# ACAT 2019

10-15 March 2019
Steinmatte conference center
Europe/Zurich timezone

## Forgotten (or not) possibility of the application of the least squares method: the analysis of the computer-based numerical results to QCD $\rm{\overline{MS}}$-on-shell mass relation at the four-loop level

Not scheduled
25m
Steinmatte conference center

#### Steinmatte conference center

Hotel Allalin, Saas Fee, Switzerland https://allalin.ch/conference/
Oral Track 3: Computations in Theoretical Physics: Techniques and Methods

### Speaker

Dr Andrei Kataev (Institute for Nuclear Research of RAS, Moscow )

### Description

The set of the four-loop numerical results for the relation between pole and running heavy quarks masses in QCD at fixed number of lighter flavors $3\leq n_l\leq 15$, which was obtained in Ref.[1] with help of the Lomonosov Supercomputer of Moscow State University, is analysed by the ordinary method of the least squares. We use its variant which allows to solve the overdetermined system of 13 linear equations and to define 4 coefficients of the polynomial of the third power in $n_l$ in the expression for the four-loop correction to the QCD $\rm{\overline{MS}}$-on-shell mass relation with corresponding uncertainties of this approach. The central values of these terms are consistent with a high degree of accuracy with the results obtained in Ref.[1]. To demonstrate the stability of the least squares method to the number of equations we also consider the situation when the number of equations is equal to 3 at $3\leq n_l\leq 5$. It is interesting that in both cases the central values of all unknown terms coincide with the previously obtained two unknown in analytical form values at $3\leq n_l \leq 15$ [2,3] while the uncertainties increase no more than 10 times. Thus the least squares method allows to check for self-consistency the results of numerical computations [1] and analytical calculations [4], obtained using different methods of evaluation of Feynman diagrams and of advanced computer-oriented programs.

References.

[1] P.~Marquard, A.~V.~Smirnov, V.~A.~Smirnov, M.~Steinhauser and D.~Wellmann,
$\overline{\rm MS}$-on-shell quark mass relation up to four loops in QCD and a general SU$(N)$ gauge group,''
Phys.\ Rev.\ D {\bf 94}, no. 7, 074025 (2016)[arXiv:1606.06754 [hep-ph]].

[2] A.~L.~Kataev and V.~S.~Molokoedov,On the flavour dependence of the $\mathcal{O}(\alpha_s^4)$ correction to the relation between running and pole heavy quark masses,'' Eur.\ Phys.\ J.\ Plus {\bf 131}, no. 8, 271 (2016)
[arXiv:1511.06898 [hep-ph]].

[3] A.~L.~Kataev and V.~S.~Molokoedov,
Multiloop contributions to the $\overline{\rm{MS}}$-on-shell mass relation for heavy quarks in QCD and charged leptons in QED and the asymptotic structure of the perturbative QCD series,'' arXiv:1807.05406 [hep-ph].

[4] R.~Lee, P.~Marquard, A.~V.~Smirnov, V.~A.~Smirnov and M.~Steinhauser,
Four-loop corrections with two closed fermion loops to fermion self energies and the lepton anomalous magnetic moment,'' JHEP {\bf 1303}, 162 (2013)
[arXiv:1301.6481 [hep-ph]]

### Primary authors

Dr Andrei Kataev (Institute for Nuclear Research of RAS, Moscow ) Mr Victor Molokoedov (Moscow Institute of Physics and Technology , Dolgoprudny )

### Presentation materials

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