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Jacob Bourjaily Adventures in Perturbation Theory Jacob Bourjaily Penn State University & The Niels Bohr Institute based on work in collaboration with Herrmann, Langer, Trnka; McLeod, von Hippel, Vergu, Volk, Wilhelm; … Organization & Outline ✦ Spiritus Movens: the surprising simplicity of QFT ✦ Loop Integrands ‣ generalized unitarity (generally speaking) ‣ building bases big-enough (for e.g. the Standard Model) ‣ non-planar power-counting (a modest proposal) ✦ Loop Integration: what makes an integral easy? ‣ integration polemics (what constitutes being integrated?) ‣ direct integration (made easy) ✦ Loop Integrals: their generic analytic structure ‣ a bestiary of Feynman integral Calabi-Yau geometries 2 Spiritus Movens the Surprising Simplicity of Scattering Amplitudes Traditional Description of QFT ✦ Quantum Field Theory: the marriage of (special) relativity with quantum mechanics ✦ Theories (can be) specified by Lagrangians—or equivalently, by Feynman rules for virtual particles ✦ Predicted probability (amplitudes) from path integrals (over virtual ‘histories’): 4 Perturbation Theory and Loops ✦ Predictions (often) made perturbatively, according to the loop expansion: [Dirac (1933)] [Feynman; Schwinger; Tomanaga (1947)] [Petermann (1957)] [Kinoshita (1990)] ✦ the most precisely tested idea in all of science! 5 derground rider is usually better o! taking a fairly simple route. lations—like an Underground rider who is willing to take very cir- Two decades later physicists extended Feynman’s technique to cuitous routes, making it hard to predict what he or she will do. Explosions of Complexitythe strong subnuclear force. By analogy with QED, the theory of Fortunately, at very short distances, including the distances rele- the strong force is known as quantum chromodynamics (QCD). vant for collisions at the LHC, the coupling diminishes in value QCD is also governed by a coupling, but as the word “strong” sug- and, for the very simplest collisions, theorists can again get away gests, its value is higher than that of the electromagnetic cou- with considering only uncomplicated Feynman diagrams. pling. On the face of it, a larger coupling increases the number of For messy collisions, though, the full complexity of the ✦ complicated diagrams that theorists must include in their calcu- Feynman technique comes rushing in. Feynman diagrams are While ultimately correct, the Feynman expansion classified by the number of external lines and WHY FEYNMAN DIAGRAMS DRIVE PHYSICISTS MAD the number of closed loops they have [see box at left]. Loops represent one of the quintessen- tial features of quantum theory: virtual parti- renders all but the most trivial predictions—Too Many to Keep Track Of cles. Though not directly observable, virtual particles have a measurable e!ect on the Each Feynman diagram strength of forces. They obey all the usual laws of nature, such as the conservation of energy A quark-quark interaction might produce more than one gluon or involve more than involving the fewest particles, at the and of momentum, with one caveat: their one virtual-particle loop, or both. The calculations quickly become unmanageable. mass can di!er from that of the corresponding “real” (that is, directly observed) particles. Zero loops One loop Loops represent their ephemeral life cycle: lowest orders of perturbation— they pop into existence, move a short distance, then vanish again. Their mass determines their life expectancy: the heavier they are, the shorter they live. The simplest Feynman diagrams ignore vir- computationally intractable One tual particles; they have no closed loops and are gluon called tree diagrams. In quantum electrody- namics, the simplest diagram of all shows two electrons repelling each other by exchanging a or theoretically inscrutable photon. Progressively more complicated dia- grams add loops one by one. Physicists refer to QUANTUM PHYSICS this additive procedure as “perturbative,” mean- ing that we start with some approximate esti- mate (represented by the tree diagrams) and gradually perturb it by adding refinements (the LOOPS, loops). For instance, as the photon travels be- tween the two electrons, it can spontaneously Two split into a virtual electron and virtual antielec- TREES gluons tron, which live a short while before annihilat- ing each other, producing a photon. The photon resumes the journey the original photon had AND THE been taking. In the next level of complexity, the electron and antielectron might themselves split temporarily. With increasing numbers of virtual particles, the diagrams describe quan- SEARCH tum e!ects with increasing precision. Even tree diagrams can be challenging. In the case of QCD, if you were brave enough to FOR NEW consider a collision involving two incoming and Three eight outgoing gluons, you would need to write gluons down 10 million tree diagrams and calculate a probability for each. An approach called recur- PHYSICS sion, pioneered in the 1980s by Frits Berends of Maybe unifying the forces of nature isn’t quite Leiden University in the Netherlands and Wal- as hard as physicists thought it would be ter Giele, now at Fermilab, tamed the problem By Zvi Bern, Lance J. Dixon and David A. Kosower for tree diagrams but had no obvious extension to loops. Worse, closed loops make the workload 34 Scientific American, May 2012 Illustration by Kenn Brown and Chris Wren, Mondolithic Studios overwhelming.6 Even a single loop causes an ex- [Bern, Dixon, Kosower,© 2012 Scientific American Scientific American (2012)] plosion in both the number of diagrams and the 38 Scientific American, May 2012 © 2012 Scientific American Needs (Once) Beyond Our Reach ✦ Background amplitudes crucial for e.g. colliders [Rev.Mod.Phys. 56 (1984)] ✦ Once considered computationally intractable 7 Needs (Once) Beyond Our Reach ✦ Background amplitudes crucial for e.g. colliders 220 diagrams—thousands of terms ✦ Once considered computationally intractable 7 Needs (Once) Beyond Our Reach ✦ Background amplitudes crucial for e.g. colliders [Nucl.Phys. B269 (1985)] ✦ Once considered computationally intractable 7 Needs (Once) Beyond Our Reach ✦ Background amplitudes crucial for e.g. colliders 7 Needs (Once) Beyond Our Reach ✦ Background amplitudes crucial for e.g. colliders 7 Discovery of Shocking Simplicity ✦ Within six months, Parke-Taylor stumbled on a simple guess—unquestionably a theorist’s delight: VoLUME 56, NUMsER 23 PHYSICAL REVIEW LETTERS 9 JUNE 1986 Amplitude for nGluon Scattering [PRL 56 (1986)] Stephen J. Parke and T. R. Taylor Fermi Xationa/Accelerator Laboratory, Batavia, Illinois 60510 (Received 17 March 1986) A nontrivial squared helicity amplitude is given for the scattering of an arbitrary number of gluons to lowest order in the coupling constant and to leading order in the number of colors. PACS QUmbcrs: 12.38.8x [van der Waerden (1929)] Computations of the scattering amplitudes for the used to improve the existing numerical programs for vector gauge bosons of non-Abelian gauge theories, the QCD jet production, and in particular for the stud- 8 besides being interesting from a purely quantum- ies of the four-jet production for which no analytic field-theoretical point of view (determination of the S results have been available so far. Before presenting matrix), have a wide range of important applications. the helicity amplitude, let us make it clear that this In particular, within the framework of quantum chro- result is an educated guess which we have compared to modynamics (QCD), the scattering of vector gauge the existing computations and verified by a series of bosons (gluons) gives rise to experimentally observ- highly nontrivial and nonlinear consistency checks. able multijet production at high-energy hadron collid- For the n-gluon scattering amplitude, there are ers. The knowledge of cross sections for the gluon (n+2)/2 independent helicity amplitudes. At the scattering is crucial for any reliable phenomenology of tree level, the two helicity amplitudes which most jet physics, which holds great promise for testing QCD violate the conservation of helicity are zero. This is as well as for the discovery of new physics at present easily seen by the embedding of the Yang-Mills theory (CERN Spp S and Fermilab Tevatron) and future (Su- in a supersymmetric theory. 2 3 Here we give an ex- perconducting Super Collider) hadron colliders. ' pression for the next helicity amplitude, also at tree In this short Letter, we give a nontrivial squared level, to leading order in the number of colors in helicity amplitude for the scattering of an arbitrary SU(N) Yang-Mills theory. number of gluons to lowest order in the coupling con- If the helicity amplitude for gluons 1, . , n, of mo- stant and to leading order in the number of colors. To menta pi, . , p„and helicities Xi, . , A, „, is our knowledge this is the first time in a non-Abelian A'„(&i, . , &„), where the momenta and helicities gauge theory that a nontrivial, on-mass-shell, squared are labeled as though all particles are outgoing, then Green's function has been written down for an arbi- the three helicity amplitudes of interest, squared and trary number of external points. Our result can be summed over color, are I2 = I P'„(+ + + + + ) c„(g,N) [0+O(g ) ], — I' = I M„( + + + + ) c„(g,N) [0+ O(g~) ], I~„(——+++ )I'= c„(q,N)[(pi p2)' x Xp[(pi p2)(p2 p3)(p3. p4) . (p„pi)] '+O(N )+O(g )], (3) where c„(g,N) =g2" 4N" 2(N2 —1)/2" 4n. The sum is over all permutations I'of 1, . , Equation (3) has the correct dimensions and symmetry properties for this n-particle scattering
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