Naturalness after the Higgs

1 July 2022

Either new particles are keeping the Higgs boson light, or the universe is oddly fine-tuned for our existence. Nathaniel Craig goes down the rabbit hole of the electroweak hierarchy problem.

Artwork from Peter Higgs’ Nobel diploma

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 KL and KS 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.

The electroweak hierarchy problem

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. 



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