Geometric picture for scattering amplitudes

Transcription

1 Geometric picture for scattering amplitudes Jaroslav Trnka Center for Quantum Mathematics and Physics (QMAP) University of California, Davis Grateful to Institute for Particle and Nuclear Physics, Charles University in Prague, and Institute for Advanced Study (IAS) in Princeton for hospitality

2 Particle experiments: our probe to fundamental laws of Nature

3 What happens during the scattering process of elementary particles?

4 What happens during the scattering process of elementary particles?

5 What happens during the scattering process of elementary particles? at least in a specific case.

6 Scattering amplitudes Outcomes of particle experiments probabilistic and given by mathematical functions: scattering amplitudes M(p, s,... ) QCD background: new physics searches At high energies proton scattering dominated by gluon interactions

7 Gluon amplitudes Early 80s: plans for new supercolliders - need for new calculations of gluon amplitudes Leading order gg! ggg Brute force calculation 24 pages of result and many others (k1 k4 )( 2 k1 )( 1 3 )( 4 5 )

8 Parke-Taylor formula Next process on the list: gg! gggg 220 Feynman diagrams 100 pages of calculations Calculation finished in 1985 Paper with 14 pages of result

9 Parke-Taylor formula Next process on the list: gg! gggg 220 Feynman diagrams 100 pages of calculations

10 Parke-Taylor formula

11 Parke-Taylor formula Within a year they realized Spinor-helicity variables h12i4 A6 = h12ih23ih34ih45ih56ih61i pµ = aµa a a (1) (2) h12i = ab a b (1) (2) [12] = a b a b

12 Parke-Taylor formula Within a year they realized 3 (p p ) A6 (p2 p3 )(p3 p4 )(p4 p5 )(p5 p6 )(p6 p1 )

13 Parke-Taylor formula Within a year they realized Spinor-helicity variables h12i4 A6 = h12ih23ih34ih45ih56ih61i pµ = aµa a a (1) (2) h12i = ab a b (1) (2) [12] = a b a b

14 Parke-Taylor formula m Fermi National Accelerator Laboratory FERMILAB-Pub-86/42-T March, 198 Within a year they realized h12i4 An = h12ih23ih34ih45i... hn1i AN AMPLITUDE FOR n GLUON SCATTERING STEPHEN 3. PARKE and T. R. TAYLOR Fermi National Accelerator P.O. Box 500, Batavia, Laboratory IL

15 Change of strategy What is the scattering amplitude? Feynman diagrams Modern methods use both: Calculate the amplitude directly Use perturbation theory Unique object fixed by physical properties Lesson from Parke-Taylor: On-shell gauge invariant objects Helicity amplitudes An,k k = 2 : e.g n Parke-Taylor formula

16 New methods for amplitudes New efficient methods of calculations Unitarity methods BlackHat collaboration QCD background for LHC (Bern, Dixon, Kosower, 1993-today) Recursion relations (Britto, Cachazo, Feng, Witten, 2005) Build amplitude recursively from simpler amplitudes Feynman diagrams Recursion relations gg! 4g gg! 5g gg! 6g

17 Amplitudes as a new field Center for Quantum Mathematics And Physics Studying and calculating scattering amplitudes became a new direction in theoretical physics AMPLITUDES QMAP University of California Davis June 11-15, 2018 Amplitudes Summer School 2018 INTERNATIONAL CONFERENCE JUNE 18TH 22ND SLAC, STANFORD, CALIFORNIA Lecturers include: Nima Arkani-Hamed Zvi Bern Jacob Bourjaily Claude Duhr Song He Eric D Hoker Yutin Huang Alexander Postnikov Experiment Solving QFT N = 4 SYM Geometry & Amplitudes LHC Realistic Theories Unified Descriptions Mathematical Structures Physical Principles Followed by Amplitudes 2018 at SLAC, June Local organizers: Lance Dixon, Enrico Herrmann, Jaroslav Trnka, Andrew Waldron Scientific Advisory Committee: Nima Arkani-Hamed, Einan Gardi, Song He, Henrik Johansson, David Kosower, Lorenzo Magnea, Pedro Vieira, Anastasia Volovich, Lauren Williams The Amplitudes 2018 Summer School will be held June at QMAP at the University of California, Davis Local Organizing Committee: Zvi Bern, Lance Dixon, Falko Dulat, Enrico Herrmann, Stefan Höche, Jaroslav Trnka Major motivations: qmap.ucdavis.edu/events/amplitudes-summer-school Questions: Mathematical sciences trnka@ucdavis.edu building, room 1147 Efficient calculations for particle colliders Use amplitudes as a probe to explore quantum field theory

18 Amplitudes as a new field Center for Quantum Mathematics And Physics Studying and calculating scattering amplitudes became a new direction in theoretical physics AMPLITUDES QMAP University of California Davis June 11-15, 2018 Amplitudes Summer School 2018 INTERNATIONAL CONFERENCE JUNE 18TH 22ND SLAC, STANFORD, CALIFORNIA Lecturers include: Nima Arkani-Hamed Zvi Bern Jacob Bourjaily Claude Duhr Song He Eric D Hoker Yutin Huang Alexander Postnikov Experiment Solving QFT N = 4 SYM Geometry & Amplitudes LHC Realistic Theories Unified Descriptions Mathematical Structures Physical Principles Followed by Amplitudes 2018 at SLAC, June Local organizers: Lance Dixon, Enrico Herrmann, Jaroslav Trnka, Andrew Waldron Scientific Advisory Committee: Nima Arkani-Hamed, Einan Gardi, Song He, Henrik Johansson, David Kosower, Lorenzo Magnea, Pedro Vieira, Anastasia Volovich, Lauren Williams The Amplitudes 2018 Summer School will be held June at QMAP at the University of California, Davis Local Organizing Committee: Zvi Bern, Lance Dixon, Falko Dulat, Enrico Herrmann, Stefan Höche, Jaroslav Trnka Major motivations: qmap.ucdavis.edu/events/amplitudes-summer-school Questions: Mathematical sciences trnka@ucdavis.edu building, room 1147 Efficient calculations for particle colliders Use amplitudes as a probe to explore quantum field theory

19 New formulation of QFT Big goal: find a new formulation of QFT where the picture of interacting particles in spacetime, locality and unitarity is replaced by other principles Hopefully it would make calculation easier Deep motivation comes from gravity: difficult to incorporate gravity in QFT We have one example now: Amplituhedron

20 Amplitude as a volume (Hodges 2009) New variables: p,! Z momentum twistors BCFW recursion relations for A6 ( ) _ _3 2 1 _ (a) _ 1 _3 5 _1 _ 2 _3 6 _ 4 5 (b) 45 _ 2 _3 det(z1 Z2 Z3 Z3 ) _ h1234i = 4 3(c) (c) h1345i h1356i Fig. 2: contributing Configurations contributing to the six-gluon Fig. 2: Configurations to the six-gluon amplitude A (1 amplitude, 2, 3, 4A, 5(1, 6, 2 )., 3, 4, 5, 6 Note (a) andby(c) by a flip and(b) a conjugation. (b) vanishes for either Note that (a) and (c) that are related a are flip related and a conjugation. vanishes for either h1234ih1245ih2345ih1235i h1235ih1256ih2356ih1236i helicity configuration of the internal line. (a) (b) 6 helicity configuration of the internal line. 6 ).

21 ell-decomposition of M. Again, [HRS] implies that SM eally, we would prefer a single 3D index for a cusped ollection of 3D indices. It is known that every two combinatorial ideal triang y a sequence of 2-3 moves [Mat87, Mat07, Pie88]. Th ndex follows from invariance under 2-3 moves. Consider two ideal triangulations T and T! with N an ove shown in Figure 1. Volume of tetrahedron in momentum twistor space! h1345i3 h1234ih1245ih2345ih1235i (Hodges 2009) New variables: p,! Z momentum twistors BCFW recursion relations for A6 ( ) Each face labeled by habcdi h1356i3 h1235ih1256ih2356ih1236i For the next proposition, a special index structure on roposition If T! admits a strict angle structure Figure 1. A 2 3 move: a bipyramid split into N tetrahed T!. Amplitude as a volume

22 ell-decomposition of M. Again, [HRS] implies that SM eally, we would prefer a single 3D index for a cusped ollection of 3D indices. It is known that every two combinatorial ideal triang y a sequence of 2-3 moves [Mat87, Mat07, Pie88]. Th ndex follows from invariance under 2-3 moves. Consider two ideal triangulations T and T! with N an ove shown in Figure 1. Volume of tetrahedron in momentum twistor space! (Hodges 2009) New variables: p,! Z momentum twistors BCFW recursion relations for A6 ( ) spurious plane h1345i3 h1234ih1245ih2345ih1235i Each face labeled by habcdi h1356i3 h1235ih1256ih2356ih1236i For the next proposition, a special index structure on roposition If T! admits a strict angle structure Figure 1. A 2 3 move: a bipyramid split into N tetrahed T!. Amplitude as a volume

23 al triangulations T of M which are a refinement of the EP is nonempty assu f M. Again, [HRS] implies that SM fer a single 3D index for a cusped manifold M, rath es. very two combinatorial ideal triangulations of a 3-man moves [Mat87, Mat07, Pie88]. Thus, topological inva variance under 2-3 moves. triangulations T and T! with N and N 1 tetrahedra r e 1. Amplitude is a volume of polyhedron (Hodges 2009) New variables: p,! Z momentum twistors BCFW recursion relations for A6 ( ) Each face labeled by habcdi Feynman diagrams is another (more complicated) triangulation If T! admits a strict angle structure structure, so does move: a bipyramid split into N tetrahedra for T and N 1 t Amplitude as a volume

24 Amplituhedron (Arkani-Hamed, JT 2013) All tree-level amplitudes of gluons Volume = scattering amplitude Position of vertices: energies and spins of particles

25 Amplituhedron (Arkani-Hamed, JT 2013) All tree-level amplitudes of gluons Volume = scattering amplitude Position of vertices: energies and spins of particles positive Grassmannian curvy space

26 Amplituhedron (Arkani-Hamed, JT 2013) All tree-level amplitudes of gluons Volume = scattering amplitude Position of vertices: energies and spins of particles 2 Definition fits on one slide

27 Amplituhedron (Arkani-Hamed, JT 2013) Loop amplitudes of gluons: can not do in QCD yet Toy model for QCD: planar N=4 super Yang-Mills It is a 4d interacting theory with hidden symmetry Analogue to integrable models: Kepler problem and Hydrogen atom Analogue of Runge-Lenz vector: dual conformal symmetry

28 Amplituhedron (Arkani-Hamed, JT 2013) Calculating amplitude is reduced to the math problem Calculate volume of certain geometric object Triangulation provides an expansion (e.g. Feynman diagrams) Can not derive Amplituhedron from QFT We can prove that the volume function satisfies all properties of scattering amplitudes: factorization etc. In this very specific case we achieved the goal of finding a new definition for the scattering amplitude

29 Step in the program Maybe this is very special and no reformulation exists in general, maybe it exists but it is something else Right/wrong: analyze theoretical data, look for new structures, make proposals and check them Step-by-step process, all steps require new ideas Masses, quarks Establish as an efficient Loop amplitudes in QCD Standard model computational tool Correlation functions Resummation, beyond perturbation theory

30 Physics vs geometry Dynamical particle interactions in 4-dimensions 2 Static geometry in high dimensional space

31 Thank you for your attention