Modern physics' two greatest puzzles have exactly the same shape ── coarse-graining fails here and only here
In Episode 4 we saw the magic of coarse-graining ── details from the floor above die as \((E/\Lambda)^n\), and the floor below closes in its own language. Hence chemists need not know about quarks. But the magic has exactly two holes. The Higgs mass and the cosmological constant. These two alone cannot shield themselves from the far-off floor of the Planck scale and take the hit square on. And the observed values are \(10^{34}\) and \(10^{120}\) times smaller than that hit ── as if the bare value and the quantum correction had agreed in advance to cancel to the 34th digit, to the 120th digit. These are modern physics' two greatest puzzles. And the important thing is ── they have the same shape. This episode reads that shape.
Start with why things are usually fine. In quantum theory every quantity receives corrections from the world at higher energies. Most of the time this is harmless because a symmetry protects it.
| Quantity | Protecting symmetry | How the correction enters | Danger |
|---|---|---|---|
| electron mass | chiral symmetry | \(\delta m \propto m\ln(\Lambda/m)\) ── multiplicatively | safe (zero stays zero) |
| photon / gluon mass | gauge symmetry | forbidden outright (zero) | safe |
| couplings such as charge | (dimension-4 operators) | slowly, as \(\ln\Lambda\) | safe |
| Higgs mass² | none | \(\delta m_H^2 \propto \Lambda^2\) ── additively | dangerous |
| cosmological constant (vacuum energy) | none | \(\delta\rho \propto \Lambda^4\) | most dangerous |
The electron row is the crucial one. If the correction is proportional to \(m\) (multiplicative), then a small starting value keeps a small correction ── so it can stay small. That is called being technically natural. If instead the correction enters additively as \(\Lambda^2\), a small starting value is swamped and smallness cannot be maintained.
As long as you trust the Standard Model, the quantum correction enters as the square of the cutoff \(\Lambda\). Taking \(\Lambda\) to be the Planck scale, \(1.22\times10^{19}\ \mathrm{GeV}\):
$$\delta m_H^2 \sim \Lambda^2 \approx 1.5\times10^{38}\ \mathrm{GeV^2}$$But the measured Higgs mass is 125 GeV, i.e.
$$m_H^2 \approx 1.6\times10^{4}\ \mathrm{GeV^2}$$The ratio is \(1.6\times10^{4}/1.5\times10^{38}\approx10^{-34}\). The bare value and the correction must therefore agree and cancel to the 34th digit. A trillionth of a trillionth of another hundred-millionth. Too painful to call a coincidence ── hence the hierarchy problem.
The vacuum has quantum-fluctuation energy too, naively of order \(\Lambda^4\):
$$\rho_{\text{theory}}\sim \Lambda^4 \approx (1.22\times10^{19})^4 \approx 2\times10^{76}\ \mathrm{GeV^4}$$The value inferred from the observed accelerating expansion is
$$\rho_{\text{observed}}\sim 10^{-47}\ \mathrm{GeV^4}$$The ratio is about \(10^{-123}\) (often quoted as roughly \(10^{-120}\), depending on how you estimate). It is sometimes called the worst theoretical prediction in the history of physics. This is the cosmological constant problem.
Here is the heart. Translated into Episode 4's language, why it is only these two becomes a single line.
| Operator dimension d | What coarse-graining does | Examples in the Standard Model |
|---|---|---|
| \(d>4\): irrelevant | shrinks as \((E/\Lambda)^{d-4}\) | almost everything (which is why the world has layers) |
| \(d=4\): marginal | drifts logarithmically | gauge couplings, Yukawa couplings |
| \(d<4\): relevant | grows as you head to low energy | exactly two of them |
Those two are ── the identity operator with \(d=0\) (the cosmological constant) and \(H^\dagger H\) with \(d=2\) (the Higgs mass²). Every other potentially relevant term (a fermion mass \(\bar\psi\psi\) with \(d=3\), a gauge-field mass \(A^2\) with \(d=2\)) is forbidden by a symmetry. So the universe has exactly two unprotected relevant operators ── and those two are precisely modern physics' two greatest puzzles.
Put differently: coarse-graining works only in the irrelevant directions. In the relevant directions, coarse-graining does not discard information ── it magnifies the floor above and brings it down. So it cannot be shielded. The hierarchy problem and the cosmological constant problem are not "new puzzles"; they are the flip side of the very fact that the world has layers. Why the magic works and why it fails in exactly two places come from the same theorem.
The easiest way to feel how strange "agreement to 34 digits" is, is to lay the digits out. The figure below places two enormous numbers ── the bare value and the quantum correction ── side by side, with a slider for "how many digits agree."
Matching digits cancel and vanish; the leftovers are the physical quantity we observe. For the Higgs you need about 34 digits, for the cosmological constant about 120, before you reach the observed value ── switch between the two problems with the button and compare how far away the mark is.
No answer yet. There are three leading positions.
| Position | Claim | Where it stands |
|---|---|---|
| ① a new symmetry protects it | supersymmetry or similar makes the \(\Lambda^2\) correction cancel against contributions from partner particles ── i.e. "no protecting symmetry" is simply wrong | the most beautiful solution, but no superpartners have shown up at the LHC in the expected mass range. Optimism has cooled |
| ② anthropics / multiverse | there are countless universes and the value varies; observers only exist where galaxies can form | Weinberg used this reasoning in 1987 to predict an upper bound on the cosmological constant, and the 1998 observation fell within it. But testing it is hard |
| ③ "naturalness" is the wrong criterion | the intuition that "matched digits are unnatural" may itself be a groundless aesthetic | a growing number of researchers take this line. But its predictive power is weak |
Established: that quantum corrections to the Higgs mass squared enter as the square of the cutoff (scalars have no symmetry protecting their mass); that fermion masses are protected by chiral symmetry and gauge-field masses by gauge symmetry; the classification of operators as relevant/marginal/irrelevant by dimension; that the unprotected relevant operators of the Standard Model are the identity operator (cosmological constant) and \(H^\dagger H\); that the ratios between the measured values (\(m_H\approx125\) GeV, the cosmological constant energy density) and naive Planck-scale estimates are around \(10^{-34}\) and \(10^{-120}\); and that Weinberg (1987) derived an anthropic upper bound on the cosmological constant which later observations fell within ── all established physics and standard understanding.
Things to be careful about: (1) The language of "\(\Lambda^2\) corrections cancelling against a bare value" is tied to cutoff regularisation. In other schemes such as dimensional regularisation the \(\Lambda^2\) terms never appear and the problem instead presents itself as a hierarchy between scales (the masses of new particles, say). The figure's "match the digits" picture is a visualisation for intuition, not the literal calculation. (2) The \(10^{-120}\) for the cosmological constant swings between roughly \(10^{-120}\) and \(10^{-123}\) depending on where you cut. (3) The very premise that "unnatural means it needs explaining" (naturalness) is not an established physical principle but a guideline, and its validity is currently under debate ── that is position ③ above. (4) This episode is therefore an organisation of an unsolved puzzle in the language of the renormalization group, not a solution.
Quantum corrections are usually protected by symmetry ── electron mass by chiral symmetry, gauge-field mass by gauge symmetry. Protected quantities take their corrections multiplicatively and can stay small. But the Higgs mass² and the vacuum energy have no protecting symmetry and take direct additive hits of \(\Lambda^2\) and \(\Lambda^4\). Matching the observed values then requires cancellations of 34 and 120 digits.
And it is no accident that there are exactly two. In Episode 4's language, the Standard Model has exactly two unprotected relevant operators (\(d<4\)) ── the \(d=0\) identity operator (the cosmological constant) and the \(d=2\) \(H^\dagger H\) (the Higgs mass²). In relevant directions, coarse-graining doesn't shield the floor above; it magnifies and delivers it. So "the world has layers" and "there is a hierarchy problem" are two sides of one coin. No answer yet (a new symmetry / anthropics / naturalness being the wrong criterion, all three still in play).
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen the slider changes how many digits match, and the button switches between the Higgs and the cosmological constant. "Snap to the observed value" jumps to the required number of digits. "See the answer" opens each solution.