A novel integral representation for the Adler function

Abstract

Abstract. New integral representations for the Adler D-function and the R-ratio of the electron-positron annihilation into hadrons are derived in the general framework of the analytic approach to QCD. These representations capture the nonperturbative information encoded in the dispersion relation for the D-function, the effects due to the interrelation between spacelike and timelike domains, and the effects due to the nonvanishing pion mass. The latter plays a crucial role in this analysis, forcing the Adler function to vanish in the infrared limit. Within the developed approach the D-function is calculated by employing its perturbative approximation as the only additional input. The obtained result is found to be in reasonable agreement with the experimental prediction for the Adler function in the entire range of momenta 0 ≤ Q 2 < ∞.

Electron–positron annihilation into hadrons at the higher-loop levels

“…(11) hereinafter, which makes Eq. (10) identical to that of both the foregoing DPT [31][32][33][34] and the so-called analytic perturbation theory 2 (APT) [54][55][56][57]. It has to be noted that, in general, the perturbative spectral function at small values of its argument may be altered by the terms of an intrinsically nonperturbative nature.…”

“…In turn, the relations (2)-(7) impose a number of strict physical inherently nonperturbative constraints on the functions Π(q 2 ), R(s), and D(Q 2 ), which should definitely be taken into account when one intends to go beyond the limits of applicability of the QCD perturbation theory. It is worthwhile to note that these nonperturbative restrictions have been merged with the corresponding perturbative input in the framework of dispersively improved perturbation theory (DPT) [31][32][33][34] (its preliminary formulation was discussed in Refs. [35][36][37][38]).…”

“…For this purpose the effects due to the masses of the involved particles can be safely neglected (a discussion of the impact of such effects 1 on the low-energy behavior of the functions Π(q 2 ), R(s), and D(Q 2 ) can be found in, e.g., Refs. [31][32][33][34][45][46][47][48][49][50][51][52][53]). Additionally, for the schemedependent perturbative coefficients β j and d j the MS-scheme will be assumed and for the uncalculated yet five-loop coefficient d 5 its numerical estimation [42] will be employed.…”

“…Starting from the third order of perturbation theory (i.e., for ≥ 3) the coefficients δ j (32) are no longer vanishing and constitute a combination of the pertinent perturbative expansion coefficients d j and β j of the first ( − 2) orders. Specifically, the third-order and fourth-order coefficients read [42,118,119] …”

“…[31]. Table 2 presents the numerical values of the first seven coefficients δ j (32), which embody the contributions of the relevant π 2 -terms (30). Tables 1 and 2 make it evident that in Eq.…”