Emergent Lorentz invariance from strong coupling ── it works for matter. And on gravity it sends us back to Episode 2
Episode 2 dismissed escape route ② (Lorentz invariance as an IR fixed point of the RG) as "logarithmic, hence not enough." That was the weak-coupling story; at strong coupling it becomes a power ── correcting that was last time's work.
This episode goes to look at the substance. The conclusion splits in two.
For the matter sector, it really works. And the mechanism is a pleasure. But for gravity, nobody has done it. Not "has not" so much as "doing it breaks things" ── and the way it breaks is exactly Episode 2, §06.
At weak coupling the Lorentz-violating couplings flow logarithmically under the RG. So no number of decades is enough ── twenty decades gives a factor \(1/\log(10^{20})=1/46\).
At strong coupling things change. Operators acquire anomalous dimensions, so the flow becomes a power. Bednik–Pujolàs–Sibiryakov built a concrete holographic example.
In the correlated case \(0\le\nu<1\), \(\epsilon\in(0,2]\). In the marginal case \(\nu=1\), \(\epsilon=0\) and we are back to logarithms.
Only particular operators are dangerous. BPS identify vector operators of dimension \(d-1\le\dim\mathcal{O}_\mu\le d\) as "dangerous" ── marginal, or slightly irrelevant. These flow to zero most slowly. The least irrelevant Lorentz-violating operator (LILVO) sets the speed of the suppression.
Kharuk–Sibiryakov obtain the same suppression in the form \((l/L)^{1-2Ml}\) and state that the maximum achievable suppression is \((l/L)^2\). The two papers give the same upper bound ── \(\epsilon\) cannot exceed 2. That 2 will matter later.
This is slightly off the main thread, but I want to write it down.
The Standard Model is made of chiral fermions. But the very definition of "chiral" relies on Weyl representations of the Lorentz group. How can chiral fermions come out of a UV that is not Lorentz invariant? Naively this looks impossible.
Kharuk–Sibiryakov's answer was just counting.
The UV theory is not Lorentz invariant, so all it has is spatial rotations, SO(3). And its spinor has two independent components.
Meanwhile a four-dimensional Weyl spinor also has two components.
They match exactly. So an SO(3) spinor can behave as a Weyl spinor in the infrared.
Episode 2 treated the Nielsen–Ninomiya no-go (chiral symmetry cannot be kept on a lattice). That is about a regular lattice. In emergence from strong coupling, chirality is generated in the infrared in the first place, so it falls outside the no-go's premises.
This series' refrain that "only a binary target carries information" appears here as a coincidence of component counts. Two and two. Off by one and it would not work.
Knowing it works, we ask over what range it works. There are two conditions.
| condition | content |
|---|---|
| from above | the mechanism's ceiling \(\epsilon\le2\) (BPS and KS agree) |
| from below | the residual violation must pass observation: \((\Lambda_{\rm conf}/M_P)^{\epsilon}\le6\times10^{-20}\) (the Crab Nebula) |
| from the side | in this mechanism the Standard Model fields emerge from strong coupling, so \(\Lambda_{\rm conf}\) is also the compositeness scale of the electron and so on. LHC compositeness bounds give \(\Lambda_{\rm conf}\gtrsim10\) TeV |
# solve (Lambda_conf/M_P)^eps <= 6e-20. M_P = 1.2209e19 GeV Lambda_conf log10(ratio) eps required corresponding nu -------------------------------------------------------- 1 TeV -16.09 1.19 0.40 10 TeV -15.09 1.27 0.36 100 TeV -14.09 1.36 0.32 1e9 GeV -10.09 1.91 0.05 3e9 GeV -9.61 2.00 0.00 <- ceiling reached 1e10 GeV -9.09 2.11 negative ── impossible # conversely, lowering eps eps = 1.0 -> Lambda_conf <= 0.73 GeV sub-GeV. excluded by compositeness eps = 0.5 -> Lambda_conf <= 4.4e-20 GeV out of the question => usable window: 10 TeV <= Lambda_conf <= 3e9 GeV 1.27 <= eps <= 2 (the top 36% of the full range (0,2])
There is a window. But it is narrow. Last time I wrote in Episode 2 that "\(\epsilon\simeq1\) suffices," but adding the compositeness bound pushes it up to \(\epsilon\gtrsim1.27\). The power has to be used nearly to its maximum.
Move the sliders and you find it surprisingly easy to leave the green. Still, the window is open ── so far, a win for escape route ②.
Here the story changes. Rereading the papers, every one of the authors puts gravity out of scope. And they do not hide it.
BPS: in both models (Randall–Sundrum, Lifshitz-type) gravity is a fixed background. They state that extending to dynamical gravity would require "an additional degree of freedom defining a preferred frame", and do not pursue it.
Kharuk–Sibiryakov: "in this scenario it is sufficient that the strong coupling occur in the matter sector alone … gravity may remain weakly coupled at all energies"
This is not a confession of weakness but a design choice. Confine the strong coupling to matter and you avoid gravity's troubles ── clever.
But there is a price. If gravity stays weakly coupled, gravity's own Lorentz violation gets no power suppression. Only logarithms. Which means the eighteen orders of Episode 2, §06 remain in the gravity sector unchanged.
One wants to think "the bounds on gravitational violation are looser, surely." They are indeed looser ──
| whose speeds are being compared | bound |
|---|---|
| photon vs electron (Crab synchrotron) | \(6\times10^{-20}\) |
| gravitational waves vs light (GW170817 + GRB 170817A) | \(-3\times10^{-15}\) to \(+7\times10^{-16}\) |
Gravity is about five orders looser. Still not enough. Logarithmic suppression buys only a factor \(1/46\) over twenty decades, so five orders of slack does not get you there.
This is the decider. "Even if violation remains in gravity, matter is protected, so what is the harm?" ── that does not hold.
The matter sector cannot be protected from symmetry breaking in the gravity sector.
Radiative corrections from quantum gravity generate Lorentz-violating couplings for abelian gauge fields (arXiv:1911.10066). Lorentz violation in quantum gravity necessarily permeates the matter sector.
What the mechanism protected was "the violation matter has in its own UV." It does not protect against "the violation gravity has." And the leakage path is exactly the radiative corrections of Episode 2, §06.
There is worse news. Work scrutinizing Hořava gravity (arXiv:1805.10299) points out fine-tuning between species ── the degree of tuning of the squared sound speeds of a U(1) gauge field and a scalar becomes more severe as the dimension rises. And it concludes that not only the gravity sector but the matter sector too must cross over to Lifshitz scaling above some scale, and the two crossover scales cannot be widely separated.
So the convenient separation "matter strongly coupled at a low scale, gravity weakly coupled far above" is structurally forbidden.
Then why not make gravity itself Lorentz violating and install emergence from strong coupling there too? There is a general obstruction here.
Explicit Lorentz violation is incompatible with general Riemannian geometry (and hence with general relativity).
So ── if Lorentz symmetry is broken, it is broken spontaneously, or gravity needs a new geometric framework.
Choose spontaneous breaking and you need something to do the breaking: a field with a vacuum expectation value that picks out a preferred frame ── an aether field. Which is exactly what BPS wrote would be needed: "an additional degree of freedom defining a preferred frame."
Here the circle closes. We used emergence in order not to create a preferred frame, and carrying it to gravity brings a preferred frame back in.
| result | |
|---|---|
| matter sector | works. power suppression. window \(10\) TeV \(\le\Lambda_{\rm conf}\le3\times10^9\) GeV, \(1.27\le\epsilon\le2\) |
| chiral fermions | they emerge. the SO(3) spinor's two components match Weyl |
| gravity sector | untouched. the authors themselves say explicitly that it "stays weakly coupled" and put it aside |
| leakage from gravity into matter | cannot be stopped. radiative corrections always permeate |
| separating the scales | not possible. the two crossover scales cannot be widely separated |
| if you break gravity | explicit breaking is incompatible with Riemannian geometry. make it spontaneous and you need an aether = a preferred frame |
For Episode 2's question "is spacetime itself a lattice," the main event is gravity. That is not filled in.
Escape route ② is upgraded from "not enough" to "works for matter; gravity is left over." Last time's revision was correct ── but the range where it works is narrower than expected.
And for the position of a universe computable locally, the options have shrunk again:
① (high-scale SUSY) ── forbids dimensions 3 and 4 including in the gravity sector. locality survives. price: SUSY.
② (emergence from strong coupling) ── matter can be saved but gravity leaks. not enough on its own.
③ (causal set) ── structurally no preferred frame. price: non-locality ── it will not become a finite difference.
If ② is not enough on its own, what remains is ① or ①＋②. That ① was the front-runner back in Episode 2 turns out, in the event, to have been right.
At strong coupling the violation dies as a power. \(\delta\sim(\Lambda_{\rm conf}/\Lambda_*)^{\epsilon}\), \(\epsilon=2(1-\nu)\). At the marginal \(\nu=1\) we return to logarithms, but in the correlated case \(0\le\nu<1\), \(\epsilon\) can run from 0 to 2. The dangerous ones are vector operators of dimension \(d-1\le\dim\mathcal{O}_\mu\le d\) (LILVO). Two papers independently give the same ceiling \(\epsilon\le2\).
Chiral fermions emerge. A non-Lorentz-invariant UV has only SO(3), and its spinor has two independent components. A 4d Weyl spinor also has two. The counts match exactly, so it can behave as a Weyl spinor in the infrared. Episode 2's Nielsen–Ninomiya is a theorem about regular lattices and does not bite here.
We measured the window ── it exists but is narrow. From above the mechanism's \(\epsilon\le2\), from below the Crab's \(6\times10^{-20}\), from the side compositeness's \(\Lambda_{\rm conf}\gtrsim10\) TeV. The result is \(10\) TeV \(\le\Lambda_{\rm conf}\le3\times10^9\) GeV, \(1.27\le\epsilon\le2\) ── the top 36% of the full range \((0,2]\). Last time's "\(\epsilon\simeq1\) suffices" was too generous; compositeness pushes it to \(1.27\).
Gravity is not included. And deliberately so. BPS puts gravity on a fixed background and states that making it dynamical requires "an additional degree of freedom defining a preferred frame," which they do not pursue. Kharuk–Sibiryakov state explicitly that "gravity may remain weakly coupled at all energies." Clever as design, but gravity's own violation gets no power suppression.
And it leaks. "The matter sector cannot be protected from symmetry breaking in the gravity sector" ── radiative corrections from quantum gravity generate Lorentz-violating couplings for gauge fields (arXiv:1911.10066). What was protected was only the violation matter has in its own UV; gravity's violation comes in by the same path as Episode 2, §06. In addition, the two crossover scales cannot be widely separated (arXiv:1805.10299).
Turning to break gravity instead runs into a no-go. Explicit Lorentz violation is incompatible with general Riemannian geometry. So it must be spontaneous, which means bringing in an aether field = a preferred frame. We used emergence in order not to create a preferred frame, and going all the way to gravity brings it back.
Accounts: ② is not enough on its own. But last time's revision ("not enough" → "works for matter") was correct. For the position of a locally computable universe, what remains is ① (high-scale SUSY) or ①＋②. Putting ① as the front-runner back in Episode 2 turns out to have been right.
This document is Episode 16 of the "Lattice We Build" series, a reading piece for high-school and university students who love physics. Where the sister series "That Clicks" explains known physics, this series shows the work itself ── including the correction that the previous episode's estimate was too generous.
Established material: the power suppression \((\Lambda_{\rm conf}/\Lambda_*)^{2(1-\nu)}\) at strong coupling; the return to logarithms at \(\nu=1\); that the dangerous operators are vector operators of dimension \(d-1\le\dim\mathcal{O}_\mu\le d\); and that gravity is treated as a fixed background with the authors stating that dynamical gravity requires an additional degree of freedom defining a preferred frame (Bednik–Pujolàs–Sibiryakov, JHEP 11 (2013) 064, arXiv:1305.0011). That the number of independent components of an SO(3) spinor matches a 4d Weyl spinor, that the maximum suppression is \((l/L)^2\), and the statement that gravity may remain weakly coupled (Kharuk–Sibiryakov, TMP 189 (2016) 1755, arXiv:1505.04130). That the matter sector cannot be protected from breaking in the gravity sector (arXiv:1911.10066). The fine-tuning of sound speeds between species and the inability to separate the crossover scales (arXiv:1805.10299). The incompatibility of explicit Lorentz violation with Riemannian geometry (Kostelecký's no-go). The \(6\times10^{-20}\) from the Crab Nebula and the \(-3\times10^{-15}\) to \(+7\times10^{-16}\) from GW170817 + GRB 170817A.
This article's window calculation (the required \(\epsilon\), the upper bound \(3\times10^9\) GeV on \(\Lambda_{\rm conf}\), and \(\epsilon\ge1.27\) when the lower bound is set at \(10\) TeV from compositeness) was computed by the author from the suppression form above and the observational limits. Setting the compositeness bound at \(10\) TeV is a rule of thumb and depends on the model. Also, "36% of the window" is a ratio of intervals in \(\epsilon\) and carries no probabilistic meaning.
The accounting that "② is not enough on its own, and taking locality leaves ① or ①＋②" is this series' author's framing. It has not been shown that installing strong-coupling emergence in the gravity sector is impossible in principle ── the accurate statement is that nobody has succeeded so far.
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