Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Feynman integrals reduction and Intersection theory Luca Mattiazzi Cortona Young 28 th May 2020 1 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Based on: – Pierpaolo Mastrolia and Sebastian Mizera Feynman integrals and Intersection theory JHEP 1902 (2019) 139 – H.Frellesvig, F.Gasparotto, S.Laporta, M.Mandal, P. Mastrolia, L.M., S.Mizera Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers JHEP 1905 (2019) 153 – H.Frellesvig, F.Gasparotto, M.Mandal, P. Mastrolia, L.M., S.Mizera Vector Space of Feynman Integrals and Multivariate Intersection Numbers Phys.Rev.Lett. 123 (2019) no.20, 201602 2 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Motivation Higgs discovery Success of Standard Model ⇓ SM as effective theory (Dark Matter, Dark Energy...) ⇓ New physics hide in subtle effects ⇓ Precision calculation needed ⇓ Scattering amplitudes are powerful means to access it 3 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Motivation Scattering amplitudes are at the core of cross sections measured in colliders very effective tool to know gra- vitational waveforms with high precision in weak field approxima- tion [Foffa,Sturani,Mastrolia,Sturm (2016)] [Foffa,Sturani,Mastrolia,Sturm,Torres Bobadilla (2019)]... 4 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Motivation Scattering amplitudes are built out of many Feynman integrals: � L � 1 d d k i I a 1 ,..., a N = 1 . . . D a N D a 1 N i =1 Higher precision ⇒ Higher loop Complexity of the calculations increases quickly State of the art calculations at 2 loop, such as requires O (10000) integrals. Needs to evaluate them all? No! [Bern, Dixon, Kosower (2012)] 5 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Integration By Parts Identities Linear relations among integrals: Integration By Parts Identities - IBPs [Chetyrkin, Tkachov (1981)] � L � � [Laporta (2001)]... � v µ ∂ d d k i = 0 ⇒ c 1 I a 1 +1 ,..., a N + · · · + c N I a 1 ,..., a N +1 = 0 ∂ k µ 1 . . . D a N D a 1 j N i =1 Integrals related by a total derivative Linear System ⇒ Gauss Elimination ⇒ Master Integrals { J i } - MIs Decomposition of an Integral in terms of MIs ν � I a 1 ,..., a N = c i J i i =1 6 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Integration By Parts Identities [Kotikov ’91, Remiddi ’97, Gehrmann & Remiddi ’00, Argeri & Mastrolia ’07, Henn ’13] [Tarasov ’96, Lee ’07] MIs as solutions of ⇒ Differential equations Built by means of IBPs Dimensional recurrence relations ν � ∂ s J k = ⇒ ∂ s J = A J A = c i J i i =1 IBPs Drawbacks: # equations grows dramatically manipulation large expressions J 1 I a 1 ,..., a N Possible bottleneck of multiloop calculations Can we directly project integrals on MI? J 2 7 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Intersection theory [Aomoto, Kita, Matsumoto, Mizera, ...] In Baikov representation [Mastrolia, Mizera (2018)] � I a 1 ,..., a N = u ( z ) ϕ ( z ) C � P i ( z ) γ i ⇒ u ( z ) multivalued function s.t. P i ( ∂ C ) = 0 u ( z ) = i ϕ ( z ) = ˆ ϕ ( z ) d z ⇒ ϕ ( z ) single valued form Total derivative translates to � � � � d u � d( u ϕ ) = d u ϕ + u d ϕ = u u + d ϕ C C C � � = u ( ω + d) ϕ = u ∇ ω ϕ = 0 P ω = { z | z is a pole of ω } C C rewriting Integration by Parts Identities as � � u ( ϕ + ∇ ω ξ ) = u ϕ ⇒ ϕ ∼ ϕ + ∇ ω ξ C C 8 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Equivalence class between forms defines the Twisted cohomology group. ω � ϕ | ≡ { ϕ |∇ ω ϕ = 0 } / {∇ ω ξ } = H n ω Key relation between IBPs and Twisted Cohomology [Aomoto (1975)] ν = dim ( H n ω ) [Lee, Pomeransky (2013)] = χ ( X ) = ( − 1) n ( n + 1 − χ ( P ω ) ) = { # of solutions of ω = 0 } Contours have similar structure � � |C ] = H ω u ϕ = u ϕ ⇒ n Twisted Homology group C C + ∂ ω g Dual integrals Feynman integrals are pairing � � cocycle � u − 1 ( ϕ + ∇ − ω ξ ) u − 1 ( z ) ϕ ( z ) = [ C| ϕ � = ���� � ϕ |C ] = u ( z ) ϕ ( z ) C C C � �� � − ω , [ C | = H − ω ⇒ | ϕ � = H n cycle n 9 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Twisted intersection number Seeking relations between integrals (forms) ⇒ Intersection number � 1 � ϕ L | ϕ R � = ι ( ϕ L ) ∧ ϕ R (2 π i ) n [Mastrolia, Mizera (2018)] [Frellesvig, Gasparotto, Laporta, Mandal, IBP built in naturally within such formalism. Mastrolia, L.M. , Mizera (2019)] � ϕ L + ∇ ω ξ | ϕ R � = � ϕ L | ϕ R + ∇ − ω ξ � = � ϕ L | ϕ R � Define basis of independent form and dual form � e 1 | , � e 2 | , · · · , � e ν | , | h 1 � , | h 2 � , · · · , | h ν � Build the matrix � ϕ L | ϕ R � � ϕ L | h 1 � � ϕ L | h 2 � . . . � ϕ L | h ν � � e 1 | ϕ R � � e 1 | h 1 � � e 1 | h 2 � . . . � e 1 | h ν � � � ϕ L | ϕ R � � � e 2 | ϕ R � � e 2 | h 1 � � e 2 | h 2 � . . . � e 2 | h ν � A ⊺ M = ≡ B C . . . . ... . . . . . . . . � e ν | ϕ R � � e ν | h 1 � � e ν | h 2 � . . . � e ν | h ν � 10 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Master Decomposition Formula � ϕ L | depends on the basis element � � � ϕ L | ϕ R � − A ⊺ C − 1 B det M = det C = 0 ⇓ ν � � ϕ L | ϕ R � = A ⊺ C − 1 B = � ϕ L | h j � ( C − 1 ) ji � e i | ϕ R � i , j =1 Since | ϕ R � is arbitrary ν ν ν � � � � ϕ L | h j � � C − 1 � C ij = δ ij � ϕ L | = ji � e i | = � ϕ L | h i � � e i | = c i J i i , j =1 i =1 i =1 � e 1 | � ϕ L | Direct projection � e 2 | 11 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Univariate computation � � � 1 �� � � � � ∇ − 1 ϕ L ∇ − 1 � ϕ L | ϕ R � = ι ( ϕ L ) ∧ ϕ R = ω ϕ L ϕ R = − − ω ϕ R Res Res (2 π i ) z = p z = p p ∈P ω p ∈P ω � � Res z = p ( ϕ L ) Res z = p ( ϕ R ) dLog = z = p ( ψ p ϕ R ) Res = Res z = p ( ω ) p ∈P ω p ∈P ω ψ p = ∇ − 1 ω ϕ L ⇒ (d + ω ) ψ p = ϕ L only local solution to ψ p needed ⇒ power series ansatz max � p τ j + O � τ max +1 � ψ ( j ) ψ p = j = min ψ p obtained by pattern matching � Sanity check: �∇ ω ξ | ϕ R � = Res z = p ( ξϕ R ) p ∈P ω 12 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Univariate computation � � � 1 �� � � � � ∇ − 1 ϕ L ∇ − 1 � ϕ L | ϕ R � = ι ( ϕ L ) ∧ ϕ R = Res ω ϕ L ϕ R = − Res − ω ϕ R (2 π i ) z = p z = p p ∈P ω p ∈P ω � � Res z = p ( ϕ L ) Res z = p ( ϕ R ) dLog = z = p ( ψ p ϕ R ) Res = Res z = p ( ω ) p ∈P ω p ∈P ω ψ p = ∇ − 1 ω ϕ L ⇒ (d + ω ) ψ p = ϕ L only local solution to ψ p needed ⇒ power series ansatz max � p τ j + O � τ max +1 � ψ ( j ) ψ p = j = min ψ p obtained by pattern matching � Sanity check: �∇ ω ξ | ϕ R � = Res z = p ( ξϕ R ) p ∈P ξϕ R 12 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Univariate computation � � � 1 �� � � � � ∇ − 1 ϕ L ∇ − 1 � ϕ L | ϕ R � = ι ( ϕ L ) ∧ ϕ R = Res ω ϕ L ϕ R = − Res − ω ϕ R (2 π i ) z = p z = p p ∈P ω p ∈P ω � � Res z = p ( ϕ L ) Res z = p ( ϕ R ) dLog = z = p ( ψ p ϕ R ) Res = Res z = p ( ω ) p ∈P ω p ∈P ω ψ p = ∇ − 1 ω ϕ L ⇒ (d + ω ) ψ p = ϕ L only local solution to ψ p needed ⇒ power series ansatz max � p τ j + O � τ max +1 � ψ ( j ) ψ p = j = min ψ p obtained by pattern matching � Sanity check: �∇ ω ξ | ϕ R � = Res z = p ( ξϕ R ) = 0 p ∈P ξϕ R 12 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Reduction on the Maximal Cut Maximal Cut ⇒ univariate integral representation � 1 � d − 5 2 4 z 2 ( s − 2 z − 1)( s − 2 z + 3) u = p ω = d log u = 0 2 sols. ⇒ 2 MIs ν = 2 The MIs chosen as J 1 = I 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1;0 = � e 1 |C ] = � 1 |C ] & J 2 = I 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1; − 1 = � e 2 |C ] = � z |C ] Decompose I 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1; − 2 = � ϕ |C ] = � z 2 |C ] Compute � ϕ | e i � i = 1 , 2 & C ij = � e i | e j � i , j = 1 , 2 Plug in the Master Decomposition Formula 2 � c 1 = − ( d − 4)( s − 1)( s + 3) c 2 = (3 d − 11)( s + 1) � ϕ | e j � � C − 1 � c i = ji 4(2 d − 7) 2(2 d − 7) j =1 agreement with SYS 13 / 20
Introduction Intersection theory Univariate Multivariate Summary and Outlook Backup Examples O (30) examples checked on the maximal cut [Frellesvig, Gasparotto, Laporta, Mandal, Mastrolia, L.M. , Mizera (2019)] 14 / 20
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