Observation of the chi _{c2}(2P) meson in the reaction gamma gamma -->DD[over -bar] at BABAR

A search for the Z ð 3930 Þ resonance in (cid:2)(cid:2) production of the D (cid:1) D system has been performed using a data sample corresponding to an integrated luminosity of 384 fb (cid:1) 1 recorded by the BABAR experiment at the PEP-II asymmetric-energy electron-positron collider. The D (cid:1) D invariant mass distribution shows clear evidence of the Z ð 3930 Þ state with a signiﬁcance of 5 : 8 (cid:3) . We determine mass and width values of ð 3926 : 7 (cid:2) 2 : 7 (cid:2) 1 : 1 Þ MeV =c 2 and ð 21 : 3 (cid:2) 6 : 8 (cid:2) 3 : 6 Þ MeV , respectively. A decay angular analysis provides evidence that the Z ð 3930 Þ is a tensor state with positive parity and C parity ( J PC ¼ 2 þþ ); therefore we identify the Z ð 3930 Þ state as the (cid:1) c 2 ð 2 P Þ meson. The value of the partial width (cid:2) (cid:2)(cid:2) (cid:3) B ð Z ð 3930 Þ ! D (cid:1) D Þ is found to be ð 0 : 24 (cid:2) 0 : 05 (cid:2) 0 : 04 Þ keV .


I. INTRODUCTION
Interest in the field of charmonium spectroscopy has been renewed with the recent discovery of numerous charmonium and charmoniumlike states [1][2][3][4][5][6][7][8][9][10][11]. However, very little is known about the first radially excited cJ ð2PÞ states which are expected to exist in the mass region from 3.9 to 4:0 GeV=c 2 , just above the D " D threshold [12]. The Belle Collaboration has observed the Zð3930Þ state in production of the D " D system [13], and this is considered a strong candidate for the c2 ð2PÞ state; indeed it is so labeled in Ref. [14]. The Belle analysis obtained a mass of m ¼ ð3929 AE 5 AE 2Þ MeV=c 2 and a total width of À ¼ ð29 AE 10 AE 2Þ MeV, with quantum numbers J PC ¼ 2 þþ preferred. The partial width À Â BðZð3930Þ ! D " DÞ was determined as ð0:18 AE 0:05 AE 0:03Þ keV, where À is the radiative width of the Zð3930Þ state, under the assumption that J ¼ 2. The observation of this state has not been confirmed so far [14].
In this paper the process ! D " D, illustrated by the Feynman diagram shown in Fig. 1, is studied in a search for the Zð3930Þ state. In Fig. 1, the initial state positron, e þ (electron, e À ), emits the virtual photon Ã 1 ( Ã 2 ), yielding the final state positron, e þ0 (electron, e À0 ); the momentum transfer to Ã 1 ( Ã 2 ) is q 1 (q 2 ). The virtual photons interact to produce the D " D final state. When the e þ0 and the e À0 are emitted along the beam directions the values of q 2 1 and q 2 2 are predominantly close to zero, and the two photons can be considered to be quasireal. Since in this case neither the e þ0 nor the e À0 is detected, the analysis is termed untagged.
The data sample used in this analysis corresponds to an integrated luminosity of 384 fb À1 recorded at the Çð4SÞ resonance (10.58 GeV) and at a center of mass (c.m.) energy of 10.54 GeV by the BABAR detector at the PEP-II asymmetric-energy e þ e À collider.
The BABAR detector is described briefly in Sec. II, and the principal criteria used in the selection of candidate twophoton-interaction events are discussed in Sec. III. The reconstruction of D " D pair events is presented in Sec. IV, and the relevant Monte Carlo simulations are detailed in Sec. V. The purity and reconstruction efficiency of the ! D " D event sample are considered in Secs. VI and VII, respectively, and the D " D signal yield and invariant mass resolution are presented in Sec. VIII; the mass and total width for the Zð3930Þ state are obtained from a fit to the D " D invariant mass distribution. The angular distribution in the D " D rest frame for the Zð3930Þ mass region is studied in Sec. IX, and the implications for the spin of the Zð3930Þ state are discussed. In Sec. X the partial radiative width of the Zð3930Þ state is extracted. Sources of systematic uncertainty are detailed in Sec. XI, and the results of the analysis are summarized in Sec. XII.

II. THE BABAR DETECTOR
The BABAR detector is described in detail elsewhere [15]. Charged particles are detected, and their momenta measured, with a combination of five layers of doublesided silicon microstrip detectors (SVT) and a 40-layer cylindrical drift chamber (DCH), both coaxial with the cryostat of a superconducting solenoidal magnet which produces a magnetic field of 1.5 T. Charged-particle identification is achieved by measurements of the energy loss dE=dx in the tracking devices and by means of an internally reflecting, ring-imaging Cherenkov detector (DIRC). Photons and electrons are detected and their energies mea- sured with a CsI(Tl) electromagnetic calorimeter (EMC), covering 90% of the 4 solid angle in the Çð4SÞ rest frame. The instrumented flux return of the magnetic field is used to identify muons and K 0 L .

III. SELECTION OF TWO-PHOTON-INTERACTION EVENTS
The selection of two-photon-interaction events for an untagged analysis is based on established procedures (see for instance Refs. [16,17]). Because of the small scattering angles involved, most of the incoming beam energy is carried away by the e þ0 and e À0 (see Fig. 1). This results in a large value of the missing mass squared where p e AE are the four-momenta of the beam electron and positron and p D , p " D are the four-momenta of the final state D and " D mesons, respectively. In addition, for these events, the resultant transverse momentum of the D " D system p t ðD " DÞ is limited to small values. In order to establish selection criteria for ! D " D events, the reaction is studied first using a data sample corresponding to an integrated luminosity of 235 fb À1 . The system X contains no additional charged particles. This reaction has been chosen because it has the same particle configuration as one of the final states we consider in this analysis. The charged kaons and pions are identified as described in detail in Sec. IV. Neutral pions are reconstructed from pairs of photons with deposited energy in the EMC larger than 100 MeV. It is required that no 0 meson candidate be found in a selected event. Two-photon production of the K À K þ þ À system should yield large values of m 2 X , the missing mass squared, m 2 X ¼ ðp e þ þ p e À À p K þ À p K À À p þ À p À Þ 2 : (3) In addition, production of the K À K þ þ À system via initial state radiation (ISR) should yield the small values of m 2 X associated with the ISR photon, for which detection is not required. The observed distribution of the K À K þ þ À invariant mass, mðK À K þ þ À Þ, resulting from the reaction of Eq. (2) is shown in Fig. 2(a).
There are clear signals corresponding to the production of c ð1SÞ, c0 ð1PÞ, and c2 ð1PÞ, and, since these states all have positive C parity, it is natural to associate them with two-photon production. Similarly, the large J=c signal observed would be expected to result from ISR production, because of the negative C parity of the J=c . For the parameters of these states, see Table I.
The distribution of m 2 X for 2:8 mðK À K þ þ À Þ 3:8 GeV=c 2 is shown in Fig. 2(b). The large peak near zero is interpreted as being due mainly to ISR production  2. (a) K þ K À þ À mass distribution for all events without any requirement on m 2 X ; (b) corresponding m 2 X distribution; (c) mðK þ K À þ À Þ with the requirement m 2 X < 10 ðGeV=c 2 Þ 2 ; (d) mðK þ K À þ À Þ with the requirement m 2 X > 10 ðGeV=c 2 Þ 2 . of the K À K þ þ À system, while two-photon-production events would be expected to occur at larger values of m 2 X . This is shown explicitly by the distributions of Figs. 2(c) and 2(d), which correspond to the requirements m 2 X < 10 ðGeV=c 2 Þ 2 and m 2 X > 10 ðGeV=c 2 Þ 2 , respectively. In Fig. 2(c) there is a large J=c signal, and a much smaller c ð2SÞ signal can also be seen. For e þ e À collisions at a c.m. energy of 10.58 GeV, the ISR production cross section for J=c is about 3 times larger than for c ð2SÞ; also BðJ=c ! K À K þ þ À Þ is approximately 9 times larger than the corresponding c ð2SÞ branching fraction value [14].
It follows that the observed J=c signal would be expected to be % 27 times larger than that for c ð2SÞ. The signals in Fig. 2(c) seem to be consistent with this expectation, and they are also in agreement with the detailed analysis of ISR production of the K À K þ þ À system in Ref. [18]. There is a c2 ð1PÞ signal in Fig. 2(c) which is comparable in size to the c ð2SÞ signal. The branching fraction for c ð2SÞ ! K À K þ þ À is % 7:5 Â 10 À4 [14], while the product Bðc ð2SÞ ! c2 ð1PÞÞ Â Bð c2 ð1PÞ ! K À K þ þ À Þ is % 7:8 Â 10 À4 [14], so that the presence of such a c2 ð1PÞ signal is consistent with the expected transition rates. For the c1 ð1PÞ, Bðc ð2SÞ ! c1 ð1PÞÞ Â Bð c1 ð1PÞ ! K À K þ þ À Þ % 4:0 Â 10 À4 , and so a c1 ð1PÞ signal of approximately half the size of the c2 ð1PÞ signal would be expected in Fig. 2(c); again the data seem to be in reasonable agreement with this expectation.
Finally, for the c0 ð1PÞ, Bðc ð2SÞ ! c0 ð1PÞÞ Â Bð c0 ð1PÞ ! K À K þ þ À Þ % 16:8 Â 10 À4 , and the corresponding signal in Fig. 2(c) would be expected to be about twice the size of the c ð2SÞ signal. The c0 ð1PÞ signal seems to be larger than that of the c ð2SÞ, but not by a factor of 2; this may be because the larger energy photon from the c ð2SÞ ! c0 ð1PÞ transition, when combined with the ISR photon, can yield a value of m 2 X which is larger than 10 ðGeV=c 2 Þ 2 . In summary, the signals observed in Fig. 2(c) appear consistent with those expected for an ISR-production mechanism, especially since there is no indication of any remnant of the large c ð1SÞ of Fig. 2(a). Furthermore, the cJ signals in Fig. 2(c) are removed by requiring that the transverse momentum of the K À K þ þ À system be less than 50 MeV=c [see discussion of Fig. 2(d) below], which indicates clearly that they do not result from two-photon production.
In Fig. 2(d), the c ð1SÞ signal of Fig. 2(a) appears to have survived the m 2 X > 10 ðGeV=c 2 Þ 2 requirement in its entirety, and the c0 ð1PÞ and c2 ð1PÞ signals have been reduced slightly, as discussed in the previous paragraph; in both Figs. 2(a) and 2(d) there is some indication of a small signal in the region of the c ð2SÞ mass. A J=c signal of about one-third of that in Fig. 2(a) is present also in Fig. 2(d). This is interpreted as being primarily due to (a) the emission of more than one initial state photon, with the consequence that values of m 2 X greater than 10 ðGeV=c 2 Þ 2 are obtained, (b) the ISR production of the c ð2SÞ with subsequent decay to J=c þ neutrals, and (c) two-photon production of the c2 ð1PÞ followed by c2 ð1PÞ ! J=c , which has a 20% branching fraction [14].
It follows from the above that the requirement m 2 X > 10 ðGeV=c 2 Þ 2 significantly reduces ISR contributions to the K À K þ þ À final state while leaving signals associated with two-photon production essentially unaffected. For this reason, the requirement that m 2 miss of Eq. (1) be greater than 10 ðGeV=c 2 Þ 2 is chosen as a principal selection criterion for the isolation of events corresponding to ! D " D. As mentioned above, it is expected that for an untagged analysis of ! D " D, the transverse momentum p t ðD " DÞ should be small. In order to quantify this statement, the data of Fig. 2(d) were divided into intervals of 50 MeV=c in the transverse momentum of the K À K þ þ À system with respect to the e þ e À collision axis, which is considered also to be the collision axis for two-photon-production events. For each interval a fit was made to the mðK À K þ þ À Þ mass distribution in the mass region 2:7 mðK À K þ þ À Þ 3:3 GeV=c 2 . The function used consists of a second-order polynomial to describe the background, a Gaussian function for the J=c signal, and a Breit-Wigner for the c ð1SÞ signal convolved with a Gaussian to account for the resolution. The p t dependence of the resulting c ð1SÞ yield is shown is Fig. 3(a), and that of the J=c yield is shown in Fig. 3(b). The shapes of the distributions are quite similar for p t > 100 MeV=c, but the interval from 50-100 MeV=c contains % 180 more c ð1SÞ signal events, and that for 0-50 MeV=c exhibits an excess of % 800 signal events. This behavior is expected for twophoton production of the c ð1SÞ. Thus, the requirement p t ðD " DÞ < 50 MeV=c is imposed as the second principal selection criterion for the extraction of ! D " D events. Since the two-photon reactions ! K À K þ þ À and ! D " D are quasiexclusive in the sense that only the final state particles e þ and e À are undetected it is required in both instances that the total energy deposits E EMC in the EMC which are unmatched to any charged-particle track be less than 400 MeV. The net effect is a small reduction in TABLE I. Charmonium states observed in the e þ e À ! K þ K À þ À X test data sample [14].  Fig. 3(c) corresponds to the K À K þ þ À candidates of Fig. 2(d) after requiring p t ðK À K þ þ À Þ < 50 MeV=c and that the EMC energy sum be less than 400 MeV. The p t criterion reduces the c ð1SÞ signal by a factor % 2, while the J=c signal is reduced by a factor % 5, as is the continuum background at 2:7 GeV=c 2 . More significantly, the continuum background at 3:7 GeV=c 2 , just below the D " D threshold, is reduced by a factor % 10.
It follows that the net effect of the three principal selection criteria described above [missing mass m 2 miss > 10 ðGeV=c 2 Þ 2 , resultant transverse momentum p t ðD " DÞ < 50 MeV=c, and total energy deposit in the calorimeter E EMC < 400 MeV] is to significantly enhance the number of two-photon-production events relative to the events resulting from ISR production, continuum production, and combinatoric background.
Concerning the histogram of Fig. 3(c), the product À ð c ð1SÞÞ Â Bð c ð1SÞ ! K À K þ þ À Þ is 1:7AE 1:0 times that for the c0 ð1PÞ state [14], and in Fig. 3(c) the c ð1SÞ signal contains % 950 events [cf. the 0-50 MeV=c interval of Fig. 3(a)], while the c0 ð1PÞ signal contains % 550 events. It follows that the signal sizes agree well with the ratio expected on the basis of a two-photon production mechanism. In a similar vein, the ratio of the partial width À ð cJ Þ Â Bð cJ ! K À K þ þ À Þ for c0 ð1PÞ and c2 ð1PÞ is 9 AE 2 [14], so that after taking into account the (2J þ 1) spin factors, the signals observed in Fig. 3(c) would be expected to be approximately in the ratio 1:8 AE 0:4. The c2 ð1PÞ signal contains % 200 events, and so is consistent with this expectation.

IV. RECONSTRUCTION OF D D EVENTS
Candidate D " D events are reconstructed in the five combinations of D decay modes listed in Table II (the use of charge conjugate states is implied throughout the text).
Events are selected by requiring the exact number of charged-particle tracks defined by the relevant final state.
Track selection requirements include transverse momentum p t > 0:1 GeV=c, at least 12 coordinate measurements in the DCH, a maximum distance of closest approach (DOCA) of 1.5 cm to the z axis, with this point at a maximum DOCA of 10 cm to the xy plane at z ¼ 0.
Kaon candidates are identified based on the normalized kaon, pion, and proton likelihood values (L K , L , and L p ) obtained from the particle identification system, by requiring L K =ðL K þ L Þ > 0:9 and L K =ðL K þ L p Þ > 0:2. Tracks that fulfill L K =ðL K þ L Þ < 0:82 and L p =ðL p þ L Þ < 0:98 are selected as pions. Additionally, in both cases the track should be inconsistent with electron identification.
Photon candidates are selected when their deposited energy in the EMC is larger than 100 MeV. Neutral pions are reconstructed from pairs of photons with combined mass within ½0:115; 0:155 GeV=c 2 and a 0 mass constraint is applied to them.
The D candidate decay products are fitted to a common vertex with a D meson mass constraint applied; candidates with a 2 fit probability greater than 0.1% are retained. Accepted D " D pairs are refitted to a common vertex consistent with the e þ e À interaction region, and those with a 2 fit probability p v ðD " DÞ greater than 0.1% are retained. . Signal yield dependence on p t ðK þ K À þ À Þ for (a) c ð1SÞ and (b) J=c . The vertical lines mark the p t ðK þ K À þ À Þ < 0:05 MeV=c region for selecting two-photon events. (c) Resulting K þ K À þ À mass distribution after applying the principal selection criteria discussed in Sec. III. TABLE II. D decay final states studied in this analysis; for channels N5, N6, and N7, inclusion of the corresponding charge conjugate combination is implied. Events with 0 candidates other than those from a D or " D decay of interest are rejected. These preselection criteria are identical for all five combinations of D decay modes.
The signal regions for accepted, unconstrained D candidates are then fitted using a multi-Gaussian signal function with free parameters 0 , r (minimal and maximal width) and m 0 ; the background is described by a polynomial. The full width at half maximum (FWHM) of the signal line shape in data is used to define each D signal region; D candidates are selected from a region of width AE1:5 FWHM around the mean mass. The mass windows are listed in Table III. From the list of accepted D " D candidates those produced in two-photon events are then selected by applying the three criteria defined in Sec. III (summarized in Table IV). These criteria are also identical for all combinations of D decay modes.
Depending on the decay mode, up to 2.5% of the events have multiple candidates which passed all selection criteria. In this case, the candidate with the best fit probability DÞ is chosen. Based on Monte Carlo (MC) studies, the correct candidate is selected in more than 99% of the cases with this method.
The resulting invariant mass spectra for D meson candidates after all selection criteria have been applied are shown in Fig. 4 for events in which the mass of the recoil " D   Table III To estimate the amount of combinatoric background in the signal region, the two-dimensional space spanned by the invariant masses of the D and " D candidates is divided into nine regions: one central signal region and eight sideband regions above and below the signal region as shown in Fig. 6 for the K À þ þ =K þ À À (C6) mode. The mass range for the signal region is AE1:5 FWHM around the mean mass. The sideband regions are 1.5 FWHM wide, leaving a gap of 1.5 FWHM between signal and sideband. No significant contribution from combinatoric background is observed in the D " D spectrum [ Fig. 5(c)].
An attempt was made to isolate the signal in Fig. 5(c) by a weighting method. This assumes that signal and background events have different angular distributions, and was successfully used in a previous BABAR analysis [19]. Simulations with a J PC ¼ 2 þþ signal (generated with its correct angular distribution) plus background showed that the method works well with high signal statistics and moderate background, but is not reliable with the limited statistics and background of the current analysis. Therefore, the method was not considered further in the present analysis.

V. MONTE CARLO STUDIES
For modeling the detector resolution, efficiency studies and the estimation of the two-photon width À of the resonance, MC events were generated which pass the same reconstruction and analysis chain as the experimental data. For each signal decay channel about 10 6 events were generated. Additional events were generated for background modes involving D Ã mesons. The GAMGAM twophoton event generator was used to simulate ! Zð3930Þ ! D " D events, while the decays of the D and " D mesons were generated by EVTGEN [20]. The detector response was simulated using the GEANT4 [21] package. The program GAMGAM uses the formalism suggested by Budnev, Ginzburg, Meledin, and Serbo (BGMS) [22]. It was developed for CLEO and was used, for example, in the analysis of c0 ð1PÞ, c2 ð1PÞ ! 4 decays [23]. GAMGAM was later adapted to BABAR and used for the analysis of c ð1S; 2SÞ ! K 0 S K AE Ç [16]. For small photon virtualities jq 1;2 j 2 (see Fig. 1) the differential cross section for the process e þ e À ! e þ e À , ! X is given by the product L Â F Â ð ! XÞ, where L is the two-photon flux. The form factor F extrapolates the process to virtual photons and is a priori not known. A plausible model is used [24], with m v being the mass of an appropriate   vector boson (, J=c , Z 0 ). In the calculations relevant to this analysis m v ¼ mðJ=c Þ was used, as the Zð3930Þ is expected to be a charmonium state. An alternative model was used in order to evaluate systematic uncertainties associated with MC simulations (see Sec. XI).
To validate the GAMGAM generator its output was compared to that of another two-photon generator (TREPS) used by Belle [25]. The cross sections for the reactions e þ e À ! e þ e À , ! c ð1S; 2SÞ were calculated in GAMGAM and compared to the Belle values [26]. In order to compare the different generators, the cross sections were calculated using the hypothetical values À Â Bð c ð1S; 2SÞ ! final stateÞ ¼ 1 keV, and q 2 1;2 was restricted to values smaller than 1 ðGeV=c 2 Þ 2 . The TREPS results were 2.11 pb for c ð1SÞ and 0.86 pb for c ð2SÞ. The corresponding GAMGAM values were 2.13 and 0.84 pb, respectively. The two generators are in agreement at the level of a few percent.
For a global check, the cross sections for the continuum reaction e þ e À ! e þ e À , ! þ À were calculated with GAMGAM for various c.m. energies and compared to QED predictions [27,28], which describe the data with high accuracy [29]. Here, the agreement was slightly worse, due to the imperfect tuning of the GAMGAM program for these reactions. Similar results were obtained when checking against calculations with nonrelativistic models for c ð1SÞ and c2 ð1PÞ [30]. Nevertheless this comparison showed that GAMGAM works properly under these conditions also. These studies lead to the assignment of a total systematic uncertainty of AE3% associated with the MC simulation (see Sec. XI).

VI. PURITY OF THE ! D D SAMPLE
The selection criteria used to enhance the two-photon content of the D " D sample were discussed in Sec. III. They were developed by investigating the reaction e þ e À ! K þ K À þ À X of Eq. (2). Figure 3(c) shows that after the selection procedure the signals associated with reactions, like that for the c ð1SÞ, are enhanced, while signals such as that for the J=c , which are typical of ISR production, are suppressed. The p t ðD " DÞ distribution is shown in Fig. 7 for events in the Zð3930Þ signal region, defined as the region from 3.91 to 3:95 GeV=c 2 . Here the p t ðD " DÞ selection criterion has not been applied. The data are fitted with a curve for events obtained from MC, plus a linear background derived from sideband studies of the D " D mass spectrum. The fit indicates that the majority of D " D candidates in the signal region result from twophoton interactions.

VII. RECONSTRUCTION EFFICIENCY
The reconstruction efficiency for each decay mode is calculated as a function of mðD " DÞ using MC events which pass the same reconstruction and selection criteria as real events and includes detector acceptance, track reconstruction, and particle identification efficiencies. The massdependent efficiency i ðmðD " DÞÞ for each channel i is fitted with a polynomial in mðD " DÞ and is found in each case to decrease with increasing D " D mass. For the combination of modes [ Fig. 5(c)], an overall weighted efficiency B ðmðD " DÞÞ, which includes the branching fractions for the D decays, is computed using as was done in Ref. [31]; N i ðmðD " DÞÞ is the number of D " D candidates in the data mass spectrum for channel i, and B i ðmðD " DÞÞ is defined as the product of the efficiency i as parametrized by the fitted polynomial and the branching fraction B i for the ith channel, as follows: The factor 1 2 originates from referring to D " D (D 0 " D 0 and D þ D À ) events, and the factor 5 from summing over the five channels. Figure 8 shows the mass dependence of B ðmðD " DÞÞ, which is parametrized by a straight line. The large uncertainties are due to the limited statistics available in the data samples. The error bars do not contain the uncertainties in the branching fractions; these will be discussed separately in Sec. XI in the context of systematic error estimation. The data are weighted by this mean efficiency, which is scaled by a constant value d to obtain weights near 1, DÞ for data in the Zð3930Þ signal region. The fitted line shape consists of the expected line shape obtained from MC plus a linear background (dotted line). The vertical line shows the p t criterion for selecting events. The histogram shows the shape of the p t ðD " DÞ distribution from simulated D Ã " D events with missing 0 or . The bump in this distribution is not seen in the data distribution, indicating that any D Ã " D background is small. signal yield obtained in the maximum likelihood fit [32]. The resulting D " D mass distribution will be discussed in Sec. VIII.

VIII. DETECTOR RESOLUTION AND SIGNAL YIELD
Monte Carlo events are used for the calculation of the mass-dependent detector resolution. The mass resolution is determined by studying the difference between the reconstructed and the generated D " D mass (Ám res ). As an example, the distribution for channel C6 is shown in Fig. 9(a). A good description of the distribution is obtained using a multi-Gaussian fit [Eq. (4)]. The parameters r and 0 ðmðD " DÞÞ were determined for every decay channel. The variation of 0 ðmðD " DÞÞ, which is parametrized by a second-order polynomial, and of the width (FWHM) of the resolution function with increasing mass are shown in Figs. 9(b) and 9(c). For channel C6, r ¼ 5:380 AE 0:137 and 0 ðmðD " DÞÞ ¼ ðÀ0:038 þ 0:018m À 0:002m 2 Þ GeV=c 2 , where mðD " DÞ is given in units of GeV=c 2 . The distributions of Fig. 9 are well described by the fitted curves shown. Comparing the generated Zð3930Þ mass with the reconstructed MC value shows that the latter is systematically low by about 0:9 MeV=c 2 , independently of the fit model. This effect is observed both in the combined fit and in fits to the individual channels. The measured J=c mass in the K þ K À þ À test sample (Sec. III) differs by the same value from the world average [14]; this offset has been seen in other studies at BABAR [16] as well. Accordingly, the mass value obtained from the fit to data will be corrected by þ0:9 MeV=c 2 . This offset value will also be used as a conservative estimate of the systematic uncertainty in the mass scale. The difference between the generated and reconstructed decay width values amounts to 0.14 MeV, and is discussed in Sec. XI with respect to systematic error estimation. In order to describe the signal structure in data around 3:93 GeV=c 2 a relativistic Breit-Wigner function BWðmÞ is used, where with m 0 as the nominal mass of the resonance; the Blatt-Weisskopf coefficients F r for different angular momentum values L are given by  is used, corresponding to the value given in Ref. [33]. The mass-dependent width is given by with À r the total width of the resonance. Here the existence of other possible decay modes is ignored. The momentum of a given D candidate in the D " D center of mass frame is denoted by p m ; p m 0 is the corresponding value for m ¼ m 0 . In the standard fit, spin J ¼ 2 (L ¼ 2) is chosen on the basis of the angular distribution analysis described in Sec. IX.
The signal function is convolved with the mass-and decay-mode-dependent resolution model parametrized as discussed previously in this section. The background is parametrized by the function which takes the D " D threshold m t into account. In the lower mass region, the line shape does not describe the background exactly. Other functional forms were tried (Sec. XI), but no improvement was obtained. The data and the curves which result from the standard (J ¼ 2) fit are shown in Fig. 10.
From the unbinned maximum likelihood fit to the five mass spectra the Zð3930Þ values m 0 ¼ ð3925:8 AE 2:7Þ MeV=c 2 and À r ¼ ð21:3 AE 6:8Þ MeV are obtained for the mass and total width, respectively (all errors in this section are statistical only). The mass is corrected by þ0:9 MeV=c 2 as described above, resulting in a final mass value of ð3926:7 AE 2:7Þ MeV=c 2 . The efficiency-corrected yield amounts to N ¼ ð76 AE 17Þ signal events. This value is based on weights around 1 as discussed in Sec. VII; taking the constant used to scale the efficiency into account [see Eq. (8)], this corresponds to a total Zð3930Þ signal of N B ¼ ð285 AE 64Þ Â 10 3 events.
The statistical significance of the peak is 5:8 and is derived from the difference Á lnL between the negative logarithmic likelihood of the nominal fit and that of a fit where the parameter for the signal yield is fixed to zero. This is then used to evaluate a p value: where fðz; n d Þ is the 2 probability density function and n d is the number of degrees of freedom, 3 in this case. We then determine the equivalent one-dimensional significance from this p value.

IX. ANGULAR DISTRIBUTION AND SPIN OF THE Zð3930Þ STATE
General conservation laws limit the possibilities for the J PC values of the Zð3930Þ state. For two-photon production the initial state has positive C parity and hence the final state must have positive C parity also. For the D " D final state, C ¼ ðÀ1Þ LþS ¼ ðÀ1Þ L since the total spin S is zero. Positive C parity then implies that the D " D system must have orbital angular momentum L which is even, and hence have even parity. It follows that for the Zð3930Þ state J PC ¼ J þþ with J ¼ 0; 2; 4; . . . In order to investigate the possible values of J, we have compared the decay angular distribution measured in the Zð3930Þ signal region to the distributions expected for J ¼ 0 and J ¼ 2; higher spin values are very unlikely for a state only 200 MeV=c 2 above threshold.
The decay angle is defined as the angle of the D meson in the D " D system relative to the D " D lab momentum vector. Figure 11 shows the Zð3930Þ signal yield obtained from fits to the D " D mass spectrum for ten regions of j cosj. The data have been weighted by a cos-dependent efficiency, which was determined in a similar manner as described in Sec. VII for the mass-dependent efficiency (Fig. 12). In these fits, the mass and width of the resonance have been  fixed to the values found in Sec. VIII, and Eq. (14) has been used to describe the background. Other background models have been tried, obtaining distributions fully consistent with Fig. 11.
The function describing the decay angular distribution for spin 2 has been calculated using the helicity formalism and has the form It has been assumed that the dominating amplitude has helicity 2. This is in agreement with previous measurements [34] and theoretical predictions [24,30]. The distribution of Eq. (16) was fitted to the experimental angular distribution, and a 2 =NDF value of 5:63=9 was obtained, with NDF indicating the number of degrees of freedom. For a flat distribution, which is expected for spin 0, a 2 =NDF ¼ 15:55=9 was obtained. It follows that the preferred J PC assignment is 2 þþ .

X. TWO-PHOTON WIDTH OF THE Zð3930Þ STATE
From the efficiency-corrected number of observed signal events, N B , we determine the total experimental cross section exp ðe þ e À ! e þ e À ; ! Zð3930Þ; Zð3930Þ ! D " where the integrated luminosity for the data sample analyzed is R Ldt ¼ ð384 AE 4Þ fb À1 and the error is only statistical.
On the other hand, the cross section for Zð3930Þ production is given by and can be calculated using GAMGAM. Here L is the twophoton flux, F is the form factor (see Sec. V), m Z (À tot ) is the resonance mass (width), and À is the two-photon width of the resonance. The kinematical factor K is given by K ¼ ðq 1 q 2 Þ 2 À q 2 1 q 2 2 (q i represent four vectors of photons). Further information can be found in Refs. [22,24,35]. The cross section depends on the spin of the resonance and on À . It is plotted for J ¼ 2 and J ¼ 0 in Fig. 13 as a function of À . From a comparison to the experimental cross section [Eq. (17)], the partial width À Â BðZð3930Þ ! D " DÞ is found to have the value ð0:24 AE 0:05Þ keV when J ¼ 2 is chosen as the most probable spin value (see Sec. IX).

XI. SYSTEMATIC ERROR ESTIMATION
Several sources of systematic uncertainty have been considered for the mass, decay width, and signal yield of the Zð3930Þ state. The yield determines the value of DÞ. The standard fit to the efficiency-corrected mass spectrum is repeated with appropriate modifications. The differences Á between the results obtained and the standard results are used as estimates of systematic uncertainty. No correlations have been taken into account. The results are summarized in Table V. Deviations for the mass (jÁmj), total width (jÁÀj), and two-photon width [jÁðÀ Â BÞj] are considered negligible if they are less than 0:05 MeV=c 2 , 0.05 MeV, and 0.0005 keV, respectively.

A. Fit parametrization
Signal line shape.-The standard fit has assumed spin J ¼ 2 for the resonance (Sec. VIII). Using different spin values and R values has no significant impact on the results [ Table V; ÁðÀ Â BÞ numbers are given for spin J ¼ 2 only].
Background description.-Different parametrizations of the background in the mðD " DÞ distribution have been used. Besides the nominal background [Eq. (14)], the following background shape was tried: the fit had a slightly worse, but still acceptable, likelihood value. The mass value changes by Ám ¼ þ0:4 MeV=c 2 , the width by ÁÀ ¼ þ3:0 MeV, the signal yield by þ9 events with respect to the standard fit, and À Â B changes accordingly by þ0:029 keV (Table V). Other background models yield consistent estimates for this source of systematic uncertainty.

B. Detector resolution
Fit precision and mass scale.-A fit of the convolution of signal line shape and resolution model to the MC sample has been performed. The mass offset observed in MC has been included by correcting the mass value by þ0:9 MeV=c 2 . As a conservative estimate, this number is also used as the systematic uncertainty for the mass scale. The deviation between the generated width and the value obtained from the fit is 0.14 MeV, and again this is used as a conservative estimate of systematic uncertainty. Based on the uncertainty of the width, a value of ÁÀ Â B ¼ 0:001 keV is derived.
Resolution model.-The parameters of the multi-Gaussian resolution model were modified. The number of steps was enlarged from 25 to 35, the total convolution range for each data point enlarged by þ0:02 MeV=c 2 , and the parameter r of the multi-Gaussian was varied within its fit uncertainty r. The corresponding shifts in the mass are Ám ¼ þ0:2, <0:05, and <0:05 MeV=c 2 . For ÁÀ, shifts of À0:2, À0:9, and À0:1 MeV are obtained; from the modified signal yield, shifts of À0:003, À0:003, and <0:0005 keV were obtained for À Â B (Table V).

C. Combined reconstruction efficiency
Parametrization.-The average mass-dependent reconstruction efficiency has been parametrized by a straight line in the standard fit (Fig. 8). Using a fit with a second- order polynomial, the width changes by À0:4 MeV; no mass shift was observed with respect to the standard fit result. For the signal yield, þ1 entry is obtained; this yields no significant shift for À Â B (Table V).
Tracking and neutrals correction.-For the tracking efficiency a correction by À0:8% is applied per chargedparticle track. This gives a correction factor of 0.968 for modes N4, N5, and 0.953 for N6, N7, and C6. The systematic uncertainty assigned to the tracking efficiency is 1.4% per track for decays with more than 5 charged-particle tracks and 1.3% otherwise. The resulting uncertainty for À Â B is 0.022 keV. Concerning efficiency corrections for neutral particles, a correction factor of 0.984 with an uncertainty of 3% per 0 is used for modes N5 and N7. The resulting uncertainty for À Â B is 0.003 keV (Table V).
Uncertainty on the D branching fractions.-The errors on the D branching fractions have been taken into account by varying the values of B i used in Eq. (6) within their standard deviations. No significant change is observed in mass and decay width. For the two-photon width ÁðÀ Â BÞ ¼ AE0:010 keV is obtained ( Table V).
Effect of angular distribution on efficiency.-The MC data sample used to obtain the efficiency and resolution was generated with a flat distribution in cos. To estimate the effect of the angular distribution on the reconstruction efficiency, a MC sample described by a sin 4 distribution has been generated and reconstructed. Comparing these reconstructed data with the nominal MC sample, the mean efficiencies differ by 8%, relatively, resulting in ÁðÀ Â BÞ ¼ AE0:018 keV.

D. Cross-section calculation from GAMGAM
Precision.-In Sec. V a relative uncertainty of AE3% was obtained for the calculated cross section. Propagating this error into the calculation of À Â B, an uncertainty ÁðÀ Â BÞ ¼ AE0:007 keV results.
Form factor.-In the standard analysis the form factor of Eq. (5) has been used with m v ¼ mðJ=c Þ. In order to estimate potential systematic effects, the cross section was evaluated using a model predicted by perturbative QCD [36] The cross section calculated with GAMGAM does not increase significantly ( % 0:1%) compared to that obtained using Eq. (5). Simultaneously the experimental efficiency decreases by 1%, so that the net effect on À Â B is small. Similar effects have been observed when data and calculations with and without q 2 selection criteria are compared [25,26,37], and also in a previous CLEO analysis [38]. As a result a systematic uncertainty of AE1% is attributed to form factor uncertainty and this yields a deviation ÁðÀ Â BÞ ¼ AE0:002 keV.

E. Other uncertainties
Particle identification (PID).-For PID studies, the pion selection criteria have been tightened significantly, and the efficiency has been recalculated accordingly. The fit to the mass spectrum yields a change of À0:4 MeV=c 2 for the mass and À1:8 MeV for the width. For ÁðÀ Â BÞ a change of À0:004 keV results.
D mass uncertainty.-The uncertainty of the D meson mass is taken into account. Both for D 0 and D AE , the uncertainty is 0:17 MeV=c 2 [14], which results in an uncertainty of AE0:34 MeV=c 2 in the mass of the Zð3930Þ state.
Integrated luminosity uncertainty.-For the integrated luminosity, an uncertainty of AE1% is assigned. From this an uncertainty ÁðÀ Â BÞ ¼ AE0:002 keV is obtained.

F. Total systematic uncertainty
The systematic uncertainty estimates discussed in Secs. XI A, XI B, XI C, XI D, and XI E are summarized in Table V. The individual estimates are combined in quadrature to yield net systematic uncertainty estimates on the Zð3930Þ mass, total width, and value of À Â BðZð3930Þ ! D " DÞ of 1:1 MeV=c 2 , 3.6 MeV, and 0.04 keV, respectively, as reported on the last line of Table V.

XII. SUMMARY
In the ! D " D reaction a signal in the D " D mass spectrum has been observed near 3:93 GeV=c 2 with a significance of 5:8 which agrees with the observation of the Zð3930Þ resonance by the Belle Collaboration [13]. The mass and total width of the Zð3930Þ state are measured to be ð3926:7 AE 2:7ðstatÞ AE 1:1ðsystÞÞ MeV=c 2 and ð21:3 AE 6:8ðstatÞ AE 3:6ðsystÞÞ MeV, respectively.
The production and decay mechanisms allow only positive parity and C parity, and an analysis of the Zð3930Þ decay angular distribution favors a tensor over a scalar interpretation. The preferred assignment for spin and parity of the Zð3930Þ state is therefore J PC ¼ 2 þþ . The product of the branching fraction to D " D times the two-photon width of the Zð3930Þ state is measured to be À Â BðZð3930Þ ! D " DÞ ¼ ð0:24 AE 0:05ðstatÞ AE 0:04ðsystÞÞ keV, assuming spin J ¼ 2. The parameters obtained are consistent with the Belle results, and with the expectations for the c2 ð2PÞ state.

ACKNOWLEDGMENTS
We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality