University Quantum Field Theory I · Interacting Fields: the Interaction Picture, the Gell-Mann-Low Formula and Wick's Theorem · 2.1

Model interactions and their coupling dimensions

Until QED arrives, most amplitudes in this course come from one of three interaction terms. This lesson writes them down and questions them before any Feynman rule exists: each coupling's dimension says how its strength changes with energy, the symmetries say what is stable and what may decay, and one model has no ground state at all.

Open the lab ↓Practice & worked examples ↓

Build the model

Connect the measurement to the mechanism.

Free fields never scatter, so field theory begins by adding to the free Lagrangian the simplest local, Hermitian, Lorentz-invariant polynomial the fields allow and asking what it changes. This course uses three: −(λ/4!)φ⁴ for one real scalar, −gψ†ψφ joining a complex scalar ψ of mass M to a real φ of mass m, and −gφψ̄ψ with ψ a Dirac field. Two pieces of bookkeeping say most of what each model can do before any diagram is drawn.

Mass dimension: with [ℒ] = 4, λ and the Dirac g are pure numbers while the scalar-Yukawa g is a mass, so that model's strength is the ratio g/E, weak at high energy and strong at low energy, whereas λ and the Dirac g change only through the logarithms loops supply. Symmetry: φ → −φ lets φ⁴ change particle number only by even amounts, and the conserved ψ number of both Yukawa models keeps ψ and ψ̄ stable and lets φ decay, to ψψ̄, only when m exceeds 2M. The models stop where their jobs stop.

They are toys, each built for one task: loops, trees and decays, spin. The scalar Yukawa potential M²ψ†ψ + ½m²φ² + gψ†ψφ is unbounded below, a fault no finite order of perturbation theory can see, so that model is trusted for amplitudes, decays and widths and never for its vacuum; the Dirac model needs scalar self-couplings of its own once fermion loops appear.

Simple definition: A model interaction is a local, Hermitian, Lorentz-invariant polynomial in the fields added to the free Lagrangian; its coupling has mass dimension 4 minus the total dimension of the fields it multiplies, and that dimension sets how the interaction's strength changes with energy.
Example: In −gψ†ψφ three scalar fields of dimension 1 leave [g] = 1, so g = 2.0 GeV means a strength g/E of 2.0 at E = 1 GeV and 0.020 at E = 100 GeV, while λ in −(λ/4!)φ⁴ is, at tree level, the same pure number at both.

φ⁴ theoryℒ = ½(∂φ)² − ½m²φ² − (λ/4!)φ⁴

Derived: [λ] = 4 − 4 = 0, so its strength is the same at every energy up to loop logarithms. Even in φ, the model changes particle number only by even amounts.

φ real, mass dimension 1 (GeV); m in GeV; [ℒ] = 4, so λ is a pure number, and λ ≥ 0 keeps the potential bounded below.

Scalar Yukawa modelℒᵢₙₜ = −g ψ†ψ φ

Derived: [g] = 4 − 3 = 1, so g is a mass and its strength at energy E is g/E, fading as E grows. Conserved ψ number keeps ψ and ψ̄ stable.

ψ a complex scalar of mass M, φ a real scalar of mass m, each of dimension 1; g, M and m in GeV.

Yukawa theory with Dirac fermionsℒᵢₙₜ = −g φ ψ̄ψ

Derived: [g] = 4 − 1 − 3/2 − 3/2 = 0, the scalar model's vertex shape with a coupling that does not fade. The Higgs boson couples to the τ this way with g = 0.0072 (quoted).

ψ a Dirac field of mass M and dimension 3/2; φ real, of mass m and dimension 1; M and m in GeV, g a pure number.

Mass dimension of an amplitude[ℳ] = 4 − n, for n external particles

Derived from topic 1.03's normalisation. ℳ = −λ for φφ → φφ is a pure number and ℳ = −g for φ → ψψ̄ is a mass: a one-vertex amplitude carries its coupling's dimension.

ℳ is defined by ⟨f|iT|i⟩ = (2π)⁴δ⁴(Pₒᵤₜ − Pᵢₙ) iℳ; each state |p⟩ has dimension −1 and δ⁴ has −4.

Width of φ → ψψ̄ in the scalar Yukawa modelΓ = (g²/(16πm)) √(1 − 4M²/m²)

Far above threshold dimensions alone force Γ ∝ g²/m, so Γ/m → (g/m)²/16π: a heavier φ is relatively narrower, which is what a coupling that is a mass does.

Γ, g, m and M in GeV; non-zero only for m above 2M. Quoted from topic 5.03, built on topic 2.09's ℳ = −g.

Scalar Yukawa potential along its valleyV = M²ρ − g²ρ²/(2m²), with ρ = ψ†ψ and φ = −gρ/m²

Derived by minimising V over φ at fixed ρ: the potential has no floor. The barrier grows as 1/g², so tunnelling through it is invisible at every finite order in g.

V in GeV⁴, ρ in GeV². The saddle sits at ρ = m²M²/g², height m²M⁴/(2g²); beyond it V falls without limit.

Three interactions, each chosen for a job

Natural units ħ = c = 1 and the metric (+, −, −, −) hold throughout, as topic 1.01 fixed them. Each model adds to Advanced QM's free Lagrangians (UAQM-09.02, 09.03 and 09.07, quoted) the simplest local, Lorentz-invariant, Hermitian polynomial its fields allow: −(λ/4!)φ⁴ for one real scalar; −gψ†ψφ for a complex scalar ψ of mass M and a real φ of mass m; −gφψ̄ψ for a Dirac ψ. With λ and g real each term is Hermitian, because ψ†ψ and ψ̄ψ are. None contains a derivative, so the canonical momenta and Unit 1's propagators are untouched and the Legendre transform gives ℋᵢₙₜ = −ℒᵢₙₜ, derived: the interaction Hamiltonian the next topics exponentiate is the potential term with its sign flipped. The factors are bookkeeping: the 4! cancels the 4! ways four identical fields meet four external legs, so the tree amplitude is ℳ = −λ (Unit 3, quoted), and the Yukawa terms need no ½ because their fields are distinct. φ⁴ is the cleanest place for loops and renormalisation, the scalar Yukawa model gives trees, decays and widths without spinor algebra, and the Dirac version adds spin for Unit 7.

Counting the dimension of each coupling

Topic 1.01 showed that a dimensionless action gives [ℒ] = 4, scalar fields dimension 1 and a Dirac field 3/2, so the coefficient of an operator of dimension n has dimension 4 − n (quoted). For φ⁴, n = 4 and [λ] = 0. For ψ†ψφ, three scalars give n = 3 and [g] = 1: the scalar Yukawa coupling is a mass, given in GeV. For φψ̄ψ, n = 1 + 3/2 + 3/2 = 4 and [g] = 0. The count depends on the spacetime dimension d: [φ] = (d − 2)/2 and, for a Dirac field, [ψ] = (d − 1)/2, so [λ] = 4 − d, the scalar Yukawa g has 3 − d/2 and the Dirac g has 2 − d/2. λ is a mass in d = 3, and the scalar g is a pure number in d = 6; Unit 6 works in d = 4 − 2ε, where λ carries dimension 2ε. Amplitudes inherit the count. Topic 1.03's ⟨p|q⟩ = 2E(2π)³δ³(p − q) gives each state dimension −1, and ⟨f|iT|i⟩ = (2π)⁴δ⁴(Pₒᵤₜ − Pᵢₙ) iℳ then gives [ℳ] = 4 − n for n external particles, derived. So ℳ = −λ for φφ → φφ is a pure number, and topic 2.09's ℳ = −g for φ → ψψ̄ is a mass, as it must be.

A coupling with dimensions is measured against the energy

A number with units cannot be small on its own; only a ratio can. Take a two-into-two tree amplitude of the scalar model, say ψψ → ψψ. Two vertices make it proportional to g², and with four legs it is dimensionless, so g² must be divided by a squared energy. Far above m and M only E² is left, so ℳ is of order (g/E)², derived up to a number: with g = 2.0 GeV that is 4 at E = 1 GeV and 4 × 10⁻⁴ at E = 100 GeV. The same count gives φ⁴ the constant ℳ = −λ, and gives the Dirac model g² times two spinor products, each of order E, over E²: energy independent again, apart from loop logarithms (Unit 6). Below E = g the ratio exceeds 1: the scalar model is strong at low energy. The static limit agrees: a slow ψ has ψ†ψ averaging n/(2M), n the number density, so topic 1.09's source energy gives V(r) = −(g²/(16πM²)) e^(−mr)/r, of strength (g/2M)²/4π; the Dirac model, with ψ̄ψ → n, gives −(g²/4π) e^(−mr)/r for every M, both derived. Positive, zero and negative coupling dimensions are called super-renormalisable, renormalisable and non-renormalisable (stated, for Unit 6).

Symmetries forbid what no coupling can switch on

Every term of the φ⁴ Lagrangian is even in φ, so φ → −φ is a symmetry. On free states it is U = (−1)^N, which flips the sign of every a and a†; U commutes with ℋᵢₙₜ = (λ/4!)φ⁴ and so with the S-operator built from it, while a state of n particles has U = (−1)ⁿ. Then ⟨f|S|i⟩ = ⟨f|U†SU|i⟩ = (−1)^{(nᵢₙ + nₒᵤₜ)}⟨f|S|i⟩, which vanishes unless the particle numbers differ by an even amount, derived, at every order: φφ → φφφφ is open above √s = 4m, φφ → φφφ never. Wick's theorem (topic 2.08) shows the same rule as counting: n external fields and 4 per vertex must pair off completely, so n is even. In both Yukawa models ψ → e^(−iα)ψ leaves every term unchanged. Its Noether charge Q = N_ψ − N_ψ̄, the ψ number, is conserved, and in a contraction ψ pairs only with ψ† or ψ̄, so ψ lines run unbroken through every vertex. Neither Yukawa model has φ → −φ: the vertex is odd in φ, so φ quanta are made and absorbed one at a time, and φ number is not conserved.

Stability and the decay threshold

Conservation laws decide decays before rates do. In either Yukawa model a lone ψ has Q = 1, and every state with Q = 1 contains at least one ψ, so its energy in the ψ rest frame is at least M, with equality only for ψ itself: ψ and ψ̄ are stable at every order, derived. The vertex also turns ψ into ψφ, but that needs M ≥ M + m, so its δ⁴ has no support (topic 2.09). A φ at rest has Q = 0 and energy m; the other states with Q = 0 are two or more φ quanta, at least 2m, or ψψ̄ pairs with anything, at least 2M. So φ decays only if m exceeds 2M, first to ψψ̄, derived; below that it too is stable. Above threshold, dimensions fix the large-m scaling: Γ is a mass proportional to g², so Γ ∝ g²/m in the scalar model and g²m in the Dirac one. Topics 5.03 and 7.07, quoted, give the widths Γ = (g²/(16πm))√(1 − 4M²/m²) and Γ = (g²m/(8π))(1 − 4M²/m²)³⁄². For g = 2.5 GeV and M = 1.0 GeV the scalar Γ/m peaks at m = √6 M = 2.45 GeV at 1.20% and falls to 0.048% by 16 GeV, while for g = 0.5 the Dirac Γ/m climbs towards g²/8π = 0.99%.

A potential with no floor

The scalar Yukawa potential is V = M²ψ†ψ + ½m²φ² + gψ†ψφ. Hold ρ = ψ†ψ fixed and minimise over φ: φ = −gρ/m², leaving V = M²ρ − g²ρ²/(2m²), which falls without limit as ρ grows, derived. The origin is only a local minimum, behind a saddle at φ = −M²/g, ρ = m²M²/g², of height m²M⁴/(2g²): for M = 1.0 GeV, m = 2.0 GeV and g = 0.50 GeV it sits at φ = −2.0 GeV and ρ = 16 GeV², 8.0 GeV⁴ high. Every scale of the saddle runs off as g → 0. Rescaling φ by M²/g and ψ by mM/g puts an overall 1/g² in front of the classical action, so any tunnelling exponent has the form M²F(m/M)/g², and e^{(−M²F/g²)} has a Taylor series in g whose every term is zero. No finite order of perturbation theory sees the fault (stated), so the model serves trees, decays and widths, never its vacuum. The other two models are safer but not finished: φ⁴ needs λ ≥ 0, and the Dirac model needs scalar self-couplings, φ⁴ among them, once fermion loops appear (stated).

Change one variable at a time

Make the relationship visible.

Interactive model

ψ mass M1.00 GeV

Scalar Yukawa g, a mass2.50 GeV

Dirac Yukawa g, a pure number0.50

Raise M: the threshold, the shaded stable band and both curves slide right together. Then compare the tails at M = 1 GeV: from m = 8 to 16 GeV the solid scalar-Yukawa ratio drops about fourfold, because its g is a mass, while the dashed Dirac ratio levels off at g²/8π.

THRESHOLD 2M2.00 GeV

SCALAR PEAK Γ/m1.20 %

SCALAR Γ/m AT m = 16 GeV0.048 %

DIRAC Γ/m AS m → ∞0.99 %

Live interpretationTHRESHOLD 2M: 2.00 GeV. SCALAR PEAK Γ/m: 1.20 %. SCALAR Γ/m AT m = 16 GeV: 0.048 %. DIRAC Γ/m AS m → ∞: 0.99 %

Catch the common trap

Explain before calculating.

Misconception check

The trap is to read a coupling's strength off its number. Put g = 0.5 in the scalar Yukawa model beside λ = 1 in φ⁴ and g looks like the weaker coupling, but the comparison means nothing: λ is a pure number, g is a mass, and g = 0.5 with no unit is not yet a value. With g = 0.5 GeV the strength is g/E: 5 at E = 100 MeV, 0.5 at 1 GeV and 0.005 at 100 GeV, so one model is strongly coupled at low energy and weakly coupled at high energy. The mirror error takes a dimensionless coupling to be small by default. The Higgs-τ coupling in this form, g = m_τ/v, is 0.0072, but the pion-nucleon coupling, a pseudoscalar cousin of the Dirac Yukawa term, has g²/4π of about 14 (quoted), far too large to expand in. Strength is always a dimensionless ratio built from the coupling and the energy of the process.

In the scalar Yukawa model, ψ is a complex scalar of mass M = 1.0 GeV, φ a real scalar of mass m = 1.5 GeV, and g = 3.0 GeV. Which statement is correct?

Choose an answer to test the model.

Practice & worked examples

Reason from the model, then test the result.

EasyA student proposes adding −(κ/3!)φ³, −(h/2)φ²ψ̄ψ and −c(ψ̄ψ)² to Yukawa theory with Dirac fermions. Find the mass dimensions of κ, h and c. Weak decays at low energy are described by a four-fermion term of the same dimension as the last, its coefficient set by the measured Fermi constant G_F = 1.166 × 10⁻⁵ GeV⁻². At what energy does G_F E² reach 1?

Topic 1.01's rule, quoted: in four dimensions [φ] = 1, a Dirac field has [ψ] = 3/2, and the coefficient of an operator of dimension n has dimension 4 − n.
φ³ has n = 3, so [κ] = 1: a mass, like the scalar Yukawa g, and its strength κ/E fades at high energy.
φ²ψ̄ψ has n = 2 + 3/2 + 3/2 = 5, so [h] = −1: an inverse mass, and its strength hE grows with energy.
(ψ̄ψ)² has n = 4 × 3/2 = 6, so [c] = −2 and the strength is cE². For c = G_F that is 1.166 × 10⁻⁵ at E = 1 GeV and 0.117 at E = 100 GeV.
G_F E² = 1 at E = 1/√G_F = 1/√(1.166 × 10⁻⁵ GeV⁻²) = 292.9 GeV. Check: [G_F E²] = −2 + 2 = 0. Well before that, at the W mass of 80.4 GeV (quoted), W exchange replaces the four-fermion term, as a coupling that grows with energy warns it must.

Answer[κ] = 1, [h] = −1, [c] = −2. G_F E² reaches 1 at E ≈ 293 GeV, above the 80.4 GeV W mass where the four-fermion description gives way.

MediumIn the scalar Yukawa model take M = 0.90 GeV, m = 2.0 GeV and g = 1.5 GeV. Decide whether φ can decay and to what, find the ψ momentum in the φ rest frame, then use topic 5.03's width Γ = (g²/(16πm))√(1 − 4M²/m²) to find Γ, the lifetime in seconds and Γ/m. Repeat Γ/m for m = 8.0 GeV at the same g and M.

ψ number is conserved and φ carries none, so its products are ψψ̄ plus anything, at least 2M = 1.80 GeV, or two or more φ quanta, at least 2m = 4.0 GeV. Only ψψ̄ fits under m = 2.0 GeV: φ decays, and only to ψψ̄.
Each daughter carries |p| = √(m²/4 − M²) = √(1.00 − 0.81) GeV = 0.436 GeV, so β = 2|p|/m = 0.436: slow daughters, close to threshold.
Γ = (1.5 GeV)² × 0.4359/(16π × 2.0 GeV) = 9.76 × 10⁻³ GeV = 9.76 MeV. Dimensions: GeV²/GeV = GeV, a mass, as a width must be.
Restore ħ: τ = ħ/Γ = 6.582 × 10⁻²⁵ GeV s/9.756 × 10⁻³ GeV = 6.75 × 10⁻²³ s, a decay length cτ = ħc/Γ = 0.19733 GeV fm/9.756 × 10⁻³ GeV = 20.2 fm. Γ/m = 4.9 × 10⁻³: a narrow state.
At m = 8.0 GeV: β = √(1 − 3.24/64) = 0.974, Γ = 5.45 MeV and Γ/m = 6.8 × 10⁻⁴, within 3% of the asymptote (g/m)²/16π = 7.0 × 10⁻⁴. Four times the mass leaves a seventh of the relative width: g is a mass, so its strength fades.

Answerφ → ψψ̄ only; |p| = 0.436 GeV; Γ = 9.76 MeV, τ = 6.75 × 10⁻²³ s (cτ = 20.2 fm) and Γ/m = 4.9 × 10⁻³. At m = 8.0 GeV, Γ/m = 6.8 × 10⁻⁴.

HardThe Higgs boson (mₕ = 125.20 GeV) couples to the τ lepton (m_τ = 1.77686 GeV) through a Dirac Yukawa term −g h τ̄τ with g = m_τ/v and v = 246.22 GeV, quoted: why g follows the mass is spontaneous symmetry breaking, beyond this course. Predict how Γ(h → τ⁺τ⁻) scales with mₕ, compute it from ℳ = −g ū(p)v(p′), and compare with experiment.

Dimensions first: g is a pure number and Γ a mass, and far above threshold mₕ is the only scale, so Γ = c g²mₕ, growing with the parent's mass; a scalar-Yukawa width would instead go as g²/mₕ.
Spin sum with Advanced QM's traces, quoted: Σ|ℳ|² = g² Tr[(p̸ + m_τ)(p̸′ − m_τ)] = 4g²(p⋅p′ − m_τ²). With p⋅p′ = (mₕ² − 2m_τ²)/2 this is 2g²mₕ²β², where β = √(1 − 4m_τ²/mₕ²).
Unit 5's two-body width, quoted, Γ = |p|Σ|ℳ|²/(8πmₕ²) with |p| = mₕβ/2, gives Γ = g²mₕβ³/(8π), so c = 1/8π. The β³ rather than β is the pair leaving in a P-wave, which parity demands of a scalar decaying to a fermion pair (stated; topic 7.07 returns to it).
Numbers: g = 1.77686/246.22 = 7.2166 × 10⁻³, g² = 5.2079 × 10⁻⁵, β² = 0.99919 and β³ = 0.99879, so Γ = 5.2079 × 10⁻⁵ × 125.20 GeV × 0.99879/(8π) = 2.59 × 10⁻⁴ GeV = 0.259 MeV.
Compare: the Standard Model's full calculation puts h → τ⁺τ⁻ at 6.3% of a 4.1 MeV total width, 0.26 MeV, and ATLAS and CMS measure the rate in agreement with it, with uncertainties near 10%. The partial lifetime is ħ/Γ = 2.54 × 10⁻²¹ s.
Limits: near threshold Γ ∝ β³ vanishes faster than the scalar model's β. Far above it, doubling mₕ at fixed g doubles Γ to 0.519 MeV, where a scalar-Yukawa width would halve.

AnswerΓ(h → τ⁺τ⁻) = g²mₕβ³/(8π) = 0.259 MeV, proportional to mₕ because g is dimensionless; it matches the Standard Model's 0.26 MeV and the measured rate.