Loop integrands in The Galileo Galilei Institute For Theoretical - PowerPoint PPT Presentation

Enrico Herrmann In collaboration with: Jaroslav Trnka + work in progress + Alex Edison, Cameron Langer, Julio Parra-Martinez Loop integrands in 
 The Galileo Galilei Institute 
 For Theoretical Physics N=4 sYM and N=8 sugra 10/31/2018


 
 
 
 
 (0) Motivation ❖ grand idea: reformulate QFT: replace unitarity & locality 
 by new mathematical principles ℙ 3 ❖ 1 hint Hodges: 6pt tree-amp = volume of polyhedron in 
 st ⟨ 1345 ⟩ 3 ⟨ 1356 ⟩ 3 ⟨ 1346 ⟩ 3 ⟨ 3456 ⟩ 3 ⟨ 5146 ⟩ 3 ⟨ 1234 ⟩⟨ 1245 ⟩⟨ 2345 ⟩⟨ 2351 ⟩ + ⟨ 1235 ⟩⟨ 1256 ⟩⟨ 2356 ⟩⟨ 2361 ⟩ = ⟨ 1234 ⟩⟨ 1236 ⟩⟨ 1246 ⟩⟨ 2346 ⟩ + ⟨ 2345 ⟩⟨ 2356 ⟩⟨ 2346 ⟩⟨ 2546 ⟩ + ⟨ 1245 ⟩⟨ 1256 ⟩⟨ 1246 ⟩⟨ 2546 ⟩

(0) Motivation ❖ fascinating interplay between physics & geometry in scattering amplitudes ❖ novel geometric structures primarily in planar N=4 sYM: ❖ Grassmannian [space of k-planes in n-dim] 
 [Arkani-Hamed,Bourjaily,Cachazo,Goncharov,Postnikov,Trnka] ❖ Amplituhedron 
 [Arkani-Hamed,Trnka] ❖ What about other theories? φ 3 ❖ -theory: Associahedron [Arkani-Hamed,Bai,He,Yan] ❖ nonplanar YM? [Bern,Litsey,Stankowicz,EH,Trnka] ❖ gravity? [EH,Trnka] + work in progress [Edison,EH,Langer,Parra-Martinez,Trnka] ❖ N<4 sYM? work in progress [EH,Langer,Trnka]

(0) Motivation ❖ comparison planar N=4 sYM, nonplanar sYM, gravity ❖ planar N=4 sYM ❖ nonplanar N=4 sYM ❖ gravity ❖ identify homogeneous properties which uniquely fix amplitude constrain UV & IR ❖ UV IR ❖ dlog-forms ❖ What are the gravity properties? ❖ no poles at infinity ❖ reformulate constraints as inequalities that define geometry ? ?

(1) Outline ❖ i) setting the stage: 
 amplitudes, integrands, cuts and on-shell diagrams ❖ ii) properties of on-shell (OS) diagrams ❖ iii) from OS-diagrams to properties of amplitudes ❖ iv) Gravity ❖ IR - properties [EH,Trnka '16] ❖ UV - properties [EH,Trnka '18] <— focus on this part ❖ Fixing the amplitude in progress [Edison,EH,Langer,Parra-Martinez,Trnka] ❖ v) Conclusions

i) loop-amplitudes ❖ loop-amplitudes in 4d: kinematic coefficients 𝒝 ( L ) = c k ∫ ℐ ∑ k d 4 ℓ 1 ⋯ d 4 ℓ L k basis integrands [Jake’s talk] ❖ generalized unitarity: match amplitude on cuts —> fix c’s

i) planar integrand ⇔ ❖ planar integrand unambiguous labels! x 3 2 ℓ 3 3 p μ i = ( x μ i +1 − x μ i ) ℓ 2 y 2 y 3 dual-variables x 4 x 2 ℓ μ i = ( y μ i − x μ i ) y 1 ℓ 1 1 4 x 1 k d 4 ℓ 1 ⋯ d 4 ℓ L = ∫ ℐ d 4 y 1 ⋯ d 4 y L c k ∫ ℐ 𝒝 ( L ) = ∑ k ❖ well-defined notion of an integrand rational function ❖ properties of integrated answer encoded in ℐ ❖

i) ambiguity in non-planar integrands ❖ no global loop-variables in nonplanar diagrams: 2 2 ℓ 2 ℓ 1 vs. 3 4 3 4 ℓ 2 ℓ 1 1 1 ❖ no global definition of an integrand —> stick with diagrams c k ∫ ℐ 𝒝 ( L ) = ∑ k d 4 ℓ 1 ⋯ d 4 ℓ L k ❖ expansion objects for: Is there a way out? ❖ non-planar YM ❖ gravity

i) cuts of loop-integrands ❖ unitarity cut: 1 = ( ℓ 1 + p 1 + p 2 ) 2 = 0 ℓ 2 ∑ 1 =0=( ℓ 1 +1+2) 2 𝒝 (1) (1234) = 𝒝 (0) L × 𝒝 (0) Res R ℓ 2 states ❖ generalized unitarity: on-shell functions ℓ 2 1 = ⋯ = ℓ 2 8 = 0 i =0 𝒝 (2) = ∑ 𝒝 (0) 1 × ⋯ × 𝒝 (0) Res 7 ℓ 2 states ❖ well-define loop-variables on cuts!

i) on-shell diagrams ❖ generalized unitarity: on-shell diagram ℓ 2 1 = ⋯ = ℓ 2 7 = 0 6 = f ( z ; λ i , ˜ ∑ i =0 𝒝 (2) (1234) = 𝒝 (0) 1 × ⋯ × 𝒝 (0) Res λ i ) ℓ 2 states ❖ elementary building blocks: 𝒝 MHV 𝒝 MHV : : 3 3 λ 1 ∼ ˜ ˜ λ 2 ∼ ˜ λ 1 ∼ λ 2 ∼ λ 3 λ 3

ii) Grassmannian and on-shell diagrams ❖ fascinating connection between physics and mathematics ❖ connection to algebraic geometry, combinatorics, …

ii) Grassmannian and on-shell diagrams ❖ planar diagrams in mathematics: building matrices with positive minors Gr ≥ ( k , n ) ≃ {[( k × n ) matrices ]/ GL ( k ) | ordered ( k × k ) minors ≥ 0} C = ( 0 − α 4 α 3 ) , 1 α 1 1 2 ↔ α i > 0 Α 1 0 α 2 1 Α 4 Α 2 k : helicity-sector / R-charge Α 3 4 3 n : # external legs ❖ connection to physics: value of N=4 sYM OS-diag is [Arkani-Hamed,Bourjaily,Cachazo,Goncharov,Postnikov,Trnka] Ω 𝒪 =4 sYM = d α 1 ⋯ d α r δ ( C ⋅ 𝒶 ) α 1 α r all external kinematics

ii) Grassmannian and on-shell diagrams ❖ non-planar diagrams —> give up positivity 3 Gr ( k , n ) ≃ {[( k × n ) matrices ]/ GL ( k )} C = ( α 2 α 2 ( α 6 + α 3 α 5 ) ( α 6 + α 3 α 5 ) 1 α 5 ) 1 α 1 + α 2 α 3 α 4 α 3 α 4 0 α 4 α 3 α 6 ↔ 2 4 5 0 α 5 α 1 α 4 k : helicity-sector / R-charge 1 n : # external legs ❖ connection to physics: value of N=8 sugra OS-diag is [EH, Trnka] Ω 𝒪 =8 sugra = [ Δ v ] δ ( C ⋅ 𝒶 ) d α 1 ⋯ d α r ∏ α 3 α 3 r 1 v

iii) from OS-diags to amplitudes ❖ planar N=4 sYM —> BCFW loop-recursion relations y ❖ e.g. 6pt NMHV y = + + = 6 = 6 ❖ amplitudes inherit properties of OS-diags! ❖ theories where BCFW-loop recursion unknown: 
 OS-diags <—> cuts of loop integrands: encode properties of amplitude

iii-1) from OS-diags to amplitudes: YM ❖ N=4 sYM (planar & non-planar) ❖ IR-property: logarithmic singularities! Ω 𝒪 =4 sYM = d α 1 ⋯ d α r δ ( C ⋅ 𝒶 ) α 1 α r all external kinematics ❖ IR-condition on analytic properties of amplitudes: dx , as x → a ( singular point ) 𝒝 ∼ x − a R ( x , . . . ) ❖ nontrivial constraints on possible local integrand basis elements!

interlude: Feynman integrals in dlog-form 2 d α 3 Ω = d α 1 d α 2 d α 4 3 × 𝒝 tree × δ ( C ⋅ 𝒶 ) α 1 4 α 1 α 2 α 3 α 4 α 4 α 2 logarithmic form in Grassmannian variables! α 3 can identify and solve for Feynman loop variables ℓ μ 4 1 Arkani-Hamed,Cachazo,Goncharov,Postnikov,Trnka: 1212.5605 ⇕ 2 3 ( ℓ − ℓ *) 2 d log ( ℓ − p 1 ) 2 ( ℓ − ℓ *) 2 d log ( ℓ − p 1 − p 2 ) 2 d log ( ℓ + p 4 ) 2 ℓ 2 Ω = d log ( ℓ − ℓ *) 2 ( ℓ − ℓ *) 2 1 ℓ 4 new representation of Feynman integrals

dlog-representation exists for more general FI Bern,EH,Litsey,Stankowicz,Trnka: 1412.8584, 1512.08591 dlog forms exist for special integrals ∙ related to UT conjecture of 𝒪 = 4 sYM ∙ basis of integrals for Henn diff. eqs. ∙ new symmetries of nonplanar theories? ∙ potential geometric interpretation?

iii-2) from OS-diags to amplitudes: YM ❖ N=4 sYM (planar & non-planar) ❖ UV-property: no poles at infinity! • planar: manifest in terms of mom. twistors • non-planar: need to check in local expansion, 
 term-by-term analysis 
 ❖ stronger than UV-finiteness, e.g. triangle integral ∼ dz ℓ μ ( z ) ∼ z , has Res @ ℓ → ∞ z ,

iii-3) uniqueness of YM ❖ non-planar N=4 sYM ❖ Combine IR- & UV-properties • term-by-term analysis • dlog-forms { • no poles at infinity 
 ❖ new non-planar symmetry? [Bern,Enciso,Ita,Shen,Zeng; Chicherin,Henn,Sokatchev] 
 𝒝 ( L ) = c k ∫ ℐ ∑ k d 4 ℓ 1 ⋯ d 4 ℓ L k ❖ fix c’s with homogeneous cuts: geometric interpretation Res 𝒝 ( L ) = 0 [Bern,EH,Litsey,Stankowicz,Trnka] 


iv) gravity [EH,Trnka] 
 ❖ Does there exist an analogous story in gravity? ❖ Gravity is nonplanar —> term-by-term analysis? • analytic properties that single out gravity? dz z ≫ 1 ∼ z 4 − L pole at infinity non-logarithmic poles at infinity drastically different properties than in YM!

iv-1) gravity in the IR [EH,Trnka] 
 ❖ Gravity on-shell diagrams: Ω 𝒪 =8 sugra = [ Δ v ] δ ( C ⋅ 𝒶 ) d α 1 ⋯ d α r ∏ α 3 α 3 r 1 v on-shell diagrams vanish in collinear region ❖ Gravity on-shell functions, i.e. more general cuts: near ⟨ ℓ 1 ℓ 2 ⟩ = 0 : ↔ 1 ∼ ℳ ∼ [ ℓ 1 ℓ 2 ] ⟨ ℓ 1 ℓ 2 ⟩ × regular ⟨ ℓ 1 ℓ 2 ⟩ [ ℓ 1 ℓ 2 ] gravity properties are “global” in nature!

iv-2) mild-IR behavior of gravity amplitudes ❖ collinear region of loop momentum: ℓ 2 = 0 ⇒ ( ℓ − p 1 ) 2 = ⟨ ℓ 1 ⟩ [ ℓ 1] ℓ 2 = 0 = ⟨ ℓ 1 ⟩ = [ ℓ 1] ⇒ ℓ μ = α p μ 1 ❖ Gravity on-shell functions vanish there! nontrivial cancelations even at L=1 ∼ ⟨ ℓ 1 ⟩ [ ℓ 1] × regular ⟨ ℓ 1 ⟩ → 0 ⟶ 0 • L=1, 4pt: 
 sum of 6 boxes homogeneous constraint! 𝒝 ( L ) ∼ 1 ℳ ( L ) ∼ 1 vs. ϵ 2 L ϵ L gravity on-shell functions vanish in collinear region <—> soft IR-behavior of Amplitude

iv-3) gravity in the UV ❖ no off-shell definition of : no invariant probe of ℓ →∞ ℓ ❖ study cuts that make well defined, then probe ℓ →∞ ℓ ❖ maximal cuts: dictate diagram scaling! ∼ dz N ∼ ( ℓ 1 ⋅ ℓ 2 ) 2 L − 6 z 4 − L

iv-3) gravity in the UV ❖ Can we do better than maximal cuts? ❖ get as close as possible to off-shell ℐ ❖ multi-unitarity cut! L+1 props on-shell + ⋯ + z →∞ ❖ interesting cancellation when as ℓ i ( z ) a < max ( b i ) → ∞ ℳ ( z ) | cut ∼ z a = z b 1 + z b 2 + z b 3 + ⋯