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Unstable-particles pair production in modified perturbation theory in NNLO

Unstable-particles pair production in modified perturbation theory in NNLO. M.L.Nekrasov. IHEP, Protvino. The problem:. A description of productions and decays of fundamental unstable particles for colliders subsequent to LH С ( ILC) generally should be made with NNLO accuracy.

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Unstable-particles pair production in modified perturbation theory in NNLO

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

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

  3. Pair production and decay, double-resonant contributions + e s1 S s2 - e MPT expansion of the cross-section in powers of 

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

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

  6. Singular integrals, scheme of calculations Dimensionless variables at given n1 and n2 : at=1/2

  7. Numerical calculations & estimate of errors •Fortran code with double precision •Simpson method for calculating absolutely convergent integrals (relative accuracy 0= 10-5) •Linear patches for resolving 0/0-indeterminacies (x/x, x2/x2, …) - actual size of the patch additional errors: (1)due to patches themselves 12 ”0/ 0 (2) due to the loss of decimals near indeterminacy points: integrand x/x:  = 10-N 2 x2/x2: 2= 10-N 2  ... 1 2 minimization of errors 1  2(numerical estimate) N = 8at D = 15 double precision 1 2  Overall error:  =0  1  2  10-3 NNLO

  8. Specific model for testing MPT • Test function Φ : _ _ 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 • Universal soft massless-particles contributions : Flux function in leading-log approximation: Coulomb singularities through one-gluon exchanges:

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

  10. Conclusion In the case of pair production and decays of unstable particles: •The existence of the MPT expansion practically has been shown (working FORTRAN code up to NNLO is written) •MPT stably works at the energies near the maximum of the cross-section and higher • At ILC energies NNLOin MPT provides a few per mile accuracy Comment: •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)

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