Interaction by particle exchange

The propagator $1/(q^2-m_X^2)$ is the single most-used factor in all of experimental particle physics: every cross section quoted at the Large Hadron Collider is, at leading order, a product of vertex couplings and propagators exactly of the form derived here, integrated over phase space. The 1983 discovery of the $W$ and $Z$ bosons at CERN's UA1 and UA2 experiments confirmed massive exchange quanta directly — the weak force's feebleness at low energy is nothing but the $1/M_W^2$ suppression of a heavy propagator, and at LHC energies where $q^2\sim M_W^2$ the "weak" force becomes comparable to electromagnetism as the propagator stops suppressing it. Deep inelastic scattering, from SLAC in the 1960s to the HERA collider, is literally this spacelike exchange at work: an electron fires a highly virtual photon (large $Q^2=-q^2$) at a proton, and the virtuality sets the spatial resolution — how the quarks were first seen. And when a mediator is tuned onto the pole $q^2=m_X^2$, as at a collider set to the $Z$ mass, the propagator diverges and the "virtual" quantum becomes a real, resonantly produced particle — the basis of the precision $Z$-factory measurements at LEP. Off the pole it is a force; on the pole it is a particle; the same denominator governs both.

Two particles approach, never touch, and yet emerge deflected. Between them a quantum is born, crosses the gap, and is absorbed — carrying momentum from one to the other. That single passed quantum is the force. But quantum mechanics forbids us to say which particle emitted and which absorbed: both time-orderings happen, and both must be added. The first figure draws the two orderings and states the punchline they add up to — the invariant amplitude that no single ordering possesses.

Notice what the sum forgets. In each separate diagram the exchanged quantum sits on its mass shell, $E_X=\sqrt{\vec{p}_X^{\,2}+m_X^2}$, but the intermediate energy $E_j$ does not match $E_i$ — energy is out of balance. When the two are added, the frame-dependent energy denominators recombine into the single invariant $q^2-m_X^2$, and the quantum goes off the mass shell instead. The mass of that quantum is the one thing the force cannot hide: it sets how far the force reaches. The second figure makes this quantitative — the Yukawa potential and its range for a real mediator.

Everything begins with the second-order term of time-ordered perturbation theory. The first-order term $\langle f|V|i\rangle$ is direct scattering in a potential; when there is no such potential, the leading process is second order, running through an intermediate state $j$ in which the exchanged quantum exists.

where: $T_{fi}$ — transition matrix element from initial state $i$ to final state $f$; $V$ — interaction Hamiltonian; $E_i,E_j$ — energies of the initial and intermediate states; the sum runs over all intermediate states $j$. Reading it: the force appears at second order, as two applications of $V$ (emit, then absorb) bridged by an energy denominator $E_i-E_j$. At each vertex three-momentum is exactly conserved, so the exchanged quantum in ordering 1 must carry $\vec{p}_X=\vec{p}_a-\vec{p}_c$. It sits on its mass shell, but energy is not balanced across the intermediate state.

where: $\vec{p}_X$ — three-momentum of the exchanged quantum; $E_X$ — its on-shell energy; $m_X$ — its mass; $E_j\neq E_i$ signals that energy conservation is not imposed on the intermediate state — the essence of a virtual particle, permitted for a time $\Delta t\sim\hbar/|E_i-E_j|$. Stripping the relativistic normalisation factors $\prod_k(2E_k)^{-1/2}$ from the vertices and summing the two orderings gives the central result of the article — a single amplitude in which every frame-dependent quantity has vanished.

where: $\mathcal{M}_{fi}$ — Lorentz-invariant amplitude; $g_a,g_b$ — scalar couplings at the two vertices (the invariant vertex factors); $q^2$ — four-momentum transfer squared. Reading it: the amplitude depends on the external particles only through the invariant $q^2$ and on the mediator only through its mass $m_X$ — this factor is the propagator. The invariant that survived is the squared four-momentum transfer, spacelike for a scattering exchange, and it is what makes equation (3) frame-independent.

where: $q$ — four-momentum transferred through the exchange; $q^2$ — its invariant square in the $(+,-,-,-)$ metric with $\hbar=c=1$. For elastic scattering $q^2<0$ (spacelike): the pole at $q^2=m_X^2$, where the mediator would go real, is never reached, so $\mathcal{M}$ is finite and the force is felt, not radiated. In the static limit the amplitude is the Fourier transform of a potential, which turns the propagator into the Yukawa form — the bridge between exchange and the everyday notion of a force law.

where: $V(r)$ — static potential between the two particles; $R$ — range, the Compton wavelength of the mediator. The exponential cutoff means a massive mediator gives a short-range force; $m_X\to0$ restores the infinite-range Coulomb law $V\propto 1/r$. Finally, Griffiths' Feynman calculus feeds this amplitude into the golden rule: the same $\mathcal{M}$ that we computed, squared and multiplied by phase space, yields the two measurable numbers of particle physics — decay rates and cross sections.

where: $\Gamma$ — decay rate of a particle of mass $m_1$; $\sigma$ — cross section; $s=(p_1+p_2)^2$ (four-momenta) — the squared centre-of-mass energy, equal to $(E_1+E_2)^2$ in the CM frame; $|\vec{p}_i|,|\vec{p}_f|$ — initial and final momenta in the CM frame; $S$ — a statistical factor for identical particles. Reading it: the "calculus" cleanly splits into dynamics ($\mathcal{M}$, from the diagram) and kinematics (phase space).

1 · The range of a force from its mediator's mass (fully worked)

Yukawa's logic in reverse: given a mediator mass, predict the range of the force it carries. Do it for the pion ($m_\pi c^2=140$ MeV), the carrier of the residual nuclear force, and for the $W$ boson ($M_W c^2=80.4$ GeV), the carrier of the charged-current weak force. Use $\hbar c=197.3$ MeV·fm.

Step 1. The range is the mediator's Compton wavelength, from equation (5): $R=\hbar/(m_X c)=\hbar c/(m_X c^2)$. Writing it with $\hbar c$ in MeV·fm makes the arithmetic a single division.

Step 2. Pion: $R_\pi=\dfrac{197.3\ \text{MeV}\,\cdot\,\text{fm}}{140\ \text{MeV}}=\mathbf{1.41\ fm}$. This is the measured reach of the nucleon–nucleon force — the number Yukawa turned around in 1935 to predict a mediator of mass $\approx 100$–$200$ MeV, a decade before the pion was found.

Step 3. $W$ boson: $R_W=\dfrac{197.3\ \text{MeV}\,\cdot\,\text{fm}}{80\,400\ \text{MeV}}=\mathbf{2.45\times10^{-3}\ fm}$. The enormous mediator mass squeezes the weak force into a range roughly a thousandth of a proton radius, which is why it looks weak at low energy: the propagator $1/(q^2-M_W^2)\approx-1/M_W^2$ is tiny, suppressing amplitudes by $1/M_W^2$ and making $\beta$ decay slow. A pion gives a 1.4 fm force; the $W$ boson gives a $2.45\times10^{-3}$ fm force — the mediator mass alone spans this factor of ~570.

2 · Photon exchange and the Rutherford angular law

Show that massless-photon exchange between two charges reproduces the famous Rutherford $\sin^{-4}(\theta/2)$ scattering, purely from the shape of the propagator.

Step 1. For photon exchange set $m_X=0$ and $g_a g_b\to e^2$ (electric charges at the vertices), so equation (3) becomes $\mathcal{M}=e^2/q^2$. In elastic scattering the energy transfer vanishes, so $q^2=-|\vec{q}|^2$ and $\mathcal{M}=-e^2/|\vec{q}|^2$.

Step 2. The momentum transfer for elastic scattering through angle $\theta$ with incoming momentum $k$ is $|\vec{q}|^2=|\vec{p}_a-\vec{p}_c|^2=2k^2(1-\cos\theta)=4k^2\sin^2(\theta/2)$. Hence $|\mathcal{M}|^2=\dfrac{e^4}{|\vec{q}|^4}=\dfrac{e^4}{16k^4\sin^4(\theta/2)}$.

Step 3. Feeding $|\mathcal{M}|^2$ into the cross-section half of equation (6), the phase-space factors are smooth in $\theta$, so the whole angular dependence comes from the propagator squared: $\dfrac{d\sigma}{d\Omega}\propto|\mathcal{M}|^2\propto\dfrac{1}{\sin^4(\theta/2)}$. The Rutherford law is the $1/q^4$ of a squared massless propagator — the same $1/r$ Coulomb tail, seen in momentum space. The steep forward peak (small $\theta$ means small $|\vec{q}|$, hence small $q^2$, hence large $\mathcal{M}$) is the signature of long-range, massless-mediated scattering; a massive mediator would tame the divergence by adding $m_X^2$ to the denominator.

Particle exchange is a chain of insights, but its modern face — the propagator that folds the time-orderings into one covariant line — belongs to Richard Feynman.

Texts. Thomson, Modern Particle Physics, Ch. 5 §5.1 (Eqs. 5.1–5.20), the time-ordered derivation followed here · Griffiths, Introduction to Elementary Particles, 2nd ed., Ch. 6 §6.1–6.2, the Feynman calculus and the ABC toy model · Halzen & Martin, Quarks and Leptons, Ch. 4, the propagator and the Yukawa potential · Peskin & Schroeder, An Introduction to QFT, Ch. 4, the covariant derivation of the same propagator from Wick's theorem · Schwartz, QFT and the Standard Model, Ch. 3–4. The key algebraic checkpoints in this article are: (1) the two energy denominators $E_a-E_c-E_X$ and $-(E_a-E_c)-E_X$ sum, after the $1/(2E_X)$ factor, to $g_a g_b/[(E_a-E_c)^2-E_X^2]$; (2) $(E_a-E_c)^2-E_X^2=q^2-m_X^2$ with $E_X^2=|\vec p_a-\vec p_c|^2+m_X^2$; (3) the Fourier pair $1/(|\vec q|^2+m^2)\leftrightarrow e^{-mr}/(4\pi r)$; (4) $R_\pi=197.3/140=1.41$ fm and $R_W=197.3/80\,400=2.45\times10^{-3}$ fm; (5) $|\vec q|^2=4k^2\sin^2(\theta/2)\Rightarrow d\sigma/d\Omega\propto\sin^{-4}(\theta/2)$ for $m_X=0$.

Prerequisites. Fermi's Golden Rule · Non-relativistic QM Review · Special Relativity Kinematics · Relativistic Wavefunction Normalisation. Leads to. Feynman Diagrams and Virtual Particles · The Yukawa Potential · Introduction to QED · The Photon Propagator.

Ask the tutor: ask me to run Yukawa's 1935 argument in reverse — predict a mediator's mass from a measured force range — or to show how the same propagator gives both the Coulomb law and the Rutherford $\sin^{-4}(\theta/2)$ scattering.