Observation of $e^+e^-$ Annihilations into the $\boldmath{C=+1}$ Hadronic Final States $\rho^0\rho^0$ and $\phi\rho^0$

We report the first observation of $e^+e^-$ annihilations into states of positive $C$-parity, namely $\rho^0 \rho^0$ and $\phi\rho^0$. The two states are observed in the $\pip\pim\pip\pim$ and $K^+K^-\pip\pim$ final states, respectively, in a data sample of 225~$fb^{-1}$ collected by the $BaBar$~experiment at the $PEP-II$ $e^+e^-$ storage rings at energies near $\sqrt{s} ~=$ 10.58~$GeV$. The distributions of $\cos\theta^*$, where $\theta^*$ is the center-of-mass polar angle of the $\phi$ meson or the forward $\rho^0$ meson, suggest production by two-virtual-photon annihilation. We measure cross sections within the range $|\cos\theta^*| ~<~ 0.8$ of $\sigma(\e^+e^- \to \rho^0 \rho^0) ~=~$$20.7 \pm 0.7 (\mathrm{stat})\pm 2.7(\mathrm{syst}) ~\mathrm{fb}$ and $\sigma(\e^+e^- \to \phi \rho^0) \~=~$$5.7\pm 0.5(\mathrm{stat}) \pm 0.8(\mathrm{syst}) ~\mathrm{fb}$.

We report the first observation of e + e − annihilations into states of positive C-parity, namely ρ 0 ρ 0 and φρ 0 . The two states are observed in the π + π − π + π − and K + K − π + π − final states, respectively, in a data sample of 225 fb −1 collected by the BABAR experiment at the PEP-II e + e − storage rings at energies near √ s = 10.58 GeV. The distributions of cos θ * , where θ * is the center-ofmass polar angle of the φ meson or the forward ρ 0 meson, suggest production by two-virtualphoton annihilation. We measure cross sections within the range | cos θ * | < 0.8 of σ(e + e − → ρ 0 ρ 0 ) = 20.7 ± 0.7(stat) ± 2.7(syst) fb and σ(e + e − → φρ 0 ) = 5.7 ± 0.5(stat) ± 0.8(syst) fb. 12.38.Aw,12.40.Vv The process e + e − → hadrons at center-of-mass (CM) energy √ s far below the Z 0 mass is dominated by annihilation via a single virtual photon with charge-conjugation parity C = −1. The high luminosity of the B factories provides an opportunity to explore rare, low multiplicity final states with C = +1 such as those produced in the two-virtual-photon annihilation (TVPA) process depicted in Fig. 1. The TVPA process has been ignored in the interpretation of the total hadronic cross section in e + e − annihilations as input to calculations [1] of the muon g−2 and the running QED coupling α. We report the first observation of the exclusive reactions e + e − → ρ 0 ρ 0 and e + e − → φρ 0 , in which the final states are even under charge conjugation, and therefore cannot be produced via single-photon annihilation.
This analysis uses a 205 fb −1 data sample of e + e − collisions collected on the Υ (4S) resonance and 20 fb −1 collected 40 MeV below with the BABAR detector at the SLAC PEP-II asymmetric-energy B factory. The BABAR detector is described in detail elsewhere [2]. Charged-particle momenta and energy loss are measured in the tracking system which consists of a silicon vertex tracker (SVT) and a drift chamber (DCH). Electrons and photons are detected in a CsI(Tl) calorimeter (EMC). An internally reflecting ring-imaging Cherenkov detector (DIRC) provides charged particle identification (PID). An instrumented magnetic flux return (IFR) provides identification of muons. Kaon and pion candidates are identified using likelihoods of particle hypotheses calculated from the specific ionization in the DCH and SVT, and the Cherenkov angle measured in the DIRC. Electrons are identified by the ratio of the energy deposited in the EMC to the momentum and by the shower shape; muons are identified by the depth of penetration into the IFR.
Events with four well-reconstructed charged tracks and a total charge of zero are selected. Charged tracks are required to have at least 12 DCH hits and a polar angle in the range 0.41 < θ < 2.54 radians. The momenta of kaon and pion candidates are required to be greater than 800 and 600 MeV/c, respectively. Among the four selected tracks, two oppositely charged tracks must be identified as pions, and the other pair must be identified as two pions or two kaons. Events in which one or more pion candidates are identified as an electron or muon are rejected (lepton veto). We fit the four tracks to a common vertex, and require the χ 2 probability to exceed 0.1%. We accept events with reconstructed invariant mass within 170 MeV/c 2 of the nominal CM energy (Fig. 2).
In the process e + e − → π + π − π + π − , there are two possible pairings of π + mesons with π − mesons. However, only one combination appears in the kinematic region of interest (m π + π − < 2 GeV/c 2 ) for both pairs. We label the pion pair with CM momentum vector pointing into the hemisphere defined by the e − beam direction π + π − f and the other as π + π − b . Figure 3(a) shows the scatter plot of the invariant masses of π + π − f and π + π − b from e + e − → π + π − π + π − events, and Fig. 3(b) the plot of invariant masses of K + K − and π + π − pairs from e + e − → K + K − π + π − events. We observe correlations of masses in Fig. 3(a) indicating the production of ρ 0 ρ 0 final states, and in Fig. 3(b) indicating the production of φρ 0 final states.
To extract the number of e + e − → ρ 0 ρ 0 and φρ 0 signal events, we perform a binned maximum-likelihood fit for nine rectangular regions (tiles) in the two-dimensional mass distributions, as shown in Fig. 3. The signal box is the central tile (tile 5), defined by the mass ranges 0.5 < m π + π − < 1.1 GeV/c 2 and 1.008 < m K + K − < 1.035 GeV/c 2 . For e + e − → K + K − π + π − , the expected number of events, n i , for each tile i can be expressed as: FIG. 2: Distributions of the invariant mass (Υ (4S) data) for the a) π + π − π + π − and b) K + K − π + π − final states. The accepted signal regions are indicated by the dashed lines.
where S is the number of φρ 0 signal events, N φ is the number of φX background events, N ρ 0 is the number of ρ 0 X background events, and B is the number of residual background events, in all nine tiles. The parameter f T i is the fraction of events of type T that contributes to tile i. The signal fractions f S i are modeled by Monte Carlo (MC) simulation [4], and f φ i and f ρ 0 i are obtained from the φX and ρ 0 X background shapes, which are estimated by fitting the projections of m K + K − and m π + π − as described later. The residual background fractions f B i are modeled by a linear function that can be expressed as where ∆x i and ∆y i are the kinematically accessible dimensions of tile i, x i and y i are at the center of tile i, and s ρ 0 and s φ are slopes obtained from the fits. A similar expression is used for the π + π − π + π − case, where φ and ρ 0 are replaced with ρ 0 f and ρ 0 b . The background fractions are obtained by mass projection fits which are confined to the central horizontal or central vertical φ or ρ 0 resonance band. The effect of neglecting the resonance width outside the central band, checked by smearing the background fractions in the central band into the adjacent tiles using the resonance widths obtained from MC, is found to be negligible. The mass projections in the central bands for π + π − recoiling against a selected ρ 0 or φ and for K + K − recoiling against a ρ 0 are shown in Fig. 4. For the ρ 0 π + π − case we fit the π + π − mass projection to the sum of a ρ 0 component, an f 2 (1270) component, and a µ + µ − background component. The ρ 0 is represented by the product of a P-wave relativistic Breit-Wigner with its width set to the Particle Data Group (PDG) [3] value, a phase space term, and a factor 1/m 2 ππ due to production via a virtual FIG. 3: Scatter plots of the invariant masses of the two oppositely charged pairs in the a) π + π − π + π − and b) K + K − π + π − final states. The dashed lines indicate K + K − /π + π − thresholds. The solid lines show the nine tiles used in the fit.
photon. The f 2 (1270) is represented by a D-wave relativistic Breit-Wigner with its mean and width set to the PDG values. The µ + µ − background shape is obtained from a sample of the related channel e + e − → ρ 0 µ + µ − isolated by requiring two oppositely charged tracks identified as muons. For the φπ + π − case, we use the same background parameterization in terms of f 2 and µ + µ − , but refit for their normalizations. For the ρ 0 K + K − case, we fit the K + K − mass projection to the sum of a Breit-Wigner with mean and width fixed to their PDG values for the φ signal, and a threshold function (q 3 )/(1 + q 3 R), where q is kaon momentum in the φ rest-frame and R is a shape parameter, for background. Assuming the masses of the two pairs to be uncorrelated and excluding the ρ 0 and φ signal contributions, the fitted functions are integrated to obtain the tile fractions f ρ 0 i , f φ i , and f ρ 0 i . The extracted ρ 0 ρ 0 and φρ 0 yields in the signal box are 1243 ± 43 and 147 ± 13 events, to be compared with total of 1508 π + π − π + π − (∼ 18% background) and 163 K + K − π + π − (∼ 10% background) events in the signal box, respectively.
To investigate the possibility of ρ 0 ρ 0 and φρ 0 production in Υ (4S) decay, we examine the data recorded at and below the Υ (4S) resonance separately. The yields below the Υ (4S) resonance are 104 ± 14 for ρ 0 ρ 0 and 14 ± 4 for φρ 0 , consistent with the expected values of 112 ± 4 and 13 ± 1 obtained by scaling the on-peak yields of 1138 ± 42 and 135 ± 13 by the relative integrated luminosities.
To investigate the production mechanism, we examine the production angle θ * , defined as the angle between the ρ 0 f (φ) direction and the e − beam direction in the CM frame. To measure the angular distributions, we subdivide the data into bins of θ * , and repeat the above fit, with linear background slopes s ρ 0 f and s ρ 0 b (s ρ 0 and s φ ) fixed to the values from the overall fit. The | cos θ * | distributions after MC efficiency correction are shown in Fig. 5. The measurements are restricted to the fiducial region | cos θ * | < 0.8, as the efficiency drops rapidly be-FIG. 4: Mass distribution for a) π + π − pairs in ρ 0 π + π − events, b) π + π − pairs in φπ + π − events, and c) K + K − pairs in ρ 0 K + K − events. The solid curves are the total fit. For the π + π − cases, the dotted curve is the µ + µ − component, while the sum of f2(1270) and µ + µ − contributions are shown as dashed. For the K + K − case, the dashed curve represents the threshold function.
The combined hardware and software trigger efficiencies for signal events in the fiducial region are 99.9% for FIG. 5: Production angle distributions, after correction for efficiency, for a) ρ 0 ρ 0 and b) φρ 0 . The solid and dashed lines are the normalized 1+cos 2 θ * 1−cos 2 θ * and 1 + cos 2 θ * distributions, respectively.
ρ 0 ρ 0 and 91.3% for φρ 0 . The lower efficiency for φρ 0 is due to an event shape cut in the software trigger. For the determination of signal cross sections, the MC cos θ * and cos θ H distributions for φ and ρ 0 are re-weighted to reproduce the expectation from TVPA. The signal efficiencies in the fiducial region of | cos θ * | <0.8 for ρ 0 ρ 0 and φρ 0 are estimated to be 26.7% and 23.2%, respectively, including corrections to MC simulations of PID, tracking, hardware and software trigger efficiencies. Initial state photon radiation is included in the MC simulation.
Systematic uncertainties due to PID and tracking efficiency are estimated based on measurements from control data samples. The related systematic uncertainties on lepton vetoes are estimated by the difference from not applying the e and µ vetoes on pions. The systematic uncertainty from background subtraction is estimated by varying assumptions about background shapes. We investigated possible feed-down background from related modes with an extra π 0 using various extrapolations from the four-particle mass sidebands. We assume that the final states are fully transversely polarized. The systematic uncertainties are summarized in Table I.
The measured cross sections are in good agreement with the calculation from a vector-dominance two-photon exchange model [7]. In summary, we have observed exclusive production of C = +1 final states in e + e − interactions. The measured C parity configuration, the signal yields in data samples on the Υ (4S) resonance and below, and the pro-  duction angle distributions support the conclusion that the production mechanism is two-virtual-photon annihilation. The Standard Model predictions of the anomalous magnetic moment of the muon and the QED coupling rely on the measurements of low-energy e + e − hadronic cross sections, which are assumed to be entirely due to single-photon exchange. We have estimated the effect due to the TVPA processes we have measured, and find it to be small compared with the current precision [1].