Comment on the X(3915) nonstandard hadron candidate

I review the experimental evidence for the $X(3915)$, the candidate nonstandard meson associated with $\omega J\psi$ resonance-like peaks in $B\rightarrow K\omega J\psi$ and $\gamma\gamma\rightarrow\omega J\psi$ near $M(\omega J\psi)=3920$~MeV, and address the conjecture that it can be identified as the $\chi_{c2}^\prime$, the radial excitation of the $\chi_{c2}$ charmonium state. Since the partial decay width for $B\rightarrow K X(3915)$ is at least an order-of-magnitude larger than that for $B\rightarrow K\chi_{c2}$, its assignment as the $\chi_{c2}^\prime$ is dubious.


Introduction
A number of meson candidates, dubbed the XYZ mesons, that contain charmed-quark anticharmed-quark (cc) pairs but do not match expectations for any of the unassigned levels of the [cc] charmonium meson spectrum, have been observed in recent experiments [1]. In some cases, the distinction between the new states that are nonstandard hadrons and conventional charmonium mesons remains controversial. This is especially the case for the X(3915) that was first observed by Belle [2] and confirmed by BaBar [3,4] as a near-threshold peak in the ωJ/ψ invariant mass distribution in exclusive B → KωJ/ψ decays (see Fig. 1a). An ωJ/ψ mass peak with similar mass and width was seen in the two-photon fusion process γγ → ωJ/ψ, again by both Belle [5] and BaBar [6] (see Fig. 1b); BaBar reported its J PC to be 0 ++ . The similar masses and widths of the peaks seen in the two production modes suggest that these are being produced a single state (i.e., the X(3915)). The Particle Data Group's (PDG) average values for the mass and width measurements from both production channels are [7]: M(X(3915)) = 3918.4 ± 1.9 MeV and Γ(X(3915)) = 20.0 ± 5.0 MeV, (1) and the product branching fraction for X(3915) production in B + meson decays is The measured γγ → ωJ/ψ production rates are used to extract the (J PC -dependent) widths: Γ γγ (X(3915)) × B(X → ωJ/ψ) = 54 ± 9 eV (0 ++ ) or 11.4 ± 2.7 eV (2 ++ ). Babar [4]. The low mass peak in the BaBar data is attributed to X(3872) → ωJ/ψ (see inset); the higher mass peak is the X(3915) → ωJ/ψ signal. The Belle analysis did not consider the possible presence of an X(3872) → ωJ/ψ signal. b) The ωJ/ψ mass spectrum for γγ → ωJ/ψ from (top) Belle [5] and (bottom) Babar [6].
2 The X(3915) is not the χ c0 charmonium state?
The Babar group's J PC determination was based on an analysis of angular correlations amongst the final-state particles in their γγ → ωJ/ψ event sample [6]. The important angles for distinguishing J = 2 + from J = 0 ± are θ * n , the angle between n, the normal to the ω → π + π − π 0 decay plane, and the γγ axis in the omega rest frame, and θ ln , the angle between n and the direction of the + from J/ψ → + − decay (see Fig. 2a). Figure 2b shows the BaBar cos θ * n distribution together with the expectation for J = 0 ± as a solid red line and J = 2 + as a dashed blue curve. There is a strong χ 2 penalty for the near-zero event likelihood near cos θ * n = ±1 for the J = 2 + hypothesis to fluctuate upward to the observed levels of ∼ 8 and ∼ 9 events, and this is the main support BaBar's J = 0 conclusion. The J = 2 hypothesis seems to fit the BaBar cos θ ln distribution (see Fig. 2c) better than that for J = 0. But in this case, the likelihood of ∼ 6 expected events near cos θ ln = ±1 to fluctuate downward to the observed 2 events is not so improbable. With J = 0 established, the 0 + vs. 0 − discrimination mostly relies on the angle θ n , which is the angle between the ω's flight path and n in the ωJ/ψ restframe. The BaBar cos θ n distribution shown in Fig. 2d favors 0 + over 0 − , mostly because of the 10 events near cos θ n = +1, where the 0 − expectation is zero.
BaBar's J PC = 0 ++ assignment led them to suggest it as a suitable candidate for the 2 3 P 0 charmonium state, commonly known as the χ c0 , and it was listed as such in the 2014 PDG tables [8]. However, this assignment had some problems and was challenged for a number of reasons [9]: the partial width for X(3915) → ωJ/ψ, which would be an OZI-suppressed decay mode for a charmonium state, was too large; the lack of evidence for X(3915) → DD, which would be the dominant mode for the χ c0 ; and the small, 9 MeV, mass splitting between the χ c2 and the X(3915), which is an order-of-magnitude lower than the smallest theoretical estimates for M χ c2 − M χ c0 [10,11]. This assignment was finally put to rest in 2017 by Belle, when they reported the observation of the X * (3860), a DD resonance with mass 3862 +47 −35 MeV in e + e − → J/ψDD annhilations with preferred spin-parity of 0 ++ [12]. These properties, particularly the strong DD decay mode, match well the expectations for the χ c0 , and the X * (3862) is clearly a much stronger candidate for this state than the X(3915).
ison of the cos θ * n distribution with J P = 0 ± (solid red) and 2 + (dashed blue) expectations. c) The corresponding plot for cos θ ln . d) The cos θ n distribution with expectations for 0 + in solid red and 0 − in dashed blue. (From ref. [6].) 3 Is it the χ c2 charmonium state?
The χ c2 was first spotted by Belle [13] and subequently confirmed by BaBar [14] as a prominent M(DD) peak in the two-photon fusion process γγ → DD that has a distinct sin 4 θ * production angle dependence that is characteristic of a J = 2 state. The mass and width [7]: are consistent with charmonium expectations for the χ c2 and there are no reasons to question this assignment. The Belle (BaBar) M(DD) and dN/d| cos θ * | distributions are shown in Fig. 3a (b). Belle and BaBar measurements of its two-photon production rate are also in good agreement and are characterized by the product BaBar's J PC = 0 ++ assignment for the X(3915) was based on a comparison to a 2 ++ scenario that only considered a helicity-2 component (h 2 ) and ignored the possibility of any helicity-0 contribution. This assumption of "helicity-2 dominance" originate from a theoretical analysis that found that in two-photon production of tensor mesons, the helicity-0 component (h 0 ) is zero in the non-relativistic limit [15]. The authors of ref. [16] point out that in the case of charmonium, the suppression of helicity-0 contributions only applies to mesons that are 100% cc, which is generally considered to be unlikely for charmonium mesons with masses above the 2m D open-charm threshold (see, e.g., ref. [17]). This is important because if the J PC of the X(3915) is 2 ++ , the mass peak identified with the X(3915) could be conceivably be due to an ωJ/ψ decay mode of the χ c2 (2P) charmonium state. The dashed lines in Fig. 4a show the ref. [16] comparison of the Belle M(DD) and | cos θ| with an h 0 1.5h 2 mixture to represent the X(3915). Figure 4b) shows BaBar's cos θ * n and cos θ ln distributions with expectations for 0 ++ , and 2 ++ with h = 0 & h = 2. With the inclusion of some h = 0 contribution, the χ 2 distinction between 0 ++ and 2 ++ angular distributions is diminished and the authors conclude that the X(3915) could be a χ c2 state that contains a sizable non-cc component.  In addition to violating helicity-2 dominance, which ref. [16] claims may not be a problem, there are other concerns with the X(3915) = χ c2 assignment. These are briefly discussed here.

Mass and width differences
Belle and BaBar measurements of the γγ → ωJ/ψ mass peak, 3915 ± 4 and 3919 ± 3 MeV, respectively, are both lower, by 2σ, than their respective χ c2 → DD mass peak measurements, 3929 ± 5 and 3927 ± 3 MeV. Since the measurements reference well known massesω and J/ψ for the X(3915) and D-meson for the χ c2 -systematic effects are small.
On the other hand, a recent LHCb report on the M(DD) distribution for inclusive D-meson pair production in high energy proton-proton collisions included observation of a distinct peak in the χ c2 mass region, shown in Fig. 5a, with mass M = 3921.9 ± 0.6 ± 0.2 MeV, 2σ below the χ c2 value listed in eqn. 4 [18]. The reported width, Γ = 36.6 ± 1.9 ± 0.9 MeV, is 2σ higher than the eqn. 4 value. The LHCb group attributes this peak to the χ c2 . Figure 5b shows recent BESIII M(ωJ/ψ) results for e + e − → Y(4220) → γωJ/ψ, where there is a strong X(3872) → ωJ/ψ signal and 3σ "evidence" for two higher mass peaks [19]. The fitted mass of the middle peak is M = 3926.4±2.5 MeV, near the Belle and BaBar results for χ c2 → DD. Thus, the current situation with mass measurements is inconclusive.

A large OZI-violating ωJ/ψ decay width for a [cc] meson
With the Γ γγ × B values listed in eqns. 3 and 5, the χ c2 assignment implies that which is large for an OZI-rule-violating decay of an above-open-charm-threshold charmonium state, and more than an order-of-magnitude higher than the measured corresponding significance [19]. c) B + → K + χ c1 and K + χ c2 signals from the full Belle data set [20].
In 2011, with their full event sample accumulated over ten years, Belle reported ∼ 3σ evidence for B + → K + χ c2 based on the 33 ± 11 event signal shown in Fig. 5c [20]. The inferred branching fraction, B(B + → K + χ c2 ) = 1.1 ± 0.4 × 10 −5 , is smaller that the product branching fraction for X(3915) → ωJ/ψ production in B + meson decays (eqn. 2). Since B(χ c2 → DD) cannot exceed unity, eqn. 6 implies B(χ c2 → ωJ/ψ) < 0.08 (90% CL). Thus, if the X(3915) produced in B → KωJ/ψ is the χ c2 , the B-meson decay width to K + χ c2 would be more than an order of magnitude larger than that to K + χ c2 . This contradicts theoretical expectations that B → K[cc] decay widths decrease with increasing radial [cc] quantum numbers [22]. Suppression of B → Kχ ( ) c2 is not unexpected. The primary mechanism for B-meson (bq) decays to K[cc] final states isb →c plus a virtual W + that, in turn, materializes as cs. The final-state c-andc-quark form the [cc] state and thes-and "spectator" q-quark form the K. This process is only allowed for J PC = 0 −+ , 1 −− and 1 ++ [cc] states, decays to [cc] states with other J PC values are higher-order and expected to be "factorization suppressed" [23]. The Belle results on B → Kχ c2 shown in Fig. 5c

Summary and conclusions
Despite its observation by different experiments in a variety of production channels, the nature of the X(3915) remains a mystery. If it is a nonstandard XYZ meson, it cannot be easily interpreted by any of the proposed models for these states. For example: its mass is too low for a QCD-hybrid [24], and not near an appropriate threshold for a molecular state or a cusp effect [25]; the lack of evidence for a ηη c decay mode [26] is problematic for a diquarkdiantiquark assignmment [27]. Thus, if it is an XYZ meson, it is a very interesting one.
The sum total of existing data on ωJ/ψ and DD production in the ∼ 3925 MeV mass region cannot be explained as being simply due to the χ c2 charmonium state. While a (tenuous) case could be made that the near-3925 MeV mass peaks seen by the LHCb in pp → DDX, Belle and BaBar in γγ → ωJ/ψ & DD and BESIII in Y(4220) → γωJ/ψ are all due to decays of the χ c2 , the existing evidence is not conclusive. Moreover, a very strong case can be made against a χ c2 interpretation of the ωJ/ψ peak seen in B → KωJ/ψ decays.
More refined mass and width measurements are needed, and reliable, separate J PC determinations for the ωJ/ψ peaks produced via γγ fusion, radiative Y(4220) transitions, and B-meson decays that eschew the helicity-2 dominance constraint are essential. The LHCb group has demonstrated that they can isolate clean B + → K + ωJ/ψ signals with good efficiency [28] and I look forward to high-statistics results from them in the near future.

Acknowledgements
I congratulate Phi-to-Psi-2018 organizers for an interesting and provocative meeting. This work is supported by the CAS President's International Fellowship Initiative.