Subject Grouping: Physics

Many of the most arbitrary aspects of the Standard Model of particle physics (SM) are intimately connected to the scalar sector of the theory. The SM comprises just one scalar particle, the Higgs boson, and assumes a specific scalar potential (the famous “Mexican hat”) to define the dynamics of electroweak (EW) interactions. But the fact that the Higgs boson acquires a non-zero vacuum expectation value that defines the mass scale of EW interactions (around 100–200 GeV) is assumed, not explained, by the SM. Indeed, why the Higgs-boson mass is constrained to be at the EW scale, while quantum corrections should push it to much higher values (the so-called naturalness problem, see Naturalness after the Higgs), is not justified by any symmetry of the SM. At the same time, the SM assumes that fermion masses are generated via arbitrary Yukawa-type interactions with the scalar field but it does not explain the hierarchy of couplings or masses that we observe, nor the specific flavour structure that arises from the presence of just one scalar field. 

Future colliders are vital to push the precision Higgs programme to the next level

The scalar sector of the SM may therefore be seen as a messenger of a more fundamental theory that replaces the SM at energies beyond the EW scale and turns apparent arbitrariness into logical consequences. After all, the mechanism of EW symmetry breaking as realised in the SM via the Brout–Englert–Higgs (BEH) field is just the simplest possible way to generate massive EW gauge bosons and fermions while preserving gauge symmetry. The scalar potential could be more complicated, for example involving multiple scalar fields, as is common in many beyond-the-SM (BSM) theories. This would result in a richer pattern of stable and metastable minima and influence the nature of the EW phase transition. A first-order phase transition, together with extra sources of CP violation beyond what is implied by the SM, could explain the origin of the matter–antimatter asymmetry of the universe via EW baryogenesis (see Electroweak baryogenesis). Understanding the origin of the EW scale is thus key to connecting very different realms of particle physics and cosmology, and the question we face while we look into the future of collider physics. 

Game changer

The discovery of the Higgs boson during Run 1 of the LHC has been a game changer in the exploration of new physics beyond the EW scale. The measurement of the Higgs-boson mass has added the last missing input parameter to precision global fits of the SM, which now provide a very powerful tool to constrain BSM scenarios. Thanks to an unprecedented level of precision reached in both theory and experiment, the measurement of Higgs-boson couplings to EW gauge bosons (W, Z) and to the first two generations of quarks and leptons (t, b, τ, µ) from Run 2 data has already constrained their deviations from SM expectations to within 5–20%, with the best accuracy reached for the couplings to the gauge bosons. Based on these results, the High-Luminosity LHC (HL-LHC) is projected to constrain the effects of new physics on Higgs-boson couplings to EW gauge bosons to 1–2%, and to heavy quarks and fermions to 3–5%. If no anomalies are found, this level of accuracy will push the lower bound on the scale of new physics into the TeV ballpark. Vice versa, the detection of possible anomalies may point to the presence of new physics at the TeV scale, possibly just around the corner.

On the other hand, testing the SM scalar potential will still be challenging even during the HL-LHC era. The shape of the BEH potential can be tested by measuring the Higgs-boson self-interactions corresponding to its cubic and quartic terms. In the SM, these interactions are strictly proportional to the Higgs-boson mass via the vacuum expectation value of the BEH field. Deviations from the SM are searched for via Higgs pair production and radiative corrections to single-Higgs measurements. Although the LHC and HL-LHC promise to provide evidence for di-Higgs production, the extraction of the Higgs self-coupling from such measurements will be statistically limited.

Future colliders are vital to push the precision Higgs programme to the next level. While the type and concept of the next collider is yet to be decided, all proposed facilities would deliver a huge number of Higgs bosons over their lifetime, operating at different and well targeted centre-of-mass energies (see “At a glance” figure). They can complement one another and, staggered over a period of the next few decades, provide the missing elements of the EW puzzle.

Among future lepton colliders under study, circular e⁺e^– colliders (CEPC, FCC-ee) are expected to operate at lower energies between 90–350 GeV with very high luminosities, while linear e⁺e^– colliders (ILC, C³, CLIC) offer both low- and high-energy phases generally with slightly lower luminosities. Combined with data from the HL-LHC, these “Higgs factories” would enable the SM, including most Higgs couplings, to be stress-tested below the per-cent level and in cases at or below the per-mille level. In particular, FCC-ee operating at the s-channel Higgs resonance (125 GeV) has the capability to provide bounds on couplings as small as the electron Yukawa coupling, while linear e⁺e^– colliders operating at 550–600 GeV and above could substantially improve on the top-quark Yukawa coupling with respect to the HL-LHC. A possible muon collider, operated either as a Higgs factory at 125 GeV or as a high-energy discovery machine at 3–10 TeV, is estimated to reach similar precisions on Higgs couplings to other particles as e⁺e^– machines. 

Finally, high-energy lepton colliders (ILC 1000, CLIC 3000 and a 3–30 TeV muon collider) and very high-energy hadron colliders (FCC-hh at 100 TeV) would reach enough statistics and energy to measure the Higgs self-coupling and investigate the nature of the BEH potential, either via di-Higgs or single-Higgs production (see “Self-coupling” figure). With an aggressive Higgs physics programme they may also reach enough sensitivity to probe the cubic and quartic terms in the BEH potential separately. 

Almost half a century after it was predicted, the LHC delivered the Higgs boson in spectacular style on 4 July 2012. Over the next 15–20 years, the machine and its luminosity upgrade will continue to enable ATLAS and CMS to make great strides in understanding the Higgs boson’s properties. But to fully exploit the discovery of the Higgs boson and explore its mysterious relation to new physics beyond the EW scale, we will need a successor collider.

Referring to the field equation of general relativity R_μν – ½ Rg_μν = κT_μν , Einstein is reported to have said that the left-hand side, constructed from space–time curvature, is “a palace of gold”; while the right-hand side, which parameterises the energy and momentum of matter, is by comparison “a hovel of wood”. Present-day physics has arrived at much more concrete ideas about the right-hand side than were available to Einstein. It is fair to say that some of it has come to look quite palatial, and fully worthy to stand alongside the left-hand side. These are the terms that involve field kinetic energy and gauge bosons, as described by the Standard Model (SM). Their form follows logically, within the framework of relativistic quantum field theory, directly from the principles of local gauge symmetry and relativity. Mathematically, they also speak the same geometric language as the right-hand side. The gauge bosons are avatars of curvature in “internal spaces”, similar to how gravitons are the avatars of space–time curvature. Internal spaces parameterise ways in which fields can vary – and thus, in effect, move – independently of ordinary motion in space–time. In this picture, the strong, weak and electromagnetic interactions arise from the influence of internal space curvature on internal space motion, similar to how gravity arises from the influence of space–time curvature on space–time motion.

The Higgs particle is the only portal connecting normal matter to such phantom fields

The other contributions to T_μν, all of which involve the Higgs particle, do not yet reach that standard. We can aspire to do better! They are of three kinds. First, there are the many Yukawa-like terms from which quark and lepton masses and mixings arise. Then there is the Higgs self-coupling and finally a term representing its mass. These contributions to T_μν contain almost two dozen dimensionless coupling parameters that present-day theory does not enable us to calculate or even much constrain. It is therefore important to investigate experimentally, through quantitative studies of Higgs-particle properties and interactions, whether this ramshackle structure describes nature accurately. 

Higgs potential

The Higgs boson is special among the elementary particles. As the quantum of a condensate that fills all space, it is metaphorically “a fragment of vacuum”. Speaking more precisely, the Higgs particle has no spin, no electric or colour charge and, at the level of strong and electromagnetic interactions, normal charge conjugation and parity. Thus, it can be emitted singly and without angular momentum barriers, and it can decay directly into channels free of colour and electromagnetically charged particles, which might otherwise be difficult to access. For these and other, more technical, reasons, the Higgs particle has the potential to reveal new physical phenomena of several kinds. 

A unique aspect of the Higgs mass term is especially promising for revealing possible shortcomings in the SM. In quantum field theory, an important property of an interaction is the “mass dimension” of the operator that implements it – a number that in an important sense indicates its complexity. Scalar and gauge fields have mass dimension 1 as do space–time derivatives, whereas fermion fields have mass dimension 3/2. More complicated operators are built up by multiplying these, and the mass dimension of a product is the sum of the mass dimensions of its factors. Interactions associated with operators whose mass dimension is greater than 4 are problematic because they lead to violent quantum fluctuations and mathematical divergences. Whereas all the other terms in the SM Lagrangian arise from operators of mass dimension 4, the Higgs mass term has mass dimension 2. Thus it is uniquely open to augmentation by couplings to hypothetical new SU(3) × SU(2) × U(1) singlet scalar fields, because the mass dimension of the augmented interaction can be 3 or 4 – i.e. still “safe”. The Higgs particle is the only portal connecting normal matter to such phantom fields.

Why is this an interesting observation? There are three main reasons: two broadly theoretical, one pragmatic. First of all, the particles that are generally considered part of the SM carry a variety of charge assignments under the gauge groups SU(3) × SU(2) × U(1) that govern the strong and electroweak interactions. For example, the left-handed up quark is charged under all three groups, while the right-handed electron carries only U(1) hypercharge. Thus it is not only logically possible, but reasonably plausible, that there could be particles that are neutral under all three groups. Such phantom particles might easily escape detection, since they do not participate in the strong or electroweak interactions. Indeed, there are several examples of well-motivated candidate particles of that kind. Axions are one. Since they are automatically “dark” in the appropriate sense, phantom particles could contribute to the astronomical dark matter, and might even dominate it, as model-builders have not failed to notice. Also, many models of unification bring in scalar fields belonging to representations of a unifying gauge group that contains SU(3) × SU(2) × U(1) singlets, as do models with supersymmetry. Only phantom scalars are directly accessible through the Higgs portal, but phantoms of higher spin, including right-handed neutrinos, could cascade from real or virtual scalars.

Mysterious values

Second, the empirical value of the Higgs mass term is somewhat mysterious and even problematic, given that quantum corrections should push it to a value many orders of magnitude higher. This is the notorious “hierarchy problem” (see Naturalness after the Higgs). Given this situation, it seems appropriate to explore the possibility that part (or all) of the effective mass-term of the SM Higgs particle arises from more fundamental couplings upon condensation of SU(3) × SU(2) × U(1) singlet scalar fields, i.e. the emergence of a non-zero space-filling field, as occurs in the Brout–Englert–Higgs mechanism.

The portal idea leads to concrete proposals for directions of experimental exploration

Third, the portal idea leads to concrete proposals for directions of experimental exploration. These are of two basic kinds: one involves the observed strength of conventional Higgs couplings, the other the kinematics of Higgs production and decay. Couplings of the Higgs field to singlets that condense will lead to mixing, altering numerical relationships among Higgs-particle couplings and masses of gauge bosons, and of fermions from their minimal SM values. Also, the Higgs-field couplings to gauge bosons and fermions will be divided among two or more mass eigenstates. Since existing data indicates that deviations from the minimal model are small, the coupling of normal matter to the “mostly but not entirely” singlet pieces could be quite small, perhaps leading to very long lifetimes (as well as small production rates) for those particles. Whether or not the phantom particles contribute significantly to cosmological dark matter, they will appear as missing energy or momentum accompanying Higgs particle decay or, through Bremsstrahlung-like processes, when they are produced. 

We introduced the term “Higgs portal” to describe this circle of ideas in 2006, triggering a flurry of theoretical discussion. Now that the portal is open for business, and with larger data samples in store at the LHC, we can think more concretely about exploring it experimentally.

Often in physics, experimentalists observe phenomena that theorists had not been able to predict. When the muon was discovered, theoreticians were confused; a particle had been predicted, but not this one. Isidor Rabi came with his famous outcry: “who ordered that?” The J/ψ is another special case. A particle was discovered with properties so different from the particles that were expected, that the first guesses as to what it was were largely mistaken. Soon it became evident that it was a predicted particle after all, but it so happened that its features were more exotic than was foreseen. This was an experimental discovery requiring new twists in the theory, which we now understand very well. The Higgs particle also has a long and interesting history, but from my perspective, it was to become a triumph for theory. 

From the 1940s, long before any indications were seen in experiments, there were fundamental problems in all theories of the weak interaction. Then we learned from very detailed and beautiful measurements that the weak force seemed to have a vector-minus axial-vector (V-A) structure. This implied that, just as in Yukawa’s theory for the strong nuclear force, the weak force can also be seen as resulting from an exchange of particles. But here, these particles had to be the energy quanta of vector and axial-vector fields, so they must have spin one, with positive and negative parities mixed up. They also must be very heavy. This implied that, certainly in the 1960s, experiments would not be able to detect these intermediate particles directly. But in theory, we should be able to calculate accurately the effects of the weak interaction in terms of just a few parameters, as could be done with the electromagnetic force. 

Electromagnetism was known to be renormalisable – that is, by carefully redefining and rearranging the mass and interaction parameters, all observable effects would become calculable and predictable, avoiding meaningless infinities. But now we had a difficulty: the weak exchange particles differed from the electromagnetic ones (the photons) because they had mass. The mass was standing in the way when you tried to do what was well understood in electromagnetism. How exactly a correct formalism should be set up was not known, and the relationship between renormalisability and gauge invariance was not understood at all. Indeed, today we can say that the first hints were already there by 1954, when C N Yang and Robert Mills wrote a beautiful paper in which they generalised the principle of local gauge invariance to include gauge transformations that affect the nature of the particles involved. In its most basic form, their theory described photons with electric charge.

Thesis topic

In 1969 I began my graduate studies under the guidance of Martinus J G Veltman. He explained to me the problem he was working on: if photons were to have mass, then renormalisation would not work the same way. Specifically, the theory would fail to obey unitarity, a quantum mechanical rule that guarantees probabilities are conserved. I was given various options for my thesis topic, but they were not as fundamental as the issues he was investigating. “I want to work with you on the problem you are looking at now,” I said. Veltman replied that he had been working on his problem for almost a decade; I would need lots of time to learn about his results. “First, read this,” he said, and he gave me the Yang–Mills paper. “Why?” I asked. He said, “I don’t know, but it looks important.”

That, I could agree  with. This was a splendid idea. Why can’t you renormalise this? I had convinced myself that it should be possible, in principle. The Yang–Mills theo­­ry was a relativistic quantised field theory. But Veltman explained that, in such a theory, you must first learn what the Feynman rules are. These are the prescriptions that you have to follow to get the amplitudes generated by the theory. You can read off whether the amplitudes are unitary, obey dispersion relations, and check that everything works out as expected.

Many people thought that renormalisation – even quantum field theory – was suspect. They had difficulties following Veltman’s manipulations with Feynman diagrams, which required integrations that do not converge. To many investigators, he seemed to be sweeping the difficulties with the infinities under the rug. Nature must be more clever than this! Yang–Mills seemed to be a divine theory with little to do with reality, so physicists were trying all sorts of totally different approaches, such as S-matrix theory and Regge trajectories. Veltman decided to ignore all that.

Solid-state inspiration

Earlier in the decade, some investigators had been inspired by results from solid-state physics. Inside solids, vibrating atoms and electrons were described by nonrelativistic quantum field theories, and those were conceptually easier to understand. Philip Anderson had learned to understand the phenomenon of superconductivity as a process of spontaneous symmetry breaking; photons would obtain a mass, and this would lead to a remarkable rearrangement of the electrons as charge carriers that would no longer generate any resistance to electric currents. Several authors realised that this procedure might apply to the weak force. In the summer of 1964, Peter Higgs submitted a manuscript to Physical Review Letters, where he noted that the mechanism of making photons massive should also apply to relativistic particle systems. But there was a problem. Jeffrey Goldstone had sound mathematical arguments to expect the emergence of massless scalar particles as soon as a continuous symmetry breaks down spontaneously. Higgs put forward that this theorem should not apply to spontaneously broken local symmetries, but critics were unconvinced.

The journal sent Higgs’s manuscript out to be peer reviewed. The reviewer did not see what the paper would add to our understanding. “If this idea has anything to do with the real world, would there be any possibility to check it experimentally?” The correct question would have been what the paper would imply for the renormalisation procedure, but this question was in nobody’s mind. Anyway, Higgs gave a clear and accurate answer: “Yes, there is a consequence: this theory not only explains where the photon mass comes from, but it also predicts a new particle, a scalar particle (a particle with spin zero), which unlike all other particles, forms an incomplete representation of the local gauge symmetry.” In the meantime, other papers appeared about the photon mass-generation process, not only by François Englert and Robert Brout in Brussels, but also by Tom Kibble, Gerald Guralnik and Carl Hagen in London. And Sheldon Glashow, Abdus Salam and Steven Weinberg were formulating their first ideas (all independently) about using local gauge invariance to create models for the weak interaction. 

I started to study everything from the ground up

At the time spontaneous symmetry breaking was being incorporated into quantum field theory, the significance of renormalisation and the predicted scalar particles were hardly mentioned. Certainly, researchers were not able to predict the mass of such particles. Personally, although I had heard about these ideas, I also wasn’t sure I understood what they were saying. I had my own ways of learning how to understand things, so I started to study everything from the ground up. 

If you work with quantum mechanics, and you start from a relativistic classical field theory, to which you add the Copenhagen procedure to turn that into quantum mechanics, then you should get a unitary theory. The renormalisation procedure amounts to transforming all expressions that threaten to become infinite due to divergence of the integrals, to apply only to unobservable qualities of particles and fields, such as their “bare mass” and “bare charge”. If you understand how to get such things under control, then your theory should become a renormalised description of massive particles. But there were complications.

The infinities that require a renormalisation procedure to tame them originate from uncontrolled behaviour at very tiny distances, where the effective energies are large and consequently the effects of mass terms for the particles should become insignificant. This revealed that you first have to renormalise the theory without any masses in them, where also the spontaneous breakdown of the local symmetry becomes insignificant. You had to get the particle book-keeping right. A massless photon has only two observable field components (they can be left- or right-rotating), whereas a massive particle with the same spin can rotate in three different ways. One degree of freedom did not match. This was why an extra field was needed. If you wanted massive photons with electric charges +, 0 or –, you would need a scalar field with four components; one of these would represent the total field strength, and would behave as an extra, neutral, spin-0 particle – the observable particle that Higgs had talked about – but the others would turn the number of spinning degrees of freedom of the three other bosons from two to three each (see “Dynamical” figure).

One question

In 1970 Veltman sent me to a summer school organised by Maurice Lévy in a new science institute at Cargèse on the French island of Corsica. The subject would be the study of the Gell–Mann–Lévy model for pions and nucleons, in particular its renormalisation and the role of spontaneous symmetry breaking. Will renormalisation be possible in this model, and will it affect its symmetry? The model was very different from what I had just started to study: Yang–Mills theory with spontaneous breaking of its symmetry. There were quite a few reputable lecturers besides Lévy himself: Benjamin Lee and Kurt Symanzik had specialised in renormalisation. Shy as I was, I only asked one question to Lee, and the same to Symanzik: does your analysis apply to the Yang–Mills case?

Both gave me the same answer: if you are Veltman’s student, ask him. But I had, and Veltman did not believe that these topics were related. I thought that I had a better answer, and I fantasised that I was the only person on the planet who knew how to do it right. It was not obvious at all; I had two German roommates at the hotel where I had been put, who tried to convince me that renormalisation of Feynman graphs where lines cross each other would be unfathomably complicated.

Veltman had not only set up detailed, fully running machinery to handle the renormalisation of all sorts of models, but he had also designed a futuristic computer program to do the enormous amount of algebra required to handle the numerous Feynman diagrams that appear to be relevant for even the most basic computations. I knew he had those programs ready and running. He was now busy with some final checks: if his present attempts to check the unitarity of his renormalised model still failed, we should seriously consider giving this up. Yang–Mills theories for the weak interactions would not work as required.

But Veltman had not thought of putting a spin-zero, neutral particle in his model, certainly not if it wasn’t even in a complete representation of the gauge symmetry. Why should anyone add that? After returning from Cargèse I went to lunch with Veltman, during which I tried to persuade him. Walking back to our institute, he finally said, “Now look, what I need is not an abstract mathematical idea, what I want is a model, with a Lagrangian, from which I can read off the Feynman diagrams to check it with my program…”. “But that Lagrangian I can give you,” I said. Next, he walked straight into a tree! A few days after I had given him the Lagrangian, he came to me, quite excited. “Something strange,” he said, “your theory isn’t right because it still isn’t unitary, but I see that at several places, if the numbers had been a trifle different, it could have worked out.” Had he copied those factors ¼ and ½ that I had in my Lagrangian, I wondered? I knew they looked odd, but they originated from the fact that the Higgs field has isospin ½ while all other fields have isospin one.

No, Veltman had thought that those factors came from a sloppy notation I must have been using. “Try again,” I asked. He did, and everything fell into place. Most of all, we had discovered something important. This was the beginning of an intensive but short collaboration. My first publication “Renormalization of massless Yang–Mills fields”, published in October 1971, concerned the renormalisation of the Yang–Mills theory without the mass terms. The second publication that year, “Renormalizable Lagrangians for massive Yang–Mills fields,” where it was explained how the masses had to be added, had a substantial impact. 

There was an important problem left wide open, however: even if you had the correct Feynman diagrams, the process of cancelling out the infinities could still leave finite, non-vanishing terms that ruin the whole idea. These so-called “anomalies” must also cancel out. We found a trick called dimensional renormalisation, which would guarantee that anomalies cancel except in the case where particles spin preferentially in one direction. Fortunately, as charged leptons tend to rotate in opposite directions compared to quarks, it was discovered that the effects of the quarks would cancel those of the leptons. 

The fourth component

Within only a few years, a complete picture of the fundamental interactions became visible, where experiment and theory showed a remarkable agreement. It was a fully renormalisable model where all quarks and all leptons were represented as “families” that were only complete if each quark species had a leptonic counterpart. There was an “electroweak force”, where electromagnetism and the weak force interfere to generate the force patterns observed in experiments, and the strong force was tamed at almost the same time. Thus the electroweak theory and quantum chromodynamics were joined into what is now known as the Standard Model.

Be patient, we are almost there, we have three of the four components of this particle’s field

This theory agreed beautifully with observations, but it did not predict the mass of the neutral, spin-0 Higgs particle. Much later, when the W and the Z bosons were well-established, the Higgs was still not detected. I tried to reassure my colleagues: be patient, we are almost there, we have three of the four components of this particle’s field. The fourth will come soon.

As the theoretical calculations and the experimental measurements became more accurate during the 1990s and 2000s, it became possible to derive the most likely mass value from indirect Higgs-particle effects that had been observed, such as those concerning the top-quark mass. On 4 July 2012 a new boson was directly detected close to where the Standard Model said the Higgs  would be. After these first experimental successes, it was of utmost importance to check whether this was really the object we had been expecting. This has kept experimentalists busy for the past 10 years, and will continue to do so for the foreseeable future. 

The discovery of the Higgs particle is a triumph for high technology and basic science, as well as accurate theoretical analyses. Efforts spanning more than half a century paid off in the summer of 2012, and a new era of understanding the particles, their masses and interactions began.

When Victor Weisskopf sat down in the early 1930s to compute the energy of a solitary electron, he had no way of knowing that he’d ultimately discover what is now known as the electroweak hierarchy problem. Revisiting a familiar puzzle from classical electrodynamics – that the energy stored in an electron’s own electric field diverges as the radius of the electron is taken to zero (equivalently, as the energy cutoff of the theory is taken to infinity) – in Dirac’s recently proposed theory of relativistic quantum mechanics, he made a remarkable discovery: the contribution from a new particle in Dirac’s theory, the positron, cancelled the divergence from the electron itself and left a quantum correction to the self-energy that was only logarithmically sensitive to the cutoff. 

The same cancellation occurred in any theory of charged fermions. But when Weisskopf considered the case for charged scalar particles in 1939, the problem returned. To avoid the need for finely-tuned cancellations between this quantum correction and other contributions to a scalar’s self-energy, he posited that the cutoff energy for scalars should be close to their observed self-energy, heralding the appearance of new features that would change the calculation and render the outcome “natural”. 

Nearly 30 years would pass before Weisskopf’s prediction about scalars was put to the test. The charged pion, a pseudoscalar, suffered the very same divergent self-energy that he had computed. As the neutral pion is free from this divergence, Weisskopf’s logic suggested that the theory of charged and neutral pions should change at around 800 MeV, the cutoff scale suggested by the observed difference in their self-energies. Lo and behold, the rho meson appeared at 775 MeV. Repeating the self-energy calculation with the rho meson included, the divergence in the charged pion’s self-energy disappeared. 

This same logic would predict something new. It had been known for some time that the relative self-energy between the neutral kaons K_L and K_S diverged due to contributions from the weak interactions in a theory containing only the known up, down and strange quarks. Matching the observed difference suggested that the theory should change at around 3 GeV. Repeating the calculation with the addition of the recently proposed charm quark in 1974, Mary K Gaillard and Ben Lee discovered that the self-energy difference became finite, which allowed them to predict that the charm quark should lie below 1.5 GeV. The discovery at 1.2 GeV later that year promoted Weisskopf’s reasoning from an encouraging consistency check to a means of predicting new physics.

Higgs, we have a problem

Around the same time, Ken Wilson recognised that the coupling between the Higgs boson and other particles of the Standard Model (SM) leads to yet another divergent self-energy, for which the logic of naturalness implied new physics at around the TeV scale. Thus the electroweak hierarchy problem was born – not as a new puzzle unique to the Higgs, but rather the latest application of Weisskopf’s wildly successful logic (albeit one for which the answer is not yet known). 

History suggested two possibilities. As a scalar, the Higgs could only benefit from the sort of cancellation observed among fermions if there is a symmetry relating bosons and fermions, namely supersymmetry. Alternatively, it could be a light product of compositeness, just as the pions and kaons are light bound states of the strong interactions. These solutions to the hierarchy problem came to dominate expectations for physics beyond the SM, with a sharp target – the TeV scale – motivating successive generations of collider experiments. Indeed, when the physics case for the LHC was first developed in the mid-1980s, it was thought that new particles associated with supersymmetry or compositeness would be much easier to discover than the Higgs itself. But while the Higgs was discovered, no signs of supersymmetry or compositeness were to be found.

In the meantime, other naturalness problems were brewing. The vacuum energy – Einstein’s infamous cosmological constant – suffers a divergence of its own, and even the finite contributions from the SM are many orders of magnitude larger than the observed value. Although natural expectations for the cosmological constant fail, an entirely different set of logic seems to succeed in its place. To observe a small cosmological constant requires observers, and observers can presumably arise only if gravitationally-bound structures are able to form. As Steven Weinberg and others observed in the 1980s, such anthropic reasoning leads to a prediction that is remarkably close to the value ultimately measured in 1998. To have predictive power, this requires a multitude of possible universes across which the cosmological constant varies; only the ones with sufficiently small values of the cosmological constant produce observers to bear witness.

An analogous argument might apply to the electroweak hierarchy problem: the nuclear binding energy is no longer sufficient to stabilise the neutron within typical nuclei if the Higgs vacuum expectation value (VEV) is increased well above its observed value. If the Higgs VEV varies across a landscape of possible universes while its couplings to fermions are kept fixed, only universes with sufficiently small values of the Higgs VEV would lead to complex atoms and, presumably, observers. Although anthropic reasoning for the hierarchy problem requires stronger assumptions than for the cosmological-constant problem, its compatibility with null results at the LHC is enough to raise questions about the robustness of natural reasoning. 

Amidst all of this, another proposed scalar particle entered the picture. The observed homogeneity and isotropy of the universe point to a period of exponential expansion of spacetime in the early universe driven by the inflaton. While the inflaton may avoid naturalness problems of its own, the expansion of spacetime and the quantum fluctuations of fields during inflation lead to qualitatively new effects that are driving new approaches to the hierarchy problem at the intersection of particle physics, cosmology and gravitation.

Perhaps the most prominent of these new approaches came, surprisingly enough, from a failed solution to the cosmological constant problem. Around the same time as the first anthropic arguments for the cosmological constant were taking form, Laurence Abbott proposed to “relax” the cosmological constant from a naturally large value by the evolution of a scalar field in the early universe. Abbot envisioned the scalar evolving along a sloping, bumpy potential, much like a marble rolling down a wavy marble run. As it did so, this scalar would decrease the total value of the cosmological constant until it reached the last bump before the cosmological constant turned negative. Although the universe would crunch away into nothingness if the scalar evolved to negative values of the cosmological constant, it could remain poised at the last bump for far longer than the age of the observed universe. 

Despite the many differences among the new approaches, they share a common tendency to leave imprints on the Higgs boson

While this fails for the cosmological constant (the resulting metastable universe is largely devoid of matter), analogous logic succeeds for the hierarchy problem. As Peter Graham, David Kaplan and Surjeet Rajendran pointed out in 2015, a scalar evolving down a potential in the early universe can also be used to relax the Higgs mass from naturally large values. Of course, it needs to stop close to the observed mass. But something interesting happens when the Higgs mass-squared passes from positive values to negative values: the Higgs acquires a VEV, which gives mass to quarks, which induces bumps in the potential of a particular type of scalar known as an axion (proposed to explain the unreasonably good conservation of CP symmetry by the strong interactions). So if the relaxing scalar is like an axion – a relaxion, you might say – then it will encounter bumps in its potential when it relaxes the Higgs mass to small values. If the relaxion is rolling during an inflationary period, the expansion of spacetime can provide the “friction” necessary for the relaxion to stop when it hits these bumps and set the observed value of the weak scale. The effective coupling between the relaxion and the Higgs that induces bumps in the relaxion potential is large enough to generate a variety of experimental signals associated with a new, light scalar particle that mixes with the Higgs.

The success of the relaxion hypothesis in solving the hierarchy problem hinges on an array of other questions involving gravity. Whether the relaxion potential can remain sufficiently smooth over the vast trans-Planckian distances in field space required to set the value of the weak scale is an open question, one that is intimately connected to the fate of global symmetries in a theory of quantum gravity (itself the target of active study in what is known as the Swampland programme).

Models abound 

In the meantime, the recognition that cosmology might play a role in solving the hierarchy problem has given rise to a plethora of new ideas. For instance, in Raffaele D’Agnolo and Daniele Teresi’s recent paradigm of “sliding naturalness”, the Higgs is coupled to a new scalar whose potential features two minima. In the true minimum, the cosmological constant is large and negative, and the universe would crunch away into oblivion if it ended up in this vacuum. In the second, local minimum, the cosmological constant is safely positive (and can be made compatible with the small observed value of the cosmological constant by Weinberg’s anthropic selection). The Higgs couples to this scalar in such a way that a large value of the Higgs VEV destabilises the “safe” minimum. During the inflationary epoch, only universes with suitably small values of the Higgs VEV can grow and expand, while those with large values of the Higgs VEV crunch away. A second scalar coupled analogously to the Higgs can explain why the VEV is small but non-zero. Depending on how these scalars are coupled to the Higgs, experimental signatures range from the same sort of axion-like signals arising from the relaxion, to extra Higgs bosons at the LHC.

Alternatively, in the paradigm of “Nnaturalness” proposed by Nima Arkani-Hamed and others, the multitude of SMs over which the Higgs mass varies occur in one universe, rather than many. The fact that the universe is predominantly composed of one copy of the SM with a small Higgs mass can be explained if inflation ends and reheats the universe through the decay of a single particle. If this particle is sufficiently light, it will preferentially reheat the copy of the SM with the smallest non-zero value of the Higgs VEV, even if it couples symmetrically to each copy. The sub-dominant energy density deposited in other copies of the SM leaves its mark in the form of dark radiation susceptible to detection by the Simons Observatory or upcoming CMB-S4 facility. 

Finally, Gian Giudice, Matthew Mccullough and Tevong You have recently shown that inflation can help to understand the electroweak hierarchy problem by analogy with self-organised criticality. Just as adding individual grains of sand to a sandpile induces avalanches over diverse length scales – a hallmark of critical behaviour, obtained without tuning parameters – so too can inflation drive scalar fields close to critical points in their potential. This may help to understand why the observed Higgs mass lies so close to the boundary between the unbroken and broken phases of electroweak symmetry without fine tuning.

Going the distance 

Underlying Weisskopf’s natural reasoning is a long-standing assumption about relativistic theories of quantum mechanics: physics at short distances (the ultraviolet, or UV) is decoupled from physics at long distances (the infrared, or IR), making it challenging to apply a theory involving a large energy scale to a much smaller one without fine tuning. This suggests that loopholes may be found in theories that mix the UV and the IR, as is known to occur in quantum gravity. 

While the connection between this type of UV/IR mixing and the mass of the Higgs remains tenuous, there are encouraging signs of progress. For instance, Panagiotis Charalambous, Sergei Dubovsky and Mikhail Ivanov recently used it to solve a naturalness problem involving so-called “Love numbers” that characterise the tidal response of black holes. The surprising influence of quantum gravity on the parameter space of effective field theories implied by the Swampland programme also has a flavour of UV/IR mixing to it. And UV/IR mixing may even provide a new way to understand the apparent violation of naturalness by the cosmological constant.

We have come a long way since Weisskopf first set out to understand the self-energy of the electron. The electroweak hierarchy problem is not the first of its kind, but rather the one that remains unresolved. The absence of supersymmetry or compositeness at the TeV scale beckons us to search for new solutions to the hierarchy problem, rather than turning our backs on it. In the decade since the discovery of the Higgs, this search has given rise to a plethora of novel approaches, building new bridges between particle physics, cosmology and gravity along the way. Despite the many differences among these new approaches, they share a common tendency to leave imprints on the Higgs boson. And so, as ever, we must look to experiment to show the way.