###### Abstract

Here, we report briefly two topics:

1) The latest version of “Analytic Perturbation Theory” (APT) devised recently for the QCD observables both in the Euclidean and Minkowskian regions.

2) Results of the APT–based calculation for some physical processes.

Analytic Perturbation Theory and New Analysis of some QCD observables

D.V. Shirkov

Bogoliubov Laboratory, JINR, 141980 Dubna, Russia

e-address:

## 1 The APT — a closed theoretical scheme

The new APT version [1, 2] mutually relates two ghost–free formulations of modified perturbation expansions for observables.

The first one, initiated about two decades ago [3, 4], changes the standard power expansion in the time-like region

into a nonpower one

Here, is a common, e.g., 3–loop QCD invariant/running coupling (usually in the scheme – see eq.(9.5a) in Ref. [5]); and , some integral images of the powers:

(1) |

The operation is a reverse to the one defined by the “Adler relation”

(2) |

and transforming a real function of a positive (time–like) argument into a real function of a positive (space–like) argument.

By operation one can define [3, 4, 6] the RG–invariant effective coupling in the time–like region. A few simple examples are in order :

— Square and cube of transform into simple expressions [3, 4]

(related by differential operation ) which are not powers of

By applying to one can “try to return” to the Euclidean domain. However, instead of powers, we arrive at some other functions analytic in the cut -plane and free of ghost singularities. At the one–loop case

(4) |

These expressions have been first obtained by other means [8, 9] at mid–90s. The first function an invariant Euclidean coupling, should now be treated as a counterpart of the invariant Minkowskian coupling [6] Both and are real monotonically decreasing functions with the same maximum value in the IR limit. All higher functions vanish, in this limit. For they oscillate in the IR region.

The same properties remain valid for a higher–loop case. Explicit expressions for and at the two–loop case can be written (see, Ref. [10]) in terms of a special Lambert function. They are presented in Figs 1a and 1b.

Here, in Fig.1a, by the dotted line we give a usual two-loop effective QCD coupling with a pole at On the other hand, the dash–dotted curves represent the one-loop APT approximations (3a) and (4). The solid APT curves are based on the exact two-loop solutions of RG equations and approximate three–loop solutions in the scheme. Their remarkable coincidence (within the 1–2 per cent) demonstrates reduced sensitivity of the APT with respect to higher–loops effects in the whole Euclidean and Minkowskian regions from IR to UV limits. Fig.1b shows higher functions calculated at the two–loop case.

Remarkably enough, the mechanism of liberation of unphysical singularities is quite different. While in the space-like domain it involves nonperturbative, power in , structures, in the time-like region, it is based only upon resummation of the “ terms”. Figuratively, (non-perturbative !) analyticization [11] in the –channel can be treated as a quantitatively distorted reflection (under ) of (perfectly perturbative) –resummation in the –channel. This effect of “distorting mirror” first discussed in [12] is clearly seen in figures.

In a real case, the procedure of the threshold matching is in use. E.g., in the scheme with it defines a “global” function

continuous in the space-like region of positive values with discontinuity of derivatives at matching points. To this there corresponds a discontinuous spectral density

(5) |

with which yields the smooth global Euclidean and spline–continuous global Minkowskian expansion functions

(6) |

## 2 The APT applications.

To illustrate a qualitative difference between our global APT scheme and common practice of data analysis, we consider a few examples.

In the usual treatment — see, e.g., Ref. [5] — the (QCD perturbative part of) Minkowskian observable, like annihilation or decay, is presented in the form

(7) |

Here, coefficients

(8) |

with reasonably decreasing coefficients and the mentioned term of being “swallowed” by

In the Euclidean channel, instead of a power expansion similar to (7), we typically have

(9) |

Here, the modification is related to nonperturbative structures like in (4).

Table 1 : Relative contributions (in %) of 1– , 2– and 3–loop terms to observables

Process | PT | APT | ||||

GLS sum rule | 65 | 24 | 11 | 75 | 21 | 4 |

Bjorken. s.r. | 55 | 26 | 19 | 80 | 19 | 1 |

Incl. -decay | 55 | 29 | 16 | 88 | 11 | 1 |

hadr. | 96 | 8 | -4 | 92 | 7 | .5 |

(at 10 GeV) | ||||||

hadr. | 98.6 | 3.7 | -2.3 | 96.9 | 3.5 | -.4 |

In Table 1, we give values of the relative contribution of the first, second, and third terms of the r.h.s. in (7),(8) and (9) for Gross–Llywellin-Smith [13] and Bjorken [14] sum rules, – decay in the vector channel [15] as well as for and inclusive cross-sections. As it follows from this Table, in the APT case, the three–loop (last) term is very small, and being compared with data errors, numerically unessential. This means that, in practice,

Using this conclusion as a hint, we reanalyzed data in the five–flavor region. Results are presented in Table 2.

Table 2 : The APT revised part () of Bethke’s [16] Table 6

Figures in brackets give the difference in the last digit between APT and common values.

loops | (s) | (s) | ||||
---|---|---|---|---|---|---|

Process | GeV | No | ref.[2] | ref.[2] | APT | APT |

-decay [5] | 9.5 | 2 | .170 | .114 | .182 | .120 (+6) |

10.5 | 3 | .200 | .130 | .198 | .129(-1) | |

22.0 | 2 | .161 | .124 | .166 | .127(+3) | |

35.0 | 2 | .145 | .123 | .149 | .126(+3) | |

42.4 | 3 | .144 | .126 | .145 | .127(+1) | |

44.0 | 2 | .139 | .123 | .142 | .126(+3) | |

58 | 2 | .132 | .123 | .135 | .125(+2) | |

had. | 91.2 | 3 | .124 | .124 | .124 | .124 (0) |

91.2 | 2 | .121 | .121 | .123 | .123(+2) | |

-”- | …. | 2 | … | … | … | … (+2) |

189 | 2 | .110 | .123 | .112 | .125(+2) |

Averaged values =

Addressing the reader interested in a more detail to our recent paper [17], we shortly comment that transition PT APT from the standard algorithm to our new one for the NLO case, as a rule, enlarges extracted values by 0.002 (or more) which results in the averaged value equal to 0.124.

At the same time it improves the correlation of events in the region. More specifically, it changes the value from 0.197 to 0.144.

## 3 Summary.

1. First, we have outlined a recently devised self-consistent scheme for analyzing data both in the space-like and time-like regions. Within this APT scheme, perturbative expressions for an observable involve expansions over the sets and that are nonpower series, free of unphysical singularities, with usual numerical coefficients obtained by calculation of the relevant Feynman diagrams.

2. Numerically, the APT calculations reveal reduced sensitivity to the NNLO effects, as it has been demonstrated in Table 1.

Table 2 summarizes our attempt to “improve” some particular data for values extracted from experiments in the Minkowskian five-flavour region. These results look encouraging. In particular, they yield the new value

quite different from the widely accepted “world average” 0.118 and even from the usual average () over the five–flavour region.

3. This result, being taken as granted, rises a physical question on mutual consistency of current data on the QCD–invariant coupling behavior in the “medium ()” and “high ” regions.

Answer to this question could be obtained by a further revised (within the APT technique) calculation.

## Acknowledgments

This work was partially supported by grants of the Russian Foundation for Basic Research (RFBR projects Nos 99-01-00091 and 00-15-96691), by INTAS grant No 96-0842 and by INTAS-CERN grant No 2000-377.

## References

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