Harmony of Scattering Amplitudes: From Quantum Chromodynamics to Supergravity BNL Physics Colloquium Nov 12, 2013 Zvi Bern, UCLA 1 Scattering amplitudes Scattering of elementary particles is fundamental to our ability to unravel microscopic laws of nature. Smooth running of the Large Hadron Collider at CERN emphasizes importance of scattering amplitudes. Vast subject. Here we give some examples of advances of past few years in understanding and calculating 4 month program at Simons Center scattering in quantum field theory. 2 Outline Will outline some new developments in understanding scattering amplitudes. Surprising harmony. 1. Scattering using Feynman diagrams. Obscures harmony. 2. Modern on-shell methods for scattering. 3. Sample applications: — LHC Physics — Reexamination of compatibility of quantum mechanics and general relativity. W q e º q g0 g g g g 3 Feynman Diagrams jets p quark p gluon jets of Higgs boson event in CMS detector hadrons Note jets of particles Quantum chromodynamics (QCD) describes the strong nuclear force binding quarks and gluons into protons and neutrons. QCD describes in great detail, scattering of protons at particle colliders. 4 Scattering Amplitudes Every graduate student in particle physics learns how to calculate scattering amplitudes via Feynman diagrams. gluon scattering Feynman propagator Coupling constant “color” factor In principle this is a complete solution for small coupling Tree – Leading Order (LO) Loop – Next-To-Leading Order (NLO) Quantum correction Feynman diagrams give us a perturbative expansion in Planck’s constant ¹h 5 Scattering Amplitudes In principle this is a complete solution for scattering with small coupling constants. Systematic! In practice not so easy: • Proper way to calculate? Many scales, large logs, infrared issues, strong coupling, non-perturbative contributions, physical states, etc. In this talk we do not address these issues. • A more practical problem is just dealing with the Feynman diagrams. In this talk we discuss this aspect. 6 State-of-the-Art Feynman Diagram Calculations In 1948 Schwinger computed anomalous magnetic moment of the electron. In 2010 typical 1-loop modern example: g : gluon -- carrier of strong force q: quark -- matter W, Z : Weak boson carrier of weak interaction 60 years later at 1 loop only 2 or 3 legs more than Schwinger if we use traditional Feynman diagrams. 7 Tree-level example: Five gluons Force carriers in QCD are gluons. Similar to photons of QED except they self interact. Consider the five-gluon amplitude: Used in calculation of pp 3 jets at CERN If you evaluate this following textbook Feynman rules you find… 8 Result of evaluation (actually only a small part of it): Messy combination of momenta and gluon polarization vectors. 9 Reconsider Five-Gluon Tree Xu, Zhang and Chang With a little Chinese magic, i.e. helicity states: and many others Use a better organization of color charges: Motivated by the color organization of open string amplitudes. 10 Mangano and Parke Example of loop difficulty Consider a tensor integral: Note: this is trivial on modern computer. Non-trivial for larger numbers of external particles. Evaluate this integral via Passarino-Veltman reduction. Result is … 11 Result of performing the integration Calculations explode for larger numbers of particles or loops. Clearly, there should be a better way! 12 W + n Jet NLO Calculations for the LHC Suppose you said “I’m going not shave until I finish calculating” W + 1 jet W + 2 jets W + 3 jets Still no Feynman diagram calculation 13 Why are Feynman diagrams difficult for high-loop or high-multiplicity processes? • Vertices and propagators involve unphysical gauge-dependent off-shell states. An important origin of the complexity. Individual Feynman diagrams unphysical Einstein’s relation between momentum and energy violated in the loops. Unphysical states! Not gauge invariant. • All steps should be in terms of gauge invariant on-shell physical states. On-shell formalism. Don’t violate Einstein’s relation! ZB, Dixon, Dunbar, Kosower (1998) 14 Heisenberg Feynman diagram loops violate on shellness because they encode the uncertainty principle. Loops: You can create new particles even with insufficient energy as long as you destroy them quickly enough. Theorem: Off-shellness or energy conservation violation is essential for getting the correct answer because of uncertainty relation. It looks like an on-shell formalism will fail to capture everything. 15 Sneaking Past Heisenberg Favorite Theorem: Theorems in physics are bound to be misleading or have major loopholes. What’s the loophole here? • Want to reconstruct the complete amplitude using only on-shell physical information. • Keep particles on shell in intermediate steps of calculation, not in final results. 16 Modern Unitarity Method on-shell ZB, Dixon, Dunbar and Kosower Two-particle cut: Systematic assembly of complete amplitudes from cuts for any number of Three-particle cut: particles or loops. ZB, Dixon and Kosower; Generalized ZB, Morgan; Britto, Cachazo, Feng; unitarity as a Ossala,Pittau,Papadopoulos; practical tool. Ellis, Kunszt, Melnikov; Forde; Badger on-shell and many others Reproduces Feynman diagrams except intermediate steps of calculation based on physical quantities not unphysical ones. 17 On-Shell Recursion for Tree Amplitudes Britto, Cachazo, Feng and Witten (BCFW) Physical scattering amplitudes constructed from on-shell physical scattering amplitudes with fewer particles! on-shell amplitude Works by making clever use of Cauchy’s residue theorem Same construction works in gravity Brandhuber, Travaglini, Spence; Cachazo, Svrcek; Benincasa, Boucher-Veronneau, Cachazo; Arkani-Hamed and Kaplan, Hall In gauge theory, extended to loops. Arkani-Hamed et al. 18 G. Salam, ICHEP 2010 19 20 G. Salam, La Thuile 2012 21 2013: NLO W+5j [BlackHat+Sherpa: Bern et al] [unitarity] BlackHat: C++ implementation of on-shell methods for one-loop amplitudes ZB, Dixon, Febres Cordero, Hoeche, Ita, Kosower, Maitre, Ozeren BlackHat is a C++ package for numerically computing one-loop matrix elements with 6 or more external particles. • Input is on-shell tree-level amplitudes. • Output is numerical on-shell one-loop amplitudes. On-shell methods used to achieve the speed and stability required for LHC phenomenology at NLO. Other (semi) on-shell packages: — Helac-NLO: Bevilacqua, Czakon, Ossola, Papadopoulos, Pittau, Worek — Rocket: Ellis, Giele, Kunszt, Melnikov, Zanderighi — SAMURAI: Mastrolia, Ossola, Reiter, Tramontano — MadLoop: Hirchi, Maltoni, Frixione, Frederix, Garzelli, Pittau — GoSam: Cullen, Greiner, Heinrich, Luisoni, Mastrolia, Ossola, Reiter, Tramontano — NGluon: Badger, Biedermann, Uwer 22 ATLAS Comparison against NLO ATLAS compared data against NLO theoretical predictions Powerful experimental confirmation of NLO approach. Validate in regions where no new physics is expected and look for discrepancies on tails of distributions where new physics may be hiding. number of jets Theory predictions from BlackHat used to aid search for supersymmetry. W +1, 2, 3, 4 jets inclusive 23 W + 5 Jets in NLO QCD Hadronic energy spectrum A triumph for on-shell methods! Recently completed. arXiv:1304.1253 LO/NLO ratio LO/NLO W q e Transverse hadronic energy º q • A new level for “state-of-the-art” LHC theory: First NLOg0 QCD 2 6 process. g g g g Examples of Harmony in Scattering Amplitudes 25 Gravity vs Gauge Theory Consider the Einstein gravity Lagrangian curvature Flat-space metric graviton field metric Infinite number of complicated interactions + … terrible mess Compare to gauge-theory Lagrangian on which QCD is based Only three and four point interactions Gravity seems so much more complicated than gauge theory. Does not look harmonious! 26 Three Vertices Standard Feynman diagram approach. Three-gluon vertex: Three-graviton vertex: About 100 terms in three vertex Naïve conclusion: Gravity is a nasty mess. Where is the harmony? 27 Simplicity of Gravity Amplitudes People were looking at gravity the wrong way. On-shell viewpoint much more powerful. On-shell three vertices contains all information: gauge theory: “square” of gravity: Yang-Mills vertex. Using modern methods, any gravity scattering amplitude constructible solely from on-shell 3 vertex. Higher-point vertices irrelevant! Harmony is revealed! 28 Gravity vs Gauge Theory Consider the Einstein gravity Lagrangian flat metric graviton field metric Infinite number of irrelevant interactions! Simple relation + … to gauge theory Compare to gauge-theory Lagrangian on which QCD is based. Only three-point interactions no Gravity seems so much more complicated than gauge theory. Does not look harmonious! 29 Duality Between Color and Kinematics momentum dependent coupling color factor kinematic factor constant Color factors based on a Lie algebra: Jacobi Identity Use 1 = s/s = t/t = u/u to assign 4-point diagram to others. n c n c n c tree = g2 s s + t t + u u A4 s t u ³ ´ Color factors satisfy Jacobi identity: Numerator factors satisfy similar identity: Color and kinematics sing the same tune! 30 Duality Between Color and Kinematics Consider five-point tree amplitude: ZB, Carrasco, Johansson (BCJ) color factor kinematic numerator factor Feynman propagators a3a4b ba5c ca1a2 a3a4b ba2c ca1a5 a3a4b ba1c ca2a5 c1 f f f ; c2 f f f ; c3 f f f ´ ´ ´ c1 c2 + c3 = 0 n1 n2 + n3 = 0 ¡ ¡ Claim: We can always find a rearrangement so color and kinematics satisfy the same algebraic constraint equations. Recent progress on unraveling mysterious relations. BCJ, Bjerrum-Bohr, Feng,