Scattering Amplitude

When two particles collide, many things can happen: they can fly past each other, be deflected, or create new particles. The scattering amplitude is a number – more precisely, a complex number with magnitude and phase – that describes how “strong” a particular outcome of this collision is. Squaring its magnitude yields the probability that exactly this outcome actually occurs. The scattering amplitude is therefore not a directly measurable quantity, but a computational step on the way to a prediction.

Scattering amplitudes as the heart of particle physics

Particle physics explains what matter is made of and how its smallest building blocks interact with each other. But theories alone are not enough – one needs numbers that can be compared with experiments. This is exactly what scattering amplitudes provide. They translate the rules of a physical theory into concrete probabilities for measurement outcomes.

Without them, it would be impossible to say how often a particular particle should be produced in a collision. If the calculated number matches what detectors at the particle accelerator measure, the theory is confirmed. If it deviates, this points to new physics – or to a calculation error.

Feynman diagrams and modern computational methods

Classically, scattering amplitudes are calculated using Feynman diagrams. These are schematic drawings that depict which particles exist before and after a collision and how they exchange force particles in between – for example, photons in the exchange of electromagnetic force. Each diagram corresponds to a mathematical term, and the sum of all terms yields the amplitude.

The problem: as precision increases, the number of required diagrams grows explosively. A higher-order collision can require millions of diagrams, which is demanding even for supercomputers. Since the 1980s, physicists have therefore been working on more elegant methods. One of these is the BCFW method: it calculates complex amplitudes recursively from simpler ones, without having to draw out every intermediate step individually. This makes it possible to obtain results for which the classical method would require pages full of terms.

Even more radical is the concept of the amplituhedron, which physicist Nima Arkani-Hamed introduced in 2013. It is a geometric figure in an abstract mathematical space whose volume directly yields the scattering amplitude – entirely without Feynman diagrams. Whether this approach will one day become practically usable remains an open question, but it shows how deeply geometry and physics can be interconnected.

Scattering amplitudes in everyday accelerator work and in AI research

At the Large Hadron Collider (LHC) at CERN in Geneva, billions of protons collide with each other every day. In order for physicists to know what to look for in the enormous detector data, they need precise predictions – and these come from scattering amplitudes. The discovery of the Higgs boson in 2012 would not have been possible at all without such calculations, because without a prediction one would not know which signal in the mass of data is even conspicuous.

More recently, the topic has also become interesting for machine learning. Researchers are training neural networks – i.e., AI systems that learn from examples – to approximate scattering amplitudes faster than classical computational methods allow. This could considerably speed up the evaluation of collision data. Conversely, some AI researchers use concepts from amplitude research to better understand structures in high-dimensional data.

Scattering amplitudes also appear in popular science reports on quantum gravity and string theory. There, attempts are made to calculate the scattering of gravitons – hypothetical particles of the gravitational force. This remains unsolved so far, but is considered a key to a unified theory of all forces of nature.