Borel–Laplace sum rules with τ decay data, using OPE with improved anomalous dimensions

Abstract

We perform numerical analysis of double-pinched Borel-Laplace QCD sum rules for the strangeless semihadronic $\tau$-decay data. The $D=0$ contribution to the theoretical contour integral in the sum rules is evaluated by the (truncated) Fixed Order perturbation theory method (FO) and by Principal Value (PV) of the Borel integration. We use for the full Adler function the Operator Product Expansion (OPE) with the terms $\sim \langle O_D \rangle$ of dimension $D=2 n$ where $2 \leq n \leq 5$ for the (V+A)-channel, and $2 \leq n \leq 7$ for the V-channel data. In our previous works [1, 2], only the (V+A)-channel data was analysed. In this work, the analysis of a new set of V-channel data is performed as well. Further, a renormalon-motivated construction of the $D=0$ part of the Adler function is improved in the $u=3$ infrared renormalon sector, by involving the recently known information on the two principal noninteger values $k^{(j)}=\gamma^{(1)}(O_6^{(j)})/\beta_0$ of the effective leading-order anomalous dimensions. Additionally, the OPE of the Adler function has now the D=6 contribution with the principal anomalous dimension ($\sim \alpha_s^{k^{(1)}}$), and terms of higher dimension (with zero anomalous dimension). Cross checks of the obtained extracted values of $\alpha_s$ and of the condensates were performed by reproduction of the (central) experimental values of several double-pinched momenta $a^{(2,n)}$. The averaged final extracted values of the (MSbar) coupling are: $\alpha_s(m_{\tau}^2) = 0.3169^{+0.0070}_{-0.0096}$, corresponding to $\alpha_s(M_Z^2)=0.1183^{+0.0009}_{-0.0012}$.

Nucleon Resonance Masses from QCD Sum Rules

“…The delta's denote the contribution of the nucleon continuum, they receive no contribution from the nucleon pole [9]. The aim is to choose P(t) which minimizes 1 ∆ and…”

“…The QCD spectral functions have been obtained in Nasrallah NF, et al [7,8], Ayala C, et al [9] and Krasnikov V, et al [10] and are conveniently expressed in Gorishnii SG, et al [11],…”

The Euclidean Adler function and its interplay with $$ \Delta {\alpha}_{\textrm{QED}}^{\textrm{had}} $$ and αs

“…Z ) = 0.115, 0.120, at O(α 5 s ), adding as perturbative uncertainties the quadratic sum of variations due to changing K 5 in a conservative interval (−125, 675) and ξ 2 ∈ (0.5, 2) [36]. The range chosen for K 5 includes the values advocated by renormalon models, Padé approximants, effective charges and conformal mappings [56][57][58][59][60][61][62], but also allows for a correction of opposite sign. The interval of variation for the renormalization scale is conventional for low-energy analyses, e.g.…”

Resurgence analysis of the Adler function at order $1/N_f^2$

“…There has been much earlier work on renormalon contributions to the Adler function [6][7][8][9][10][11][12][13], as well as their connection to its operator production expansion [14,15]. The Adler function appears in more recent renormalon studies [16][17][18][19][20], including a study whether field theories can have renormalons beyond the usual first Borel plane [21]. The Adler function has recently been studied using resurgence techniques [22,23], based on renormalization group considerations [24,25].…”