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

A precise measurement of the cross section for the process $e^{+}{e}^{-}\to K^{+}{K}^{-}(\gamma )$ from threshold to an energy of 5 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 measurement uses the effective ISR luminosity determined from the $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}(\gamma ){\gamma }_{\mathrm{ISR}}$ process with the same data set. The corresponding lowest-order contribution to the hadronic vacuum polarization term in the muon magnetic anomaly is found to be $a_{\mu }^{KK,\mathrm{LO}}=(22.93\pm {0.18}_{\mathrm{stat}}\pm {0.22}_{\mathrm{syst}})\times {10}^{-10}$. The charged kaon form factor is extracted and compared to previous results. Its magnitude at large energy significantly exceeds the asymptotic QCD prediction, while the measured slope is consistent with the prediction.

Figure 1

The two-dimensional ${\chi }^2$ distribution for the $KK(\gamma ){\gamma }_{\mathrm{ISR}}$ data sample in the $[0.98–5]\mathrm{GeV}/c^2$ range of the fitted $KK$ mass, where different interesting regions are defined. The line labeled “no add. photon” corresponds to events with no detected additional photon, which are characterized by the ${\chi }_{\mathrm{ISR}}^2$ value only.

Figure 2

The $K^{+}{K}^{-}$ invariant mass spectrum for the data sample, after the tight ${\chi }^2$ selection: $\varphi$ mass region (left), masses above $m_{\varphi }$ (right).

Figure 3

Distributions of the absolute value of the cosine of the angle between the ISR photon and the charged tracks in the $KK$ center-of-mass, for data (black points) and MC (blue histogram). The $KK$ mass range is from 1.01 to $1.03\mathrm{GeV}/c^2$. The MC is normalized to the number of events in the data.

Figure 4

Fit of the data/MC correction for the tracking efficiency (per event, i.e., for the two tracks) as a function of $\delta \varphi$. The function for the fit is a constant plus two Gaussians. The central values of the Gaussians are fixed at 0.1 rad. The red band indicates the errors computed from the covariance matrix of the fit parameters.

Figure 5

The data/MC correction for the tracking efficiency as a function of $m_{KK}$. The red error bars show the (small) statistical errors from the sampling, whereas the blue ones show the total errors (including the errors from the fit). The figure on the right is a zoom of the figure on the left in the $\varphi$ resonance region.

Figure 6

The data/MC correction for $K$-ID efficiency as a function of $m_{KK}$. The red error bars show the statistical errors from the sampling, whereas the blue ones show the total errors (including the errors from the fit). The plots correspond to a sampling with MC events in the tight ${\chi }^2$ region. The figure on the right is a zoom in the $\varphi$ resonance region.

Figure 7

Data/MC correction for the $\mu \mu \to ‘KK’$ (red points) and $\pi \pi \to ‘KK’$ (blue triangles) mis-ID fractions as a function of $m_{KK}$.

Figure 8

${\chi }_{\mathrm{ISR}}^2$ distribution for the kaon ISR subsample (left) and muon ISR subsample (right) in data (points) and MC (histogram). The plots correspond to $0.95<m_{KK}<1.1\mathrm{GeV}/c^2$ and $m_{\mu \mu }<1\mathrm{GeV}/c^2$, respectively, for events satisfying a loose ${\chi }^2$ criterion. The MC is normalized to the number of events in the data.

Figure 9

Energy distribution of the additional ISR photon in the c.m. frame, in the $KK$ ISR subsample in background-subtracted data (points) and MC (histogram). The plot corresponds to $0.95<m_{KK}<1.1\mathrm{GeV}/c^2$. The MC is normalized to the data luminosity. The sharp cutoff at 2.3 GeV in MC is caused by the $m_{KK{\gamma }_{\mathrm{ISR}}}>8\mathrm{GeV}/c^2$ requirement set at generation.

Figure 10

Angle (degrees) between the additional photon and the closest kaon for data after background subtraction (points) and $KK$ MC (histogram). The plot corresponds to the “FSR” sample in the $[0.95–1.1]\mathrm{GeV}/c^2$ mass region. The MC is normalized to the data luminosity. The fit for the data/MC comparison for the amount of FSR events is also shown (solid line).

Figure 11

Data/MC correction for the efficiency of the tight ${\chi }^2$ selection of $\mu \mu (\gamma ){\gamma }_{\mathrm{ISR}}$ events, as a function of the $KK$ mass computed using muon-track momenta with assigned kaon mass. The bins in the $J/\psi$ and $\psi (2S)$ vicinity are removed (see text).

Figure 12

Distribution of the larger of the two transverse distances of closest approach to the interaction point (${\mathrm{doca}}_{xy}^{\max }$), for muons (black points) and kaons (blue histogram on the left, blue circles on the right) for MC (left) and data (right) in the intermediate ${\chi }^2$ region. The ${\mu }^{+}{\mu }^{-}$ plots are normalized to the $K^{+}{K}^{-}$ results in the region of ${\mathrm{doca}}_{xy}^{\max }<0.05\mathrm{cm}$.

Figure 13

${\chi }_{\mathrm{ISR}}^2$ distribution of $KK\gamma$ events in data (points) and MC (histogram) with $0.98<m_{KK}<1.04\mathrm{GeV}/c^2$ and $0.15<{\mathrm{doca}}_{xy}^{\max }<0.5\mathrm{cm}$. The MC is normalized to the data luminosity and corrected for data/MC differences in secondary interactions.

Figure 14

Total data/MC correction for the efficiency of the tight ${\chi }^2$ selection as a function of the $KK$ mass. The figure on the right is a zoom of the figure on the left in the $\varphi$ resonance region.

Figure 15

The $\mu \mu \to ‘KK’$ (left) and $\pi \pi \to ‘KK’$ (right) backgrounds corrected for data/MC differences in mis-ID, as a function of $m_{KK}$.

Figure 16

The data $\ln ({\chi }_{\mathrm{ISR}}^2+1)$ distributions (black squares) in the $[0.98–1.1]\mathrm{GeV}/c^2$ (left), $[1.1–3]\mathrm{GeV}/c^2$ (middle) and $[3–5]\mathrm{GeV}/c^2$ (right) $KK$ mass regions, after subtracting the $\pi \pi$ and $\mu \mu$ backgrounds. The (red) open points show the contributions of the remaining backgrounds normalized as described in the text. The solid line represents the result of the fit to the data distributions in the second and third $m_{KK}$ regions, where the signal shape is taken from the $[0.98–1.1]\mathrm{GeV}/c^2$ region, and the background shape from MC.

Figure 17

Total estimated background (${\mu }^{+}{\mu }^{-}\gamma$ and ${\pi }^{+}{\pi }^{-}\gamma$ backgrounds not included) with the tight ${\chi }^2$ selection. The error bars are dominated by the correlated systematic errors due to the normalization factors of the $q\overline{q}$ MC and the $K^{+}{K}^{-}\eta \gamma$ MC.

Figure 18

Relative difference between data and reconstructed MC in the tight ${\chi }^2$ region, at the first step (blue histogram), after one iteration (dotted green line) and after a second iteration (dashed black line). The diagonal elements of the error matrix are indicated by the red histograms. The bottom plot is a zoom of the top plot in the $[1–1.1]\mathrm{GeV}/c^2$ $m_{KK}$ range.

Figure 19

Transfer matrix for events in the tight ${\chi }^2$ region. The plot on the right is a zoom of the left plot in the $[1–1.1]\mathrm{GeV}/c^2$ $m_{KK}$ range.

Figure 20

Relative correction applied to the data spectrum at the first step of the unfolding (blue histogram), after one iteration (dotted green line) and after a second iteration (dashed black line). The diagonal elements of the error matrix are indicated by the red histograms. The bottom plot is a zoom of the top plot in the $[1–1.1]\mathrm{GeV}/c^2$ $m_{KK}$ range.

Figure 21

The full correction to the $KK\gamma /\mu \mu \gamma$ acceptance ratio to account for data/MC differences for additional ISR and secondary interaction effects. The vertical black error bars show the small but fully correlated errors coming from the data/MC correction of secondary interactions. The green error bars show the total error in each bin.

Figure 22

Global acceptance ${\epsilon }_{KK\gamma }$ computed with AfkQed. A zoom on the low-mass region is shown in the right plot.

Figure 23

The measured $e^{+}{e}^{-}\to K^{+}{K}^{-}(\gamma )$ bare cross section (including FSR). Systematic and statistical uncertainties are shown, i.e., the diagonal elements of the total covariance matrix. The contributions of the decays of the $J/\psi$ and $\psi (2S)$ resonances to $K^{+}{K}^{-}$ have been subtracted.

Figure 24

The measured $e^{+}{e}^{-}\to K^{+}{K}^{-}$ bare cross section in the $[1–1.04]\mathrm{GeV}$ (top), $[1.04–1.6]\mathrm{GeV}$ (bottom left), and $[1.6–2.1]\mathrm{GeV}$ (bottom right) mass intervals, together with results published by previous experiments. Systematic and statistical uncertainties are shown, i.e., the diagonal elements of the total covariance matrices.

Figure 25

Relative variations of the $KK(\gamma )$ cross section due to mass calibration (left) and resolution (right) uncertainties. The solid black (dashed blue) histogram indicates the effect corresponding to a $+1$ ($-1$) standard-deviation variation of the given parameter.

Figure 26

Fit of the squared BABAR charged kaon form factor with a model based on a sum of resonances (see text), in the energy interval from threshold up to 2.4 GeV (left) and $[1–1.04]\mathrm{GeV}$ (right). Systematic and statistical uncertainties are shown for data points, i.e., the diagonal elements of the total covariance matrices.

Figure 27

Relative difference between the charged kaon squared form factor from BABAR data and the 19-parameter phenomenological fit in three mass regions. Systematic and statistical uncertainties are included for data (diagonal elements of the total covariance matrices). The width of the band shows the propagation of statistical errors in the fit and the quoted systematic uncertainties, added quadratically.

Figure 28

Different contributions to the fit of the squared BABAR charged kaon form factor (black line) in the energy interval $[0.99–1.1]\mathrm{GeV}$. The dominant contribution under the overwhelming $\varphi$ resonance is from the interference between the $\rho +\omega$ and $\varphi$ amplitudes (dotted blue line).

Figure 29

Relative difference between the charged kaon squared form factor from CMD2 (left) and SND (right) data, and the BABAR phenomenological fit in the $\varphi$ mass region. Only the statistical uncertainties are included for data (diagonal elements of the covariance matrix). The width of the band shows the propagation of statistical errors in the BABAR fit and the quoted systematic uncertainties, added quadratically. The solid line shows a fit of the relative difference, with $\varphi$ masses different by $\Delta m$ (see text).

Figure 30

Relative difference between the charged kaon squared form factor from SND and OLYA (left), and DM1 and DM2 (right), and the BABAR phenomenological fit in different mass regions. Systematic and statistical uncertainties are included for data (diagonal elements of the covariance matrix). The width of the band shows the propagation of statistical errors in the fit and the quoted systematic uncertainties, added quadratically.

Figure 31

Fit (green band) of the squared BABAR charged kaon form factor in the high-mass region, using a function that has the shape of the QCD prediction (blue curve; see text). The extrapolation of the fit at low energy is indicated by the dotted green line. We also indicate measurements from CLEO data (red squares), close to the $\psi (2S)$ mass and above. Systematic and statistical uncertainties are shown for data points, i.e., the diagonal elements of the total covariance matrices.

Figure 32

Left: $K^{+}{K}^{-}$ mass spectrum in the data in the $J/\psi$ resonance region. Right: Distribution of the difference between the generated and the fitted $K^{+}{K}^{-}$ mass in MC, for events with a generated mass between 3 and $3.2\mathrm{GeV}/c^2$. The solid lines represent the results of fits by a Gaussian plus a constant term.

Figure 33

Left: $K^{+}{K}^{-}$ mass spectrum in data in the $\psi (2S)$ resonance region. Right: Distribution of the difference between the generated and the fitted $K^{+}{K}^{-}$ mass in MC, for events with a generated mass between 3.6 and $3.8\mathrm{GeV}/c^2$. The solid lines represent the results of fits by a Gaussian plus a constant term.