Tord Riemann. DESY, Zeuthen, Germany

Transcription

1 1/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Algebraic tensor Feynman integral reduction Tord Riemann DESY, Zeuthen, Germany Based on work done in collaboration with Jochem Fleischer Corfu Summer Institute, Aug 29 Sep, 2010, Corfu, Greece

2 Introduction 1-loop 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 2/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece

3 3/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece A simple example 1-loop self-energy: I µ 2 = d d k k µ iπ d/2 [k 2 M1 2] [(k + p)2 M2 2] = p µb 1 Solve: p µi µ 2 = p 2 B 1 (p, M 1, M 2 ) d d k pk d d k pk = iπ d/2 [k 2 M1 2] [(k + p)2 M2 2] = iπ d/2 D 1 D 2 [ d d k D2 (p 2 M2 2 = M2 1 ) D ] 1 iπ d/2, D 1 D 2 1 [ ] B 1 (p, M 1, M 2 ) = 2p 2 A 0 (M 1 ) A 0 (M 2 ) (p 2 M2 2 M2 1 )B 0(p, M 1, M 2 ) A tensor Feynman integral is expressed in terms of scalar Feynman integrals.

4 / v :2 T. Riemann Tensor reduction Corfu 2010, Greece Systematic approach: Passarino, Veltman 1978 [1] Need in addition a library of scalar functions: thooft, Veltman 1979 [2] State of the art: Hahn, LoopTools/FF [3, ]

5 / v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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. Need Extensions: Reduction of n-point functions with n > Need Improvements: Avoid the break-down in certain kinematical configurations Recent developments in the Fleischer-Davydychev-Tarasov approach get tensor reduction such that one may : kill pentagon Gram determinants treat sub-diagram Gram determinants

6 6/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Outline [] 1991 Davydychev,... Reducing Feynman diagrams to scalar integrals [6] 1996 Tarasov, Connection [of] Feynman integrals [with] different... space-time dimensions [7] 1999 Fleischer et al., Algebraic reduction of one-loop Feynman graph amplitudes 1 Introduction 2 Recursions 3 Simplifying recursions Numbers: D 111 Summary 6 Backup transparencies References: [8] Diakonidis et al., PRD 80 (2009) [9] Diakonidis et al., PLB 683 (2010) 69 [10] J. Fleischer, T. Riemann, PoS(ACAT2010)07 [arxiv: ] and unpubl. work

7 7/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Notations: I {µ 1, },s n 1 etc. The tensor integral I {µ 1, },s n 1,ab is obtained from the integral I {µ 1, } n by shrinking line s raising the powers of inverse propagators a, b s a + b + I {µ 1, } n = I {µ 1, },s n 1,ab (3) The operators i ±, j ±, k ± act by shifting the indices ν i, ν j, ν k by ±1.

8 8/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Notations: Gram and modified Cayley determinant, signed minors [Melrose:196] Gram determinant G n: G n = 2q i q j, i, j = 1,... n () 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, () with matrix elements Y ij = (q i q j ) 2 + mi 2 + mj 2, (i, j = 1... N) (6) For a choice q n = 0, both determinants are related: () N = G N 1 The Gram determinant () N does not depend on the masses.

9 9/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Notations: signed minors [Melrose:196] signed minors of () N are constructed by deleting m rows and m columns from () N, and multiplying with a sign factor: ( ) j1 j 2 j m k 1 k 2 k m N ( 1) l (j l +k l ) rows j sgn {j} sgn 1 j m deleted {k} columns k 1 k m deleted 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. Example: (7) ( 0 0 ) N Y 11 Y Y 1N Y 12 Y Y 2N Y 1N Y 2N... Y NN, (8)

10 10/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Example: Getting a -point function from a six-point function I e + Z µ µ t eµ e e Z d W u d t ed 0 ν µ 0 0 M Z 0 0 s µνu s νu Figure: A six-point topology (a) leading to four-point functions (b) with realistically vanishing Gram determinants.

11 11/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Example: Getting a -point function from a six-point function II The example is taken from [11]. The corresponding -point tensor integrals are, in LoopTools [3, 12] notation: D0i(id, 0, 0, s νu, t ed, tēµ, s µ νu, 0, MZ 2, 0, 0). (9) The Gram determinant is: () = 2tēµ [s 2 µ νu + s νut ed s µ νu(s νu + t ed tēµ )], (10) It vanishes if: t ed t ed,crit = sµ νu(sµ νu s νu + tēµ ) s µ νu s νu. (11) In terms of a dimensionless scaling parameter x, t ed = (1 + x)t ed,crit, (12)

12 12/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Example: Getting a -point function from a six-point function III the Gram determinant becomes: () = 2 x s µ νutēµ (s µ νu s νu + tēµ ). (13) We will also need the modified Cayley determinant: ( 0 0) = 2M 2 Z M 2 Z M 2 Z sµ νu M2 Z M 2 Z 0 s νu M 2 Z MZ 2 sµ νu s νu 0 t ed MZ 2 tēµ t ed 0 = s 2 µ νut 2 ēµ + 2 M2 Z tēµ [ 2s νut ed + s µ νu(s νu + t ed tēµ )] + M Z (s2 νu + (t ed tēµ ) 2 2s νu(t ed + tēµ )).

13 13/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Recursions for hexagons Express any hexagon by pentagons [Fleischer:1999,Binoth:200,Denner:200,Diakonidis:2008 [7, 13, 1, 8] ] I µ 1...µ R 1 ρ 6 = 6 s=1 I µ 1...µ R 1,s Q ρ s. (1) auxiliary vectors Q ρ s = 6 q ρ i i=1 ( 0s ) 0i 6 ( 0 0) 6, s = (1)

14 1/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Dimensional shifts and recurrence relations for pentagons (I) Following [Davydychev:1991 []] Replace tensors by scalar integrals in higher dimensions: Example R = 3: I µνλ = = d 2ɛ k iπ d/2 i,j,k=1 r=1 and n ijk = (1 + δ ij )(1 + δ ik + δ jk ). c 1 r k µ k ν k λ (16) q µ q ν i j qk λ n ijk I [d+]3 + 1 n 1 (g µν q,ijk i λ + g µλ qi ν + g νλ q µ )I [d+]2, 2 i,i i=1 [d+] l = 2ɛ + 2l, and for definition of I [d+]2,i etc. see (3).

15 1/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Dimensional shifts and recurrence relations for pentagons (II) Naive, direct approach just perform dimensional recurrences Following [Tarasov:1996,Fleischer:1999 [6, 7]] apply recurrence relations, relating scalar integrals of different dimensions, in order to get rid of the dimensionalities [d+] l = 2ɛ + 2l: (d [ ( ) ( ν j j + I [d+] ) 1 j = () 0 i=1 + [( ) ν i + 1)I [d+] = 1 0 () 0 ( ) ] j k I (17) k k=1 ( ) ] 0 k I, (18) k where the operators i ±, j ±, k ± act by shifting the indices ν i, ν j, ν k by ±1. k=1

16 16/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Dimensional shifts and recurrence relations for pentagons (III) Represent a pentagon tensor of rank R: After repeated use of the recurrence relations, all the higher dimensional scalar integrals disappear A representation by the simple scalar functions in d dimensions is achieved: self-energies B 0 vertices C 0 boxes D 0 For the tensor rank R one gets ) R inverse powers of Gram determinants:( 1 () The algebraic derivations have to be re-organized in order to cancel in a controlled way these inverse powers of Gram determinants

17 The result of simplifying manipulations [indicated in the backup slides (mark 8)] and collecting all contributions, our final result for e.g. the tensor of rank R = 3 can be written as follows: I µ ν λ = i,j,k=1 q µ i q ν j q λ k E ijk + k=1 g [µν q λ] k E 00k, (19) with: E 00j = s=1 1 ( 0 0) [ 1 ( 0s ) I [d+],s d 1 ( s ) ] I [d+]2,s, (20) 2 0j 3 j E ijk = s=1 1 ( 0 0) {[ (0j ) I [d+]2,s sk ] + (i j),i no scalar -point integrals in higher dimensions no inverse Gram det. () We have yet: scalar -point integrals in higher dimensions: I [d+]2,s inverse Gram det. ( 0 0 ) (),ij etc. + ( 0s ) } ν ij I [d+]2,s. (21) 0k,ij 17/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece

18 18/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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 (22) I [d+],s,i, I [d+]2,s,i I [d+]2,s,ij boxes with higher indices (23) Application of dimension-shifting recurrence relations produces powers of 1/(). They will be the unwanted and unavoidable sub-gram-determinants (). Next and last two steps: Reduce the I [d+],s,i, I [d+]2,s,i, I [d+]2,s,ij etc. to non-indexed scalars Then look at the non-indexed scalars

19 19/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Reduce I [d+]l,s,ij to I [d+]l,s plus simpler objects I By nontrivial manipulations we get e.g.: [ ( ) I [d+],s 1 0s,i = ) is ( 0s 0s (d 3)I [d+],s + t=1 ( ) ] 0st I3 st 0si (2) ν ij I [d+]2,ij = ( 0 i ) ( 0 0) ( 0 j ) ( 0 0) ( 0 j ) ( 0 0) d 2 ( 0 0) (d 2)(d 1)I [d+]2 + t=1 ( 0i ) 0j ( 0 0) ( ) 0t I [d+],t 3 + ( 1 0i 0 0) t=1 I [d+] ( ) 0t I [d+],t 3,i (2) 0j These equations are free of inverse Gram determinants (). But they contain yet the generic -point and (partly indexed) 3-point functions in higher dimensions, I [d+],s, I [d+],t 3, etc.

20 20/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Last step: evaluate the I [d+],s, I [d+],t 3, etc. I Several strategies are now possible: Just evaluate them analytically in d + 2l 2ɛ dimensions if you may do that Just evaluate them numerically in d + 2l 2ɛ dimensions Reduce them further by recurrences buy the towers of 1/() apply (18) Make a small Gram determinant expansion apply (18) another way round Last two items are done here.

21 21/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Reduction of scalars I D to the generic dimension Id = D 0, I d 3 = C 0 I Non-small -point Gram determinants: Direct, iterative use of (18) yields e.g.: [( 0 ( I [d+]l = 0) I [d+] l 1 t 0) I [d+] 3 () () I [d+]l,t 3 = ( 0t ) 0t ( t t) I [d+]l 1,t 3 t=1 u=1,u t And we are done. This works fine if () is not small. l 1,t ] ( ut ) 0t ( t I t) [d+]l 1,tu 2 1 d + 2l (26) 1 d + 2l (27)

22 22/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Make a small Gram expansion I Again use (18): () (d i=1 If () = 0, then it follows (n = ): ν i + 1)I [d+] = [ ] (0 ) ( 0 I I3 0 k) k k=1 I D n = ( n 0 k) n ( 0 I k 0) D,k n 1 (28) n If () << 1, re-write (18), as follows: I D n = ( n 0 k) n ( 0 I k 0) D,k n 1 ( ()n 0 [(D + 1) n 0) n n i ν i ]I D+2 n. (29) Effectively we may evaluate In D in terms of simpler functions I D,k n 1 with a small correction depending on In D+2.

23 23/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece We may go a step further, and insert into (29) for In D+2 the rhs. of (28), taken now at D = D + 2: I D n = ( n 0 k) n ( 0 I k 0) D,k n 1 n ()n ( 0 0) [(D + 1) n [ n ( 0 k) n ( 0 I k 0) D+2,k n 1 n n ν i ] i ()n ( 0 0) n [(D + 3) n i ν i ]I D+ n The terms proportional to [() ( 0) n/ 0 n ]a, a = 0, 1 may be evaluated at the correct kinematics. They depend on three-point functions, and their reduction by normal recurrences will not introduce the unwanted powers of 1/(). The last term, suppressed by the factor [() ( 0 n/ 0 )n ]2, depends on In D+. It may either be taken approximately at () n = 0, where it can also be represented by 3-point functions (and their reductions), or it may be evaluated more correctly by another iteration based on (28). And so on and so on In the numerical example next section we worked out up to 10 stable iterations. ].

24 2/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece A quite similar attempt to perform such a series of approximations was undertaken in [1] (see equation () there), where a specific example, forward light-by-light scattering through a massless fermion loop, was studied. The approach was then not further followed. W. Giele, E. W. N. Glover, and G. Zanderighi, in: Proceedings of Loops ans Legs 200: Numerical evaluation of one-loop diagrams near exceptional momentum configurations,

25 2/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece An example from A. Denner [11]: -point tensor of rank 3 D 111 Few 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

26 26/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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!

27 27/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Following Davydychev, [], one gets I µνλ = d k µ k ν k λ n r=1 cr = n i,j,k=1 q µ i q ν j q λ k ν ijk I [d+]3 n,ijk n i=1 g [µν q λ] I [d+]2 (30) i n,i We identify the tensor coefficient D 111 a la LoopTools: D 111 ν ijk I [d+]3,ijk for ijk = 222

28 28/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Rank R = tensor D 1111 Numerics with dimensional recurrences From (29) we see that a small Gram determinant expansion will be useful when the following dimensionless parameter becomes small: R = () ( 0 0) where s is a typical scale of the process, e.g. we will choose s = s µ νu. Following [11], we further choose: s µ νu = 2 10 GeV 2, s νu = 1 10 GeV 2, tēµ = 10 GeV 2, s, (31) and get t ed,crit = 6 10 GeV 2. For x=1, the Gram determinant becomes () = GeV 3. The small expansion parameter R(x) and D 1111 are shown in figure 2. R x D

29 x Re D 1111 Im D 1111 Re D 0. [exp0] E E [exp2] E E [exp] E E [exp6] E E [pade] E E [direct] E E [exp6] E E [pade] E E [direct] E E [exp6] E E [pade] E E [fit] E [direct] E E [LoopT] E E [exp6] E E [pade] E E [direct] E E [LoopT] E E-10 29/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Rank R = tensor D 1111 Numerics with dimensional recurrences

30 30/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Rank R = 3 D 111 Numerics with dimensional recurrences x Re D 111 Im D [exp0] E E [exp1] E E [exp] E E [exp6] E E [pade] E E E E [exp6] E E [pade] E E E E [exp6] E E [pade] E E E E [exp6] E E [pade] E E E E E E E E

31 31/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Rank R = 2 D 11 etc. Numerics based on the dimensional recurrences x Re D 11 Im D 11 R 0 [exp0] E E [exp0] E E [exp2] E E [exp6] E E [pade] E E E E [exp6] E E [pade] E E E E [exp6] E E [pade] E E E E [exp6] E E [pade] E E E E E E E E

32 32/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Summary Recursive treatment of hexagon and pentagon tensor 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 = Proper isolation of inverse Gram determinants of subdiagrams of the type ( ) s s, which cannot be completely n avoided Some numerics, so far in Mathematica, and a numerical C++ package (together with V. yundin) under way

33 References I G. Passarino and M. Veltman, One loop corrections for e + e annihilation into µ + µ in the Weinberg model, Nucl. Phys. B160 (1979) 11. G. t Hooft and M. Veltman, Scalar one loop integrals, Nucl. Phys. B13 (1979) T. Hahn and M. Perez-Victoria, Automatized one-loop calculations in four and d dimensions, Comput. Phys. Commun. 118 (1999) 13, [hep-ph/98076]. G. J. van Oldenborgh, FF: A Package to evaluate one loop Feynman diagrams, Comput. Phys. Commun. 66 (1991) 1 1. A. I. Davydychev, A simple formula for reducing Feynman diagrams to scalar integrals, Phys. Lett. B263 (1991) O. 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, J. Gluza, K. Kajda, T. Riemann, and J. Tausk, A complete reduction of one-loop tensor - and 6-point integrals, Phys. Rev. D80 (2009) , [arxiv: ]. T. Diakonidis, J. Fleischer, T. Riemann, and J. B. Tausk, A recursive reduction of tensor Feynman integrals, Phys. Lett. B683 (2010) 69 7, [arxiv: ]. 33/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece

34 References II J. Fleischer and T. Riemann, Some variations of the reduction of one-loop Feynman tensor integrals, PoS RADCOR2010 (2010) 07, [arxiv: ]. A. Denner, Techniques and concepts for higher order calculations, Introductory Lecture at DESY Theory Workshop on Collider Phenomenology, Hamburg, 29 Sep - 2 Oct T. Hahn, LoopTools 2. User s Guide, LT2Guide.pdf. 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]. A. Denner and S. Dittmaier, Reduction schemes for one-loop tensor integrals, Nucl. Phys. B73 (2006) 62 11, [hep-ph/00911]. W. Giele, E. W. N. Glover, and G. Zanderighi, Numerical evaluation of one-loop diagrams near exceptional momentum configurations, Nucl. Phys. Proc. Suppl. 13 (200) , [hep-ph/007016]. 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. 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 / v :2 T. Riemann Tensor reduction Corfu 2010, Greece

35 References III 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) 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) , [ ]. R. K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP 02 (2008) 002, [arxiv: ]. 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]. 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]. 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/ ]. 3/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece

36 36/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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

37 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) 37/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece

38 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. 38/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece

39 39/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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

40 0/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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

41 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 10 1/ 2 v : E 11 + i E 11 T. Riemann Tensor reduction Corfu 2010, Greece

42 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 [16]]. 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) 2/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece

43 3/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Algebraic simplifications, 1st step With the identity ( ) ( ) 0 s 0 i = ( ) ( ) ( ) 0s 0 s () + 0i i 0 (32) 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 0) ( 0 0) s=1 ] I µ 1...µ R 1,s Q µ 0 s=1 I µ 1...µ R 1,s Q µ s (33) The auxiliary vectors Q µ s were introduced already for n = 6: Q µ 0 = q µ i i=1 ( 0 i ) () while Q µ s = q µ i i=1 ( 0s ) 0i ( 0 0) (3)

44 / v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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 (3) Example: For R = 3 pentagons need rank 2 tensor: [ (0 ) T µν = I µν 0 s=1 ( ) ] s I µν,s 0 (36)

45 / v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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 (37)

46 6/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Davydychev s higher dimensional integrals The second term with I µν is a typical example of [Davydychev:1991 []] : 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 (38)

47 7/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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:1978jh [1]] Examples: LO (Lowest order) of e.g. Z e + µ is one-loop [Riemann:1981 [17], Mann:1983 [18]] NLO: one-loop corrections to e.g. H τ + τ, WW, ZZ [Fleischer:1980 [19]] NNLO: e.g. radiative loop corrections e + e e + e γ (here with -point functions)

48 Some opensource packages package FF [vanoldenborgh:1990 []] package LoopTools/FF v.2 [Hahn:1998,2006 [3]] 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 [16]] for n 6, but only massless propagators Mathematica package hexagon.m [Diakonidis:2008 [20, 8]] for n 6, rank R package for all n scalar integrals: QCDloop [Ellis:2007 [21]] see also: review A.Denner, DESY TH workshop 2009 Our approach: Package hexagon.m by Kajda et al. Package olotic.f by Diakonidis et al. In preparation: C++ package fry by V. Yundin et al. 8/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece

49 9/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Crucial contributions [of course, list is incomplete... ] [Campbell:1996 [22]] [Denner:2002,200 [23, 1]] [Binoth:1999,200 [2, 13]] [Bern:1993 [2]] [Ossola:2006 [26] ] In the following, I will describe recent developments in the Fleischer-Davydychev-Tarasov approach. [Davydychev:1991, Tarasov:1996, Fleischer:1999,Diakonidis:2008,2009 [, 6, 7, 8, 9]] get tensor reduction kill pentagon-gram det s treat sub-gram det s

50 0/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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 (39) ( ) ( ( ( ) 0 ( ) ( ) i i s j 0i s j) j) = +, g µν = 2 q µ q ν i () sj 0 () () i j (0) i,j=1 ( ) ( ) ( ) ( ) ( ) ( ) s 0s s 0s s 0s = 0 is i 0s s 0i (1) ( ) ( ) ( ) ( ) ( ) ( ) s ts s ts s ts = 0 js j 0s s 0j (2)

51 1/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece Algebraic simplifications, 2nd step Use of identities for the determinants work!!! ( ) ( ) ( ) ( ) ( ) ( ) s is s 0s s 0i = + 0 js i js s sj (3) ( ) ( ) ( ) ( ) ( ) s 0st 0s st ts 2 = s 0st 0s st 0s () [ ( ) ( ) ( ) ( ) ] (s ) [ ( ) ( ) ( ) ( ) ] (st ) ts ust ts ust ts us ts us = 0s jst js 0st s 0s js js 0s st () ( ) ts = 0 (6) is t=1

52 2/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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 (7) I µ ν = E ij = i,j=1 1 ( 0 0) q µ i q ν j E ij + g µν E 00 (8) s=1 E 00 = 1 1 ( 2 0 0) [( ) 0i I [d+],s sj + ) s=1 ( s 0 ( ) ] 0s I [d+],s 0j,i (9) I [d+],s (0)

53 3/ v :2 T. Riemann Tensor reduction Corfu 2010, Greece 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 (1) {[( ) 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 (2) (3)

54 / v :2 T. Riemann Tensor reduction Corfu 2010, Greece OLD 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