A scattering amplitude describes the probability for a specific scattering process to happen. The probability can in turn be measured by a high-energy experiment (like the detectors at the LHC). This makes scattering amplitudes the link between theory and experiment. To test a theory means to check whether the scattering amplitudes computed from it match the measured probabilities. Computing scattering amplitudes, therefore, has a very practical purpose and computing them efficiently is important in order to get results to a precision matching that of modern day experiments. However, scattering amplitudes are also interesting from a theoretical perspective and modern computational methods have revealed surprisingly simple structures that perhaps can tell us something about the underlying theory.
Modern Computational Methods
There exists a well-known method for computing scattering amplitudes in Quantum Field Theories: Feynman rules. There are two steps to using Feynman rules:
One begins by drawing all possible diagrams associated with the relevant scattering process (called Feynman diagrams).
Then one computes the mathematical quantities associated with each diagram and sum them up.
Feynman rules have been known for decades and remain the standard textbook method for computing scattering amplitudes. So why are we interested in other methods?
It turns out that Feynman rules can be rather cumbersome. The number of Feynman diagrams can be huge and yet the result be simple. When a result is much simpler that the calculation leading up to it, it is a hint that there is an easier way.
The modern methods for computing scattering amplitudes try to make the calculations as simple as possible. Often these methods exploit previously found results, e.g. by constructing recursion relations, such that amplitudes involving n+1 particles are written in terms of amplitudes involving n or less particles. The methods have both practical and theoretical implications. One can try to implement the methods into computer algorithms that calculate probabilities to compare with experiments. Or one can use them to expose previously unknown symmetries in some of the more well-behaved models that theorists are interested in.
Gauge theories are essential to our understanding of the fundamental interactions of nature. Quantum Electrodynamics and Quantum Chromodynamics are examples of realistic gauge theories relevant to experiments while N=4 super Yang-Mills is an example of a very simple theory, that theorists study because of its many unique features.
The modern approach to scattering amplitudes can be useful to both the realistic models as well as the simple ones. In the case of N=4 super Yang-Mills, it has been used to find new symmetries in the scattering amplitudes.
Gravity theories are some of those theories where cumbersome Feynman diagram calculations can lead to surprisingly simple results. Even very simple scattering processes require long calculations. This is an area where modern computational methods have lead to some remarkable progress.
One key insight is that gravity theories behave like the square of gauge theories. This has made it possible to compute scattering amplitudes in gravity theories to a precision unattainable through standard Feynman rules. One of the goals of such calculations it to understand the occurrence of divergences and whether one of these quantum field theories of gravity is free of divergences.