The Belle Collaboration

The charmonium-like state $X$(3872) has been observed in various interactions since its first observation in $B\rightarrow KJ/\psi\pi^{+}\pi^{-}$ decays [1]. Its spin, parity, and charge conjugation are determined to be $1^{++}$ [2], but its internal structure is still a puzzle [3, 4]. Subsequent to the spin-parity determination, the $X(3872)$ has not been searched for in two-photon interactions because axial-vector particles are forbidden to decay to two real photons ¹¹1 The $X(3872)$ was searched for in two-photon interactions before its spin-parity determination: S. Dobbs et al. (CLEO Collaboration), Phys. Rev. Lett. 94, 032004 (2005).. However, mesons with $J^{PC}=1^{++}$ can be produced if one or both photons are highly virtual [6]—denoted as $\gamma^{*}$.

We perform the first search for a $1^{++}$ charmonium state in two-photon interactions using $e^{+}e^{-}\rightarrow e^{+}e^{-}X(3872)$, 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 ²²2We use “electron” to denote both electron and positron.. Such events are called single-tag events. The $X$(3872) is reconstructed via its decay to $J/\psi\pi^{+}\pi^{-}$ ($J/\psi\rightarrow\ell^{+}\ell^{-}$). By measuring the momentum of the tagging electron, we measure the $Q^{2}$ dependence of $X(3872)$ production, where $-Q^{2}$ is the invariant mass-squared of the virtual photon. If the $X(3872)$ has a molecular component in its structure, it must have a steeper $Q^{2}$ dependence than the regular $c\bar{c}$ 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 $X(3872)$. Early attempts to calculate such decay widths for charmonium-like exotic states have been reported in Ref. [8]. We also search for the $X(3915)$ in the same final state through the $G$-parity-violating $J/\psi\rho^{0}$ ($\rho^{0}\rightarrow\pi^{+}\pi^{-}$) channel, as well as $J/\psi\omega$ ($\omega\rightarrow\pi^{+}\pi^{-}$) decay[9].

We use 825 fb^-1 of data collected by the Belle detector operated at the KEKB $e^{+}e^{-}$ asymmetric collider [10, 11]. The data were taken at the $\Upsilon(nS)$ resonances ($n\leq 5$) and nearby energies, 9.43 GeV$<\sqrt{s}<$ 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 $e^{+}e^{-}\rightarrow e^{+}e^{-}X(3872)$ events in which the $X(3872)$ decays to $J/\psi\pi^{+}\pi^{-}$ and $J/\psi$ decays leptonically. For simulating radiative $J/\psi$ decays, we use PHOTOS [16, 17]. A GEANT3-based 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 (IP) and having $p_{\rm T}>0.1$ GeV/$c$, with two or more having $p_{\rm T}>0.4$ GeV/$c$, where $p_{\rm T}$ is the transverse momentum with respect to the $e^{+}$ direction.

$J/\psi$ candidates are reconstructed by their decays to $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$. A charged track is identified as an electron if its electron likelihood ratio, ${\cal L}_{e}/({\cal L}_{e}+{\cal L}_{\pi})$, is greater than 0.66 and as a muon if it is not selected as an electron and if its muon likelihood ratio, ${\cal L}_{\mu}/({\cal L}_{\mu}+{\cal L}_{\pi}+{\cal L}_{K})$, is greater than 0.66; ${\cal L}_{x}$ 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 $e^{+}e^{-}$ pair, we include the four-momenta 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 $p_{\rm T}>0.4$ GeV/$c$. The electron momentum includes the momenta of radiated photons, using the same requirements as for the electrons from $J/\psi$ decays.

We identify a charged track as a pion if it satisfies the likelihood ratio criteria of ${\cal L}_{\pi}/({\cal L}_{\pi}+{\cal L}_{K})>0.2$, ${\cal L}_{\mu}/({\cal L}_{\mu}+{\cal L}_{\pi}+{\cal L}_{K})<0.9$, ${\cal L}_{e}/({\cal L}_{e}+{\cal L}_{\pi})<0.6$, and its $E/p$ is less than 0.8 [21]. Events should have no photons with energy above 0.4 GeV or $\pi^{0}$ candidates with $\chi^{2}$ from the mass-constrained fit less than 4.0.

As the $X(3872)$ 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 ($\pi\pm 0.1$) rad.

The total visible transverse momentum of the event, $p^{*}_{\rm T}$ ³³3 The $e^{+}e^{-}$ center-of-mass quantities are indicated by asterisks., should be less than 0.2 GeV/$c$. We also require that the measured energy of the $J/\psi\pi^{+}\pi^{-}$ system, $E^{*}_{\rm obs}$, be consistent with the expectation, $E^{*}_{\rm exp}$, calculated from the momentum of the tagging electron and the direction and invariant mass of the $J/\psi\pi^{+}\pi^{-}$ 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 $p^{*}_{\rm T}$ vs. $E^{*}_{\rm obs}/E^{*}_{\rm exp}$ 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 $e^{+}$-tagging events.

We search for $X(3872)$ and $X(3915)$ by looking for events in the $J/\psi\pi^{+}\pi^{-}$ mass distribution, $M(J/\psi\pi^{+}\pi^{-})$. 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 $X(3872)$ and 3.895–3.935 GeV/$c^{2}$ for the $X(3915)$. The former accommodates the $X(3872)$ with a known mass of $3871.69\pm 0.17$ MeV/$c^{2}$ and a decay width less than $1.2$ MeV ⁴⁴4 Recent measurements of the decay width show $\Gamma^{\rm BW}_{X(3872)}=0.96^{+0.19}_{-0.18}\pm 0.21$ MeV [3] and $\Gamma^{\rm BW}_{X(3872)}=1.39\pm 0.24\pm 0.10$ MeV [4].; the latter accommodates the $X(3915)$ with a known mass of $3918.4\pm 1.9$ MeV/$c^{2}$ and a decay width of $20\pm 5$ MeV. We constrain the $J/\psi$ mass to 3.09690 GeV/$c^{2}$ when we calculate $M(J/\psi\pi^{+}\pi^{-})$ [24].

The dominant background, centered at 3.686 GeV/$c^{2}$, arises from radiatively produced $\psi(2S)$, $e^{+}e^{-}\rightarrow e^{+}e^{-}\psi(2S)$, with $\psi(2S)\rightarrow J/\psi\pi^{+}\pi^{-}$. Figure 2 shows the $M(J/\psi\pi^{+}\pi^{-})$ distribution in data in the vicinity of $\psi(2S)$. Although the width of the $\psi(2S)$ 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 $J/\psi\pi^{+}\pi^{-}$ produced by initial-state radiation (ISR) [25]. To remove $\psi(2S)$ events, we veto events within 0.03 GeV/$c^{2}$ of the $\psi(2S)$ mass, 3.686 GeV/$c^{2}$. Figure 3 shows the $Q^{2}$ distribution after removing those events, where $Q^{2}=2(p_{\rm in}\cdot p_{\rm out}-m_{e}^{2}c^{2})$ and $p_{\rm in}$ and $p_{\rm out}$ 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 MC is pure $X(3872)$. Since two-photon processes are strongly suppressed at high $Q^{2}$, we require $Q^{2}<25$ GeV²/$c^{2}$ to reduce non-two-photon background. Our measurement is insensitive for $Q^{2}<1.5$ GeV²/$c^{2}$ due to low reconstruction efficiency.

Figure 4 shows the observed events in the $Q^{2}$ vs. $M(J/\psi\pi^{+}\pi^{-})$ plane. Three events are in the $X(3872)$ signal region; no events are in the $X(3915)$ region. The masses of the events in the $X(3872)$ signal region are 3.8726, 3.8701 and 3.8742 GeV/$c^{2}$, averaging to 3.8723$\pm$0.0012 GeV/$c^{2}$, where the uncertainty is statistical. At masses below the $X(3872)$ region, 3.716-3.867 GeV/$c^{2}$, there are six events, presumably from $\psi(2S)$ events; at masses above the $X(3872)$, there are no events below 4.266 GeV/$c^{2}$, in region of the $Y(4260)$ mass. A similar distribution was seen in the Belle ISR study [25]. The $J/\psi\pi^{+}\pi^{-}$ 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 $J/\psi\pi^{+}\pi^{-}$, like $\psi(2S)$, while the $C$-even $X(3872)$ peak can only be produced from the two-photon process. The absence of a prominent $Y(4260)$ enhancement in our data argues against non-negligible contribution from the $C$-odd process through the decay $\gamma^{*}\rightarrow Y(4260)\rightarrow\gamma X(3872)$ [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 GeV/$c^{2}$ $<M<$ 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 (3-5)$\times 10^{-2}$/(10 MeV/$c^{2}$) in the region between 3.8 GeV/$c^{2}$ and 4.2 GeV/$c^{2}$. This explains the absence of events between the $X(3872)$ and 4.26 GeV/$c^{2}$.

To estimate the background level in the $X(3872)$ signal region, we fit a linear function

to the data in the region $\pm 0.156$ GeV/$c^{2}$ centered at the $X(3872)$ 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 $X(3872)$ and the upper boundary, $3.716$ GeV/$c^{2}$, of the $\psi(2S)$ vetoed region. Using an unbinned extended maximum-likelihood fit, we obtain $a=-345\pm 195$ /(GeV/$c^{2}$)² and $b=10.5\pm 10.1$ /(GeV/$c^{2}$). This yields $n_{\rm b}=0.11\pm 0.10$ background events in the $X(3872)$ 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, $a^{\prime}/(M(J/\psi\pi^{+}\pi^{-})-b^{\prime})^{c^{\prime}}$ with $b^{\prime}$ set to 2.4 GeV/$c^{2}$; the fit is insensitive to the value of $b^{\prime}$. This gives $n_{\rm b}=0.096\pm 0.068$. The other is a linear function with a break at 3.800 GeV/$c^{2}$, $a^{\prime\prime}(M(J\psi\pi^{+}\pi^{-})-3.800~{}{\rm GeV}/c^{2})+b^{\prime\prime}$ for $M(J\psi\pi^{+}\pi^{-})<$3.800 GeV/$c^{2}$ and $b^{\prime\prime}$ for $M(J\psi\pi^{+}\pi^{-})\geq$3.800 GeV/$c^{2}$, based on the shapes of the $M(J/\psi\pi^{+}\pi^{-})$ distributions in the ISR [25, 28] and the $e^{+}e^{-}$ annihilation studies [29, 30]. This gives $n_{\rm b}=0.122\pm 0.095$. From the variations of $n_{\rm b}$ in the three forms, we derive $\pm 0.013$ for the systematic uncertainty. This is negligible compared to the statistical uncertainty. The estimated number of background events is $0.11\pm 0.10$, including statistical and systematic uncertainties.

With this background, the significance of three events is 3.2$\sigma$. For the $X(3872)$ signal, with three observed and 0.11 expected background events, we calculate the number of signal events, $N_{\rm sig}=2.9{+2.2\atop-2.0}(\text{stat.})\pm 0.1(\text{syst.})$, at 68% confidence level (C.L.). For the $X(3915)$ signal, with zero observed and 0.3 expected background events, we obtain $N_{\rm sig}<2.14$ 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]

where $L_{\gamma^{*}\gamma}$ is the single-tag luminosity function, $M$ is the resonance mass, $-Q^{2}$ is the invariant mass squared of the virtual photon, $\Gamma_{\gamma^{*}\gamma}(Q^{2})$ is the $\gamma^{*}\gamma$ decay width, $W$ is the invariant mass of the $\gamma^{*}\gamma$ system, and $J$ is the resonance spin. The factor of two comes from the existence of two production modes: $e^{-}\gamma^{*}$ and $e^{+}\gamma^{*}$ scattering.

For a $J{=}1$ resonance, spin-parity conservation forbids production at $Q^{2}=0$. To remove the $Q^{2}$-dependence from $\Gamma_{\gamma^{*}\gamma}(Q^{2})$, we use the reduced $\gamma\gamma$ decay width $\tilde{\Gamma}_{\gamma\gamma}$ defined as [6, 33]

using its $Q^{2}$ dependence near zero; $\Gamma_{\gamma^{*}\gamma}^{\rm LT}$ is the $\gamma^{*}\gamma$ 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

for $Q^{2}\ll M^{2}$, where an extra factor of two 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), $\epsilon$ is the ratio $L^{\rm LT}/L^{\rm TT}$, where $L^{\rm LT}$ is the luminosity function for the production of one longitudinally polarized photon and one transversely polarized photon and $L^{\rm TT}$ is that for two transversely polarized photons. Using the Schuler-Berends-Gulik (SBG) model [6]⁵⁵5 As a validation of the SBG model at higher $Q^{2}$, Ref. [32] provides measurements of single-tag to no-tag ratios for the $\gamma\gamma$ decay widths for $\chi_{c0}$ and $\chi_{c2}$, which agree with the predictions of this model. for $q\bar{q}$-type axial-vector mesons, this can be extended to higher $Q^{2}$ [33]:

where

accounting for contributions from helicity 0 and 1. The SBG model, based on $c\bar{c}$, 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, $\tilde{\Gamma}_{\gamma\gamma}$, we use Eqs. (6) and (7) assuming the $X(3872)$ is a pure $c\bar{c}$ state [6],

where $\varepsilon_{\rm eff}(Q^{2})$ is the $Q^{2}$-dependent reconstruction efficiency, $L_{\rm int}$ is the integrated luminosity, ${\cal B}(X\rightarrow J/\psi\pi^{+}\pi^{-})$ is the branching fraction of the $X(3872)$ to $J/\psi\pi^{+}\pi^{-}$, and ${\cal B}(J/\psi\rightarrow\ell^{+}\ell^{-})=0.1193$ is the branching fraction of $J/\psi$ to lepton pairs [24]. We estimate the reconstruction efficiency from MC, in which we model the $X(3872)$ decay as $X(3872)\rightarrow J/\psi\rho^{0}$ with $J/\psi\rightarrow\ell^{+}\ell^{-}$ and $\rho^{0}\rightarrow\pi^{+}\pi^{-}$ and with all daughter particles isotropically distributed in the rest frames of their parents. The decay model via $\rho$ is motivated by the measured mass distributions [1, 35, 36]. It has a reconstruction efficiency 12% higher than that for non-resonant $\pi^{+}\pi^{-}$; we include a 6% systematic uncertainty to account for this. The angular distribution of the decay products of the $X(3872)$ negligibly affects the reconstruction, as confirmed by simulating with an alternative model with decay angles of daughters from a $J^{P}=1^{+}$ resonance with helicities 0 and 1.

Detection efficiencies range from 4% to 8% for $Q^{2}$ between 3 GeV${}^{2}/c^{2}$ and 25 GeV${}^{2}/c^{2}$ and have smaller values for $Q^{2}<3$ GeV${}^{2}/c^{2}$. They are estimated for our three center-of-mass energies on the $\Upsilon(2S)$, $\Upsilon(4S)$, and $\Upsilon(5S)$ resonances and average the values weighted by their corresponding integrated luminosities. We also average over the four detection modes given the two tagging charges ($e^{+}$ and $e^{-}$) and the two $J/\psi$ decay modes ($e^{+}e^{-}$ and $\mu^{+}\mu^{-}$).

The luminosity functions for our beam energies are calculated as functions of $Q^{2}$ using TREPSBSS. We set $\epsilon=1$ as a convention for the present application of Eq. (7)[6]. After performing the $Q^{2}$ integration in Eq. (8), from $Q_{\rm min}^{2}=1.5$ GeV²/$c^{2}$ to $Q_{\rm max}^{2}=25$ GeV²/$c^{2}$, we obtain

including the total systematic uncertainty from the integration.

The dominant systematic uncertainty on $\tilde{\Gamma}_{\gamma\gamma}{\cal B(}X\rightarrow J/\psi\pi^{+}\pi^{-})$ is from the reconstruction efficiency, primarily due to differences between MC and data. The largest uncertainty, 7%, is in the $J/\psi$ selection from the uncertainty of the $e^{+}e^{-}$ background level. We estimate the total systematic uncertainty to be 13%.

From $N_{\rm sig}$, we determine

To set a limit on $\tilde{\Gamma}_{\gamma\gamma}$, we need ${\cal B}(X\rightarrow J/\psi\pi^{+}\pi^{-})$. We derive an upper limit, using the measured products of $B$-meson decay branching fractions and the $X(3872)$ decay branching fractions, ${\cal B}(B^{+}\rightarrow K^{+}X){\cal B}(X\rightarrow J/\psi\pi^{+}\pi^{-}~{}{\rm and~{}other~{}specific~{}final~{}states})$ ⁶⁶6 From ${\cal B}(B^{+}\rightarrow K^{+}X){\cal B}(X\rightarrow J/\psi\pi^{+}\pi^{-})=(8.6\pm 0.6)\times 10^{-6}$ and the sum over the measured products of the branching fractions, ${\cal B}(B^{+}\rightarrow K^{+}X){\cal B}(X\rightarrow J/\psi\pi^{+}\pi^{-},J/\psi\gamma,\psi(2S)\gamma,D^{0}\bar{D^{0}}\pi^{0},\bar{D^{*0}}D^{0})=(1.4\pm 0.4)\times 10^{-4}$, where we exclude $\bar{D}^{*0}\rightarrow\bar{D}^{0}\pi^{0}$, we obtain that ${\cal B}(X\rightarrow J/\psi\pi^{+}\pi^{-})<0.061$ using the Bayesian method at 90% C.L. This limit is consistent with C. Li and C.-Z. Yuan, Phys. Rev. D 100, 094003 (2019).. With the measured lower limit [35, 38, 24], this gives $0.032<{\cal B}(X\rightarrow J/\psi\pi^{+}\pi^{-})<0.061$ at 90% C.L. Using the Feldman-Cousins method for three observed events and 0.11 background, we obtain $0.995<N_{\rm sig}<7.315$ at 90% C.L. This, with Eq. (9), divided by ${\cal B}(X\rightarrow J/\psi\pi^{+}\pi^{-})$, gives the $\tilde{\Gamma}_{\gamma\gamma}$ range: 20-500 eV. This is consistent with values predicted for the $c\bar{c}$ model [6, 8]. For a comparison of experimental results with non-$c\bar{c}$ models, we must wait for improved calculations in the future.

No events consistent with $X(3915)\rightarrow J/\psi\pi^{+}\pi^{-}$ are observed. This, combined with past measurements [9, 39], indicates no excess of $G$-parity-violating decays of $X(3915)$.

In summary, we find the first evidence for $X(3872)$ production in two-photon, $\gamma^{*}\gamma$, interactions. We observe three $X(3872)$ candidates with a significance of 3.2$\sigma$ and an estimated yield of $2.9{+2.2\atop-2.0}~{}(\text{stat.})~{}\pm 0.1~{}(\text{syst.})$. From this, we obtain $\tilde{\Gamma}_{\gamma\gamma}{\cal B}(X(3872)\rightarrow J/\psi\pi^{+}\pi^{-})=5.5{+4.1\atop-3.8}(\text{stat.})\pm 0.7(\text{syst.})$ eV, assuming the $Q^{2}$ dependence of a $c\bar{c}$ meson model. With future advances in calculations of $\tilde{\Gamma}_{\gamma\gamma}$ for non-$c\bar{c}$ states and higher luminosities accumulated by Belle II, we expect this method will clarify our understanding of the $X(3872)$.