The Higgs field
and the origin of mass

Gauge theory provides the starting point for understanding the Higgs mechanism. It connects a freedom in how we describe particles to the fields that carry forces. Electromagnetism gives a concrete example.

An electron's wavefunction has a size and a phase at each point in space, like the length and angle of a clock hand. The size squared gives the probability density for finding the electron there. Phase differences affect how waves reinforce or cancel each other, but rotating every hand by the same angle changes no prediction.

That is a symmetry, written ψ → e^iαψ: ψ is the field, and multiplying by e^iα rotates its phase by the angle α. Noether's theorem connects this continuous symmetry of the charged field to conservation of electric charge.

Now allow a different rotation at every point, ψ(x) → e^iα(x)ψ(x), and require the physics to stay unchanged. Think of choosing where each clock face marks zero. This freedom to choose the phase convention locally is a gauge symmetry.

This creates a problem. The equations compare the field at neighboring points using derivatives. If the two points use different phase conventions, that comparison picks up an extra term from the changing convention:

The extra term tells us what is missing: a field that connects the phase conventions at neighboring points. To preserve the symmetry, this field must change whenever those conventions change, canceling the extra term. Call it A_μ. Including it in the derivative makes the equations work again:

We can now work out what kind of field this must be. It interacts with electric charge. A direct mass term M²A_μA^μ would spoil the symmetry, so that term is forbidden. And the simplest relativistic kinetic term for this field, proportional to F_μνF^μν, gives Maxwell's equations.

These are the equations of electromagnetism. A_μ is the electromagnetic potential; F_μν contains the electric and magnetic fields. Its particles are massless photons, its waves travel at c, and its force between stationary charges follows the inverse-square law. Starting with the freedom to choose a phase convention locally, and giving the required field its simplest relativistic dynamics, we have recovered the familiar electromagnetic interaction.

The weak force will pose a further problem: its carriers have mass. We need to explain how that can happen while keeping the gauge symmetry that constrains their interactions.

The corrected derivative D_μ is called the covariant derivative. Here ∂_μ means a derivative with respect to time or one spatial coordinate; μ labels which. The symbol i is the imaginary unit, and e sets the electromagnetic coupling strength. The symmetry leaves the numerical value of e to be measured.

The Standard Model gauge groups

For electromagnetism, we rotated a phase. We can also consider symmetries that mix several components of a field. Quarks, for example, have three components called colors. These are labels for strong-interaction charge, unrelated to visible color. The transformations that mix them form a mathematical group called SU(3).

Allow those transformations to vary from point to point, and the same problem returns: ordinary derivatives pick up extra terms. This time there are eight independent ways to make a small transformation, so we need eight compensating gauge fields. Their particles are the eight gluons. For weak isospin, the group SU(2) acts on pairs of components. It has three independent transformations and requires three W fields.

There is a further consequence. Two phase rotations can be performed in either order with the same result. For SU(2) and SU(3), changing the order can change the result; such groups are called non-abelian. Preserving these local symmetries requires interactions among the gauge fields themselves. Gluons therefore interact with other gluons, and the W fields interact with one another.

The Standard Model (SM) combines SU(3), SU(2) and a phase symmetry called U(1). In the table, a generator is one independent direction of a small transformation; each needs its own gauge field.

The U(1) in SU(3)×SU(2)×U(1) describes hypercharge. Its field B combines with one of the W fields to form the photon and Z once the Higgs background is present; section 06 shows how. The symmetry principle tells us how to build the interactions once we choose the groups and matter fields. Experiment tells us which choices describe nature.

This creates a problem for the weak interaction. Gauge symmetry forbids direct mass terms for the gauge fields. A term quadratic in A_μ fails to stay unchanged under a gauge transformation. The photon is massless, so it needs no such term. Gluons also have no direct mass term. In quantum chromodynamics (QCD), the theory of the strong interaction, confinement binds quarks and gluons into composite particles, preventing them from propagating freely over long distances. The W and Z have measured masses of about 80 and 91 GeV. Their large masses account for the weak force's short range. The Higgs mechanism must explain those masses while preserving the gauge symmetry of the equations.

The gauge fields in section 01 gave us the forces, but their equations have no direct mass terms. That works for the photon. The W and Z are massive, so the theory still owes us an explanation.

For a scalar field, whose particles have spin zero, a term proportional to the field squared can give those particles a rest mass. Trying the same thing with a gauge field runs into the freedom we just introduced. A gauge transformation shifts the field's value. Squaring that shifted value changes the proposed mass term, even though we intended the transformation to change only our description.

The rest of the gauge theory is built so that these changes cancel. The direct mass term has nothing to cancel its change. Adding it therefore breaks the gauge symmetry of the equations. A massive-vector theory can still be useful as an approximation at low energies, but its high-energy scattering calculations eventually need additional contributions.

The Higgs proposal supplies mass through an interaction that obeys the symmetry. Add a scalar field carrying weak isospin and hypercharge so that it interacts with the electroweak gauge fields. If it has a constant background value, that interaction can contribute to the force carriers' rest energy and behave as a mass term.

This proposal depends on something we have not yet explained: the scalar field must remain nonzero even in empty space. The Higgs field's potential energy makes that possible.

Removing every particle does not make every field zero. Empty space is the lowest-energy state available, and for the Higgs field, zero would cost more energy than a nonzero value.

Picture the potential energy of an ordinary scalar field as a bowl. Its lowest point is at zero, so a disturbance pulls the field back toward zero. The Higgs potential has a hill at the center instead. A field balanced there is unstable: a small disturbance moves it toward the surrounding valley.

Two terms produce this shape. The negative term lowers the energy as the field moves away from zero. The positive fourth-power term eventually dominates and turns the energy upward again. Between them lies a valley at a definite, nonzero magnitude:

The field value at the bottom is its vacuum expectation value, or vev, written ⟨H⟩. In this convention its magnitude is v/√2, about 174 GeV; the parameter v is about 246 GeV. Both numbers describe the same background. In the potential, |H| is the field magnitude, and the positive parameters μ and λ set the depth and steepness of the valley.

Once the field settles in the valley, changing its magnitude costs energy. Small disturbances oscillate around the minimum. Quantizing those oscillations gives the Higgs boson; the curvature of the valley determines its mass. The phrase “negative mass squared” refers to the instability at the hilltop. The particle we detect is an excitation around the stable minimum.

A scalar field has no spacetime direction attached to its value. The Higgs is a doublet: two complex components, equivalent to four real numbers. Each component is a spacetime scalar. This distinction matters: its vacuum value can select an orientation among the internal components without selecting an axis in ordinary space. A nonzero electric field, by comparison, points along a spatial direction.

The potential's parameters are inferred from measurements within the Standard Model. v = 246.22 GeV is determined from the weak-interaction strength measured in muon decay. At leading order, meaning before quantum corrections are included, the Higgs mass measures the curvature of the potential near its minimum: m_h² = 2λv². This gives λ = 0.129. Measuring the self-interactions separately tests whether this assumed potential is correct.

The right panel shows a two-dimensional slice through the four-component field. Since the potential depends only on |H|², it gives every orientation with the same magnitude the same energy. Changing the magnitude moves up the sides of the valley and produces a restoring force. Changing orientation follows the valley without changing the potential energy. The full four-dimensional field has three such orientation directions; the drawing shows only one.

The photon carries a long-range electromagnetic force. The massive W and Z carry a short-range weak force. Yet both come from the same electroweak theory. The Higgs background explains how that theory produces such different particles.

We have established why the Higgs field remains nonzero in empty space. To understand why this breaks a symmetry, return to the circular valley in the previous diagram.

The valley has rotational symmetry: every point around it lies at the same height. A point at the center stays put when we rotate the picture. A point on the valley floor moves to another position. The potential has a symmetry that a particular minimum does not share. The field cannot stay at the center, because that is the unstable hilltop.

For the Higgs, these rotations act among its four real components. Zero in every component stays zero when we mix them. A nonzero value in one component can move into another. We can describe the vacuum by putting its background value in one component and setting the others to zero. Transformations that move this value between components no longer leave that chosen background unchanged.

This is why the breaking is called spontaneous. The potential treats all those orientations equally, but its minimum requires a nonzero field. The background follows from the lowest-energy state of the theory. We have added no symmetry-violating mass term of the kind that failed in section 02.

The orientation is a gauge convention: gauge-related choices predict the same masses and interactions. The physical distinction here is between zero and nonzero Higgs magnitude. The gauge symmetry of the full equations remains intact.

Four real components went into the Higgs field, but only one appears as a Higgs boson. What happens to the other three explains how the W and Z can become massive.

The potential diagram gives us a way to keep track of them. One direction changes the field's magnitude, climbing the sides of the valley. Three directions change its orientation, following the valley floor. An independent way for a field to fluctuate is called a mode. The magnitude mode gives the Higgs boson. The three orientation modes are called Goldstone modes.

Now consider what a force carrier needs in order to be massive. A massless spin-1 particle has two independent spin states, called polarizations. A massive one has three. Its extra state is longitudinal, with zero spin projection along its motion. The W⁺, W⁻ and Z each need one such state: three in total.

The Higgs field supplies exactly those three states. Through its gauge interactions, the orientation modes become the longitudinal polarizations of the massive W and Z. This is what physicists mean when they say the bosons “eat” the Goldstone modes. The modes are now part of the massive force carriers' description, so they do not appear as three additional scalar particles.

The diagram follows each mode into its new role. G⁺, G⁻ and G⁰ label the three Goldstone modes; h labels the Higgs boson.

There are twelve physical degrees of freedom in the electroweak gauge and Higgs fields, both before and after symmetry breaking. DOF below means degrees of freedom. Read the two halves of the table as separate inventories; corresponding rows do not indicate which field becomes which particle.

A wave can involve more than one field varying together. When those variations keep a fixed proportion, they form a single pattern of motion. This is what a “combination of fields” means here.

We started with W¹, W², W³ and B. The Higgs interacts with these fields, and its background couples some of their motions. To identify the particles, we look for wave patterns that propagate with a definite mass. W¹ and W² form the charged W⁺ and W⁻. The neutral fields W³ and B form two other patterns.

Imagine a wave in which W³ and B rise and fall together, with their amplitudes in a particular ratio. Both fields interact with the Higgs background, but their contributions to that interaction have opposite signs. At the right ratio, the contributions cancel. This wave pattern receives no mass from the background. Its particles are photons.

In the other pattern, W³ and B vary with opposite signs and a different amplitude ratio. Their contributions to the Higgs interaction reinforce each other. The background gives this pattern a rest energy, hence a mass. Its particles are Z bosons. The two patterns can be superposed to describe any motion of the original neutral fields.

The photon couples to electric charge. For the Higgs component with the background value, the weak-isospin contribution T³ = −1/2 and the hypercharge Y = +1/2 add to zero: Q = T³ + Y = 0. This is why that component is electrically neutral. Electromagnetic phase changes rotate charged fields, so they leave this neutral background unchanged. This is what it means for electromagnetic symmetry to survive.

The interaction strengths fix those amplitude ratios. Calling the weak-isospin coupling g and the hypercharge coupling g′, we can express the ratios through one weak mixing angle, θ_W. The sine and cosine below are the weights of the original fields in each combination:

The same angle also relates the W and Z masses, as the triangle below shows. Using the measured masses gives about 28.2°. Its other name, the Weinberg angle, recognizes one of the physicists who developed the electroweak theory.

The mixing predicts a relation between masses that could otherwise have been independent. The W mass is set by the Higgs background and the coupling g; the Z mass involves both g and g′, making it larger. The formulas are M_W = gv/2 and M_Z = v√(g²+g′²)/2. Their ratio is cosθ_W at leading order, before quantum corrections.

This mass relation gives experiments another check. An additional symmetry of the Higgs sector, called custodial symmetry, protects the leading-order relation. Quantum corrections and new fields can change it. Comparing precise W and Z measurements therefore tests the Higgs sector even without producing a Higgs boson.

Why the weak force is weak at everyday energies

A massive force carrier makes the force fall off rapidly with distance. The W and Z are so heavy that their range is about 0.002 femtometers, roughly a thousandth of a proton's size. The massless photon allows electromagnetism to act over long distances. Weak interactions are therefore strongly suppressed in low-energy processes, even though their underlying coupling strengths are comparable to the electromagnetic coupling. At collision energies comparable to the W and Z masses, that mass suppression becomes much less important.

The Higgs carries no color charge, so its background gives no mass to the gluons. Their confinement, discussed in section 01, has a different origin.

The Higgs also gives mass to electrons and quarks. For these matter particles, or fermions, the obstacle involves two different kinds of field component. We need to distinguish chirality from the more familiar handedness of spin before following how the Higgs connects them.

Helicity

Helicity describes the direction of a particle's spin relative to its motion. For an electron, spin aligned with its momentum gives right-handed helicity; spin opposed to its momentum gives left-handed helicity. Spin is intrinsic angular momentum, so these arrows are a way to represent a quantum property.

For a massive particle, helicity depends on the observer. Picture an electron flying away from you with right-handed helicity. If you overtake it, from your new frame the electron is moving backwards. Nothing has touched its spin — you only changed your own speed — but its direction of travel has reversed, so its spin is now opposed to its momentum. You measure left-handed helicity. The two observers measure different helicities for the same electron.

A massless particle travels at c, so an observer cannot overtake it. Changes of inertial reference frame preserve its helicity. Observers can disagree about its direction in their coordinate systems while agreeing on the spin's orientation relative to that direction.

Chirality

Chirality classifies the components of a fermion field. The Standard Model assigns different weak charges to the left-chiral and right-chiral components. These assignments hold regardless of how fast an observer moves.

Mathematically, the two chiralities transform differently when we change reference frame. A rotation or a change of observer speed preserves a field component's chirality label. A mass term, however, couples the two types of component in the equations of motion.

For massless fermion particles, chirality and helicity agree. For a massive electron with definite helicity, both chiral components contribute, and their relative sizes depend on the observer's frame. The antiparticle convention needs care: a left-chiral fermion field creates massless antiparticles of right-handed helicity.

The two chiral components

We write the electron's chiral fields as e_L and e_R. They have the same electric charge but different weak charges. After the Higgs field generates mass, they are coupled components of the physical electron. They are not physical halves of it.

The left-chiral fields carry weak isospin and form pairs called doublets: the electron with its neutrino, or an up quark with a down quark. A W interaction can convert one member into the other. The right-chiral fields have no such partner and are called singlets. This difference is what the L in SU(2)_L records. Right-chiral fields can still carry hypercharge and interact with the Z after symmetry breaking.

A mass term connects e_L and e_R. But the weak symmetry changes these fields in different ways, so their direct connection changes too. That is why simply adding an electron mass would spoil the symmetry. The same obstacle appears for the other charged leptons and quarks.

The Higgs already has the charges needed to compensate for this mismatch: it is a weak-isospin doublet with hypercharge +1/2. An interaction joining the left-chiral field, the Higgs and the right-chiral field can therefore preserve the symmetry.

With the Higgs background present, this interaction keeps linking the electron's two chiral components. In the equations, the Higgs background multiplied by the interaction strength becomes the electron's mass. The interaction strength is called its Yukawa coupling. We have obtained mass from an allowed interaction, with the gauge symmetry intact.

The top quark and the electron sit in the same Higgs background, yet the top is about 340,000 times heavier. Their different interaction strengths with the Higgs account for that difference.

The same interaction that links the electron's chiral components also links the top quark's, but with a much larger Yukawa coupling. It therefore produces a much larger rest energy and mass. Each remains a single elementary particle. Neither loses energy to friction as it moves through the uniform background.

This gives us a way to test the explanation. The same Yukawa interaction couples the fermion to both the steady background and its fluctuations, the Higgs bosons. Once we know the fermion's mass and the background value, we can predict how strongly it should interact with a Higgs boson. A particle's mass and its measured Higgs interaction must agree:

The subscript f labels a fermion species. A loop is a quantum contribution drawn as a closed chain of virtual intermediate particles in a Feynman diagram. Such diagrams represent terms in a scattering or decay calculation.

Experimental papers often use capital H for the detected Higgs boson, while the field formulas above use h for its fluctuation. In channel names, ttH means producing a Higgs with a top–antitop pair; H → bb̄ means decay to a bottom–antibottom pair. A branching fraction is the share of decays in a channel. κ is the measured coupling divided by the Standard Model value, so κ = 1 matches the prediction. The statistical labels are explained in section 11.

The range of couplings remains unexplained. Between the top quark and the electron the couplings differ by a factor of 340,000, and every one of those numbers is an input rather than a result. This is the flavor puzzle, and the Higgs mechanism does not address it. It converts a list of unexplained masses into a list of unexplained couplings.

The predicted relation appears in the plot below. Plot each particle's coupling against its mass on logarithmic axes and the prediction is a line of slope one, with no parameter available to tilt it or shift it. Each equal step on a logarithmic axis represents the same multiplicative change. The dots show predicted values; the caption distinguishes them from measurements. V in the plot labels a vector boson, W or Z.

Even before the Higgs was discovered, colliding W bosons gave theorists a reason to expect something new. Keep the W's measured mass but leave out the Higgs, and the usual scattering calculation eventually violates conservation of probability.

The trouble involves the longitudinal polarization, the extra spin state a massive W possesses. As collision energy rises, its contribution to scattering grows. The W fields' own interactions cancel the fastest-growing terms, but some growth remains. Around 1.2 TeV, the simplest approximation no longer respects the bounds that probabilities must obey. This is where that approximation fails; it tells us that the description is missing important contributions.

The Higgs supplies a contribution with precisely the needed behavior. The colliding W bosons can exchange a Higgs, and this contributes with the opposite sign to the remaining energy growth. With the Standard Model coupling, the growing terms cancel. The result stays within the probability bound at high energy.

This gave the Higgs proposal a test beyond explaining masses: its interaction with W bosons must also repair their high-energy scattering. Other new physics could in principle do that job. Finding the Higgs and measuring its couplings lets us test this particular solution.

Conservation of total probability is called unitarity. The graph plots a₀, the angular-momentum-zero part of the scattering amplitude; an amplitude's squared magnitude enters the probability. For this real, leading-order approximation, the bound is |a₀| ≤ 1/2. Here W_L means a longitudinal W.

Testing the cancellation

At high energies, the longitudinal W states retain the scattering behavior of the Goldstone modes described in section 05. Physicists can use the simpler scalar interactions to calculate that behavior, with corrections that shrink as the energy increases. This connection is called the Goldstone equivalence theorem.

The cancellation also makes small deviations useful. If the Higgs coupling to W and Z differs from the Standard Model value, some energy growth remains. A more precise coupling measurement then tells us how far the present description can work before additional contributions become necessary. High-energy weak-boson scattering tests this directly.

Imagine a world with no Higgs background. The elementary particle masses simplify dramatically, but ordinary atoms cannot survive. Much of the mass associated with the strong force remains.

This is a hypothetical change to the vacuum: section 03 showed why a zero background is unstable with our Higgs potential. First remove the Higgs contributions to particle masses, then consider what the strong interaction would still do.

The mass contributions from the Higgs background vanish, so quarks and charged leptons lose their Higgs-generated masses. The Yukawa interactions with Higgs fluctuations can still exist. With no mass term coupling them, the left-chiral and right-chiral fields propagate independently. Neutrino masses require an extension of the minimal Standard Model, so their behavior depends on that extension.

Before allowing for effects from the strong force, the natural gauge-field basis is W¹, W², W³ and B. The Higgs background no longer selects a massive Z and a massless photon. At this stage the electroweak fields have no Higgs-generated mass.

The strong force adds a further effect. It binds quarks and gluons into massive composite particles even when the quarks have no Higgs-generated mass. It also causes quark–antiquark pairs to develop a collective background, called a condensate. Because the quarks carry weak charges, this background can itself give the W and Z mass.

The mechanism is the one we have already met, at a much smaller scale. Modes associated with the quark condensate supply the extra W and Z polarizations, taking the place of the Higgs Goldstone modes. The weak bosons would therefore retain masses of roughly tens of MeV, thousands of times below their observed masses. The weak force would reach across a nucleus.

These modes are related to pions, light quark–antiquark excitations. The illustrative estimate g f_π/2 ≈ 30 MeV uses the pion decay constant f_π to set the condensate scale. A femtometer (fm) is one quadrillionth of a meter. These are QCD-scale estimates, not a full recalculation of the hypothetical vacuum.

Most of the mass of ordinary matter comes from the energy of quarks and gluons inside protons and neutrons. Higgs-generated quark masses contribute a much smaller share. Yet the electron mass sets atomic length scales, and the up–down quark mass difference helps make the neutron heavier than the proton. Removing these small masses would destroy the conditions that let ordinary atoms exist.

Finding a new particle near 125 GeV was the first step. To identify it as the Higgs, experiments had to test its spin and its interactions with other particles. The measured pattern agrees with the Standard Model within the uncertainties so far; several rarer interactions remain difficult to measure.

Mass and width from the 2025 Particle Data Group (PDG) review; the 246 GeV value is fixed by the measured muon lifetime, not by a Higgs measurement. Decay width is inversely related to lifetime. Sources and dates are in the footer.

How the LHC makes and sees them

A Higgs boson decays long before it could reach a detector. Experiments reconstruct it from the particles left behind. The most common decay, into a bottom quark and antiquark, is difficult to distinguish from the many ordinary collisions producing the same particles. Much rarer decays into two photons or four charged leptons give cleaner evidence: an excess of events whose reconstructed mass clusters near 125 GeV.

At 13.6 TeV the predicted Higgs production cross section is roughly 59 picobarns. A cross section measures how likely a process is; multiplying it by the accumulated collision exposure gives the expected count. About 90% comes from gluon fusion, mainly through a top-quark loop. The estimated Run 2 and Run 3 total is roughly 25 million Higgs bosons produced at each of ATLAS and CMS, only a fraction of which can be identified reliably.

In the channel labels, γ is a photon, ℓ is an electron or muon, and ν is a neutrino. The asterisk on W* or Z* means the intermediate boson is off-shell: its decay products have a combined invariant mass different from the usual W or Z mass. A mass peak is an excess of reconstructed events near one mass, above the smooth background from other processes.

Run 3 ended on 27 June 2026 and Long Shutdown 3 began on 29 June. The LHC delivered close to 550 fb⁻¹ across its three runs, nearly twice the original design target. Beams return in June 2030 as the High-Luminosity LHC, aiming for more than 3 ab⁻¹ (one inverse attobarn is 1,000 inverse femtobarns) — about six times everything collected so far. Analyses of the full Run 2 + Run 3 dataset will keep appearing throughout the shutdown.

The main steps from gauge theory to Higgs measurements. These dates and the glossary that follows provide reference material; section 14 returns to the open questions.

Thirty-one terms used in Higgs research.

The discovery and measured couplings support the Higgs mechanism. Open questions concern its self-interactions, the origin of its parameters, and whether additional fields contribute.

During the next four years, analyses of the remaining Run 2 + Run 3 data can tighten the self-coupling bound, the muon coupling and the charm limit without new collisions. Further ATLAS–CMS combinations should improve these constraints.