Search for short strings in~$e^+e^-$-annihilation

In this work the behavior of power corrections to Adler finction in operator product expansion (OPE) is studied, in particular the possible contribution of operator of dimension 2. The OPE terms of dimension 4 and 6 are taken into account. Various experimental data on reactions of $e^+e^-$-annihilation to pion channels ($\pi^+\pi^-$, $2\pi^+2\pi^-$, $\pi^+\pi^-2\pi^0$, $3\pi^+3\pi^-$, $2\pi^+2\pi^-2\pi^0$) are used. The high-precision fits of the experimental data are obtained and used. The method based on Borel transform of Adler function is applied. It is shown that the contribution of operator of dimension 2 is negative being compatible to zero at three standard deviations level. The strong (anti)corellation between short string and local gluon condensate is found.


Introduction
The problem of the existence of the operator with dimension 2, whose contribution to QCD sum rules [1] is proportional to 1/Q 2 , and search for relevant OPE corrections are performed already for quite a long time [2,3,4].
In the pioneering paper [2] a concept of short strings leading to corrections with dimension 2 was suggested. In Cornell potential [5] V (r) ≈ − 4α s (r) 3r + kr the term kr describes string potential (connected to the phenomenon of confinement) at short distances, leading 1 to the correction ∼ k/Q 2 . In paper [3] the correlation between short strings and perturbation theory order was established 2 . The contribution of the operator with dimension 2 to the e + e − data was studied later [4,7] and it was found to be compatible to zero with large errors. Our analysis is based on the use of Adler function being a simple two-point correlator which is convenient to compare to experimental data. We use a large number of them and perform an accurate numerical analysis. It consists of construction of the fitting model, calculating R-ratio and D-function in dispersional form, applying Borel transform (BT), construction of sum rule and extraction of the OPE coefficients.
The main purpose is to verify whether the operator with dimension 2 exists.
For description of the data on squared pion form-factor |F π (s)| 2 depending on energy √ s of reaction e + e − → π + π − we use the three-resonance (Gounaris-Sakurai) model [8], [27], where the form-factor of each resonance V is calculated using Breit-Wigner formula: The full pion form-factor with resonances ρ, ω and ρ has the following expression: Performing the χ 2 -minimisation we get the fit of the experimental data on |F π | 2 (s) taken from work For cross-sections of the processes e + e − → 2π + 2π − , e + e − → π + π − 2π 0 , e + e − → 2π + 2π − 2π 0 , and e + e − → 3π + 3π − the description in the form of sum of three Gaussian curves, describing wide resonances, is assumed: The results are shown in table 1.

R-ratio, Adler function and sum rule
The full R-ratio is the sum of R-ratios of particular processes. R(s) = i σ e + e − →hadrons of type i (s) σ e + e − →µ + µ − (s) .  Extracting the current with isospin I = 1 (j µ = 1 2 (ūγ µ u −dγ µ d)) from full electromagnetic current, we obtain the formula for D-function with isospin I = 1 in the framework of OPE (for example, see work [29], eq. 3.4): where a 2n are OPE coefficients, we take into account three first power corrections. The another form of D-function, dispersional, is the following: where R exp (s) is our fitting result (shown by black solid curve on Fig. 1) and R th (s) is the one-loop approximation of perturbative QCD (shown by red dashed curve on Fig. 1). The continuum threshold √ s 0 = 1.57 GeV is chosen to guarantee that R exp (s) and R th (s) and even their first derivatives take similar values (under the statistical uncertainties of experimental data), as it is seen on Fig. 1 (vertical dashed line). The function R exp (s) decreases above √ s 0 = 1.57 GeV, which is explained by the absence of the data on e + e − -annihilation to 8π channels. Equating the both forms of Adler function: by OPE (1) and dispersional (2), and applying BT, we get the sum rule:

The sum rule analysis
Finally, let us turn to the discussion of obtained sum rule (3). The Borel mass M 2 is varied in interval 0.75 ÷ 4 GeV 2 which is divided to 20 points, then the coefficients C 2 and C 4 are determined by χ 2 -minimisation, and C 6 = − 448 π 3 27 α s qq 2 = −0.121 GeV 6 , which can be expressed in terms of quark condensate [29], is fixed [30]. The coefficient C 4 can be expressed in terms of gluon condensate C 4 = 2π 2 3 αs GG π [1]. Our result is shown on Fig. 2; the regions corresponding to 1, 2 and 3 σ-levels are marked in red, blue and yellow. The minimal value is χ 2 min = 0.483, and corresponding gluon condensate and C 2 are found to be: (α s /π) GG = 0.025 GeV 4 , C 2 = −0.086 GeV 2 . The allowed intervals for C 2 and gluon condensate at one σ-level are: At one σ-level our results do not contradict with the well-known results [29,30,31], but at the same time C 2 is not compatible with zero. One can see that C 2 = 0 is possible at 3 σ-level.
Furthermore, by the form of regions corresponding our results (see Fig. 2) we can suppose that there is some (anti)correlation between C 2 and C 4 . A rough estimation of this connection is expressed by formula: One can compare the received values with the existing values of the gluon condensate [29,31,30].

Conclusions and outlook
The resonance contribution fitting model is developed, the Adler function with Borelization is obtained and its precise numerical analysis is performed. C 2 has negative sign and is compatible with zero only at 3 σ-level. At 1 σ-level: C 2 = −0.086 ± 0.050 GeV 2 , α s /π GG = 0.025 ± 0.012 GeV 4 . Strong (anti)correlation between short strings and local gluon condensate is found: We plan to perform an analogous analysis in the framework of the analytic perturbation theory [32,33,34]. O. P. Solovtsova for useful discussions and comments. M. K. is thankful to Yu. L. Ryzhykau for valuable advices and assistance in numerical analysis. The work is supported in part by RFBR Grant-140100647.