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An Afternoon with Bern, Dixon and Kosower Breakthroughs in Quantum Field Theory and Gravity: an Afternoon With Bern, Dixon and Kosower Queen Mary University of London 7 November 2019 This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 764850 (SAGEX) Gravity as a Double Copy of Gauge Theory and its Applications November 7, 2019 SAGEX Meeting, QMUL Zvi Bern This project has received funding from the European Union’s Horizon 2020 research and innovation2 programme under the Marie Skłodowska-Curie grant agreement No. 764850 (SAGEX) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA Outline There are many interesting problems in gravity that require peturbative expansions in Newton’s contant. Here I’ll describe some ideas for dealing with complexity 1. Complications with gravity perturbation theory. 2. Antidotes: — Unitarity method. — Double copy and color-kinematics duality. 3. Applications: • Web of theories. • Gravitational wave physics. • Nonrenoralizability properties of gravity. 4. Outlook. 3 Quantum Field Theory and Scattering Amplitudes Scattering amplitudes give us quantum mechanical description of events at particle colliders. particle scattering Large Hadron Collider ATLAS Detector quark gluon Higgs boson event Feynman diagram loop diagram higher order 4 Complications with Gravity 5 Perturbative Gravity Consider the Einstein gravity Lagrangian curvature Flat-space metric graviton field metric Infinite number of complicated interactions + … Very nonlinear 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. Gauge and gravity theories seem rather different. 6 Three-Point Interactions Standard perturbative approach: Three-gluon vertex from strong interactions: Three-graviton vertex: About 100 terms in three vertex Naïve conclusion: Gravity is a nasty mess. 7 Feynman Diagrams for Gravity We will be talking about high order processes ~1020 No surprise it has 3 loops never been TERMS calculated via − Calculations to settle Feynman diagrams. this seemed utterly hopeless! − Seemed destined for 4 loops ~1026 dustbin of undecidable TERMS questions. 31 5 loops ~10 More terms than TERMS atoms in your brain! At present there is only one basic approach to carry out such computations, which I will outline. Antidotes to Complexity 9 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. • Use gauge invariant on-shell physical states. • On-shell formalism. • Don’t violate Einstein’s relation! ZB, Dixon, Dunbar, Kosower (1998) 10 Scattering Amplitudes Revolution Z. Bern, L. Dixon, D. Kosower, Over the years we have developed tools for calculating May 2012 Scientific American quantum scattering amplitudes that are “impossibly complicated” Don’t use Lagrangians or Feynman diagrams. 1. Generalized unitarity. Complicated amplitudes assembled from simpler ones. Loops from trees. 2. Double-copy relations. Gravity amplitudes built from much simpler gauge-theory ones. Key theme: Recycling! Many more advances, involving also beautiful mathematics most of which I won’t discuss here.. 11 From Tree to Loops: Generalized Unitarity Method No Feynman rules; no need for virtual particles. E2 = p~ 2 + m2 on-shell ZB, Dixon, Dunbar and Kosower (1994) Two-particle cut: • Systematic assembly of complete loop amplitudes from tree amplitudes. Three-particle cut: • Works for any number of particles or loops. on-shell ZB, Dixon and Kosower; ZB, Morgan; Generalized Britto, Cachazo, Feng; Ossala,Pittau,Papadopoulos; unitarity as a Ellis, Kunszt, Melnikov; practical tool Forde; Badger; ZB, Carrasco, Johansson, Kosower for loops. and many others Idea used in the “NLO revolution” in QCD collider physics. Want to apply it to gravitational wave problem. 12 Simplicity of Gravity Amplitudes People were looking at gravity amplitudes the wrong way. On-shell viewpoint much more powerful. 2 2 On-shell three vertices contains all information: E ~k =0 <latexit sha1_base64="1oOyxI0ag7pl27Rf3v1BdEvi88w=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyyCG8tMEXQjFEVwWcE+oB2HTHqnDc1khiRTKLULf8WNC0Xc+hvu/BvTaRFtPXDh5Jx7yb0nSDhT2nG+rIXFpeWV1dxafn1jc2vb3tmtqTiVFKo05rFsBEQBZwKqmmkOjUQCiQIO9aB3NfbrfZCKxeJODxLwItIRLGSUaCP59v61z+5L+AS3+kBxL3tcYMe3C07RyYB/iDtLCmiKim9/ttoxTSMQmnKiVNN1Eu0NidSMchjlW6mChNAe6UDTUEEiUN4w23+Ej4zSxmEsTQmNM/X3xJBESg2iwHRGRHfVrDcW//OaqQ7PvSETSapB0MlHYcqxjvE4DNxmEqjmA0MIlczsimmXSEK1iSxvQpg7eZ7USkXXKbq3p4Xy5TSOHDpAh+gYuegMldENqqAqougBPaEX9Go9Ws/Wm/U+aV2wpjN76A+sj29e+ZO8</latexit> i − i Yang-Mills gauge theory: “square” of Einstein Yang-Mills gravity: vertex. Very simple interactions. 13 KLT Relation Between Gravity and Gauge Theory KLT (1985) Kawai-Lewellen-Tye string relations in low-energy limit: gravity gauge-theory color ordered Generalizes to explicit all-leg form. ZB, Dixon, Perelstein, Rozowsky 1. Gravity is derivable from gauge theory. Standard Lagrangian methods offers no hint why this is possible. 2. It is very generally applicable. 14 Duality Between Color and Kinematics ZB, Carrasco, Johansson (2007) momentum dependent coupling constant color factor kinematic factor 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 Color factors⇣ satisfy Jacobi identity:⌘ Numerator factors satisfy similar identity: Proven at tree level ZB, Carrasco, Johansson; Kiermaier; Bjerrum-Bohr, Damgaard, Sondergaard, Vanhove; Cachazo, etc 15 Duality Between Color and Kinematics Consider five-point tree amplitude: ZB, Carrasco, Johansson (BCJ) color factor 15 kinematic numerator factor tree cini 5 = A p2 Feynman propagators i=1 ↵i ↵i X Q a3a4b ba1c ca2a5 a3a4b ba5c ca1a2 a3a4b ba2c ca1a5 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. Progress on unraveling relations. BCJ, Bjerrum-Bohr, Feng, Damgaard, Vanhove, ; Mafra, Stieberger, Schlotterer; Tye and Zhang; Feng, Huang, Jia; Chen, Du, Feng; Du, Feng, Fu; Naculich, Nastase, Schnitzer O’Connell and Montiero; Bjerrum-Bohr, Damgaard, O’Connell and Montiero; O’Connell, Montiero, White; Du, Feng and Teng, Song and Schlotterer, etc. 16 Higher-Point Gravity and Gauge Theory ZB, Carrasco, Johansson color factor kinematic numerator gauge theory factor (QCD): Feynman propagators c n i ! i sum over diagrams Einstein gravity: with only 3 vertices Gravity and gauge theory kinematic numerators are the same! Same ideas conjectured to hold at loop level. Cries out for a unified description of gravity with gauge theory, presumably along the lines of string theory. 17 Gravity From Gauge Theory Here we consider only simplest constructions: N = 8 sugra: (N = 4 sYM) (N = 4 sYM) N = 5 sugra: (N = 4 sYM) (N = 1 sYM) N = 4 sugra: (N = 4 sYM) (N = 0 sYM) Spectrum controlled by simple tensor product of gauge theories. More sophisticated lower-susy cases: QCD, magical supergravities, Einstein-YM with and without Higgsing, twin supergravities. Anastasiou, Bornsten, Duff; Duff, Hughs, Nagy; Johansson and Ochirov; Carrasco, Chiodaroli, Günaydin and Roiban; ZB, Davies, Dennen, Huang and Nohle; Nohle; Chiodaroli, Günaydin, Johansson, Roiban. A. Anastasiou, L. Borsten, M.J. Duff, M.J. Hughes, Marrani, Nagy, Zoccali. Many other theories in double-copy story, including open and closed string theory, NLSM, Dirac Born Infeld, Galileon and Z theory. Cachazo, He, Yuan; Chen Du, Broedel, Schlotterer and Stieberger; Carrasco, Mafra, Schlotterer; Hard to keep track of theories where double copy holds. 18 Web of Theories ZB, Carrasco, Chiodaroli, Johansson, Roiban arXiv:1909.01358, Section 5. Double copy links various theories through their component theories. 19 Double Copy for Classical Solutions Goal is to formulate gravity solutions directly in terms of gauge theory Variety of special cases: • Schwarzschild and Kerr black holes. • Solutions with cosmological constant. • Radiation from accelerating black hole. • Maximally symmetric space times. • Plane wave background. • Gravitational radiation. Luna, Monteiro, O’Connell and White; Luna, Monteiro, Nicholsen, O’Connell and White; Ridgway and Wise; Carrillo González, Penco, Trodden; Adamo, Casali, Mason, Nekovar; Goldberger and Ridgway; Chen; Still no general understanding. Luna, Monteiro, Nicholson, Ochirov; But plenty of examples. Bjerrum-Bohr, Donoghue, Vanhove; O'Connell, Westerberg, White; Kosower, Maybee, O’Connell, etc 20 Gravitational Waves See also David Kosower’s talk 21 Outline Era of gravitational wave astronomy has begun. For an instant brighter in gravitational radiation than all the stars in the visible universe are in EM radiation! 22 Can Particle Theory Help with Gravitational Waves? What does particle physics have to do with classical dynamics of astrophysical objects? Black holes and neutron stars are point particles as far as long wavelength radiation is concerned. EFT approach: Goldberger, Rothstein, Porto; Vaydia, Foffa , Porto, Rothstein, Sturani; etc Will explain that the tools that we use in particle physics are ideally suited to push the state of the art in gravitational-wave
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