What can radiative decays of the X(3872) teach us about its nature?

Starting from the hypothesis that the X(3872) is a $D\bar D^*$ molecule, we discuss the radiative decays of the X(3872) into $\gamma J/\psi$ and $\gamma\psi'$ from an effective field theory point of view. We show that radiative decays are very weakly sensitive to the long-range structure of the X(3872). In particular, contrary to earlier claims, we argue that the experimentally determined ratio of the mentioned branching fractions is not in conflict with a wave function of the X(3872) that is dominated by the $D\bar D^*$ hadronic molecular component.


Introduction
The X(3872) was discovered by the Belle Collaboration in 2003 [1]. It has a mass extremely close to the D 0D * 0 threshold, and thus it has been regarded as one of the most promising candidates for a hadronic molecule, which can be either an S-wave bound state [2][3][4][5][6][7] or a virtual state in the DD * system [8]. Its quantum numbers were determined by the LHCb Collaboration to be J P C = 1 ++ [9] 10 years after the discovery.
Other models exist in addition to the hadronic molecule interpretation, which include a radial excitation of the P -wave charmonium χ c1 (2P ) [10], a tetraquark [11], a mixture of an ordinary charmonium and a hadronic molecule [12,13], or a state generated in the coupled-channel dynamical scheme [14,15]. It was claimed in Ref. [16] that the radiative decays of the X(3872) into the γJ/ψ and γψ ′ (here and in what follows ψ ′ denotes ψ(2S)) are very sensitive to its structure. Especially, using vector meson dominance and a quark model, in Ref. [16] it was predicted that the ratio is about 4 × 10 −3 , if the X(3872) is a hadronic molecule with the dominant component D 0D * 0 plus a small admixture of the ρJ/ψ and ωJ/ψ. Various quark model calculations R Ref. [23] 11 64 5.8 Ref. [10] 70 180 2.6 Ref. [24] 50-70 50-60 0.8 ± 0.2 predict this ratio in a range as wide as from approximately 0.6 and up to about 6 assuming a cc nature for the X(3872), see a few paradigmatic examples collected in Table 1. Such a large uncertainty in the predictions as well as the fact that the values R > 1 are preferred should not come as a surprise, since the quark model assignment for the X(3872) is that of a radially excited χ c1 (2P ) cc state while a radiative decay matrix element is proportional to the overlap integral of the initial state and the final state wave functions. Thus, on the one hand, such an overlap is very sensitive to the details of the wave functions, in particular to the position of their nodes. On the other hand, the overlap of the radially excited χ c1 (2P ) charmonium wave function with the one-node ψ ′ wave function is expected to be larger than its overlap with the nodeless J/ψ wave function. The ratio R was first measured by the BABAR Collaboration with the result 3.4±1.4 [17]. Later on, the Belle Collaboration reported a negative result for the γψ ′ mode, and set the upper limit R < 2.1 at 90% confidence level [18]. Very recently, the LHCb Collaboration reported the ratio [19] R = 2.46 ± 0.64 ± 0.29, which is consistent with both previous measurements. Based on the mentioned claim of Ref. [16] this result was interpreted as a strong evidence that the X(3872) cannot be a hadronic molecule. However, it was found in Ref. [20], which updates earlier calculations in Refs. [21,22], in a phenomenological study allowing for both a molecular as well as a compact component of the X(3872) that an enhanced decay of the X(3872) into γψ ′ compared to γJ/ψ is fully compatible with a predominantly molecular nature of X(3872). An admixture of 5-12% of a cc component was sufficient to explain the data. In this paper we critically re-investigate the validity of the claim of Ref. [16] from an effective field theory point of view.
In particular we demonstrate that, contrary to earlier claims, radiative decays do not allow one to draw conclusions on the nature of X(3872) and therefore confirm qualitatively the findings of Ref. [20] that the observed ratio is not in conflict with a predominantly molecular nature of the X(3872).

Generalities
According to Ref. [25] one may define the molecular component of a bound state by the probability to find the continuum component in the wave function of the physical state. This definition provides a close link between the significance of hadronic loops and hadronic molecules. In studies of quarkonia, the importance of hadronic loops is case dependent. In some transitions as discussed in, for example, Refs. [26][27][28], they are expected to give sizeable contributions. In contradistinction hadronic molecules have large effective coupling constants to the continuum as follows straightforwardly from the analysis of Ref. [25], and a pure molecule only couples to its constituents. As a consequence, in their decays hadronic loops are by definition a leading order effect. Because of this, hadronic molecules leave unique imprints in some properly chosen observables, but not in all: as we discuss in this paper in detail, in order to quantitatively control (ratios of) transition rates, additional information, not at all linked to the nature of the state under investigation, on the matrix element that connects the continuum state to the final state might be necessary. In addition, not all observables are related to the long-range tail of the wave function of a molecular state. In particular, we demonstrate that radiative decays are sensitive to the short-range parts of the X(3872) wave function and therefore are blind to the long-distance nature of X(3872).
The situation is analogous to that of the D * s0 (2317): when being treated as a cs state meson loops appear in the effective field theory only at subleading orders and give a small contribution to the decays [29]. On the other hand, if the assumed structure is a DK molecule, meson loops are a leading order effect [30]. However, this does not imply that all observables allow one to distinguish between the two scenarios: in Ref. [30] it was argued that while the strong decays are sensitive to the nature of the state the radiative decays are not because there are short-range contributions present already at the leading order.
The decay mechanisms for X(3872) into the γψ, with ψ denoting J/ψ or ψ ′ , are shown in Fig. 1. The charge conjugated diagrams are not depicted but are taken into account in the decay amplitude. We use the diagrams shown in Fig. 1 to calculate the X(3872) radiative decay widths employing a covariant approach. The details are presented in the next section. Heavy quark spin symmetry (HQSS) is used wherever appropriate to relate the vector and pseudoscalar charmed mesons. Our phase convention for the charge conjugation of the charmed mesons is Under this convention, the wave function of the X(3872) as a pure hadronic molecule may be written as

Formalism
The loop amplitude for the decay is given by the sum of the diagrams (a)-(e) depicted in Fig. 1.
where the values of the coupling constants of the X(3872) to the charged and neutral charmed mesons are similar, see, for example, Ref. [31,32], so in what follows we do not distinguish 3 between them and set x c = x 0 = x. For convenience, we relate the relativistic coupling to the nonrelativistic one, where m, m * , and m X are the D-meson, D * -meson, and the X(3872) mass, respectively. We do not distinguish between the neutral and charged D-and D * -meson masses. The value of x nr was extracted from the binding energy of the X(3872) in the hadronic molecule picture in Ref. [33]. Then the X σ (p) → DD * ν (k) vertex takes the form The vertices relating the vector field ψ (ψ = J/ψ, ψ ′ ) to the D and D * mesons follow from the Lagrangian where the couplings are related via the heavy-quark symmetry. With the help of the nonrelativistic Lagrangians from Refs. [26,27,34] one finds with the nonrelativistic coupling constant g 2 used in Refs. [26,27]. Thus, the Next, we need the couplings of the photon to the open-charm states. The leading electric couplings emerge from gauging the kinetic terms for the charged heavy mesons. The importance of the charged component of the X(3872) wave function for the radiative decays was stressed in Ref. [35].
where e is the electric charge. The Both vertices satisfy the appropriate Ward identities, where the D-meson propagator is and the D * propagator and its inverse form are Finally, the vertex (11) gives rise to a four-point vertex DD * ψγ after gauging, see diagram (e) in Fig. 1.
To extract the magnetic vertices we use the covariant generalisation of the nonrelativistic Lagrangian in Refs. [36,37], which reads where v µ is the four-velocity of the heavy quark with v µ v µ = 1, Q = diag(2/3, −1/3) is the light quark charge matrix, and m c and Q c are the charmed quark mass and its charge, Q c = 2/3, respectively. In the above Lagrangian, the terms proportional to Q c /m c come from 5 the magnetic moment of the charm quark, and the β-terms are from the nonperturbative light-flavour cloud in the charmed meson. Then the magnetic D * a and Γ (m)ab µλ respectively. Notice that both vertices (19) and (20) are manifestly transversal with respect to the photon momentum q λ . With the given ingredients the expression for the loop amplitude of the radiative decay where The tensor J µνλ (k) acquires contributions from diagram (a)-(e) in Fig. 1, where the superscripts e and m label the electric and magnetic contributions, respectively. The individual contributions read In the expressions above the heavy-quark four-velocity is substituted by the X(3872) fourvelocity and the contribution of the conjugated loops is taken into account explicitly. The amplitude (22) is gauge invariant. This follows from the transversality of the magnetic vertices of Eqs. (19) and (20) as well as from the Ward identities of Eq. (15). It is easy to verify that the loop integral in the amplitude (22) is divergent. Therefore, to render the result well defined, one needs to include in addition to the loop amplitude (22) described above the Xγψ counterterm amplitude (diagram (f) in Fig. 1), which is also manifestly gauge invariant. The strength of the contact interaction λ is subject to renormalisation to absorb the divergence of the loops, so that the contact amplitude (30) with the renormalised strength λ r provides a finite contribution to the total decay width. In the next section two different ways are presented on how to estimate the size of this finite contribution. The necessity to include a contact term at leading order shows that the radiative decays are not only sensitive to the long-range parts of the matrix element entering via the loops but also to the short-range structure of the wave function which is not known. Because of that, as a matter of principle, the radiative decays of X(3872) cannot be used as a source of information on its long-distance structure.
In the calculations we used the following values for the masses [38]: For the magnetic coupling of the charmed mesons, the parameters are [39] β −1 = 379 MeV, m c = 1876 MeV.
The value of the coupling constant x nr is very uncertain because the value of its mass, or the binding energy, is not known precisely. In Ref. [33], it was extracted to be |x nr | = 0.97 +0. 40 −0.97 GeV −1/2 . The coupling constants for J/ψ and ψ ′ to the charmed mesons, g 2 and g ′ 2 , cannot be measured directly and are badly known. We observe, see Eq. (22), that the width in the pure molecular picture is proportional to |x nr g 2 | 2 . In order to give definite values for the partial widths, we set the finite part of the counterterms to zero, λ (′) r = 0, and define ratios where |x nr | = 0.97 GeV −1/2 [33] and |g (0) 2 | = 2 GeV −3/2 is taken from model estimates [27,34].
The integrals are evaluated using dimension regularisation with the MS subtraction scheme at the scale µ = m X . Numerical calculations are performed with the help of the FeynCalc [40] and the LoopTools [41] packages for Mathematica.

Results and discussion
Our numerical results for the partial radiative decay widths for the X(3872) are displayed in the middle column of Table 2. To arrive at these results, the contact terms were set to zero (λ (′) r = 0) and the couplings were chosen as explained in the previous section. 7 Table 2: The calculated radiative decay widths Γ(X → γψ) for ψ = J/ψ, ψ ′ and their ratio R. Here, g 2 (g ′ 2 ) are the spin symmetric coupling constants of the J/ψ(ψ ′ ) to the charm meson-antimeson pair, see Eq. (9), r x and r Although the badly known constant x nr responsible for the X(3872) coupling to the charmed mesons drops from the ratio R, still additional assumptions are necessary in order to connect the results from Table 2 to the actual data, namely (i) to fix the ratio of the coupling constants of the ψ ′ and J/ψ, g ′ 2 /g 2 , which has nothing to do with the nature of the X(3872), and (ii) to estimate the size of the contact interaction contribution to the width, which is only sensitive to the short-range structure of the X.
First one needs to fix the ratio g ′ 2 /g 2 . If both couplings were equal and λ (′) r = 0, then indeed the γψ ′ channel would be suppressed, although a lot less than claimed in Ref. [16]. However, already a value of g ′ 2 /g 2 ∼ 3 lying in a natural range is sufficient to bring R in accordance with the experiment also for a purely molecular X(3872), see Eq. (2). In this context it is interesting to observe that Ref. [20] finds g ′ 2 /g 2 ∼ 2 referring to the analysis presented in Ref. [34].
Second, so far the counterterms were set to zero. From the information we have available there is no way to fix their strength λ (′) r (we do not agree to the claim of Ref. [35] that the strength can be taken directly from a model of the formation of the X(3872), since different scales are involved in scattering and decay). One way to estimate the size of the contact term contribution to the width is to vary the regularisation scale used in the evaluation of the loops. The impact of a variation of µ from m X /2 to values as large as 2m X is also presented in Table 2. Since any physical amplitude should be independent of µ the variation displayed in Table 2 should be compensated by a corresponding variation in the counterterm. In this sense the observed variation in the width is a measure of the size of the counterterm. This confirms the claim made earlier in this paper that for the radiative decays of X(3872) short-range contributions are of similar importance as their long-range counter parts.
Alternatively one could estimate the size of the counterterms by employing a model. Indeed, since the counterterms parametrise short-range physics they may be modelled by a heavy quark loop (cf. discussion in Ref. [20]). We consider a few paradigmatic examples of the estimates for the radiative decay widths of the 2 3 P 1 cc charmonium found in the literature, which are collected in Table 1. These estimates for the γJ/ψ mode appear to be 2-3 times larger than the result quoted in Table 2 if the ratios defined in Eq. (32) are taken to be unity, while for the γψ ′ mode they exceed the molecule estimate by more than one order of magnitude. Unfortunately, it is very difficult to estimate the uncertainties of the results of quark models. Thus, as an anchor, we take the averaged values Γ(X(cc) → γJ/ψ) ≃ 50 keV and Γ(X(cc) → γψ ′ ) ≃ 100 keV with the ratio R ≃ 2. However, these numbers cannot be used directly as an estimate for the counterterm contributions since both the size of the hadron loops presented in Table 2 as well as the size of the quark loops presented in Table 1 are based on the condition that the normalisation of the wave function is saturated by the hadronic loops or the quark loop, respectively. Therefore, in order to add both results one needs to multiply the widths from the hadronic loops by (1 − Z) and those from the quark loop by Z, where Z denotes the probability to find the compact component in the physical wave function of the X(3872) [25] 1 . Accordingly, Z = 0 refers to a pure molecule while Z = 1 points to a purely elementary state. Then Z ∼ 0.1 brings the quark loop contribution to the same order of magnitude with that of the hadron loops, which clearly allows one to fit the data. This observation is in line with that of Ref. [20]. Specifically, for Z ∼ 0.3-0.4 found in Ref. [43] from the combined data analysis on the D 0D0 π 0 and π + π − J/ψ decay modes of the X, the ratio g ′ 2 /g 2 needed to bring R in accordance with the experiment is reduced to approximately 2 in line with the ratio of couplings found in Ref. [20] referring to the analysis of Ref. [34].
It should be stressed, however, that these findings have to be interpreted with caution, not only since the numbers presented in the tables are highly uncertain: as a matter of principle it is not possible to identify in a hadronic effective field theory the physics of the short-range contributions. The latter could as well be, for example, higher momentum components of the hadronic wave function. At this point all one can conclude is that the radiative decays of X(3872), and especially their ratio R, is not sensitive to the long-range structure of the X, and thus they cannot be used to rule out the picture that the X(3872) is dominantly a hadronic molecule. In order to make statements on the hadronic molecule structure of the X, one needs to measure its decays which are sensitive to the long-distance physics, such as X → DDγ or X → DDπ, see, for example, Refs. [8,32,[43][44][45][46].