Precise measurement of the $e^{\mathbf{+}}{e}^{\mathbf{-}}\to {\pi }^{\mathbf{+}}{\pi }^{\mathbf{-}}\mathbf{(}\gamma \mathbf{)}$ cross section with the initial-state radiation method at BABAR

A precise measurement of the cross section of the process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}(\gamma )$ from threshold to an energy of 3 GeV is obtained with the initial-state radiation (ISR) method using $232{\mathrm{fb}}^{-1}$ of data collected with the BABAR detector at $e^{+}{e}^{-}$ center-of-mass energies near 10.6 GeV. The ISR luminosity is determined from a study of the leptonic process $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}(\gamma ){\gamma }_{\mathrm{ISR}}$, which is found to agree with the next-to-leading-order QED prediction to within 1.1%. The cross section for the process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}(\gamma )$ is obtained with a systematic uncertainty of 0.5% in the dominant $\rho$ resonance region. The leading-order hadronic contribution to the muon magnetic anomaly calculated using the measured $\pi \pi$ cross section from threshold to 1.8 GeV is $(514.1\pm 2.2(\mathrm{stat})\pm 3.1(\mathrm{syst}))\times {10}^{-10}$.

Figure 1

The generic Feynman diagrams for the processes relevant to this study with one or two real photons: lowest-order (LO) ISR (top left), LO FSR (top right), next-to-leading order (NLO) ISR with additional ISR (bottom left), NLO with additional FSR (bottom right).

Figure 2

The 2D-${\chi }^2$ distribution for $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ (data) for $0.5<m_{\pi \pi }<1.0\mathrm{GeV}/c^2$, where different interesting regions are defined.

Figure 3

The 2D-${\chi }^2$ distribution for $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ (data) for $0.5<m_{\mu \mu }<1.0\mathrm{GeV}/c^2$, where the signal and background regions are indicated.

Figure 4

The 2D-${\chi }^2$ distributions in $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ data: (left) below the central $\rho$ region ($m_{\pi \pi }<0.5\mathrm{GeV}/c^2$); (right) above the central $\rho$ region ($1.<m_{\pi \pi }<2.\mathrm{GeV}/c^2$). The line indicates the boundary for the tight ${\chi }^2$ selection.

Figure 5

The data/MC event trigger and filter correction $C_{\mathrm{trig}}$ for the $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ (top) and $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ (bottom) cross sections as a function of the $\mu \mu$ and $\pi \pi$ masses, respectively. Statistical errors only.

Figure 6

The data/MC event tracking correction $C_{\mathrm{track}}$ for the $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ (top) and $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ (bottom) cross sections as a function of the $\mu \mu$ and $\pi \pi$ masses, respectively.

Figure 7

The data/MC event correction $C_{\mathrm{PID}}$ for muon-ID efficiency as a function of $m_{\mu \mu }$.

Figure 8

The data/MC event correction $C_{\mathrm{PID}}$ for ${\pi }_{\mathrm{ID}}$ efficiency as a function of $m_{\pi \pi }$.

Figure 9

Global PID test on data (see text). (top): The $m_{\pi \pi }$ spectrum of all $xx\gamma$ events (no PID applied, data points) compared to $d{N}_{xx\mathrm{pred}}/d{m}_{\pi \pi }$ (histogram). (middle): The different components $N_{xx}$ of the histogram in the top plot, with $xx$ labels indicated, and their sum (top histogram with dots). (bottom): The relative difference between the two spectra in the top plot: predicted/measured $-1$. The independently estimated systematic uncertainty is shown by the band.

Figure 10

The ratio of the efficiencies for ’$\pi {\pi }_h$’ identification of $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ events for data and MC, the line is the linear fit result in the $\rho$ region.

Figure 11

Distributions of $\ln ({\chi }_{\mathrm{ISR}}^2+1)$ for $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ (left) and $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ (right) events in the $0.2–1\mathrm{GeV}/c^2$ ($0.5–1\mathrm{GeV}/c^2$) mass region for muons (pions), selected with $\ln ({\chi }_{\mathrm{FSR}}^2+1)>\ln ({\chi }_{\mathrm{ISR}}^2+1)$ and $E_{\gamma }^{*}{}_{\mathrm{add}.\mathrm{ISR}}>0.2\mathrm{GeV}$ for data (black points with errors) and MC (blue histogram). The distributions are normalized to the data luminosity.

Figure 12

Distributions of $E_{\gamma }^{*}{}_{\mathrm{add}.\mathrm{ISR}}$ for $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ (left) and $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ (right) events in the $0.2–1\mathrm{GeV}/c^2$ ($0.5–1\mathrm{GeV}/c^2$) mass region for muons (pions), selected with $\ln ({\chi }_{\mathrm{FSR}}^2+1)>\ln ({\chi }_{\mathrm{ISR}}^2+1)$ and $E_{\gamma }^{*}{}_{\mathrm{add}.\mathrm{ISR}}>0.2\mathrm{GeV}$ for data (black points with errors) and MC (blue histogram). The distributions are normalized to the data luminosity.

Figure 13

Distributions of $\ln ({\chi }_{\mathrm{FSR}}^2+1)$ for $\mu \mu \gamma {\gamma }_{\mathrm{ISR}}$ (left) and $\pi \pi \gamma {\gamma }_{\mathrm{ISR}}$ (right) events, selected with $\ln ({\chi }_{\mathrm{FSR}}^2+1)<\ln ({\chi }_{\mathrm{ISR}}^2+1)$ and $E_{\gamma \mathrm{add}.\mathrm{FSR}}>0.2\mathrm{GeV}$ for data (black points with errors) and MC (blue histogram). The mass regions are chosen to be similar for muons ($0.2–1\mathrm{GeV}/c^2$) and pions ($0.5–1\mathrm{GeV}/c^2$); the MC distribution is normalized to the number of events in data for muons and with $\ln ({\chi }_{\mathrm{FSR}}^2+1)<2$ for pions.

Figure 14

The additional “FSR” photon angular distribution with respect to the closer outgoing muon for the $\mu \mu \gamma {\gamma }_{\mathrm{ISR}}$ (left) and background-subtracted $\pi \pi \gamma {\gamma }_{\mathrm{ISR}}$ (right) events with $\ln ({\chi }_{\mathrm{FSR}}^2+1)<\ln ({\chi }_{\mathrm{ISR}}^2+1)$, $E_{\gamma \mathrm{add}.\mathrm{FSR}}>0.2\mathrm{GeV}$, $\ln ({\chi }_{\mathrm{FSR}}^2+1)<2.5$, and in the mass intervals $0.2<m_{\mu \mu }<1\mathrm{GeV}/c^2$, $0.5<m_{\pi \pi }<1\mathrm{GeV}/c^2$ (data: black points with errors, MC: blue histogram). The mass regions are chosen to be similar for the muon and pion samples and the MC is normalized to data luminosity.

Figure 15

The additional FSR photon energy distributions for the $\mu \mu \gamma {\gamma }_{\mathrm{ISR}}$ (left) and background-subtracted $\pi \pi \gamma {\gamma }_{\mathrm{ISR}}$ (right) events with $\ln ({\chi }_{\mathrm{FSR}}^2+1)<\ln ({\chi }_{\mathrm{ISR}}^2+1)$, $E_{\gamma \mathrm{add}.\mathrm{FSR}}>0.2\mathrm{GeV}$, $\ln ({\chi }_{\mathrm{FSR}}^2+1)<2.5$, in the mass intervals $0.2<m_{\mu \mu }<1\mathrm{GeV}/c^2$, $0.5<m_{\pi \pi }<1\mathrm{GeV}/c^2$, and ${\theta }_{\mu {\gamma }_2}<20°$, ${\theta }_{\pi {\gamma }_2}<10°$ (data: black points with errors, MC: blue histogram). The mass regions are chosen to be similar for the muon and pion samples and the MC is normalized to data luminosity.

Figure 16

The ${\chi }^2$ efficiency (top) for $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ data (after background subtraction), MC (AfkQed), and the ratio $C_{{\chi }^2}$ of data to MC (bottom), as a function of $m_{\mu \mu }$. The gap at $3.0–3.2\mathrm{GeV}/c^2$ corresponds to the excluded $J/\psi$ window (Sec. 6a).

Figure 17

(top): The distribution of the largest of the two transverse distances of closest approach to the interaction point ${\mathrm{doca}}_{\mathrm{xy}}^{\max }$ for pions and muons in data, for the intermediate ${\chi }^2$ region. (bottom): Same distributions in simulation.

Figure 18

The rate of $\mu \mu$ events misidentified as ’$\pi {\pi }_h$’ in data relative to the MC expectation, measured in mass intervals below and above the $\rho$ region. The curve with the error band is a second-order polynomial fit to the data points, used to interpolate through the $\rho$ mass region ($0.6–0.9\mathrm{GeV}/c^2$).

Figure 19

The $m_{KK}$ mass distribution in the ’$\pi {\pi }_h\gamma$’ data sample. The solid line represents the result of the fit (see text).

Figure 20

The $|\cos {\theta }_{\pi }^{*}|$ distribution of ’$\pi {\pi }_h$’ data in the $0.28–0.32\mathrm{GeV}/c^2$ $m_{\pi \pi }$ range, fitted (black curve) to two free components: $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ from MC (blue dashed line) and $ee\gamma$ background from downscaled radiative Bhabha events (red dotted line). The small $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ contribution is subtracted out.

Figure 21

The ${\gamma }_{\mathrm{ISR}}\gamma$ mass distribution for data events in a background enriched region. The ${\pi }^0$ signal is fitted with a Gaussian while additional contributions are represented by a linear term. The histogram is the $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ MC distribution. Contribution from $\tau \tau$ events has been subtracted.

Figure 22

2D-${\chi }^2$ distribution of $q\overline{q}$ MC events normalized to the data luminosity for $0.5<m_{\pi \pi }<1\mathrm{GeV}/c^2$. The solid broken line indicates the loose ${\chi }^2$ criterion, while the dashed line defines a “sleeve” in the signal region where most of the background is concentrated.

Figure 23

The $m_{\pi \pi }$ distribution in the background-rich “sleeve” region. The solid line represents a fit to the data with $\pi \pi$ signal and multihadron background components, with their shapes both taken from simulation.

Figure 24

The initial mass-transfer matrix $A_{ij}$ from the simulation giving the number of events generated with a (true) mass $\sqrt{s^{\prime }}$ in a bin $j$ and reconstructed with a (measured) mass $m_{xx}$ in a bin $i$: $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ (top), $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ with tight ${\chi }^2$ criterion (middle) and with loose ${\chi }^2$ criterion (bottom). For the latter case only the relevant range $0.5–1.0\mathrm{GeV}/c^2$ is shown. The $\sqrt{s^{\prime }}$ dependence comes from QED and a model of the pion form factor used in the AfkQed generator, respectively.

Figure 25

(top): The difference between the ${\pi }^{+}{\pi }^{-}$ mass distributions of data and reconstructed MC before unfolding ($\mathrm{\text{data}}-\mathrm{rMC}$) and after one iteration ($\mathrm{\text{data}}-\mathrm{rMCm}$), for the loose ${\chi }^2$ selection used in the central $\rho$ region. The data statistical errors ($\pm 1\sigma$) are shown for comparison. The correction to the initial MC distribution is small, but significant in the peak and tail regions. (bottom): The difference between the result of the first unfolding (UR1) and the initial data for the same loose ${\chi }^2$ criterion. It exceeds the data statistical error (band) only in the $\rho -\omega$ interference region. No significant improvement is observed between the first (UR1) and second (UR2) unfolding results.

Figure 26

Fit of the ’$\mu \mu$’ mass distribution in the $J/\psi$ region including the QED-$J/\psi$ interference as a momentum calibration test.

Figure 27

Breakdown of the full simulation acceptance with respect to the generated events in the ISR photon angular range in the $e^{+}{e}^{-}$ center of mass ${20}^0–{160}^0$ for $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ events. The numbers refer to the sequential application of the selection requirements: (1) trigger $+$ acceptance selection for reconstructed ISR photon and tracks; (2) preselection of ISR events $+$ $E_{\gamma }^{*}>3\mathrm{GeV}$; (3) $p>1\mathrm{GeV}/c$ for both tracks; (4) tracks in IFR active area; (5) tracks in DIRC active area; (6) $‘\mu \mu ’\mathrm{\text{−}}\mathrm{ID}+{\chi }^2\mathrm{\text{selection}}+J/\psi \mathrm{\text{rejection}}+\mathrm{\text{minor selections}}$.

Figure 28

The ratio of the $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ acceptances determined in Phokhara and AfkQed at the generator level with fast simulation.

Figure 29

The comparison between the distributions of data (points with errors) and simulation corrected for data/MC differences in PID (blue histogram), for ${\theta }_{\gamma }$ in radians in the $m_{\mu \mu }$ intervals $0.5–1\mathrm{GeV}/c^2$ (top), $1.5–2\mathrm{GeV}/c^2$ (middle), $2.5–3\mathrm{GeV}/c^2$ (bottom).

Figure 30

The comparison between the distributions of data (points with errors) and simulation corrected for data/MC differences in PID (blue histogram), for $\pm p_{{\mu }^{\pm }}$ in $\mathrm{GeV}/c$ in the $m_{\mu \mu }$ intervals $0.5–1\mathrm{GeV}/c^2$ (top), $1.5–2\mathrm{GeV}/c^2$ (middle), $2.5–3\mathrm{GeV}/c^2$ (bottom).

Figure 31

The comparison between the distributions of data (points with errors) and simulation corrected for data/MC differences in PID (blue histogram), for $|\cos {\theta }_{\mu }^{*}|$ in the $m_{\mu \mu }$ intervals: (from top to bottom) $0.20–0.25\mathrm{GeV}/c^2$, $0.25–0.30\mathrm{GeV}/c^2$, and $0.5–1\mathrm{GeV}/c^2$.

Figure 32

The ratio of the $\mu \mu$ mass spectrum in data over the absolute prediction from QED using the BABAR luminosity. The NLO QED prediction is obtained from the data-corrected (for detector simulation) and Phokhara-corrected (for NLO effects) AfkQed mass spectrum. The band is drawn around the fit of the $0.2–3.5\mathrm{GeV}/c^2$ region to a free constant, with a width given by $\pm$ the total expected systematic uncertainty.

Figure 33

The FSR correction ${\delta }_{\mathrm{FSR}}^{\mu \mu }$ obtained with AfkQed.

Figure 34

The effective ISR luminosity for the $\pi \pi$ analysis: the data points give ${\mathcal{L}}_{\mathrm{ISR}}^{\mathrm{eff}}$ in $\Delta \sqrt{s^{\prime }}=50\mathrm{MeV}$ bins. The conditions for the detected/identified ISR photon are $E_{\gamma }^{*}>3\mathrm{GeV}$ and $20°<{\theta }_{\gamma }^{*}<160°$ in the $e^{+}{e}^{-}$ c.m. frame, while one additional ISR photon is allowed without any restriction. The superimposed histogram is the lowest-order ISR prediction following Eq. (23). The $J/\psi$ mass region is removed for the luminosity determination.

Figure 35

The total background fraction for the $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ sample in the central $\rho$ region (loose ${\chi }^2$ selection).

Figure 36

The $m_{\pi \pi }$ spectrum of $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ events in the $\rho$ region, in $2\mathrm{MeV}/c^2$ bins.

Figure 37

The fractions of different backgrounds in the physical sample with the tight 2D-${\chi }^2$ criterion and strengthened ’$\pi {\pi }_h$’-ID (as used in the $\rho$ tails region) as a function of the $\pi \pi$ mass. (top left): Multihadrons, including $\tau \tau$. (top right): $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ ($\mathrm{\text{data}}+\mathrm{\text{measured mis}}\mathrm{\text{−}}\mathrm{\text{ID}}$). (bottom left): $KK(\gamma ){\gamma }_{\mathrm{ISR}}$ ($\mathrm{\text{data}}+\mathrm{\text{measured mis}}\mathrm{\text{−}}\mathrm{\text{ID}}$). (bottom right): $p\overline{p}{\gamma }_{\mathrm{ISR}}$ (MC).

Figure 38

The $m_{\pi \pi }$ spectrum of $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ events selected with ’$\pi {\pi }_h$’ identification and the tight $\ln ({\chi }_{\mathrm{ISR}}^2+1)<3$ criterion, from threshold to $3\mathrm{GeV}/c^2$ in $50\mathrm{MeV}/c^2$ mass intervals.

Figure 39

The angular pion distribution in the $\pi \pi$ system with respect to the ISR photon direction as function of $|\cos {\theta }_{\pi }^{*}|$ for background-subtracted $\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}$ data (points) in the $\rho$ central region ($0.5<m_{\pi \pi }<1\mathrm{GeV}/c^2$). The blue histogram is the shape obtained in the simulation, normalized to the data.

Figure 40

The full correction to the ${\epsilon }_{\pi \pi (\gamma ){\gamma }_{\mathrm{ISR}}}/{\epsilon }_{\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}}$ acceptance ratio for noncanceling effects (see text).

Figure 41

The ratio of the corrected and unfolded mass spectra (data points) for loose over tight 2D-${\chi }^2$ selection in the central $\rho$ region fitted in $100\mathrm{MeV}/c^2$ bins, compared to the band of independently estimated uncertainties (solid lines). The MC mass-matrix resolution correction is shown as the dashed histogram.

Figure 42

The measured cross section for $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}(\gamma )$ over the full mass range. Systematic and statistical uncertainties are shown, but based only on the diagonal elements of the covariance matrix (see text).

Figure 43

The measured cross section for $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}(\gamma )$ in the lower mass range. Systematic and statistical uncertainties are shown, but based only on the diagonal elements of the covariance matrix (see text).

Figure 44

The measured cross section for $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}(\gamma )$ in the $\rho$ region. Systematic and statistical uncertainties are shown, but based only on the diagonal elements of the covariance matrix (see text).

Figure 45

The pion form factor-squared measured by BABAR as a function of $\sqrt{s^{\prime }}$ from 0.3 to 3 GeV and the VDM fit described in the text.

Figure 46

The pion form factor-squared measured by BABAR as a function of $\sqrt{s^{\prime }}$ and the VDM fit from 0.3 to 3 GeV described in the text. (top): Low-mass region (0.3–0.5 GeV). (bottom): $\rho$ peak region with $\rho -\omega$ interference (0.70–0.82 GeV).

Figure 47

The relative difference between the pion form factor-squared from BABAR data and the 18-parameter phenomenological fit in three mass regions. Systematic and statistical uncertainties are included for data (diagonal errors). The width of the band shows the propagation of statistical errors from the fit and the quoted systematic uncertainties, added quadratically.

Figure 48

The relative difference of pion form factor-squared from the BABAR fit in the 0.5–1 GeV region with CMD-2 (left) and with SND (right). Systematic and statistical uncertainties are included in the data points. The width of the BABAR band shows the propagation of statistical errors from the fit and the quoted systematic uncertainties, added quadratically.

Figure 49

The relative difference of pion form factor-squared from the BABAR fit in the $\rho -\omega$ mass region with CMD-2 (left) and with SND (right). Systematic and statistical uncertainties are included in the data points. The width of the BABAR band shows the propagation of statistical errors from the fit and the quoted systematic uncertainties, added quadratically.

Figure 50

The relative difference of pion form factor-squared from KLOE and the BABAR fit in the 0.5–1 GeV region. Systematic and statistical uncertainties are included in the data points. The width of the BABAR band shows the propagation of statistical errors from the fit and the quoted systematic uncertainties, added quadratically.

Figure 51

The measured pion form factor-squared compared to published results from other experiments. Systematic and statistical uncertainties are shown for all results, with the diagonal elements of the BABAR covariance matrix.

Figure 52

The measured cross section for $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}(\gamma )$ compared to published results from CMD-2 up to 1.4 GeV and DM2 above. Systematic and statistical uncertainties are shown for all results, with the diagonal elements of the BABAR covariance matrix.

Figure 53

The relative difference of pion form factor-squared from CMD-2 and SND and the BABAR fit in the region below 0.5 GeV. Systematic and statistical uncertainties are included in the data points. The width of the BABAR band shows the propagation of statistical errors from the fit and the quoted systematic uncertainties, added quadratically.

Figure 54

The relative difference of pion form factor-squared from CMD-2 and the BABAR fit in the region above 1 GeV. Systematic and statistical uncertainties are included in the data points. The width of the BABAR band shows the propagation of statistical errors from the fit and the quoted systematic uncertainties, added quadratically.

Figure 55

The relative difference of the form factor-squared from the $\tau$ data of ALEPH (top), CLEO (middle) and Belle (bottom) with respect to the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ BABAR measurements in the 0.5–1 GeV region. Systematic and statistical uncertainties are included in the data points. The width of the BABAR band shows the propagation of statistical errors from the fit and the quoted systematic uncertainties, added quadratically. The $\tau$ data are normalized to the value of $B_{\pi \pi }$ measured by each experiment independently.

Figure 56

The LO hadronic VP $2\pi$ contributions to the muon magnetic anomaly, evaluated in the $2m_{\pi }-1.8\mathrm{GeV}$ range from the present analysis and other analyses using $e^{+}{e}^{-}$ data [7] and $\tau$ data [6]. The vertical bands show the values obtained for each data set by combining the respective spectral functions.