Measurement of the e+e− → π+π− process cross section with the SND detector at the VEPP-2000 collider in the energy region 0.525 <s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s} $$\end{document}< 0.883 GeV

The cross section of the process e+e− → π+π− has been measured in the Spherical Neutral Detector (SND) experiment at the VEPP-2000 e+e− collider VEPP-2000 in the energy region 525 <s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s} $$\end{document}< 883 MeV. The measurement is based on data with an integrated luminosity of about 4.6 pb−1. The systematic uncertainty of the cross section determination is 0.8% at s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqrt{s} $$\end{document}> 0.600 GeV. The ρ meson parameters are obtained as mρ = 775.3 ± 0.5 ± 0.6 MeV, Γρ = 145.6 ± 0.6 ± 0.8 MeV, Bρ→e+e− × Bρ→π+π− = (4.89 ± 0.02 ± 0.04) × 10−5, and the parameters of the e+e− → ω → π+π− process, suppressed by G-parity, as Bω→e+e− × Bω→π+π− = (1.32 ± 0.06 ± 0.02) × 10−6 and and ϕρω = 110.7 ± 1.5 ± 1.0 degrees.


Introduction
SND [1,2] is a general purpose nonmagnetic detector operating at the VEPP-2000 e + e − collider in the center-of-mass energy range from 0.2 to 2.0 GeV [3]. Experimental studies include measurements of the cross sections of the e + e − annihilation processes into hadrons. These measurements are largely motivated by the need for high-precision calculation of the hadronic contribution to the anomalous magnetic moment of the muon (g − 2)/2 [4]. In particular, the e + e − → π + π − cross section in the energy region below 1 GeV gives the dominant contribution to this value and should be measured with accuracy better than 1% [5].
The cross section of the e + e − → π + π − process in the energy region √ s < 1000 MeV can be described within the vector meson dominance model (VMD) framework and is determined by the transitions V → π + π − of the light vector mesons (V = ρ, ω, ρ , ρ ). The main contribution in this energy region comes from the ρ → π + π − and from the Gparity violating ω → π + π − transitions. Studies of the e + e − → π + π − reaction allow us to determine the ρ and ω meson parameters, provide information on the G-parity violation mechanism and ρ, ρ , ρ mixing [6].

Experiment
The SND is operated at the VEPP-2000 collider since 2010 till present day. It consists of a tracking system based on cylindrical drift and proportional chambers placed in a common gas volume, aerogel threshold counters [35], a three-layer spherical electromagnetic calorimeter based on NaI (Tl) crystals and a muon system which includes two layers of proportional tubes and scintillation counters (figure 1). The calorimeter energy and angular resolutions depend on the photon energy E as σ E /E(%) = 4.2%/ 4 E(GeV) and σ φ,θ = 0.82 • / E(GeV) ⊕ 0.63 • . Its total solid angle is 95% of 4π. The solid angle of the tracking system is 94% of 4π. Its angular resolution is 0.45 • and 0.8 • for the azimuthal and polar angles, respectively. The threshold Cherenkov counters are based on aerogel with the refractive index of 1.05. The threshold momenta for e/µ/π are approximately equal to 1.6/330/436 MeV/c, respectively. This system covers 60% of the total solid angle.
The VEPP-2000 collider beam energy is determined using a beam-energy-measurement system based on the Compton back-scattering of laser photons on the electron beam. The accuracy of the beam-energy measurement is about 30 keV [36,37].

Analysis
The cross section of the process e + e − → π + π − is measured as follows.
2. The selected events are sorted into the two classes: e + e − and π + π − , µ + µ − using the energy depositions in the calorimeter crystals.
3. The luminosity is determined from the number of e + e − → e + e − events: Here N ee , ε ee and σ ee are the number of events, detection efficiency and cross section of the process e + e − → e + e − respectively. To obtain the number of e + e − → π + π − events, the number of e + e − → µ + µ − events is calculated using theoretical cross section as and then subtracted from the total number of π + π − and µ + µ − events. Here ε µµ and σ µµ are the detection efficiency and cross section of e + e − → µ + µ − , respectively.
4. The Born cross section of the process e + e − → π + π − is calculated using formula: Here 1 + δ r is a radiative correction, N ππ and ε ππ are the number of events and the detection efficiency for the process e + e − → π + π − .
The detection efficiency for each process is derived from the Monte Carlo simulation based on GEANT4 [38,39]. Apparatus effects such as electronics noise, signal pile-up, actual time and amplitude resolutions of electronics channels, the bad channels are taken into account in the simulation.
Generation of e + e − → e + e − , µ + µ − and π + π − events is performed by the MCGPJ [40] generator. It is based on formulae from [41,42]. The generator takes into account initial and final state radiation (ISR and FSR), as well as Coulomb interaction in the final state. It allows one to calculate cross sections and radiative corrections with accuracy σ rad = 0.2%. The simulation of the process e + e − → e + e − is performed with the cut on the polar angles of the final electron and positron 30 • < θ e ± < 150 • .
The e + e − → e + e − , µ + µ − and π + π − events have different distributions of the energy deposition over calorimeter crystals. In e + e − → e + e − events the electrons and positrons produce electromagnetic showers, with the most probable energy losses of about 0.92 of the initial particle energy. Muons lose their energy by ionization of the calorimeter material through which they pass. The charged pions lose energy due to ionization and nuclear interaction with the detector material. The separation parameter of e + e − → e + e − and  Figure 2. The distribution of the separation parameter R for all collinear events (e + e − → e + e − , π + π − and µ + µ − ) at the energy √ s =778 MeV. The insert depicts the same histograms in the region between the peaks. Dots -experiment, histogram -simulation. Histogram for MC simulation is sum of distributions for e + e − → µ + µ − , e + e − and π + π − events. The contribution of each process to the histogram was calculated according to cross sections used in MCGPJ generator [40].
e + e − → π + π − events (R) in the energy region √ s = 0.5-1.0 GeV is based on the differences in the energy deposition profiles. It was developed using machine learning method [43]. The distribution of the separation parameter R is shown in figure 2. The e + e − → e + e − events are located in the region R < 0, while e + e − → π + π − , µ + µ − events are located at R > 0.

Events selection
During the data taking, the first-level trigger selects events with one or more tracks in the drift chamber and with the total energy deposition in the calorimeter greater than 100 MeV. During processing of the experimental data, event reconstruction is performed [1]. For the further analysis the collinear events are selected using the following criteria.
1. The number of charged particles N cha ≥ 2. An event can also contain additional neutral particles due to beam background, nuclear interaction of charged pions, splitting of electromagnetic showers and initial and final state radiation.
where θ 1,2 and φ 1,2 are the polar and azimuthal angles of charged particles with the largest energy deposition (particles in the event are ordered by the energy deposition), respectively.
5. |r 1,2 | < 1 cm, where r i is the distance between the track of the ith particle and the beam axis.
6. |z 1,2 | < 8 cm, where z i is the coordinate of the ith particle vertex (point of the track closest to the beam axis) along the beams axis.
7. The muon system veto is used for suppressing the cosmic background.
3.2 Subtraction of e + e − → π + π − π 0 and cosmic background In the event sample selected under these conditions, one has e + e − → e + e − , π + π − , µ + µ − events, residual cosmic background, and a small contribution from e + e − → π + π − π 0 reaction at √ s ≈ m ω . The number of background events from the process e + e − → π + π − π 0 is estimated as where M 3π is a number of simulated e + e − → π + π − π 0 events selected using the nominal conditions for collinear events, described above, n 3π and m 3π are the number of data and simulated events, respectively, selected under conditions: 2. The number of neutral particles N neu ≥ 2.
is the χ 2 of the kinematic fit of the event under e + e − → π + π − π 0 hypothesis.
It is found that the e + e − → π + π − π 0 background is maximal in the energy point √ s = 782.9 MeV, where its fraction is less than 0.15%, corresponding to 37 background events.
The cosmic events are suppressed by the muon system. The z coordinate distribution of the production point for collinear events is shown in figure 3. The e + e − annihilation events have a Gaussian distribution peaked at z = 0, while the cosmic distribution is nearly uniform. As figure 3 shows, the muon subsystem veto (veto = 1) separates cosmic muons from the e + e − annihilation events.
The number of the residual cosmic events is estimated as follows where N veto=1 data is the number of collinear events selected using the nominal selection criteria, but with veto = 1, N veto=0 cosm and N veto=1 cosm are the numbers of cosmic events with veto = 1 and veto = 0, respectively. Two types of cosmic events are used: 1. Collinear events with additional cuts: |r 1,2 | > 0.5 cm and |z 1,2 | > 5 cm.

Events recorded in special cosmic runs satisfying the nominal selection criteria.
In both cases, the ratio N veto=0 cosm /N veto=1 cosm is found to be equal to 2.5% ± 0.1%.

Detection efficiency
The ∆φ and ∆θ distributions for the e + e − → e + e − and e + e − → π + π − events are shown in figures 4, 5, 6 and 7. There are small differences in the shapes of the data and simulated spectra.
The following values are used as a measure of the systematic uncertainty due to the ∆θ and ∆φ cuts: Here  1.8 Figure 6. The ∆φ distribution for e + e − → π + π − events at √ s = 778 MeV. The solid histogram represents simulation, while the dotted histogram shows data. Their ratio depicted below. The ratio of the θ 0 distributions for the e + e − → π + π − and e + e − → e + e − events is shown in figure 8. There are some differences between these ratios for data and simulated distributions. To estimate the systematic error due to the θ 0 cut, the following ratio is used: are the numbers of e + e − → π + π − and e + e − → e + e − events summed over all energy points in experiment and simulation with θ x < θ 0 < 180 • − θ x . The largest deviation of δ θ 0 from unity is equal to 0.005 ( figure 9). This value is taken as a systematic error σ θ associated with 50 • < θ 0 < 130 • cut. Imperfection in simulation of pion nuclear interactions implies that the cut on the particle energy deposition leads to an inaccuracy in the detection efficiency of the e + e − → π + π − process. To take this inaccuracy into account, the detection efficiency is multiplied by the correction coefficient. The correction coefficient is obtained by using pseudo ππ events, which are constructed using events of the processes e + e − → ω(φ) → π + π − π 0 and e + e − → π + π − [43]. The corrections obtained using different types of pseudo events differ less than 0.005, they do not depend on the pion energy and their average is equal to 0.992. As a result, the correction coefficient is set equal to 0.992, and the difference is taken as a systematic error σ E = 0.005.
In the tracking system, the particle track can be lost due to reconstruction inefficiency. The probabilities ε data ππ and ε data ee to find two tracks in the e + e − → π + π − and e + e − → e + e − events are determined using experimental data. Their ratio to probabilities derived from simulated events can vary significantly in the different energy points. But the ratio R ππ /R ee , which contributes to the measured cross section, is energy independent and equal to unity with error 10 −4 ( figure 10). Pions can be lost due to the nuclear interaction in the detector material before the tracking system. The probability of pion loss is studied using e + e − → π + π − π 0 events. It was found that the difference between these values in data and simulation is 0.002, which is taken as a systematic error σ nucl = 0.2%. The use of the muon system veto for event selection (veto = 0) leads to inaccuracy in the determination of the measured cross section due to the uncertainty in the simulation of the muons and pions traversing the detector. To obtain the necessary corrections, the events close to the median plane 0 where the cosmic background is minimal, are used. The correction is the ratio of the e + e − → π + π − cross sections measured with (veto = 0) and without (veto ≥ 0) using the muon system: In the case of veto ≥ 0, a contribution of the residual cosmic muons background is estimated from the fit to the (z 1 + z 2 )/2 spectrum with a sum of the Gaussian and uniform distributions. The δ veto does not dependent on energy and its average value is consistent with 1 ( figure 11). This indicates the absence of the systematic error related to the condition veto = 0. Relatively high χ 2 /n.d.f in figures 11 (1.71) and 10 (1.6) is due to the large devitions of 2-3 energy points. It's caused by background contamination of the control samples (events with veto ≥ 0 or only one reconstructed track) used in δ veto and R ππ /R ee calculations.
Trigger efficiency is greater than 99.9% for all types of collinear events due to the energy deposition cuts E 1,2 > 40 MeV. These cuts provide performance of the energy deposition threshold. Therefore systematic uncertainty from trigger inefficiency is considered to be negligible.
Uncertainties in simulation of energy depositions in the calorimeter can lead to an inaccuracy in e/π discrimination. The identification efficiency and related systematic error were studied in [43] using pseudo-ππ and pseudo-ee events. It varies with energy from JHEP01(2021)113 0.996 to 0.998 for the e + e − → e + e − events and from 0.994 to 0.998 for the e + e − → π + π − events. The systematic error σ PID of the e + e − → π + π − cross section measurement due to cuts R< 0.0 and R> 0.0 does not exceed 0.002 at √ s > 650 MeV. Below 650 MeV, the σ PID value increases with decrease of energy and reaches 0.005 at √ s = 525.1 MeV.

Calculation of the cross section
The number of selected events in the regions R >0 and R <0 are: where index a = 1, 2 indicates the events with 0< R and R >0 respectively; σ jj and ε a jj are physical cross section and detection efficiency of the process with jj = π + π − , µ + µ − , e + e − in the final state, N a nc is a number of e + e − → π + π − π 0 and cosmic background events, IL is the integrated luminosity. The detection efficiencies ε a jj take into account the correction coefficients described above. Using the formula for N a , the e + e − → π + π − cross section is calculated as .

(3.15)
Subtraction of the e + e − → µ + µ − background leads to additional contribution to the systematic error, which is estimated as follows: The Born cross section for the e + e − → π + π − process is calculated from as . (3.17) The radiative correction δ rad (s i ), which takes into account the initial and final states radiation, is calculated using the MCGPJ generator. The value of δ rad (s) depends on the σ 0 ππ (s) cross section at lower energies, and it is therefore calculated iteratively. The iteration stops when its value changes by less than 0.05% in consecutive iterations. The correction for the center of mass energy spread is taken into account also. The spread does not exceed 0.3 MeV in the energy region below 1 GeV, and the correction is less than 0.1%.
The measured cross section σ 0 ππ is presented in table 1. The systematic errors of the cross section determination are listed in table 2.

Fit to the measured cross section
In the framework of the vector meson dominance model, the cross section of the e + e − → π + π − process is σ 0 ππ (s) = 2 3  Table 1. The results of the e + e − → π + π − cross section measurements. σ ππ , σ 0 ππ and F (s) are the physical, Born cross sections of the process e + e − → π + π − , and pion form factor, calculated with formula in ref. [25]. σ bare is the e + e − → π + π − undressed cross section without vacuum polarization, but with the final state radiative correction. Both statistical and systematic errors are shown.
where P ππ (s) is the phase space factor: The transition amplitudes are given by where Here, f denotes the final state of the vector meson V decay, m V is the vector meson mass, Γ V = Γ V (m V ) and φ ρV is the relative interference phase between the vector mesons V and ρ, and, hence, φ ρρ = 0.
The following forms of the energy dependence of the vector meson total widths are used: Here, q πγ = (s − m 2 π )/2 √ s, W ρπ (s) is the phase space factor for the ρπ → π + π − π 0 final state [45], B V →f is the branching fraction of the vector meson decay to the final state f . In the energy dependence of the ρ and ρ mesons widths only the V → π + π − decays are taken into account. Such approach is justified in the energy region √ s < 1000 MeV. The relative decay probabilities are calculated as follows The fit to the measured cross section σ 0 ππ is performed with the following free parameters m ρ , Γ ρ , σ(ρ → π + π − ), σ(ω → π + π − ), σ ( ρ → π + π − ) and φ ρω . The values of m ω , Γ ω , m ρ , Γ ρ are taken from [46]. The relative phase φ ρρ is fixed at π according to ref. [25]. Only uncorrelated errors of the cross section are taken into account in the fit. The results of the fit (figure 12) together with the results of the SND measurements at the VEPP-2M collider [26] are presented in table 3. The products are also presented in table 3. The ratio between the measured cross section and the fit curve is shown in figure 13. The systematic errors of m ρ and Γ ρ are related to the model uncertainty. It is estimated by comparison of the central values of these parameters presented in table 3 with the results of the fit with a model based on the Gounaris-Sakurai parametrization [27,42]. If the m ω and Γ ω are free parameters of the fit, their values are in agreement with those presented in PDG [46], and the value of φ ρω is shifted by 1 • . This difference is taken as the systematic uncertainty of the φ ρω .  Figure 13. The relative difference between the measured e + e − → π + π − cross section and the fit curve.

Contribution to the a µ
The contribution to the anomalous magnetic moment of the muon due to π + π − (γ) intermediate state in the vacuum polarization is calculated via the dispersion integral where K(s) is the known kernel [47] and σ(e + e − → µ + µ − ) = 4πα 2 3s .  Figure 14. The relative difference between the e + e − → π + π − cross section measured by BaBar [32] and the fit to the SND data. The error bars take into account both statistic and systematical errors of BaBar data. The shaded area corresponds to the quadratic sum of the systematic and statistical errors of the SND.
Here σ bare (table 1) is the bare cross section of the process e + e − → π + π − (the cross section without vacuum polarization contribution but taking into account the final state correction): where Π(s) is the polarization operator calculated according to the ref. [41] from the known e + e − → hadrons cross section [48]. The last factor a(s) takes into account the final state radiation for the point-like pion [49]. The integral (3.29) is evaluated by using the trapezoidal rule. As a result it is obtained a µ (ππ, 525 MeV ≤ √ s ≤ 883 MeV) = (409.79 ± 1.44 ± 3.87) × 10 −10 .
The difference 1.8 × 10 −10 between this value and one, calculated with a fit curve, is taken as additional source of the systematics.

Discussion
The comparison of the e + e − → π + π − cross section obtained in this work with the results [25,28,32,50] is shown in figure 14, 15, 16. The difference of 3% between SND and BABAR data is observed in the energy region 0.62 ≥ √ s ≤ 0.7 GeV, while outside it the SND and BABAR data are consistent ( figure 14). The deviation between the KLOE and SND data is 1-3% at √ s ≥0.7 GeV. Below 0.7 GeV, the measurements are consistent ( figure 15). The results obtained in this work and in experiments at VEPP-2M are in agreement ( figure 16).
The parameters of the ρ and ω mesons obtained in this analysis are consistent with the results of [26] (table 3) Figure 15. The relative difference between the e + e − → π + π − cross section measured by KLOE [50] and the fit to the SND data. The error bars take into account both statistic and systematical errors of KLOE data. The shaded area corresponds to the quadratic sum of the systematic and statistical errors of the SND.  Figure 16. The relative difference between the e + e − → π + π − cross sections measured by SND [26] and CMD-2 [28] at VEPP-2M and the fit to the SND data at VEPP-2000. The error bars take into account both statistic and systematical errors of VEPP-2M data. The shaded area corresponds to the quadratic sum of the systematic and statistical errors of the SND at VEPP-2000. experiments [26,28,32,51] (figure 17). Its width Γ ρ agrees with results of refs. [26,28,51] and contradict to the value reported by BaBar [32] ( figure 18). To understand the source of the latter difference, we perform BABAR cross section fit in the energy region 0.525-0.883 GeV using our model (3.18). The obtained ρ meson width is 147.38 MeV ± 0.47 MeV.  Figure 18. The ρ meson width Γ ρ measured in this work and in refs. [26,28,32,51]. The shaded area shows the PDG value [46].

JHEP01(2021)113
We conclude that the discrepancy can be partially explained by difference between the fitting models.
The result of this work is in agreement with VEPP-2M measurements, but is in conflict with BaBar and KLOE measurements. The π + π − (γ) contribution to the anomalous magnetic moment of the muon has been derived from the measured cross section: a µ (ππ, 525 MeV ≤ √ s ≤ 883 MeV) = (409.79 ± 1.44 ± 3.87) × 10 −10 .