Higgs couplings & properties

The Higgs boson is the one particle in the Standard Model whose mass the theory refuses to predict — $m_H=\sqrt{2\lambda}\,v$ depends on a self-coupling $\lambda$ that must be measured, not derived. Yet the very same theory fixes, with no freedom at all, every one of its interactions. The rule is a single line: the Higgs couples to each particle in proportion to that particle's mass. For a fermion the vertex carries the factor $-i\,m_f/v$ , equivalently $-i\,(m_f/2m_W)g_W$ ; for the weak bosons the couplings are $g_W m_W$ for $W^+W^-$ and $(g_W/\cos\theta_W)m_Z$ for $ZZ$ . This is not an accident to be memorised but a consequence to be derived: the field that gives a particle its mass is the very field the Higgs is a ripple of, so mass and coupling come from one term in the Lagrangian. That single principle organises the entire phenomenology. Because each decay amplitude is proportional to the coupling, fermionic partial widths scale as $m_f^2$ , so a $125$ GeV Higgs decays most often to the heaviest pair it can afford — bottom quarks, $\mathrm{BR}(H\to b\bar b)\approx 57.8\%$ — even though $\tau^+\tau^-$ , $c\bar c$ , $WW^*$ and $ZZ^*$ are all open. The would-be strongest coupling of all, to the top quark, cannot appear as a decay at all: two tops weigh far more than one Higgs. Instead the top rules production and hides inside loops, driving the flagship loop-induced decays $H\to gg$ and $H\to\gamma\gamma$ that occur despite the final particles being massless. This article assembles the whole coupling structure from the Lagrangian and shows how it dictates what the Higgs does.

Key takeaways

The Higgs couples to mass: the $Hf\bar f$ vertex factor is $-i\,m_f/v = -i\,(m_f/2m_W)g_W$ , and the gauge couplings are $g_W m_W$ and $(g_W/\cos\theta_W)m_Z$ — all set by mass.
The reason is structural: after symmetry breaking the mass term and the Higgs coupling come from the same term, so $m_f=y_f v/\sqrt2$ and the coupling is $y_f/\sqrt2=m_f/v$ .
Fermionic partial widths scale as $\Gamma\propto N_c\,m_f^2\,m_H$ , so for $m_H=125$ GeV the dominant decay is $H\to b\bar b$ at $\approx 57.8\%$ , with $\Gamma(H\to b\bar b)\approx 2.4$ MeV.
The top has the strongest coupling ( $y_t\approx1$ ) but $2m_t>m_H$ forbids $H\to t\bar t$ ; the top instead dominates Higgs production and the loop decays.
Loop-induced $H\to gg$ ( $8.6\%$ ) and $H\to\gamma\gamma$ ( $0.2\%$ ) proceed through heavy virtual top and $W$ loops even though gluons and photons are massless.
Key idea: there is one organising principle behind every Higgs interaction — the coupling strength is proportional to the mass of the particle it couples to, so the Higgs talks loudest to the heaviest things it can reach.

Where this lives today

Every measured property of the $125$ GeV boson is now a test of the coupling-mass rule, and so far the rule holds. At the LHC the Higgs is made overwhelmingly through the top: gluon fusion, a top-quark loop turning two gluons into a Higgs, accounts for about $87\%$ of production, with vector-boson fusion ( $\sim7\%$ ), associated $WH/ZH$ production ( $\sim4\%$ ), and $t\bar tH$ ( $\sim1\%$ ) making up the rest — a production table dictated, like the decays, by which particles couple most strongly. ATLAS and CMS report an overall signal strength $\mu=\sigma/\sigma_{\rm SM}\approx1.0$ within a few percent, and the coupling modifiers $\kappa_t,\kappa_W,\kappa_Z,\kappa_b,\kappa_\tau$ all sit on unity within uncertainties. The observation of $H\to b\bar b$ and $t\bar tH$ in 2018 confirmed the dominant decay and the top coupling directly, and the $\sim3\sigma$ evidence for $H\to\mu\mu$ in 2020 reached down to a second-generation coupling — the first sign the Higgs talks to the lighter families as the rule demands. The open frontier is the couplings the LHC cannot yet pin. The Higgs self-coupling $\lambda$ — the one interaction not proportional to another particle's mass — is being chased through di-Higgs production, still statistics-limited and a headline goal of the High-Luminosity LHC. The first-generation couplings ( $H\to ee$ , and the Higgs to up and down quarks) are far too small to see and would take a future machine. Proposed "Higgs factories" — the FCC-ee, ILC, CEPC and CLIC — would produce millions of Higgs bosons in the clean $e^+e^-$ environment to measure the couplings to sub-percent precision, where any deviation from the mass-proportional line would be an unambiguous signal of physics beyond the Standard Model. Whether the Higgs couples to dark matter, and whether its potential is exactly the Standard-Model shape, are questions that ride on those measurements.

The physical picture

Picture the Higgs field as a medium filling all of space with a constant value $v=246$ GeV. A particle acquires mass by dragging through that medium, and the Higgs boson is a local ripple in it. The two facts are inseparable: the strength with which a particle drags — its mass — is the strength with which it couples to a ripple. So every Higgs vertex is a mass in disguise. A fermion attaches with amplitude $-i\,m_f/v$ ; a $W$ or $Z$ attaches with amplitude proportional to $m_V^2/v$ . Massless particles — the photon and gluon — have no direct handle at all, which is why their couplings are absent at tree level and must be built from loops of heavy virtual particles. The first figure is the whole vertex zoo with the numbers that set it.

Figure

The Higgs vertices, all fixed by mass. Fermions attach with factor −i m_f/v; the W and Z attach with i g_W m_W g^μν = i 2m_W²/v g^μν and i 2m_Z²/v g^μν. Photons and gluons have no tree coupling — H → γγ and H → gg run through loops of heavy top quarks and W bosons. With v = 246 GeV the top Yukawa is y_t = √2 m_t/v ≈ 0.99, the largest coupling of all

The consequence of "coupling proportional to mass" is a prediction so clean it can be drawn as a single straight line. Define each particle's reduced coupling — for a fermion, $g_{Hff}=m_f/v$ ; for a boson, $\sqrt{g_{HVV}/2v}=m_V/v$ — and plot it against the particle's mass. The Standard Model says every point must fall on the line $\kappa=m/v$ , a diagonal of slope one that runs unbroken from the muon at $0.1$ GeV to the top quark at $173$ GeV, more than three decades in mass. This is the most direct test of the whole idea, and when ATLAS and CMS measured the couplings a decade after discovery, the points landed on the line. The second figure is that line, with the measured particles on it.

Figure

The flagship test: reduced Higgs coupling κ = m/v against particle mass, on log axes. The Standard Model predicts a single straight line of slope one, κ = m/(246 GeV), spanning the muon, tau, bottom, W, Z, and top. Every measured particle sits on it — the coupling really is proportional to mass, over more than three decades from 0.1 GeV to 173 GeV

The core equations

The mass itself is the one thing the theory does not fix. It is set by the quartic self-coupling $\lambda$ of the Higgs potential and the vacuum value $v$ , both of which enter as inputs — measuring $m_H$ measures $\lambda$ .

m_H=\sqrt{2\lambda}\,v,\qquad \lambda=\frac{m_H^2}{2v^2}\approx0.13 \qquad (1)

where: $m_H=125$ GeV — the measured Higgs mass; $v=246$ GeV — the Higgs vacuum expectation value, fixed by $v=(\sqrt2\,G_F)^{-1/2}$ ; $\lambda$ — the quartic self-coupling. Reading it: unlike every coupling below, $m_H$ is not predicted — it is the single free parameter the Higgs sector adds, and it fixes $\lambda\approx0.13$ . The central rule of the whole subject is the fermion vertex. It is proportional to the fermion mass, and it can be written three equivalent ways that each expose a different piece of physics.

-i\,\frac{m_f}{v}=-i\,\frac{m_f}{2m_W}\,g_W=-i\,\frac{y_f}{\sqrt2},\qquad m_f=\frac{y_f v}{\sqrt2} \qquad (2)

where: $m_f$ — fermion mass; $v=246$ GeV; $g_W=2m_W/v\approx0.65$ — the weak coupling; $y_f$ — the Yukawa coupling. Reading it: the same $y_f$ that sets the mass $m_f=y_f v/\sqrt2$ sets the Higgs coupling $y_f/\sqrt2=m_f/v$ — mass and coupling are two faces of one number, which is why the coupling is proportional to mass. The gauge bosons couple through their masses squared, not linearly. The two forms below make explicit that the coupling strength $g_{HVV}$ is $2m_V^2/v$ .

g_{HWW}=g_W m_W=\frac{2m_W^2}{v},\qquad g_{HZZ}=\frac{g_W}{\cos\theta_W}\,m_Z=\frac{2m_Z^2}{v} \qquad (3)

where: $m_W=80.4$ GeV, $m_Z=91.19$ GeV; $\theta_W$ — the weak mixing angle, $g_Z=g_W/\cos\theta_W$ . Reading it: bosons couple as $m_V^2$ while fermions couple as $m_f$ — the difference is why, on the flagship plot, one takes the square root of the boson coupling to put $W$ and $Z$ on the same line as the fermions. Squaring the fermion vertex and folding it into two-body phase space gives the fermionic partial width — the master formula that turns "coupling proportional to mass" into "width proportional to mass squared".

\Gamma(H\to f\bar f)=\frac{N_c\,m_f^2\,m_H}{8\pi v^2}\left(1-\frac{4m_f^2}{m_H^2}\right)^{3/2} \qquad (4)

where: $N_c=3$ for quarks, $1$ for leptons; the factor $(1-4m_f^2/m_H^2)^{3/2}=\beta^3$ is the phase-space velocity factor, cubed because a scalar decays into a p-wave fermion pair. Reading it: away from threshold $\beta\to1$ and the width is simply $\propto N_c m_f^2 m_H$ , so the ratio of any two fermionic channels is $N_c m_f^2$ against $N_{c}' m_{f'}^2$ — pure mass-squared scaling. The total width is the sum over every open channel, and each channel's share is its branching ratio. The Higgs is extraordinarily narrow, which is itself a consequence of its couplings all being small.

\Gamma_H=\sum_i\Gamma_i\approx4.1\ \mathrm{MeV},\qquad B_i=\frac{\Gamma_i}{\Gamma_H},\qquad \tau_H=\frac{\hbar}{\Gamma_H}\approx1.6\times10^{-22}\ \mathrm{s} \qquad (5)

where: the sum runs over $b\bar b$ , $WW^*$ , $gg$ , $\tau^+\tau^-$ , $c\bar c$ , $ZZ^*$ , $\gamma\gamma$ and smaller modes; $\Gamma_H\approx4.1$ MeV is minuscule for a $125$ GeV particle. Reading it: the tiny width follows directly from the couplings — no channel is large, so the Higgs lives a hundred times longer than the $Z$ despite being heavier. Finally, experiment reports coupling modifiers $\kappa_i$ : the ratio of the measured coupling to its Standard-Model value. The "coupling proportional to mass" hypothesis is the statement that every $\kappa_i=1$ , i.e. the reduced coupling equals $m/v$ .

\kappa_i=\frac{g_i}{g_i^{\rm SM}},\qquad g_{Hff}=\frac{m_f}{v},\qquad \sqrt{\frac{g_{HVV}}{2v}}=\frac{m_V}{v} \qquad (6)

where: $\kappa_i=1$ in the Standard Model; deviations would signal new physics. Reading it: plotting $g_{Hff}$ (fermions) and $\sqrt{g_{HVV}/2v}$ (bosons) against mass must give the single line $\kappa=m/v$ of the second figure — the cleanest possible display of the article's thesis.

Derivation AWhy the coupling is proportional to mass

The goal is to prove the article's central claim — that the $Hf\bar f$ coupling equals $m_f/v$ — directly from the Lagrangian, showing it is not a fitted rule but a forced consequence of how fermions get mass. We start from the Yukawa term that couples the fermion field to the Higgs doublet, substitute the symmetry-broken form of the doublet, and read off the mass term and the interaction term side by side, so that their shared origin becomes visible in a single line of algebra.

Step 1. Write the gauge-invariant Yukawa coupling of a fermion to the Higgs doublet $\phi$ : $\mathcal L_Y=-y_f\,\bar\psi_L\,\phi\,\psi_R+\text{h.c.}$ Before symmetry breaking there is no fermion mass term allowed — a bare $m_f\bar\psi_L\psi_R$ would violate the gauge symmetry — so all of the fermion's mass must come from this one interaction. Why: this is the crux — mass is forbidden by the symmetry and can only be manufactured by coupling to the Higgs, which ties mass to coupling before any number is computed.

Step 2. In unitary gauge the doublet takes the value $\phi=\tfrac{1}{\sqrt2}\binom{0}{\,v+H\,}$ , where $v$ is the constant vacuum value and $H$ is the physical Higgs field. Substitute: $\mathcal L_Y=-\tfrac{y_f}{\sqrt2}(v+H)\,\bar\psi\psi=-\tfrac{y_f v}{\sqrt2}\,\bar\psi\psi-\tfrac{y_f}{\sqrt2}\,H\,\bar\psi\psi$ . Why: the single term splits into two — one with $v$ , one with $H$ — and they carry the identical coefficient $y_f/\sqrt2$ , which is the whole secret.

Step 3. Identify the pieces. The first term is a Dirac mass, so $m_f=y_f v/\sqrt2$ . The second is the Higgs interaction, with coupling $y_f/\sqrt2$ . Eliminate $y_f$ using the first to write the coupling as $y_f/\sqrt2=m_f/v$ , giving the vertex Feynman rule $-i\,m_f/v$ . Why: because both terms share $y_f/\sqrt2$ , dividing one by the other cancels the unknown Yukawa entirely and leaves a coupling fixed purely by the already-measured mass — a parameter-free prediction.

Step 4. Recast in weak-coupling language using $v=2m_W/g_W$ : $m_f/v=(m_f/2m_W)g_W$ , the form in which the vertex appears in electroweak calculations. Check the extreme cases: the top ( $m_t=173$ GeV) gives $y_t=\sqrt2\,m_t/v=0.99$ , a coupling of order one; the electron gives $y_e\approx3\times10^{-6}$ , essentially decoupled. Why: the same formula that makes the top nearly maximally coupled makes the electron almost invisible to the Higgs — the enormous dynamic range of fermion masses is exactly the range of Higgs couplings, which is the content of the flagship line.

Go deeper · Derivation B · 1 minThe fermionic partial width from the vertex

Derivation BThe fermionic partial width from the vertex

The objective is to derive the master formula (4), $\Gamma(H\to f\bar f)=\tfrac{N_c m_f^2 m_H}{8\pi v^2}\beta^3$ , by squaring the scalar vertex, summing over final-state spins with the Dirac completeness relations, and integrating over two-body phase space. The calculation is short because the vertex has no gamma matrix, but it hides an important feature — the cubed velocity factor $\beta^3$ that marks a scalar, as opposed to a vector, parent.

Step 1. The amplitude for $H\to f(p_1)\bar f(p_2)$ is $\mathcal M=-\tfrac{m_f}{v}\,\bar u(p_1)v(p_2)$ , a pure scalar bilinear with no Lorentz index because the Higgs is spin-0. Sum over final spins: $\sum|\mathcal M|^2=\tfrac{m_f^2}{v^2}\,\mathrm{Tr}\big[(\slashed p_1+m_f)(\slashed p_2-m_f)\big]$ . Why: with no gamma matrix between the spinors, the spin sum collapses to a single trace of two propagator numerators — the simplest matrix element in the Standard Model.

Step 2. Evaluate the trace: $\mathrm{Tr}[(\slashed p_1+m_f)(\slashed p_2-m_f)]=4\,p_1\!\cdot p_2-4m_f^2$ . Use $m_H^2=(p_1+p_2)^2=2m_f^2+2p_1\!\cdot p_2$ to write $p_1\!\cdot p_2=\tfrac12(m_H^2-2m_f^2)$ , so $\sum|\mathcal M|^2=\tfrac{m_f^2}{v^2}\,(2m_H^2-8m_f^2)=\tfrac{2m_f^2 m_H^2}{v^2}\big(1-\tfrac{4m_f^2}{m_H^2}\big)$ . Why: the combination $1-4m_f^2/m_H^2=\beta^2$ appears here from the trace itself — one power of the velocity factor is already built into the squared amplitude.

Step 3. Insert into the two-body decay rate $\Gamma=\dfrac{|\vec p\,|}{8\pi m_H^2}\sum|\mathcal M|^2$ with final momentum $|\vec p\,|=\tfrac{m_H}{2}\beta$ , and multiply by the colour multiplicity $N_c$ : $\Gamma=N_c\,\dfrac{(m_H/2)\beta}{8\pi m_H^2}\cdot\dfrac{2m_f^2 m_H^2}{v^2}\beta^2=\dfrac{N_c m_f^2 m_H}{8\pi v^2}\,\beta^3$ . Why: one power of $\beta$ from phase space times $\beta^2$ from the amplitude gives $\beta^3$ — the signature of a scalar decaying to fermions, distinct from the $\beta$ of a vector, and in principle a way to confirm the Higgs is spin-0.

Step 4. Take the realistic limit $m_f\ll m_H$ (true even for $b$ : $2m_b/m_H\approx0.07$ ), so $\beta^3\to1$ and $\Gamma\to N_c m_f^2 m_H/8\pi v^2$ . For $b\bar b$ with the running mass $m_b(m_H)\approx2.8$ GeV, $N_c=3$ : $\Gamma=3(2.8)^2(125)/[8\pi(246)^2]=1.9$ MeV, rising to $\approx2.4$ MeV after QCD corrections. Why: the width depends on $m_f^2$ , so it is dominated by the heaviest accessible fermion and is why $b\bar b$ , not $\tau^+\tau^-$ or $c\bar c$ , is the leading decay of the Higgs.

Go deeper · Derivation C · 1 minThe gauge-boson couplings from the covariant kinetic term

Derivation CThe gauge-boson couplings from the covariant kinetic term

The aim is to show that the $HWW$ and $HZZ$ couplings of equation (3), $g_{HVV}=2m_V^2/v$ , arise from the same term that gives the $W$ and $Z$ their masses — the covariant-derivative kinetic term of the Higgs doublet — so that the "coupling proportional to mass" pattern extends to the gauge sector, but as mass squared. We expand the kinetic term about the vacuum and match the quadratic and cubic pieces.

Step 1. The Higgs kinetic term is $|D_\mu\phi|^2$ with $D_\mu$ the electroweak covariant derivative. For the neutral component this contains $\tfrac14 g_W^2(v+H)^2\,W^+_\mu W^{-\mu}$ plus the analogous $Z$ piece. Expand $(v+H)^2=v^2+2vH+H^2$ . Why: the boson fields couple to $\phi$ quadratically through the covariant derivative, so the single scalar factor $(v+H)^2$ generates mass, single-Higgs, and double-Higgs couplings all at once.

Step 2. The $v^2$ term is the $W$ mass: $\tfrac14 g_W^2 v^2\,W^+W^-\equiv m_W^2\,W^+W^-$ , so $m_W=\tfrac12 g_W v$ . Why: mass comes from the constant background $v$ , exactly as the fermion mass did — the same vacuum value sources every mass in the theory.

Step 3. The $2vH$ term is the $HWW$ coupling: $\tfrac14 g_W^2\cdot2vH\,W^+W^-=\tfrac12 g_W^2 v\,H\,W^+W^-$ . Substituting $v=2m_W/g_W$ gives the coefficient $\tfrac12 g_W^2 v=g_W m_W=2m_W^2/v$ , the vertex factor $i\,(2m_W^2/v)\,g^{\mu\nu}$ . The identical computation with $g_Z=g_W/\cos\theta_W$ gives $g_{HZZ}=2m_Z^2/v$ . Why: because the mass came from $v^2$ and the coupling from $2vH$ in the same expansion, the coupling is locked to the mass — here as $m_V^2$ , since the boson enters the kinetic term quadratically.

Step 4. Contrast with fermions. Fermions sit in a Yukawa term linear in $\phi$ , so their coupling scales as $m_f$ ; bosons sit in a term quadratic in $\phi$ , so theirs scales as $m_V^2$ . This is why the flagship plot puts $g_{Hff}=m_f/v$ but $\sqrt{g_{HVV}/2v}=m_V/v$ — the square root undoes the boson's quadratic origin so all particles share one line. Why: recognising the different power of mass for bosons versus fermions is essential to reading the coupling-mass plot correctly, and it is a direct fingerprint of where each particle sits in the Lagrangian.

Limiting cases

Case	Result	Meaning
Top quark	$y_t=\sqrt2\,m_t/v\approx1$ , but $2m_t>m_H$	strongest coupling of all, yet $H\to t\bar t$ is forbidden — the top rules production and loops instead
Bottom quark	$\Gamma\propto3\,m_b^2$ , use running $m_b(m_H)\approx2.8$ GeV	the largest open channel, $\mathrm{BR}\approx57.8\%$ — heaviest kinematically accessible fermion
Light fermion (electron)	$y_e\approx3\times10^{-6}$ , $\mathrm{BR}(H\to ee)\sim5\times10^{-9}$	negligible — the Higgs essentially ignores the light generations
Massless particle ( $\gamma$ , $g$ )	no tree coupling; loop only	$H\to\gamma\gamma$ , $H\to gg$ via heavy virtual top and $W$ — massless finals, massive messengers
Off-shell $WW^*$	one $W$ virtual, $q^2<m_W^2$	still $\approx21.6\%$ : the large $g_W m_W$ coupling beats the propagator suppression
Higgs self-coupling	$\lambda=m_H^2/2v^2\approx0.13$ , vertex $HHH\propto3m_H^2/v$	the only coupling not proportional to another particle's mass — a target for the HL-LHC

Common trapH → WW is forbidden because m_H < 2m_W, and loop decays to massless particles are impossible

Both halves of this are wrong, and they are the classic traps of Higgs phenomenology. First, $H\to WW$ is not forbidden by $m_H<2m_W$ : the decay proceeds with one $W$ off its mass shell, written $H\to WW^*$ , where the virtual $W$ carries $q^2<m_W^2$ and immediately decays to a fermion pair. The propagator does cost a suppression, but the $HWW$ coupling $g_W m_W\approx53$ GeV is so large that $WW^*$ still claims the second biggest branching ratio, $\approx21.6\%$ — larger than $\tau\tau$ , $c\bar c$ , or $\gamma\gamma$ . Kinematic thresholds bound on-shell two-body decays, not decays through virtual intermediaries. Second, $H\to\gamma\gamma$ and $H\to gg$ are not blocked by the photon and gluon being massless. There is indeed no tree-level $H\gamma\gamma$ or $Hgg$ vertex — the Higgs has no direct coupling to a massless particle — but the decay runs through a loop of heavy charged or coloured particles (the top quark and, for $\gamma\gamma$ , the $W$ boson) that couple both to the Higgs and to the photon or gluon. Because those loop particles are heavy and strongly Higgs-coupled, the loops are large enough to make $H\to gg$ an $8.6\%$ channel and $H\to\gamma\gamma$ the golden $0.2\%$ discovery mode. The rule "coupling proportional to mass" applies to direct couplings; loops route around it.

Worked examples

1 · The dominant decay width $H\to b\bar b$ (the flagship number)

Compute $\Gamma(H\to b\bar b)$ for $m_H=125$ GeV using equation (4), decide which $b$ -quark mass to use, and compare with the value implied by the measured branching ratio.

Step 1. The velocity factor is negligible: $\beta^2=1-4m_b^2/m_H^2$ . Even with $m_b\approx3$ GeV, $4m_b^2/m_H^2=4(9)/(125^2)=0.0023$ , so $\beta^3\approx0.9965\approx1$ . The width is essentially $\Gamma=N_c m_b^2 m_H/8\pi v^2$ with $N_c=3$ .

Step 2. Choose the mass. Using the pole mass $m_b=4.18$ GeV gives $\Gamma=3(4.18)^2(125)/[8\pi(246)^2]=\mathbf{4.3\ MeV}$ — too large. The correct choice is the running $\overline{\text{MS}}$ mass evaluated at the Higgs scale, $m_b(m_H)\approx2.8$ GeV, which resums the large QCD logarithms. This gives $\Gamma=3(2.8)^2(125)/[8\pi(246)^2]=\mathbf{1.9\ MeV}$ .

Step 3. Add the residual QCD correction. Gluon radiation multiplies the width by $[1+5.67\,\alpha_s(m_H)/\pi+\dots]\approx1.2$ with $\alpha_s(m_H)\approx0.11$ , raising it to $\Gamma(H\to b\bar b)\approx\mathbf{2.3\ MeV}$ .

Step 4. Cross-check against experiment. The total width is $\Gamma_H\approx4.1$ MeV and $\mathrm{BR}(H\to b\bar b)=57.8\%$ , so the measured partial width is $0.578\times4.1=\mathbf{2.37\ MeV}$ — in agreement. The leading Higgs decay is $H\to b\bar b$ at $\approx2.4$ MeV, and getting it right demands the running $b$ mass, not the pole mass — a factor of two on the width.

2 · Testing "coupling proportional to mass" (the straight line)

Evaluate the reduced coupling $\kappa=m/v$ for the top, $W$ , $Z$ , $b$ , $\tau$ , and $\mu$ , and show they span a single line of slope one.

Step 1. Use $v=246$ GeV throughout. For the unambiguous leptons and bosons, take pole masses. Top: $g_{Htt}=173/246=\mathbf{0.70}$ . $Z$ : $\sqrt{g_{HZZ}/2v}=m_Z/v=91.19/246=\mathbf{0.37}$ . $W$ : $80.4/246=\mathbf{0.33}$ .

Step 2. The lighter fermions: $\tau$ : $1.777/246=\mathbf{0.0072}$ . $\mu$ : $0.1057/246=\mathbf{0.00043}$ . The bottom needs the running mass for the coupling: $m_b(m_H)\approx2.8$ gives $\kappa_b\approx0.011$ (the pole mass $4.18$ would give $0.017$ ).

Step 3. Notice the range. From $\mu$ at $4.3\times10^{-4}$ to $t$ at $0.70$ the coupling spans a factor of $1600$ , exactly tracking the mass ratio $m_t/m_\mu=173/0.106=1630$ . On log-log axes the six points fall on the line $\kappa=m/v$ of slope one.

Step 4. Interpret. The Standard Model predicts every coupling modifier $\kappa_i=1$ ; the measured ATLAS/CMS values scatter around $1$ within uncertainties, and the fitted line reproduces slope one to a few percent. Over three decades in mass the Higgs coupling grows in lockstep with the mass — the single most direct confirmation that the Higgs is the origin of mass.

3 · Why $H\to\gamma\gamma$ survives at all: the loop competition

The diphoton amplitude is dominated by a $W$ loop and a top loop that interfere. Estimate the interference and explain why the channel is both rare and precious.

Step 1. In the heavy-particle limit the two loop amplitudes are, in standard units, $A_W\approx-8.3$ (the $W$ boson) and $A_t=N_c Q_t^2\times\tfrac43=3\cdot\tfrac49\cdot\tfrac43\approx+1.8$ (the top, with colour $N_c=3$ and charge $Q_t=\tfrac23$ ). They carry opposite sign — the fermion and boson loops interfere destructively.

Step 2. Add them: $A_W+A_t=-8.3+1.8=\mathbf{-6.5}$ . The rate goes as the square, so the top reduces the $W$ -only rate by $(6.5/8.3)^2=\mathbf{0.62}$ — the top loop cuts the diphoton rate by about $38\%$ .

Step 3. Put it in context. The diphoton branching ratio is only $\approx0.2\%$ because it is a loop and a factor $\alpha^2$ down, but the final state — two high-energy photons whose invariant mass reconstructs to a sharp peak at $125$ GeV — is measured with exquisite resolution against a smooth background.

Step 4. This is why $\gamma\gamma$ was a discovery channel despite its rarity. $H\to\gamma\gamma$ is a loop decay of a scalar into massless photons that would be "impossible" by the naive coupling rule, yet its clean two-photon peak was one of the two channels that revealed the Higgs in 2012 — and the $W$ –top interference makes its rate a sensitive probe of both couplings at once.

Physics of the portrait

The rule that the Higgs couples to mass, and the massive scalar that carries it, were written down by one man in a two-page paper that a journal first rejected. He spent the rest of his life somewhat bemused that a particle bore his name.

Origins · 1 minPeter Higgs and how this was discovered

Peter Higgs

1929–2024 · the 1964 prediction of a massive scalar boson coupled to mass

Peter Higgs was a British theoretical physicist at the University of Edinburgh who, in the summer of 1964, published two short papers on how a gauge field could acquire mass without spoiling the underlying symmetry. Several groups reached the mechanism nearly simultaneously — Robert Brout and François Englert in Brussels, and Gerald Guralnik, Carl Hagen and Tom Kibble in London — but Higgs's second paper contained something the others did not spell out: the mechanism necessarily leaves behind a massive scalar boson, a physical particle that could in principle be produced and detected. His manuscript was initially rejected by Physics Letters; he added the crucial sentence pointing to the new boson and resubmitted to Physical Review Letters. The deep structure — that the field giving mass to the $W$ , $Z$ , and every fermion must couple to each of them in proportion to that mass — followed once Weinberg and Salam embedded the mechanism in the electroweak theory in 1967. For decades the boson was a theorist's conjecture, its mass unknown and its couplings a target rather than a measurement. Higgs, a modest and private man who disliked the fame, lived to see the particle found. When ATLAS and CMS announced a $125$ GeV boson on 4 July 2012, he wept; the following year he shared the Nobel Prize with Englert. He died in April 2024, by which time the coupling-mass line his mechanism predicted had been measured and confirmed across three decades of mass.

History

1964

— Higgs, and independently Brout–Englert and Guralnik–Hagen–Kibble, publish the mass-generating mechanism; Higgs alone notes a massive scalar boson

1967

— Weinberg and Salam embed the mechanism in electroweak theory, fixing every Higgs coupling proportional to mass

1976

— Ellis, Gaillard and Nanopoulos catalogue the boson's expected production and decays in "A Phenomenological Profile of the Higgs Boson"

2012

— ATLAS and CMS discover a 125 GeV boson in the γγ and ZZ → 4ℓ channels

2013

— Higgs and Englert share the Nobel Prize in Physics

Going further

Texts. Thomson, Modern Particle Physics, §17.6 (properties of the Higgs boson), Eqs. (17.49)–(17.52), Table 17.1 · Halzen & Martin, Quarks and Leptons, §15.6 (the Higgs boson) · Griffiths, Introduction to Elementary Particles, 2nd ed., §11.7–11.9 · Peskin & Schroeder, An Introduction to Quantum Field Theory, §20.1–20.2 (spontaneous breaking and Higgs couplings) · Schwartz, Quantum Field Theory and the Standard Model, Ch. 29 (electroweak symmetry breaking).

Provenance. The vertex $-i\,m_f/v$ and the mass relation $m_f=y_f v/\sqrt2$ are owned by aq-pp-higgs-mechanism and aq-pp-spontaneous-symmetry-breaking; the two-body phase-space measure by aq-pp-particle-decay-rates; the Dirac spinor completeness relations by aq-pp-dirac-solutions. This article synthesises them into the coupling structure and decay pattern. A verifier should confirm: (1) $\lambda=m_H^2/2v^2=0.13$ and $v=246$ GeV; (2) the vertex $-i\,m_f/v=-i(m_f/2m_W)g_W$ from the Yukawa term with $y_t=\sqrt2\,m_t/v=0.99$ ; (3) $g_{HWW}=g_W m_W=2m_W^2/v=52.6$ GeV; (4) $\Gamma(H\to f\bar f)=N_c m_f^2 m_H\beta^3/8\pi v^2$ and the trace evaluation; (5) $\Gamma(H\to b\bar b)\approx1.9$ MeV with running $m_b=2.8$ GeV, $\approx2.4$ MeV with QCD, matching $\mathrm{BR}=57.8\%$ of $\Gamma_H=4.1$ MeV; (6) branching ratios $b\bar b\,57.8$ , $WW^*\,21.6$ , $gg\,8.6$ , $\tau\tau\,6.3$ , $c\bar c\,2.9$ , $ZZ^*\,2.6$ , $\gamma\gamma\,0.2$ percent; (7) the reduced couplings $m/v$ on one line.

Prerequisites. The Higgs Mechanism · Spontaneous Symmetry Breaking · Dirac Spinor Solutions · Particle Decay Rates and Phase Space. Leads to. The Higgs Discovery · Electroweak Radiative Corrections · Open Questions of the Standard Model.

Ask the tutor: Want the diphoton loop amplitude with the $W$ –top interference worked out line by line, or how gluon-fusion production is the same top loop running in reverse?

Why does the Higgs couple to mass rather than to charge or something else?

Because mass and Higgs coupling come from the same term in the Lagrangian. A fermion is forbidden by the gauge symmetry from having a bare mass, so its mass can only be generated by coupling to the Higgs field. That single Yukawa term, once the Higgs field is split into its constant vacuum value $v$ and its fluctuation $H$ , produces two pieces with the identical coefficient: a mass $m_f=y_f v/\sqrt2$ and an interaction $H\bar\psi\psi$ with coupling $y_f/\sqrt2=m_f/v$ . Divide one by the other and the unknown Yukawa cancels, leaving a coupling equal to $m_f/v$ — fixed entirely by the mass. So it is not that the Higgs "chooses" to couple to mass; mass simply is the strength of the Higgs coupling, expressed in energy units.

If the Higgs couples to mass, how can it decay to massless gluons and photons?

Through loops. There is no direct $H\gamma\gamma$ or $Hgg$ vertex — the Higgs genuinely has no tree-level coupling to a massless particle. But the Higgs can couple to a heavy particle that is itself electrically charged or colour-charged, and that particle can emit the photons or gluons. For $H\to gg$ the loop is a top quark (the only strongly-coupled, heavy, coloured particle); for $H\to\gamma\gamma$ it is a $W$ boson loop and a top loop, which interfere. Because the loop particles are heavy, their Higgs coupling is large, and the loops are big enough that $H\to gg$ reaches $8.6\%$ and $H\to\gamma\gamma$ reaches $0.2\%$ . The naive rule "coupling proportional to mass" is a statement about direct couplings; quantum loops route the Higgs to massless final states indirectly.

Why is H → b b̄ the biggest decay if the top has the strongest coupling?

The top does have the strongest coupling — its Yukawa is nearly one — but a $125$ GeV Higgs cannot decay to two tops, because two top quarks weigh $346$ GeV, almost three times the Higgs mass. The channel is kinematically shut. So the Higgs decays to the heaviest pair it can afford, which is bottom quarks at $2m_b\approx8$ GeV, well below $125$ GeV. Since the width scales as the mass squared times the colour factor, $\Gamma\propto3m_b^2$ , and the bottom is the heaviest open fermion, $H\to b\bar b$ wins with a $57.8\%$ branching ratio. The top is not absent from Higgs physics, though — it dominates Higgs production (via gluon fusion) and hides inside the loops for $H\to gg$ and $H\to\gamma\gamma$ .

Why does H → WW* dominate over H → ττ even though m_H < 2m_W?

Because the coupling wins over the kinematic penalty. The Higgs is lighter than two on-shell $W$ bosons, so one $W$ must be virtual (off mass shell), written $H\to WW^*$ ; the virtual $W$ immediately decays to a fermion pair. Going off-shell does cost a propagator suppression. But the $HWW$ coupling is enormous — $g_W m_W\approx53$ GeV, because bosons couple as mass squared — and that large coupling more than compensates, leaving $WW^*$ with the second-largest branching ratio, $\approx21.6\%$ , ahead of $\tau^+\tau^-$ at $6.3\%$ . The lesson is that a kinematic threshold bounds only genuine on-shell two-body decays; a decay through a virtual intermediary is always available, and whether it matters depends on the size of the coupling.

Why must you use the running b-quark mass, not the pole mass, in the width?

Because the decay $H\to b\bar b$ happens at an energy scale of $125$ GeV, and the effective mass that enters the coupling runs with that scale. The QCD interactions that dress the bottom quark shrink its effective (running $\overline{\text{MS}}$ ) mass from the low-energy pole value of $\approx4.18$ GeV down to $m_b(m_H)\approx2.8$ GeV at the Higgs scale. Since the width scales as $m_b^2$ , using the pole mass would overestimate it by $(4.18/2.8)^2\approx2.2$ — more than a factor of two. The running mass automatically resums the large logarithms $\ln(m_H/m_b)$ that would otherwise appear as huge QCD corrections, leaving only a modest residual correction. This is a general feature of Higgs decays to quarks and a common stumbling block in first calculations.

Does the Higgs couple to itself, and why is that coupling special?

Yes — the Higgs is the one particle that couples to itself, through the cubic and quartic terms of its potential. The trilinear coupling is $g_{HHH}\propto3m_H^2/v$ , fixed by the same $\lambda$ that sets the mass. It is special because it is the only Higgs coupling not proportional to some other particle's mass — it is proportional to the Higgs's own mass and to $\lambda$ directly, so measuring it is a direct probe of the shape of the Higgs potential. That potential is what determines the vacuum structure of the universe, including whether our vacuum is truly stable. The self-coupling is extremely hard to measure — it requires producing two Higgs bosons at once — and pinning it down is one of the central goals of the High-Luminosity LHC and any future Higgs factory.

Is the Higgs coupling the same thing as "the Higgs gives everything mass"?

Not quite — and the distinction matters. The Higgs field, with its constant vacuum value $v$ , gives the elementary particles ( $W$ , $Z$ , quarks, charged leptons) their masses. The Higgs boson is the ripple in that field, and it couples to each particle in proportion to the mass the field gave it. So "the Higgs gives mass" refers to the field's vacuum value, while "Higgs couplings" refers to interactions with the physical boson — two aspects of one object. Crucially, the Higgs does not give mass to everything: most of the mass of ordinary matter is the binding energy of quarks and gluons inside protons and neutrons (from QCD), not the Higgs. The Higgs is responsible for the small intrinsic masses of the quarks and for the masses of the electron, the $W$ , and the $Z$ — not for the bulk of the mass of an atom.

Higgs couplings & properties

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The dominant decay width $H\to b\bar b$ (the flagship number)

2 · Testing "coupling proportional to mass" (the straight line)

3 · Why $H\to\gamma\gamma$ survives at all: the loop competition

Physics of the portrait

Going further

Before this

Where this leads

Higgs couplings & properties

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The dominant decay width H→bbˉH\to b\bar bH→bbˉ (the flagship number)

2 · Testing "coupling proportional to mass" (the straight line)

3 · Why H→γγH\to\gamma\gammaH→γγ survives at all: the loop competition

Physics of the portrait

Going further

Before this

Where this leads

1 · The dominant decay width $H\to b\bar b$ (the flagship number)

3 · Why $H\to\gamma\gamma$ survives at all: the loop competition