Comment on the X(3915) nonstandard hadron candidate

TL;DR

This paper reviews experimental evidence for the X(3915) meson candidate, questioning its classification as a charmonium state due to decay width inconsistencies and suggesting it may not be a standard meson.

Contribution

It critically assesses the X(3915) as a nonstandard meson candidate and challenges its identification as the radially excited _{c2} state based on decay width analysis.

Findings

The decay width for BK X(3915) is much larger than for BK _{c2}.

The evidence does not support X(3915) as the _{c2}^ state.

X(3915) likely not a standard charmonium meson.

Abstract

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.

Equations

$$M(X(3915))$$

$$\mathcal{B}(B^{+}\to K^{+}X(3915))\times \mathcal{B}(X\to \omega J/\psi )$$

$${\Gamma }_{\gamma \gamma }(X(3915))\times \mathcal{B}(X\to \omega J/\psi )$$

$$M({\chi }_{c2}^{\prime })$$

$${\Gamma }_{\gamma \gamma }({\chi }_{c2}^{\prime })\times \mathcal{B}({\chi }_{c2}^{\prime }\to D\bar{D})$$

$$\frac{\mathcal{B}({\chi }_{c2}^{\prime }\to \omega J/\psi )}{\mathcal{B}({\chi }_{c2}^{\prime }\to D\bar{D})}$$

Full text

1 Introduction

A number of meson candidates, dubbed the $XYZ$ mesons, that contain charmed-quark anticharmed-quark ($c\bar{c}$) pairs but do not match expectations for any of the unassigned levels of the $[c\bar{c}]$ charmonium meson spectrum, have been observed in recent experiments Olsen:2017bmm . 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 Abe:2004zs and confirmed by BaBar Aubert:2007vj ; delAmoSanchez:2010jr as a near-threshold peak in the $\omega J/\psi$ invariant mass distribution in exclusive $B\rightarrow K\omega J/\psi$ decays (see Fig. 1a). An $\omega J/\psi$ mass peak with similar mass and width was seen in the two-photon fusion process $\gamma\gamma\rightarrow\omega J/\psi$, again by both Belle Uehara:2009tx and BaBar Lees:2012xs (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 Tanabashi:2018oca :

[TABLE]

and the product branching fraction for $X(3915)$ production in $B^{+}$ meson decays is

[TABLE]

The measured $\gamma\gamma\rightarrow\omega J/\psi$ production rates are used to extract the ($J^{PC}$-dependent) widths:

[TABLE]

2 The $X(3915)$ is not the $\chi_{c0}^{\prime}$ charmonium state?

The Babar group’s $J^{PC}$ determination was based on an analysis of angular correlations amongst the final-state particles in their $\gamma\gamma\rightarrow\omega J/\psi$ event sample Lees:2012xs . The important angles for distinguishing $J=2^{+}$ from $J=0^{\pm}$ are $\theta_{\rm n}^{*}$, the angle between $\vec{\rm n}$, the normal to the $\omega\rightarrow\pi^{+}\pi^{-}\pi^{0}$ decay plane, and the $\gamma\gamma$ axis in the omega rest frame, and $\theta_{\rm ln}$, the angle between $\vec{\rm n}$ and the direction of the $\ell^{+}$ from $J/\psi\rightarrow\ell^{+}\ell^{-}$ decay (see Fig. 2a). Figure 2b shows the BaBar $\cos\theta_{\rm n}^{*}$ distribution together with the expectation for $J=0^{\pm}$ as a solid red line and $J=2^{+}$ as a dashed blue curve. There is a strong $\chi^{2}$ penalty for the near-zero event likelihood near $\cos\theta_{\rm n}^{*}=\pm 1$ for the $J=2^{+}$ hypothesis to fluctuate upward to the observed levels of $\sim 8$ and $\sim 9$ events, and this is the main support BaBar’s $J=0$ conclusion. The $J=2$ hypothesis seems to fit the BaBar $\cos\theta_{\rm ln}$ distribution (see Fig. 2c) better than that for $J=0$. But in this case, the likelihood of $\sim 6$ expected events near $\cos\theta_{\rm ln}=\pm 1$ to fluctuate downward to the observed $\simeq 2$ events is not so improbable. With $J=0$ established, the $0^{+}$ vs. $0^{-}$ discrimination mostly relies on the angle $\theta_{\rm n}$, which is the angle between the $\omega$’s flight path and $\vec{\rm n}$ in the $\omega J/\psi$ restframe. The BaBar $\cos\theta_{\rm n}$ distribution shown in Fig. 2d favors $0^{+}$ over $0^{-}$, mostly because of the $\simeq 10$ events near $\cos\theta_{\rm n}=+1$, 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 $\chi_{c0}^{\prime}$, and it was listed as such in the 2014 PDG tables Agashe:2014kda . However, this assignment had some problems and was challenged for a number of reasons Guo:2012tv : the partial width for $X(3915)\rightarrow\omega J/\psi$, which would be an OZI-suppressed decay mode for a charmonium state, was too large; the lack of evidence for $X(3915)\rightarrow D\bar{D}$, which would be the dominant mode for the $\chi_{c0}^{\prime}$; and the small, $\simeq 9$ MeV, mass splitting between the $\chi_{c2}^{\prime}$ and the $X(3915)$, which is an order-of-magnitude lower than the smallest theoretical estimates for $M_{\chi_{c2}^{\prime}}-M_{\chi_{c0}^{\prime}}$ Wang:2014voa ; Olsen:2014maa . This assignment was finally put to rest in 2017 by Belle, when they reported the observation of the $X^{*}(3860)$, a $D\bar{D}$ resonance with mass $3862^{+47}_{-35}$ MeV in $e^{+}e^{-}\rightarrow J/\psi D\bar{D}$ annhilations with preferred spin-parity of $0^{++}$ Chilikin:2017evr . These properties, particularly the strong $D\bar{D}$ decay mode, match well the expectations for the $\chi_{c0}^{\prime}$, and the $X^{*}(3862)$ is clearly a much stronger candidate for this state than the $X(3915)$.

3 Is it the $\chi_{c2}^{\prime}$ charmonium state?

The $\chi_{c2}^{\prime}$ was first spotted by Belle Uehara:2005qd and subequently confirmed by BaBar Aubert:2010ab as a prominent $M(D\bar{D})$ peak in the two-photon fusion process $\gamma\gamma\rightarrow D\bar{D}$ that has a distinct $\sin^{4}\theta^{*}$ production angle dependence that is characteristic of a $J=2$ state. The mass and width Tanabashi:2018oca :

[TABLE]

are consistent with charmonium expectations for the $\chi_{c2}^{\prime}$ and there are no reasons to question this assignment. The Belle (BaBar) $M(D\bar{D})$ and $dN/d|\cos\theta^{*}|$ 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

[TABLE]

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 Krammer:1977an . The authors of ref. Zhou:2015uva point out that in the case of charmonium, the suppression of helicity-0 contributions only applies to mesons that are 100% $c\bar{c}$, which is generally considered to be unlikely for charmonium mesons with masses above the $2m_{D}$ open-charm threshold (see, e.g., ref. Pennington:2007xr ).

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 $\omega J/\psi$ decay mode of the $\chi_{c2}(2P)$ charmonium state. The dashed lines in Fig. 4a show the ref. Zhou:2015uva comparison of the Belle $M(D\bar{D})$ and $|\cos\theta|$ with an $h_{0}\simeq 1.5h_{2}$ mixture to represent the $X(3915)$. Figure 4b) shows BaBar’s $\cos\theta_{\rm n}^{*}$ and $\cos\theta_{\rm ln}$ distributions with expectations for $0^{++}$, and $2^{++}$ with $h=0$ & $h=2$. With the inclusion of some $h=0$ contribution, the $\chi^{2}$ distinction between $0^{++}$ and $2^{++}$ angular distributions is diminished and the authors conclude that the $X(3915)$ could be a $\chi_{c2}^{\prime}$ state that contains a sizable non-$c\bar{c}$ component.

3.1 Other aspects of the $X(3915)=\chi_{c2}^{\prime}$ assignment

In addition to violating helicity-2 dominance, which ref. Zhou:2015uva claims may not be a problem, there are other concerns with the $X(3915)=\chi_{c2}^{\prime}$ assignment. These are briefly discussed here.

3.1.1 Mass and width differences

Belle and BaBar measurements of the $\gamma\gamma\rightarrow\omega J/\psi$ mass peak, $3915\pm 4$ and $3919\pm 3$ MeV, respectively, are both lower, by $\simeq 2\sigma$, than their respective $\chi_{c2}^{\prime}\rightarrow D\bar{D}$ mass peak measurements, $3929\pm 5$ and $3927\pm 3$ MeV. Since the measurements reference well known masses – $\omega$ and $J/\psi$ for the $X(3915)$ and $D$-meson for the $\chi_{c2}^{\prime}$– systematic effects are small.

On the other hand, a recent LHCb report on the $M(D\bar{D})$ distribution for inclusive $D$-meson pair production in high energy proton-proton collisions included observation of a distinct peak in the $\chi_{c2}^{\prime}$ mass region, shown in Fig. 5a, with mass $M=3921.9\pm 0.6\pm 0.2$ MeV, $2\sigma$ below the $\chi_{c2}^{\prime}$ value listed in eqn. 4 Aaij:2019evc . The reported width, $\Gamma=36.6\pm 1.9\pm 0.9$ MeV, is $2\sigma$ higher than the eqn. 4 value. The LHCb group attributes this peak to the $\chi_{c2}^{\prime}$.

Figure 5b shows recent BESIII $M(\omega J/\psi)$ results for $e^{+}e^{-}\rightarrow Y(4220)\rightarrow\gamma\omega J/\psi$, where there is a strong $X(3872)\rightarrow\omega J/\psi$ signal and $3\sigma$ “evidence” for two higher mass peaks Ablikim:2019zio . The fitted mass of the middle peak is $M=3926.4\pm 2.5$ MeV, near the Belle and BaBar results for $\chi_{c2}^{\prime}\rightarrow D\bar{D}$. Thus, the current situation with mass measurements is inconclusive.

3.1.2 A large OZI-violating $\omega J/\psi$ decay width for a $[c\bar{c}]$ meson

With the $\Gamma_{\gamma\gamma}\times{\mathcal{B}}$ values listed in eqns. 3 and 5, the $\chi_{c2}^{\prime}$ assignment implies that

[TABLE]

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 ratio for $\psi^{\prime\prime}\rightarrow\pi^{+}\pi^{-}J/\psi$ and $D\bar{D}$. If $\chi_{c2}^{\prime}\rightarrow D\bar{D}$ and $D\bar{D}^{*}$ are the dominant decay modes and $\Gamma_{\chi_{c2}^{\prime}}(D\bar{D}^{*})\approx\Gamma_{\chi_{c2}^{\prime}}(D\bar{D})$ (as predicted in ref. Barnes:2005pb ), then $\Gamma_{\chi_{c2}^{\prime}}(\omega J/\psi)>200$ keV (at the $\sim$90% CL), and much larger than any measured OZI-violating width for a charmonium state.

3.1.3 ${\mathcal{B}}(B\rightarrow K\chi_{c2}^{\prime})>>{\mathcal{B}}(B\rightarrow K\chi_{c2})$ ?

In 2011, with their full event sample accumulated over ten years, Belle reported $\sim 3\sigma$ evidence for $B^{+}\rightarrow K^{+}\chi_{c2}$ based on the $33\pm 11$ event signal shown in Fig. 5c Bhardwaj:2011dj . The inferred branching fraction, ${\mathcal{B}}(B^{+}\rightarrow K^{+}\chi_{c2})=1.1\pm 0.4\times 10^{-5}$, is smaller that the product branching fraction for $X(3915)\rightarrow\omega J/\psi$ production in $B^{+}$ meson decays (eqn. 2). Since ${\mathcal{B}}(\chi_{c2}^{\prime}\rightarrow D\bar{D})$ cannot exceed unity, eqn. 6 implies ${\mathcal{B}}(\chi_{c2}^{\prime}\rightarrow\omega J/\psi)<0.08$ (90% CL). Thus, if the $X(3915)$ produced in $B\rightarrow K\omega J/\psi$ is the $\chi_{c2}^{\prime}$, the $B$-meson decay width to $K^{+}\chi_{c2}^{\prime}$ would be more than an order of magnitude larger than that to $K^{+}\chi_{c2}$. This contradicts theoretical expectations that $B\rightarrow K[c\bar{c}]$ decay widths decrease with increasing radial $[c\bar{c}]$ quantum numbers Bodwin:1992qr .

Suppression of $B\rightarrow K\chi_{c2}^{(^{\prime})}$ is not unexpected. The primary mechanism for $B$-meson ($\bar{b}q$) decays to $K[c\bar{c}]$ final states is $\bar{b}\rightarrow\bar{c}$ plus a virtual $W^{+}$ that, in turn, materializes as $c\bar{s}$. The final-state $c$- and $\bar{c}$-quark form the $[c\bar{c}]$ state and the $\bar{s}$- and “spectator” $q$-quark form the $K$. This process is only allowed for $J^{PC}=0^{-+},1^{--}\ {\rm and}\ 1^{++}$ $[c\bar{c}]$ states, decays to $[c\bar{c}]$ states with other $J^{PC}$ values are higher-order and expected to be “factorization suppressed” Beneke:1999br . The Belle results on $B\rightarrow K\chi_{c2}$ shown in Fig. 5c demonstrate that for $J^{PC}=2^{++}$ $[c\bar{c}]$ states, factorization suppression is very effective: ${\mathcal{B}}(B\rightarrow K\chi_{c2})<0.04\times{\mathcal{B}}(B\rightarrow K\chi_{c1})$ (90% CL).

4 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 Liu:2012ze , and not near an appropriate threshold for a molecular state or a cusp effect Olsen:2018ikz ; the lack of evidence for a $\eta\eta_{c}$ decay mode Vinokurova:2015txd is problematic for a diquark-diantiquark assignmment Lebed:2016yvr . Thus, if it is an $XYZ$ meson, it is a very interesting one.

The sum total of existing data on $\omega J/\psi$ and $D\bar{D}$ production in the $\sim 3925$ MeV mass region cannot be explained as being simply due to the $\chi_{c2}^{\prime}$ charmonium state. While a (tenuous) case could be made that the near-3925 MeV mass peaks seen by the LHCb in $pp\rightarrow D\bar{D}X$, Belle and BaBar in $\gamma\gamma\rightarrow\omega J/\psi$ & $D\bar{D}$ and BESIII in $Y(4220)\rightarrow\gamma\omega J/\psi$ are all due to decays of the $\chi_{c2}^{\prime}$, the existing evidence is not conclusive. Moreover, a very strong case can be made against a $\chi_{c2}^{\prime}$ interpretation of the $\omega J/\psi$ peak seen in $B\rightarrow K\omega J/\psi$ decays.

More refined mass and width measurements are needed, and reliable, separate $J^{PC}$ determinations for the $\omega J/\psi$ peaks produced via $\gamma\gamma$ 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 $B^{+}\rightarrow K^{+}\omega J/\psi$ signals with good efficiency Andreassia:2014phd and I look forward to high-statistics results from them in the near future.

5 Acknowledgements

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