New Particle Collision Math May Offer Quantum Clues
The Mathematical Structure of Particle Collisions Comes Into View | Quanta Magazine

    When particle physicists try to model experiments, they confront an impossible calculation — an infinitely long equation that lies beyond the reach of modern mathematics. 
Fortunately, they can generate largely accurate predictions without seeing this arcane math all the way through. By cutting the calculation short, scientists at CERN’s Large Hadron Collider in Europe make forecasts that match events they actually observe when they send subatomic particles barreling toward each other around a nearly 17-mile track.
Unfortunately, the era of agreement between forecast and observation may be ending. As measurements grow more precise, the approximation schemes theorists use to make predictions may not be able to keep up.
“We’re getting close to exhausting what can be done,” said Claude Duhr, a particle physicist at CERN. 
But three recent papers from a group of physicists led by Pierpaolo Mastrolia of the University of Padua in Italy and Sebastian Mizera of the Institute for Advanced Study in Princeton, New Jersey, have revealed an underlying mathematical structure in the equations. The structure provides a new way of collapsing interminable terms into just dozens of essential components. Their method may help bring about new levels of predictive accuracy, which theorists desperately need if they are to move beyond the leading but incomplete model of particle physics.
“They have delivered lots of proof-of-concept results which show that this is a very promising technique,” Duhr said.
There could be a bigger payoff than improved predictions. 
The new method skirts the traditional mathematical slog by directly computing “intersection numbers,” which some hope could eventually lead to a more elegant description of the subatomic world. 
“This is something that’s not just mathematics,” said Simon Caron-Huot of McGill University, a quantum theorist who is studying the implications of Mastrolia and Mizera’s work. “It’s something that’s deeply baked into quantum field theory.” 
An Infinite Loop
When physicists model particle collisions they use a tool called a Feynman diagram, a simple schematic invented by Richard Feynman in the 1940s.
To get a feel for these diagrams, consider a simple particle event: Two quarks streak in, exchange a single gluon as they “collide,” then bounce away on their separate trajectories.
In a Feynman diagram the quarks’ paths are represented by “legs,” which join to form “vertices” when particles interact. Feynman developed rules for turning this cartoon into an equation which calculates the probability that the event actually takes place: You write a specific function for each leg and vertex — generally a fraction involving the particle’s mass and momentum — and multiply everything together. For straightforward scenarios like this one, the calculation might fit on a cocktail napkin. 

        
            
            Samuel Velasco/Quanta Magazine
        
    

But the golden rule of quantum theory is to consider all possibilities, and exchanging a simple gluon represents just one among a vast landscape of scenarios that could unfold when two quarks collide. The exchanged gluon might momentarily split into a “virtual” quark pair, for instance, before reconstituting itself in a flash. Two quarks enter and two quarks leave, but a lot can happen in the middle. A full accounting, implying a perfect prediction, would demand an infinite number of diagrams. No one expects perfection, but the key to improving a calculation’s precision is getting further along in the infinite line of events. 
And that’s where physicists are getting stuck. 
Zooming in to that hidden center involves virtual particles — quantum fluctuations that subtly influence each interaction’s outcome. The fleeting existence of the quark pair above, like many virtual events, is represented by a Feynman diagram with a closed “loop.” Loops confound physicists — they’re black boxes that introduce additional layers of infinite scenarios. To tally the possibilities implied by a loop, theorists must turn to a summing operation known as an integral. These integrals take on monstrous proportions in multi-loop Feynman diagrams, which come into play as researchers march down the line and fold in more complicated virtual interactions. 
Physicists have algorithms to compute the probabilities of no-loop and one-loop scenarios, but many two-loop collisions bring computers to their knees. This imposes a ceiling on predictive precision — and on how well physicists can understand what quantum theory says. 

