NEXT year a machine that promises to change forever our understanding of matter will be started up. Straddling the border between France and Switzerland, the Large Hadron Collider will smash protons together at ridiculously high energies. Particle physicists everywhere are hoping it will reveal the origin of mass, uncover the first evidence of supersymmetry and even expose the nature of dark matter. There’s just one fly in the ointment…
It is becoming clear that these exotic new processes will leave calling cards in the form of splashes of energy and trails of charge that are distressingly similar to those of run-of-the mill processes. The maths simply isn’t up to distinguishing between supersymmetry, for example, and ordinary collisions. Without accurate calculations to tell them how many of each reaction to expect, particle physicists at the LHC won’t be able to work out what they have found. Extra dimensions and mini black holes might be forming right under their noses without anyone noticing.
All is not lost, however, because an old idea called twistor theory may yet come to the rescue. Theorists now believe that with “twistors” they can vastly simplify the calculations needed to predict the results of experiments at the LHC. “Things that were much too hard to do even with a supercomputer, now you need just a good postdoc and a piece of paper,” explains Zvi Bern, a theoretical physicist at the University of California, Los Angeles.
Advertisement
“Twistor theory may yet come to the rescue”
In spite of Albert Einstein’s famous declaration that god does not play dice, modern physicists believe that is exactly what he does. A collider experiment is nothing more than a chance to watch god roll the dice many billions of times and in this way deduce the properties of the dice. At the LHC, the dice are proton-proton collisions that lead to the production of myriad conventional particles and perhaps even something new.
Take the infamous Higgs boson, the hypothetical particle that could explain how most subatomic particles get their mass. According to current theories, there are many ways to make a Higgs at the LHC. Among them are the fusion of two gluons to produce a Higgs and the fusion of two heavyweight particles called W and Z bosons.
The Higgs itself is ephemeral, living for a fleeting moment before decaying in one of numerous different ways. Each leaves behind its own signature in sprays of particles called jets, for example, or ghostly neutrinos that leave no trace at all. And, of course, there are lots of ways that the protons could interact and not produce a Higgs. Because high-energy physics is a game of probabilities, you need to calculate both the frequency of these normal “background” events and the expected frequency of rare events involving the Higgs particle.
One way to test for the presence of a Higgs involves comparing how frequently you observe a certain configuration of jets with the rate predicted by the standard model of particle physics. Any discrepancy between observations and the theory is taken as a sign of a possible Higgs. This means that physicists need to understand the standard model’s predictions in great detail before they can make claims about any new discoveries.
Such predictions rely on being able to calculate what is called the scattering amplitude of each possible set of incoming and outgoing particles. The amplitude represents the probability of observing a particular configuration of particles in an experiment.
For more than 50 years, physicists have been computing amplitudes using Feynman diagrams, named after the physicist Richard Feynman. These depict moving particles as straight lines and the forces that act between them as wavy lines.
The stick-like illustrations are more than just pictures, of course: Feynman diagrams keep track of all the possible ways of getting from the observed incoming particles to the observed outgoing particles. So to work out how frequently a collision can produce a certain outcome, the trick is to write down all the possible Feynman diagrams, calculate their amplitudes and then add them all together. Feynman himself called the idea a “sum over all possible histories”.
His approach works well for collisions between electrons and their antimatter counterparts, positrons. Yet it is almost impossible for proton-proton collisions. This is because a proton is a much more complicated particle than an electron: it is made up of three quarks bound together very strongly by gluons. “The problem with protons is that they are like Swiss watches,” says Lance Dixon, a theoretical physicist at the Stanford Linear Accelerator Center in California.
All of this would be fine if experimenters were only going to look at processes that produce four gluons or fewer. But because of its high collision energy, interactions of up to 10 gluons will be the norm for the LHC. As the number of particles involved in the collision increases, the number of Feynman diagrams (or histories) skyrockets. For an interaction involving four gluons, there are only four possible histories. For six gluons, there are 220 possible histories, and for 10 gluons, there are more than 10 million.
It gets worse. The rules of quantum weirdness allow for a pair of particles to spontaneously appear and then annihilate each other, creating a loop in a Feynman diagram. When loops are included, the number of possible histories gets even more out of control. At some point, the brute-force method used to calculate amplitudes simply runs out of steam. Even with the best supercomputers, theorists have failed to compute amplitudes for more than 10 gluons in the no-loop case, six gluons in the one-loop case, and four gluons in the two-loop case.
Now, from out of the mathematical dustbin, comes a tool that may vastly decrease the amount of labour required to crunch the necessary amplitudes. The technique, called twistor analysis, exploits some hidden symmetries in the Feynman diagrams that first showed up in equations as early as 1986. That year, Stephen Parke and Tomasz Taylor of Fermilab in Batavia, Illinois, discovered that the calculations became much simpler when they took into account spin, a quantum mechanical property of gluons that is always either up or down.
Parke and Taylor found that gluons cannot interact if five or six of them have their spins pointing in the same direction. And if four of the gluons have the same spin, the calculations give a simple answer that renders most of the Feynman diagrams unnecessary.
At the time, no one really understood why. Then in 2003, Edward Witten, a string theorist at the Institute for Advanced Study in Princeton, New Jersey, unlocked the secret of Parke and Taylor’s work. By combining ideas from twistor theory and string theory, he showed that gluons obeyed simple algebraic rules that short-circuited pages and pages of calculations.
Twistor theory and string theory are equally revolutionary, yet inherently different. Both were candidate “theories of everything”, devised to unite the disparate physics of gravity and subatomic particles. Twistor theory was invented in the late 1960s by Roger Penrose at the University of Oxford, and before strings came along it was the front runner for a unified theory (New Scientist, 31 July 2004, p 26). The trouble was that twistor theory made too many predictions that violated reality as we know it, so it crashed and burnt as a physics theory in the 1970s. Yet it lived on as a mathematical theory (see “Six dimensions with a twist”).
Although they started off with the same aim, string and twistor theories differ from one another in their descriptions of the fundamental nature of space. String theory treats space-time as real and adds six extra dimensions to it, while twistor theory replaces four-dimensional space-time with six-dimensional “twistor space”. In twistor space, ordinary space-time becomes a sort of hallucination that we cannot perceive directly because quantities such as momentum are described by complex numbers.
“Ordinary space-time becomes a hallucination”
Nevertheless, in his 2003 paper, Witten managed to combine the two ideas. He envisioned gluons as twistors, but assumed an extra property called supersymmetry, which adds four dimensions to twistor space. In supersymmetric twistor space, he discovered that processes involving large numbers of gluons and vast numbers of Feynman diagrams boil down to a simple algebraic-geometric arrangement. This translates into simple rules for computing amplitudes.
Change of fortune
The surprising thing was that by marrying two of the most abstract, visionary theories in physics, Witten had produced an offspring relevant to next year’s LHC experiments. “I’m mystified how Witten saw this,” says Bern. “He’s the only person in the world who could have linked two sides of physics in this way.”
Strictly speaking, Witten’s ideas only apply to interactions where all but two of the gluons have the same spin. But there are strong hints that similar relations will hold for other cases as well. Witten’s breakthrough has unleashed a flood of alternative, twistor-inspired methods that have already gone way past the Parke-Taylor formula. Among them are the insights gained by Dixon and his colleagues.
Dixon turned to twistor theory after becoming frustrated by the lack of progress in string theory. Last year he showed that it is possible to replace 220 Feynman diagrams that describe collisions involving six gluons with just three histories in twistor space ().
Dixon has also been working with theorists Simon Badger, Nigel Glover and Valentin Khoze at the University of Durham in the UK. Together they have been able to calculate the amplitudes of one of the most important reactions at the LHC. In this process, two gluons chipped from the colliding protons fuse to produce a massive Higgs and three extra gluons. It is an important step forward, both for experimentalists hunting the Higgs and for theorists. Until now, theorists have preferred to work with massless particles such as gluons because the equations are much simpler. Although no one has ever seen a Higgs it is far from massless: experiments at CERN and Fermilab suggest that it must be at least 125 times heavier than a proton.
Dixon and his collaborators have also calculated what happens when two gluons collide to leave four gluons behind. Among the many possible ways this reaction can occur is one in which the gluons spontaneously split into gluon pairs which then merge again, creating a loop in the Feynman diagram.
While these are important steps forward, Dixon points out that twistors have a long way to go before they beat Feynman diagrams. So far, they have only been able to partially describe processes at the LHC.
There are two main obstacles to be surmounted before it will be possible to make complete and concrete predictions for the LHC. Frustratingly, either one can be overcome individually, but not both at the same time.
The first roadblock is loops – the pairs or groups of particles that spontaneously spring into existence and then annihilate themselves. Twistor theory works perfectly for diagrams without loops. Yet it has come along just a little too late to make a practical difference for such diagrams. Even though Feynman diagrams are horrendously inefficient, computers can labour through tens of thousands of them faster than a theorist can calculate dozens of diagrams in twistor space.
The real issue is how to compute amplitudes involving loops. This is where a theorist with pencil and paper has a real chance to win. No computer in existence can calculate the scattering of seven gluons with a loop. However, Dixon and his colleagues have shown how to handle six gluons and they are now working on seven.
But gluons are massless particles and this simplifies the calculations enormously. Unfortunately, the basic formulae of twistor theory break down when they are applied to massive particles. They predict negative probabilities, which are impossible, and a hypothetical particle called the conformal graviton whose properties don’t comply with the usual laws of physics. This is what torpedoed twistor theory in the 1970s.
“At the moment we can do loops or we can have massive particles, but we cannot do both,” says Bern. As Dixon and his colleagues have shown with the Higgs calculations, the presence of mass is not insurmountable when there are no loops. With loops, though, the bogus gravitons can suddenly pop into existence and throw the calculations seriously out of whack. Nevertheless, Bern is optimistic about the future. “Mass is not a solved problem, but you can take it to the bank that it will be solved,” he predicts. “In five years’ time, there will be a complete solution for one-loop interactions.”
For many of the interactions on the experimenters’ wish list, pencil-and-paper calculations may not catch up with supercomputers before 2007. Yet in some ways the two approaches may be complementary. The twistor-inspired methods work best with the assumption of supersymmetry and with massless particles. Meanwhile, Feynman diagrams may continue to work better for other cases. Also, as the twistor-inspired methods become better understood, it is almost certain that they will be programmed into computers and thus become a new standard.
Even if particle physicists continue to use computers to churn out scattering amplitudes, twistor-inspired methods will provide one thing that no computer output can: a sense of understanding. For that reason, it seems unlikely that the twistor craze will die out again. “For sure we’re on the downhill side of the problem, not the uphill side,” says Bern. “The problems in front of us are much easier than the ones we have overcome.”
Six dimensions with a twist
As a mathematical theory, twistors have been extremely fertile. Many of the differential equations used in physics are much easier to solve in twistor space than in normal space-time, as the theory’s inventor Roger Penrose realised.
In ordinary space, these equations would only be solvable with a computer; in twistor space, they turn into algebraic problems that can be solved with nothing more sophisticated than a pencil and paper.
According to Claude LeBrun of the State University of New York at Stony Brook, twistors “unlocked a huge reservoir of information from algebraic geometry that could be applied to physics.”
The central precept of twistor theory is that the four-dimensional space-time we perceive is not the most fundamental reality. Space-time is derived from a more basic, six-dimensional universe called twistor space.
Twistors overturn centuries of tradition dating back to Euclid that makes a point in space the fundamental, indivisible unit of geometry. The indivisible unit of twistor geometry is instead a twisting ray of light – something that to us looks infinitely long, stretching far back into the past and continuing far into the future.At the same time, a point in space-time that seems to be irreducible to us is a derived quantity in twistor geometry. According to the theory it is the set of all light rays that intersect at that point.
For calculations, though, mathematicians need an easier way to think of them. It turns out that every twistor can be represented with four complex numbers, which can be written in the form a + ib where i is the square root of -1 and a and b are real numbers. One of these complex numbers turns out to be redundant and because each remaining coordinate depends on two real numbers, twistor space is six-dimensional. In fact, this space was well known to mathematicians over a century ago, though Penrose was the first person to grasp its profound importance to physics.
