The charmoniumlike state Xð3872Þ has been observed in various interactions since its first observation in B → KJ=ψπ þ π -decays [1]. Its spin, parity, and charge conjugation are determined to be 1 þþ [2], but its internal structure is still a puzzle [3,4]. Subsequent to the spinparity determination, the Xð3872Þ has not been searched for in two-photon interactions because axial-vector particles are forbidden to decay to two real photons [5]. However, mesons with J PC ¼ 1 þþ can be produced if one or both photons are highly virtual [6]-denoted as γ Ã .

We perform the first search for a 1 þþ charmonium state in two-photon interactions using e þ e -→ e þ e -Xð3872Þ, where one of the final-state electrons, referred to as a tagging electron, is observed, and the other scatters at an extremely forward (backward) angle and is not detected [7]. Such events are called single-tag events. The Xð3872Þ is reconstructed via its decay to J=ψπ þ π -(J=ψ → l þ l -). By measuring the momentum of the tagging electron, we measure the Q 2 dependence of Xð3872Þ production, where -Q 2 is the invariant mass squared of the virtual photon. If the Xð3872Þ has a molecular component in its structure, it must have a steeper Q 2 dependence than the regular cc state. Hence, the single-tag two-photon interactions provide information on the structure of this state. The value of the two-photon decay width, obtained from this measurement, is sensitive to the internal structure of the Xð3872Þ. Early attempts to calculate such decay widths for charmoniumlike exotic states have been reported in Ref. [8]. We also search for the Xð3915Þ in the same final state through the G-parity-violating J=ψρ 0 (ρ 0 → π þ π -) channel, as well as J=ψω (ω → π þ π -) decay [9].

We use 825 fb -1 of data collected by the Belle detector operated at the KEKB e þ e -asymmetric collider [10,11]. The data were taken at the ϒðnSÞ resonances (n ≤ 5) and nearby energies, 9.43 < ffiffi ffi s p < 11.03 GeV. The Belle detector is a general-purpose magnetic spectrometer [12,13]. Charged-particle momenta are measured by a silicon vertex detector and a cylindrical drift chamber. Electron and charged-pion identification relies on a combination of the drift chamber, time-of-flight scintillation counters, aerogel Cherenkov counters, and an electromagnetic calorimeter made of CsI(Tl) crystals. Muon identification relies on resistive plate chambers in the iron return yoke.

For Monte Carlo (MC) simulations, used to set selection criteria and derive the reconstruction efficiency, we use TREPSBSS [14,15] to generate single-tag e þ e -→ e þ e -Xð3872Þ events in which the Xð3872Þ decays to J=ψπ þ π -and J=ψ decays leptonically. For simulating radiative J=ψ decays, we use PHOTOS [16,17]. A GEANT3based program simulates the detector response [18].

Since one final-state electron is undetected, we select events with exactly five charged tracks, each coming from the interaction point and having p T > 0.1 GeV=c, with two Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP 3 .

or more having p T > 0.4 GeV=c, where p T is the transverse momentum with respect to the e þ direction.

J=ψ candidates are reconstructed by their decays to e þ e -or μ þ μ -. A charged track is identified as an electron if its electron likelihood ratio L e =ðL e þ L π Þ is greater than 0.66 and as a muon if it is not selected as an electron and if its muon likelihood ratio

is greater than 0.66; L x is the likelihood for a particle to be of species x [19,20]. We require the mass of the lepton pair to be in the range 3.047-3.147 GeV=c 2 . In the calculation of the invariant mass of an e þ e -pair, we include the fourmomenta of radiated photons, having energy less than 0.2 GeV and angle relative to an electron direction of less than 0.04 rad.

The tagging electron must have an electron likelihood ratio greater than 0.95 or E=p greater than 0.87, where E is the energy measured by the electromagnetic calorimeter and p is the momentum of the particle. We require that the tagging electron have momentum above 1 GeV=c and p T > 0.4 GeV=c. The electron momentum includes the momenta of radiated photons, using the same requirements as for the electrons from J=ψ decays.

We identify a charged track as a pion if it satisfies the likelihood ratio criteria of

E=p is less than 0.8 [21]. Events should have no photons with energy above 0.4 GeV or π 0 candidates with χ 2 from the mass-constrained fit less than 4.0.

As the Xð3872Þ should be back to back with the tagging electron projected in the plane perpendicular to the beam axis, we require the difference between their azimuthal angles be in the range ðπ AE 0.1Þ rad.

The total visible transverse momentum of the event p à T [22] should be less than 0.2 GeV=c. We also require that the measured energy of the J=ψπ þ π -system E à obs be consistent with the expectation E à exp calculated from the momentum of the tagging electron and the direction and invariant mass of the J=ψπ þ π -system, imposing energy-momentum conservation. Since the energy and total transverse momentum are correlated, we impose a two-dimensional criterion

Figure 1 shows the distribution of events and these selection criteria in the p Ã

T vs E Ã obs =E Ã exp plane. Finally, we place a requirement on the missing momentum of the event, equal to the momentum of the unmeasured electron that goes down the beam pipe. We require the missing-momentum projection in the e -beam direction in the center-of-mass frame be less than -0.4 GeV=c for e --tagging events and greater than 0.4 GeV=c for e þ -tagging events.

We search for Xð3872Þ and Xð3915Þ by looking for events in the

The reconstructed mass resolution is expected to be 2.5 MeV=c 2 from the MC simulation. We define two signal regions: 3.867-3.877 GeV=c 2 for the Xð3872Þ and 3.895-3.935 GeV=c 2 for the Xð3915Þ. The former accommodates the Xð3872Þ with a known mass of 3871.69 AE 0.17 MeV=c 2 and a decay width less than 1.2 MeV [23]; the latter accommodates the Xð3915Þ with a known mass of 3918.4 AE 1.9 MeV=c 2 and a decay width of 20 AE 5 MeV. We constrain the J=ψ mass to 3.09690 GeV=c 2 when we calculate MðJ=ψπ þ π -Þ [24].

The dominant background, centered at 3.686 GeV=c 2 , arises from radiatively produced ψð2SÞ, e þ e -→ e þ e -ψð2SÞ, with ψð2SÞ → J=ψπ þ π -. Figure 2 shows the MðJ=ψπ þ π -Þ distribution in data in the vicinity of ψð2SÞ. Although the width of the ψð2SÞ peak is 2.7 MeV=c 2 , it has a tail on the higher mass side. This feature was also seen in previous studies of J=ψπ þ π -produced by initial-state radiation (ISR) [25]. T and E Ã obs =E Ã exp ; events below the line are accepted. To remove ψð2SÞ events, we veto events within 0.03 GeV=c 2 of the ψð2SÞ mass, 3.686 GeV=c 2 . Figure 3 shows the Q 2 distribution after removing those events, where

and p in and p out are the four-momenta of the incoming (beam) and outgoing (tagging) electrons, and m e is the electron mass. In Fig. 3, data are dominated by background events, while the MC simulation is pure Xð3872Þ. Since two-photon processes are strongly suppressed at high Q 2 , we require Q 2 < 25 GeV 2 =c 2 to reduce non-two-photon background. Our measurement is insensitive for Q 2 < 1.5 GeV 2 =c 2 due to low reconstruction efficiency.

Figure 4 shows the observed events in the Q 2 vs MðJ=ψπ þ π -Þ plane. Three events are in the Xð3872Þ signal region; no events are in the Xð3915Þ region. The masses of the events in the Xð3872Þ signal region are 3.8726, 3.8701, and 3.8742 GeV=c 2 , averaging to 3.8723 AE 0.0012 GeV=c 2 , where the uncertainty is statistical. At masses below the Xð3872Þ region, 3.716-3.867 GeV=c 2 , there are six events, presumably from ψð2SÞ events; at masses above the Xð3872Þ, there are no events below 4.266 GeV=c 2 , in the region of the Yð4260Þ mass. A similar distribution was seen in the Belle ISR study [25]. The J=ψπ þ π -events can also originate from t-channel photon exchange with the emission of a virtual photon, which we call internal bremsstrahlung (IB) [26]. Both processes produce C-odd J=ψπ þ π -, like ψð2SÞ, while the C-even Xð3872Þ peak can only be produced from the two-photon process. The absence of a prominent Yð4260Þ enhancement in our data argues against nonnegligible contribution from the C-odd process through the decay γ Ã → Yð4260Þ → γXð3872Þ [27]. To estimate the background from IB, which has the same final-state particle configuration as our process and is hence difficult to separate, we use the ISR data [25]. By fitting the ISR data to our data in the region 3.5 < M < 4.5 GeV=c 2 , corrected for the differences in the diagrams of s and t channels, we estimate the number of background events to be ð3-5Þ × 10 -2 =ð10 MeV=c 2 Þ in the region between 3.8 and 4.2 GeV=c 2 . This explains the absence of events between the Xð3872Þ and 4.26 GeV=c 2 .

To estimate the background level in the Xð3872Þ signal region, we fit a linear function maxð0

to the data in the region AE0.156 GeV=c 2 centered at the Xð3872Þ mass, excluding the signal region; a and b are free in the fit. The width of 0.156 GeV=c 2 is determined by the distance between the Xð3872Þ and the upper boundary, 3.716 GeV=c 2 , of the ψð2SÞ vetoed region. Using an unbinned extended maximum-likelihood fit, we obtain a ¼ -345 AE 195=ðGeV=c 2 Þ 2 and b ¼ 10.5 AE 10.1=ðGeV=c 2 Þ. This yields n b ¼ 0.11 AE 0.10 background events in the Xð3872Þ signal window, where the uncertainty is statistical only.

To derive the systematic uncertainty due to background modeling, we test two modified fitting functions. One is a power function, a 0 =½MðJ=ψπ þ π -Þb 0 c 0 with b 0 set to 2.4 GeV=c 2 ; the fit is insensitive to the value of b 0 . This gives n b ¼ 0.096 AE 0.068. The other is a linear function with a break at 3.800 [25,28] and the e þ e -annihilation studies [29,30]. This gives n b ¼ 0.122 AE 0.095. From the variations of n b in the three forms, we derive AE0.013 for the systematic uncertainty. This is negligible compared to the statistical uncertainty. The estimated number of background events is 0.11 AE 0.10, including statistical and systematic uncertainties.

With this background, the significance of three events is 3.2 σ. For the Xð3872Þ signal, with three observed and 0.11 expected background events, we calculate the number of signal events, N sig ¼ 2.9 þ2.2 -2.0 ðstatÞ AE 0.1ðsystÞ, at 68% confidence level (C.L.). For the Xð3915Þ signal, with zero observed and 0.3 expected background events, we obtain N sig < 2.14 at 90% C.L. The Feldman-Cousins method is used in both cases [31].

The differential cross section for the production of a resonance (X) in a single-tag two-photon interaction is expressed as [32] dσ ee ðXÞ

where L γ Ã γ is the single-tag luminosity function, M is the resonance mass, -Q 2 is the invariant mass squared of the virtual photon, Γ γ Ã γ ðQ 2 Þ is the γ Ã γ decay width, W is the invariant mass of the γ Ã γ system, and J is the resonance spin. The factor of 2 comes from the existence of two production modes: e -γ Ã and e þ γ Ã scattering. For a J ¼ 1 resonance, spin-parity conservation forbids production at Q 2 ¼ 0. To remove the Q 2 dependence from Γ γ Ã γ ðQ 2 Þ, we use the reduced γγ decay width Γγγ defined as [6,33] Γγγ ≡ lim

using its Q 2 dependence near zero; Γ LT γ Ã γ is the γ Ã γ decay width corresponding to a formation of the resonance from a longitudinal (virtual) photon and a transverse (real) photon. Substituting this expression into Eq. (3), we obtain

, where an extra factor of 2 comes from the difference in the number of spin degrees of freedom: the longitudinal component has one degree of freedom and the transverse component has two with unpolarized incident photons. In Eq. ( 5), ϵ is the ratio L LT =L TT , where L LT is the luminosity function for the production of one longitudinally polarized photon and one transversely polarized photon, and L TT is that for two transversely polarized photons. Using the Schuler-Berends-Gulik (SBG) model [6,34] for q q-type axial-vector mesons, this can be extended to higher Q 2 [33],

dσ ee ðXÞ dQ 2 ¼ Γγγ FðM; Q 2 ; ϵÞ

where

accounting for contributions from helicity 0 and 1. The SBG model, based on cc, is the only model available at present that can reliably extend Eq. ( 5) to the higher Q 2 region: Eq. ( 7).

To relate the number of signal events and the decay width Γγγ , we use Eqs. ( 6) and (7), assuming the Xð3872Þ is a pure cc state [6],

is the branching fraction of the Xð3872Þ to J=ψπ þ π -, and

1193 is the branching fraction of J=ψ to lepton pairs [24]. We estimate the reconstruction efficiency from MC simulation, in which we model the Xð3872Þ decay as Xð3872Þ → J=ψρ 0 with J=ψ → l þ l - and ρ 0 → π þ π -and with all daughter particles isotropically distributed in the rest frames of their parents. The decay model via ρ is motivated by the measured mass distributions [1,35,36]. It has a reconstruction efficiency 12% higher than that for nonresonant π þ π -; we include a 6% systematic uncertainty to account for this. The angular distribution of the decay products of the Xð3872Þ negligibly affects the reconstruction, as confirmed by simulating with an alternative model with decay angles of daughters from a J P ¼ 1 þ resonance with helicities 0 and 1. Detection efficiencies range from 4% to 8% for Q 2 between 3 and 25 GeV 2 =c 2 and have smaller values for Q 2 < 3 GeV 2 =c 2 . They are estimated for our three centerof-mass energies on the ϒð2SÞ, ϒð4SÞ, and ϒð5SÞ resonances and average the values weighted by their corresponding integrated luminosities. We also average over the four detection modes given the two tagging charges (e þ and e -) and the two J=ψ decay modes (e þ e -and μ þ μ -).

The luminosity functions for our beam energies are calculated as functions of Q 2 using TREPSBSS. We set ϵ ¼ 1 as a convention for the present application of Eq. ( 7) [6]. After performing the Q 2 integration in Eq. ( 8), from

including the total systematic uncertainty from the integration.

PHYSICAL REVIEW LETTERS 126, 122001 (2021)

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The dominant systematic uncertainty on Γγγ BðX → J=ψπ þ π -Þ is from the reconstruction efficiency, primarily due to differences between MC simulation and data. The largest uncertainty, 7%, is in the J=ψ selection from the uncertainty of the e þ e -background level. We estimate the total systematic uncertainty to be 13%.

From N sig , we determine Γγγ B(Xð3872Þ → J=ψπ þ π -) ¼ 5.5 þ4.1 -3.8 ðstatÞ AE 0.7ðsystÞ eV:

To set a limit on Γγγ , we need BðX → J=ψπ þ π -Þ. We derive an upper limit, using the measured products of B-meson decay branching fractions and the Xð3872Þ decay branching fractions BðB þ → K þ XÞBðX → J=ψπ þ π -and other specific final states) [37]. With the measured lower limit [24,35,38], this gives 0.032 < BðX → J=ψπ þ π -Þ < 0.061 at 90% C.L. Using the Feldman-Cousins method for three observed events and 0.11 background, we obtain 0.995 < N sig < 7.315 at 90% C.L. This, with Eq. ( 9), divided by BðX → J=ψπ þ π -Þ, gives the Γγγ range: 20-500 eV. This is consistent with values predicted for the cc model [6,8]. For a comparison of experimental results with non-cc models, we must wait for improved calculations in the future.

No events consistent with Xð3915Þ → J=ψπ þ π -are observed. This, combined with past measurements [9,39], indicates no excess of G-parity-violating decays of Xð3915Þ.

In summary, we find the first evidence for Xð3872Þ production in two-photon γ Ã γ interactions. We observe three Xð3872Þ candidates with a significance of 3.2σ and an estimated yield of 2.9 þ2.2 -2.0 ðstatÞ AE 0.1ðsystÞ. From this, we obtain Γγγ B(Xð3872Þ → J=ψπ þ π -) ¼ 5.5 þ4.1 -3.8 ðstatÞ AE 0.7ðsystÞ eV, assuming the Q 2 dependence of a cc meson model. With future advances in calculations of Γγγ for non-cc states and higher luminosities accumulated by Belle II, we expect this method will clarify our understanding of the Xð3872Þ.