The experimental discovery of the Higgs boson at the Large Hadron Collider (LHC) of CERN in Geneva around the mass value of 125 GeV, will remain as one of the major scientific discoveries of our time, and a great success of the Standard Model of fundamental physics. Yet, its far-reaching implications are only just beginning to be understood. For example, some smart people in the particle physics community noticed that the “Planck’s mass” corresponds to a value *m _{P }*= 4,34 10

^{-8}kg, obtained by combining the values of Planck’s constant

*h*, speed of light

*c*, and the gravitational constant

*G*. If such a mass were to be concentrated in a sphere with surface about equal to the “Planck’s surface”,

*S*=

_{P}*hG*/

*c*

^{3}=1.65 10

^{-19}m

^{2}, it would become a black hole, since the radius of the sphere

*S*is very close to the Schwarzschild radius

_{P}*R*=2

_{S}*Gm*/

_{P}*c*

^{2}for the Planck mass

*m*. This is the limit at which gravity and quantum mechanics merge. In practice, this means that the Planck’s mass is the maximum mass that a single particle can have in our universe, that is 43,4 micro-grams, or about 10

_{P}^{19}GeV. Then, why is the Higgs boson mass so much smaller than Planck’s mass? What should exist between the two values? Already Paul Dirac, around 1938, had been puzzled by the fact that the masses of common particles (electron, proton, neutron) are so much smaller than

*m*. In modern particle physics the notion of “naturalness” has become popular, to indicate that typical values of physical quantities should be close to each other, on a sort of “natural” scale that puts the values in ascending, or “hierarchical” order. This comes as a more general feeling in the whole of physics, which I translate to my students by saying my (in)famous phrase: “In physics we have only three numbers: zero, one and infinity”. Meaning that a quantity can be either negligible, or too large to bother: we are only interested in something that is of order-1, in a sort of “anthropic principle”. (Although several examples exist, of quantities varying over tens of orders of magnitude, such as viscosities of liquids, yield strength of materials, radioactive decay constants…)

_{P}The essence of the hierarchy problem can be understood by looking at a well-known example from classical electrodynamics. The electron is surrounded by an electric field that diverges as 1/*r* , at small distances from its point-like charge. The energy in this classical field is called the *electron self-energy*, and it must necessarily contribute to the electron’s mass. But if we take the radius of the electron, *r _{e}*~10

^{-18}cm, the calculated self-energy contribution to the mass turns out greater than 100 GeV, a million times larger than the experimental electron’s rest mass of 511 keV. We could guess that the electron’s “bare” mass – the mass not coming from the electric field – somehow cancels almost all the self-energy contribution, but that kind of cancellation would require a precision of one part in a million, something which looks quite “unnatural”. It is like finding two kids with exactly the same weight, so exact to put them on the two ends of a seesaw and keep the balance at better than a few milligrams difference. However, when considering the quantum nature of the electron field with its spontaneous creation and annihilation of electron-positron pairs, such quantum loops screen the bare electron charge, and the self-energy becomes “naturally” close to the observed low value. The “naturalness” strategy suggests to always look further to find the solution (in the case of the electron, it is a breaking of the chiral symmetry at energies below 511 keV).

But if all particles get their masses from Higgs, this particle should not be the *heaviest* of them all? As it turns out, about 20 years before, in 1995, physicists at the Tevatron measured the mass of the *top* quark for the first time: it was a huge 174 GeV, 40 times bigger than the second-heaviest quark, and most importantly, more massive than the Higgs’ boson (predicted) value. So there goes this argument, that the Higgs’ boson could have a mass close to the maximum allowed value (and that is the Planck’s mass) but its actual value is about 125 GeV (that is, 17 orders of magnitude smaller). Are we in a situation similar to that of the electron self-energy? Should we look further to find some “new physics” that solves the puzzle? As we learned in the first two episodes of the series, the Higgs’ boson is the simplest known particle, a “fragment of vacuum” with no charge or spin. Its interaction with a *heavier* particle like the top quark, however, makes for quantum-loop corrections. Summing up such quantum corrections, a value of the Higgs’ mass much closer to *m _{p}* would be obtained: apart from putting it beyond the reach of any conceivable experiment, such a heavy-weight Higgs would not allow the universe as we know it, to have formed. Stephen Hawking once said that “the Higgs’ boson could make the Universe collapse”. What he actually meant (despite the stupid journalists’ hype that followed) was that the breaking of symmetry that led to the ”Mexican hat” potential, a few fractions of a second after the Big Bang, could not be the last one, and that it is theoretically possible that other, lower minima of the Higgs’ field exist, waiting to be reached by another spontaneous symmetry breaking. Theorists express this concept by saying that the Higgs’ mass is “unstable to quantum fluctuations”. (Un)fortunately, the reality tells a different story, and the Higgs’ mass remains there, at its lonely value of 125 GeV. The LHC has been going way further in its exploration of the mass-energy domain until 2012, when it was shut off for a major overhaul; when it restarted shooting beams of protons against antiprotons again, in the summer of 2015, it arrived at exploring energies up to 14 TeV, that is 100 times bigger than Higgs’ mass and – at least up to now – it found just… nothing.

LHC collisions to produce Higgs’ bosons are very infrequent: for every billion proton-proton collisions, only one Higgs boson is created. To further complicate the picture, Higgs bosons decay very quickly into other particles (quark-antiquark, W^{+}W^{–}, or Z^{0}Z^{0} pairs), and it is only by measuring the characteristics of these particles that the prior existence of the Higgs boson can be inferred. That’s the proverbial “needle in the haystack” problem. Such problems were discussed during a week-long workshop, Exotic Approaches to Naturalness, hosted by the CERN Theoretical Physics department, from January 30 to February 3, 2023. As it can be easily imagined, the workshop spurred a flurry of ideas, from the simplest to the weirdest: maybe the Higgs boson is made of yet more basic entities held together by some new, very strong force; maybe space-time possesses additional “supersymmetric” dimensions, generalized symmetries, ultraviolet/infrared mixing; some also went along the path of weak-gravity conjectures, and “magic zeroes”, including those who reject naturalness altogether; maybe the “unnatural” mass of the Higgs boson might even be linked to the non-zero value of the cosmological constant, which is responsible for the accelerating expansion of the universe. And it may as well be that the difficulty of finding a “natural” solution to the hierarchy problem is symptomatic of something bigger: the omission of gravity in the standard model. Perhaps the Higgs mass problem will disappear in a future theory that unifies quantum mechanics and gravity.

In reading about the Higgs’ boson, it is also possible that you stumble into a notion that the whole story is just *trivial*. However, as usual, the meaning of this word in physics is quite different from its ordinary meaning. This is just another technicality that hides another theoretical difficulty. (Okay, I remember once I saw a t-shirt saying: “A physicist is someone who solves a problem you didn’t know you had, in a way you don’t understand.”) Wikipedia cryptically states the following: *A quantum field theory is said to be trivial when the renormalized coupling goes to zero, after removing the ultraviolet cutoff. Consequently, the propagator becomes that of a free particle and the field is no longer interacting.* Easy, huh?

We can better try to get an intuitive meaning of “triviality” by thinking of the trajectories of (quantum-field) particles and their collisions (interactions). Generally speaking, in *d* dimensions, two geometric objects of the same dimensionality *k*, typically intersect in sets of dimension 2*k-d*. For example, two curves (*k*=1) in a plane (*d*=2) intersect in a set of 2−2 = 0-dimensional objects, that is at discrete points. Two surfaces (*k*=2) in the tridimensional space (*d*=3) intersect in 4−3 = 1-dimensional objects, that is along curves. Now, an “interaction” means that particles only collide if their spacetime trajectories touch somewhere. The quantum particle trajectories in a field path-integral are, in fact, random walks, and these things are known to have Hausdorff dimension 2 (the idea of this comparison comes from an old paper by Giorgio Parisi). Hence, two quantum random walks in three dimensions will typically intersect in a set of dimension 4−3 = 1 … lots of interactions therefore. In the four dimensions of space-time, however, 4−4=0, and the particles can only intersect at isolated points: not so many interactions. And in more than four dimensions the trajectories of randomly walking particles do not intersect at all, no matter how strong the interaction fields could be, nothing happens: so the theory is free, or “trivial”.

Unfortunately, it turns out that the classical theory in which the Higgs boson has spin-0 could not exist as a quantum-field theory, unless it is non-interacting. The problem is that the famous “Mexican hat” potential, with a term proportional to the fourth power of the field, or φ^{4}, diverges at high energies unless the interaction parameter goes to zero (*this* is the actual meaning of the cryptic Wikipedia statement above). Another old, 1981 paper by Michael Aizenman, at that time assistant professor at Princeton, showed that this is rigorously true for dimensions 5 and higher. So, we may think, since we live in four dimensions there is no reason to worry. The point is that Aizenman left the question in 4 dimensions open. Recently, triviality of the φ^{4} continuous model has been proved in 3+1 dimensions, whereas numerical simulations of the discrete-lattice version of the same model seems to indicate non-triviality. However, numerical simulations cannot probe the extreme high-energy domain (UV-cutoff), then the question remains unsolved. So, does Higgs particle exist or not? And how does it yield mass to other particles, if it cannot interact with them? The LHC people swear that the Higgs boson is there, and the Nobel committee stamped it. Therefore, in some way or another, the scalar part of of the Standard Model theory must properly work. Or else.

Physics-after-Higgs is today wandering in a foggy swampland, with lots of questions open (dark matter and dark energy, inflation, cosmological constant, multiverses, naturalness…) and claims that the “standard” models, both in high-energy physics and cosmology, could be the wrong track despite their striking successes. Physics seems on the verge of another “period of crisis” and waiting for another “change of paradigm”, in the language of Thomas Kuhn, similar to what happened at the beginning of XX century when the otherwise successful classical physics was replaced by quantum mechanics. However, the biggest difference is that the paradigm change of quantum mechanics was triggered by many unexpected experimental discoveries, whereas physics today seemingly suffers from a *lack* of expected experimental discoveries.