CONFORMAL TRANSFORMATIONS THAT CLICKBONUS ⑦ (finale) ／ Episode 8's homework, used two ways

The conformal anomaly has two completely different uses — demand that it vanish and you get a dimension; demand that it be positive and you get a direction.

The anomaly fixes the dimension and the direction Episode 8 showed that the symmetry breaks when you quantise. Demand that the breaking vanish
and the dimension of spacetime falls out; demand that it be positive and the renormalisation
group acquires a direction. And the tool for proving that is episode 4's dilaton.

What you need: addition, inequalities, and a feel for "information only decreases" $a_{\rm UV}>a_{\rm IR}$

Episode 8 showed that conformal symmetry breaks on quantisation, that the size of the breaking is measured by the $\beta$ function, and that this is what "$1/137$ becomes $1/128$" really was. This time the breaking itself is the protagonist. In four dimensions the anomaly has two coefficients, and those two have two uses — demand that it vanish and the dimension of spacetime is fixed; demand that it be positive and the renormalisation group acquires a direction. The tool used to prove the second is precisely the dilaton we built in episode 4 when we promoted the scale factor to a field. And finally the quantity turns out to be an entanglement entropy, joining bonus ②'s universe-as-a-finite-resource-computer. Here the series closes.

01The anomaly has two coefficients

In episode 8 we wrote the breaking of conformal symmetry as "a trace of the stress tensor survives". In curved space and four dimensions, it takes this form.

The four-dimensional conformal anomaly $$T^\mu{}_\mu = c\,W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}\ -\ a\,E_4$$

$W$ is the Weyl tensor (the star of bonus ⑤) and $E_4$ the Euler density (a topological quantity). There are two coefficients, $c$ and $a$.

In two dimensions there is only one, and it is the central charge $c$ — the central object of string theory and of two-dimensional critical phenomena. As bonus ⑥ showed, two dimensions is the special case where conformal symmetry becomes infinite-dimensional, and $c$ carries correspondingly more weight there.

These two coefficients have two entirely different uses. In order.

02Demand that it vanish — and the dimension of spacetime falls out

A string is described by the two-dimensional worldsheet that a one-dimensional object sweeps through spacetime. And that raises a problem: how you label the worldsheet must not affect the physics. But quantise, and an anomaly appears there.

So the anomaly has to vanish. That is string theory's consistency condition.

Just counting — two lines

Bosonic string: $D$ spacetime coordinates ($c=1$ each) plus ghosts ($c=-26$)

$$c_{\rm total}=D-26=0\qquad\Longrightarrow\qquad \boxed{D=26}$$

Superstring: $D$ bosons plus $D$ fermions ($c=1/2$ each) plus ghosts ($-26$ and $+11$)

$$c_{\rm total}=\frac{3D}{2}-15=0\qquad\Longrightarrow\qquad \boxed{D=10}$$

Conclusion of section 02

A single equation, anomaly = 0, fixes the dimension of spacetime.

The quantity that in episode 8 was "the source of 99% of your mass" is here an equation for a dimension. Same $c$.

In two dimensions, $c$ is the theory's name tag The solvable two-dimensional CFTs (the minimal models) line up in a discrete list, $c=1-\dfrac{6}{m(m+1)}$.
$m=3$: $c=1/2$ (2d Ising) ／ $m=4$: $c=7/10$ (tricritical Ising) ／ $m=5$: $c=4/5$ (three-state Potts).
A theory is named by a single dimensionless number. Together with bonus ⑥'s list of $\Delta$s, that is the entire content of a 2d CFT.

◇ ◇ ◇

03Demand that it be positive — and the flow acquires a direction

The second use asks not about zero but about which is bigger.

The renormalisation group is the operation of deciding not to look at fine detail and moving to a coarser description (episode 8). It flows from the ultraviolet (fine) to the infrared (coarse). Whether that flow has a direction — whether you can ever go back — was unknown for a long time.

the c-theorem (Zamolodchikov, 1986)$c_{\rm UV}>c_{\rm IR}$; the central charge always decreases along the flow

the a-theorem (Komargodski & Schwimmer, 2011)$a_{\rm UV}>a_{\rm IR}$; proved twenty-five years later

the F-theorem (Casini, Huerta and others)the free energy on a sphere decreases

Conclusion of section 03

$$a_{\rm UV}\ >\ a_{\rm IR}$$

The renormalisation group has a direction. It only flows towards the coarse. And the scale that measures that direction is episode 8's anomaly coefficient.

What is striking is $c$. In four dimensions $c$ has no monotonicity at all — it can decrease or increase. Of the two coefficients, only one carries a direction.

04The tool of the proof was episode 4's dilaton

Here is the most satisfying part. How was the a-theorem actually proved?

Like this. Along a renormalisation group flow the conformal symmetry is broken — so add a field that compensates the breaking. That is, promote $\Omega$ to a field.

This is exactly episode 4's move

Treat the conformal factor $\Omega$ not as a property of spacetime but as a field living on it — the dilaton.

The whole theory then becomes formally Weyl invariant, and every trace of the breaking is pushed into the dilaton's interactions. Compute two-dilaton scattering and —

The bones of the proof

The difference appears in the $s^2$ coefficient of the forward amplitude

$$\mathcal{A}(s)\ \supset\ \frac{a_{\rm UV}-a_{\rm IR}}{f^4}\,s^2$$

Unitarity (the optical theorem) forces that coefficient to be positive

$$a_{\rm UV}-a_{\rm IR}\ >\ 0$$

Nothing but "probabilities are not negative" — and the renormalisation group acquires a direction.

In episode 4 we wrote that the scale factor is not a property of spacetime but a field living on it. That was meant as a picture; in fact the very same move is the technique that proves four-dimensional renormalisation group flows have a direction. Bonus ⑤'s "the only way to make the gravitational anomaly bite is to make $\theta$ a field" has the same shape. This series has been unknowingly repeating a single trick.

05Counting for the Standard Model — the staircase of $a$

For free fields, $a$ is a simple sum. The ratios are

$$\text{real scalar}:\text{Weyl fermion}:\text{vector}\ =\ 1:\tfrac{11}{2}:62$$

Checking the normalisation You can verify these ratios with $\mathcal{N}=4$ super Yang–Mills, a theory known to satisfy $a=c$ exactly, whose content (per generator) is 6 real scalars, 4 Weyl fermions and 1 vector. The ratios above give $6+5.5\times4+62=90$; the $c$ ratios ($1:6:12$ counting Dirac fermions, times 3) give $(6+12+12)\times3=90$. They agree.

Now count the Standard Model: 4 real scalars (a complex doublet), 45 Weyl fermions, 12 vectors (8 gluons + 3 W + 1 B).

energy	$a$ ∝	what drops out there
Planck – 173 GeV	995.5	the whole Standard Model (scalars 0.4% / fermions 24.9% / gauge fields 74.7%)
below 173 GeV	772.5	top, Higgs, W and Z
below 4.2 GeV	739.5	bottom
below 1.3 GeV	695.5	tau, charm
below 0.2 GeV	89.5	QCD confines (8 gluons and the light quarks)
below 0.511 MeV	78.5	the electron
below 0.05 eV	62.0	the neutrinos — only the photon is left

Overall $a_{\rm UV}/a_{\rm IR}=16.06$. And the deepest single step is QCD confinement, which alone divides it by 7.77. The eight gluons account for $8\times62=496$ — 49.8% of the Standard Model's $a$.

What that says

Half of everything the universe lost, it lost at the moment QCD confined.

Episode 8 said that 99% of your body weight is a product of the anomaly. The same confinement that made that mass also erased half of the anomaly coefficient.

Figure: energy scale along the horizontal axis (Planck down to $10^{-12}$ GeV, logarithmic), $a$ on the vertical. Drag the knob from right to left and particles drop out one by one, stepping the staircase down. It never steps up — that is the a-theorem. The deepest step is QCD confinement.

Energy scale log₁₀(μ/GeV)2.50

log₁₀ μ = 2.50

a (degrees of freedom left) current scale QCD confinement

06$a$ turns out to be an entanglement entropy

So what is $a$ counting? The answer is known — draw a sphere, cut inside from outside, and measure how entangled they are.

The universal part of the entanglement entropy

Two dimensions (an interval of length $\ell$)

$$S=\frac{c}{3}\log\frac{\ell}{\epsilon}+\text{const}$$

Four dimensions (a sphere of radius $R$)

$$S\ \supset\ -4a\,\log\frac{R}{\epsilon}$$

The coefficient of the part that depends on neither the shape nor the cutoff $\epsilon$ is exactly $c$ and $a$.

The one line of the finale

The direction of the renormalisation group is set by the loss of information.

$a_{\rm UV}>a_{\rm IR}$ restated is "coarse-graining reduces entanglement". "Deciding not to look at the fine detail" is, literally, throwing information away — and that has been proved as a theorem.

07Where this joins bonus ②

Bonus ② tried to write the universe as a computer with finite resources: $10^{122}$ bits of memory, 140 clock ticks, right at the Landauer limit. Its conclusion was that we had been constraining the wrong thing — what should be constrained is not $a(t)$ but the information on a light sheet.

That "information" now has a theorem attached to it.

bonus ②

an estimate that the universe's memory is finiteholographic bounds, the Landauer limit, $2.96\times10^{122}$ bits

bonus ⑦

a theorem that coarse-graining loses informationthe a-theorem; $a$ is the coefficient of the sphere entanglement entropy

One asks how many bits the universe has; the other asks how many bits you lose by looking more coarsely. Neither uses a single dimensionful quantity. A bit is dimensionless. Exactly as episode 10's procedure demands, information was in the right-hand column from the start.

08Sixteen episodes on one table

To close the series, we do the sorting into two columns one last time, all the way to the end.

ep.	what moved (bookkeeping)	what remained (physics)
3	the choice of time coordinate (this is what changes the speed of light)	$1+z=t_0/t_e$, the recombination ratio 52.6, $\alpha$
4	the gauge $s$ (splitting expansion between metric and field)	galaxy separation ÷ atomic radius
5	the apparent potential	the equation of state $w$, the triple boundary at $w=-1/3$
6	the curvature $R$ (you can even set $\tilde R=0$)	$N=mc^2t/\hbar$, the Planck time
7	the ruler's calibration (1 of the metric's 10 components)	the light cone = causal structure (the other 9)
8	──	the $\beta$ function, $1/137\to1/128$, 99% of the proton mass
9	──	the conformal factor problem (no gauge erases it)
bonus ④	the magnitude $\|m\|$, individual phases $\theta_i$	mass ratios, $\arg\det M$, $\bar\theta<10^{-10}$
bonus ⑤	Newton's $G$, the Planck mass $M_{\rm Pl}$	$\alpha_G=(m/M_{\rm Pl})^2$, $R\tilde R\propto\mathrm{Im}(\Psi_2^2)$
bonus ⑥	dimensionful quantities do not exist at all	$\Delta$, the critical exponents, $\alpha+2\beta+\gamma=2$
bonus ⑦	──	$c$ and $a$: the dimensions $D=10,26$ and the direction of the flow

The left column is about how we choose to write things. Only the right column says anything about the world. Sixteen episodes of work amounted to separating those two columns accurately.

The honest line

The form $T^\mu{}_\mu=cW^2-aE_4$ in section 01 has sign and normalisation conventions that vary between sources. The counting in section 02 (ghosts at $-26$ and $+11$) is standard in covariant quantisation, but it is a consistency condition inside string theory — an unverified framework — and there is no observation that spacetime is ten-dimensional.

The three theorems in section 03 are all real results, but the a-theorem's proof concerns flows whose ultraviolet and infrared ends are both CFTs. That $c$ has no monotonicity in four dimensions is also standard.

The staircase in section 05 is a schematic built from free-field counting. Strictly, $a$ is defined only at conformal fixed points, and there is no such thing as "the current $a$" at intermediate energies. The treatment of massive vectors (W and Z) and of confinement is crude here. The steps necessarily go down because we are removing fields, so this is not a test of the a-theorem — it is a map of where things drop and by how much.

The entanglement entropy formulas in section 06 depend on normalisation conventions, and in four dimensions shape-dependent non-universal terms are present. The "junction" with bonus ② in section 07 is this article's reading; the a-theorem says nothing about the universe's total bit count. The table in section 08 is this series' own organisation, not a standard textbook classification.

Exercises

In one line, why is the bosonic string's dimension 26?
Show the answer
The worldsheet conformal anomaly must vanish; $D$ spacetime coordinates give $c=D$ and the ghosts $c=-26$, so $D=26$. The condition that episode 8's anomaly cancel is itself an equation for the dimension. For the superstring, fermions add $c=1/2$ each and the ghosts change, giving $3D/2-15=0$, i.e. $D=10$.
State the difference between $c$ and $a$.
Show the answer
Both are coefficients of the four-dimensional conformal anomaly, but only $a$ is monotonic. $a_{\rm UV}>a_{\rm IR}$ is the a-theorem; $c$ can go either way along a flow. In two dimensions there is only one coefficient, the central charge $c$, and Zamolodchikov's c-theorem makes it decrease.
Why introduce a dilaton to prove the a-theorem?
Show the answer
Because conformal symmetry is broken along the flow, so you cannot argue in the language of symmetry as it stands. So you add a field that compensates the breaking and make the theory formally Weyl invariant. All the information about the breaking then sits in the dilaton's interactions and you can study its scattering amplitude instead. Exactly the move of episode 4, "the scale factor is a field living on spacetime".
Where does $a$ drop the most in the Standard Model, and by what fraction?
Show the answer
QCD confinement ($\sim0.2$ GeV): from 695.5 to 89.5, a factor of 7.77. The eight gluons alone are $8\times62=496$, which is 49.8% of the Standard Model's total $a$. Half of everything the universe lost disappeared in that single step.
(Harder) If $a$ is "information", how would you restate the fact that the renormalisation group has a direction?
Show the answer
$a$ is the coefficient of the universal part of the sphere entanglement entropy, so $a_{\rm UV}>a_{\rm IR}$ says "coarse-graining reduces entanglement". That is, "deciding not to look at fine detail" is literally discarding information, and discarded information does not come back. The renormalisation group is irreversible because it is forgetting. The same side of physics as Landauer's principle, that erasing information has a cost — the junction with bonus ②.

SummaryOne quantity, two uses

The "size of the symmetry breaking" measured in episode 8 has two coefficients in four dimensions, $c$ and $a$, and they have two uses. Demand that it vanish and the dimension of spacetime is fixed — $D-26=0$ for the bosonic string, $3D/2-15=0$ for the superstring, giving 26 and 10. Demand that it be positive and the renormalisation group acquires a direction — $a_{\rm UV}>a_{\rm IR}$, the a-theorem. Of the two coefficients, only $a$ carries a direction.

And the tool that proved it was episode 4's dilaton. Add a field to compensate the broken conformal symmetry, make the theory formally Weyl invariant, and $a_{\rm UV}-a_{\rm IR}$ appears as the $s^2$ coefficient of that field's scattering amplitude, positive by unitarity. The single move of "promote $\Omega$ to a field", which this series made in episode 4 and repeated in bonus ⑤, is the technique that proves the flow has a direction.

Counting the Standard Model gives $a_{\rm UV}\propto995.5$, and 62 in the deep infrared where only the photon is left — a ratio of 16.06. The deepest step is QCD confinement, where the eight gluons take away 496: half of everything the universe lost went in that one step. The same confinement that episode 8 credited with 99% of your body weight also erased half of the anomaly coefficient.

Finally, what $a$ is. It is the coefficient of the entanglement entropy across a sphere. So $a_{\rm UV}>a_{\rm IR}$ restated is "coarse-graining reduces entanglement", and the renormalisation group is irreversible because it is forgetting. Bonus ② said that to write the universe as a finite-resource computer you should constrain the information on a light sheet. That information now has a theorem. A bit is dimensionless — it was in the right-hand column all along.

The series closes here "Cosmology That Clicks" began with "light used to be faster". "Conformal Transformations That Click" tracked down what that construction line really was; bonus ①②③ judged $c\cdot t=\text{const}$ from three directions; ④ went looking for a phase in mass; ⑤ followed it as far as gravity; ⑥ filled in the other half of the series' own title ($\Delta$); and ⑦ closed episode 8's homework. Sixteen episodes of work amounted to one thing: separating the two columns, bookkeeping and physics, accurately. If it moves it is physics, if it does not it is bookkeeping — the procedure is simple and can always be applied. But as episode 10 already said, the procedure never tells you which dimensionless quantity to build. Asking a good question remains, permanently, a human job.

This document is bonus episode ⑦ (the finale) of the series "Conformal Transformations That Click", written for high-school and university students who enjoy physics. That the four-dimensional conformal anomaly takes the form $T^\mu{}_\mu=cW^2-aE_4$ (with sign and normalisation conventions varying between sources), and that in two dimensions there is a single coefficient, the central charge $c$, are standard. That cancelling the worldsheet conformal anomaly gives $D=26$ for the bosonic string and $D=10$ for the superstring is a standard result in covariant quantisation, but string theory is an observationally unverified framework. The minimal-model formula $c=1-6/(m(m+1))$ is standard. The c-theorem is due to Zamolodchikov (1986), the four-dimensional a-theorem to Komargodski & Schwimmer (2011), and the three-dimensional F-theorem to Casini, Huerta and collaborators. That the a-theorem's proof rests on the dilaton effective action and the positivity of the $s^2$ coefficient in 2→2 scattering (dispersion relations, the optical theorem) is the content of that paper. The free-field ratios for $a$, real scalar : Weyl fermion : vector $=1:11/2:62$, are standard; this article checked the normalisation using $a=c$ in $\mathcal{N}=4$ super Yang–Mills (both give 90). The staircase in section 05, $a_{\rm UV}/a_{\rm IR}=16.06$, the factor 7.77 at QCD confinement and the 49.8% gluon contribution are schematic free-field computations made for this article; $a$ is properly defined only at CFT fixed points, and the treatment of massive vectors and of confinement is crude. That the universal coefficient of the entanglement entropy is $c/3$ in two dimensions and proportional to $a$ in four is standard, but normalisations and signs are convention-dependent and shape-dependent non-universal terms appear in four dimensions. The connection to bonus ② in section 07 is this article's reading; the a-theorem makes no claim about the universe's total bit count. The table in section 08 is this series' own organisation. The academic standard is the $\Lambda$CDM model including inflation, together with the Standard Model. ── To print, use your browser's Print → Save as PDF (sliders freeze and answers are hidden in the print version).

Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lowers the energy scale and you can watch the staircase of $a$ drop. "Show the answer" opens each solution.