X (3915) as a tensor D ∗ ¯ D ∗ molecule

. Two-photon decays of a tensor D ∗ ¯ D ∗ molecule are studied and the suggested approach is applied to the X (3915) charmonium-like state under the assumption of the latter being a 2 ++ molecule — a spin partner of the X (3872). It is argued that the existing experimental data disfavour such an identiﬁcation of this state. Therefore, it is suggested that either the X (3915) has a di ﬀ erent exotic nature or it has to be identiﬁed as a scalar.

Since there exist in total four independent, gauge invariant two-photon structures, the amplitude M µνρσ from (1) can be expanded as in a complete set of four tensors {e µνρσ In particular, for real photons, a possible choice of such tensors is given by [17] e µνρσ It can be demonstrated that, due to the properties of the polarisation tensor ε ρσ (p), only the coefficient C 4 and the linear combination (C 2 √ 2 − C 3 ) contribute to the helicity-2 and helicity-0 amplitude, respectively, of the two-photon annihilation process which is assumed to proceed through the formation of the tensor state X 2 [17]. Then, for any final state f , the angular distribution takes a universal form, where the functions f ( f ) 0 and f ( f ) 2 depend on the particular final state and are given by the normalised helicity-0 and helicity-2 distribution, respectively -see, for example, the distributions derived in [13,17,19]. The dynamics of the reaction is encoded in the helicity amplitudes A 0 and A ±2 trivially related to the aforementioned combinations of the coefficients C n , It also proves convenient to introduce a ratio of the amplitudes, which defines the relative strength of the helicity-0 and helicity-2 contributions to the probability. In particular, the total cross section can be presented in the form where σ 0 corresponds to the total cross section evaluated solely for the helicity-0 amplitude. If the helicity-2 component is dominant, like for the genuine tensor quarkonium [16], R 1.

Ratio of the helicity amplitudes from data
If one assumes the X(3915) to be an S -wave molecular state with the quantum numbers 2 ++ then the only plausible candidate for such a resonance is the D * D * molecule. Then, its dominating production/decay mechanism proceeds through the S -wave X 2 → D * D * vertex preceded/followed by D ( * ) -meson loops, as depicted in Fig. 1. It is important to note that, while each individual coefficient C n defined in (7) and given by a loop integral diverges, the combination C 2 √ 2 − C 3 (and, therefore, the helicity-0 contribution to the two-photon width of the X(3915)) appears to be finite [17], so that one needs only one observable to completely fix the model.
However, before we proceed to the full calculation, we make an instructive exercise and study the dependence of the helicity-0 contribution to the two-photon width on the mass of the resonance. To make things even more transparent, we evaluate the ratio Γ 0 (M X 2 )/Γ ph 0 , where Γ ph 0 ≡ Γ 0 (M ph X 2 ) with M ph X 2 = 3915 MeV. In this ratio, the coupling constant g X 2 D * D * drops out since it scales both helicity components and, therefore, the result is insensitive to the details of the molecule coupling to its constituents and as such is a prediction of the model, although it is known to carry a significant mass dependence itself, g 2 X 2 D * D * ∝ M th − M X 2 , for near threshold states [20,21].
The result of the calculations is depicted in Fig. 2 and demonstrates that, near the threshold, the helicity-0 part to the width vanishes as a result of strong cancellations between different contributions to the amplitude. In other words, near the threshold, helicity-2 dominance holds also for the D * D * molecule, in analogy to regular charmonia. However, since the mass of the X(3915) is significantly below the D * D * threshold, further considerations appear to be necessary.
In order to proceed to the calculation of the two-photon width of the X(3915) one needs to estimate the coupling g X 2 D * D * . Typically, as was mentioned above, the coupling constant of a molecule to its constituents can be evaluated through its binding energy [20,21], however, in case of the X(3915) as a D * D * molecule, the binding energy appears to be as large as about 100 MeV that entails uncontrolled finite-range corrections to the coupling obtained this way. We choose to bypass this problem by employing an additional assumption that the X(3915) is a spin-2 partner of the X(3872) treated as the DD * molecule. Under this assumption, the couplings g X 1 DD * and g X 2 D * D * are related by the Heavy-Quark Spin Symmetry (HQSS) which is expected to be accurate up to the corrections 1 governed by a fairly small parameter Λ QCD /m c 1/5. Since the binding energy of the X(3872) is tiny [10], where m 0 and m * 0 are the masses of the neutral D and D * mesons, respectively, the finite-range corrections to the result for the coupling g X 1 DD * derived from value (17) [22] are also tiny, and the main uncertainty comes from the accuracy of the binding energy extraction from the data -see the error in (17). For the reasons explained below we choose the maximal value E B = 0.21 MeV consistent with the experimental result (17). This allows us to arrive at the following theoretical prediction for the two-photon decay width of the X(3915): where, as before, Γ 0 (X 2 → γγ) denotes the width evaluated solely for the helicity-0 amplitude [17], We may now employ the result from (18) to extract the ratio R from the data on Γ(X(3915) → γγ) for which we take the value Γ exp (X(3915) → γγ) 0.18 keV (20) consistent with various measurements performed by Belle and BaBar [10] -see a detailed discussion of the issue in [17]. Therefore, in order to reconcile the theoretical result from (18) with the experimental one contained in (20) one needs as large ratios R as and it is important to notice that smaller values of the X(3872) binding E B would result in even larger values of R in (21) that justifies our above-mentioned choice of the maximal E B consistent with the data.

Conclusions
In conclusion, let us summarise the results obtained in this work and give an outlook. 1 The role of HQSS violating effects for the molecular partners in the c-and b-quark sectors is discussed, for example, in [23,24]. Motivated by the claim made in [13], we investigate the possibility that the X(3915) charmoniumlike state may be identified as a tensor D * D * molecule and as such may have a prominent helicity-0 component in its wave function which would allow one to treat the X(3915) as a helicity-0 realisation of the χ c2 (3930) state. 2 Thus, we study the two-photon annihilation processes proceeding through the X(3915) and, in particular, evaluate the two-photon decay width of this state. We find that, for the tensor D * D * molecule, the helicity-0 component vanishes as the binding energy tends to zero while the helicity-2 piece (which contains an unknown counter term) is expected to be finite, unless the X(3915) completely decouples from the γγ channel in this limit. Thus, it appears natural that the X(3915), as a shallow bound state, shares the feature of a helicity-2 dominance with regular charmonia.
As a next step, we employ the Occam's razor principle to identify the tensor X(3915) state with the hypothetical spin-2 partner of the X(3872) treated as an axial-vector DD * molecule. This allows us to use the HQSS constraints to relate the coupling constant of the X(3915) to the D * D * pair with the experimentally measured binding energy of the X(3872). Then we evaluate the contribution of the helicity-0 amplitude to the two-photon decay width of the X(3915), which is finite and comes out as a prediction of the model, and argue that the experimental data presently available favour a scenario in which the contribution of the helicity-2 amplitude dominates over the helicity-0 one, similarly to the case of the genuine 2 ++ charmonium. Therefore, if the X(3915) is indeed dominated by the helicity-0 contribution of a nearby tensor state, as suggested in [13], its nature would require some exotic interpretation related neither with regular quarkonia nor with the D * D * spin partner of the X(3872). However, an important disclaimer for the results and conclusions obtained in this work is that the analysis performed is subject to several uncertainties which are difficult to quantify, given the present status of the experimental data as well as our theoretical understanding of the charmonium spectrum above the open-flavour thresholds. We expect that these uncertainties could be reduced in future studies.
An alternative scenario for the X(3915) would be to assign it the quantum numbers 0 ++ but, to avoid the criticisms contained in [11,12], to treat it as an exotic scalar. For example, in [25], the X(3915) was proposed to be a scalar D sDs molecule. The scalar D * D * molecule option also looks plausible for the X(3915), however, it requires an additional experimental input to fix the model. This work is in progress and will be reported elsewhere.