Feynman integral relations from parametric annihilators

Transcription

1 Feynman ntegral relatons from parametrc annhlators Erk Panzer All Souls College Oxford) 3rd May 218 Loops & Legs, St. Goar jont work wth Thomas Btoun, Chrstan Bogner, René Pascal Klausen [arxv: ]

2 An ntegral famly s defned by a set of denomnators D 1,..., D N that are quadratc or lnear) forms n loop momenta l 1,..., l L : Example Iν 1, ν 2 ; d) = L Iν 1,..., ν N ; d) = k=1 d d l 1 π d/2 l 2 ) ν 1 l + p) 2 ) ν 2 d d ) l k N π d/2 Da νa a=1 p k 2 = l+p k 1 = l p A famly s also descrbed by a matrx Λ, vectors Q and a scalar J such that N L L x a D a = Λ j l l j ) + 2Q l ) + J a=1,j=1 =1 Assocated polynomals: U := det Λ, F := U Q Λ 1 Q + J )

3 In terms of ω := ν ν N L d 2 and G := U + F Lee-Pomeransky), ) Iν) = Γ d 2 N x ν ) 1 ) dx Γ d 2 ω G d/2 Γν ) Example Γ d 2 Iν 1, ν 2 )= ) Γd ν 1 ν 2 ) =1 x ν dx 1 x ν dx 2 Γν 1 ) Γν 2 ) x 1 + x }{{ 2 p 2 x } 1 x }{{ 2 } U F ) d 2

4 In terms of ω := ν ν N L d 2 and G := U + F Lee-Pomeransky), ) Iν) = Γ d 2 N x ν ) 1 ) dx Γ d 2 ω G d/2 Γν ) Example Γ d 2 Iν 1, ν 2 )= ) Γd ν 1 ν 2 ) =1 x ν dx 1 x ν dx 2 Γν 1 ) Γν 2 ) The Melln transform of a functon f : R N + C s N M{f } ν) := =1 ) dx f x 1,..., x N ), Γν ) x ν 1 whenever ths ntegral exsts. Specal case: Iν) = x 1 + x }{{ 2 p 2 x } 1 x }{{ 2 } U F Γd/2) Γd/2 ω)ĩν) for Ĩν) = M { G d/2} ν). ) d 2

5 Speer Such ntegrals converge n a non-empty, open doman wrt d, ν). They have a unque, meromorphc extenson to C 1+N. The poles are smple and located on ratonal hyperplanes. To prove relatons between regularzed Feynman ntegrals, we may assume convergent values of the parameters. Example Iν 1, ν 2 ) = p 2 ) d/2 ν 1 ν 2 Γd/2 ν 1 )Γd/2 ν 2 )Γν 1 + ν 2 d/2) Γν 1 )Γν 2 )Γd ν 1 ν 2 ) Poles: {d/2 ν 1 = k} {d/2 ν 2 = k} {ν 1 + ν 2 d/2 = k}; k Z If d s the only regulator ν Z N ), poles coalesce and cease to be smple.)

6 Propertes of the Melln transform 1 M{αf + βg} ν) = αm{f } ν) + βm{g} ν) α, β C) 2 M{x f } ν) = ν M{f } ν + e ) x ν 1 dx x f ) = Γν ) 3 M{ f } ν) = M{f } ν e ) x ν 1 dx f ) = Γν ) ν x ν dx ν Γν ) f = [ ] x ν 1 Γν ) f + x = ν x ν +1) 1 Γν + 1) dx f x ν 2 dx Γν 1) f

7 Propertes of the Melln transform 1 M{αf + βg} ν) = αm{f } ν) + βm{g} ν) α, β C) 2 M{x f } ν) = ν M{f } ν + e ) =: î+ M{f }) v) x ν 1 dx x f ) = Γν ) 3 M{ f } ν) = M{f } ν e ) =: Shft operators: x ν 1 dx f ) = Γν ) ν x ν dx ν Γν ) f = ) M{f } ν x ν +1) 1 Γν + 1) v) [ ] x ν 1 Γν ) f + x = dx f x ν 2 dx Γν 1) f F )ν) := F ν e ) n F )ν) = ν F ν) for î+ F )ν) := ν F ν + e ) n := î+

8 Propertes of the Melln transform 1 M{αf + βg} ν) = αm{f } ν) + βm{g} ν) α, β C) 2 M{x f } ν) = ν M{f } ν + e ) =: î+ M{f }) v) x ν 1 dx x f ) = Γν ) 3 M{ f } ν) = M{f } ν e ) =: Shft operators: x ν 1 dx f ) = Γν ) ν x ν dx ν Γν ) f = ) M{f } ν x ν +1) 1 Γν + 1) v) [ ] x ν 1 Γν ) f + x = dx f x ν 2 dx Γν 1) f F )ν) := F ν e ) n F )ν) = ν F ν) for î+ F )ν) := ν F ν + e ) n := î+

9 Gven any dfferental operator P A N [d] n the Weyl algebra A N [d] := C[d] x 1,..., x N, 1,..., N [, x j ] = δ,j such that P G d/2 = annhlator), the substtutons x î +,, x n defne a shft operator M{P} S N [d] n the shft algebra S N [d] := C[d] ˆ1 +,..., ˆN +, 1,..., N [ j, î + ] = δ,j { such that M{P} M G d/2} { = M P G d/2} = relaton). Example G = x 1 + x 2 p 2 x 1 x 2 ) [ ] 1 p 2 ) d/2 x )x 1 + d/2 x 1 1 x 2 2 ) G d/2 = 2 p 2 ) d/2 + n 1 + 1)ˆ1 +Ĩ = d/2 + n 1 + n 2 )Ĩ 3 p 2 )ν 1Ĩν 1 + 1, ν 2 ) = d/2 + ν 1 + ν 2 d/2 + ν Ĩν 1, ν 2 )

10 M{ } { } P A N [d]: P G d/2 = annhlator) { P S N [d]: P Ĩ = } shft relaton) M 1 { } The nverse Melln transform of f ν) := M{f } ν) s f x) = M 1 {f N } x) = =1 σ +R ) Γν ) dν 2π) x ν f ν). Therefore, every shft relaton comes from an annhlator.

11 Open problems A fnte lst of generators for all annhlators.e. IBP relatons) can sometmes be computed wth Sngular. Queston 1 Is the annhlator of G d/2 lnearly generated? For a full set of ISPs, the acton of q j on the momentum space q ntegrand leads to IBP relatons that map to lnear annhlators Õ j. Queston 2 Do the momentum space IBP s generate all annhlators relatons)? No counterexamples found, but only few cases tested.

12 We defne the number of master ntegrals of {G Ĩν) = M d/2} ν) as C G) := dm Cd,ν) Cd, ν) Ĩν + n) n Z N

13 We defne the number of master ntegrals of Ĩν) = M {G d/2} ν) as C G) := dm Cd,ν) Cd, ν) Ĩν + n) n Z N C G) dm Cd) n Z N Cd) Ĩn) ) no symmetres not modulo subtopologes exactly computable! Usng the Melln transform, θ := x = M 1 {n }, ) C G) = dm Cd,θ) Cd, θ) C[d,θ] A N [d]g d/2 }{{} M Here, A N d)g d/2 s a holonomc D-module, and M s a holonomc system of fnte dfference equatons [Loeser & Sabbah 91].

14 Theorem 1) N C G) = χ ) ) C N \ {x 1 x N G = } = χ C ) N \ {G = } mples fnteness [Smrnov & Petukhov] The Euler characterstc χx) = 1) dm H X) s a fundamental nvarant and can be computed wth many dfferent tools, for example: χx) = χx \ Z) + χz) χx Y ) = χx) χy ) χe) = χb) χf ) for fbratons F E B D-modules and Groebner bases e.g. Sngular) [Oaku & Takayama] algorthms by M. Helmer CharacterstcClasses n Macaulay2) Kouchnrenko/Khovansk s theorem: For non-degenerate G, C G) = N! Vol NP G)

15 For some nfnte famles one can prove explct formulas: C C = C = 2 L+1 1 = LL + 1) 2 [Kalmykov & Knehl] Plenty of further computatons agreed wth predctons by Azurte, e.g. Graph G C G) massless C G) massve

16 Massve one-loop sunrse U = x 1 + x 2 F = x 1 + x 2 ) 2 + x 1 x 2 In Macaulay2, the Euler characterstc C G) = 3 can be computed wth load "CharacterstcClasses.m2" R=QQ[x,x1,x2] I=dealx*x1*x2*x1+x2)*x+x1+x2)ˆ2+x1*x2)) EulerI) The ndvdual cohomology groups can also be obtaned wth load "Dmodules.m2" R=QQ[x1,x2] f=x1*x2*x1+x2+x1+x2)ˆ2+x1*x2) derham f H X) = Q, H 1 X) = Q 3, H 2 X) = Q 5 χx) = = 3 The same can be done n Sngular.

17 Thanks Thank you for your attenton! The Melln transform translates IBP relatons to annhlators [Tkachov, Bakov, Lee, Pomeransky]. Algorthms for computatons wth D-modules are avalable. Applcaton: The number of master ntegrals, for free ν s, s C G) = 1) N χc ) N \ {G = }) < Goal: Extend IBP reducton from ν Z N to free ν.

18 Thanks Thank you for your attenton! The Melln transform translates IBP relatons to annhlators [Tkachov, Bakov, Lee, Pomeransky]. Algorthms for computatons wth D-modules are avalable. Applcaton: The number of master ntegrals, for free ν s, s C G) = 1) N χc ) N \ {G = }) < Goal: Extend IBP reducton from ν Z N to free ν.

19 Parametrc representatons ω := ν ν N L d 2 N x ν ) 1 dx e F/U Iν 1,..., ν N ) = =1 Γ ν ) U d/2, N x ν ) 1 dx δ 1 ) N j=1 x j Iν 1,..., ν N ) = Γω) =1 Γ ν ) U d/2 ω F ω ) Iν 1,..., ν N ) = Γ d 2 N x ν ) 1 ) dx Γ d 2 ω G d/2. Γν ) =1

20 ) 2 = ε ε

21 ) 2 = ε ε 3ε

22 Mscountng n Azurte p 3 3 p 4 G = p 2 1 p 1 G/ {1, 2} = p 4 p G = 3 4 5