Feynman diagrams & virtual particles

Feynman diagrams are the daily working tool of the entire field. Every Standard Model prediction quoted with a theoretical uncertainty is a truncated sum of diagrams: the electron's anomalous magnetic moment $a_e$ is computed through five loops — tens of thousands of diagrams — and agrees with experiment to better than one part in $10^{12}$, the most precisely verified prediction in science. Automated tools such as MadGraph generate the millions of diagrams needed to model LHC collisions, where the $s$-channel resonance shape of a virtual propagator is exactly what ATLAS and CMS scan across to weigh the Z, discover the Higgs at $125$ GeV, and hunt for new mediators. The off-shell propagator is visible in data too: the running of $\alpha(q^2)$ is the imprint of virtual loops correcting the photon propagator. Whenever a physicist says "this goes through a virtual photon" or "the W propagator suppresses the rate", they are reading equation (1) off a picture Feynman first drew in 1948.

Fix the external particles and ask when the exchange happened. In the first story $a$ emits the mediator $X$ and turns into $c$; the quantum flies across and is absorbed by $b$, which becomes $d$. In the second the sequence is reversed. Momentum is conserved at each vertex in both, but energy is not: the intermediate state, with a real on-shell $X$ of energy $E_X=\sqrt{\mathbf{p}_X^2+m_X^2}$, has more (or less) energy than the initial state, surviving only for the interval the energy–time uncertainty relation allows. Neither story is Lorentz invariant alone — observers disagree on which came first — but their sum is invariant, and that sum is what the Feynman diagram draws.

Once the sum is done, the propagator $1/(q^2-m_X^2)$ governs the physics you measure. Its behaviour splits by the sign of $q^2$: in $t$-channel exchange the four-momentum is spacelike, $q^2<0$, the denominator never vanishes, and the amplitude grows smoothly toward small angles (the forward $1/t$ peak of Rutherford and Bhabha scattering); in $s$-channel annihilation it is timelike, $q^2=s>0$, and when $\sqrt{s}$ is tuned to $m_X$ the denominator hits zero and a real resonance forms. In the static limit the same propagator Fourier-transforms into the Yukawa potential, so the mass in its denominator becomes the range of the force — which the second figure makes quantitative.

The central result is the amplitude for single-quantum exchange — what the whole time-ordering sum evaluates to. Every Feynman-rule calculation elaborates it: replace $g$ by the vertex factors and the scalar denominator by the correct tensor propagator, and this is QED, QCD, and the weak interaction.

where: $\mathcal{M}_{fi}$ — Lorentz-invariant matrix element for $a+b\to c+d$; $g_a,g_b$ — couplings at the two vertices; $m_X$ — mass of the exchanged quantum; $q$ — four-momentum transferred. Reading: the numerator is dynamics (how strongly the particles couple), the denominator is the propagator (how the virtual quantum carries momentum between them). The transferred four-momentum is fixed entirely by the external particles, and its square is a Lorentz scalar — the same in every frame, and exactly the Mandelstam $t$. Its sign decides whether the exchange is spacelike or timelike.

where: $p_a,p_c$ — four-momenta of the same particle before and after the vertex. Reading: because $q$ is built from measured external momenta, $q^2$ is not free, and there is no reason for it to equal $m_X^2$ — the statement that the exchanged particle is off mass-shell. The propagator factor is the heart of the Feynman rules. For a spin-0 mediator it is the reciprocal below; its pole at $q^2=m_X^2$ is where the internal line would become a real, on-shell particle.

where: $g^{\mu\nu}$ — metric tensor carrying the mediator's spin structure. Reading: a massless photon gives $1/q^2$ and a $1/r$ potential; a heavy W gives a denominator dominated by $m_W^2$ at low $q^2$, which is why the weak interaction looks like a feeble contact force. In the static limit ($q^0\to0$, so $q^2\to-|\mathbf{q}|^2$) the exchange amplitude is the Fourier transform of a potential. Transforming $-g^2/(|\mathbf{q}|^2+m_X^2)$ to position space gives the Yukawa potential, whose exponential fall-off defines a range set by the mediator mass.

where: $g^2/4\pi$ — dimensionless coupling; $R$ — range. Reading: $m_X\to0$ recovers the infinite-range Coulomb potential; a heavy mediator screens the force beyond $R=\hbar/m_X c$. Finally, the amplitude is not itself observable — it must be squared and folded with phase space. Griffiths' Feynman calculus factorises every rate into $|\mathcal{M}|^2$ (dynamics, from the diagram) and the density of final states (kinematics), joined by Fermi's golden rule.

where: $\Gamma$ — transition rate; $\rho(E_f)$ — density of final states; $\tau$ — mean lifetime; $\mathcal{L}$ — luminosity; $\sigma$ — cross section. Reading: the diagram supplies $\mathcal{M}$, phase space supplies $\rho$, and their product is what the detector registers.

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

Electromagnetism reaches across the galaxy; the nuclear force dies within a nucleus; the weak force barely reaches across a proton. Using $R=\hbar/(m_X c)=\hbar c/(m_X c^2)$ with $\hbar c=197.3$ MeV·fm, find the range set by a massless photon, a 139.6 MeV pion, and an 80.4 GeV W boson.

Step 1. Photon. With $m_\gamma=0$ the propagator is $1/q^2$ and $R=\hbar c/0\to\infty$; the static limit is the Coulomb potential $V\propto1/r$, which never cuts off. Electromagnetism is infinite-range.

Step 2. Pion. The lightest hadron, Yukawa's mediator of the nuclear force: $R=197.3\ \mathrm{MeV\,fm}/139.6\ \mathrm{MeV}=\mathbf{1.41\ fm}$ — the observed reach of the nucleon–nucleon attraction, from which Yukawa predicted a $\sim140$ MeV quantum a decade before the pion was found.

Step 3. W boson. The weak mediator is enormously heavy: $R=197.3\ \mathrm{MeV\,fm}/80\,380\ \mathrm{MeV}=\mathbf{2.45\times10^{-3}\ fm}$, about a thousandth of a proton radius.

Step 4. The three ranges span nearly six orders of magnitude, and the only input that changed was the mediator mass in the denominator. A force is long-ranged if and only if its mediator is (nearly) massless; the weak force is short-ranged because the W and Z are heavy, not because its coupling is small — as the next example shows.

2 · Why the weak force is "weak" (fully worked)

The weak coupling $g_W\approx0.65$ is actually larger than the electromagnetic charge $e=\sqrt{4\pi\alpha}\approx0.30$ — so why does beta decay look so much feebler? Compare the weak and EM amplitudes at a nuclear momentum transfer $q^2=0.01\ \mathrm{GeV}^2$, then at $q^2=m_W^2$. Take $m_W=80.4$ GeV.

Step 1. Write the two amplitudes. Electromagnetic (massless photon): $\mathcal{M}_{\mathrm{EM}}\sim e^2/q^2$. Weak (heavy W): $\mathcal{M}_{\mathrm{weak}}\sim g_W^2/(q^2-m_W^2)\to -g_W^2/m_W^2$ when $q^2\ll m_W^2$.

Step 2. Form the ratio at low $q^2$: $\dfrac{\mathcal{M}_{\mathrm{weak}}}{\mathcal{M}_{\mathrm{EM}}}\approx\dfrac{g_W^2/m_W^2}{e^2/q^2}=\dfrac{g_W^2}{e^2}\cdot\dfrac{q^2}{m_W^2}$. With $g_W^2/e^2\approx4.6$ and $q^2/m_W^2=0.01/6464=1.5\times10^{-6}$, the ratio is $\approx\mathbf{7\times10^{-6}}$.

Step 3. So at nuclear energies the weak interaction is roughly a million times weaker than electromagnetism — but this is entirely the factor $q^2/m_W^2$ from the heavy propagator, not the coupling. The low-energy contact strength is the Fermi constant, $\dfrac{G_F}{\sqrt2}=\dfrac{g_W^2}{8m_W^2}$, giving $G_F=\sqrt2\,(0.65)^2/(8\times80.4^2)=\mathbf{1.17\times10^{-5}\ GeV^{-2}}$, the measured value.

Step 4. Now go to $q^2=m_W^2$: the ratio becomes $g_W^2/e^2\approx\mathbf{4.6}$ — the weak amplitude is now comparable to the electromagnetic one. The weak force is weak only because we usually probe it at $q^2\ll m_W^2$; at momentum transfers of order $m_W$ it stands shoulder to shoulder with electromagnetism — the content of electroweak unification.

3 · How off-shell is a virtual photon?

In $e^+e^-\to\mu^+\mu^-$ at $\sqrt{s}=10$ GeV the annihilation proceeds through a single $s$-channel virtual photon. By how much is it off mass-shell, and how does that compare to a $t$-channel photon in Møller scattering?

Step 1. In the $s$-channel the virtual photon carries the total four-momentum, $q=p_{e^+}+p_{e^-}$, so $q^2=s=\mathbf{100\ GeV^2}$. A real photon has $q^2=0$, so this one is off-shell by the full $\mathbf{100\ GeV^2}$ — timelike, which for a photon is impossible on-shell.

Step 2. In $t$-channel Møller scattering the exchanged photon carries $q^2=t=-\tfrac{s}{2}(1-\cos\theta^*)\le0$, which is spacelike — even further from any physical photon, since no real particle can have $q^2<0$.

Step 3. Interpret. The same photon field appears as a wildly off-shell internal line — timelike in annihilation, spacelike in exchange — and only its coupling to on-shell external legs is measurable; the propagator $1/q^2$ merely records how momentum flows through.

The diagram, the propagator, and the whole calculational machine of modern particle physics trace to one person, who drew his first such picture at a fractious 1948 conference.

Texts. Thomson, Modern Particle Physics, §5.1–§5.2 (interaction by particle exchange; Feynman diagrams and virtual particles) and §5.2.1 (scattering in a potential) · Griffiths, Introduction to Elementary Particles, 2nd ed., Ch. 6 (the Feynman calculus), §6.1–§6.2 · Halzen & Martin, Quarks and Leptons, Ch. 4 · Peskin & Schroeder, An Introduction to Quantum Field Theory, §4.6–§4.7 (Feynman rules and diagrams) · Schwartz, Quantum Field Theory and the Standard Model, Ch. 7.

Provenance. The collapse of the two time orderings into $\mathcal{M}_{fi}=g_a g_b/(q^2-m_X^2)$ follows Thomson §5.1 and is re-derived above; the Yukawa potential and range follow Thomson §5.2.1 and Yukawa, Proc. Phys.-Math. Soc. Japan 17, 48 (1935). The space-time diagram method is Feynman, Phys. Rev. 76, 769 (1949), systematised by Dyson, Phys. Rev. 75, 486 (1949). Numeric anchors — the pion range $1.41$ fm, the W range $2.45\times10^{-3}$ fm, the Fermi constant $G_F=1.17\times10^{-5}\ \mathrm{GeV}^{-2}$ from $g_W^2/8m_W^2$, and the weak/EM amplitude ratios $7\times10^{-6}$ at $q^2=0.01\ \mathrm{GeV}^2$ versus $4.6$ at $q^2=m_W^2$ — were each checked directly.

Prerequisites. Interaction by particle exchange · Four-vector notation · Mandelstam variables · Fermi's golden rule. Leads to. Feynman rules for QED · The photon propagator · The Yukawa potential · Interaction cross sections.

Ask the tutor: Name any exchange process — a force, a decay, or a collision — and I will draw its Feynman diagram, write the propagator, say whether the virtual particle is timelike or spacelike, and read off the range or resonance it implies.