Some variations of the reduction of one-loop Feynman tensor integrals

Transcription

1 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 1 / 6 Some variations of the reduction of one-loop Feynman tensor integrals Tord Riemann DESY, Zeuthen in collaboration with Jochem Fleischer [and T. Diakonidis, B. Tausk, J. Gluza, K.Kajda] 13th International Workshop on Advanced Computing and Analysis Techniques in Physics Research, February 22-27, 2010 Jaipur, India version presented

2 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 2 / 6 Outline 1 Introduction 2 Recursions 3 Simplifying recursions Numbers: D 111 Summary 6 Backup: 6- and -pt numbers

3 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers Introduction n-point tensor integrals of rank R: (n,r)-integrals I µ 1 µ R d d k R r=1 n = k µr iπ d/2, (1) n j=1 cν j j d = 2ɛ and denominators c j have indices ν j and chords q j c j = (k q j ) 2 m 2 j + iε (2) k q1 k q2 k k q3 k qn 1 tensor integrals due to: fermion propagators three-gauge boson couplings e.g. unitary gauge propagators 3 / 6

4 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers / 6 Reduction of tensor integrals express them by a (very) small set of scalar integrals Presently needed for massive processes: n 6 and rank R n For box diagrams and simpler ones: Use of the conventional Passarino-Veltman reduction [Passarino:1978 [1]] Examples: LO (Lowest order) of e.g. Z e + µ is one-loop [Riemann:1981,Mann:1983 [2][3]] NLO: one-loop corrections to e.g. H τ + τ, WW, ZZ [Fleischer:1980 []] NNLO: e.g. radiative loop corrections e + e e + e γ (here with -point functions)

5 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers / 6 Status of opensource packages - hopefully complete package FF [vanoldenborgh:1990 []], package LoopTools/FF [Hahn:1998,2006 [6]] covers also -point functions, rank R 1/ɛ 2 not covered, and we observed sometimes problems in certain configurations with light-like external particles package Golem9 [Binoth:2008 [7]] for n 6, but only massless propagators Mathematica package hexagon.m [Diakonidis:2008 [8, 9]] for n 6, rankr package for all n scalar integrals: QCDloop [Ellis:2007 [10]] see also: review A.Denner, DESY TH workshop 2009 So, if you want to evaluate something less trivial, you have to create your own tensor reduction library.

6 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 6 / 6 This talk: derive efficient reduction formulae in the algebraic Fleischer-Davydychev-Tarasov approach The original Passarino-Veltman reduction allows to express tensor integrals by a small set of scalar -,3-,2-,1-point functions integrals in d dimensions. Extensions: Need of reduction of n-point functions with n > Improvements: Avoid the break-down in certain kinematical configurations A crucial step was to avoid for n = pentagons the towers of inverse Gram determinants. They may get small or vanishing and may appear by reducing algebraically the tensors to scalars.

7 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 7 / 6 Crucial contributions [of course, list is incomplete... ] [Campbell:1996 [11]] [Denner:2002,200 [12, 13]] [Binoth:1999,200 [1, 1]] [Bern:1993 [16]] [Ossola:2006 [17] ] In the following, I will describe recent developments in the Fleischer-Davydychev-Tarasov approach. [Davydychev:1991,Tarasov:1996,Fleischer:1999,Diakonidis:2008,2009 [18, 19, 20, 9, 21]] get tensor reduction kill pentagon-gram det s treat sub-gram det s

8 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 8 / 6 Recursions for hexagons I Express any hexagon by pentagons [Fleischer:1999,Binoth:200,Denner:200,Diakonidis:2008 [20, 1, 13, 9] ] I µ 1...µ R 1 ρ 6 = 6 s=1 I µ 1...µ R 1,s Q ρ s. (3) auxiliary vectors Q ρ s = 6X q ρ i i=1 `0s 0i 6 `0 0 6, s = () Scalar case [N=: Melrose:196 [22]] `0 6X s N I N = `0 s=1 0 N IN 1 s, N =, 6 ()

9 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 9 / 6 Notations: I {µ 1, },s n 1 etc. I {µ 1, },s n 1,ab is obtained from by shrinking line s I {µ 1, } n raising the powers of inverse propagators a, b.

10 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 10 / 6 Notations: modified Cayley determinant [Melrose:196] Modified Cayley determinant () N of a diagram with N internal lines and chords q j : () N Y 11 Y Y 1N 1 Y 12 Y Y 2N Y 1N Y 2N... Y NN, (6) with matrix elements Y ij = (q i q j ) 2 + mi 2 + mj 2, (i, j = 1... N) (7) Gram determinant G n: G n = 2q i q j, i, j = 1,... n For a choice q n = 0, both determinants are related: () N = G N 1 The determinant () N does not depend on the masses.

11 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 11 / 6 Notations: signed minors [Melrose:196] We also need signed minors of () N, constructed by deleting m rows and m columns from () N, and multiplying with a sign factor: P ( 1) l (j l +k l ) sgn {j} sgn {k} «j1 j 2 j m k 1 k 2 k m N rows j 1 j m deleted columns k 1 k m deleted (8) where sgn {j} and sgn {k} are the signs of permutations that sort the deleted rows j 1 j m and columns k 1 k m into ascending order.

12 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 12 / 6 Dimensional shifts and recurrence relations for pentagons I Naive, direct approach. Following [Davydychev:1991 [18]] replace tensors by scalar integrals in higher dimensions: Example R = 3: I µνλ = Z = d 2ɛ k π d/2 X i,j,k=1 Y r=1 and n ijk = (1 + δ ij )(1 + δ ik + δ jk ). c 1 r k µ k ν k λ (9) q µ q ν i j qk λ n ijk I [d+]3 + 1 n 1 X (g µν q,ijk i λ + g µλ qi ν + g νλ q µ )I [d+]2, 2 i n,i i=1

13 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 13 / 6 Following [Tarasov:1996,Fleischer:1999 [19, 20]] apply recurrence relations, some of them relating scalar integrals of different dimensions, in order to get rid of the dimensionalities D = d 2ɛ + 2l: (d nx i=1 "! ν j j + I [d+] = 1 j + () 0 "! ν i + 1)I [d+] n = 1 0! 0 0 ν j j + I = () n X k=1 0! 0j 0k n! # X j k I (10) k k=1! # nx 0 k I n, n = here(11) k k=1 n " # X d ν i (k i + + 1) I (12) where the operators i ±, j ±, k ± act by shifting the indices ν i, ν j, ν k by ±1. i=1

14 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 1 / 6 At the end one may arrange a representation of e.g. I µνλ in terms of a collection of scalar 1-point to -point functions in generic dimension d: A 0, B 0, C 0, D 0. A problem for applications are the powers of the inverse Gram determinants 1/ () in recursions with dimensional shifts.

15 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 1 / 6 Alternative: Recursions for pentagons I Express any (, R) pentagon by a (, R 1) pentagon plus (, R 1) boxes [Fleischer et al., Diakonidis:2010 [23]] I µ 1...µ R 1 µ = I µ 1...µ R 1 Q µ 0 s=1 I µ 1...µ R 1,s Q µ s (13) auxiliary vectors with inverse Gram determinants ( s ) Q s µ = q µ i i, s = 0,..., (1) () i=1 For e.g. R = 3, again [1/() ] 3 will occur.

16 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers Interlude: Further recursion steps The recursions may be set forth, see [Diakonidis:2009 [21]] ; they get additional terms, e.g.: I µνλρ = I µνλ Q ρ 0 X t=1 with the additional tensor and vector components: X T µν = I µ,[d+] Q0 ν I µ,[d+],t 3 Qt ν t=1 G µν I [d+]2 (16) `i G µλ = 1 X 2 gµλ q µ q λ j i j () i,j=1 I µνλ,t 3 Q ρ t G µρ T νλ G νρ T µλ G λρ T µν, (1) I µ I µν I µνλρ I µνλρσ, s I µνλρ I µνλ, s, st I µνλ I µνλ 3, s I µν, st I µν, stu I µν 3 2, s I µ, st I µ, stu I µ, stuv 3 2 I µ 1 E0 D0 C0 B0 A0 Fleischer s triangle 16 / 6

17 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 17 / 6 Algebraic simplifications, 1st step With the identity ««0 s 0 i = «««0s 0 s () + 0i i 0 (17) we eliminate the inverse Gram determinant from all terms with exclusion of Q µ 0 : " I µ 1...µ R 1 µ = I µ 1...µ R 1 `s X 0 `0 s=1 0 # I µ 1...µ R 1,s Q µ 0 X s=1 I µ 1...µ R 1,s Q µ s (18) The auxiliary vectors Q µ s were introduced already for n = 6: X Q µ 0 = q µ i i=1 `0 X i while Q µ 0 () = i=1 q µ i `0 i Q µ s = () X q µ i i=1 `0s 0i `0 0 (19)

18 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 18 / 6 Algebraic simplifications, 2nd step I Have to show for the product T µ 1...µ R 1 Q µ 0 that the Gram determinant cancels. This came out to be a complicated task. [ (0 ) T µ 1...µ R 1 = I µ 1...µ R 1 0 s=1 ( ) ] s I µ 1...µ R 1,s 0 (20) Example: For R = 3 pentagons need rank 2 tensor: [ (0 ) T µν = I µν 0 s=1 ( ) ] s I µν 0 (21)

19 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 19 / 6 Algebraic simplifications, 2nd step I Example R = 3: the building blocks are here I µν and I µν : I µν = work!!! q µ i qj ν i,j=1 q µ i qj ν i,j=1 ] [ [(1 + δ ij )I [d+]2,ij + g µν 1 ] 2 I[d+] 1 ( 0 0) [ +g µν 1 1 ( 2 0 0) s=1 s=1 [( ) 0i I [d+],s sj + ] ( ) s I [d+],s 0 See: The I µν had already been made free of 1/(). ( ) ] 0s I [d+],s 0j,i (22)

20 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 20 / 6 Davydychev s higher dimensional integrals The second term with I µν is a typical example of [Davydychev:1991 [18]] : tensor scalars in d + 2l I µν = q µ i qj ν i,j=1 ] [ [(1 + δ ij )I [d+]2,ij + g µν 1 ] 2 I[d+] Further, at this point, we have to reduce the scalar integrals etc. to generic dimension d with Tarasov s recurrence relations, see next slide. The I µν is naturally free of 1/(). I [d+]2,ij (23)

21 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 21 / 6 Algebraic simplifications, 2nd step Work out the red part in ( [ (0 0 0) Iµνλ = 0) Iµν ( s s=1 0 work!!! ) Iµν,s Use of identities for the determinants «««««0 s 0s 0 s = () + 0 i 0i i 0 ] Q0 λ s=1 Iµν,s Q 0,λ s (2) «0 ««i i s j 0i s j X j = +, g µν = 2 q µ q ν i () sj 0 () () i j (2) i,j=1 ««««««s 0s s 0s s 0s = 0 is i 0s s 0i (26) ««««««s ts s ts s ts = 0 js j 0s s 0j (27)

22 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 22 / 6 Algebraic simplifications, 2nd step Use of identities for the determinants work!!! ««««««s is s 0s s 0i = + 0 js i js s sj (28) «««««s 0st 0s st ts 2 = s 0st 0s st 0s (29) " ««««# s «" ««««# st «ts ust ts ust ts us ts us = 0s jst js 0st s 0s js js 0s st (30) X «ts = 0 (31) is t=1

23 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers Express pentagons I µ, Iµν, Iµνλ etc. by d-shifted scalar boxes I Intermediate result with I [d+],s, I [d+]2,s,ij etc. I µ = [ 1 ( 0 0) i=1 s=1 ( ) ] 0i I s q µ 0s i (32) I µ ν = E ij = i,j=1 1 ( 0 0) q µ i q ν j E ij + g µν E 00 (33) s=1 E 00 = 1 1 ( 2 0 0) [( ) 0i I [d+],s sj + ) s=1 ( s 0 ( ) ] 0s I [d+],s 0j,i (3) I [d+],s (3) 23 / 6

24 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 2 / 6 Express pentagons I µ, Iµν, Iµνλ etc. by d-shifted scalar boxes II I µ ν λ = i,j,k=1 E ijk = 1 ( 0 0) E 00j = 1 ( 0 0) q µ i q ν j q λ k E ijk + s=1 s=1 [ 1 2 k=1 g [µν q λ] k E 00k (36) {[( ) 0j I [d+]2,s sk,i + (i j) ( ) 0s I [d+],s 0j d 1 3 ] + ( ) ] s I [d+]2,s j These presentations are evidently free of inverse Gram determinants. ( ) } 0s ν ij I [d+]2,s 0k,ij (37) (38)

25 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 2 / 6 Isolation of inverse sub-gram det s () I We have now two kinds of objects in higher dim s to be evaluated: I s, I[d+],s, I [d+]2,s boxes (39) I [d+],s,i, I [d+]2,s,i I [d+]2,s,ij boxes with higher indices (0) Application of dimension-shifting recurrence relations produces powers of 1/(). They are the unwanted sub-gram-determinants (). Simplest: In I µ (32) appears Is.

26 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 26 / 6 Isolation of inverse sub-gram det s () II One of Tarasov s recurrence relations (11) applies (n = ): () n (d 3)I [d+] = So that we express: I D = k=1 [ (0 ) 0 k=1 ( ) ] 0 k I (1) k ( k 0) ( 0 I 0) D,k 3, for() n = 0 (2) If the dimension D = d + 2l > d, one has to reduce the dimension of I D,k 3 by Tarasov s dim-recurrences, but now without 1/(), because now 1/() 3 appear.

27 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 27 / 6 Isolation of inverse sub-gram det s () III For the other boxes, with higher indices, the game is more involved. Try to have inverse Gram determinants only in front of boxes I, and hold the I 3, I 2, I 1 free of them Introduce: ) lim I d,s ( s = s) 0 t=1 ( ts 0s ( 0s ) 0s A typical example appearing in (3) is I [d+],s,i. With the aid of relations like 0s ts 0s ts is 0s 0s is I d,st 3 Z d,s (3) s = s 0st, () 0si

28 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 28 / 6 Isolation of inverse sub-gram det s () IV one may cancel out the factor `s from the combination s `0s `ts X is `s Z s is `s I3 st = = 1 X 0st `0s I3 st () 0si s t=1 s 0s t=1 This allows to rewrite the recurrence relation ( 0s ) ( ts ) I [d+],s is,i = ( s I s) s + is ( s s) t=1 into one with 1/ ( ) s s = 1/() ONLY in front of boxes: ( ) [ I [d+],s 0s I s,i = Z s ] ( is s + s) 1 ) ( 0s 0s t=1 I st 3 ( ) 0st I3 st (6) 0si

29 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 29 / 6 Isolation of inverse sub-gram det s () V This is a nice expression, because it contains an explicit difference quotient for the case that ( ) s s becomes zero or if it becomes small but finite. The numerical strategy will be: The () 0: Just evaluate what you like. The () = 0: Replace [ I s Z s ] ( s s) d 3 ( 0 0) I [d+],s and the I [d+],s may be written, in that case () = 0, by t c ki [d+],st 3. Further reducing to scalars in d dimensions, no 1/() will re-appear.

30 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 30 / 6 Isolation of inverse sub-gram det s () VI This is, what one may do if not explicitely work on expansions of the scalar integrals for small () n.

31 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 31 / 6 Example: -point tensor of rank 3 I Following Davydychev, [18], one gets I µνλ = Z d k µ k ν k λ Q n r=1 cr = nx i,j,k=1 q µ i q ν j q λ k ν ijk I [d+]3 n,ijk nx i=1 g [µν q λ] I [d+]2 (7) i n,i We identify the tensor coefficient D 111 : D 111 ν ijk I [d+]3,ijk for ijk = 222

32 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 32 / 6 An example from A. Denner: D 111 Next figures copied from: A.Denner, plenary talk DESY Theory Workshop 2009, p.69 (backup transparency) box integral 0 t e d 0 tēµ M Z s µ νu e + s νu appears, e.g., in subgraph of diagram Gramdet.: (N) 0 if t e d t crit s µ νu(s µ νu s νu + tēµ ) s µ νu s νu e e Z Z µ d W µ ν µ u d

33 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 33 / 6 The figure demonstrates the effects of careful treatment of vanishing Gram determinant (). Gramdet.: (N) 0 if t e d t crit s µ νu(s µ νu s νu + tēµ ) s µ νu s νu numerical comparison: maximal tensor rank = 6 (similar to ee f application) Absolute predictions D 1 D 111 D 0 D 11 (etc.) Passarino Veltman reduction improved by Gram expansion Z Relative deviations from best Passarino--Veltman region x t e d t crit 1 s µ νu = GeV 2 s νu = GeV 2 tēµ = 10 GeV 2 t crit = 6 10 GeV 2 PV reduction breaks down, but Gram exp. stable for (N) 0!

34 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 3 / 6 D 111 numerics with our reductions Now rewrite: ν ijk I [d+]3,ijk ν ij ν ijk I [d+]3,s,reg,ijk ν ij ν ijk I [d+]3,s,ijk 0 t i j k = I [d+] 2 X k + I [d+] 2,t k + I [d+] 2 k + I [d+] 2 (),ij () 3,ij t=1,t i,j (),j (),i 0s» ks = ν s ij I s [d+]2,s Z [d+]2,s,ij,ij ««0si I [d+]2,s 0sj + I [d+]2,s X «0st + ν 0s 0sk,j 0sk,i ij I [d+]2,st (9) 0sk 3,ij 0s t=1,t i,j «k i j = d (d 3)(d + 1)I [d+]3 1 0i + (d 3) I 0 0j 0 0 [d+]2 + 2ν ij I [d+]3,ij 0 3 d 16 j X «0t I 0 0 0i 0 0 [d+]2,t + d 3 X «0t I 3 0 0j t=1 0 [d+]2,t 3,i t= ««0i I 0 0k 0 [d+]2 0j X «+ I [d+]2 0t + ν,j 0k,i ij I [d+]2,t (0) 0k 3,ij t=1,t i,j (8)

35 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 3 / 6 Numerics: LoopTools versus our approach Using LoopTools call and our math numerics (preliminary): x D111-7 : I D0i[dd111] I Zd30,Zd20,Iid20-6 : I D0i[dd111] I Zd30,Zd20,Iid20 - : I D0i[dd111] I Zd30,Zd20,Iid20 x< -: LoopTools dies out - : I D0i[dd111] I flei x < -3: loss of accuracy -3 : I D0i[dd111] I flei -2 : I D0i[dd111] I flei -1 : I D0i[dd111] I flei

36 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 36 / 6 Summary Recursive treatment of pnetagon tensors integrals of rank R in terms of pentagons and boxes of rank R 1 Systematic derivation of expressions which are explicitely free of inverse Gram determinants () until pentagons of rank R = Properly isolation of inverse Gram determinants of subdiagrams of the type ( ) s s, which cannot be completely n avoided. Some [preliminary] numerics, so far Mathematica, and C++ under way

37 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers References I G. Passarino and M. J. G. Veltman, One loop corrections for e + e annihilation into µ + µ in the Weinberg model, Nucl. Phys. B160 (1979) 11. T. Riemann, G. Mann, and D. Ebert. Nonconservation of lepton number in Z decay, in: F. Kaschluhn (ed.), Proc. XVth Int. Symp. Ahrenshoop on Special Topics In Gauge Field Theories, Nov -12, 1981, Ahrenshoop, GDR, AdW, Zeuthen (1981) PHE 81-07, pp G. Mann and T. Riemann, Effective flavor changing weak neutral current in the standard theory and Z boson decay, Annalen Phys. 0 (198) 33. J. Fleischer and F. Jegerlehner, Radiative Corrections to Higgs Decays in the Extended Weinberg-Salam Model, Phys. Rev. D23 (1981) G. J. van Oldenborgh, FF: A Package to evaluate one loop Feynman diagrams, Comput. Phys. Commun. 66 (1991) 1 1. T. Hahn and M. Perez-Victoria, Automatized one-loop calculations in four and d dimensions, Comput. Phys. Commun. 118 (1999) 13, [hep-ph/98076]. T. Binoth, J. P. Guillet, G. Heinrich, E. Pilon, and T. Reiter, Golem9: a numerical program to calculate one-loop tensor integrals with up to six external legs, Comput. Phys. Commun. 180 (2009) , [ ]. T. Diakonidis, J. Fleischer, J. Gluza, K. Kajda, T. Riemann, and J. Tausk, On the tensor reduction of one-loop pentagons and hexagons, Nucl. Phys. Proc. Suppl. 183 (2008) , [ ]. 37 / 6

38 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 38 / 6 References II T. Diakonidis et al., A complete reduction of one-loop tensor - and 6-point integrals, Phys. Rev. D80 (2009) , [ ]. R. K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP 02 (2008) 002, [ ]. J. Campbell, E. W. N. Glover, and D. Miller, One-loop tensor integrals in dimensional regularisation, Nucl. Phys. B98 (1997) 397 2, [hep-ph/961213]. A. Denner and S. Dittmaier, Reduction of one-loop tensor -point integrals, Nucl. Phys. B68 (2003) , [hep-ph/021229]. A. Denner and S. Dittmaier, Reduction schemes for one-loop tensor integrals, Nucl. Phys. B73 (2006) 62 11, [hep-ph/00911]. T. Binoth, J. P. Guillet, and G. Heinrich, Reduction formalism for dimensionally regulated one-loop N-point integrals, Nucl. Phys. B72 (2000) , [hep-ph/991132]. T. Binoth, J. Guillet, G. Heinrich, E. Pilon, and C. Schubert, An algebraic / numerical formalism for one-loop multi-leg amplitudes, JHEP 10 (200) 01, [hep-ph/00267]. Z. Bern, L. J. Dixon, and D. A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B12 (199) , [hep-ph/930620]. G. Ossola, C. Papadopoulos, and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B763 (2007) , [hep-ph/ ].

39 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers References III A. I. Davydychev, A Simple formula for reducing Feynman diagrams to scalar integrals, Phys. Lett. B263 (1991) O. V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D (1996) , [hep-th/ ]. J. Fleischer, F. Jegerlehner, and O. Tarasov, Algebraic reduction of one-loop Feynman graph amplitudes, Nucl. Phys. B66 (2000) 23 0, [hep-ph/ ]. T. Diakonidis, J. Fleischer, T. Riemann, and J. B. Tausk, A recursive reduction of tensor Feynman integrals, Phys. Lett. B683 (2010) 69 7, [arxiv: ]. D. B. Melrose, Reduction of Feynman diagrams, Nuovo Cim. 0 (196) T. Diakonidis, J. Fleischer, T. Riemann, and B. Tausk, A recursive approach to the reduction of tensor Feynman integrals, PoS RADCOR2009 (2009) 033, [ ]. 39 / 6

40 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 0 / 6 Numbers (I) Pentagons Randomly chosen phase space point with massive and massless internal particles p E E E E+00 p E E E E+00 p E E E E+00 p E E E E+01 p E E E E+01 m 1 = 0.0, m 2 = 2.0, m 3 = 3.0, m =.0, m =.0 p 3 p 1 + p 2 m 3 p 1 + p 2 m p1 m2 p2 m 1 m m 1 m 3 p 1 + p 2 + p 3 p + p m3 m1 p m p3 m p m p p + p m p 3 m 1 p

41 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers Selected pentagon components Shown are the constant terms of the tensor components E 2 E 12 E 232 E 0321 E Pentagon.F ( E-0, E-0) ( E-06, E-06) ( E-0, E-0) ( E-06, E-0) ( E-0, E-0) Box.F LoopTools ( E-03, E-03) ( E-03, E-03 ) ( E-03, E-02) ( E-03, E-02) ( E-03, E-03) ( E-03, E-03) ( E-0, E-0) ( E-0, E-0) Triangle.F LoopTools C 2 ( E-0, E-03) ( E-0, E-03) C 01 ( E-02, E-02) ( E-02, E-02) C 133 ( E-02, E-02) ( E-02, E-02) Bubble.F LoopTools B 3 ( E+00, E+00) ( , ) B 12 ( E+00, E-02) ( , E-02) 1 / 6

42 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers here some text 1. p 3 p 2 k+q 2 k+q 3 m 3 m p k+q 1 m 2 m k+q p 1 m 1 m 6 p k+q 0 k+q p 6 Figure: Momenta flow for the massive six-point topology. here some text 2. 2 / 6

43 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers 3 / 6 Numbers (II) Hexagons p E E+03 p E E+03 p E E E E+02 p E E E E+02 p E E E E+01 p E E E E+01 m 1 = 110.0, m 2 = 120.0, m 3 = 130.0, m = 10.0, m = 10.0, m 6 = F E 18 i E 19 µ F µ E 17 + i E E 17 i E E 17 + i E E 16 + i E 17 µ ν F µν E 1 i E E 1 + i E E 1 i E E 1 i E E 1 i E E 16 + i E E 1 i E E 1 i E E 1 + i E E 1 + i E 1

44 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers / 6 µ ν λ F µνλ E 11 i E E 13 i E E 13 + i E E 12 + i E E 12 + i E E 13 i E E 13 i E E 12 + i E E 13 + i E E 12 + i E E 13 + i E E 13 i E E 12 i E E 13 + i E E 1 + i E E 13 + i E E 13 i E E 12 i E E 13 i E E 12 i E 12 Table: Tensor components for a massive rank R = 3 six-point function

45 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers µ ν λ ρ F µνλρ E 09 + i E E 10 + i E E 10 i E E 10 i E E 10 i E E 11 + i E E 11 + i E E 10 i E E 11 i E E 10 i E E 10 i E E 11 i E E 10 + i E E 11 + i E E 11 i E E 11 i E E 11 i E E 10 + i E E 11 + i E E 09 + i E E 10 + i E E 12 i E E 11 i E E 11 + i E E 11 + i E E 10 + i E E 12 i E E 11 i E E 12 i E E 11 i E E 10 + i E E 11 + i E E 10 + i E E 11 + i E 11 / 6

46 Introduction Recursions Simplifying recursions Numbers: D 111 Summary Backup: 6- and -pt numbers p p p p p p 6 = (p 1 + p 2 + p 3 + p + p ) Table: Phase space point of massless six-point functions taken from [Binoth:2008 [7]]. Golem9: Binoth, Guillet, Heinrich, Pilon, Reiter [arxiv:hep-ph/ ] Shown are only the constant terms of the tensor components. Hexagon.F Golem9 F ( E+00, E-01 ) ( E+00, E-01 ) F ( E+01, E+00 ) ( E+01, E+00) F ( E+02, E+02 ) ( E+02, E+02 ) F ( E+00, E+00 ) ( E+00, E+00 ) F ( E+00, E+00) ( E+00, E+00) 6 / 6