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§ Phase xiii · Feynman Diagrams · Chapter 02

Electron-electron scattering

Two electrons fly toward each other in an empty vacuum. They never quite touch. Somewhere between them a photon flickers in and out of existence, an unseen messenger that carries the kick from one charge to the other. In the spring of 1932 a young Dane in Copenhagen wrote down the formula for that handshake, and the calculation he produced is still the first real problem any graduate student in quantum electrodynamics is asked to solve.

Picture the simplest possible particle-physics experiment. You have two electrons. They are both negatively charged, so they repel. You aim them at each other on a near-miss trajectory, fire them off, and a moment later they come out the other side, moving in directions different from where they started. That is the whole event. Nothing complicated happens. No new particle is created. No mass is converted into energy. There is no exotic flash. Two electrons go in and two electrons come out, slightly redirected. The previous chapter taught the language of Feynman diagrams in the abstract. This chapter is about reading the very first diagram any working physicist reads, the one that describes this scenario in full.

In classical physics the calculation is straightforward and ancient. You write down the Coulomb potential between two point charges, integrate the equations of motion, and out comes the Rutherford scattering formula that Ernest Rutherford himself worked out in 1911 to explain alpha particles bouncing off a gold foil. It is a beautiful piece of mathematics. It is also wrong in detail. It treats the electromagnetic field as something rigid and instantaneous, a fixed potential that just sits there between the charges. It treats the electron as a tiny billiard ball with a definite position. It misses spin entirely. And it misses, completely, the small but crucial fact that electromagnetic influence does not propagate instantly. It must be carried, and the thing that carries it must itself be quantum.

By the early 1930s the new quantum mechanics had absorbed Paul Dirac’s relativistic equation for the electron and was beginning to take seriously the idea that the electromagnetic field, too, was quantum. The carrier of the field, the photon, was not just a fiction useful for explaining the photoelectric effect; it was a particle in the proper sense, with definite energy and momentum, that could be emitted and absorbed in elementary acts. If you had two electrons, then, the way they pushed on each other was not via some abstract potential. It was via the exchange of a photon. One electron emits the photon, the other absorbs it, and that exchange is the entire mechanism of the force. The young theorist who first turned this picture into a working calculation was a twenty-eight-year-old Dane named Christian Møller, working under Niels Bohr in Copenhagen.

The Møller calculation is conceptually clean. You start with two incoming electrons, with four-momenta p₁ and p₂, flying toward each other. You end with two outgoing electrons, with four-momenta p₃ and p₄, flying apart. Conservation of energy and momentum tells you that p₁ + p₂ = p₃ + p₄. You then ask: what is the probability that this scattering happens at any given pair of final angles? In quantum field theory you do not compute the probability directly. You compute an amplitude, a complex number whose squared magnitude is the probability density, and you do it by summing over every Feynman diagram that begins with two electrons and ends with two electrons.

At the lowest order of approximation, the so-called tree level, there are only two diagrams. The clean modern way to draw them is to place time running left to right, with the two incoming electrons coming in from the left and the two outgoing electrons going out to the right. The first diagram has electron one emit a photon, which then travels across the page and is absorbed by electron two. The second diagram is similar, but with the two outgoing electrons crossed: electron one ends up where electron two would have gone, and vice versa. Both diagrams must be included because the electrons are identical particles, and quantum mechanics has no way to tell which outgoing electron came from which incoming one. The total amplitude is the sum of the two contributions, with a relative minus sign that arises from the antisymmetry of fermions under exchange. The probability is the absolute square of that sum.

That quote is the modern, textbook version of what Møller worked out in 1932 without the diagrams. In Møller’s paper there were no pictures. There were Dirac spinors, gamma matrices, and Lorentz-invariant traces, page after page of them. He proceeded by writing down the matrix element of the relativistic Coulomb interaction between two electron currents, then carefully constructed the corresponding scattering cross-section using the Born approximation extended to relativistic kinematics. The result, when squared and integrated over the appropriate phase space, gave a definite prediction for how many electrons would scatter into any given angular bin per unit time per unit incident flux. The formula was complicated. It depended on the centre-of-mass energy, the scattering angle, and the spin of the incoming and outgoing electrons. But it was, as far as anybody could tell, the right answer.

What Feynman did sixteen years later, in his 1949 paper Space-Time Approach to Quantum Electrodynamics, was to take the same physical input and replace the pages of algebra with a picture. Every Feynman diagram is a recipe. The two external solid lines coming in from the left say “put in a Dirac spinor for each incoming electron.” The two external solid lines going out to the right say “put in a Dirac spinor for each outgoing electron.” The internal wavy line in the middle says “put in a photon propagator, the function that describes how a virtual photon carries momentum between two points.” The two vertices, where the wavy line touches the solid lines, each say “multiply by −ie γ^μ, the coupling constant times the gamma matrix that mediates the electron-photon interaction.” Multiply the pieces together, contract the Lorentz indices, and you have the contribution of that diagram to the amplitude. Add the two diagrams, square, and integrate. The result is identical to Møller’s, but the steps are now a sort of pictorial bookkeeping rather than a thicket of algebra.

The relative minus sign between the two diagrams is the most important rule that comes from quantum statistics. Electrons are fermions. If you swap any two identical fermions in the wavefunction of a system, the wavefunction picks up a minus sign. The two outgoing legs of the second Møller diagram are exactly such a swap of the legs of the first. So when you add their amplitudes, the second comes in with a minus sign relative to the first. Square the sum and you get an interference pattern: a piece that looks like the t-channel by itself, a piece that looks like the u-channel by itself, and a cross-term that comes from the two diagrams interfering with each other. The cross-term is what makes electron-electron scattering different from any classical Coulomb calculation. You cannot get it without taking the quantum-statistical antisymmetry seriously.

So far we have looked at two electrons, two negative charges. What about an electron and a positron? The positron is the electron’s antiparticle, with the same mass and the same spin but the opposite charge. The first observation of the positron, by the American physicist Carl Anderson in 1932, came barely a year before Møller’s paper, and it left an immediate question hanging in the literature: what does the scattering formula look like for an electron and a positron? The answer was worked out in 1936 by the young Indian physicist Homi Jehangir Bhabha, then a research scholar at Caius College, Cambridge. The process he treated, e⁻ + e⁺ → e⁻ + e⁺, is called Bhabha scattering, and it is the natural sibling of Møller scattering in every modern QED course.

The first thing you notice is that the kinematics are almost identical. An incoming electron with momentum p₁ and an incoming positron with momentum p₂ collide and produce an outgoing electron with momentum p₃ and an outgoing positron with momentum p₄. The conservation rules are the same. The Feynman rules for the vertices and the photon propagator are the same. There is again a t-channel diagram, in which the electron and the positron exchange a virtual photon and continue on their way without changing identity. But because the incoming particles are now distinct, an electron and a positron, the u-channel “swap the outgoing electrons” trick of Møller no longer applies. The outgoing electron is the only one that could have come from the incoming electron line; the outgoing positron is the only one that could have come from the incoming positron line. There is nothing to swap.

What Bhabha realised, however, was that Bhabha scattering has a new diagram of its own. The electron and the positron can annihilate each other. The energy of the collision becomes, for one fleeting instant, a virtual photon. That photon then re-creates an electron-positron pair, which separates and flies out as the two final-state particles. This is called the s-channel diagram, because the squared four-momentum carried by the virtual photon is now s = (p₁ + p₂)², the total centre-of-mass energy squared. The two original diagrams of Møller scattering plus this one new annihilation diagram make three diagrams in total. The total tree-level Bhabha amplitude is the sum of all three, and the squared sum gives the cross-section that you can measure in a real collider.

This is the rhythm of every QED calculation. You list the diagrams. You write the amplitude that each one stands for. You add them, with the appropriate signs from quantum statistics. You square the sum and integrate over the bits of phase space that the experiment does not resolve. The output is a cross-section, a number with the dimensions of an area, that tells you how big an effective target each electron presents to the other. At low energies the Møller cross-section reduces, beautifully, to the Mott formula and from there to the Rutherford formula in the non-relativistic limit. At high energies the relativistic and spin effects kick in and the prediction deviates from the classical answer. In every regime the prediction can be checked against experiment, and in every regime it works to roughly one percent at tree level.

−i g^μν / q²

iℳ_t  =  ū(p₃) (−ie γ^μ) u(p₁)  ·  ( −i g_μν / t )  ·  ū(p₄) (−ie γ^ν) u(p₂)

iℳ_u  =  −  ū(p₄) (−ie γ^μ) u(p₁)  ·  ( −i g_μν / u )  ·  ū(p₃) (−ie γ^ν) u(p₂)

dσ/dΩ  =  α² / (2 s)  ·  [ (s² + u²) / t²  +  (s² + t²) / u²  +  2 s² / (t u) ]

The pattern that holds for Møller and Bhabha holds, with minor adjustments, for almost every elementary process in QED. Compton scattering of a photon off an electron has two tree diagrams. Pair production of an electron-positron pair by two photons has two tree diagrams. Bremsstrahlung, the radiation a charged particle emits when it accelerates in an external field, has two tree diagrams. In each case you draw the picture, apply the rules, sum the amplitudes, square, integrate. At leading order the answers come out within roughly one percent of experiment, which is already astonishing given that we started by writing down two pictures.

If you have followed the chapter this far, you have done something subtle. You have stopped thinking of an electromagnetic force as a vague push and started thinking of it as an exchange of quanta, each one represented by an internal line in a diagram you can actually draw. You have learned that the diagram is not a literal picture of a flight path but a mnemonic for a piece of arithmetic. You have seen that identical fermions force a minus sign on you, and that the minus sign produces a real, measurable interference term in the cross-section. You have seen that a particle and its antiparticle can annihilate and re-emerge, which is the first proper hint of what quantum field theory really is: a theory in which particle number is not a conserved input but a dynamical variable that can change at vertices.

The Møller and Bhabha calculations are not just exercises. They are the workhorses of every electron-positron collider on Earth. Bhabha scattering is the standard luminosity monitor at any e⁻e⁺ machine, because its cross-section is known from QED to a precision the experimentalists could not hope to compute from first principles in any other way. You count the Bhabha events at very small forward angles, you compare to the QED prediction, you extract the luminosity. Everything else the collider measures, every new-physics result that the LEP collaborations and BaBar and Belle and CESR have ever reported, has been calibrated, in the end, against the cross-section that Christian Møller and Homi Bhabha worked out before the war. That is what it means for a calculation to be a workhorse: not that the theorist celebrates it, but that the experimentalist quietly trusts it, day in and day out, every time the beam comes on.

What we have not yet done is touch a force other than electromagnetism. In the very next chapter we will redraw essentially the same kind of diagram for a process where the exchanged particle is not a photon but a W boson, and the incoming and outgoing particles are not all electrons. The geometry will look familiar. The bookkeeping will follow the same rules, with the propagator and the vertex factor swapped out. The result will explain something that had been a mystery for forty years before Feynman drew a picture of it: why a free neutron sitting in empty space lives for fifteen minutes and then quietly turns into a proton, an electron, and a thing nobody had yet learned how to see.