From Tensor Integral to IBP

Transcription

1 From Tensor Integral to IBP Mohammad Assadsolimani, in collaboration with P. Kant, B. Tausk and P. Uwer 11. Sep Mohammad Assadsolimani From Tensor Integral to IBP 1

2 Contents Motivation NNLO Tensor Integral Tarasov s method Projection method Applications Heavy Quark Form Factors Single Top Quark Production Conclusions Mohammad Assadsolimani From Tensor Integral to IBP 2

3 Multi loop calculation Multi loop calculation More precision in calculated results Mohammad Assadsolimani From Tensor Integral to IBP 3

4 More precision Ex.: Total cross section for Higgs production in gluon fusion σ(pp H+X) [pb] s = 2 TeV σ(pp H+X) [pb] s = 14 TeV NNLO NLO LO M H [GeV] 10 NNLO NLO LO M H [GeV] [R. Harlander, W. Kilgore Nov. 02] Perturbative convergence LO NLO( 70%) and NLO NNLO( 30%) Mohammad Assadsolimani From Tensor Integral to IBP 4

5 Multi loop calculation Multi loop calculation More precision in calculated results New effects Mohammad Assadsolimani From Tensor Integral to IBP 5

6 New effects Ex.: forward backward charge asymmetry of the top quark 6 σ[pb] p p t t+jet+x s = 1.96TeV p T,jet > 20GeV NLO (CTEQ6M) LO (CTEQ6L1) 0.04 A t FB p p t t+jet+x s = 1.96TeV p T,jet > 20GeV µ/m t NLO (CTEQ6M) LO (CTEQ6L1) 1 µ/m t 10 [S. Dittmaier, P. Uwer, S. Weinzierl Apr. 08] Mohammad Assadsolimani From Tensor Integral to IBP 6

7 Multi loop calculation Multi loop calculation More precision in calculated results New effects Exact NLO or NNLO calculations ofσ hard needed because of: Mohammad Assadsolimani From Tensor Integral to IBP 7

8 Multi loop calculation Multi loop calculation More precision in calculated results New effects Exact NLO or NNLO calculations ofσ hard needed because of: Accurate and reliable predictions of parton level observables. Mohammad Assadsolimani From Tensor Integral to IBP 7

9 Multi loop calculation Multi loop calculation More precision in calculated results New effects Exact NLO or NNLO calculations ofσ hard needed because of: Accurate and reliable predictions of parton level observables. Backgrounds for New Physics Searches Mohammad Assadsolimani From Tensor Integral to IBP 7

10 NNLO When do we need NNLO? When NLO corrections are large Mohammad Assadsolimani From Tensor Integral to IBP 8

11 NNLO When do we need NNLO? When NLO corrections are large When truly high precision is needed Mohammad Assadsolimani From Tensor Integral to IBP 8

12 NNLO When do we need NNLO? When NLO corrections are large When truly high precision is needed When the precision of NLO calculation has to be verified Mohammad Assadsolimani From Tensor Integral to IBP 8

13 NNLO When do we need NNLO? When NLO corrections are large When truly high precision is needed When the precision of NLO calculation has to be verified For instance single top quark production : Mohammad Assadsolimani From Tensor Integral to IBP 8

14 NNLO Single Top Quark Production Single top quark production Process S σlo (pb) σ NLO (pb) t channel 2.0 TeVpp TeVpp [B.Harris, E. Laenen, L.Phaf, Z. Sullivan, S. Weinzierl 02] Mohammad Assadsolimani From Tensor Integral to IBP 9

15 NNLO Single Top Quark Production Single top quark production Process No colour exchange at NLO: S σlo (pb) σ NLO (pb) t channel 2.0 TeVpp TeVpp [B.Harris, E. Laenen, L.Phaf, Z. Sullivan, S. Weinzierl 02] W W Only vertex corrections contribute: W Mohammad Assadsolimani From Tensor Integral to IBP 9

16 NNLO Single Top Quark Production Single top quark production Process No colour exchange at NLO: S σlo (pb) σ NLO (pb) t channel 2.0 TeVpp TeVpp [B.Harris, E. Laenen, L.Phaf, Z. Sullivan, S. Weinzierl 02] Colour exchange at NNLO: W W W W 0 Only vertex corrections contribute: W W 0 W Mohammad Assadsolimani From Tensor Integral to IBP 9

17 NNLO Single Top Quark Production The true uncertainty due to missing higher order correction may be greater because new colour exchange diagrams first contribute at NNLO. A significant effect on kinematical distributions Important for studies of the V A structure Source of polarised top quarks Access to the b quark pdfs Mohammad Assadsolimani From Tensor Integral to IBP 10

18 Tensor integral In the NNLO corrections occur tensor integrals: I(d,a 1,,a n )[1,k µ 1,kν 2, ] = d d k 1 d d ij k kµ i 1 kν j 2 2 P a 1 1 Pa n n Possibilities to reduce tensor integrals to scalar integrals: Mohammad Assadsolimani From Tensor Integral to IBP 11

19 Tensor integral In the NNLO corrections occur tensor integrals: I(d,a 1,,a n )[1,k µ 1,kν 2, ] = d d k 1 d d ij k kµ i 1 kν j 2 2 P a 1 1 Pa n n Possibilities to reduce tensor integrals to scalar integrals: By Schwinger parametrization By projection method [O. V. Tarasov, Phys. Rev. 96, Nucl. Phys. 81] [T. Binoth, E.W.N. Glover, P. Marquard and J.J. van der Bij 02; E.W.N. Glover 04] Mohammad Assadsolimani From Tensor Integral to IBP 11

20 Tensor integral In the NNLO corrections occur tensor integrals: I(d,a 1,,a n )[1,k µ 1,kν 2, ] = d d k 1 d d ij k kµ i 1 kν j 2 2 P a 1 1 Pa n n Possibilities to reduce tensor integrals to scalar integrals: By Schwinger parametrization By projection method [O. V. Tarasov, Phys. Rev. 96, Nucl. Phys. 81] [T. Binoth, E.W.N. Glover, P. Marquard and J.J. van der Bij 02; E.W.N. Glover 04] Tensor reduction various scalar integrals with the same structure of the integrand with different powers of propagators Mohammad Assadsolimani From Tensor Integral to IBP 11

21 Integration by Parts Possible approaches: Mohammad Assadsolimani From Tensor Integral to IBP 12

22 Integration by Parts Possible approaches: solve each integral individually, Mohammad Assadsolimani From Tensor Integral to IBP 12

23 Integration by Parts Possible approaches: solve each integral individually, express all scalar integrals as a linear combination of some basic master integrals, Integration by parts (IBP). [Chetyrkin, Tkachov 81] Mohammad Assadsolimani From Tensor Integral to IBP 12

24 Integration by Parts Possible approaches: solve each integral individually, express all scalar integrals as a linear combination of some basic master integrals, Integration by parts (IBP). Reduction techniques: [Chetyrkin, Tkachov 81] Mohammad Assadsolimani From Tensor Integral to IBP 12

25 Integration by Parts Possible approaches: solve each integral individually, express all scalar integrals as a linear combination of some basic master integrals, Integration by parts (IBP). Reduction techniques: Laporta: efficient algorithm to solve linear system of IBP Identities [Chetyrkin, Tkachov 81] Mohammad Assadsolimani From Tensor Integral to IBP 12

26 Integration by Parts Possible approaches: solve each integral individually, express all scalar integrals as a linear combination of some basic master integrals, Integration by parts (IBP). Reduction techniques: Laporta: efficient algorithm to solve linear system of IBP Identities [Chetyrkin, Tkachov 81] AIR [Anastasiou, Lazopoulos 04] FIRE [Smirnov 08] Crusher [Marquard, Seidel (to be published)] REDUZE 1&2 [Studerus 09; Manteuffel, Studerus 12] Mohammad Assadsolimani From Tensor Integral to IBP 12

27 Tarasov s method Tensor reduction leads to a very large number of scalar integrals which are shifted in dimension and have other powers of propagators I(d,a 1,,a n )[k µ 1 kν 2, ] gµν i I(d +x i,a i 1,,ai n )[1] Example for two loop corrections to Axial Vector Form Factors I(d,1,1,1,1,1,1)[1,k µ 1 1 k µ 2 1 kν 1 2 kν 2 2 ] I(2+d,2,1,1,1,1,2)+ +I(4+d,1,1,1,2,3,1) + +I(8+d,3,3,3,2,1,2) Mohammad Assadsolimani From Tensor Integral to IBP 13

28 Tarasov s method Shift in the dimension Mohammad Assadsolimani From Tensor Integral to IBP 14

29 Tarasov s method Shift in the dimension An arbitrary scalar Feynman integral: I (d) N ({s i },{ms 2}) j=1 c j 0 0 dα j α a j 1 j [D(α)] d 2 [ Q({si e i },α) ] N D(α) l=1 α l (m l 2 iǫ) Mohammad Assadsolimani From Tensor Integral to IBP 14

30 Tarasov s method Shift in the dimension An arbitrary scalar Feynman integral: I (d) N ({s i },{ms 2}) j=1 c j 0 0 dα j α a j 1 j [D(α)] d 2 [ Q({si e i },α) ] N D(α) l=1 α l (m l 2 iǫ) Mohammad Assadsolimani From Tensor Integral to IBP 14

31 Tarasov s method Shift in the dimension An arbitrary scalar Feynman integral: I (d) N ({s i },{ms 2}) j=1 c j 0 0 dα j α a j 1 j [D(α)] d 2 [ Q({si e i },α) ] N D(α) l=1 α l (m l 2 iǫ) D ( ) m (polynomial differential operator) obtained from D(α) by j 2 substituting α i j / mj 2. The application ofd( i) to the scalar integral: I (d 2) ({s i },{m 2 s }) D( j) I (d) ({s i },{m 2 s }), Mohammad Assadsolimani From Tensor Integral to IBP 14

32 Tarasov s method Shift in the dimension An arbitrary scalar Feynman integral: I (d) N ({s i },{ms 2}) j=1 c j 0 0 dα j α a j 1 j [D(α)] d 2 [ Q({si e i },α) ] N D(α) l=1 α l (m l 2 iǫ) D ( ) m (polynomial differential operator) obtained from D(α) by j 2 substituting α i j / mj 2. The application ofd( i) to the scalar integral: I (d 2) ({s i },{m 2 s }) D( j) I (d) ({s i },{m 2 s }), apply this to master integrals I master (d 2,a 1,,a n ) = i c i I(d,a i 1,,a i n), Mohammad Assadsolimani From Tensor Integral to IBP 14

33 Tarasov s method all scalar integrals in rhs. of that equation have to be replaced by master integrals. i.e. I master (d 2,a 1,,a n ) = j D j I master (d,a j 1,,aj n), Mohammad Assadsolimani From Tensor Integral to IBP 15

34 Tarasov s method all scalar integrals in rhs. of that equation have to be replaced by master integrals. i.e. I master (d 2,a 1,,a n ) = j D j I master (d,a j 1,,aj n), we have for all master integrals: I d 2 1. Il d 2 master = D ll wherelis the number of master integrals. I d 1. I d l master Mohammad Assadsolimani From Tensor Integral to IBP 15

35 Projection method By this method we get scalar products between loop momenta and external momenta and no shift in the dimension of integrals I(d,a 1,,a n )[1,k µ 1,kµ 2, ] gµν ij I(d,a 1,,a n )[1]k i p j Example for two loop corrections to Axial Vector Form Factors I(d,1,1,1,1,1,1)[1,k µ 1 1 k µ 2 1 kν 1 2 kν 2 2 ] I(d, 2,1,1,1,1,1) + +I(d, 1,0,1,1,1,1) + +I(d,0,1,1,1, 2, 2) Mohammad Assadsolimani From Tensor Integral to IBP 16

36 Projection method The general tensor structure for the amplitude A: A = n B i (t,u,s)s i, i=1 where t,u and s are the Mandelstam variables and S i are the Dirac structures. Projectors for the tensor coefficients: S j A = n i=1 B i (t,u,s) ( S j S i }{{} M ji ) Bi (t,u,s) = i M 1 ( ij S j A) Mohammad Assadsolimani From Tensor Integral to IBP 17

37 Projection method The general tensor structure for the amplitude A: A = n B i (t,u,s)s i, i=1 where t,u and s are the Mandelstam variables and S i are the Dirac structures. Projectors for the tensor coefficients: S j A = n i=1 B i (t,u,s) ( S j S i }{{} M ji ) Bi (t,u,s) = i M 1 ( ij S j A) essential for this method : to be able to calculate the inverse matrix M 1 ij Mohammad Assadsolimani From Tensor Integral to IBP 17

38 Tarasov s method vs. Projection method Tarasov s method Positive powers for propagators (the sum of the powers of all propagators is large ) Calculate the inverse matrix in order to shift back the dimension Mohammad Assadsolimani From Tensor Integral to IBP 18

39 Tarasov s method vs. Projection method Tarasov s method Positive powers for propagators (the sum of the powers of all propagators is large ) Calculate the inverse matrix in order to shift back the dimension Projection method Negative powers for propagators Calculate the inverse matrix for the projector coefficients Mohammad Assadsolimani From Tensor Integral to IBP 18

40 Tarasov s method vs. Projection method Tarasov s method Positive powers for propagators (the sum of the powers of all propagators is large ) Calculate the inverse matrix in order to shift back the dimension Projection method Negative powers for propagators Calculate the inverse matrix for the projector coefficients Mohammad Assadsolimani From Tensor Integral to IBP 18

41 Two loop corrections to Heavy Quark Form Factors We implemented both methods to calculate the two loop corrections to Heavy Quark Vector and axial Vector Form Factors: [W. Bernreuther,R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia, E. Remiddi 04] [J. Gluza, A. Mitov, S. Moch, T. Riemann 09 ] Mohammad Assadsolimani From Tensor Integral to IBP 19

42 Two loop corrections to Heavy Quark Form Factors q 1 p 2 Mohammad Assadsolimani From Tensor Integral to IBP 20

43 Two loop corrections to Heavy Quark Form Factors q 1 p 2 There are 6 Dirac structures (Heavy Quark Vector and axial Vector Form Factors): S 1 = ū(q 1 )(1+γ 5 )u(p 2 )p µ 2 S 2 = ū(q 1 )(1 γ 5 )u(p 2 )p µ 2 S 3 = ū(q 1 )(1+γ 5 )u(p 2 )q µ 1 S 4 = ū(q 1 )(1 γ 5 )u(p 2 )q µ 1 S 5 = ū(q 1 )(1+γ 5 )γ µ u(p 2 ) S 6 = ū(q 1 )(1 γ 5 )γ µ u(p 2 ) Mohammad Assadsolimani From Tensor Integral to IBP 20

44 Two loop corrections to Heavy Quark Form Factors q 1 p 2 There are 6 Dirac structures (Heavy Quark Vector and axial Vector Form Factors): S 1 = ū(q 1 )(1+γ 5 )u(p 2 )p µ 2 S 2 = ū(q 1 )(1 γ 5 )u(p 2 )p µ 2 S 3 = ū(q 1 )(1+γ 5 )u(p 2 )q µ 1 S 4 = ū(q 1 )(1 γ 5 )u(p 2 )q µ 1 S 5 = ū(q 1 )(1+γ 5 )γ µ u(p 2 ) S 6 = ū(q 1 )(1 γ 5 )γ µ u(p 2 ) Projection Tarasov number of integrals max sum of powers of propagators 6 14 max sum of negative powers of propagators 4 0 reduction time 7500 s s Mohammad Assadsolimani From Tensor Integral to IBP 20

45 Two loop corrections to Heavy Quark Form Factors q 1 p 2 There are 6 Dirac structures (Heavy Quark Vector and axial Vector Form Factors): S 1 = ū(q 1 )(1+γ 5 )u(p 2 )p µ 2 S 2 = ū(q 1 )(1 γ 5 )u(p 2 )p µ 2 S 3 = ū(q 1 )(1+γ 5 )u(p 2 )q µ 1 S 4 = ū(q 1 )(1 γ 5 )u(p 2 )q µ 1 S 5 = ū(q 1 )(1+γ 5 )γ µ u(p 2 ) S 6 = ū(q 1 )(1 γ 5 )γ µ u(p 2 ) Projection Tarasov number of integrals max sum of powers of propagators 6 14 max sum of negative powers of propagators 4 0 reduction time 7500 s s Now one may come to the conclusion: Mohammad Assadsolimani From Tensor Integral to IBP 20

46 Two loop corrections to Heavy Quark Form Factors q 1 p 2 There are 6 Dirac structures (Heavy Quark Vector and axial Vector Form Factors): S 1 = ū(q 1 )(1+γ 5 )u(p 2 )p µ 2 S 2 = ū(q 1 )(1 γ 5 )u(p 2 )p µ 2 S 3 = ū(q 1 )(1+γ 5 )u(p 2 )q µ 1 S 4 = ū(q 1 )(1 γ 5 )u(p 2 )q µ 1 S 5 = ū(q 1 )(1+γ 5 )γ µ u(p 2 ) S 6 = ū(q 1 )(1 γ 5 )γ µ u(p 2 ) Projection Tarasov number of integrals max sum of powers of propagators 6 14 max sum of negative powers of propagators 4 0 reduction time 7500 s s Now one may come to the conclusion: projection method is an alternative method for the multi loop calculation! Mohammad Assadsolimani From Tensor Integral to IBP 20

47 Two loop corrections to Heavy Quark Form Factors q 1 p 2 There are 6 Dirac structures (Heavy Quark Vector and axial Vector Form Factors): S 1 = ū(q 1 )(1+γ 5 )u(p 2 )p µ 2 S 2 = ū(q 1 )(1 γ 5 )u(p 2 )p µ 2 S 3 = ū(q 1 )(1+γ 5 )u(p 2 )q µ 1 S 4 = ū(q 1 )(1 γ 5 )u(p 2 )q µ 1 S 5 = ū(q 1 )(1+γ 5 )γ µ u(p 2 ) S 6 = ū(q 1 )(1 γ 5 )γ µ u(p 2 ) Projection Tarasov number of integrals max sum of powers of propagators 6 14 max sum of negative powers of propagators 4 0 reduction time 7500 s s Now one may come to the conclusion: projection method is an alternative method for the multi loop calculation! but Mohammad Assadsolimani From Tensor Integral to IBP 20

48 Two loop corrections to single Top Quark Production There are three topological families: W W W t t t Mohammad Assadsolimani From Tensor Integral to IBP 21

49 Two loop corrections to single Top Quark Production There are three topological families: W W W t t t and 11 Dirac structures: S 1 = u(q 1 ) γ 7 u(p 2 )u(q 2 ) γ 6 γ q1 u(p 1 ) S 2 = u(q 1 ) γ 6 γ p1 u(p 2 )u(q 2 ) γ 6 γ q1 u(p 1 ) S 3 = u(q 1 ) γ 6 γ µ1 u(p 2 )u(q 2 ) γ 6 γ µ1 u(p 1 ) S 4 = u(q 1 ) γ 7 γ µ1 γ p1 u(p 2 )u(q 2 ) γ 6 γ µ1 u(p 1 ) S 5 = u(q 1 ) γ 7 γ µ1 γ µ2 u(p 2 )u(q 2 ) γ 6 γ µ1 γ µ2 γ q1 u(p 1 ) S 6 = u(q 1 ) γ 6 γ µ1 γ µ2 γ p1 u(p 2 )u(q 2 ) γ 6 γ µ1 γ µ2 γ q1 u(p 1 ) S 7 = u(q 1 ) γ 6 γ µ1 γ µ2 γ µ3 u(p 2 )u(q 2 ) γ 6 γ µ1 γ µ2 γ µ3 u(p 1 ) S 8 = u(q 1 ) γ 7 γ µ1 γ µ2 γ µ3 γ p1 u(p 2 )u(q 2 ) γ 6 γ µ1 γ µ2 γ µ3 u(p 1 ) S 9 = u(q 1 ) γ 7 γ µ1 γ µ2 γ µ3 γ µ4 u(p 2 )u(q 2 ) γ 6 γ µ1 γ µ2 γ µ3 γ µ4 γ q1 u(p 1 ) S 10 = u(q 1 ) γ 6 γ µ1 γ µ2 γ µ3 γ µ4 γ p1 u(p 2 )u(q 2 ) γ 6 γ µ1 γ µ2 γ µ3 γ µ4 γ q1 u(p 1 ) S 11 = u(q 1 ) γ 6 γ µ1 γ µ2 γ µ3 γ µ4 γ µ5 u(p 2 )u(q 2 ) γ 6 γ µ1 γ µ2 γ µ3 γ µ4 γ µ5 u(p 1 ) Mohammad Assadsolimani From Tensor Integral to IBP 21

50 Two loop corrections to single Top Quark Production Vertex corrections: both methods Mohammad Assadsolimani From Tensor Integral to IBP 22

51 Vertex corrections + + [ = N C 2 [ TC F g4 s {S t +t t +44t 2 1 +O(ǫ) 44t m5 2ǫ(t 1)3 12(t 1)3 t 3m3 t (t 1)3 +O(ǫ) 2n f 1 +O(ǫ) t +t t +41t 2 + +O(ǫ) 3m3 t (t 1) m3 t 3ǫ(t 1)3 18(t 1)3 2n + f 3m t (t 1) +O(ǫ) + 12t ( ) ] 2m m t (t 1)3 +O(ǫ) t (t +1) +O(ǫ) 3(t 1) +S 3 [ 1 m4 t 11+12t +13t 2 24ǫ(t 1)2 1 ( t ) m2 3ǫ +O(ǫ) 9t t + 1 m2 t n f (5 19t) 36(t 1) t 324t t3 +76t4 24t5 72(t 1)3t n f (t +1) ( ) + n f /9+n f /(3ǫ)+O(ǫ) ) +( 2/t t/3+o(ǫ) m2 t (3t2 +19t +28) +O(ǫ) m2 t 6ǫ(t 1) +O(ǫ) 1 + m2 t +N C 2 { } C2 +N F C 2 { }] C F C 1 ( ) A m2 t t m2 W +O(ǫ) 2t 3ǫ(t 1) + 2t(1+5t 14t 2 ) 9(t 1)3 n + f t n f (3+t) 3ǫ(t 1) 9(t 1) +O(ǫ) + (1+3t +3t 2 t3 ) +O(ǫ) 3(t 1)2 m2 t t(3t2 +4t +1) +O(ǫ) 9(t 1) +O(ǫ) 5t 2 +12t ǫ(t 1)2 + 29t 2 86t 87 72(t 1)2 ]} +O(ǫ) Mohammad Assadsolimani From Tensor Integral to IBP 23

52 Two loop corrections to single Top Quark Production Vertex corrections: both methods Mohammad Assadsolimani From Tensor Integral to IBP 24

53 Two loop corrections to single Top Quark Production Vertex corrections: both methods Planar double boxes : projection method Mohammad Assadsolimani From Tensor Integral to IBP 24

54 Two loop corrections to single Top Quark Production Vertex corrections: both methods Planar double boxes : projection method problem: build the inverse matrix M ji = S j S i Mohammad Assadsolimani From Tensor Integral to IBP 24

55 Two loop corrections to single Top Quark Production Vertex corrections: both methods Planar double boxes : projection method problem: build the inverse matrix M ji = S j S i try with common computer algebra system, e.g. Mathematica or Maple runtime 1 month with 64GB RAM Mohammad Assadsolimani From Tensor Integral to IBP 24

56 Two loop corrections to single Top Quark Production Vertex corrections: both methods Planar double boxes : projection method problem: build the inverse matrix M ji = S j S i try with common computer algebra system, e.g. Mathematica or Maple runtime 1 month with 64GB RAM We calculated M 1 ji and all Planar double boxes diagrams Mohammad Assadsolimani From Tensor Integral to IBP 24

57 Two loop corrections to single Top Quark Production Vertex corrections: both methods Planar double boxes : projection method problem: build the inverse matrix M ji = S j S i try with common computer algebra system, e.g. Mathematica or Maple runtime 1 month with 64GB RAM We calculated M 1 ji and all Planar double boxes diagrams Non Planar double boxes: a challenge! Mohammad Assadsolimani From Tensor Integral to IBP 24

58 Status of the calculation for single Top Quark Production Topology # Diagrams reduction performed checks Vertex corrections 29 Planar double boxes 6 work in progress Non Planar double boxes 12 work in progress There are two most complicated topologies, which could not be reduced completely until now : t t W W t t t t Mohammad Assadsolimani From Tensor Integral to IBP 25

59 Conclusions We have seen two possibilities to reduce tensor integrals to scalar integrals The choice of reductions method determines how difficult the next step (IBP) is As a test of our setup, we have calculated the O(α 2 s) contributions to the Heavy Quark Vector and Axial Vector Form Factors, confirming the results of Bernreuther et al. and Gluza et al. We have also calculated the two loop vertex corrections to single Top Quark Production Mohammad Assadsolimani From Tensor Integral to IBP 26