Bhabha & Moller scattering

Bhabha scattering ( $e^+e^- \to e^+e^-$ ) and Møller scattering ( $e^-e^- \to e^-e^-$ ) are the two simplest elastic reactions in quantum electrodynamics, and together they are the workhorses of the whole subject: they are the first cross sections a student computes with real Dirac spinors, and they are the numbers a collider physicist trusts to five significant figures every time a beam is delivered. Both are built from a single QED vertex — an electron, a positron, and a photon — but the way that vertex is wired up differs by the charges involved, and that difference is the entire story. Two like charges ( $e^-e^-$ ) can only exchange a photon, and because the two electrons are identical the amplitude must be antisymmetrised, giving a t-channel and a u-channel diagram with a relative minus sign. An electron and its antiparticle ( $e^+e^-$ ) can do something extra: they can annihilate into a virtual photon that reappears as a fresh pair, adding an s-channel diagram alongside the t-channel exchange. The interference between exchange and annihilation is what makes Bhabha scattering distinctive — and its steep forward peak, calculable from pure QED with almost no theoretical uncertainty, is why every $e^+e^-$ collider from PETRA to LEP measured its luminosity by counting small-angle Bhabha events. This article assembles both cross sections from the Feynman rules, squares the amplitudes with the trace technology recalled from aq-pp-trace-techniques, and reads off what the resulting angular distributions mean for experiment.

Key takeaways

Møller scattering $e^-e^- \to e^-e^-$ has two diagrams — t-channel and u-channel photon exchange — added with a relative minus sign because the two final electrons are identical fermions.
Bhabha scattering $e^+e^- \to e^+e^-$ has a t-channel exchange diagram plus an s-channel annihilation diagram ( $e^+e^-\to\gamma^*\to e^+e^-$ ), again with a relative minus sign.
The unpolarised squared amplitudes are compact functions of the Mandelstam invariants $s,t,u$ ; Bhabha follows from Møller by the crossing $s\leftrightarrow u$ .
The t-channel photon makes both cross sections diverge in the forward direction like Rutherford scattering, $d\sigma/d\Omega\propto 1/\sin^4(\theta/2)$ .
That forward divergence is a feature: small-angle Bhabha scattering is the luminosity monitor at $e^+e^-$ colliders, precise to better than $0.1\%$ .
Key idea: Bhabha and Møller are the same QED vertex wired two ways — like charges exchange a photon (t and u channels), a particle–antiparticle pair can also annihilate (s channel) — and the coherent sum of those diagrams, squared, gives forward-peaked angular distributions that are among the most precisely known predictions in physics.

The core insight

Bhabha and Møller scattering are the two elastic QED reactions built from one $e$ – $e$ – $\gamma$ vertex: identical charges can only exchange a photon (t- and u-channel), while a particle and its antiparticle can additionally annihilate through a virtual photon (s-channel), and the coherently summed, spin-averaged amplitude gives forward-peaked cross sections calculable to extraordinary precision.

Where this lives today

Bhabha scattering is not a historical curiosity — it is a live instrument. Every $e^+e^-$ collider ringed a dedicated low-angle calorimeter around the beampipe to catch the forward Bhabha peak, and the counting rate there, divided by the QED cross section, is the collider's luminosity. At LEP the calculation and measurement were pushed to the $0.1\%$ level, and that number set the absolute scale for the entire electroweak precision programme — including the measurement of the number of light neutrino species from the invisible $Z$ width, which depends directly on the luminosity normalisation. Modern facilities continue the tradition: BaBar and Belle at the B-factories, BESIII at Beijing, and the proposed FCC-ee and CEPC all rely on small-angle Bhabha counting, now computed to two-loop QED accuracy by generators such as BabaYaga and BHLUMI. Møller scattering, meanwhile, has become a precision polarimeter and a probe of new physics: because the like-charge cross section is exquisitely sensitive to the electrons' spin correlations, fixed-target Møller experiments at SLAC (E158) and Jefferson Lab (the MOLLER experiment) measure the tiny parity-violating asymmetry induced by $Z$ – $\gamma$ interference to extract the weak mixing angle $\sin^2\theta_W$ at low energy, hunting for deviations that would signal physics beyond the Standard Model. The same two 1930s cross sections thus anchor both the luminosity of the highest-energy lepton colliders and the search for new forces at the intensity frontier.

The physical picture

Start from the one interaction QED allows: at a single spacetime point an electron line, a positron line, and a photon line meet, with strength $e$ . Every diagram in this article is two copies of that vertex joined by one virtual photon. What changes between Møller and Bhabha is not the vertex but the topology the charges permit. Two electrons cannot annihilate — charge and lepton number forbid it — so they can only trade a photon, and because the outgoing electrons are indistinguishable the amplitude must be antisymmetric under swapping them, producing two exchange diagrams. An electron and a positron carry opposite charge and can annihilate outright into a photon, so alongside the exchange diagram Bhabha gains an annihilation diagram. The first figure lays the four diagrams side by side; the second shows what they add up to.

Figure

The four tree diagrams. Bhabha e⁺e⁻ → e⁺e⁻ runs on t-channel photon exchange plus s-channel annihilation, while Møller e⁻e⁻ → e⁻e⁻ runs on t-channel plus u-channel exchange with a relative minus sign from Fermi statistics; the massless Mandelstam invariants obey s + t + u = 0

Adding the diagrams as amplitudes — coherently, before squaring — is the whole game, and interference is where the physics hides. In Bhabha scattering the t-channel photon can be exchanged with vanishingly small momentum transfer when the electron barely deflects, so its propagator $1/t$ blows up and the cross section soars into the forward direction. The s-channel annihilation term has a fixed $1/s$ and gives only a gentle, broad angular floor. The result is a wildly asymmetric distribution — a towering forward peak with a nearly flat back — utterly unlike the symmetric pattern of Møller scattering, whose two exchange channels make it identical under $\theta\to\pi-\theta$ . The second figure plots the real Bhabha distribution at a typical energy: the number a luminosity monitor lives on.

Figure

Bhabha differential cross section dσ/dΩ at √s = 10 GeV on a log scale versus centre-of-mass angle θ: the t-channel photon drives the rate up as 1/sin⁴(θ/2) into the forward direction — about 3500 nb/sr at 10° against 0.47 nb/sr at 90° — which is why counting small-angle Bhabha events measures collider luminosity

The core equations

Everything below is expressed in natural units $\hbar=c=1$ , with the coupling $e^2=4\pi\alpha$ and the metric $g^{\mu\nu}=\mathrm{diag}(+,-,-,-)$ . The two Mandelstam invariants that carry all the angular information are $s=(p_1+p_2)^2$ (the squared centre-of-mass energy), $t=(p_1-p_3)^2$ (the momentum transfer), and $u=(p_1-p_4)^2$ ; for massless electrons they satisfy $s+t+u=0$ . We label the incoming momenta $p_1,p_2$ and the outgoing $p_3,p_4$ . The Møller amplitude is the antisymmetrised sum of two photon-exchange diagrams. The Feynman rules of aq-pp-feynman-rules-qed attach a vertex $-ie\gamma^\mu$ at each end of a photon propagator $-ig_{\mu\nu}/q^2$ ; the relative minus sign is forced by Fermi statistics for the identical outgoing electrons.

i\mathcal{M}_{ee}=-ie^2\!\left[\frac{[\bar u_3\gamma^\mu u_1][\bar u_4\gamma_\mu u_2]}{t}-\frac{[\bar u_4\gamma^\mu u_1][\bar u_3\gamma_\mu u_2]}{u}\right] \qquad (1)

where: $u_i\equiv u(p_i)$ — Dirac spinor for the electron of momentum $p_i$ (from aq-pp-dirac-solutions); $\bar u=u^\dagger\gamma^0$ ; the first term is t-channel exchange, the second u-channel; the minus sign is the antisymmetry of a two-electron final state. Swapping $p_3\leftrightarrow p_4$ (i.e. $t\leftrightarrow u$ ) flips the overall sign, as an amplitude of identical fermions must. The Bhabha amplitude replaces the u-channel exchange with an s-channel annihilation: the $e^+e^-$ pair fuse into a timelike virtual photon of invariant mass $\sqrt s$ , which then materialises as the outgoing pair. Here $v_i$ are the positron (antiparticle) spinors of aq-pp-antiparticles.

i\mathcal{M}_{e\bar e}=-ie^2\!\left[\frac{[\bar u_3\gamma^\mu u_1][\bar v_2\gamma_\mu v_4]}{t}-\frac{[\bar v_2\gamma^\mu u_1][\bar u_3\gamma_\mu v_4]}{s}\right] \qquad (2)

where: $u_1$ — incoming electron; $v_2$ — incoming positron; $u_3$ — outgoing electron; $v_4$ — outgoing positron; the first term is t-channel scattering ( $1/t$ ), the second is s-channel annihilation ( $1/s$ ). The relative minus sign again encodes fermion antisymmetry, now between the electron and the annihilating positron line. Squaring with Casimir's trick — averaging over $2\times2$ initial spins and summing over final spins — collapses the spinor products into Dirac traces, evaluated in aq-pp-trace-techniques. In the ultra-relativistic (massless) limit the Møller result is a symmetric rational function of $s,t,u$ .

\overline{|\mathcal{M}_{ee}|^2}=2e^4\!\left[\frac{s^2+u^2}{t^2}+\frac{s^2+t^2}{u^2}+\frac{2s^2}{tu}\right] \qquad (3)

where: the first term is $|{\rm t}|^2$ , the second $|{\rm u}|^2$ , the third the t–u interference; the whole expression is symmetric under $t\leftrightarrow u$ , reflecting the indistinguishability of the two final electrons. The bar denotes the spin average $\tfrac14\sum_{\rm spins}$ . The Bhabha squared amplitude follows from Møller by the crossing substitution $s\leftrightarrow u$ , which turns a u-channel exchange into an s-channel annihilation — a shortcut that saves recomputing every trace.

\overline{|\mathcal{M}_{e\bar e}|^2}=2e^4\!\left[\frac{s^2+u^2}{t^2}+\frac{t^2+u^2}{s^2}+\frac{2u^2}{st}\right] \qquad (4)

where: the first term is t-channel ( $\propto 1/t^2$ , the forward-peaking piece), the second s-channel annihilation ( $\propto 1/s^2$ , the flat floor), the third their interference. Unlike (3) this is not symmetric under $t\leftrightarrow u$ , because an $e^+e^-$ final state is distinguishable from its swap. The differential cross section in the centre-of-mass frame packages the flux and phase-space factors; for massless kinematics $|\mathbf p_f|=|\mathbf p_i|$ and it reduces to a single clean prefactor.

\frac{d\sigma}{d\Omega}=\frac{\overline{|\mathcal{M}|^2}}{64\pi^2 s}=\frac{\alpha^2}{2s}\times[\text{Mandelstam bracket}] \qquad (5)

where: the $1/64\pi^2 s$ is the massless two-body phase space and flux; using $e^2=4\pi\alpha$ turns $2e^4$ into $32\pi^2\alpha^2$ , cancelling to the tidy $\alpha^2/2s$ . In terms of angle, with $t=-\tfrac{s}{2}(1-\cos\theta)$ and $u=-\tfrac{s}{2}(1+\cos\theta)$ , the brackets become the compact forms used to draw the figures: Bhabha $=\dfrac{1+\cos^4\frac\theta2}{\sin^4\frac\theta2}-\dfrac{2\cos^4\frac\theta2}{\sin^2\frac\theta2}+\dfrac{1+\cos^2\theta}{2}$ and Møller $=\dfrac{2(3+\cos^2\theta)^2}{\sin^4\theta}$ .

Derivation AFrom the Feynman rules to the two-diagram amplitude

The goal is to build the Møller amplitude from the QED Feynman rules and to see exactly why two diagrams appear where the spin-0 toy model had one, and why they enter with a relative minus sign. We attach a photon propagator between two electron currents, enforce the antisymmetry demanded by Fermi–Dirac statistics for the identical outgoing electrons, and read off the t- and u-channel structure on which every subsequent trace computation rests.

Step 1. Draw the lowest-order connected diagram for $e^-(p_1)e^-(p_2)\to e^-(p_3)e^-(p_4)$ . With only the $e$ – $e$ – $\gamma$ vertex available, the two electrons must exchange a single virtual photon. Assign the outgoing electron $p_3$ to the line that started as $p_1$ : this is the t-channel, with momentum transfer $q=p_1-p_3$ and propagator $-ig_{\mu\nu}/q^2=-ig_{\mu\nu}/t$ . Why: no vertex converts two electrons into anything else, so a single exchanged photon is the only order- $e^2$ process — the "direct" pairing of in and out lines.

Step 2. Write the two electron currents. Each vertex contributes $-ie\gamma^\mu$ sandwiched between the spinors of that line, giving the current $[\bar u_3\gamma^\mu u_1]$ on the upper line and $[\bar u_4\gamma_\mu u_2]$ on the lower. Contracting them through the propagator yields the t-channel amplitude $\mathcal{M}_t=e^2[\bar u_3\gamma^\mu u_1][\bar u_4\gamma_\mu u_2]/t$ . Why: the photon propagator's $g_{\mu\nu}$ ties the Lorentz index of one current to the other — this is the QED statement that the two electrons interact by their electromagnetic currents.

Step 3. Because the two final electrons are identical, quantum mechanics does not let us say which one "is" $p_3$ . There is a second diagram in which $p_1$ emerges as $p_4$ : the u-channel, obtained from the t-channel by the swap $p_3\leftrightarrow p_4$ , with propagator $1/u$ and currents $[\bar u_4\gamma^\mu u_1][\bar u_3\gamma_\mu u_2]$ . The total amplitude must be antisymmetric under exchanging the two identical fermions, so the u-channel enters with a relative minus sign, giving equation (1). Why: Fermi–Dirac statistics demand that the amplitude change sign when identical fermions are interchanged; the minus sign is not optional bookkeeping but the deep reason like-spin electrons scatter less at $90°$ than distinguishable particles would.

Step 4. For Bhabha, repeat with an incoming positron. The t-channel exchange survives (an electron and positron can trade a photon), but now the "crossed" diagram is the annihilation channel: $e^+e^-\to\gamma^*\to e^+e^-$ , with the timelike propagator $1/s$ . Crossing symmetry relates it to Møller's u-channel by $u\leftrightarrow s$ , and the same antisymmetry fixes the relative sign, giving equation (2). Why: replacing an incoming electron by an outgoing positron (crossing) turns a spacelike u-channel exchange into a timelike s-channel annihilation — the single most important structural difference between the two processes.

Go deeper · Derivation B · 2 minSquaring the amplitude with Casimir's trick and the trace technology

Derivation BSquaring the amplitude with Casimir's trick and the trace technology

The goal is to turn the amplitude of the previous derivation into an unpolarised cross section by summing over final spins and averaging over initial spins — Casimir's trick — which converts spinor bilinears into Dirac traces, and to evaluate those traces in the massless limit to reach the compact Mandelstam form. We work the pure t-channel piece in full and quote the crossing-related u-channel and interference by symmetry.

Step 1. Form the unpolarised square $\overline{|\mathcal{M}|^2}=\tfrac14\sum_{\rm spins}|\mathcal{M}|^2$ , averaging over the $2\times2=4$ initial spin states and summing over final ones. For the pure t-channel term this is $\tfrac14\,\tfrac{e^4}{t^2}\sum[\bar u_3\gamma^\mu u_1][\bar u_1\gamma^\nu u_3]\,[\bar u_4\gamma_\mu u_2][\bar u_2\gamma_\nu u_4]$ , where each bracket has been paired with its Hermitian conjugate. Why: cross sections average over unknown initial polarisations and sum over unresolved final ones, so no spinor is left dangling — every $u$ meets a matching $\bar u$ .

Step 2. Apply the completeness relations $\sum_s u(p)\bar u(p)=\slashed p+m$ and $\sum_s v(p)\bar v(p)=\slashed p-m$ . In the massless limit ( $m\to0$ ) both become $\slashed p$ , and the two spin sums turn the spinor sandwich into a product of two Dirac traces: $\tfrac14\tfrac{e^4}{t^2}\,\mathrm{Tr}[\slashed p_3\gamma^\mu\slashed p_1\gamma^\nu]\,\mathrm{Tr}[\slashed p_4\gamma_\mu\slashed p_2\gamma_\nu]$ . Why: the completeness relation is precisely what converts "sum over spin states" into "close the spinor line into a trace" — this is the mechanism that makes QED cross sections computable without ever solving for individual spinors.

Step 3. Evaluate each trace with $\mathrm{Tr}[\gamma^\alpha\gamma^\mu\gamma^\beta\gamma^\nu]=4(g^{\alpha\mu}g^{\beta\nu}-g^{\alpha\beta}g^{\mu\nu}+g^{\alpha\nu}g^{\mu\beta})$ . Contracting the two traces, the $g^{\mu\nu}$ cross terms cancel and one is left with $\tfrac{8e^4}{t^2}\big[(p_1\!\cdot\!p_2)(p_3\!\cdot\!p_4)+(p_1\!\cdot\!p_4)(p_2\!\cdot\!p_3)\big]$ . Why: the four-gamma trace theorem is the workhorse of aq-pp-trace-techniques; the cancellation of the metric terms is what leaves a clean sum of dot products.

Step 4. Convert dot products to Mandelstam invariants using the massless relations $s=2p_1\!\cdot\!p_2=2p_3\!\cdot\!p_4$ , $t=-2p_1\!\cdot\!p_3$ , $u=-2p_1\!\cdot\!p_4=-2p_2\!\cdot\!p_3$ . Then $(p_1\!\cdot\!p_2)(p_3\!\cdot\!p_4)=s^2/4$ and $(p_1\!\cdot\!p_4)(p_2\!\cdot\!p_3)=u^2/4$ , so the pure t-channel term is $2e^4(s^2+u^2)/t^2$ — the first term of equation (3). The u-channel follows by $t\leftrightarrow u$ , and the interference term $4e^4 s^2/(tu)$ from the same trace machinery, completing the bracket. Why: expressing everything in $s,t,u$ makes the crossing relation to Bhabha ( $s\leftrightarrow u$ ) manifest and reduces the whole calculation to a rational function that any limiting case can be read off directly.

Go deeper · Derivation C · 1 minThe forward divergence and why Bhabha scattering measures luminosity

Derivation CThe forward divergence and why Bhabha scattering measures luminosity

The goal is to extract the small-angle behaviour of the Bhabha cross section, show that the t-channel photon makes it diverge as $1/\theta^4$ exactly like Rutherford and Mott scattering, and explain why that divergence — far from a defect — is precisely what makes small-angle Bhabha scattering the absolute luminosity calibration at every $e^+e^-$ collider.

Step 1. Isolate the t-channel term of equation (4), $\overline{|\mathcal{M}|^2}\supset 2e^4(s^2+u^2)/t^2$ . In the forward limit $\theta\to0$ the momentum transfer $t=-\tfrac{s}{2}(1-\cos\theta)\to-\tfrac{s\theta^2}{4}$ vanishes while $u\to-s$ , so this term behaves as $2e^4\cdot2s^2/t^2\propto 1/\theta^4$ . Why: a massless photon propagator $1/t$ has no scale to cut off small momentum transfers, so nearly-undeflected scattering is overwhelmingly the most probable outcome — the same long-range Coulomb tail that gives Rutherford scattering its $1/\theta^4$ .

Step 2. Feed this into equation (5). The differential cross section becomes $\dfrac{d\sigma}{d\Omega}\approx\dfrac{\alpha^2}{2s}\,\dfrac{1}{\sin^4(\theta/2)}\approx\dfrac{8\alpha^2}{s\,\theta^4}$ , a towering forward peak. The s-channel and interference terms are finite here and merely add a small correction. Why: this reproduces the Mott cross section — relativistic Rutherford scattering — confirming that at small angles Bhabha scattering is just an electron Coulomb-scattering off a positron, with QED spin corrections.

Step 3. Integrate the peak over a narrow forward ring between polar angles $\theta_{\min}$ and $\theta_{\max}$ : $\sigma\approx\dfrac{16\pi\alpha^2}{s}\Big(\dfrac{1}{\theta_{\min}^2}-\dfrac{1}{\theta_{\max}^2}\Big)$ . This is large, sharply defined by the ring geometry, and computable from pure QED to fractions of a percent — the electroweak $Z$ contribution is negligible at small $|t|$ . Why: a cross section that is simultaneously huge (good statistics), geometry-dominated (insensitive to modelling), and theoretically pristine is the ideal "standard candle" for normalising a detector's event rates.

Step 4. Define luminosity operationally. Any process has rate $R=\mathcal{L}\,\sigma$ ; measuring the small-angle Bhabha rate $R_{\rm B}$ and dividing by the calculated $\sigma_{\rm B}$ gives the instantaneous luminosity $\mathcal{L}=R_{\rm B}/\sigma_{\rm B}$ , which then normalises every other cross section the experiment reports. Why: this is exactly how LEP reached per-mille luminosity precision — dedicated forward calorimeters counted Bhabha electrons, turning a QED calculation into the absolute scale for the entire physics programme.

Limiting cases

Case	Result	Meaning
Ultra-relativistic ( $s\gg m_e^2$ )	$s+t+u=0$ ; brackets (3),(4) hold	the massless limit used above; electron mass negligible above a few MeV
Forward ( $\theta\to0$ )	$d\sigma/d\Omega\to 8\alpha^2/(s\theta^4)$	t-channel Coulomb peak (Mott/Rutherford); the luminosity signal — diverges, so no finite total
Right angle ( $\theta=90°$ )	Bhabha $=\tfrac{9}{2}\tfrac{\alpha^2}{2s}$ ; Møller $=18\,\tfrac{\alpha^2}{2s}$	clean benchmark; Møller is $4\times$ Bhabha here — extra u-channel and interference
Møller symmetry	$d\sigma/d\Omega$ invariant under $\theta\to\pi-\theta$	identical final electrons: forward and backward are the same event, count each pair once
Non-relativistic ( $v\to0$ )	$d\sigma/d\Omega\to$ Mott $\propto 1/(v^4\sin^4\tfrac\theta2)$	reduces to spin-corrected Coulomb scattering; mass terms restored
Near $s=M_Z^2$	pure-QED (4) fails; add $Z$ propagator	annihilation channel resonates on the $Z$ — Bhabha becomes an electroweak, not QED, process

Common trapthe two diagrams are separate processes you can add up

The single most common error is to compute the t-channel "rate" and the s- or u-channel "rate" separately and add them. That is wrong twice over. First, amplitudes add, not probabilities: you must sum the diagrams coherently ( $\mathcal{M}=\mathcal{M}_t-\mathcal{M}_{s/u}$ ) and only then square, which produces an interference term — the third term in equations (3) and (4) — that can be as large as either "direct" piece and is negative over much of the angular range. Second, the relative sign between the diagrams is fixed by Fermi–Dirac antisymmetry and is physically real: dropping it, or getting it wrong, changes the $90°$ cross section by tens of percent. A related trap is thinking Møller is just "Bhabha with the positron relabelled" — the crossing $s\leftrightarrow u$ moves a divergent forward-peaking channel into a finite annihilation channel, so the two angular distributions look nothing alike.

Worked examples

1 · Bhabha cross section at a B-factory energy (confidence builder)

A symmetric $e^+e^-$ collider runs at $\sqrt s = 10\ \mathrm{GeV}$ (so $s=100\ \mathrm{GeV}^2$ , safely below the $Z$ ). Find the differential cross section for $e^+e^-\to e^+e^-$ at $\theta=90°$ in nb/sr, and compare it to the forward rate at $\theta=10°$ .

Step 1. Fix the prefactor. With $\alpha=1/137.036$ , $\alpha^2=5.325\times10^{-5}$ , so $\dfrac{\alpha^2}{2s}=\dfrac{5.325\times10^{-5}}{200\ \mathrm{GeV}^2}=2.66\times10^{-7}\ \mathrm{GeV^{-2}}$ . Converting with $(\hbar c)^2=0.3894\ \mathrm{mb\,GeV^2}$ and $1\ \mathrm{mb}=10^6\ \mathrm{nb}$ gives $\dfrac{\alpha^2}{2s}=\mathbf{0.104\ nb/sr}$ . Every angular number is this times the Bhabha bracket.

Step 2. Evaluate the bracket at $\theta=90°$ , where $\theta/2=45°$ and $\sin^2\tfrac\theta2=\cos^2\tfrac\theta2=\tfrac12$ , $\cos\theta=0$ : the t-channel term is $\dfrac{1+1/4}{1/4}=5$ , the interference is $-\dfrac{2(1/4)}{1/2}=-1$ , the s-channel is $\dfrac{1+0}{2}=0.5$ , summing to $\mathbf{4.5}$ . Hence $\dfrac{d\sigma}{d\Omega}\big|_{90°}=0.104\times4.5=\mathbf{0.47\ nb/sr}$ .

Step 3. Now $\theta=10°$ : the forward approximation $\dfrac{1}{\sin^4(\theta/2)}=\dfrac{1}{\sin^4 5°}=\dfrac{1}{(0.0872)^4}\approx1.73\times10^4$ dominates the bracket (the full value is $3.41\times10^4$ once the $u$ -numerator is kept). Thus $\dfrac{d\sigma}{d\Omega}\big|_{10°}\approx0.104\times3.41\times10^4\approx\mathbf{3.5\times10^{3}\ nb/sr}$ .

Step 4. The ratio is the punchline: the forward rate exceeds the $90°$ rate by a factor $\approx 3540/0.47\approx\mathbf{7600}$ . The Bhabha distribution is forward-peaked by nearly four orders of magnitude between $10°$ and $90°$ — which is why a small-angle detector, not a $90°$ one, is used to count luminosity events.

2 · Møller versus Bhabha at right angles

Compare the two elastic cross sections at $\theta=90°$ and $\sqrt s=10\ \mathrm{GeV}$ , and explain the difference physically.

Step 1. The Møller bracket at $\theta=90°$ is $\dfrac{2(3+\cos^2\theta)^2}{\sin^4\theta}=\dfrac{2(3)^2}{1}=18$ , so $\dfrac{d\sigma}{d\Omega}\big|_{ee}=0.104\times18=\mathbf{1.87\ nb/sr}$ . Equivalently, in Mandelstam form each of the three terms of (3) equals $5,5,8$ (t, u, interference) at $\theta=90°$ , summing to $18$ .

Step 2. Bhabha gave $4.5$ (from Step 2 above), i.e. $0.47$ nb/sr. The ratio is $18/4.5=\mathbf{4}$ : Møller scattering is exactly four times larger than Bhabha at $90°$ for the same energy.

Step 3. Physically, Møller has two large exchange channels (t and u), each contributing a " $5$ ", plus a strongly constructive interference (" $+8$ "), because both channels peak at the edges and reinforce in between. Bhabha replaces the u-channel with a modest annihilation floor (" $+0.5$ ") and the interference is destructive (" $-1$ "), pulling the total down. The identical-particle structure of Møller is what makes it the bigger cross section away from the forward direction, while both share the same divergent forward Coulomb peak.

Physics of the portrait

The two processes carry the names of the men who first computed them on Dirac's brand-new theory of the electron and positron — Christian Møller for the like-charge case in 1932, and Homi Bhabha for the electron–positron case in 1936. Bhabha's calculation, done as a young theorist in Cambridge, was the first to treat the annihilation channel and its interference, and the forward-peaked distribution he derived is still measured billions of times a year in collider luminosity monitors.

Origins · 1 minHomi J. Bhabha and how this was discovered

Homi J. Bhabha

1909–1966 · QED scattering and the founding of Indian physics

Homi Jehangir Bhabha computed the electron–positron scattering cross section in 1936, in a paper titled "The Scattering of Positrons by Electrons with Exchange on Dirac's Theory of the Positron," written while he held a research studentship at Cambridge under Ralph Fowler and worked alongside Dirac and Kapitza. His achievement was conceptual as much as computational: Dirac's positron had been experimentally confirmed by Anderson only in 1932, and Bhabha was among the first to take its scattering seriously, recognising that an electron and a positron could interact both by exchanging a photon and by annihilating into one — and that the two amplitudes must be added coherently, interference and all. The forward-peaked angular distribution that now bears his name is the everyday tool of collider physics. Bhabha went on, with Walter Heitler in 1937, to explain the cascade multiplication of cosmic-ray showers, quantitatively predicting the number of secondary electrons and photons as a function of depth — work that underlies every calorimeter built since. Returning to India, he founded the Tata Institute of Fundamental Research in 1945 and the Atomic Energy Establishment at Trombay (later renamed the Bhabha Atomic Research Centre) in 1954, effectively architecting Indian physics as an institution. He died in the 1966 crash of Air India Flight 101 on Mont Blanc. The process he calculated remains, ninety years on, one of the most precisely known predictions in all of physics.

History

1928

— Dirac's relativistic equation implies a positive-energy antiparticle for the electron

1929

— Mott derives the relativistic Coulomb scattering of electrons (the forward-peak limit)

1932

— Møller publishes the e⁻e⁻ elastic scattering cross section from Dirac's theory

1936

— Bhabha computes e⁺e⁻ scattering including the annihilation channel and interference

1949

— Feynman's diagram rules make the coherent two-diagram structure systematic

1989

— LEP adopts small-angle Bhabha scattering as its absolute luminosity standard

Going further

Texts. Peskin & Schroeder, An Introduction to Quantum Field Theory, §5.4 and Problem 5.3 (massless Bhabha and Møller) · Halzen & Martin, Quarks and Leptons, Ch. 6 (Bhabha and the QED scattering catalogue) · Griffiths, Introduction to Elementary Particles, 2nd ed., Ch. 7–8 · Thomson, Modern Particle Physics, Ch. 6 · Schwartz, Quantum Field Theory and the Standard Model, Ch. 13.

Provenance. The Dirac spinors $u,v$ and their normalisation come from aq-pp-dirac-solutions and aq-pp-dirac-equation; the gamma-matrix algebra $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ from aq-pp-gamma-matrices-clifford-algebra; the vertex and propagator factors from aq-pp-feynman-rules-qed; the antiparticle spinors from aq-pp-antiparticles; and the spin-sum-to-trace machinery of Derivation B is owned by aq-pp-matrix-element-spin-averaging and aq-pp-trace-techniques (recalled here, not re-derived). A verifier should confirm: (1) equations (3) and (4) are related by $s\leftrightarrow u$ ; (2) the pure t-channel trace gives $2e^4(s^2+u^2)/t^2$ ; (3) $s+t+u=0$ massless with $t=-\tfrac{s}{2}(1-\cos\theta)$ ; (4) the numeric prefactor $\alpha^2/2s=0.104\ \mathrm{nb/sr}$ at $\sqrt s=10$ GeV, giving Bhabha $0.47$ and Møller $1.87$ nb/sr at $90°$ .

Prerequisites. The Dirac Equation · Gamma Matrices and the Clifford Algebra · Solutions to the Dirac Equation · Feynman Rules for QED · Antiparticles. Leads to. $e^+e^-\to\mu^+\mu^-$ · Spin Averaging of Matrix Elements · Trace Techniques · Compton Scattering.

Ask the tutor: Want me to walk through the full trace algebra for the Bhabha interference term, or show how the small-angle formula becomes a working luminosity measurement?

Why does Bhabha have an annihilation channel but Møller does not?

Because charge and lepton number must be conserved at every vertex. In Møller scattering both particles are electrons, carrying the same charge and the same lepton number, so there is no way for them to annihilate — a two-electron state cannot turn into a neutral photon. They can only exchange a photon, and since the two outgoing electrons are identical there are two ways to do that (t- and u-channel). In Bhabha scattering the positron is the electron's antiparticle: together they carry zero net charge and zero net lepton number, exactly the quantum numbers of a photon, so they can annihilate into a virtual photon that reappears as a fresh pair. That extra s-channel diagram — and its interference with the t-channel — is the entire difference between the two processes.

Where exactly does the relative minus sign between the diagrams come from?

From the antisymmetry of fermionic amplitudes under exchange of identical particles. In field theory the electron field anticommutes with itself, so when you compute the matrix element the two ways of contracting identical external fermions differ by a sign. In Møller scattering the two final electrons are identical, so the u-channel (where the roles of $p_3$ and $p_4$ are swapped) enters with a minus relative to the t-channel. In Bhabha scattering the sign sits between the t-channel and the s-channel, and it traces to the same anticommuting statistics applied to the electron and the annihilating positron line. It is not a bookkeeping convention: the sign makes the interference term negative over much of the angular range and changes measurable rates by tens of percent.

Why does the total cross section diverge, and is that a problem?

The t-channel photon is massless, so its propagator $1/t$ has no scale to tame small momentum transfers. As the scattering angle goes to zero, $t\to0$ and $d\sigma/d\Omega\propto 1/\theta^4$ blows up; integrating over all angles gives infinity. This is not a pathology of QED but a genuine feature of long-range Coulomb forces — classical Rutherford scattering has the same divergent total cross section. It is never a problem in practice because no real detector covers $\theta=0$ : there is always a minimum angle set by the beampipe, and the measured cross section, integrated from that angle outward, is perfectly finite. Indeed the divergence is what makes the forward rate so large and hence so useful as a luminosity monitor.

How is Bhabha scattering actually used to measure luminosity?

Luminosity $\mathcal{L}$ is the proportionality between a process's rate and its cross section, $R=\mathcal{L}\sigma$ . To find $\mathcal{L}$ you need a process whose $\sigma$ you know exactly and whose $R$ you can count cleanly. Small-angle Bhabha scattering is ideal: its cross section is pure QED, calculable now to two loops with sub-per-mille uncertainty, and its forward peak is enormous, so a dedicated ring of calorimeters just outside the beampipe records millions of events. Dividing that measured rate by the calculated cross section gives $\mathcal{L}$ , which then normalises every other measurement the experiment makes. At LEP this technique reached $0.1\%$ precision and set the absolute scale for the electroweak precision tests, including the count of light neutrino generations.

How do these differ from e⁺e⁻ → μ⁺μ⁻?

The muon-pair process $e^+e^-\to\mu^+\mu^-$ has only the s-channel annihilation diagram: an electron and positron cannot exchange a photon with a muon in the final state, because the incoming and outgoing particles are different species, so there is no t-channel. That makes it the cleanest QED reaction of all — a single diagram, no interference, and the famous $d\sigma/d\Omega\propto(1+\cos^2\theta)$ distribution with total cross section $\sigma=4\pi\alpha^2/3s$ . Bhabha scattering contains that same annihilation term (its s-channel piece and the $\tfrac{1+\cos^2\theta}{2}$ floor in the compact formula) but adds the t-channel exchange and its interference. So $e^+e^-\to\mu^+\mu^-$ is, in effect, Bhabha scattering with the exchange channel switched off — which is exactly why it is taught first, in aq-pp-ee-to-mumu.

Does the electron mass ever matter for these cross sections?

For most collider applications, no — at centre-of-mass energies of even a few GeV, $s\gg m_e^2$ and the massless formulae (3) and (4) are accurate to parts in $10^{8}$ . The mass corrections scale as $m_e^2/s$ and are utterly negligible. Two situations restore the mass. First, in the exact forward and backward directions the massless propagators would need regulating, and the finite electron mass provides a physical (if tiny) cutoff. Second, at low energy — non-relativistic Møller scattering, or in atomic and condensed-matter contexts — the mass dominates the kinematics and the cross section reverts to the Mott formula, $d\sigma/d\Omega\propto 1/(v^4\sin^4\tfrac\theta2)$ , with explicit $m_e$ dependence. The massless limit is a statement about high energy, not a universal truth.

Why does the pure-QED prediction fail near the Z resonance?

Because the s-channel virtual photon is not the only neutral boson an $e^+e^-$ pair can annihilate into — it can also form a virtual $Z^0$ . Far below the $Z$ mass ( $\sqrt s\ll 91$ GeV) the $Z$ propagator $1/(s-M_Z^2)$ is small and its contribution is a negligible electroweak correction, so pure QED suffices. But as $\sqrt s$ approaches $M_Z$ , the $Z$ propagator resonates and the annihilation channel is dominated by the $Z$ , not the photon; the cross section rises by orders of magnitude and acquires parity-violating spin dependence. There Bhabha scattering is no longer a QED process but a full electroweak one, and computing it requires the $Z$ diagram alongside the photon. This is precisely the sensitivity that low-energy Møller experiments exploit to measure the weak mixing angle.