Документ взят из кэша поисковой машины. Адрес оригинального документа : http://qfthep.sinp.msu.ru/talks2011/nekrasov.pdf Дата изменения: Wed Oct 5 14:32:12 2011 Дата индексирования: Mon Oct 1 19:39:36 2012 Кодировка:

Results of numerical simulations for pair production of unstable particles in MPT in NNLO
M.L.Nekrasov
IHEP, Protvino


The problem:
A description of productions and decays of fundamental unstable particles
for colliders subsequent to LH (

generally should be made with NNLO accuracy
(i) gauge cancellations and unitarity; (ii) enough high accuracy of computation of resonant contributions

ILC)

Existing methods:
· Pole expansion/DPA:
Laurent expansion around complex poles + conventional PT for residues / LEP1, LEP2 /

· Complex mass scheme (CMS): complex-valued renormalized mass

NLO

complex-valued Weinberg angle etc.

/ A.Denner, S.Dittmaier, M.Roth, etc. / /J.Papavassiliou, A.Pilaftsis, D.Binosi, etc. /

· Pinch-technique method huge volume of extra calculations
Modified perturbation theory (MPT):

direct expansion of the cross-section in powers of the coupling constant with the aid of distribution-theory methods Asymptotic expansion in gauge invariance should be maintained The accuracy of description of resonant contributions = ?
To clear up this question, I do numerical simulation in the MPT up to the NNLO

/

M.Nekrasov, Mod.Phys.Lett. A 26 (2011) 223,1807

/

2


The problem:
A description of productions and decays of fundamental unstable particles
for colliders subsequent to LH (

generally should be made with NNLO accuracy
(i) gauge cancellations and unitarity; (ii) enough high accuracy of computation of resonant contributions

ILC)

Existing methods:
· Pole expansion/DPA:
Laurent expansion around complex poles + conventional PT for residues / LEP1, LEP2 /

· Complex mass scheme (CMS): complex-valued renormalized mass

NLO

complex-valued Weinberg angle etc.

/ A.Denner, S.Dittmaier, M.Roth, etc. / /J.Papavassiliou, A.Pilaftsis, D.Binosi, etc. /

· Pinch-technique method huge volume of extra calculations
Modified perturbation theory (MPT):

direct expansion of the cross-section in powers of the coupling constant with the aid of distribution-theory methods Asymptotic expansion in gauge invariance should be maintained The accuracy of description of resonant contributions = ?
To clear up this question, I do numerical simulation in the MPT up to the NNLO

/

M.Nekrasov, Mod.Phys.Lett. A 26 (2011) 223,1807

/

2


Pair production and decay, double-resonant contributions

e+

S
e-

s s

1

2

Expansion in powers of in the sense of distributions
3


Basic ingredients of MPT

·

Asymptotic expansion of BW factors in powers of

/ F.Tkachov,1998 /

Taylor in



Polynomial in


3-loop

NNL O :

·

Analytic regularization of the kinematic factor
/ M.Nekrasov,2007 /

analytic calculation of "singular" integrals

·

Conventional-perturbation-theory for "test" function
4


Coefficients cn()
NNLO:

OMS

conventional

:

OMS :

/ M.Nekrasov, 2002 / M = pole mass, gauge-invariant and scheme-independent (observable mass)

(pole scheme) / B.Kniel & A.Sirlin, 2002 /

Unitarity:
5


Singular integrals, scheme of calculations
Dimensionless variables

at given n1 and n2 :

at = 1/2

6


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
actual size of the patch integrand
0

(x/x, x2/x2, ...)

additional errors:
1: due to patches themselves
2

1 "0/

2: due to the loss of decimals near indeterminacy points:

x/x: x2/x2:
1 2 1 2

= 10-

N

2 2
double precision NNLO

2 = 10-

N

...
minimization of errors 1 2 (numerical estimate)

N = 8 at D = 15
-3

Overall error:

= 0 1 2 < 10

7


Specific models for testing MPT

·

Test function :
e+ e e+e
-

(, Z) (, Z)

tt
W +W
-

W+b W -b 4f

Born approximation

· ·

Breigt-Wigner factors :
= 1+ 22+ 3
3

three-loop contributions to self-energy

Universal soft massless-particles contributions :
Flux function in leading-log approximation:

Coulomb singularities through one-gluon/photon exchanges:

8


Results of calculations. The case of top-quarks
Total cross-section (s) :
Mt = 175 GeV MW = 80.4 GeV Mb = 0

s

[GeV]



[pb]



LO



NLO



NNLO

500 1000 1500

100%

0.6724 0.2255

84.6%

0.5687 0.1821

94.3%

0.6344 0.2124

99.6%

0.6698 0.2240

100% 100%

80.8% 77.3%

94.2% 93.8%

99.3% 99.2%

0.1122

0.0867

0.1053

0.1113

9


Results of calculations. The case of W-bosons
Total cross-section (s) :
MW = 80.40 GeV MZ = 91.19 GeV

3 00 0

s
200 500 1000 3000



[pb]



LO



NLO



NNLO

15.258
100% 100%

116.92% 109.09% 105.12% 91.33%

17.839

15.175

99.46%

15.235
99.85%

6.9355 2.8286 0.61023

7.5657

99.91%

6.9294 2.8263 0.60625

99.98%

6.9342 2.8285

100% 100%

2.9733

99.92%

100.00%

0.55733

99.35%

0.61026
100.00%

10


Conclusion
In the case of pair production and decays of unstable particles:
· The existence of MPT expansion practically has been shown
(working FORTRAN code up to NNLO is presented)

· MPT stably works at the energies near the maximum of the cross-section and higher · At ILC energies NNLO in MPT provides accuracy of
- a few 0.1%, in the case of top quarks *) - less than 0.1%, in the case of W bosons

MTV is a good candidate for support at the ILC the pair production and decay of fundamental unstable particles

*)

The higher precision is possible if proceeding to NNNLO or if using NNLO of MPT for calculati on of loop contributions only,
on the analogy of actual practice of application of DPA

11