Thinking in Waves Episode 32 ／ Part XIII · Frontier physics as waves

What was the negative mass term?

The Higgs as waves ──
the Higgs boson is AM, the Goldstone is FM Flip the sign of the mass term and the cutoff frequency turns imaginary ── the wave does not oscillate, it grows exponentially.
★ The time-domain version of Episode 28's "a wave that cannot travel is an exponential". Nonlinearity stops it ── but does not choose a vacuum. Dissipation does.
★★ What is left are two modes: radial ＝ amplitude modulation, angular ＝ phase modulation. AM and FM, exactly as in audio.

Prerequisites: Episodes 6, 12, 18, 23, 28Computation: kenshou/higgs.js

Episode 6 read it this way ── mass ＝ cutoff frequency. Klein–Gordon's $\omega^2=c^2k^2+(mc^2/\hbar)^2$ was word for word a waveguide.
Which turns the Higgs question into this ── ★ how do you give empty vacuum a cutoff frequency?

01★ A negative mass term ＝ an imaginary cutoff frequency

The Higgs potential has the mass term's sign reversed $$V(\phi)=-\mu^2|\phi|^2+\lambda|\phi|^4\qquad\Longrightarrow\qquad \omega^2=c^2k^2-\mu^2$$

All we did was put $m^2=-\mu^2$. For $k<\mu$, $\omega^2<0$ ── $\omega$ is purely imaginary

$k$ [GeV]	$\omega^2$ [GeV²]	$\omega$	What it is
0	−7843.78	$i\cdot88.5651$	★ grows exponentially (fastest)
50	−5343.78	$i\cdot73.1012$	★ grows exponentially
88.565	0.00	0	★ critical
120	6556.22	80.9705	a travelling wave

The heart of §01

With $\mu=m_H/\sqrt2=88.565$ GeV, every mode with $k<\mu$ grows. The fastest is $k=0$, with time constant $\hbar/\mu=7.43\times10^{-27}$ s and doubling time $5.15\times10^{-27}$ s.

★★★ This is the same story as Episode 28.
Episode 28: a wave that cannot travel is an exponential ($k^2<0\to e^{\pm\kappa x}$, in space)
★ Here　　: a wave that cannot travel is an exponential ($\omega^2<0\to e^{\pm\mu t/\hbar}$, in time)
── The tachyonic instability is the time-domain evanescent wave.

02Nonlinearity stops it ── but ★ does not choose a vacuum

What stops the growth is the nonlinear term (Episode 23) $$\ddot\phi=\mu^2\phi-\lambda\phi^3,\qquad \lambda=\frac{m_H^2}{2v^2}=0.129384,\quad v=\frac{\mu}{\sqrt\lambda}=246.2197\ \text{GeV}$$

Starting from a seed $\phi_0=10^{-6}$ GeV, first measure the growth.

$t$ [1/GeV]	$\phi$ [GeV]	Slope of $\ln\phi$	Predicted $\mu$
0.080	5.9709e−4	88.5604	88.5651
0.140	1.2140e−1	88.5651	88.5651

Next, where the nonlinearity turns it around (no friction).

The turning point $$\text{measured }348.2072\ \text{GeV}\qquad\text{vs}\qquad \sqrt2\,v=348.2072\ \text{GeV}\quad(\text{rel. diff. }1.2\times10^{-10})$$

$V(0)=V(\sqrt2v)=0$, so it sails past the valley and climbs to the same height on the far side

★★ A prediction that failed

The field does not settle at $v$. It oscillates forever between $-\sqrt2v$ and $+\sqrt2v$ (energy is conserved).
★ It passes through the valley, climbs all the way to the opposite vacuum, and comes back ── choosing neither.

Settling requires dissipation (decay into other fields, the friction of cosmic expansion) ── $\ddot\phi+\gamma\dot\phi=\mu^2\phi-\lambda\phi^3$.

Friction $\gamma$ [GeV]	$\phi$ at $t=3$ [GeV]	Rel. diff. from $v$	Result
0	−0.041235	1.0e+0	★ keeps swinging
1	265.018954	7.6e−2	still ringing
10	246.219546	4.2e−7	★ settles into the vacuum
50	246.219650	1.4e−14	★ settles into the vacuum

The heart of §02 ── ★★★ this episode's yield

★ Nonlinearity halts the runaway but does not pick a vacuum.
"Symmetry breaking" requires dissipation.

★ In cosmology that friction is Hubble friction $3H\dot\phi$ plus decay into other fields. The dissipation Episode 12 located in $\sin(\alpha\pi/2)$ is here doing the job of choosing the vacuum.

03★★ The Higgs boson is amplitude modulation, the Goldstone is phase modulation

Look at small fluctuations around the vacuum $v$. The field is complex, so there are two directions ── $\phi=(v+h)e^{i\theta}$. $V$ depends only on $|\phi|$.

Two directions $$\text{radial}:\ V''(v)=-\mu^2+3\lambda v^2=2\mu^2\ \Rightarrow\ \omega=\sqrt2\,\mu\qquad\qquad \text{angular}:\ V\ \text{is flat}\ \Rightarrow\ \omega=0$$

Direction	Measured numerically	Prediction
Radial	$\omega=125.263$ GeV	$\sqrt2\mu=m_H=125.250$ (rel. diff. 1.0e−4)
Angular	$\|\phi\|$ varies by $1.9\times10^{-11}$ GeV	no restoring force ＝ zero mass

The heart of §03

★★★ Radial fluctuation ＝ amplitude modulation (AM) → the Higgs boson (125 GeV)
Angular fluctuation ＝ phase modulation (FM) → the Nambu–Goldstone boson (massless)

Wobble the carrier's size, or wobble its phase ── that is the whole difference.

04Comparing AM and FM with an actual FFT

AM and FM have completely different spectra. Computed at $f_c=1000$ Hz, $f_m=100$ Hz, $m=0.5$, $\beta=3$.

Frequency [Hz]	$n$	AM amplitude	FM amplitude	$\|J_n(3)\|$
700	−3	──	0.309063	0.309063
800	−2	──	0.486091	0.486091
900	−1	0.250000	0.339059	0.339059
1000	0	1.000000	0.260052	0.260052
1100	+1	0.250000	0.339059	0.339059
1400	+4	──	0.132034	0.132034

The heart of §04

AM has three lines (carrier plus one sideband each side). FM has a whole row of them, with amplitudes matching the Bessel functions $|J_n(\beta)|$ to 6 digits.
★ Back in physics ── the Higgs (AM) has mass and is observable as a countable particle; the Goldstone (FM) is massless and is never seen on its own. ★ It is "eaten" by the gauge field and becomes the longitudinal polarization.

05The Goldstone is eaten ── polarizations go from 2 to 3

Field	Before mass	After mass
Gauge field (W, Z)	2 (transverse)	★ 3 (＋longitudinal)
Goldstone	1 (phase)	★ 0 (gone)
Total	3	3

★ The count does not change. The phase fluctuation simply turned into the longitudinal wave. In audio terms ── the carrier absorbed the FM sidebands.

Quantity	From the formula	Measured [GeV]
$m_W=gv/2$	80.3770	80.3770
$m_Z=\sqrt{g^2+g'^2}\,v/2$	91.1876	91.1876
$\sin^2\theta_W$	0.223052	0.2231 (on-shell)

★ $\sin^2\theta_W$ depends on the scheme ── the on-shell definition $1-m_W^2/m_Z^2$ gives 0.2231, while the $\overline{\rm MS}$ value at $M_Z$ is 0.2312. The 3.5 % gap is not an error but a difference of definition (the same caution as $\Lambda_{\rm QCD}$ in Episode 25 and running masses in Episode 28).

06Yukawa couplings ── five decades of mass from a single vacuum

Particle	Mass [GeV]	Yukawa $y=\sqrt2\,m/v$
Top	172.69	0.99188 ── ★ essentially 1
Bottom	4.18	2.40e−2
Tau	1.777	1.02e−2
Muon	0.1057	6.07e−4
Electron	5.11e−4	2.94e−6

The heart of §06

Only the top has $y=0.9919$ ── essentially 1. The ratio to the electron is $3.4\times10^5$. Five-plus decades of mass, all from one vacuum $v$.
★ The same conclusion as Episode 18 ── the Higgs does not "make" mass; it distributes one dimensionful quantity. The ratios it distributes in are not explained.

07What is explained and what is not

Item	Verdict	Reason
Why the wave stops	◎ explained	exponential growth → nonlinear saturation
The form of the Higgs mass	◎ explained	$\sqrt2\mu$ ＝ the radial restoring force
Why the Goldstone is massless	◎ explained	no potential in the angular direction
Polarizations 2 → 3	◎ explained	the phase freedom becomes longitudinal
★ Why $\mu^2<0$	× not explained	the sign is an input
★ The 5-decade Yukawa hierarchy	× not explained	all inputs. Episode 18's wall
★ Where $v=246$ GeV comes from	× not explained	a dimensionful quantity (Episode 15's wall)

The heart of §07

Seeing it as waves explains the mechanism, not the values.
★ Episode 15's "dimensionless quantities land, dimensionful ones miss" applies here unchanged.

Exercises

A "tachyon" is not a faster-than-light particle. What is it, as a wave?
Show the answer
An unstable mode. With $\omega^2=c^2k^2-\mu^2$ and $k<\mu$, $\omega$ is purely imaginary, so $e^{-i\omega t}\to e^{\mu t/\hbar}$ ── not oscillation but exponential growth.
★ Not "too fast" but "does not travel at all, and grows where it stands" ── the time-domain version of Episode 28's below-cutoff evanescent wave ($e^{-\kappa x}$).
★ Indeed the group velocity $d\omega/dk$ is imaginary for $k<\mu$, so the word "speed" loses its meaning.
If the Higgs boson is AM, why does it "have mass"?
Show the answer
Because there is a restoring force in the amplitude direction. $V''(v)=2\mu^2>0$, so displacing the amplitude pulls it back ── and how fast it pulls back is exactly the cutoff frequency ＝ the mass (Episode 6).
The angular direction has a flat $V$, no restoring force, hence zero cutoff frequency ＝ zero mass.
★ "Mass" is the strength of that mode's restoring force: we are measuring the curvature of the Mexican hat's cross-section, nothing more.
§02 found that without dissipation no vacuum is chosen. Is that a problem in the real universe?
Show the answer
No ── the universe supplies friction. Expansion automatically brings Hubble friction $3H\dot\phi$, and the Higgs field can decay into other particles (it has Yukawa couplings).
★ The lesson runs the other way: breaking does not follow from the potential alone ── the potential only offers possible vacua; which one is taken is settled by dissipation.
★ This is exactly the structure of inflationary "slow roll" (the friction-dominated regime), and it is why cosmology always writes the $3H\dot\phi$ term.

Summary　Make the cutoff imaginary, let it grow, then stop it

★ A negative mass term is an imaginary cutoff frequency. With $\omega^2=c^2k^2-\mu^2$, every mode below $k=\mu$ grows, fastest at $k=0$ with time constant $\hbar/\mu=7.4\times10^{-27}$ s. The time-domain counterpart of Episode 28's spatial exponential.

Numerically the slope of $\ln\phi$ matched $\mu=88.5651$ to 6 digits, and the nonlinear turning point is $\sqrt2\,v=348.2072$ GeV (agreeing to $1.2\times10^{-10}$).

★★ But a prediction failed ── the field does not settle into a vacuum. It swings forever between $\pm\sqrt2v$. Settling needs dissipation; at $\gamma\ge10$ GeV it converges on $v$ to $10^{-7}$. ★ Nonlinearity halts the runaway but does not choose. "Symmetry breaking" requires dissipation ── which is exactly what cosmology's Hubble friction provides.

★★★ What remains are two directions ── radially, a restoring force $2\mu^2$ giving $\omega=\sqrt2\mu=m_H$ (matched to $10^{-4}$); angularly, no restoring force at all ($|\phi|$ varying by $2\times10^{-11}$). In audio terms, amplitude modulation and phase modulation: by FFT, AM has two sidebands while FM has an unending row following $J_n(\beta)$ (6-digit agreement).

★ The Higgs boson is AM; the Nambu–Goldstone boson is FM.
And the FM sidebands are absorbed by the gauge carrier as its longitudinal wave (2 → 3 polarizations, the count unchanged).

The vacuum expectation value gives $m_W$ and $m_Z$, and the Yukawa couplings $y=\sqrt2m/v$ put only the top at 0.9919 ≈ 1, a factor $3.4\times10^5$ above the electron. ── But nothing explains why $\mu^2<0$, nor $\lambda$, nor the Yukawa hierarchy, nor $v=246$ GeV. Seeing it as waves explains the mechanism, not the values (Episode 15).

Next episode The Yukawa hierarchy went unexplained. ★ So is there no relation at all among the lepton masses?
★ Next time, the Koide formula ── $\dfrac{m_e+m_\mu+m_\tau}{(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2}=\dfrac23$, reread as a wave.
★★ Treat the three $\sqrt m$ as a sinusoid sampled at 120° intervals and ── the Koide formula says the AC and DC parts have equal RMS.

This document is Episode 32 of the "Thinking in Waves" series. The computations are in kenshou/higgs.js. On the numbers here: the dispersion table, the field's time evolution (with and without friction), the small-oscillation periods, and the AM/FM FFT against Bessel functions were all computed for this article. $v,\ m_H,\ m_W,\ m_Z,\ m_{\rm top}$ and the rest are PDG measurements; $\mu$ and $\lambda$ follow from them by definition ($\lambda=m_H^2/2v^2$, $\mu=m_H/\sqrt2$). The Higgs mechanism, the Nambu–Goldstone theorem, the Goldstone becoming the longitudinal mode, and the Bessel-function structure of FM sidebands are all well-known results and not discoveries of this article ── what this article did was translate them into the language of cutoff frequency, exponential growth and modulation, and check the correspondence numerically. The dynamics in §02 is a one-component classical toy model; the real Higgs field is a complex doublet, and cosmological evolution also involves expansion (the friction $\gamma$ is a crude stand-in). §03's "the angular direction is massless" is the global-symmetry case; gauging it gives §05's longitudinal absorption. The Yukawa couplings in §06 are tree-level definitions and become scheme-dependent once run (Episode 28). ── To print, use your browser's Print → Save as PDF.