The scale that may not be where it looks

The length no experiment can resolve computes how far away the Planck scale is and the answer is bleak: the LHC resolves six thousand million million Planck lengths, and a century of accelerators has covered seven of the twenty decades between. Assembling the scale from the three constants is what puts it there in the first place.

Every number in that computation shares one assumption. It takes Newton’s constant as measured in a laboratory, combines it with $\hbar$ and $c$, and gets a length — and that procedure is correct only if the gravitational field that laboratory measurement probes is the same field, in the same number of dimensions, all the way down.

Why weakness might be geometry #

Gravity is extraordinarily weak, and that weakness is what puts the Planck scale twenty decades away. The ratio of the gravitational to the electrical attraction between two protons is $10^{-36}$; the ratio of the proton’s mass to the Planck mass is $10^{-19}$. Nothing explains either number.

The arrangement proposed by Arkani-Hamed, Dimopoulos and Dvali in 1998 explains it as a dilution. Suppose there are $n$ additional spatial dimensions, compact, of size $R$, and suppose that everything except gravity is confined to the ordinary three — stuck to a surface, in the sense the theory makes precise. Gravity is not: its field lines spread into all of them.

Then at distances below $R$ the field falls as $1/r^{2+n}$, because the flux is spreading through a $(2+n)$-dimensional surface. At distances above $R$ the extra dimensions are used up, the field goes back to an inverse square, and its strength is what it is because the flux has already been diluted into the extra volume. The observed Newton constant is not fundamental; it is the fundamental one divided by the volume of the hidden dimensions.

The relation that follows is one line:

$$M_{\text{Pl}}^2 = M_*^{2+n}R^n$$

with $M_*$ the true scale. Gravity is weak because its field goes somewhere, and the twenty decades between a proton and the Planck mass are a statement about a volume rather than about a coupling. Which is the same move that reads a field’s falloff as a statement about the shape of its source, run in reverse: an exponent that does not match the geometry assumed is evidence that the geometry assumed is wrong.

Solved for $R$ at $M_* = 1$ TeV, it gives the hero figure, and the steepness of that curve is the whole reason the idea is testable rather than merely elegant.

What each case would mean #

One extra dimension would have to be $3\times10^{13}$ metres — about two hundred astronomical units. Gravity would have departed from the inverse square across the whole solar system, the planets’ orbits would not be what they are, and this case was excluded the moment it was written down.

Two would have to be about a millimetre. That is the case that made the paper famous: nothing in 1998 had verified the inverse-square law of gravity below about a centimetre, because gravity is so feeble that measuring the attraction between two small nearby masses is very hard. A millimetre-sized departure from Newton’s law was, astonishingly, still permitted by experiment, and it was permitted at the very distance a laboratory could reach. Gravity’s feebleness is the reason for that gap: between two protons it is 10⁻³⁶ of the electrical attraction, so any measurement of it between small nearby masses is a measurement of a residue after everything else has been cancelled — and the electrical forces that have to be cancelled include the ones a neutral body still has, which is why an electrostatic shield is the hardest part of the apparatus.

Three or more require $R$ below ten nanometres, falling to forty femtometres at six — which is inside a nucleus, and is the region a scattering experiment probes with energy rather than with a balance. No gravitational measurement reaches those distances or ever will, because at ten nanometres the electrical forces between the test masses exceed the gravitational ones by twenty decades.

So the number of dimensions decides not only what the answer is but whether the question is empirical, and the cases that are empirical are the ones with the fewest.

What a torsion balance saw #

The departure from an inverse square is written conventionally as a Yukawa addition: the potential picks up a term $\alpha\,e^{-r/\lambda}$, with $\lambda$ of order the compactification size and $\alpha$ of order the number of extra dimensions. Measuring gravity at a separation and comparing with Newton bounds the pair $(\alpha, \lambda)$.

The experiments are torsion balances and they are beautiful pieces of work. Two closely spaced discs with matched holes in them, one hung from a fibre and one rotating beneath it, so that the gravitational torque oscillates at a frequency set by the rotation and the pattern of holes; an electrostatic shield between them; and a great deal of care about everything that is not gravity, which at these separations is everything. The Eöt-Wash group has verified the inverse-square law down to fifty-two micrometres, and the two-extra-dimension case — a millimetre — is excluded by a wide margin.

What the second figure is really about is how the sensitivity dies. An exponential does not fall off; it falls over. A range above the measured distance is barely suppressed and is excluded outright; a range an order of magnitude below it is suppressed by four decades and is invisible. There is no frontier creeping downwards — each experiment tests roughly its own shortest distance and almost nothing shorter, and the gap between fifty micrometres and the ten nanometres the three-dimension case needs is not a gap that better torsion balances close.

And what the colliders saw #

The other half of the programme is at accelerators, and it tests the same arrangement from the energy side.

If the true scale is a TeV, then at a TeV gravity is strong, and three things should happen at a collider. Gravitons should be produced and escape into the extra dimensions, giving events with momentum missing in a characteristic spectrum. Virtual gravitons should modify the rates of ordinary processes. And above the scale, collisions should make microscopic black holes, which would evaporate at once into a spray of everything with a multiplicity far above any ordinary hard collision — a signature that gets simpler as the energy rises, which is the opposite of everything particle physics has ever seen and would be unmistakable.

None of it has been seen. The LHC’s searches put the fundamental scale above five to ten TeV depending on the number of dimensions and the assumptions, which does not close the arrangement but removes the region it was invented for: the point of putting the scale at a TeV was that a TeV is the electroweak scale, and a scale at ten TeV reintroduces a tuning of a hundred.

What else has to be checked when a force law changes #

A proposal that changes gravity below a millimetre has to survive more than a torsion balance, and the other constraints are worth collecting because they come from places nobody would think to look.

Stars. If gravitons can escape into extra dimensions they carry energy out of a hot dense object, and the object cools faster than it should. Supernova 1987A’s neutrino burst lasted about ten seconds, which is what a proto-neutron star cooling by neutrinos alone predicts; a competing graviton channel would have shortened it. That single observation constrains the two-dimension case more tightly than any laboratory experiment, and the three-dimension case appreciably.

Neutron stars. Gravitons emitted and then gravitationally captured accumulate around a neutron star and decay, producing gamma rays. The absence of that glow from known neutron stars is a further bound, and for two dimensions it is severe.

And the early universe. The extra dimensions would have to be stabilised at their present size before nucleosynthesis, and the graviton emission from the hot early plasma must not have overclosed the universe. That is a constraint on the whole cosmological history rather than on the force law.

The pattern across all four is worth noticing. A proposal that weakens a bound in one regime almost always strengthens one somewhere else, because the mechanism that makes gravity strong at short distances is the same mechanism that lets energy escape from a hot dense place. The strongest limits on large extra dimensions come from astrophysics rather than from the experiment designed to test them, which is common and is a reason to look before building.

The problem it was invented to remove #

That last sentence needs the problem stated, because the whole arrangement is a response to it and its size is a specific number.

The Higgs boson’s squared mass receives quantum corrections proportional to the square of whatever energy the theory remains valid up to. If that is the Planck energy, the correction is $10^{32}$ times the observed value, and the bare parameter must cancel it to thirty-two decimal places.

That is not a contradiction. A cancellation to any number of places is arithmetically possible and the theory is perfectly consistent with one. It is an unnaturalness, and the reason it is taken seriously is historical: every other small dimensionless number in physics has turned out to be protected by something — the electron’s mass by chiral symmetry, the photon’s by gauge symmetry — and a number that has to be dialled in by hand has always eventually been explained. A symmetry is the only mechanism physics has for making a quantity exactly anything, and the absence of one here is what the word unnatural is doing.

Lowering the cutoff to a TeV removes the tuning entirely, which is the whole argument. If gravity becomes strong at a TeV then the theory is not valid above a TeV, the correction is of order the Higgs mass itself, and nothing needs to cancel.

What that does not do is answer the question. The tuning has been traded for a different unexplained number: why are the extra dimensions the size they are? For the two-dimension case, $R$ is a millimetre and the fundamental length is $10^{-19}$ metres, so the compactification radius is sixteen orders of magnitude larger than the only length in the theory — which is a hierarchy of exactly the kind that was supposed to be explained. Several mechanisms for stabilising such a radius have been proposed and none is compelling.

Which is worth setting against the distance the four-dimensional version leaves, because the two figures are the same statement with and without the arrangement in it.

The one thing that would have been decisive #

It is worth being concrete about what a positive result would have looked like, because the arrangement’s virtue is that it is not vague.

A collider above the true scale does not probe shorter distances. It makes black holes — and the black hole it makes gets larger with energy, so the apparent size of whatever is being hit increases as the machine is pushed harder. Such a hole would have a lifetime of order a Planck time at the true scale, which is $10^{-27}$ seconds, and would evaporate thermally into every species there is in roughly equal numbers.

The signature is therefore an event with an enormous multiplicity, a nearly isotropic distribution, democratic among particle species, with no jet structure and no memory of what went in — arriving above a threshold energy and then rising rapidly in rate. Nothing in the standard model produces anything resembling it.

That is an unusually clean prediction, and it is what makes the null result meaningful rather than merely inconclusive. A search that could not have seen a signal tells nothing; this one could have, in a form that no background imitates, and it did not.

Leading order, a convention, and a problem left standing #

The relation between the Planck mass and the compactification volume is a leading-order one. It assumes all the extra dimensions are the same size, that they are flat, and that the compactification is simple. Warped geometries — where the extra dimension has a strongly curved metric — produce the same weakness with a single dimension of ordinary size, by a quite different mechanism, and are not constrained by the figures here at all.

The Yukawa parameterisation is a convention. The true short-distance form of the force depends on the shape of the compact space and is a sum over its modes rather than a single exponential. The convention is used because it is what experiments report bounds in, and translating a bound into a statement about a particular compactification is an extra step with its own assumptions.

The collider bounds depend on what else is assumed. Graviton emission rates, black-hole production cross-sections and the interpretation of missing energy all involve extrapolating a semiclassical treatment to the scale where it fails, which is the same discomfort the discomfort recorded about black-hole production. What such a hole does once made is an evaporation whose temperature rises as its mass falls, which is why the lifetime is a Planck time and not something observable. The bounds are real and the factor by which they are conservative is not known.

And the scenario does not address the vacuum energy. Lowering the cutoff to a TeV reduces the zero-point estimate from $10^{121}$ times the measured density to about $10^{54}$, which is a large improvement and leaves a discrepancy larger than any other in physics. The two problems are not solved by the same move.

Why the number of dimensions is not a free dial #

The first figure draws a size against a number of dimensions and treats both as free. They are not independent in any actual theory: string constructions produce particular numbers of dimensions with particular shapes and particular ways of stabilising them, and a plot over $n$ from one to six is a survey of the arithmetic rather than of the possibilities.

The deviation figure draws a single exponential and a single bound, which flattens the experimental situation considerably. The real exclusion is a region in a plane of $\alpha$ against $\lambda$, assembled from many experiments each sensitive over a different range, and the boundary of that region has a shape with structure in it. Reducing it to one number at one distance is honest about the order of magnitude and hides what the programme actually produced.

And the hierarchy figure draws a required precision as though the cancellation were between two numbers. It is a cancellation between a bare parameter, which has no independent meaning, and a sum of loop corrections that depends on the regularisation scheme — so the thirty-two decimal places are a scheme-dependent statement about an unphysical quantity. That criticism is made seriously and it is not obviously wrong; what survives it is the observation that the sensitivity of the observable mass to the theory’s short-distance details is enormous, however the bookkeeping is arranged.

Still open: whether naturalness was the right thing to want #

The arrangement in this essay was one of a family of responses to one judgement — that a theory requiring a parameter to be tuned to thirty-two places is saying something. Supersymmetry was another, technicolour a third. All of them predicted new physics at about a TeV, all of them have been searched for, and none has been found.

That is a real result and the field has not settled on what it means. One reading is that the new physics is a little heavier and the LHC has not reached it, which is possible and is getting less comfortable. Another is that naturalness was a heuristic rather than a principle — that the Higgs mass is tuned, that the vacuum energy is tuned far worse, and that both are what they are because in a large enough ensemble of possibilities, observers arise only where the numbers permit. A third is that the whole framing is wrong and that the bare parameters of an effective theory are not the sort of thing to have intuitions about.

There is one more consideration that belongs here and is easy to leave out. The negative results do not merely fail to support the arrangement; they have moved the scale they were testing. A fundamental scale pushed to ten TeV reintroduces a tuning of a hundred in the Higgs mass, which is a hundredth of the problem the arrangement was invented to solve rather than none of it. So the position is not “not yet found” but “found to be less useful than it was proposed to be”, and those are different states for an idea to be in.

What can be said with confidence is small and worth saying. The four-dimensional Planck scale is a computed quantity that rests on an assumption about geometry; that assumption is testable and has been tested over a range; the tests have found nothing; and the range they cover is fifty micrometres down, which excludes the cases that would have been most interesting and cannot reach the rest.

The habit worth carrying away is the one the hero figure is. When a quantity is computed from measured constants, ask what the computation assumes about the regime it is being extrapolated into. The Planck length is $\sqrt{\hbar G/c^3}$, and $G$ is measured at centimetres. Extending it by twenty decades assumes that nothing about gravity changes over those twenty decades, which is a physical claim rather than an arithmetic one — and it is the only claim in the whole subject that an experiment on a bench has ever been able to touch.