Mass That ClicksEpisode 3 / How the finite floor wells up

The Higgs, and confinement ── even the same "mass" wells up in utterly different ways

Mass wells up in two ways In Episode 2: "a finite floor wells up when symmetry breaks." So how does it actually well up?
The electron from the Higgs, the proton from confinement ── and 99% of the universe's visible mass is the latter.

Tools you need: Episode 1's floor, Episode 2's curvature at the bottom, multiplication Hook equations: $m_f=y_f\,v/\sqrt2$ / proton $\sim$ a few $\times\Lambda_{\rm QCD}$

In Episode 1, mass was "an energy floor you can't erase"; in Episode 2, "the zero floor is protected by symmetry, and a finite floor wells up when symmetry breaks." This time we see how it wells up. But ── there isn't just one way; there are two. Elementary particles like the electron receive it directly from the Higgs, while composite particles like the proton are born from the energy of confinement. And astonishingly, 99% of your body weight is the work of the latter, not the famous Higgs.

01Way 1: the Higgs ── "grabbing on" to a field that fills the vacuum

Let's push Episode 2's "mass² = the curvature at the bottom of the potential" figure one step further. The Higgs field's potential has the coefficient of the bottom made negative ── the so-called Mexican hat. Then the vacuum (the valley bottom) shifts sideways off the origin, and the field takes on a value $v\neq0$ everywhere. This is "spontaneous symmetry breaking."

Figure: the Higgs potential. Make the knob μ² negative and the origin becomes a "hill," shifting the vacuum to $\phi=v\neq0$ (spontaneous symmetry breaking). The field $v$ filling the vacuum gives particles a floor of mass.

Symmetry knob μ² (more negative to the left = broken)-1.00

Move the knob and the position of the vacuum changes.

Higgs potential V(φ) Vacuum value v

Particles couple to this field that fills the vacuum. Being coupled to it, they carry an energy floor even at rest ── this is Episode 1's "floor" = mass. The height of the floor is set by the field value $v$ and the coupling strength (the Yukawa coupling $y_f$).

The mass the Higgs gives

$$m_f=\frac{y_f\,v}{\sqrt2},\qquad v\approx 246\ \text{GeV}$$

With the same $v$, the coupling $y_f$ differs by orders of magnitude, so the masses do too. The top quark has $y\approx1$ and is about 173 GeV; the electron has $y\approx3\times10^{-6}$ and is about 0.5 MeV. The "personality" of a mass was the strength of its coupling to the vacuum.

A connecting voice (paying off Episodes 1 and 2) When it breaks, Episode 2's "flat directions" (Goldstones) appear in the potential. The $W,Z$ particles eat these flat directions, acquire the longitudinal mode we spoke of in Episode 1, and become heavy. The combination left unbroken is the massless photon. Episode 1 (the longitudinal mode), Episode 2 (the flat direction), and this episode (mass switching on) shake hands in this single hat.

◇ ◇ ◇

02Way 2: confinement ── energy becomes mass directly

The other way is something else entirely. The proton is a composite particle in which three quarks (two up, one down) are bound tightly by the strong force (gluons). Yet adding up all the "bare masses" of the quarks inside (the Higgs-derived part) gives only a small fraction of the proton's weight.

Let's try it ── the breakdown of the proton's weight $$\frac{\text{bare mass of 3 quarks}}{\text{proton mass}}\approx\frac{2.2+2.2+4.7\ \text{MeV}}{938\ \text{MeV}}\approx\frac{9}{938}\approx 1\%$$

The remaining ~99% is the energy of the gluon field confining the quarks, plus the kinetic energy of the confined quarks. That turns directly into mass via $E=mc^2$.

This mechanism is completely unrelated to the Higgs. Even if you switched the Higgs off and made all quarks massless, the proton would remain at almost the same weight. Because the source of its mass is not the weight of the parts but the energy of confinement.

The mass confinement gives (a scale wells up)

$$m_{\text{proton}}\ \sim\ \text{a few}\times\Lambda_{\rm QCD}\qquad(\Lambda_{\rm QCD}\approx 0.2\ \text{GeV})$$

From a theory that supposedly has no scale at all, a ruler for weight called $\Lambda_{\rm QCD}$ wells up ── the true nature of this mystery comes in Episode 4.

03So 99% of your body weight is not the Higgs

Here's this episode's biggest reveal. It's often said "the Higgs gives everything its mass," but that's overstating it. What the Higgs gives is only the "bare mass" of elementary particles. The bulk of the weight of the protons and neutrons that make up your body is the QCD confinement energy. Let's line up the two mechanisms.

Higgs (bare mass of elementary particles)

Applies to: the rest mass of quarks, leptons, $W,Z$.
Mechanism: spontaneous symmetry breaking. Coupling to the field $v$ that fills the vacuum.
Equation: $m_f=y_f v/\sqrt2$.
Contribution to visible mass: about 1%.

Confinement (mass of composite particles)

Applies to: the bulk of the mass of hadrons like protons and neutrons.
Mechanism: confinement by the strong force. Field energy via $E=mc^2$.
Equation: $m\sim$ a few $\times\Lambda_{\rm QCD}$.
Contribution to visible mass: about 99%.

04Two floors, one backbone

However they well up, they arrive at the same place ── Episode 1's "floor you can't erase." The Higgs floor is made by coupling to the field that fills the vacuum. The confinement floor is born from field energy turning into mass. Episode 2's "mass² = the curvature at the bottom" holds literally on the Higgs side (the mass of the Higgs particle itself = the radial curvature of the hat). The confinement side is a different kind of welling-up, and it is exactly there that a door opens where this series' watchword $c\cdot t=\text{constant}$ works legitimately for the first time.

The honest line

The metaphor "the Higgs is like molasses that gives resistance and makes things heavy" is inaccurate. Resistance shouldn't act at constant velocity ── what actually happens is "coupling to the field that fills the vacuum gives a floor of energy at rest = mass." Episode 1's "floor" is the correct picture.

This time too, the lens $c\cdot t=\text{constant}$ stays out of the spotlight. The Higgs mechanism is standard physics unrelated to $c\cdot t$ (not forcing it to work = the lesson of the sister series' Bonus ③). But the confinement side's "a scale wells up" is different. In the next episode, Episode 4, this lens really works. The "99%" wobbles a bit with how you define the quarks' bare masses, but as an order of magnitude, 99% is a solid fact.

Practice problems (solvable with this episode's content)

In one line, why is "the Higgs gives everything its mass" an overstatement?
See the answer
What the Higgs gives is only the "bare mass" of elementary particles (quarks, leptons, W, Z). The bulk of the visible mass (protons, neutrons), about 99%, comes from QCD confinement energy, and the Higgs contribution is only about 1%.
Using $m_f=y_f v/\sqrt2$, explain why the top quark is overwhelmingly heavier than the electron.
See the answer
The vacuum value $v$ is the same, but the Yukawa coupling $y_f$ differs by orders of magnitude (top $\approx1$, electron $\approx3\times10^{-6}$). Mass is set by the strength of the coupling to the vacuum, so the top, with a large coupling, is heavy, and the electron, with a small one, is light.
If you switched the Higgs off and made all quarks massless, what would happen to the proton's weight?
See the answer
Almost nothing would change. About 99% of the proton's mass comes from confinement energy and barely depends on the quarks' bare masses (about 1%). The proton's weight is set almost independently of the Higgs.

Episode 3 summaryThe same "floor" had two ways of welling up

A finite mass (floor) wells up in two ways. The Higgs, through spontaneous symmetry breaking, fills the vacuum with a field $v$ and gives elementary particles a floor according to the coupling $y_f$ ($m_f=y_f v/\sqrt2$). Confinement has the energy of the strong force's field turn directly into mass via $E=mc^2$, making up about 99% of the proton. The bulk of the universe's visible mass was the work of confinement, not the Higgs.

The Higgs side was drawn cleanly with Episode 2's "curvature at the bottom." But the confinement side's "a ruler for weight, $\Lambda_{\rm QCD}$, wells up out of nothing" can't be drawn yet. This emergence of a scale is the star of the next episode ── and, at last, the place where the $c\cdot t=\text{constant}$ lens works legitimately.

Next time ── Episode 4 "A scale wells up from a scaleless theory." We chase the true nature of the confinement side's "99%" with the running coupling and dimensional transmutation. A theory that classically has no ruler (= a conformally invariant arena where $c\cdot t$ works cleanly) breaks that invariance through quantum effects and gives rise to $\Lambda_{\rm QCD}$. This is also the true nature of the "mass gap." From here, the $c\cdot t$ lens truly begins to work.

This document is Episode 3 of the "Mass That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. The Higgs mechanism (spontaneous breaking of the electroweak symmetry, the vacuum expectation value $v\approx246$ GeV, the fermion mass $m_f=y_f v/\sqrt2$, and $W,Z$ acquiring mass by absorbing Nambu–Goldstone degrees of freedom), and the fact that the bulk of the mass of protons and neutrons (about 99%) comes not from the quarks' rest masses but from the dynamical energy of quantum chromodynamics (the gluon field and the motion of the confined quarks), set by the scale $\Lambda_{\rm QCD}\approx0.2$ GeV via dimensional transmutation, are all established physics. The "Higgs = resistance like molasses" metaphor is inaccurate; correctly, coupling to the field that fills the vacuum gives the rest energy (mass). The figure is a schematic cross-section of $V=\tfrac12\mu^2\phi^2+\tfrac14\phi^4$. ── To print, use "Print" in your browser and choose "Save as PDF" (in the printed version the slider and answers are frozen and hidden).

Print / save as PDF: Ctrl+P (⌘+P on Mac). On screen you can move the symmetry breaking (the shift of the vacuum) with the slider. "See the answer" opens the solutions.

Mass wells up in two ways In Episode 2: "a finite floor wells up when symmetry breaks." So how does it actually well up?The electron from the Higgs, the proton from confinement ── and 99% of the universe's visible mass is the latter.

01Way 1: the Higgs ── "grabbing on" to a field that fills the vacuum

02Way 2: confinement ── energy becomes mass directly

03So 99% of your body weight is not the Higgs

04Two floors, one backbone

Episode 3 summaryThe same "floor" had two ways of welling up

Mass wells up in two ways In Episode 2: "a finite floor wells up when symmetry breaks." So how does it actually well up?
The electron from the Higgs, the proton from confinement ── and 99% of the universe's visible mass is the latter.