Once found, these unfathomable rules have helped particle physicists calculate scattering amplitudes with much higher levels of precision than they could achieve with the traditional approach. The restructuring also allowed Dixon and his collaborators to discover the hidden connection between the two seemingly unrelated scattering amplitudes.
Antipode Map
The core of duality is the ‘antipode map’. In geometry, an antipode map takes a point on a sphere and reverses its coordinates, sending you straight through the center of the sphere to a point on the other side. It’s the mathematical equivalent of digging a hole from Chile to China.
In scattering amplitudes, the antipodal map that Dixon found is a bit more abstract. It reverses the order of the letters used to calculate the amplitude. Apply this antipode map to all terms in the scattering amplitude for two gluons becoming four, and (after a simple change of variables) it yields the amplitude for two gluons becoming one gluon plus a Higgs.
In the DNA analogy of Dixon, duality is like reading a genetic sequence backwards and realizing that it encodes a totally new protein that is unrelated to the protein encoded by the original sequence.
“We used to be convinced that the antipode map was useless. It didn’t seem to have any physical meaning, or do anything meaningful,” said Matt von Hippel, an amplitude specialist at the Niels Bohr Institute in Copenhagen who was not involved in the study. “And now there’s a totally inexplicable duality that uses it, which is pretty wild.”
Not quite our world
There are now two big questions. First, why does duality exist? And second, will a similar connection be found in the real world?
The 17 known elementary particles that make up our world adhere to a series of equations called the Standard Model of Particle Physics. According to the Standard Model, two gluons, the massless particles that glue atomic nuclei together, easily interact with each other to double their own number, making them four gluons. However, to produce one gluon and one Higgs boson, colliding gluons must first turn into a quark and an antiquark; these then transform into a gluon and a Higgs through a force other than the force that determines the mutual interactions of gluons.
These two scattering processes are so different, with one encompassing an entirely different sector from the Standard Model, that a duality between the two would be very surprising.
But the antipodal duality is also unexpected, even in the simplified model of particle physics that Dixon and his colleagues studied. Their toy model controls fictitious gluons with additional symmetries, allowing for more accurate calculations of scattering amplitudes. The duality links a scattering process involving these gluons and a process requiring an external interaction with particles described by another theory.
Dixon thinks he has a very vague idea of where duality comes from.
Think of those inexplicable rules Volovich and her colleagues found that govern what combinations of words are allowed in a scattering amplitude. Some rules seem to arbitrarily limit which letters can appear next to each other in the amplitude from two gluons to gluon plus Higgs. But assign those rules to the other side of duality, and transform them into a set of established rules that provide causality – guarantee that the interactions between incoming particles take place before the outgoing particles appear.
For Dixon, this is a small hint at a deeper physical connection between the two amplitudes, and a reason to think something similar might hold in the Standard Model. “But it’s rather weak,” he said. “It’s like second-hand information.”
Other dualities between disparate physical phenomena have already been found. The AdS-CFT correspondence, for example, in which a theoretical world without gravity is double to a world with gravity, has fueled thousands of research papers since its discovery in 1997. But this duality also only exists for a gravitational world with a distorted geometry as opposed to that of the actual universe. Yet the fact that multiple dualities almost exist in our world suggests to many physicists that they could scratch the surface of an all-encompassing theoretical structure in which these surprising connections manifest. “I think they’re all part of the story,” Dixon said.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to advance public understanding of science by covering research developments and trends in mathematics and the natural and life sciences.