Hierarchy Puzzle

Prerequisites: Standard Model of Particle Physics, Renormalization. Full path (35 pages)

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Concrete Naturally Speaking: The Naturalness Criterion and Physics at the LHC

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

According to the Higgs mechanism, space is filled with a uniform density of Higgs substance. But because of the complexity of the quantum vacuum, the calm sea of Higgs substance is continually disturbed by the rapid production and annihilation of all sorts of virtual particles. The constant buzz of virtual particles affects the density of the space-filling Higgs substance. Just as ghosts flickering in and out of the netherworld leave vivid memories in the minds of impressionable individuals, so virtual particles leave their indelible imprint in the Higgs substance. The swirling of virtual particles effectively gives a gigantic contribution to the density of the Higgs substance, which becomes extremely thick. Theoretical calculations show that this contribution to the density of the Higgs substance is proportional to the maximum energy carried by virtual particles. Since virtual particles can carry huge amounts of energy, the molasses-like Higgs substance becomes thicker than mud or even as hard as rock when quantum-mechanical effects are taken into account. Ordinary particles moving inside this medium should feel a tremendous resistance or, in more precise physical terms, should acquire enormous masses. Calculations based on a simple extrapolation of the Standard Model all the way down to the Planck length yield the result that electrons should be one million billion times more massive than what we observe – as heavy as prokaryote bacteria. But since electrons are obviously not as heavy as bacteria, we are confronted with a puzzle: why is the Higgs substance so dilute in spite of the natural tendency of virtual particles to make it grow thicker? This dilemma is usually referred to as the naturalness problem. The density of the Higgs substance determines how far the W and Z particles can propagate the weak force; in other words, it determines the weak length. Therefore, when virtual particles make the Higgs substance thicker, they effectively make the weak length shorter. The naturalness problem then refers to the conflict between, on one side, the tendency of virtual particles to make the weak length as small as the Planck length and, on the other side, the observation that the two length scales differ by the enormous factor of $10^{17}$ . Let us rephrase the problem with an analogy. Suppose that you insert a piece of ice into a hot oven. After waiting for a while, you open the oven and you discover that the ice is perfectly solid and hasn’t melted at all. Isn’t it puzzling? The air molecules inside the hot oven should have conveyed their thermal energy to the piece of ice, quickly raising its temperature and melting it. But they did not. The naturalness problem is equally puzzling. The energetic virtual particles are like the hot air molecules of the oven analogy, and the Higgs substance is like the piece of ice. The frenzied motion of virtual particles is communicated to the Higgs substance, which should become as hard as rock. And yet, it remains very dilute. The weak length should become as small as the Planck length. And yet, the two lengths differ by a factor of $10^{17}$ . Just as in the inside of a hot oven nothing can remain much cooler than the ambient temperature, so in the quantum vacuum virtual particles do not tolerate that the weak length remain much larger than the Planck length. Thus, the real puzzle is that no hierarchy between weak and gravitational force should exist at all, let alone there being a difference by a factor of $10^{17}$ . The essence of the naturalness problem is that the anarchic behaviour of virtual particles does not tolerate hierarchies. At this point, a very important warning should be issued. The naturalness problem is not a question of logical consistency. As the word says, it is only a problem of naturalness. Virtual particles provide one part of the energy stored in the Higgs substance. Nothing forbids the possibility of nature carefully choosing the initial density of the Higgs substance in such a way as to nearly compensate the effect from virtual particles. Under these circumstances, the enormous disparity between the weak and Planck lengths could be just the result of a precise compensation among various effects. Although this possibility cannot be logically excluded, it seems very contrived. Most physicists have difficulties accepting such accurate compensations between unrelated effects, and regard them as extremely unnatural.

"A Zeptospace Odyssey" by Guidice

Concrete

The Higgs potential of the standard model is

$V(H) = -\mu^2 H^\dagger H + \lambda (H^\dagger H)^2 .$

Its minimum is at $\langle H^\dagger H\rangle = v^2/2$ with $v^2 = \mu^2/\lambda$ , and expanding around it gives a Higgs boson of mass $m_h^2 = 2\mu^2 = 2\lambda v^2$ . The measured values are $v \approx 246\,\mathrm{GeV}$ and $m_h \approx 125\,\mathrm{GeV}$ . The mass parameter $\mu^2$ is the only dimensionful parameter of the standard model, and it sets the scale of all masses of the W, Z, quarks and charged leptons.

Quantum corrections. Loops of particles that couple to the Higgs shift $\mu^2$ . With a momentum cutoff $\Lambda$ , the largest one-loop contribution comes from the top quark,

$\delta\mu^2 \approx \frac{3y_t^2}{8\pi^2}\Lambda^2 ,$

with $y_t \approx 1$ (the fermion loop lowers the coefficient $-\mu^2$ of $H^\dagger H$ ). If the standard model is valid up to $\Lambda = 10\,\mathrm{TeV}$ , this is about $4\times10^6\,\mathrm{GeV}^2$ , compared with $\mu^2 = m_h^2/2 \approx 8\times10^3\,\mathrm{GeV}^2$ . The bare parameter must then cancel the correction to about one part in a few hundred. If $\Lambda$ is the Planck scale, about $10^{19}\,\mathrm{GeV}$ , the cancellation must hold to about 32 decimal places.

The cutoff itself is not physical, and in dimensional regularization the $\Lambda^2$ term does not appear. The problem comes back as soon as there is any heavy particle of mass $M$ that couples to the Higgs, for example through a term $\lambda_S\,H^\dagger H\,S^\dagger S$ with a heavy scalar $S$ . Its loop gives a physical shift of order $\lambda_S M^2/(16\pi^2)$ (times a logarithm), so the Higgs mass is sensitive to the heaviest scale in the theory. Grand unification or quantum gravity would provide such scales.

Comparison with the electron. The electron mass receives the QED correction $\delta m_e = \frac{3\alpha}{4\pi}m_e\ln(\Lambda^2/m_e^2)$ . It is proportional to $m_e$ itself, because for $m_e = 0$ the theory has a chiral symmetry that forbids a mass term, and it grows only logarithmically. A small electron mass is therefore stable under quantum corrections. For the Higgs mass no symmetry of the standard model appears when $m_h \to 0$ , and the corrections are quadratic in the heavy scale. This difference is what 't Hooft's criterion of technical naturalness captures (see also the discussion of naturalness under "Why is it interesting?").

Abstract

In the Wilsonian picture of renormalization, a quantum field theory with cutoff $\Lambda$ is a point in the space of couplings, and the renormalization group flow describes how the couplings change as modes are integrated out. Near the free (Gaussian) fixed point, the operator $H^\dagger H$ has scaling dimension close to $2$ in four dimensions, so it is relevant. Its dimensionless coupling $\hat\mu^2(k) = \mu^2(k)/k^2$ grows approximately like $(\Lambda/k)^2$ as the scale $k$ is lowered from $\Lambda$ . Obtaining $m_h \ll \Lambda$ at low energies requires the bare coupling at the scale $\Lambda$ to lie close to the critical surface, the set of theories that flow to a massless theory. The required distance is of order $m_h^2/\Lambda^2$ . This is the same tuning that in statistical mechanics is needed to bring a system close to a critical point, where a correlation length much larger than the lattice spacing appears.

Two standard ways to quantify the tuning:

't Hooft's criterion: a small parameter is technically natural if setting it to zero enlarges the symmetry of the theory. Fermion masses (chiral symmetry) and gauge boson masses (gauge symmetry) satisfy it, the Higgs mass parameter does not.
The Barbieri-Giudice measure $\Delta = \max_p\left|\partial\ln m_Z^2/\partial\ln p\right|$ over the fundamental parameters $p$ of a model counts how sensitively the electroweak scale depends on them.

Proposed solutions add structure that makes $\mu^2$ technically natural or lowers the effective cutoff:

Supersymmetry relates the Higgs to fermions. Bosonic and fermionic loops cancel, and the remaining corrections are proportional to the supersymmetry breaking scale, which must then be near the TeV scale.
In composite Higgs models the Higgs is a pseudo-Goldstone boson of a new strong interaction, and the shift symmetry protects its mass up to the compositeness scale.
In models with extra dimensions the fundamental scale of gravity can be much lower than $10^{19}\,\mathrm{GeV}$ .
Other ideas give up a symmetry explanation: cosmological relaxation of $\mu^2$ by a dynamical field, or anthropic selection.

The LHC has not found the new particles that the first two classes predict near the TeV scale. Whether naturalness is a good guide for model building is therefore debated, and the hierarchy problem is closely related to the cosmological constant problem, where the same kind of argument applies to an operator of dimension zero.

Fine Tuning Problem and the Renormalization Group C. Wetterich

Why is it interesting?

Since long ago [1, 2] physicists have been reluctant to accept small (or large) numbers without an underlying dynamical explanation, even when the smallness of a parameter is technically natural in the sense of ’t Hooft [3]. One reason for this reluctance is the belief that all physical quantities must eventually be calculable in a final theory with no free parameters. It would be strange for small numbers to pop up accidentally from the final theory without a reason that can be inferred from a low-energy perspective.

https://arxiv.org/pdf/1610.07962.pdf

Look at the Higgs field φ responsible for breaking electroweak theory. We don’t know its renormalized or physical mass precisely, but we do know that it is of order $M_{EW}$ . Imagine calculating the bare perturbation series in some grand unified theory - the precise theory does not enter into the discussion —starting with some bare mass $\mu_0$ for φ. The Weisskopf phenomenon tells us that quantum correction shifts $μ_0^2$ by a huge quadratically cutoff dependent amount $δμ_0^2 ∼ f^2\Lambda^2 \sim f^2 M_{GUT}^2$ , where we have substituted for $\Lambda$ the only natural mass scale around, namely $M_{GUT}$ , and where $f$ denotes some dimensionless coupling. To have the physical mass squared $μ^2 = μ_0^2 + δμ_0^2$ come out to be of order $M_{EW}$ , something like 28 orders of magnitude smaller than $M_{GUT}$ , would require an extremely fine-tuned and highly unnatural cancellation between $\mu_0$ and $δμ_0$ . How this could happen “naturally” poses a severe challenge to theoretical physicists.

Naturalness The hierarchy problem is closely connected with the notion of naturalness dear to the theoretical physics community. We naturally expect that dimensionless ratios of parameters in our theories should be of order unity, where the phrase “order unity” is interpreted liberally between friends, say anywhere from $10^{-2}$ or $10^{-3}$ to $10^2$ or $10^3$ . Following ’t Hooft, we can formulate a technical definition of naturalness: The smallness of a dimensionless parameter η would be considered natural only if a symmetry emerges in the limit η → 0. Thus, fermion masses could be naturally small, since, as you will recall from chapter II.1, a chiral symmetry emerges when a fermion mass is set equal to zero. On the other hand, no particular symmetry emerges when we set either the bare or renormalized mass of a scalar field equal to zero. This represents the essence of the hierarchy problem.

page 419 in QFT in a Nutshell by A. Zee

Hierarchy Puzzle

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Concrete

Abstract

Why is it interesting?

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