Systematic approximation of multi-scale Feynman integrals arXiv:1804.06824 Daniel Hulme In collaboration with: Thomas Gehrmann and Sophia Borowka Amplitudes in the LHC era, Galileo Galilei Institute 29.10.2018 Daniel Hulme arXiv:1804.06824 29.10.2018 1
Motivation Feynman integrals are often the bottleneck for multi-scale multi-loop calculations, especially with massive propagators! Analytic evaluation of very complicated Feynman integrals not generally understood, although progress is being made! Numerical evaluation often the only option. Producing accurate numerical results over full phase space in an automated way is difficult - divergent nature of loop integrals. An algorithm to analytically approximate Feynman integrals - TayInt ! Daniel Hulme arXiv:1804.06824 29.10.2018 2
Philosophy Three aims: to produce an algebraic integral library with full phase space validity for any kind of integral The idea - the integrand has to be Taylor expanded in the Feynman parameters - otherwise you can’t integrate it - the kinematics are not to be touched. The algorithm brings an integral into a form optimised for an accurate Taylor expansion with validity in all kinematic regions. Divide and rule - take all the nastiness in the Feynman integral - distribute it so that it does not hinder a TAYLOR EXPANSION IN THE FEYNMAN PARAMETERS. Daniel Hulme arXiv:1804.06824 29.10.2018 3
Setup I A generic Feynman loop integral G in an arbitrary number of dimensions D at loop level L with N propagators, wherein the propagators P j with mass m j can be raised to arbitrary powers ν j , � L � k µ 1 α 1 · · · k µ R � α R � G µ 1 ...µ R d D κ α α 1 ...α R ( { p } , { m } ) = j =1 P ν j � N j ( { k } , { p } , m 2 j ) α =1 d D κ α = µ 4 − D d D k α , P j ( { k } , { p } , m 2 j ) = q 2 j − m 2 j + i δ , D i π 2 The q j are linear combinations of external momenta p i and loop momenta k α . Henceforth scalar integrals considered. Daniel Hulme arXiv:1804.06824 29.10.2018 4
Setup II Rewrite scalar integrals in terms of Feynman parameters t j , j = 1 ... N . Integrate the loop momenta to give, � ∞ N N ( − 1) N ν t l ) U N ν − ( L +1) D / 2 dt j t ν j − 1 � � G = δ (1 − , j � N F N ν − LD / 2 j =1 Γ( ν j ) 0 j =1 l =1 U and F are the first and second Symanzik polynomials. Daniel Hulme arXiv:1804.06824 29.10.2018 5
Summary of the method U1: reduce the Feynman Integral to a quasi-finite basis U2: perform an iterated sector decomposition below threshold above threshold BT1: t j → y j OT1: t j → θ j , generate K BT2: Taylor expand OT2: find Θ o (0) ,... o ( J − 1) and integrate OT3: perform one fold integrations find θ ∗ OT4: j and the optimum Θ o (0) ,... o ( J − 2) OT5: determine P j OT6: Taylor expand and integrate Daniel Hulme arXiv:1804.06824 29.10.2018 6
Diagrams p 2 + p 3 p 1 m m 1 m m 1 p p m p 4 p 1 + p 2 m 2 m 1 m m p 2 p 1 (a) S14 01220 (b) T41 (c) I10 m 1 p 2 + p 3 p 2 + p 3 m 1 p 1 p 4 m 1 m 1 m 1 m 2 m 2 m 1 p 4 p 4 m 1 m 1 m 2 m 1 m 1 m 1 m 1 p 1 p 1 p 2 p 3 (d) I21 (e) I246 (f) I39 The finite sunrise S14 01220 and the triangle T41 are used to illustrate TayInt . Integrals I10, I21, I246 and I39 enter the Higgs+jet calculation - are computed with TayInt . Daniel Hulme arXiv:1804.06824 29.10.2018 7
Method - U1 Universal step 1 (U1) - Feynman integral G expressed as a superposition of finite Feynman integrals G F multiplying poles in ǫ Quasi-finite basis: (von Manteuffel, Panzer, Schabinger, arXiv:1411.7392),(Panzer, arXiv:1401.4361) these integrals contain no divergences from integration of Feynman parameters - all divergent parts restricted to prefactors. The G F have a shifted number of dimensions or dotted propagators or both. Automated shell script used to direct all required Reduze (von Manteuffel, Panzer, Schabinger, arXiv:1411.7392),(von Manteuffel, Studerus, arXiv:1201.4330) jobs towards generating the quasi-finite basis. Daniel Hulme arXiv:1804.06824 29.10.2018 8
Illustration of Method - U1 The divergent sunrise S14 01110 in terms of the finite integrals S14 01220 , S14 01320 and the tadpole S6 30300 , 8 m 2 ( p 2 − 4 m 2 )( p 2 + 2 m 2 ) S14 01110 = ( − 3 + D )( − 8 + 3 D )( − 10 + 3 D ) · S14 01320 + ((4 − D ) p 4 + ( − 5 + D )8 m 4 + (18 − 5 D )4 p 2 m 2 ) · S14 01220 ( − 3 + D )( − 8 + 3 D )( − 10 + 3 D ) 16 m 4 (( − 4 + D ) p 2 + 2 ( − 24 + 7 D ) m 2 ) ( − 3 + D )( − 4 + D ) 2 ( − 8 + 3 D )( − 10 + 3 D ) · S5 30300 , − Poles in ǫ : ( − 4 + D ) − 1 terms. Daniel Hulme arXiv:1804.06824 29.10.2018 9
Method - U2 Universal step 2 (U2) - decompose integrals G F to iterated sectors using version 3 of SecDec (Borowka, Heinrich et al., arXiv:1703.09692) Iterated sectors: � 1 U N ν − ( L +1) D / 2 N � � � t j d t j t A l − B lj ǫ � G F l l = , j F N ν − LD / 2 0 j =2 l where l = 1 , . . . , r , and r is the number of iterated sector integrals. A k and B k are numbers independent of the regulator ǫ . Remap so that j runs from 0 to J − 1. Iterated sector integrals G F l - building blocks of TayInt . Daniel Hulme arXiv:1804.06824 29.10.2018 10
Illustration of the method - U2 O ( ǫ 0 ) coefficient of an I10 iterated sector, � 1 2 1 � I10 1 = dt j (1 + t 0 + t 1 + t 2 + t 1 t 2 ) 0 j =0 · [ t 0 ( − u − m 2 t 1 ) + m 2 1 (1 + t 2 0 + t 2 + t 2 1 (1 + t 2 )+ t 1 (2 + 2 t 2 ) + t 0 (2 + t 2 + t 1 (2 + t 2 )))] − 1 , Three Feynman parameters after iterated decomposition, three kinematic scales, m 1 , m 2 and u . Daniel Hulme arXiv:1804.06824 29.10.2018 11
Method - BT1 Below Threshold step 1 (BT1) - maximise distance to nearest point of non-analyticity The iterated sectors G F l - still have non-analytic points outside the integration region. To move these as far away as possible, import G F l into Mathematica (Wolfram), apply conformal mappings, t j = ay j + b cy j + d . Thus far: optimum mapping found. Daniel Hulme arXiv:1804.06824 29.10.2018 12
Illustration of the method - BT1 3.2 · 10 -5 2.4 · 10 -5 0.002 1.6 · 10 -5 I10 11 (t 0 ,t 1 =0,t 2 =0) 8.0 · 10 -6 0 0.5 1 0 -0.002 -4 -3 -2 -1 0 1 2 3 4 t 0 0.0003 3.2 · 10 -5 2.4 · 10 -5 0.0002 1.6 · 10 -5 8.0 · 10 -6 I10 11 (y 0 ) 0.0001 -1 -0.75 -0.5 0 -0.0001 -4 -3 -2 -1 0 1 2 3 4 y 0 Plot of a one dimensional integrand, I10 1 ( t 0 , t 1 = 0 , t 2 = 0) = 1 / ((1 + t 0 )( − m 2 t 0 + m 2 1 (1 + 2 t 0 + t 2 0 ))) , before and after a conformal mapping y 0 = − 1 − t 0 . t 0 Daniel Hulme arXiv:1804.06824 29.10.2018 13
Method - BT2 and BT3 Below Threshold steps 2 and 3 (BT2-3) - Taylor expand and integrate in the Feynman parameters BT2 - Taylor expand the integrand in the re-mapped Feynman parameters y j . BT3 - integrate over the y j . Done in FORM (Kuipers, Ueda, Vermaseren, arXiv:1310.7007). Daniel Hulme arXiv:1804.06824 29.10.2018 14
Illustration of the method - BT2 and BT3 1.012 w/o conformal map with conformal map 1.01 SecDec/Taylor 1.008 1.006 1.004 1.002 1 0 30000 60000 90000 u (Gev 2 ) Ratio of SecDec and TayInt calculation of the ǫ 0 coefficient of I10. Daniel Hulme arXiv:1804.06824 29.10.2018 15
Going over threshold (OT1-OT6) Above lowest threshold of an integral - discontinuities on the real axis. A Taylor expansion won’t converge! TayInt returns to the result of U2, the iterated sector integrands G F l ( t j ). The Feynman + i δ prescription is implemented in Mathematica. TayInt determines the contour configuration in the complex plane to avoid the discontinuities. Over threshold part of TayInt is fully automated in Mathematica. Daniel Hulme arXiv:1804.06824 29.10.2018 16
Method - OT1 Over Threshold step 1 (OT1) - generate all possible contour configurations for each iterated sector integrand The first Over Threshold step (OT1) - transform the Feynman parameters of the J − 1 iterated sectors, t j → 1 2 + 1 2 exp ( i θ j ). Generate representative sample of the kinematic region. A nested list of values K = {{ s 1 , . . . , s β } 1 , . . . , { s 1 , . . . , s β } γ } = {K 1 , . . . , K γ } for a β scale integral, sample size of γ points. Daniel Hulme arXiv:1804.06824 29.10.2018 17
Method - OT2 Over Threshold step 2 (OT2) - select contour configuration optimised for a Taylor expansion OT2 - calculate the mean absolute value of the θ j derivatives (MAD) of the G F l ( θ j ). Kinematic scales first set to the mean of the sample, MAD calculated at the edges. Kinematic scales then set to each sample value, MAD calculated over bulk. MAD calculated for all possible contour configurations, Θ o (1) ,... o ( J − 1) - o ( j ) = ± is the orientation of the j th contour in the θ j . Contour configuration which minimises the MAD selected. Daniel Hulme arXiv:1804.06824 29.10.2018 18
Illustration of the method - OT1 and OT2 Slice of I10 2 without a complex mapping... Daniel Hulme arXiv:1804.06824 29.10.2018 19
Illustration of the method - OT1 and OT2 with a complex mapping, contour orientation { o (1) , ... o ( J − 1) } determined via TayInt . Daniel Hulme arXiv:1804.06824 29.10.2018 20
Recommend
More recommend
