Although I cannot yet understand most of the content of this article, the exciting information it conveys made me decide to repost it. The article suggests that unifying the various forces of nature might not be as difficult as physicists originally thought.
Written by: Zvi Bern, Lance J. Dixon, David A.
Kosower
Translated by: Kao Yong-Chyuan (Professor, Department
of Physics, National Taiwan University)
Provided by: Scientific American (Traditional Chinese
Edition)
Key Points
Physicists’ understanding of particle collisions has recently undergone a quiet revolution. The concepts introduced by the famous physicist Richard Feynman have reached their limits for many applications. The authors and their collaborators have developed new methods.
Using these new methods, physicists can more reliably describe the behavior of ordinary particles under the extreme conditions of the Large Hadron Collider (LHC), which will help experimentalists search for new particles and new forces.
The new method also has more profound applications: it has breathed new life into a unified theory that was abandoned by physicists in the 1980s; gravity appears to act like a "double" strong nuclear force.
On a sunny spring day, author Dixon entered the Maida Vale station of the London Underground, intending to head to Heathrow Airport. With 3 million passengers daily on the London Underground, he looked at a stranger and idly wondered: what is the probability that this fellow will leave the subway at Wimbledon station? Since this person could take any subway line, how should this probability be calculated? After thinking for a while, he realized that this problem is actually very similar to the trouble faced by particle physicists: how to predict the consequences of particle collisions in modern high-energy experiments.
The Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN) is the most important exploratory experiment of our time; it makes protons collide at nearly the speed of light and then studies the debris after the collision. We know that building the collider and detectors requires the most cutting-edge technology; however, what is less known is that interpreting the findings of the detectors is also an extremely difficult challenge. At first glance, it shouldn’t be that difficult, because the Standard Model of elementary particles has long been established, and theorists have been using this model to predict experimental results. These theoretical predictions rely on the calculation techniques developed by the famous physicist Richard P. Feynman more than 60 years ago. Every particle physicist learns Feynman’s techniques during their graduate studies; every popular science book and article about particle physics also borrows Feynman’s concepts.
However, Feynman’s techniques have actually become ineffective for current problems! Although they provide an intuitive, approximate way to grasp the simplest processes, they are hopelessly complex for more complicated processes or more precise calculations. Compared to predicting where a subway passenger will go, predicting the outcome of a particle collision is much harder. Even for a very ordinary collision in the LHC, we cannot calculate the result even if we use all the computers in the world. If theorists cannot make precise predictions about known physical laws and known forms of matter, how would we know even if the collider really produced something new? For all we know, the LHC may have already found the answers to some mysteries of nature, but we are still kept in the dark because our solutions to the Standard Model equations are not precise enough.
In recent years, the three of us and our collaborators have developed a new approach: the unitarity method, to analyze particle reaction processes. It avoids the complexity of the Feynman method and is essentially equivalent to a highly simple method for predicting where a subway passenger will go. The key lies in recognizing that whenever a subway passenger faces a choice, his options are actually quite limited, so we can decompose the desired answer into a series of probabilities of actions. This new method has allowed us to solve many theoretical problems in particle physics that were originally unsolvable, so we can more deeply understand the predictions of current elementary particle theories and identify new discoveries. This method can also be used for another interesting model and yield many results; this model is similar to the Standard Model but describes an idealized world that physicists are interested in because it is seen as a stepping stone on the path to the ultimate theory.
The unitarity method is not just a useful calculation technique; it suggests that the theory of particle interactions actually has a brand-new form, which is controlled by unexpected symmetries, meaning that the Standard Model has exquisite aspects that have not yet been appreciated. In particular, physicists have long been committed to combining quantum theory with Einstein’s general relativity into a theory of quantum gravity, and the unitarity method has revealed a very strange twist in this regard. Before the 1970s, physicists always assumed that the nature of gravity was similar to other basic interactions, so they tried to generalize existing theories to include gravity. But when they applied Feynman’s techniques to such theories, they found that they either got absurd answers or were trapped by complex mathematics; so, gravity seemed completely different from other forces after all. Therefore, frustrated physicists turned to more revolutionary ideas, such as supersymmetry and later string theory.
But the unitarity method allows us to actually perform calculations that were attempted in the 1980s but were impossible to do, and the results show that some originally expected contradictions do not actually exist. That is to say, gravity does look similar to other forces, but in an unexpected way: the behavior of gravity is like "double" the strong nuclear force that binds nucleons together. Since the strong nuclear force is transmitted by gluons, and gravity should be transmitted by particles called gravitons, the new image provided by the unitarity method is that each graviton is like two gluons sewn together. This concept is quite strange, and even experts cannot yet properly imagine its meaning. Regardless, this "double" property provides a brand-new perspective to explore how gravity can be combined with other forces.
From a Tree to a Forest
Why is Feynman’s technique so convincing and useful? The key is that it gives us a graphical rule to deal with extremely complex calculations. The core of the Feynman method is a type of diagram (often called a Feynman diagram) that allows us to visually view the collision or mutual scattering of two or more particles. In every research institution exploring elementary particle physics, you will surely see blackboards covered with such diagrams. When theorists make quantitative predictions, they first draw a set of diagrams, each representing a possible way for the particle collision process to proceed, just like the various paths a London Underground passenger might take. As long as theorists follow a set of detailed rules set by Feynman, Freeman Dyson, and others, they can assign a number to each diagram, representing the probability of the event occurring as shown in the diagram.
The disadvantage of the Feynman method is that there are too many diagrams we can draw—in principle, infinitely many. However, for the purpose for which Feynman originally developed this method, this disadvantage was not very important. He was studying Quantum Electrodynamics (QED) at the time, aiming to describe the interaction between electrons and photons. This interaction is controlled by a quantity called the "coupling constant," which is approximately 1/137. Since this electromagnetic coupling constant is very small, more complex diagrams account for a smaller portion of the calculation and can often be ignored. Just as for a subway passenger, choosing a simpler route is usually more advantageous.
Twenty years later, physicists generalized Feynman’s techniques to the strong nuclear force. Based on the analogy with QED, the theory of the strong nuclear force is called Quantum Chromodynamics (QCD). QCD is also controlled by a strong coupling constant, but from the word "strong," we can expect that the strong coupling constant of QCD is much larger than the electromagnetic coupling constant of QED. On the surface, if the coupling constant becomes larger, the number of complex diagrams that theorists must include in the calculation must increase. This is like if a subway passenger is willing to choose a very circuitous route, it is difficult for us to predict his next step. Fortunately, however, at very short distances (including the distance scales involved in particle collisions in the LHC), the strong coupling constant actually becomes smaller, so theorists only need to consider uncomplicated Feynman diagrams when calculating the simplest collisions.
However, if messy collisions are to be handled, the complexity involved in the Feynman method is unavoidable. Feynman diagrams have "external lines" and "loops." We can classify Feynman diagrams by the number of external lines and the number of loops. Loops represent one of the most core features of quantum theory: virtual particles. Although virtual particles are not directly observable, they have observable effects on the strength of the force (the size of the coupling constant). Virtual particles follow all ordinary laws of nature, such as conservation of energy and momentum, with one exception: the mass of a virtual particle is not the same as the corresponding "real particle" (which can be directly observed). Loops represent the extremely short life cycle of virtual particles: they suddenly appear, travel a very short distance, and then disappear. The mass of virtual particles determines their lifespan—the larger the mass, the shorter the lifespan.
The simplest Feynman diagrams ignore virtual particles, meaning there are no loops, and are therefore called tree diagrams. In QED, the simplest diagram is two electrons repelling each other by exchanging a photon. Next, we can add loops one by one to get more complex diagrams. Physicists call this superposition step "perturbation theory," meaning we start from an approximation (represented by a tree diagram) and then gradually perturb the initial approximation by adding corrections (represented by loops). For example, when a photon travels between two electrons, it can spontaneously split into a virtual electron and a virtual positron. This pair of electron and positron exists for a very short time and then annihilates each other to turn back into a photon, which then continues the original photon’s journey. If we consider the next order of correction, the electron and positron themselves may also temporarily split. The more virtual particles in a Feynman diagram, the more accurately the Feynman diagram can describe quantum effects.
But even tree diagrams are troublesome. In QCD, if you bravely consider more complex collisions, such as two gluons coming in and eight gluons going out, you must draw about 10 million tree-like Feynman diagrams and calculate the corresponding probability for each diagram. There is a method called recursion, first developed in the 1980s by Frits Berends of Leiden University and Walter Giele, now at Fermilab, which can handle tree diagrams but cannot be generalized to loop diagrams; worse, loops bring an unbearable amount of calculation. Even if only one loop is involved, the number of Feynman diagrams and the complexity of each diagram will increase significantly, and the mathematical formulas are enough to fill an encyclopedia. Using more computers to brute-force these diagrams can only cope for a while; once the number of external lines and loops increases, we can only surrender.
Worse still, Feynman diagrams, which were originally a concrete way to visualize the microscopic world, have now made the microscopic world blurred: a single Feynman diagram is already too complex to grasp, and there are so many diagrams to handle that we simply cannot figure out where the key lies. But what is truly shocking is that when we add all the diagrams together, the final result is quite simple! Some diagrams cancel each other out, so that a mathematical expression that originally had millions of terms sometimes simplifies to just one term! These cancellations mean that Feynman diagrams are not the right tool for dealing with the problem, just like using a feather to drive a nail. There must be a better way.
A Better Method than Feynman Diagrams
Over the years, physicists have tried many calculation techniques, each a little better than the previous one. Gradually, a method to replace Feynman diagrams took shape. We began to participate in the early 1990s, when Bern and Kosower among us proved that string theory techniques could be used to integrate all relevant Feynman diagrams into a single formula, simplifying QCD calculations. Then the three of us used this formula to analyze a particle reaction that had remained misunderstood: two gluons scattering into three gluons through a virtual particle loop; this process would be very complex if handled by traditional methods, but we could describe it with an extremely simple formula (which can fit on a piece of paper).
This formula was so simple that we and David Dunbar, then at the University of California, Los Angeles, discovered that by using a principle called "unitarity," this scattering process could be almost completely understood. Unitarity means that the sum of the probabilities of all possible outcomes (of a quantum process) must be 100%. (Strictly speaking, we are dealing with the square root of the probability, not the probability itself, but the difference is not important here.) In Feynman’s technique, although unitarity also holds, it is often obscured by the complexity of the calculation, so we developed another method to highlight the central position of unitarity. In fact, the idea of basing calculations on unitarity appeared as early as the 1960s, but it did not gain favor. It is common in science for abandoned ideas to sometimes return in a new guise.
The key to the success of the unitarity method lies in avoiding the direct use of virtual particles, which is the main reason why Feynman diagrams become so complex. These particles have real effects and false effects; by definition, false effects must cancel each other out in the final result, so they are extra mathematical baggage that physicists are naturally happy to discard.
We can use the previously mentioned subway analogy to understand this method: in the complex London Underground system, there are many paths between any two subway stations; suppose we want to know the probability of a person entering the subway from Maida Vale station and leaving from Wimbledon station. The Feynman method is equivalent to adding up the probabilities of all possible paths. "All" here is real, meaning that in addition to going through corridors and tunnels, Feynman diagrams include paths through rocks (where there are no subway tracks or walkways); these impractical paths are like false contributions from virtual particle loops. Although they will eventually cancel out, none can be omitted in the calculation steps. In the unitarity method, we only consider those paths that have practical meaning. We decompose the problem to calculate the probability of a passenger taking a certain path: how high is the probability that this person passes through a turnstile and takes this or that track? This can significantly reduce the amount of calculation required.
There is no right or wrong between the Feynman method and the unitarity method; both present the same basic physical laws and will eventually yield the same probability, but they represent different levels of description. A complex collision involves tens of thousands of Feynman diagrams, one of which is like a molecule in a drop of liquid; although in principle you can track all molecules to determine the behavior of the fluid, doing so is only applicable to microscopic, extremely small drops of liquid. Generally speaking, doing so is extremely laborious and unhelpful for understanding fluid behavior; a fluid may rush down a slope, but it is difficult for us to know this from the perspective of molecules; considering properties at the next level (such as fluid velocity, density, pressure) may be more helpful. Similarly, instead of imagining particle collisions as being constructed from individual Feynman diagrams, it is better to view them holistically; we should focus all our attention on those properties that control the entire process, such as unitarity and those special symmetries emphasized in the unitarity method. In some special cases, we can make completely accurate theoretical predictions. If the Feynman method were used, this would require infinitely many diagrams and infinite time!
The benefits of the unitarity method do not stop there. After we developed the unitarity method for virtual particle loop diagrams, Ruth Britto, Freddy Cachazo, Bo Feng, and Edward Witten, then at the Institute for Advanced Study in Princeton, proposed a complementary idea: they reconsidered tree diagrams. For example, the probability of a collision involving five particles can be derived from the probability of a collision of four particles followed by a process in which one particle splits into two particles. This is a surprising result because a five-particle collision usually looks very different from the aforementioned two sequential collision and splitting processes. In this way, we can use many methods to decompose troublesome particle problems into simpler ones.
Searching for New Physics in Collisions
Proton collisions in the LHC are extremely complex. Feynman once compared such collisions to smashing Swiss watches together to understand their internal structure; his technique was used to track what happened during the collision. Protons are not elementary particles but small balls held together by the strong nuclear force binding quarks and gluons. When protons collide, the quarks inside can collide with quarks, quarks can collide with gluons, and gluons can also collide with gluons, while quarks and gluons can also split into more quarks and gluons. Finally, they gather together to become composite particles, which are ejected from the collider in very narrow jets.
New particles, new symmetries, or new spacetime dimensions that humans have never seen before may be hidden in the messy jets, but it is not easy to screen these new things out. For detection instruments, the difference between new particles and ordinary particles is extremely small and easily ignored. With the unitarity method, we can describe known physics very accurately, making unusual physics stand out.
For example, Joe Incandela of the University of California, Santa Barbara, who is currently the spokesperson for the Compact Muon Solenoid (CMS) experiment team at the LHC, approached us to ask about the problems his team encountered while searching for new particles that make up the dark matter of the universe. Astronomers believe that this mysterious matter exists in the universe, but physicists have not yet found it. If the LHC produces such particles, CMS cannot capture them; we will only notice that some energy seems to have disappeared. Unfortunately, the disappearance of energy cannot prove that the LHC produced dark matter. For example, the LHC often produces Z bosons, and each Z boson has a 1/5 probability of decaying into two neutrinos. Neutrinos are also not captured by detectors (because their interaction with ordinary matter is extremely weak), so we will also find that energy is missing. So how many "Standard Model particles with effects similar to dark particles" will the LHC produce?
Incandela’s team proposed a prediction method: infer the number of events involving neutrinos from the number of photons recorded by CMS to see if the missing energy can be explained. If not, the LHC may have produced dark matter. Such an inference method is actually quite typical because experimentalists cannot directly observe certain types of particles and have to infer them indirectly. However, for this method to succeed, Incandela’s team must accurately estimate the relationship between the number of photons and the number of neutrinos; unless they are quite clear about this, this inference will fail. Therefore, we and several collaborators studied this problem with new theoretical tools and concluded that the estimation of Incandela’s team was quite accurate. With such assurance, the CMS team used their method to set the strictest limits on the properties of dark matter particles. So our technique is useful!
This success encouraged us to challenge even more difficult calculations. Our partners include Fernando Febres Cordero of the Simón Bolívar University in Venezuela, Harald Ita of Tel Aviv University and the University of California, Los Angeles, Daniel Maître of Durham University in the UK, Stefan Höche of the SLAC National Accelerator Laboratory, and Kemal Ozeren of the University of California, Los Angeles; they come from all over the world, which is very common in modern particle physics research. Together, we accurately calculated the probability of an LHC collision producing a pair of neutrinos and four jets. If Feynman diagrams were used, even a large physics team working hard for 10 years with the latest computers would find these calculations too difficult. The unitarity method allowed us to complete the calculation within a year. Another LHC experiment, the ATLAS team, has already compared our predictions with their experimental data, and so far they match very well, which makes us very happy. Next, they will use these results to search for new physics.
The unitarity method also helps in the search for the long-awaited Higgs particle. One sign of finding the Higgs particle is the production of an electron, a pair of jets, and a neutrino after a collision, and the invisible neutrino also makes us think that energy has disappeared. But other particle reactions that do not involve the Higgs particle may also have the same products, so one of the original uses of the unitarity method was to accurately calculate the probability of these confusing reactions occurring.
Returning to Gravity
The unitarity method has an even more amazing application, which is to explore quantum gravity. If physicists are to develop a complete theory of nature, they must integrate gravity into the framework of quantum mechanics. If the behavior of gravity is similar to other types of forces, it should be transmitted by gravitons. Gravitons, like other particles, will collide and scatter, and we can also draw corresponding Feynman diagrams. However, when physicists tried to quantize Einstein’s theory in the simplest way in the mid-1980s to describe the scattering of gravitons, they got unreasonable results, such as some quantities that are clearly impossible to be infinite being predicted as infinite. Infinite quantities themselves are not actually a problem because they may appear in the calculation process; even in a theory like the Standard Model that has no problems, infinities may appear. But for any observable quantity, the infinities appearing in the calculation process should all cancel out; for gravity, the infinities do not cancel out. Specifically, this means that the quantum fluctuations of space and time (which the late quantum gravity pioneer John Wheeler called "quantum foam") will become more and more intense, completely out of control.
One possible explanation is that nature contains undiscovered particles that can harness these intense quantum effects. Such an idea has been concretely presented in the so-called "supergravity" theory and was also deeply studied in the 1970s and early 1980s. But when indirect arguments suggested that unreasonable infinities would still arise from Feynman diagrams with more than two loops, everyone’s excitement cooled down. Supergravity seemed destined to fail.
The disappointment led many to study string theory. String theory is very different from the Standard Model: according to string theory, particles such as quarks, gluons, and gravitons are no longer tiny points but oscillations of one-dimensional strings. The interaction of particles is spread out everywhere on the string, not concentrated at a single point, which automatically avoids the generation of infinities. But string theory has its troubles; for example, it cannot make clear theoretical predictions for observable quantities.
The Comeback of Supergravity Theory
In the mid-1990s, the famous physicist Stephen Hawking of the University of Cambridge advocated giving supergravity theory another chance. He pointed out that researchers in the 1980s took shortcuts, so their conclusions were suspicious. However, Hawking couldn’t convince anyone because it made sense for people to take shortcuts: complete calculations were impossible, and their difficulty was beyond the reach of even the smartest mathematical wizards. To know if a Feynman diagram with three virtual graviton loops would produce an infinite quantity, we must calculate 10^{20} terms; if we consider a five-loop Feynman diagram, we must calculate 10^{30} terms; such a large number is approximately equal to the number of atoms in an LHC detector. No wonder complete calculations were destined to be hopeless.
The unitarity method has turned the whole situation around. We used the unitarity method to re-evaluate supergravity theory to see if it could be vindicated. For the work that originally required calculating 10^{20} terms, we now only need to calculate a few dozen terms. Collaborators on this work include Radu Roiban of Pennsylvania State University and John Joseph Carrasco and Henrik Johansson, who were then graduate students at the University of California, Los Angeles. Our conclusion is that the guess in the 1980s was wrong: the quantities that originally seemed infinite are actually finite. Supergravity is not as unbearable as physicists thought, which means that the quantum fluctuations of space and time in supergravity are much milder than previously imagined. If readers keep bringing us good wine, they might come across a certain supergravity theory we are speculating about, which might be the long-sought theory of quantum gravity.
More interestingly, the interaction of three gravitons is very similar to "double" the interaction of three gluons. No matter how many particles are scattering, no matter how many virtual particle loops are involved, this "double" property seems to hold, which means that gravity seems to be the square of the strong interaction. Translating the aforementioned mathematical discovery into physical insight and testing whether it holds in all situations will take some time; but the most critical point at present is that gravity may not be different from other interactions.
It is common in science that after each debate settles, another controversy arises. After we calculated the three-loop diagram, someone immediately suspected that the four-loop diagram might have problems. As long as there is a controversy, someone will bet; someone bet good wine on the results of the calculation: Italian Barolo red wine against California Napa Valley Chardonnay white wine. When we completed the calculation, we found no signs of trouble, thus ending this debate (and opening a bottle of Barolo).
So will supergravity theory never encounter infinities? Or is the high degree of symmetry in the theory only functional in suppressing infinities when the number of loops is small? If the latter is correct, then trouble should quietly appear in the five-loop diagram, and by the seven-loop diagram, quantum effects will be strong enough to produce infinities. If the seven-loop diagram has no infinities, David Gross of the University of California, Santa Barbara, is willing to offer a bottle of California Zinfandel red wine; to determine this bet, someone has already begun the calculation. If the seven-loop diagram is finite, skeptics will not only be very surprised but may finally accept that supergravity is a consistent theory. But even so, this theory does not capture the so-called non-perturbative effects, which are too subtle to be seen in the perturbative method we use to calculate loop by loop; they may still require a deeper theory (perhaps string theory) to handle.
Physicists like to think of new theories as coming from the bold application of new principles (such as relativity, quantum mechanics, symmetry). But sometimes, new theories arise from a careful examination of known principles. Our understanding of particle collisions has improved significantly. This quiet revolution has enabled us to calculate the results of the Standard Model in an incredibly precise way, thus significantly increasing our ability to discover new physics beyond the Standard Model. Even more surprisingly, this method has allowed us to explore undeveloped corners of old physics, including a path that was once ignored but may lead to the unification of gravity and other known forces. From many perspectives, the journey of studying how elementary particles scatter is not like taking a predictable London Underground trip, but more like a Knight Bus journey in Harry Potter—you can never be completely sure what will happen next!
(This article was provided by Scientific American and originally published in Scientific American, Issue 125, June 2012.)
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