1. Unstable-particles pair production in modified perturbation theory in NNLO M.L.Nekrasov IHEP, Protvino

2. The problem: Modified perturbation theory (MPT) implies direct expansion of the cross-section in powers of the coupling constant A description of productions and decays of fundamental unstable particles for colliders subsequent to LHС ( ILC) generally should be made with NNLO accuracy To clear up this question, I do numerical simulation in the MPT up to the NNLO Gauge invariance should be maintained Accuracy of description of resonant contributions = ? Existing methods: DPA, CMS,... NLO Pinch-technique method huge volume of extra calculations (i) gauge cancellations and unitarity; (ii) enough high accuracy of computation of resonant contributions / M.Nekrasov, arXiv:0912.1025 / with the aid of distribution-theory methods

3. e e + - S s1s1 s2s2 Pair production and decay, double-resonant contributions MPT expansion of the cross-section in powers of 

4. Asymptotic expansion of BW factors in powers of  / F.Tkachov,1998 / NNLO : Basic ingredients of MPT Analytic regularization of the kinematic factor Conventional-perturbation-theory for “test” function Φ Polynomial in  analytic calculation of “singular” integrals Taylor in  / M.Nekrasov,2007 / 3-loop

5. Coefficients c n () NNLO: OMS : (pole scheme) / M.Nekrasov, 2002 / / B.Kniel & A.Sirlin, 2002 / Unitarity: OMS conventional : M – observable (pole) mass gauge invariant

6. Singular integrals, scheme of calculations Dimensionless variables at given n 1 and n 2 : at = 1/2

7. Numerical calculations & estimate of errors Fortran code with double precision Linear patches for resolving 0/0-indeterminacies ( x/x, x 2 /x 2, …) Simpson method for c alculating absolutely convergent integrals ( relative accuracy  0 = 10 - 5 ) additional errors: (1) due to patches themselves  1   2 ” 0 / 0 (2) due to the loss of decimals near indeterminacy points: x/x:  = 10 -N  2   - actual size of the patch integrand x 2 /x 2 :  2 = 10 -N  2  N = 8 at D = 15 double precision minimization of errors  1   2 (numerical estimate) Overall error:  =  0   1   2  10 -3 NNLO  22 11   1   2...

8. Specific model for testing MPT Universal soft massless-particles contributions : Flux function in leading-log approximation: Test function Φ : Coulomb singularities through one-gluon exchanges: e + e - , Z t t W + b W - b Born approximation Breigt-Wigner factors :  =  1 +  2  2 +  3  3 three-loop contributions to self- energy __

9. Results  s [ GeV]  [pb]  LO  NLO  NNLO __ __ Total cross-section  (s) : 500 0.6724 0.5687 0.6344 0.6698(7) 1000 0.2255 0.1821 0.2124 0.2240(2) 100% 84.6% 94.3% 99.6(1)% 100% 80.8% 94.2% 99.3(1)% M t = 175 GeV M W = 80.4 GeV M b = 0

10. Conclusion MPT stably works at the energies near the maximum of the cross-section and higher The higher precision is possible if proceeding to NNNLO or if using NNLO of MPT for calculation loop contributions only (on the analogy of actual practice of application of DPA) In the case of pair production and decays of unstable particles: Comment: At ILC energies NNLO in MPT provides a few per mile accuracy The existence of the MPT expansion practically has been shown (working FORTRAN code up to NNLO is written)