Gravituhedron Jaroslav Trnka Center for Quantum Mathematics and - PowerPoint PPT Presentation

Towards the Gravituhedron Jaroslav Trnka Center for Quantum Mathematics and Physics (QMAP) University of California, Davis QCD meets Gravity, UCLA, December 2019

What is the scattering amplitude? New picture? ?

Various approaches s n o i t a l e r n o i s r physical u c e r principles , y t i r a t i n u

Various approaches s n o i t a l e r n o i s r physical u c e r principles , y t i r a t i n u color-kinematics, CHY,… unified picture for amplitudes

Various approaches s n o i t a l e r n o i s r physical u c e r principles , y t i r a t i n u color-kinematics, CHY,… unified picture for amplitudes

Various approaches s n o i t a l e r n o i s r physical u c e r principles , y t i r a t i n u color-kinematics, CHY,… unified picture positive geometry for amplitudes amplitude is form/volume

Various approaches s n o i t a l e r n o i s r physical u c e r principles , y t i r a t i n u color-kinematics, CHY,… unified picture different sort of geometry, positive geometry for amplitudes D=4 important worldsheet geometry amplitude is combinatorics of cubic graphs form/volume

Geometry of gluon amplitudes

T ree-level gluon amplitudes ✤ Simplest amplitude: MHV amplitude (Parke, Taylor) color-ordered (Nair) δ ( Q ) amplitudes: A n = h 12 ih 23 ih 34 ih 45 i . . . h n 1 i cyclic symmetry ✤ General N=4 SYM tree-level amplitude dual conformal symmetry A n,k = A n × R n,k ( Z j ) momentum twistors geometric interpretation

Amplitude as volume ✤ Focus on 6pt NMHV amplitude 1 − 2 − 3 − 4 + 5 + 6 + (Hodges 2009) 3-d polytope in Z momentum twistor space A 6 = dV P “face” of 4-d polytope superamplitude

Amplitude as volume

Amplitude as volume face Z 4

Amplitude as volume face Z 6

Amplitude as volume edge 13 ≡ Z 1 Z 3

Amplitude as volume vertices correspond to poles in the amplitude h 2356 i h 1256 i h 2345 i h 6123 i h 1245 i h 1234 i “2” is special h ii + 1 jj + 1 i ⇠ s i +1 ...j 1 − 2 − 3 − 4 + 5 + 6 +

Amplitude as volume vertices correspond to poles in the amplitude s 345 [61] [34] [12] s 234 [23]

Amplitude as volume vertices correspond to poles in the amplitude s 345 [61] [34] [12] s 234 [23] only edges, faces correspond to consistent factorizations / soft limits s 234 = [23] = 0

Amplitude as volume How to calculate amplitude from this picture?

T riangulation 135

T riangulation numerator fixed by weights 135 h 1345 i 3 h 1245 ih 2345 ih 1234 ih 1235 i Volume of tetrahedron poles are vertices

T riangulation 135 Volume of tetrahedron h 1356 i 3 h 1256 ih 6123 ih 2356 ih 1235 i

T riangulation 135 h 1345 i 3 h 1245 ih 2345 ih 1234 ih 1235 i - R 6 , 3 = h 1356 i 3 h 1256 ih 6123 ih 2356 ih 1235 i Volume of polyhedron

T riangulation 135 h 1345 i 3 h 1245 ih 2345 ih 1234 ih 1235 i - R 6 , 3 = h 1356 i 3 h 1256 ih 6123 ih 2356 ih 1235 i Volume of polyhedron spurious pole: cancels

T riangulation 135 h 1345 i 3 h 1245 ih 2345 ih 1234 ih 1235 i - R 6 , 3 = h 1356 i 3 h 1256 ih 6123 ih 2356 ih 1235 i In momentum space h 1 | 2 + 3 | 4] 3 h 3 | 4 + 5 | 6] 3 A 6 , 3 = + s 234 [23][34] h 56 ih 61 ih 5 | 3 + 4 | 2] s 345 [61][12] h 34 ih 45 ih 5 | 3 + 4 | 2]

T riangulation 135 h 1345 i 3 h 1245 ih 2345 ih 1234 ih 1235 i - R 6 , 3 = h 1356 i 3 h 1256 ih 6123 ih 2356 ih 1235 i In momentum space h 1 | 2 + 3 | 4] 3 h 3 | 4 + 5 | 6] 3 A 6 , 3 = + s 234 [23][34] h 56 ih 61 ih 5 | 3 + 4 | 2] s 345 [61][12] h 34 ih 45 ih 5 | 3 + 4 | 2]

Amplituhedron (Arkani-Hamed, JT 2013) (Arkani-Hamed, Thomas, JT 2017) ✤ This volume picture does not generalize dual A = volume A = dlog form ✤ Amplituhedron: generalization to Grassmannians, beyond ✤ Search for “dual Amplituhedron” still ongoing (Arkani-Hamed, Hodges, JT 2015) (Herrmann, Langer, Zheng, JT, in progress)

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ

BCFW recursion relations ✤ Calculate 6pt amplitude using BCFW recursion relations 1 − 2 − 3 − 4 + 5 + 6 + many different shifts λ e λ We always get one of two formulas: R 6 , 3 = (1) + (3) + (5) = (2) + (4) + (6) R-invariants make manifest R [ a, b, c, d, e ] = ( h abcd i η e + h bcde i η a + · · · + h eabc i η d ) 4 dual conformal h abcd ih bcde ih cdea ih deab ih eabc i (and also Yangian) symmetry where (1) = R [2 , 3 , 4 , 5 , 6] etc

R-invariants (Drummond, Henn, Korchemsky, Sokatchev) (Arkani-Hamed, Cachazo, Cheung, Kaplan) (Mason, Skinner) R 6 , 3 = (1) + (3) + (5) = (2) + (4) + (6) h 1356 i 3 h 1345 i 3 h 1256 ih 6123 ih 2356 ih 1235 i + h 1245 ih 2345 ih 1234 ih 1235 i h 3456 i 3 h 1456 i 3 h 1346 i 3 h 2345 ih 2356 ih 2346 ih 2456 i + h 1245 ih 1256 ih 2456 ih 1246 i + h 1234 ih 6123 ih 1246 ih 2346 i 1 − 2 − 3 − 4 + 5 + 6 + For helicity amplitude

R-invariants (Drummond, Henn, Korchemsky, Sokatchev) (Arkani-Hamed, Cachazo, Cheung, Kaplan) (Mason, Skinner) R 6 , 3 = (1) + (3) + (5) = (2) + (4) + (6) h 1356 i 3 h 1345 i 3 h 1256 ih 6123 ih 2356 ih 1235 i + h 1245 ih 2345 ih 1234 ih 1235 i h 3456 i 3 h 1456 i 3 h 1346 i 3 h 2345 ih 2356 ih 2346 ih 2456 i + h 1245 ih 1256 ih 2456 ih 1246 i + h 1234 ih 6123 ih 1246 ih 2346 i For helicity amplitude 1 − 2 − 3 − 4 + 5 + 6 + Two different triangulations

Gluon recap Amplitude = volume of geometry

Gluon recap Amplitude = volume of geometry Triangulation: BCFW recursion: = spurious vertices spurious poles rigid formulas = fixed by geometry manifest dual conformal symmetry

Graviton amplitudes

Graviton amplitudes ✤ No momentum twistors, use spinor helicity variables ✤ MHV amplitude: non-trivial, does not factorize Other formulas det H A n = (Berends, Giele, Kuijf 1988) (Hodges 2012) (Bern, Dixon, Perelstein, Rozowsky 1999 ( abc )( def ) (Mason, Skinner 2008) (Spradlin, Volovich, Wen 2009) ✤ Natural proposal: study BCFW recursion relations I dz A ( z )( a + bz ) Problem: good large-z behavior = 0 A n ( z ) ∼ 1 z z 2 Many different formulas, no rigidity, they all look different

Example of BCFW formula ✤ Explicit example: (34) shift (Cachazo, Svrcek) flip flip A (1 − , 2 − , 3 − , 4 + , 5 + , 6 + ) = D 1 + D + D 2 + D 3 + D + D 6 . � | � 1 3 ↔ ⟨ 23 ⟩⟨ 1 | 2 + 3 | 4] 7 � � ⟨ 1 | 2 + 3 | 4] ⟨ 5 | 3 + 4 | 2][51] + [12][45] ⟨ 51 ⟩ p 2 234 D 1 = ⟨ 15 ⟩⟨ 16 ⟩ [23][34] 2 ⟨ 56 ⟩ p 2 234 ⟨ 1 | 3 + 4 | 2] ⟨ 5 | 3 + 4 | 2] ⟨ 5 | 2 + 3 | 4] ⟨ 6 | 3 + 4 | 2] ⟨ 6 | 2 + 3 | 4] � | � + (1 ↔ 2) . ⟨ 13 ⟩ 7 ⟨ 25 ⟩ [45] 7 [16] D 2 = − ⟨ 16 ⟩ [24][25] ⟨ 36 ⟩ p 2 245 ⟨ 1 | 2 + 5 | 4] ⟨ 6 | 2 + 5 | 4] ⟨ 3 | 1 + 6 | 5] ⟨ 3 | 1 + 6 | 2] � | � ↔ + (1 ↔ 2) + (5 ↔ 6) + (1 ↔ 2 , 5 ↔ 6) . ⟨ 13 ⟩ 8 [14][56] 7 � � ⟨ 23 ⟩⟨ 56 ⟩ [62] ⟨ 1 | 3 + 4 | 5] + ⟨ 35 ⟩ [56] ⟨ 62 ⟩⟨ 1 | 3 + 4 | 2] D 3 = ⟨ 14 ⟩ [25][26] ⟨ 34 ⟩ 2 p 2 134 ⟨ 1 | 3 + 4 | 2] ⟨ 1 | 3 + 4 | 5] ⟨ 1 | 3 + 4 | 6] ⟨ 3 | 1 + 4 | 2] ⟨ 3 | 1 + 4 | 5] ⟨ 3 | 1 + 4 | 6] � | � + (1 ↔ 2) . ⟨ 12 ⟩ [56] ⟨ 3 | 1 + 2 | 4] 8 D 6 = 124 ⟨ 5 | 1 + 2 | 4] ⟨ 6 | 1 + 2 | 4] ⟨ 3 | 5 + 6 | 1] ⟨ 3 | 5 + 6 | 2] . [21][14][24] ⟨ 35 ⟩⟨ 36 ⟩⟨ 56 ⟩ p 2

Reminder: Yang-Mills formula h 1 | 2 + 3 | 4] 3 h 3 | 4 + 5 | 6] 3 Rewrite: + s 234 [23][34] h 56 ih 61 ih 5 | 3 + 4 | 2] s 345 [61][12] h 34 ih 45 ih 5 | 3 + 4 | 2] h 1 | 2 + 3 | 4] 3 X A 6 , 3 = s 234 · [23][34] · h 56 ih 61 i · h 5 | 3 + 4 | 2] S relabeling sum multiparticle spurious holomorphic (123 , 456) → (321 , 654) anti-holomorphic Only adjacent indices appear