-energy mate, emergent property of matter. modes that, following Nambu and Goldstone, we 2. There is a contingent but clear conceptual path identify as representing pseudoscalar mesons. magnitude v leading from qualitative aspects of the quantum dy- In additionU → δ we might consider space-time variation namics of elementary quanta to Newtonian mass in the of . Thus we consider imposing as an approximate property of bulk matter. Thus k k (with qjLq) R v σ x; t δj the issue of the origin of mass, for standard matter, reduces to understanding the properties of the

¯ = + ( ) (20) relevant elementary quanta. σ on field histories, and forming an effective action. In 3. The elementary quantau thatd give quantitatively im- the language of the old model [14], [15] we rep- portant contributions to the mass of standard matter resent these excursions in and around the vacuum are color gluons, and quarks, and to a much s manifoldv x; t asU x; t v σ iτ~ · π~ ; lesser extent electrons and photons. Everything else is basically negligible. (The contribution of the quark is tricky to separate from the gluons, ( ) ( ) =τ~ ( + ) exp ( ) (21)π~ since most of its effect can be absorbed into a renor- σ x; t malization of the coupling.) with Pauli matrices and a triplet of pion fields . Since variations in ( ) take us outside the vac- 4. The equations for massless fields exhibit several uum manifold, we should not expect them to have kinds of enhanced symmetry: scale invariance, chi- vanishing energy, even in the limit of long wave- ral symmetry, gauge symmetry, that are spoiled by length.σ It may be possible to associate an observed non-zero mass. By first assumingorigin those symmetries, excitation in QCD, the broad scalar meson reso- and then explaining how they are transcended, we nance (500) around 500 MeV, with variations of can claim sharp insight into the of mass. this kind. v x; t U x; t 5. We have superb calculations accounting for the ori- For later comparison with the minimal Higgs sector, gin of nucleon masses, and the masses of hadrons an alternative notation for ( ) ( ) is signifi- generally, based on QCD. A highly symmetric, cant. We can write φ φ vU parameter-free version of QCD that incorporates −φ∗ φ∗ scale, chiral, and gauge symmetry already provides 0 1 ! a good approximation to the relevant hadronic spec- = 1 0 (22) trum, and accounts for most of the mass of ordinary 1030 Frank Wilczek

matter. Within this framework, we find very con- and substitute this value into the covariant derivative crete understanding of how each mass-forbidding terms, we find a term µ symmetry is transcended. L ⊃ m AµA ; I would like to emphasize that this theory of the origin of m q 2v ; the mass of standard matter makes no reference to symme- 2 2 2 try breaking in the electroweak sector, nor to the Higgs = (26) condensate, nor of course to the Higgs particle. Those are important subjects, but they are different subjects, to 2 which we now turn. that corresponds to the indicated (mass) for the vector 7. Mass and symmetry breaking in field. φ superconductivity After Bardeen, Cooper,q ande Schrieffer, we understand that in superconductors the -field represents the density of 7.1.Cooper Gauged pairs, with broken= 2 . symmetry, NOT broken gauge symmetry Photons in empty space have zero mass, for a profound reason, connected to (unconfined) gauge symmetry. Yet the equations for photons inside superconductors describe a massive particle. That is a deep interpretation of the The common usage “broken gauge symmetry” to describe Meissner effect, implicit in the work of London and of superconductivity and its electroweak analogue is unfor- Landau and Ginzburg. In the context of superconductiv- tunate. As I’ve already had occasion to recall, local gauge ity, those authors were primarily focused on the classi- invariance cannot be broken, since it is a statement about cal behavior of the electromagnetic fields and their effect theory construction. Yet the common usage was not cho- on external probes, not the appearance of field quanta sen whimsically, or perversely; clearly there is something to imaginary observers immersed in the superconducting in it. What? material. Nevertheless their equations, analyzed from that perspective, unambiguously correspond to a massive pho- In a weakly coupled gauge theory, we are invited to con- ton. sider the theory we would get without the gauge fields, The Landau-Ginzburg framework is perfectly adapted to and then add in the gauge interactions as a perturbation. re-interpretation and generalization. In particular, the This is what we actually do, in both superconductivity fields that appear in their construction can be promoted to and in the minimal electroweak theory. In the absence relativistic quantum fields, and the gauge symmetry can be of gauge interactions, we have a model of broken global taken to be nonabelian. Those generalizations, and the in- symmetry; then we couple in the gauge fields, and com- terpretation of the field quanta as massive vector particles, pute the spectrum perturbatively, finding massive gauge were developed in the work of several authors (including bosons. Higgs [16, 17]; Brout and Englert [18]; Guralnick, Hagen, (Optional digression: An amusing point is that since or- and Kibble [19]; ’tHooft [20]) whose separate contributions dinary superconductivity arises, ultimately, primarily from I will not attempt to disentangle here. The essence of the electromagnetic forces, the concept of “turning off” the matter can be captured in a few strokes. We consider the gauge field in that context appears paradoxical. Within interaction of the electromagnetic (or more general gauge) the Landau-Ginzburg framework, of course, no paradox field with a complex (or non-singlet) scalar field described arises. The potential has a life of its own, with its µν ∗ µ ∗ by theL relativistic− Fµν F Lagrangian∇µφ ∇ φ − V φ φ ; dependence on the underlying microscopic forces suppressed. Presumably some separation of the instanta- 1 V= + ( ) ( ) ( ) (23) neous Coulomb interaction, which should be treated non- 4 perturbatively, from the perturbative propagating trans- ∇µφ ∂µφ − iqAµφ where is an invariant potential and verse field could help here, or a renormalization group derivation of effective interactions whose strengths are = V (24) regarded, for conceptual purposes, as independently ad- is the covariant derivative. If we suppose that is mini- justable parameters.) mized away from theV originv V : Thus we should properly speak of gauged broken sym- 2 v 6 min metry, as opposed to broken gauge symmetry. That dis- = tinction might appear pedantic, but it clarifies a profound = 0 (25) conceptual unification, as will appear momentarily. 1031 Origins of mass

8. Mass for W and Z bosons W Z confinement for hadrons, and the Higgs mechanism for W Z and (which is furthermore closely related to su- In the context of SU(2)xU(1) electroweak gauge theory perconductivity). Superficially those mechanisms appear we get a good account of the origin of and SUboson quiteSU different, but at a fundamental level theyV are es- masses andφ mixings along theseY/ lines,/ using a condensate sentially the same. Indeed, as we imaginatively turn up having the quantum numbers of a component an (2) the (2) gauge coupling, keeping the potential fixed, doublet with hypercharge 2 = 1 2. The details are we will go from dynamics most conveniently described textbook material [21], which I’ll forego here. It is appro- as “gauged broken symmetry” to dynamics most conve- priate to mention a conceptually important feature of the niently described as “confinement”. But as long as the derivation, however: The new structure it introduces into non-photonic spectrum remains massive throughout this the theory is quite minimal. One must specify the gauge thought-experiment there can be no sharp transition. quantum numbers of the condensate, and its magnitude; If we ignore the gauge couplings of the standardσ model that is all. Since the ratio of output to inputfor the is large W and and Z altogether, we find that the minimal Higgs sector has impressive,bosons I think it is fair to claim that this mechanism precisely the same structure as the old modelSU (inL its× greatly illuminates the origin of mass, SU × U linearSU R incarnation), an effective theory for low-energy

. The historical sequence of eventsGF was likewise pionSO physics,∼ inSU the idealization× SU of exact chiral (2) impressive: Since both the strength of the (2) (1) (2) . The ungauged Higgs sector likewiseσ features gauge couplings and the Fermi constant , which sets an (4) (2) (2) symmetry, that is sponta- theW mass scale,Z had been measured, it was possible to neously broken to the diagonal subgroup. In modelσ lan- form precise expectations for the masses and properties of guage, this spontaneous breaking produces three massless the and bosons prior to their experimental discovery. Nambu-Goldstone bosons, the pions, plus the meson. The known particles of the standard model, that is the When we turn on the gauge couplings the pions combine quarks, leptons, and gauge bosons, do not by themselves with the gauge fields, providingH the longitudinal modes for produce the required condensate. The quark-antiquark now massive gauge bosons.σ We see, from this correspon- condensate of QCD actually has the right quantum num- dence, that the Higgs particle is quite closely related, bers for the job, but it is far too small quantitatively. To conceptually,σ to the meson ofbe QCD.H Indeed, if the Higgs produce the required dynamics, the simplest procedure is condensate were dominated by its quark-antiquark con- to introduce an elementary scalar doublet with the re- tribution, the meson would . quired quantum numbers. This can be done using a direct The concepts here briefly reviewed play a prominent role extrapolation of the Landau-Ginzburg Lagrangian. An im- in technicolor model building, where hypothetical new portant practical virtue of this construction is that itV can higher-scale analogues of QCD are introduced to produce be carried out in the framework of weak coupling, renor- a dominant contribution to the electroweak condensate. malizable quantum field theory, provided we restrict to Whatever the fate of such models, the deep connection the form V φ∗φ − µ φ∗φ λ φ∗φ : between our two best ideas for understanding the origins of mass in different contexts remains a satisfying insight. 2 2 ( ) = + ( ) (27) 9. Mass for quarks and leptons

SU × U In this was we get quite a definite theory, whose consequences can be calculated accurately. I will discuss it In the (2) (1) electroweak theory left- and right- under the heading “The Minimal Model” in the following handed fermions couple differently. Therefore mass terms, section. whichW connectZ left and right, are not gauge invariant. Of course, as a logicale.g. matter reality need not be so sim- Fortunately, the same doublet condensate that works for ple. The condensate might arise from several distinguish- and can also serve here, to connect the differently able contributions, from several distinct doublets, or handed chiral fermions. In detail, for quarks: from other representations, or from composites beyond the L ⊃ − yjk U Q φα − zjk D Q αβ φ∗ : : 8.1.known, Continuitysmall quark-antiquark of confinement portion. and the Higgs j αk j αk β mechanism (and superconductivity) ¯ ¯ + h c (28) φ

When the appropriate component of acquires a vacuum I’ve discussed two mechanisms for explaining the origin expectation value, and we insert that value into Eq. (28), of mass, that operate in diﬀerent contexts – QCD and then we obtain a quadratic expression in the quark ﬁelds. 1032 Frank Wilczek

H By some complicated field redefinitions (even more com- light quarks, photons,W electrons,Z and gluons that we have plicated, if we take into account possible family struc- easiest access to. exchange has small but measurable ture in the kinetic Dirac Lagrangian) we can diagonalize effects on the and masses, and on some other very that expression, to obtain physical masses of the indepen- accurately measured electroweak parameters. The con- dently propagating quanta. The relative rotations neces- straint from these estimates, according to the LEP Elec- sary for diagonalization encode the weak mixing angles, troweak Working Group, is shown in the following Figure: or Cabibbo-Kobayashi-Maskawa (CKM) matrix. In this construction parameters proliferate, unconstrained by theoretical understanding. There is a similar construction for leptons, with the additional complication that (in theφ minimal model) neutrino masses arise from non-renormalizable dimension five interactions, that are quadratic in . To call this mess “the origin of mass”, even for the elementary quanta, is to aim too low. The measured masses and mixings of quarks and leptons are in need of elucidation, but this constructionµ manifestly shirks the burden of explaining them. It is, at best, an accommodation. Simi- larly, the quantity that appears in Eq. (µ27), which has dimensions of mass, appears as a free parameter, and is no way elucidated or explained. Yet this is intimately related to the mass-scale of the condensate and it governs the physical mass of the classic Higgs particle itself quite directly. 10. The message of Higgs particle phenomenology Figure 2. Constraint on Higgs particle mass, based on analysis of radiative corrections, as analyzed by the 10.1. The minimal model LEP Electroweak Working Group. Figure from http://lepewwg.web.cern.ch/LEPEWWG/, where more detailed information may be found.

Now let me turn, at last, to the recent excitement in Higgs particle phenomenology. For the sake of organizing the discussion, I will ﬁrst consider as default hypotheses: The message that emerges from these exquisite calculations and experiments, that involve every part of the stan-

• First hypothesis: The minimal version of the standard modelm andH deep, intensive use of quantum field the- dard model, with just one doublet, is a good approx- < ory, is that agreementm withH ∼ the minimal model is possible, imation to reality (in the relevant energy range). but only if is below 140 GeV or so. Since the LEP direct search excluded 115 GeV, one had only a nar- • Second hypothesis: All indications from the rele- row window for consistency, even prior to the LHC work. vant LHC (and LEP, Tevatron, ...) experiments are The situation at LHC is developing rapidly, so this is a correct. dangerousH timem toH interpret≈ the situation. Nevertheless...! The immediate question then becomes: Do these hypothe- As we see in the Figure 3, there are significant hints of ses hang together? an signal at 125 GeV. As mentioned previously, a virtue of the first hypothesis The signal recorded there has been distilled from a com- is that it gives us a definite, quantitative, one-parameter bination of several channels, and the details are rather theory, that covers an enormous range of phenomena. De- complicated. I’d like briefly to discuss one (dominant) spite years of hard effort to poke holes in that theory, so contribution to the signal, which is particularly clean and far itm hasH withstood every test. Most of its predictions,H of has some special theoretical interest. It is the process course, areH not very sensitive to the value of the param- sketched in Figure 4. eter , which after all affects only virtual particles, since the mostly interacts feebly, especially with the 1033 Origins of mass

Figure 3. Preliminary hints of signal consistent with minimal standard model Higgs particle, mH ≈ GeV. Figure from M. Jørgenson at http://pdg2.lbl.gov/atlasblog/?p=1211. 125

Figure 5. Origin of the eﬀective coupling between Higgs particle and gluons. The annotations are explained in the text.

u; d; e; γ g

heavy!),H while the and color gluon ﬁeldsH have tiny or zero mass. Thus theg direct, classical couplings of to ordinary matter are highly suppressed. can, however, communicate with pairs through a quantum eﬀect,H as indicated by a loop graph. The gluons communicate with virtual top quarks, which in turn communicate with . Ordinarily one might expect that processes involving virtual heavy particles are suppressed by inverse powers of the mass, but this process is an exception. Indeed, by power counting (taking into account gauge invariance, which brings∝ out/M two externalM momenta for the gluons) the loop integral converges linearly, and so the large momentaH contribute∝ M 1 , whereM is the mass of the virtualHgg particles. However the basic, Yukawa coupling at the ver-

tex is , so the dependence cancels.mQ ThusmH the coupling gets essentially equalmH contributions from all suf- ficiently heavy quarks. (Quarks with will not Figure 4. Stylized Feynman graph illustrating the production of a contribute, however, since provides an infrared cutoff.) Higgs particle through gluon fusion and its decay through photon fission. Time advances upwards. In any case, the main conclusion, as tentative as it is wel-

come, has to be that our twomH default≈ hypotheses do indeed hang together. The simplest, minimal implementation of the standard model, with 125 GeV, ﬁts all the facts γγ brilliantly.

The experimentalmH signature is a resonant enhancement in It is appropriate toH remark that both the successful theo- production, when the invariant massH of the photons retical calculation of rates and backgrounds for observable matches . One sees that this process is basically a consequences of , and the experimental detection of its doubled version of the coupling of to gauge bosons, rare subtle signals in an extreme, complex environment displayed in Figure 5. will be, if confirmed, scientific achievements of the highest I’m particularly fond of this coupling, which for a brief 10.2.order. Beyond the minimal model shiningH moment following its discovery [22] was called the “the Wilczek vertex”, but somehow devolved to “gluon fusion”. has a difficult time coupling to ordinary matter. Indeed, it prefers to couple to the quanta of heavy fields (or The tight fit between experiment and a minimal, weakly- rather, the quanta of fields it couples to thereby become coupled theory is clearly bad news for speculations about 1034 Frank Wilczek

focus point new strongly interacting sectors at the electroweak scale, Severalsupersymmetry theorists have considered this scenario to be nat- including technicolor or large extra dimensions in most or ural, and studied it in detail, under the heading φ all of their many variants. [24]. More interesting, I think, are the implications for low- In detail, supersymmetry requires at least two doublet energy supersymmetry. Low-energy supersymmetry can fields. The physical spectrumH± of spin-0 “Higgs” particles be implemented in the framework of weak coupling. It includes one that mimics the minimal H, plus 2 other neu- also features many cancellations among virtual boson and trals, and a charged . Of course these particles, like fermion loops. Thus, despite the large number of super- the particles of the minimal standard model, all come with partners it requires, supersymmetry can hide itself pretty superpartners! efficiently. We shall see... Lest we forget, let me remind you, with one figure, that 11. Conclusion: Mass in the uni- there is an excellent, quantitative reason [23] to suspect verse that low-energy supersymmetry is relevant to the description of Nature:

As reviewed here, we have achieved profound insight into the origin of mass for standard matter, andW we mayZ be set to crown, with the discovery of the Higgs particle, a compelling account of the origin of mass for and bosons. Those origins are distinct, though there is an attractive conceptual connection between their mechanisms, and between both mechanisms and superconductivity. That’s the good news. The bad news is that nothing in these ideas explains the origin of the mass of the Higgs particle itself, nor do Figure 6. Quantitative comparison of unification of couplings with- they greatly elucidate the observed complicated struc- out (dotted lines) and with (solid lines) contributions from low-energy supersymmetry. Figure from H. Abe, ture of quark and lepton masses and mixings, nor the http://www.sci.waseda.ac.jp/english/researchprofiles/ ad- associated physical phenomena of CP violation, neutrino vanced/subject01_16.html. oscillations,... . The ugly news is that the origin of most of the mass in the universe, that in dark matter and dark energy, remains mH ≈ deeply mysterious. If theθ dark matter is in axions, the dynamical origin of its mass will be clear, as it arises from If 125 GeV, we willm haveH another. For while the standard model itself has nothing much toH say, theoret- knownF effects in QCD ( term), although the magnitude of that mass is tied up with yet another mass scale, the scale ically, about the value of ,m tastefulW m implementationsZ of Peccei-Quinn symmetry breaking. If the dark matter of low-energymH supersymmetry producei.e. an -like particle is some kind of supersymmetric particle, the question of whose mass is closelymH; : ≤ tiedmZ to and . Indeed the value of inferred classically – ignoring virtual the origin of its mass will be tied up with the larger ques- cl loops – satisfies , and has long been excluded tion of supersymmetry breaking, concerning which there are many speculations, none compelling. by experiment.mH mH Fortunately loop∼ corrections, especially from stop loops, We’ve passed some milestones, but the end of the road is can raise . = 125 GeV fits most comfortably with not in sight. quite heavy ( 10 TeV?) stop masses. 12. Update: Higgs particle discov- The idea that the superpartnersT of quarks and leptons ery might have such heavy masses had been anticipated on the basis of flavor violation and (non-)phenomenology. All current observations are consistent with no additional contributions beyond what arises in the minimal standard (Added 18 August 2012.) model. Exchange of superpartners, on the other hand, can On July 4 2012 the ATLAS and CMS experimental groups, induce many effects of that kind. The simplest way to dis- reporting work at the Large Hadron Collider (LHC) at pose of these dangerous possibilities is simply to suppress CERN, announced the discovery of a remarkable new par- the exchanges, by making the superpartners heavy. ticle. Their results have now been codified in scientific 1035 Origins of mass

γγ papers [25, 26]. Perhaps the simplest and most easily heavy particles. (But note that complete cancella- interpreted reported signatureM ≈ is an enhancement in tion of mass factors between vertex and loop inte- pair production, over standard model backgrounds, in the gral,h indicated in Figure5, only occurs for heavy invariant mass region 125 GeV. particles whose sole source of mass is coupling Although significant deviations are not yet excluded, the toportal;.) hidden sectors observationsM ≈ so far are close enough to calculated expectations for a minimal standard model Higgs particle, with 3. : mass 125 GeV, that that hypothesis must be consid- The Higgs field, because it features the onlySU super-× ered the default interpretation. A pre-existing cluster of renormalizableSU × U couplings in the standard model, theoretical and phenomenological arguments, reviewed in is uniquely open to contamination from (3) earlier sections, all pointing in the same direction, pow- (2) (1) singlet “hidden sectors” [28]. Such erfully reinforce that interpretation. mixings could lead to overall diminution of the pro- Nothing from my earlier discussion needs revision in light duction rate, and/or invisible decay channels. For of the experimental discoveries (on the contrary, they con- this reason, the Higgs particle is said to open a firm its logic), so I’ve left it unchanged. portalsupersymmetry into hidden sectors. Here I’ll add some brief but pointed comments about the hgg significance of the discovery in a broader context, espe-h 4. hγγ : h − H cially in its implications for future work. To avoid possible Supersymmetry supplies both fodder0 for and confusion, in this addendum I will use lower-case to loops, and possibilities for mixing. denoteh the observed Higgs particle. (In the literature of Acknowledgements supersymmetric phenomenology,H ;A;H± it is conventionalH to useA ± for the light, standard-model0 H like scalar;0 one also hash a secondH doublet of CP even and CP odd neutral,0 and charged spin 0 particles; furthermore I’d like to thank Jonathan Feng for helpful comments on and canh mix.) the manuscript, and Jesse Thaler for post-discovery ori- Since the rates of many processes involving the Higgs entaton. This work is supported by the U.S. Department particle can be measuredh [27], we can look forward to of Energy under cooperative research agreement Contract detailed quantitative comparisons betweenWW the observedZZ tt Number DE-FG02-05ER41360. bbvaluesττ ofgg fundamentalγγ couplings and standard model expectations, for several different channels: , , ¯ , References ¯ , ¯ , , , and “invisible” (missing energy). The results will be revealing, since there are serious reasons to anticipatemass to deviations: mess [1] I. Newton, Principia (Royal Society, London, 1687) 1. : h [2] E. Wigner, Ann. Math. 40, 149 (1939) As reviewed above (and emphasized already in [22]) [3] F. Wilczek, Rev. Mod. Phys. 71, 85 (1999) the coupling of to fermions is not deeplyW rootedZ [4] I. Newton, Opticks, 4th edition (William Innys, Lon- in profound principles, and theoretically it is open don, 1731) to easy modification. Specifically: If the and [5] J. C. Maxwell, In: Encyclopedia Britannica, 9th edi- get partv 0 of their mass fromh a doublet that does not tion, vol. 3 (Adam and Charles Black, Edinburgh, couple to all fermions, then the vacuum expectation 1875) 36 0 value associatedyff ∝ mf /v with could be smaller than [6] C. E. Detar, Phys. Rev. D 17, 323 (1978) the canonical value, and the correspondingW YukawaZ [7] R. L. Jaffe, F. E. Low, Phys. Rev. D 19, 2105 (1979) ¯ couplings larger. More generally, one [8] H. A. Lorentz, Archives néerlandaises des sciences can preserve theh successful account of and exactes et naturelles 25, 363 (1892) masses and neutral currents in the Standard Model [9] H. A. Lorentz, Theory of Electrons (Teubner, Leipzig, while allowing to be a mixture of several singlets 1916) virtual heavies and doublets, with perturbed couplings. [10] S. Dürr et al., Science 322, 1224 (2008) [11] J. Beringer et al. (Particle Data Group), Phys. Rev. D 2. : gg 86, 010001 (2012) Alsoγγ as emphasized already in [22], and reviewed [12] K. Wilson, Phys. Rev. D 10, 2445 (1974) above, the virtual loops that underlie the and [13] H. Nielsen, M. Ninomiya, Phys. Lett. B 105, 219 channel can be sensitive to otherwise unknown (1981) 1036 Frank Wilczek

[14] Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 [22] F. Wilczek, Phys. Rev. Lett. 39, 1304 (1977) (1961) [23] S. Dimopoulos, S. Raby, F. Wilczek, Phys. Rev. D 24, [15] M. Gell Mann, M. Lévy, Nuovo Cimento 16, 705 1681 (1981) (1960) [24] J. L. Feng, K. T. Matchev, T. Moroi, Phys. Rev. D 61, [16] P. Higgs, Phys. Rev. Lett. 13, 508 (1964) 075005 (2000) [17] P. Higgs, Phys. Rev. 145, 1156 (1964) [25] G. Aad, et al. (The ATLAS Collaboration), Phys. Lett. [18] F. Englert, R. Brout, Phys. Rev. Lett. 13, 321 (1964) B 716, 1 (2012) [19] G. Guralnick, C. Hagen, T. Kibble, Phys. Rev. Lett. [26] CMS collaboration, arXiv:1207:7235 13, 585 (1964) [27] M. Peskin, arXiv:1207.2516 [20] G. ’t Hooft, Nucl. Phys. B 35, 167 (1971) [28] F. Wilczek, Czech. J. Phys. 54, A415 (2004) [21] T.-P. Cheng, L.-F. Li, Gauge Theory of Elementary Particle Physics (Clarendon Press, Oxford, 1984)

1037