Thinking in Waves Episode 18 ／ Part VIII · Differentiating the effective constants

99 % of your body weight is the energy of confined waves

Where does mass come from? ──
put it in a box, or soak it in a medium Of the proton's 938 MeV, the quark masses sum to just 9 MeV.
The remaining 99 % is ── the energy of a confined field.
★ In wave language: a wave confined in a box has a lowest frequency.

Prerequisites: Episodes 5, 6, 16, 17Computation: kenshou/dokoka.js

Episode 17 established that mass is the intercept and the curvature of a dispersion relation. So who makes that intercept?
We looked, and found two ways of making it ── which, in wave language, turned out to be the same thing.

01The proton's mass budget

Particle	Mass [MeV]	Note
Proton	938.2721	uud
u quark	2.16	$\overline{\rm MS}$, 2 GeV
d quark	4.67	$\overline{\rm MS}$, 2 GeV
u + u + d	8.99	★ 0.96 % of the proton
Remainder	929.28	★ 99.04 %

The heart of §01

99 % of the proton's mass is not quark mass. What is it, then ── the energy of a confined field.

★ In wave language ──

A wave confined in a box has a lowest frequency.
That lowest frequency appears as mass.

Episode 6's "mass = cutoff frequency" doubles as the explanation of where mass comes from.

02Put it in a box and mass appears

Boundary conditions cut off the wavenumber from below $$\text{1D box}:\ k=\frac{\pi}{L}\ \Rightarrow\ E=\frac{\pi\hbar c}{L},\qquad \text{spherical box (lowest)}:\ kR=2.04\ \Rightarrow\ E=\frac{2.04\,\hbar c}{R}$$

$\hbar c=197.327$ MeV·fm

Confinement size	1D $\pi\hbar c/L$	Sphere $2.04\hbar c/R$	Comparison
0.50 fm	1239.8 MeV	805.1 MeV
0.84 fm	738.0 MeV	479.2 MeV	★ the proton charge radius
1.00 fm	619.9 MeV	402.5 MeV	nucleon scale
5.00 fm	124.0 MeV	80.5 MeV	a heavy nucleus

Confine anything to 1 fm and you get hundreds of MeV ── the same order as the proton's 938.

Invert it ── if three massless quarks carried the whole mass $$3\times\frac{2.04\,\hbar c}{R}=938.3\ \text{MeV}\quad\Longrightarrow\quad R=1.287\ \text{fm}$$

The measured charge radius is 0.841 fm ── off by a factor of 1.53

Being honest about it

A factor of 1.5. The order of magnitude is right, and that is all.
★ This is a sanity check on "confinement produces mass", not a calculation of the proton mass. The real thing is lattice QCD, which now reproduces it to 1 %.
What we did was see that "why hundreds of MeV?" is set by $\hbar c/$(1 fm).

03The other source ── coupling to a condensate

In Episode 17 the photon acquired mass inside a superconductor. The same thing happens for elementary particles ── with the Higgs field as the condensate.

Invert the couplings from the masses $$m_f=\frac{y_f\,v}{\sqrt2},\qquad v=246.22\ \text{GeV}\quad\Longrightarrow\quad y_f=\frac{\sqrt2\,m_f}{v}$$

Particle	Mass [MeV]	Yukawa coupling $y$	× the electron
electron	0.511	$2.94\times10^{-6}$	1
u quark	2.16	$1.24\times10^{-5}$	4.2
muon	105.7	$6.07\times10^{-4}$	207
tau	1776.9	$1.02\times10^{-2}$	3478
b quark	4180	$2.40\times10^{-2}$	8180
t quark	172690	0.9919	$3.4\times10^5$

§03 ── only the top has $y\approx1$

$y_{\rm top}=0.9919$ ── almost exactly one. Everything else is smaller.

And from electron to top the couplings span five and a half orders ($3.4\times10^5$).
Why so wide? Nobody knows.

★ This is the flavour hierarchy problem, a door as open as Episode 15's "where does the value of $\alpha$ come from?".

04Mass runs too ── literally an integral here

In Episode 5 the couplings ran with $\ln\mu$. So does mass.

At one loop it has a closed solution $$\frac{d\ln m}{d\ln\mu}=-\gamma(\alpha_s)\quad\Longrightarrow\quad m(\mu)=m(\mu_0)\Bigl[\frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}\Bigr]^{\gamma_0/2\beta_0},\quad \gamma_0=8,\ \beta_0=11-\tfrac{2n_f}{3}$$

Scale $\mu$ [GeV]	$\alpha_s(\mu)$	$m_b(\mu)$ [GeV]	$m_\tau$ [GeV]	$m_b/m_\tau$
4.18	0.21185	4.1800	1.7769	2.3525
91.19 ($m_Z$)	0.11790	3.1072	1.7769	1.7487
$10^3$	0.08969	2.6576	1.7769	1.4957
$2.79\times10^7$	──	1.7769	1.7769	1.0000
$10^{16}$	0.02247	1.2051	1.7769	0.6782

§04 ── "constants are integrals" holds for mass too

$m_b(m_Z)=3.107$ GeV (published about 2.83 GeV, with two loops and beyond). One loop is 9 % off, but the run from 4.18 to 3.1 is there.

★ A caveat ── $m_b(\mu)$ is dimensionful. By Episode 7's rule it should be bookkeeping. What is observable is a ratio.

So we looked at $m_b/m_\tau$ (the tau does not run under QCD, so the ratio moves).
★ At $\mu=2.8\times10^7$ GeV the b quark and the tau have the same mass, and at the grand-unification scale $10^{16}$ GeV, $m_b/m_\tau=0.68$ ── close to 1, but not equal.

This is a known result (b–τ unification fails in the Standard Model and improves with supersymmetry, as the standard argument goes). We merely reproduced it at one loop.

05Inertial and gravitational mass ── the same thing, as waves

Experiment	Bound on $\eta=2\|a_1-a_2\|/(a_1+a_2)$
Eötvös (1908)	$5\times10^{-9}$
Dicke (1964)	$1\times10^{-11}$
Eöt-Wash (2008)	$2\times10^{-13}$
MICROSCOPE (2022)	$1\times10^{-15}$

§05 ── the equivalence principle, said as a wave

Gravity is "clocks run at different rates in different places" (Episode 16's $c_{\rm eff}$).
And a dispersion relation is written against that clock.

★ So every wave is affected identically, whatever it is made of.
That is "gravitational mass = inertial mass", in wave language.

Conversely ── the equivalence principle can only fail if different kinds of wave see different clocks. Which is why one searches for a fifth force.

06The hierarchy of masses ── 29 orders

Particle	Mass	× the electron	Note
Neutrino (bound)	$\sim10^{-7}$ MeV	$2\times10^{-7}$	cosmological bound on the sum, $\sim0.1$ eV
Electron	0.511 MeV	1
Proton	938.3 MeV	1836	★ 99 % confinement
W boson	80377 MeV	$1.6\times10^5$
t quark	172690 MeV	$3.4\times10^5$	★ $y\approx1$
Planck mass	$1.22\times10^{22}$ MeV	$2.4\times10^{22}$	★ $\sqrt{\hbar c/G}$

29 orders from neutrino to Planck mass, and $7.1\times10^{16}$ from top to Planck alone ── the hierarchy problem.

07Three sources ── and two of them are the same

Source	Mechanism	Where it dominates
① Confinement	boundary conditions create a lowest frequency	99 % of the proton
② Condensate	coupling to a background field creates an intercept	quarks and leptons
③ Running	the part that depends on scale	$m_b:4.18\to3.1$ GeV

§07 ── the main finding of this episode

★ ① and ② are the same thing in wave language ──

both create an intercept in the dispersion relation.
① does it with boundary conditions, ② with a background field.

Episode 6 said mass is a cutoff frequency. What this episode adds is ── ★ there are two ways to make that cutoff.
Put it in a box, or soak it in a medium.

Episode 17's superconductor was case ② (a condensate giving the photon an intercept); Episode 16's waveguide was case ① (boundaries giving the photon an intercept).
★ The photon in a waveguide, the photon in a superconductor, the proton and the electron all acquire mass by the same mechanism.

Exercises

Answer in one sentence: what is 99 % of your body weight?
Show answer
Gluon fields and kinetic energy confined inside protons and neutrons.
Almost all your weight is nucleon mass, and 99 % of nucleon mass is confinement energy ── the Higgs contribution (the bare quark masses) is only about 1 %.
★ One often hears "the Higgs gives things mass"; for the mass around you, that is a statement about 1 %.
What do a photon in a waveguide, a photon in a superconductor and a proton have in common?
Show answer
All three have an intercept in their dispersion relation.
Waveguide: boundary conditions give $\omega_c=\pi c/a$ (Episode 6)
Superconductor: a condensate gives $\omega_c=c/\lambda_L$ (Episode 17; 5.06 eV in niobium)
Proton: confinement gives $\omega_c\sim2.04c/R$ (this episode)
★ The only difference is boundary conditions versus background field; what comes out is the same "mass".
Did Episode 3's hypothesis "mass is an integral" come out right for mass?
Show answer
Half right ── but the integration variable is $\ln\mu$, not time.
$\ln m(\mu)=\ln m(\mu_0)-\int\gamma(\alpha_s)\,d\ln\mu$ is literally an integral (§04).
★ With Episode 7's proviso: $m(\mu)$ is dimensionful, hence bookkeeping, and what is observable is a ratio like $m_b/m_\tau$.
Exactly the structure of Episode 5's couplings: the running is real, but only the dimensionless part means anything.

Summary　There are only two ways to make an intercept

Of the proton's 938.27 MeV, the quark masses sum to 8.99 MeV ── 0.96 %. The other 99 % is the energy of a confined field. So is 99 % of your body weight.

In wave language it is simple ── a wave in a box has a lowest frequency. $\pi\hbar c/L$ is 620 MeV at 1 fm, the same order as the proton. Inverting gives $R=1.29$ fm against a measured 0.841 ── a factor of 1.5, which is the crudeness of the model.

The other source is coupling to a condensate. Inverting the Yukawa couplings, only $y_{\rm top}=0.9919$ is of order one, with five and a half orders down to the electron ── an unexplained hierarchy.

And mass runs ($m_b:4.18\to3.11$ GeV at one loop). At $\mu=2.8\times10^7$ GeV the b and the tau coincide, and at $10^{16}$ GeV $m_b/m_\tau=0.68$. "Constants are integrals" holds for mass, with $\ln\mu$ as the variable.

★ In wave language, confinement and condensate are the same thing ──
both create an intercept in the dispersion relation, one by boundaries, one by a background field.

The photon in a waveguide (Episode 6), the photon in a superconductor (Episode 17), the proton and the electron ── all four acquire mass by the same mechanism.

Next time Episode 3 said that anything with a period longer than the age of the universe is indistinguishable from a constant.
So what about a time constant that sits right at the age of the universe? ── $^{232}$Th has a half-life of 14.05 Gyr; the universe is 13.79 Gyr old. They overlap.
★ And surprisingly ── a decay constant can be measured even when it is longer than the age of the universe. A single gram gives 4000 counts a second. Why is no full period needed?

This document is Episode 18 of the series "Thinking in Waves". The computation is in kenshou/dokoka.js. On the numbers: the quark masses are PDG $\overline{\rm MS}$ values at 2 GeV, and "the mass of a quark" is itself scheme-dependent (in a constituent-quark model it would be about 300 MeV) ── the statement "99 % is confinement" refers to current quark masses. The lowest spherical mode $kR=2.04$ is the MIT bag value, but our inversion ($R=1.29$ fm) ignores bag pressure and zero-point corrections and is not a calculation of the proton mass. The Yukawa couplings are inverted from the tree-level relation $m=yv/\sqrt2$, ignoring scale dependence (including for $y_{\rm top}$). The running mass is one loop only and comes out 9 % above the published $m_b(m_Z)\simeq2.83$ GeV ── two loops and proper quark thresholds improve it. We held $m_\tau$ fixed in the $m_b/m_\tau$ discussion, though it runs slightly under QED. The assessment of b–τ unification summarises the standard literature and is not our claim. The equivalence-principle bounds are representative values from each experiment. ── To print, use your browser's Print → Save as PDF.