Deep into the Amplituhedron Jaroslav Trnka Center for Quantum - PowerPoint PPT Presentation

Deep into the Amplituhedron Jaroslav Trnka Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, USA work with Nima Arkani-Hamed, Hugh Thomas, Cameron Langer, Akshay Yelleshpur Srikant, Enrico Herrmann, Minshan Zheng, Ryota Kojima KEK Theory Workshop, December 2019

How to define/calculate the perturbative S-matrix in QFT?

How to define/calculate the perturbative S-matrix in QFT?

New picture? In this talk: 2 1 3 6 7 5 4 Positive geometry

Motivation ✤ New efficient methods to calculate S-matrix, understand details of perturbation theory: analytic behavior, function space ✤ New picture for the S-matrix Standard: local evolution in spacetime Search for alternative point of view: get the same result from other principles New approach to deal with quantum gravity

Unexpected simplicity ✤ Practical need for new understanding: simplicity in scattering amplitudes invisible in Feynman diagrams ✤ Famous example: 2->4 gluon amplitudes in QCD 120 Feyman diagrams ( k 1 · k 4 )( ✏ 2 · k 1 )( ✏ 1 · ✏ 3 )( ✏ 4 · ✏ 5 ) 100 pages

Unexpected simplicity ✤ Practical need for new understanding: simplicity in scattering amplitudes invisible in Feynman diagrams ✤ Famous example: 2->4 gluon amplitudes in QCD (Parke, Taylor 1985) M 6 (1 − 2 − 3 + 4 + 5 + 6 + ) Helicity amplitude Tr( T a 1 T a 2 . . . T a 6 ) A 6 (123456) X Color ordering M 6 = Maximal-helicity h 12 i 4 A 6 = ( MHV ) violating h 12 ih 23 ih 34 ih 45 ih 56 ih 61 i amplitude

Modern methods ✤ Very rich playground of ideas Use of physical constraints: unitarity methods, recursion relations Calculating loop integrals, study mathematical functions, symbols Symmetries of N=4 SYM, UV of N=8 SUGRA, string amplitudes ✤ Connection between amplitudes and geometry Canonical example is the geometry of worldsheet Fascinating development of recent years: write QFT amplitudes on worldsheet - CHY formula (Cachazo, He, Yuan 2013) 0 1 Z dz 1 . . . dz n s ab @X A I n A n = Vol[ SL (2 , C )] δ z a − z b b 6 = a

Positive geometry ✤ Geometric space defined using a set of inequalities F k ( x i ) ≥ 0 polynomials parametrize kinematics ✤ Define the differential form on this space Ω ( x i ) Special form: logarithmic singularities on the boundaries Ω ( x i ) ∼ dx i near boundary x i = 0 x i

Simple examples ✤ Example: 1d interval F 1 ( x ) = x − x 1 > 0 F ( x ) = x > 0 F 2 ( x ) = x 2 − x > 0 x = 0 x = x 1 x = x 2 x = ∞ form: Ω = dx normalization: singularities are unit x ≡ dlog x ✓ x − x 1 ◆ dx ( x 1 − x 2 ) Ω = ( x − x 1 )( x − x 2 ) = dlog x − x 2

Simple examples ✤ Example: 2d region x > 0 y > 0 x > 0 y > 0 1 − x − y > 0 x = (0 , ∞ ) y = (0 , ∞ ) x = (0 , 1 − y ) y = (0 , ∞ ) Ω = dx dy ( y − 1) dx y = ( y − 1) dx dy x ( x + y − 1) ∧ dy Ω = x y xy ( x + y − 1) ✤ General positive geometry: more than just boundaries

Positive Grassmannian ✤ Consider space of (2x4) matrices modulo GL(2) Grassmannian ✓ ◆ G (2 , 4) a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 describes a line in P 3 ✤ Positive Grassmannian G + (2 , 4) ( ij ) > 0 not all of them are boundaries All (2x2) minors

Positive Grassmannian ✤ Consider space of (2x4) matrices modulo GL(2) Grassmannian ✓ ◆ G (2 , 4) a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 describes a line in P 3 ✤ Positive Grassmannian G + (2 , 4) ( ij ) > 0 not all of them are boundaries All (2x2) minors Shouten identity (13)(24) = (12)(34) + (14)(23) ( product positive all positive (13) , (24) > 0 (13) , (24) < 0

Positive Grassmannian ✤ Positive Grassmannian G + (2 , 4) Fix GL(2): choose ✓ ◆ 1 0 x − y x, y, z, w > 0 parametrization 0 1 w z ✤ Boundaries: (12) , (23) , (34) , (14) = 0 ✤ Logarithmic form: d 2 × 4 C 1 Ω = dx dy dz dw w = vol[ GL (2)] (12)(23)(34)(14) x y z

Positive geometry for amplitudes ✤ Amplituhedron: planar N=4 SYM (Arkani-Hamed, JT) (Arkani-Hamed, Thomas, JT) Tree-level and all-loop integrand ✤ Associahedron: biadjoint scalar at tree-level (Arkani-Hamed, Bai, He, Yan) Connection to CHY, recent work on 1-loop talk by Song He ✤ More: cosmological polytopes, CFT, EFT (Arkani-Hamed, Benincasa, Huang, Shao) ✤ Gravituhedron: tree-level GR??? (JT, in progress) Note: at the moment, no work on the final (integrated) loop amplitudes space of functions is too complicated beyond 1-loop

Amplituhedron (Arkani-Hamed, JT 2013) (Arkani-Hamed, Thomas, JT 2017)

Amplitudes in planar N=4 SYM ✤ Large N limit: only planar diagrams, cyclic ordering ✤ superfield: ⌘ A e ⌘ B e ⌘ C e ⌘ D G − N = 4 Φ = G + + e ⌘ A Γ A + · · · + ✏ ABCD e n − 2 Component amplitudes ✤ Superamplitudes: X A n = A n,k η 4 k with power ˜ k =2 Contains A n ( − − · · · − + + · · · +) ( ✤ Tree-level + loop integrand k conformal invariant ( Yangian broken after integration dual conformal invariant PSU(2,2|4) due to IR divergencies

Amplitudes as volumes of polytopes ✤ The simplest example is the 6pt NMHV amplitude pioneered by Andrew Hodges in 2009 Z A 6 = dV 3d projection P Volume in dual 2 momentum twistor space Later found this is equal to logarithmic form on the “cyclic polytope” in P 4 (Arkani-Hamed, Bourjaily, Cachazo, Hodges, JT 2010)

T riangulation (Hodges 2009) ✤ Calculation: triangulation in terms of elementary building blocks Divide into two simplices by cutting the polyhedron with (1235) plane 2 The first only depends on (12345) and second on (12356) A 6 = [12345] + [12356] each simplex is associated with “R-invariant” this correctly reproduces amplitude

From kinematics to geometry (Arkani-Hamed, Thomas, JT 2017) ✤ Change of kinematics: p i , ✏ j → Z k ∈ P 3 k = 1 , 2 , . . . , n points in projective space A n,k, ` ` j → ( Z A Z B ) j ∈ P 3 lines in projective space positive geometry in P 3+ k Definition of the Amplituhedron projection Form with logarithmic kinematical space singularities on the boundaries P 3 = amplitude

Back to 6pt NMHV amplitude ✤ Definition of the space: (5x5) determinants convex hall of points h Z 1 Z 2 Z 3 Z 4 Z 5 i , h Z 1 Z 2 Z 3 Z 4 Z 6 i , · · · > 0 Y : P 4 → P 3 projection: positive geometry in h z i z i +1 z j z j +1 i > 0 such that P 4 convex Z j projection these are boundaries ∼ ( p i +1 + p i +2 + . . . p j ) 2 kinematical space In this case (boundaries)>0 P 3 completely specifies the z j projection, hence the space in P 3

Back to 6pt NMHV amplitude ✤ Triangulation -> differential form -> amplitude Ω 6 = [12345] + [12356] two simplicies project to P 3 change variables: x i → z k Ω = dx 1 dx 2 dx 3 dx 4 Logarithmic form: x 1 x 2 x 3 x 4 [12345] = ( h 1234 i dz 5 + h 2345 i dz 1 + h 3451 i dz 2 + h 4512 i dz 3 + h 5123 i dz 4 ) 4 h 1234 ih 2345 ih 3451 ih 4512 ih 5123 i [12356] = ( h 1235 i dz 6 + h 2356 i dz 1 + h 3561 i dz 2 + h 5612 i dz 3 + h 6123 i dz 5 ) 4 h 1235 ih 2356 ih 3561 ih 5612 ih 6123 i where h 1234 i ⌘ h z 1 z 2 z 3 z 4 i

Back to 6pt NMHV amplitude ✤ Triangulation -> differential form -> amplitude A 6 = [12345] + [12356] two simplicies project to P 3 Differential form Replace: Superfunction dz j → η j [12345] = ( h 1234 i η 5 + h 2345 i η 1 + h 3451 i η 2 + h 4512 i η 3 + h 5123 i η 4 ) 4 h 1234 ih 2345 ih 3451 ih 4512 ih 5123 i [12356] = ( h 1235 i η 6 + h 2356 i η 1 + h 3561 i η 2 + h 5612 i η 3 + h 6123 i η 5 ) 4 h 1235 ih 2356 ih 3561 ih 5612 ih 6123 i where h 1234 i ⌘ h z 1 z 2 z 3 z 4 i

TECHNICAL SLIDE Definition of Amplituhedron (Arkani-Hamed, Thomas, JT 2017) ✤ Constraints on positive geometry and the projection h Z a 1 Z a 2 Z a 3 . . . Z a k +3 i > 0 convex hall where h . . . i is (k+3)x(k+3) determinant positive geometry in Y Z a → z a P 3+ k − ∈ ∈ P k +3 P 3 projection h z i z i +1 z j z j +1 i > 0 { h 1234 i , h 1235 i , . . . , h 123 n i } has k sign flips kinematical space P 3 analogue of (13)>0 for G + (2 , 4)

TECHNICAL SLIDE Definition of Amplituhedron (Arkani-Hamed, Thomas, JT 2017) ✤ Tree-level A n,k, ` =0 h Z a 1 Z a 2 Z a 3 . . . Z a k +3 i > 0 h z i z i +1 z j z j +1 i > 0 { h 1234 i , h 1235 i , . . . , h 123 n i } has k sign flips form 4 k Ω n,k ( z j ) → A n,k ( z j , e η j ) tree-level amplitude ✤ Loop integrand A n,k, ` for each line (loop momentum): h ( AB ) j z i z i +1 i > 0 { h ( AB ) j 12 i , h ( AB ) j 13 i , . . . h ( AB ) j 1 n i } has (k+2) sign flips for each pair of lines: h ( AB ) j ( AB ) k i > 0 Ω n,k, ` ( z j , ( AB ) k ) → I n,k, ` ( z j , e η j , ( AB ) k ) 4 k + 4 ` form loop integrand

From geometry to amplitudes ✤ Amplituhedron: space of points and lines in projective space triangulate the space into “simplices” = elementary regions for which the form is trivial dlog form Ω = dx 1 dx 2 . . . dx 4 k +4 ` x k = f ( z i , ( AB ) j ) where x 1 x 2 x 4 k +4 ` ✤ Turn the physics problem of particle interactions, Feynman diagrams to a math problem of triangulations ✤ All the physics properties of scattering amplitudes are consequences of positivity geometry of Amplituhedron

Exploring Amplituhedron (Arkani-Hamed, JT 2013) (Arkani-Hamed, Thomas, JT 2017) (Arkani-Hamed, Langer, Yelleshpur Srikant, JT 2018) (Rao 2017,2018) (Kojima 2018) (Langer, Kojima to appear) (Herrmann, Langer, Zheng, JT, to appear)