Higher derivative corrections from mixing color and kinematics - PowerPoint PPT Presentation

Higher derivative corrections from mixing color and kinematics Laurentiu Rodina 
 IPhT, CEA Saclay with J.J. Carrasco, Z. Yin, S. Zekioglu December 12, 2019 
 QCD meets Gravity V

Amplitude bootstrap without unitarity gauge Adler soft UV color/ Locality + invarianc e zero theorems scaling kinematics Yang-Mills X X X Gravity X X X Einstein-Yang-Mills X X X Bi-adjoint scalar X X X NLSM X X X X DBI X X special Galileon X X X Born-Infeld X conformal dilaton X [Arkani-Hamed, LR, Trnka ’16] [LR ’16, ’18] [Carrasco, LR ’19] 2


 What can black holes teach us about amplitudes? Proposed that soft particles carry black hole information 
 [Hawking, Perry, Strominger ’16] 
 How much BH information can soft particles carry anyway? 
 How much amplitude information can soft particles carry? 3


 
 
 
 
 
 IR vs UV • Naive answer: IR and UV are disjoint p = zp , z → 0 A n +1 = 1 z A ( − 1) + z 0 A (0) + zA (1) + z 2 A (2) + … = ( ∞ z S 0 + z 0 S 1 ) A n + 1 ∑ z i A ( i ) i =2 = IR (soft theorem satisfying) + UV (soft theorem avoiding) • In fact, soft theorems completely constrain amplitudes! • The UV info of one particle is hidden in the IR of a di ff erent one zp 1 → 0 ⟶ O ( z ) e 1 ⋅ e 2 e 3 ⋅ p 1 e 4 ⋅ p 1 p 1 . p 2 zp 2 → 0 ⟶ O (1/ z ) 4


 
 Adler zero • NLSM, scalar theory of Nambu-Goldstone bosons 
 p 1 ⋅ p 3 p 4 ⋅ p 6 A 6 = ( p 1 + p 2 + p 3 ) 2 + … A 4 = p 1 ⋅ p 3 • Adler zero: 
 p i → zp i , z → 0 ⇒ A nlsm → O ( z ) • Adler zero uniquely fixes NLSM amplitudes (DBI, sGal) 
 [Cheung, Kampf, Novotny, Trnka, ’14][Arkani-Hamed, LR, Trnka, ’16][LR ’16] • Implies Yang-Mills uniqueness from soft theorems n +1 → ( n +1 → ( z S 0 + z 0 S 1 ) A n + O ( z ); z S 0 + z 0 S 1 ) A n + O ( z ) 1 1 A YM B YM A YM n +1 − B YM n +1 → O ( z ) 5


 “Adler 1/0” • UV scaling under two particle BCFW shifts fixes YM, GR [LR ’16] • GR integrands via UV [Edison, Herrmann, Parra-Martinez, Trnka ’19] • Scalar theories? 
 p i → z p i , z → ∞ • Fixes NLSM, sGal, bi-adjoint scalar [Carrasco, LR ’19]

Possibility: IR/UV duality • IR or UV behavior ⇒ unique S-matrix solution • Conformal symmetry? [Loebbert, Mojaza, Plefka ’18] 7


 
 Color-kinematic duality 𝒝 = ∑ c i n i d i • Color-dual representation: satisfy antisymmetry and Jacobi c , n • 1. Double copy 
 c → n ⇒ ℳ = ∑ n i n i d i • 2. Amplitude relations 
 k 12 A (2134... n ) + k 13 A (2314... n ) + . . . + k 1, n − 1 A (234...1 n ) = 0

Bootstrapping via color/kinematics • Test subject: NLSM • Immensely simpler: BCJ relations + cyclic inv. 
 k 12 A (2134... n ) + k 13 A (2314... n ) + . . . + k 1, n − 1 A (234...1 n ) = 0 • BCJ relations + Locality uniquely fix NLSM amplitudes 
 [Carrasco, LR ’19] • Add Unitarity => Higher derivative NLSM • But (h.d.) NLSM = abelian Z-theory… 9


 Higher derivatives • Open superstring? (Z-theory) (SYM) 
 ⊗ [Carrasco, Mafra, Schlotterer ’17][Mafra, Schlotterer ’17] • Open bosonic string? (Z-theory) ( DF ) 2 (YM+ ) 
 ⊗ [Huang, Schlotterer, Wen ’16][Azevedo, Chiodaroli, Johansson, Schlotterer ’18] A 4 = u 2 + 2 s t A 4 = u 2 − s t • O ( p 4 ) NLSM? ; [LR ’18][Low, Yin ’19] • Natural question: local building blocks that reach all orders in mass dim? 𝒝 = ∑ c i n i d i ? Yes! Very few building blocks are needed

Mixing color with kinematics 𝒝 = ∑ c i n i d i • Traditionally color and kinematics separated • Allow mixing: c = a 1 s 12 Tr( T 1 T 2 T 3 T 4 ) + a 2 s 23 Tr( T 1 T 4 T 3 T 2 ) + … • New solutions possible! • Constructive approach?

Building blocks and a composition rule • Kinematic permutation invariants spanned by 
 σ 2 = s 2 + t 2 + u 2 
 σ 3 = s 3 + t 3 + u 3 = stu • Composition rule (New Jacobi from Old Jacobi) J s ( k , j ) = k t j t − k u j u • Simplest kinematic numerator (linear): n ss => YMS s = t − u • Next simplest (quadratic): n nl s = J s ( n ss , n ss ) = s ( t − u ) => NLSM • Next: J s ( n nl , n ss ) = n ss … s σ 2 • In general scalar kin: n = n ss f ( σ 2 , σ 3 ) + n nl g ( σ 2 , σ 3 )

̂ 
 ̂ ̂ 
 
 
 
 ̂ 
 
 ̂ ̂ ̂ Modifying color factors • c s ≡ c s σ Y 2 σ X 3 • c ss s = J s ( n ss , c ) σ X 2 σ Y 3 = ( n ss t c t − n ss u c u ) σ Y 2 σ X 3 d abcd = ∑ Tr(T a T b T c T d ) • Color permutation invariant: • c nl , d = d abcd n nl s σ X 2 σ Y s 3 C = ∑ C s n s C t n t C u n u c ss + ̂ c nl , d ) ⟶ 𝒝 = ( ̂ c + ̂ + + s t u X , Y • String field theory limit can rearranged to manifest building blocks 
 (see our paper!) 


̂ 
 ̂ 
 ̂ ̂ ̂ ̂ ̂ Corrections to supergravity amplitudes C s n ym C t n ym C u n ym 𝒝 ym = s t u • + + s t u n ym = ̂ • Replace color with kinematics c → c | c → n ym n ym = n ym σ Y • Di ff . inv. only for c = c σ Y 2 σ X 2 σ X 3 → 3 GR hd = GR ∑ a X , Y σ Y 2 σ X 3

̂ 
 Non-susy h.d. A = ∑ c i n i • Non-susy h.d.? Need to modify n ym d i • Only 7 gauge invariant tensor structures 
 [Bern, Edison, Kosower, Parra-Martinez ’17] Bosonic string = (Z theory) ⊗ (YM + ( DF ) 2 ) n F 3 , n ( F 3 ) 2 + F 4 , n d 2 F 4 , n d 4 F 4 σ Y • Only 4 new building blocks ( 2 σ X ), , with C 3 [Broedel, Dixon ’12][Garozzo, Queimada, Schlotterer ’18][Azevedo, Chiodaroli, Johansson, Schlotterer ’18] • conjectured to be fixed by gauge inv., unitarity, BCJ relations, and ( DF ) 2 scaling in the limit 
 α ′ � → ∞

Summary and Outlook • Just a few building blocks required to capture the full expansion of open superstring, a few extra to capture bosonic string • This was only single trace color structure, can be extended to double trace [Low, Yin ’19] • Composition rules exist at higher multiplicity • General building blocks, loop-level building blocks? Stay tuned!