KAHIBARO
Table of Contents
How particles get mass
In the Standard Model, the masses of many elementary particles are not inserted in a simple, arbitrary way. Instead, they arise through interaction with the Higgs field. The key idea of this chapter is that a particle can behave as if it has mass because it couples to a field that fills all of space.
This does not mean that the Higgs field is like a sticky substance that slows particles down in an ordinary mechanical sense. Rather, mass appears in the mathematical description of the particle as a result of its interaction with the field after the symmetry of the theory is broken.
The Higgs field in empty space
The Higgs field has a nonzero value even in vacuum. This vacuum value is called the vacuum expectation value, usually written as $v$. Its approximate value is
$$
v \approx 246\ \text{GeV}.
$$
Because the vacuum already contains this nonzero Higgs field, particles that interact with it can acquire mass everywhere, even in otherwise empty space.
Important idea: in the Standard Model, the vacuum is not "nothing". The Higgs field has a nonzero vacuum value, and this is what allows several particles to have mass.
Fermion masses
Quarks and charged leptons obtain mass through Yukawa interactions with the Higgs field. A Yukawa coupling is a number that tells us how strongly a given fermion interacts with the Higgs field.
After the Higgs field takes its vacuum value, the fermion mass becomes
$$
m_f = \frac{y_f v}{\sqrt{2}},
$$
where $m_f$ is the fermion mass, $y_f$ is its Yukawa coupling, and $v$ is the Higgs vacuum expectation value.
This formula shows that different fermions have different masses because they have different Yukawa couplings. A stronger coupling to the Higgs field means a larger mass.
For example, the electron has a small Yukawa coupling, so it has a small mass. The top quark has a coupling close to 1, so it is very massive compared with most other known fermions.
A comparison of fermion masses
The Higgs mechanism explains the pattern in a structural way, but it does not by itself predict the numerical values of all Yukawa couplings. Those couplings are input parameters of the Standard Model.
| Particle | Symbol | Approximate mass |
|---|---|---|
| Electron | $e$ | $0.511\ \text{MeV}/c^2$ |
| Muon | $\mu$ | $105.7\ \text{MeV}/c^2$ |
| Tau | $\tau$ | $1.777\ \text{GeV}/c^2$ |
| Up quark | $u$ | a few $\text{MeV}/c^2$ |
| Strange quark | $s$ | about $100\ \text{MeV}/c^2$ |
| Bottom quark | $b$ | about $4.2\ \text{GeV}/c^2$ |
| Top quark | $t$ | about $173\ \text{GeV}/c^2$ |
These values span a huge range. Understanding why the Yukawa couplings take these particular values is one of the open questions beyond the Standard Model.
Gauge boson masses
The $W$ and $Z$ bosons also gain mass through the Higgs field. Before symmetry breaking, the electroweak theory treats the relevant gauge fields in a symmetric way. After the Higgs field takes its vacuum value, the $W$ and $Z$ bosons acquire mass, while the photon remains massless.
Their masses are
$$
m_W = \frac{gv}{2},
$$
and
$$
m_Z = \frac{\sqrt{g^2 + g'^2}\, v}{2},
$$
where $g$ and $g'$ are electroweak coupling constants.
The fact that the photon has zero mass is crucial. A massless photon leads to the long range of electromagnetic interactions.
Key result: the Higgs mechanism gives mass to the $W$ and $Z$ bosons, but not to the photon.
Why a mass term matters
In relativistic physics, mass is not just a measure of "amount of matter". It appears in the energy and momentum relation,
$$
E^2 = p^2 c^2 + m^2 c^4.
$$
A particle with nonzero mass can be at rest and has rest energy $E = mc^2$. A massless particle must always move at the speed of light in vacuum.
This is why the difference between massive and massless particles is so important physically. Electrons, quarks, and weak bosons behave very differently from photons because of their masses.
Neutrinos and mass
In the simplest version of the Standard Model, neutrinos are massless. However, experiments show that neutrinos do have very small masses, because neutrino oscillations occur. This means the Standard Model must be extended in some way to account for neutrino mass.
So the Higgs mechanism explains many particle masses, but the neutrino sector suggests that the full story is not yet complete.
Mass of composite particles
It is important to distinguish the mass of elementary particles from the mass of composite objects such as protons and neutrons. A proton is made of quarks, but most of the proton's mass does not come simply from adding the Higgs generated masses of its quarks.
Instead, most of the proton mass comes from the energy of the strong interaction inside it. According to relativity, energy contributes to mass.
Important distinction: the Higgs field gives mass to elementary quarks, leptons, and weak bosons, but most of the mass of ordinary matter, such as the proton, comes from strong interaction energy.
The Higgs boson and particle masses
The Higgs boson is the observable quantum excitation of the Higgs field. Because particle masses come from coupling to the Higgs field, the Higgs boson couples more strongly to heavier particles.
This is why Higgs boson decays and production rates are closely related to particle masses. Heavier particles usually have stronger Higgs interactions, if those decays are kinematically allowed.
Visual picture
A simple schematic picture is that particles move through a vacuum that already contains the Higgs field. Different particles interact with that background with different strengths, and this difference appears as different masses.
What the Standard Model explains, and what it does not
The Standard Model explains how masses can arise without simply inserting them by hand in a way that would spoil the gauge structure of the theory. It gives a mechanism for mass generation through the Higgs field.
But it does not explain why the Yukawa couplings have the values they do, why particle masses span such a huge range, or fully how neutrino masses arise.
Summary formula for fermions:
$$
m_f = \frac{y_f v}{\sqrt{2}}
$$
Larger Yukawa coupling $\Rightarrow$ larger particle mass.
Final perspective
Particle masses in the Standard Model are deeply connected to the Higgs field. Electrons, quarks, and the $W$ and $Z$ bosons are massive because they interact with a Higgs field that has a nonzero vacuum value. The amount of mass depends on the strength of that interaction. This idea is one of the central achievements of modern particle physics.