Decay widths of the spin-2 partners of the X(3872)

We consider the $X(3872)$ resonance as a $J^{PC}=1^{++}$ $D\bar D^*$ hadronic molecule. According to heavy quark spin symmetry, there will exist a partner with quantum numbers $2^{++}$, $X_{2}$, which would be a $D^*\bar D^*$ loosely bound state. The $X_{2}$ is expected to decay dominantly into $D\bar D$, $D\bar D^*$ and $\bar D D^*$ in $d$-wave. In this work, we calculate the decay widths of the $X_{2}$ resonance into the above channels, as well as those of its bottom partner, $X_{b2}$, the mass of which comes from assuming heavy flavor symmetry for the contact terms. We find partial widths of the $X_{2}$ and $X_{b2}$ of the order of a few MeV. Finally, we also study the radiative $X_2\to D\bar D^{*}\gamma$ and $X_{b2} \to \bar B B^{*}\gamma$ decays. These decay modes are more sensitive to the long-distance structure of the resonances and to the $D\bar D^{*}$ or $B\bar B^{*}$ final state interaction.


I. INTRODUCTION
In the infinite quark mass limit, heavy quark spin-flavor symmetry (HQSFS) implies that the dynamics involving heavy quarks are independent of their spin or flavor. In this way, charm and bottom spectra can be related, up to corrections suppressed by 1/m Q with m Q the heavy quark mass. It should also be possible, in a given heavy flavor sector, to relate states with different spins.
These relations are very useful in the study of composites that mix heavy and light quarks. In this work, we focus on hadronic molecular states composed by a heavy-light meson and a heavy-light antimeson (P ( * )P ( * ) , P = D,B). These molecular states were predicted in the mid 70s [1,2]. So far, the best experimental candidate to fit this molecular description is the X(3872) resonance, first observed by the Belle Collaboration in 2003 [3], that can be thought as a DD * bound state with J P C = 1 ++ (quantum numbers confirmed later on in Ref. [4]). Since then, many other new XY Z states which are good candidates to be exotic hadrons have been experimentally observed [5,6].
Within the molecular description of the X(3872), the existence of a X 2 [J P C = 2 ++ ] s-wave D * D * bound state was predicted in the effective field theory (EFT) approach of Refs. [7,8]. As a result of the heavy quark spin symmetry (HQSS), the binding energy of the X 2 resonance was found to be similar to that of the X(3872), i.e., The existence of such a state was also suggested in Refs. [9][10][11][12][13][14]. Both the X(3872) and the X 2 , to be denoted by X 2 (4013) in what follows, have partners in the bottom sector [15], 1 which we will call X b and X b2 , respectively, with masses approximately related by It is worthwhile to mention that states with 2 ++ quantum numbers exist as well as spin partners of the 1 ++ states in the spectra of the conventional heavy quarkonia and tetraquarks. However, the mass splittings would only accidentally be the same as the fine splitting between the vector and pseudoscalar charmed mesons, see Eq. (1). 2 For instance, the mass splitting between the first radially excited charmonia with 2 ++ and 1 ++ in the well-known Godfrey-Isgur quark model is 30 MeV [17], which is much smaller than the value in Eq. (1). In a quark model calculation with screened potential, the 2 ++ − 1 ++ mass splitting for the 2P charmonia is around 40 MeV [18]. As 1 In Ref. [15], the bottom and charm sectors are connected by assuming the bare couplings in the interaction Lagrangian, see Eq. (A6), to be independent of the heavy quark mass. This assumption will also be used throughout this work. 2 Were these states due to threshold cusps, the splittings would be the same as those of the hadronic molecules.
However, it was shown in Ref. [16] that narrow near threshold peaks in the elastic channel cannot be produced by threshold cusps.
for the tetraquark states, the corresponding mass splitting predicted in Ref. [19] is 70 MeV, which is again much smaller than M D * − M D . Notice that it is generally believed that the χ c2 (2P ) has been discovered [20,21], and its mass is much lower than 2M D * . Therefore, we conclude that a possible discovery of a 2 ++ charmonium-like state with a mass around 4013 MeV as a consequence of HQSS [22] would provide a strong support for the interpretation that the X(3872) is dominantly a DD * hadronic molecule. It is thus very important to search for such a tensor resonance, as well as the bottom analogues, in various experiments and in lattice QCD (LQCD) simulations.
Some exotic hidden charm sectors on the lattice have been recently studied [23][24][25][26][27], and evidence for the X(3872) from DD * scattering on the lattice has been found [24], while the quark mass dependence of the X(3872) binding energy was discussed in Refs. [28,29]. The 2 ++ sector has not been exhaustively addressed yet in LQCD, though a state with these quantum numbers and a mass of (m ηc + 1041 ± 12) MeV= (4025 ± 12) MeV, close to the value predicted in Refs. [7,8], was reported in Ref. [23]. The simulation used dynamical fermions, novel computational techniques and the variational method with a large basis of operators. The calculations were performed on two lattice volumes with pion mass ≃ 400 MeV. There exists also a feasibility study [30] of future LQCD simulations, where the EFT approach of Refs. [7,8] was formulated in a finite box.
On the other hand, despite the theoretical predictions on their existence, none of these hypothetical particles has been observed so far. Nevertheless, they are being and will be searched for in current and future experiments such as BESIII, LHCb, CMS, Belle-II and PANDA. It is thus of paramount importance to provide theoretical estimates on their production rates in various experiments, as well as the dominant decay modes and widths. 3 The production of these states in hadron colliders and electron-positron collisions has been studied in Refs. [31,32]. In this work, we will investigate the dominant decay modes of the spin-2 partners of the X(3872), i.e. the X 2 (4013) and X b2 , and provide an estimate of their decay widths.
Besides, we will also discuss the radiative X 2 → DD * γ and X b2 →BB * γ transitions. These decay modes are more sensitive to the long-distance structure of the resonances and might provide valuable details on their wave-functions. The situation is similar to that of the X(3872) → D 0D0 π 0 decay studied in Refs. [33,34]. Also here, the widths will be affected by the DD * or BB * final state interaction (FSI). FSI effects are expected to be large because they should be enhanced by the presence of the isovector Z c (3900) and Z b (10610) resonances located near the DD * and BB * thresholds, respectively. Besides, FSI corrections will be also sensitive to the negative Cparity isoscalar DD * orBB * interaction. Eventually, precise measurements of these radiative 3 If a resonance is too broad, say Γ 200 MeV, it would be very difficult to be identified since it is highly nontrivial to distinguish the signal for a broad resonance from various backgrounds.
decay widths might provide valuable information on the interaction strength in this sector, which would be important in understanding the P ( * )P ( * ) system and other exotic systems related to it through heavy quark symmetries [15,35].
The structure of the paper is as follows. First in Sect. II, we briefly discuss the relation of the charm and bottom 2 ++ states with the X(3872) resonance, and in Sect. III we present our predictions for the X 2 → DD, DD * hadron decays and the X b2 → BB, BB * ones in the bottom sector. In Sect. IV, the X 2 and X b2 radiative decays are investigated, paying special attention to the loop mechanisms responsible for the FSI contributions. The conclusions of this work are outlined in Sect. V and in addition, there are three Appendices. In the first one (Appendix A), we collect different heavy meson Lagrangians used through this work, while the validity of the perturbative treatment of the DD for the X 2 is discussed in the second one (Appendix B). Finally, in Appendix C, we give some details on the evaluation of different three-point loop functions that appear in the computation of the hadronic and radiative decays.

II. HQSFS, THE X(3872) RESONANCE AND THE CHARM AND BOTTOM X 2 STATES
A. X(3872) As mentioned, we start assuming the X(3872) to be a positive C-parity DD * bound state, with quantum numbers J P C = 1 ++ . At very low energies, the leading order (LO) interaction between pseudoscalar and vector charmed (D 0 , D + , D * 0 , D * + ) and anti-charmed (D 0 , D − ,D * 0 , D * − ) mesons can be described just in terms of a contact-range potential, which is constrained by HQSS [7,8,15].
Pion exchange and particle coupled-channel 4 effects turn out to be sub-leading [7,36]. For the case of the X(3872), isospin breaking is important [38] as this bound state is especially shallow. The energy gap between the D 0D * 0 and D + D * − channels is around 8 MeV, which is much larger than the X(3872) binding energy with respect to the D 0D * 0 threshold. As a consequence, the neutral (D 0D * 0 ) and charged (D + D * − ) channels should be treated independently. The coupled-channel 5 contact potential in the 1 ++ sector is given by [8] (see also Appendix A 1) with C 0X and C 1X low energy constants (LECs) that need to be fixed from some input. This interaction is used as kernel of the Lippmann-Schwinger equation (LSE) in the coupled channel 4 We do not refer to charge channels, but rather to the mixing among the DD, DD * , D * D * pairs in a given IJC (isospin, spin and charge conjugation) sector. 5 Actually, positive C-parity combinations in both the neutral D 0D * 0 and charged D +D * − channels are being considered. space in the 1 ++ sector, with M 1 and M 2 the masses of the involved mesons, µ −1 12 = M −1 1 + M −1 2 , E the center of mass (c.m.) energy of the system and p ( p ′ ) the initial (final) relative three momentum of the DD * pair in the c.m. frame. The used normalization is such that above threshold [E > (M 1 + M 2 )], the single channel elastic unitary condition is Im The discussion is similar for any other J P C sector. Due to the use of contact interactions, the LSE shows an ill-defined ultraviolet (UV) behaviour, and requires a regularization and renormalization procedure. We employ a standard Gaussian regulator (see, e.g. [37]) with C IX any of the LECs of Eq. (3) in the case of the X(3872), or the relevant ones for any other J P C sector. We take cutoff values Λ = 0.5 − 1 GeV [7,8], where the range is chosen such that Λ will be bigger than the wave number of the states, but at the same time it will be small enough to preserve HQSS and prevent that the theory might become sensitive to the specific details of short-distance dynamics. 6 The dependence of the results on the cutoff, when it varies within this window, provides a rough estimate of the expected size of sub-leading corrections. Bound states correspond to poles of the T -matrix below threshold on the real axis in the first Riemann sheet (RS) of the complex energy, while the residues at the pole give the s-wave couplings of the state to each channel (D 0D * 0 and D + D * − in the case of the X(3872) resonance 7 ).
The LECs C 0X and C 1X can in principle be determined [8] from M X(3872) = (3871.69 ± 0.17) MeV (mass average quoted by the PDG [5]) and the isospin violating ratio of the decay amplitudes for the X(3872) → J/ψππ and X(3872) → J/ψπππ, R X(3872) = 0.26 ± 0.07 [39]. We use m D 0 = (1864.84 ± 0.07) MeV, m D + = (1869.61 ± 0.10) MeV, m D * 0 = (2006.96 ± 0.10) MeV and m D * + = (2010.26 ± 0.07) MeV [5]. Note that m D 0 + m D * 0 = (3871.80 ± 0.12) MeV, and the uncertainty in the value of this lowest threshold affects the precision of the X(3872) binding energy. We have taken into account this effect by adding in quadratures the PDG error of the X(3872) mass and that of the neutral channel threshold and assign this new error to the mass of the resonance, that now reads M X(3872) = (3871.69 ± 0.21) MeV. For the LECs, we obtain: 6 However, as will be shown later on, the situation is more complicated in the two-body d-wave hadronic decays. 7 For instance, in the case of the X(3872), we have where Tij are the matrix elements of the T -matrix solution of the UV regularized LSE.
for Λ = 0.5(1.0) GeV. Errors, at the 68% confidence level (CL), have been obtained from a Monte Carlo (MC) simulation assuming uncorrelated Gaussian distributions for the two inputs (M X(3872) , R X(3872) ). In the simulation, we have rejected MC samples for which the X(3872) turned out to be unbound, since the scheme of Ref. [8] only allows to determine the properties of the resonance when it is bound.
B. X 2 (4013): J P C = 2 ++ , charm sector HQSS predicts that the s-wave D * D * interaction in the 2 ++ sector is, up to corrections suppressed by the charm quark mass, identical to that in the X(3872) sector (1 ++ ) and given by Eq. (3). Thus, in the 2 ++ sector, the potential in the (D * 0D * 0 ), (D * + D * − ) coupled channel space reads [7,8] (see also Appendix A 1) with the same structure and involving the same LECs that in the X(3872) channel. Besides, in the above equation m c ∼ 1.5 GeV is the charm quark mass and q ∼ Λ QCD , a scale related to the light degrees of freedom. Taking Λ QCD ∼ 300 MeV [5], corrections of the order of 20% to the interaction predicted by HQSS cannot be discarded. Nevertheless, it seems natural to expect a 2 ++ D * D * loosely bound state (X 2 ), the HQSS partner of the X(3872), and located in the vicinity of the D * 0D * 0 threshold (∼ 4014 MeV) [7,8,15]. This is illustrated in the X 2 binding energy distributions depicted in Fig. 1 Ref. [15] was simpler, because there we worked in the isospin symmetric limit and used the averaged masses of the heavy mesons, which are larger than those of the physical D 0 and D * 0 mesons.
For later use, we also need the couplings of the X 2 to its neutral (D * 0D * 0 ) and charged (D * + D * − ) components, g X 2 0 and g X 2 c , respectively. They turn out to be slightly different because the X 2 distributions are first generated and are used to evaluate the X 2 mass. X(3872) mass trials above threshold are rejected. To evaluate the red shaded histogram, and to account for the HQSS 20% uncertainty in the 2 ++ interaction, each of the members of any MC sample (C 0X + C 1X , C 0X − C 1X ) pair is multiplied by independent N(µ = 1, σ = 0.2) Gaussian distributed random quantities r ± . resonance is an admixture of isospin 0 and 1, since its binding energy is much smaller than the energy difference between the two thresholds [8,38]. Considering the HQSS uncertainties, we find: for Λ = 0.5(1.0) GeV.
C. X b2 : J P C = 2 ++ , bottom sector Owing to the heavy flavor symmetry, the LO 2 ++ B * B * interaction is given by Eq. (7) as well, and thus we should also expect a 2 ++ B * B * bound state (X b2 ), the HQSFS partner of the X(3872), located close to the B * B * threshold (∼ 10650 MeV) [15]. The X b2 binding energy distributions are heavier than their charmed cousins, the expected X b2 binding energy is significantly larger than in the charm sector, around a few tens of MeV, and thus we do not expect any significant isospin breaking effects and the X b2 resonance would be a pure isoscalar (I = 0) state.
As can be seen in Fig. 2, in this case we have a robust prediction even when HQSS uncertainties (20%) are taken into account. We obtain the mass and the coupling from the residue at the pole for Λ = 0.5 (1.0) GeV: 8 This bound state, being isoscalar, equally couples to the neutral and charged components and, therefore: Our predictions in Eq. (10), both for the mass and the B * B * coupling of the resonance show some dependence on the UV cutoff, which is to some extent diminished when HQSS uncertainties are taken into account. Nevertheless, this Λ dependence might hint to non-negligible sub-leading corrections (among others, pion exchange and coupled channel effects [7], which can be larger here than in the charm sector due to the larger binding energy and larger meson masses). We will compute the decay widths for both UV regulators, and the spread of results will account for this source of uncertainty. III. THE HADRONIC X 2 AND X b2 DECAYS The quantum numbers, J P C = 2 ++ , of these resonances constrain their possible decay channels.
In this work, for hadronic decays we only consider the decays into two heavy hadrons: X 2 → DD 8 There appear small differences in the central value of the resonance mass with respect to value quoted in [15] due to small differences in the used hadron masses. without pion-exchange FF with pion-exchange FF Λ = 0.5 GeV Λ = 1 GeV Λ = 0.5 GeV Λ = 1 GeV   Fig. 1 (red shaded) and the g X2 0 and g X2 c couplings given in Eqs. (8) and (9).
Note that the procedure takes into account 20% HQSS uncertainties and the correlations between X 2 masses (binding energies) and g X2 0 and g X2 c couplings. Errors on the widths provide 68% CL intervals. and X 2 → DD * (D * D ), and the analogous processes X b2 → BB and X b2 → BB * (B * B ). We expect that these d-wave decay modes should largely saturate the widths of these states. Because the DD couples in a d-wave to the 2 ++ system, its contribution to the mass renormalization of the X 2 is of higher order (see Appendix B). We thus did not include the DD as a coupled channel in the MeV, there appears only one additional D * Dπ coupling (g). We take g = 0.570 ± 0.006 as inferred from the new value of Γ = (83.4 ± 1.8) keV for the D * + decay width quoted by the PDG [5]. This is mostly determined by the recent BABAR Collaboration measurement [44] of this width, which is approximately a factor 12 times more precise than the previous value, Γ = (96 ± 4 ± 22) keV by the CLEO Collaboration [45]. Thus, we end up with an uncertainty of the order 1% for g. Though the hadronic X 2 and X b2 widths evaluated in this section will be proportional to g 4 , this source of error (∼ 4%) will be much smaller than others and it will be ignored in what follows.
A. Charm decays We will first consider the X 2 (4013) → D + D − (D 0D0 ) decay, which proceeds through the Feynman diagrams depicted in Fig. 3. We treat charm mesons non-relativistically, and neglect p D * ,D * /m D * terms and the temporal components in the D * ,D * propagators. We obtain for the X 2 (4013) → D + D − process, in the resonance rest frame and with q and k the 4-momenta of the D where ǫ ij (λ) is the symmetric spin-2 tensor with λ denoting the polarization of the X 2 state and accounts for the normalization of the heavy meson fields 9 and some additional factors needed when the couplings g X 2 c,0 , as determined from the residues at the pole of the EFT T -matrix, are used for the X 2 D * D * vertex. For the neutral and charged pion masses, we have used the values quoted by the PDG [5] and heavy meson isospin averaged masses to compute N . Besides, I ij is a three-point loop function, the detailed evaluation of which is relegated to the Appendix C 1. 10 The loop is seemingly logarithmically divergent. However, since the X 2 polarization is traceless, the divergent part which comes with a Kronecker delta does not contribute. This is because the decay occurs in a d-wave, thus the loop momentum is converted to external momenta, and the remaining part of the integral is convergent. Nevertheless, we will include two different form factors in the computation of the three-point loop function. One is inherited from the UV regularization/renormalization procedure sketched in Eq. (5) and employed to make the LSE Tmatrix finite. In addition, we will include a second form factor to account for the large virtuality of the pion in the loop. We will discuss this at length below and in Appendix C 1.
are similar, with the appropriate changes of the exchanged pion charges. 9 We use a non-relativistic normalization for the heavy mesons, which differs from the traditional relativistic one by a factor √ MH . 10 In the computation of I ij , we are consistent with the former approximations, and we use non-relativistic charm meson propagators.
Analogously, the X 2 (4013) → D 0D0 amplitude is, The two-body decay width in the X 2 rest-frame reads [5]: The sum over the X 2 polarizations can be easily done in the c.m. frame, As discussed in Appendix C 1, the three-point loop function has a tensor structure of the type The I 1 term carries the UV divergence, which however does not contribute to the width, because it vanishes after the contraction with the traceless spin-2 polarization tensor, as mentioned above.
Therefore, only the I 0 term is relevant, which is free of UV divergences. Moreover, the contraction (14)] leads to a factor of 2 q 4 /3. Thus, the integration over the solid angle dΩ(q) trivially gives 4π, and the width scales like | q | 5 as expected for a d-wave process.
Our predictions for the X 2 (4013) → D + D − , D 0D0 decays are compiled in the Table I. If we look at the first two columns of results in the table, we find widths of the order of a few MeV, with asymmetric errors that favour larger values. This is mostly due to the similar asymmetry of the uncertainties quoted for the g X 2 0 and g X 2 c couplings in Eqs. (8) and (9).
Our scheme is based on a low-energy EFT, in which the momenta should be smaller than a hard scale which serves as a momentum cutoff [see Eq. (5)]. The high-momentum modes are out of control in the low-energy EFT. Therefore in the computation of the width, we include a Gaussian regulator at the D * D * X 2 vertex, as discussed in Eq. (C6). The cutoff should be the same as the one used in generating the X 2 as it is related to the same unitary cut in the D * D * system. In Fig. 4, we display, as an example, the dependence of the ] on the pion loop momentum. In the left plot we see that in spite of including the Gaussian X 2 form factor, large momenta above 1 GeV provide a sizable contribution to the integral (≃ 14%, 30% and 45% for Λ = 0.5 GeV, 1 GeV and ∞, respectively), which is an unwanted feature within the low-energy EFT scheme and signals a sizeable short-distance contribution. Indeed, the momentum of the exchanged pion, peaks at around 750 MeV, which is a somehow large value in the sense that the hard scale for the chiral expansion which controls the pionic coupling is Λ χ ∼ 1 GeV.
We see that below the peak, the curves for both cutoff values are very close to each other, and they are also close to the curve corresponding to the case without any regulator. This is the region where the low-energy expansion works and thus model-independent conclusions can be made. The curves start deviating from each another after the peak, that is in the region with a pion momentum Λ χ .
Because the loop integrals are not completely dominated by momentum modes well below Λ χ , the widths of interest will bear an appreciable systematic uncertainty. This is reflected in the fact that the widths in the second column in Table I are larger than those in the first column by a factor around 2. 11 On the other hand, the fact that the pion could be quite far off-shell should be reflected in the D * Dπ vertex, which should be corrected, similarly as it is done in the case of the N N π one. Thus, to give an estimate of the hadronic decay widths, in the spirit of the Bonn potential [46], we have integrand is shown in the right plot of Fig. 4. The large pion momenta contribution (| l | > 1 GeV), which is not reliable in a low-energy EFT calculation, is reduced now to 6.5%, 13% and 16% for Λ = 0.5 GeV, 1 GeV and ∞, respectively. This makes also more appropriate the non-relativistic treatment of the charmed mesons adopted here. Besides, the dependence of the width on the UV Gaussian regulator is significantly softer, though the widths are further reduced by almost an order of magnitude.
We believe that the most realistic estimates are those obtained with the inclusion of the pionexchange form factor and the spread of results compiled in the Table I give a conservative estimate of the systematic uncertainties, beyond the mere existence of the X 2 (4013) state, as discussed in Sect. II B. We remind here that because of the additional 20% HQSS uncertainty, approximately a 23% (32%) for Λ = 0.5 GeV (Λ = 1 GeV) of the X(3872) events [(M X(3872) , R X(3872) ) MC samples] do not produce the X 2 as a bound state pole, since the strength of the resulting interaction in the 2 ++ sector is not attractive enough to bind the D * D * .
Nevertheless, assuming the existence of the X 2 state, and in view of the results given in Table I, we estimate the X 2 → DD partial width (including both the charged and neutral channels) to be where the first error accounts for the dependence on the UV Gaussian regulator used in the D * D * X 2 vertex, while the second one is obtained from the uncertainties given Table I. This latter error 11 Note that the coupling constants obtained with both cutoffs are similar, see Eqs. (8) and (9), and thus the difference should come mainly from the loop integration.
With pion form-factor (Λ π ∼ 1 GeV) Integrand includes both some additional systematic (HQSS violations) and statistical (X(3872) input used to fix the properties of the X 2 resonance) uncertainties. Notice that, as discussed above, the calculation is probably already beyond the valid range of the EFT due to the large contribution of high-momentum modes. We thus have adopted a more phenomenological strategy and used the pion-exchange form factor with a cutoff of 1 GeV to make an estimate of the decay widths. The values presented in Eq. (16) refer only to the last two columns in Table I that include the effect of the pion-exchange form factor.
Here, we will study the D + D * − , and D 0D * 0 channels, which proceeds through the Feynman diagrams depicted in Fig. 5. This is also a d-wave decay so that both angular momentum and parity are conserved. The decay widths are expected to be comparable to those found for the X 2 (4013) → DD decays, despite the phase space is considerably more reduced. This extra enhancement is caused by the extra multiplicity due to the spin of the final D * (D * ) meson.
As commented before, we treat charm mesons non-relativistically and obtain the decay amplitude for the X 2 (4013) → D + D * − process as transition are similar, with the appropriate changes of the exchanged pion charges.
in the resonance rest frame. Here, q is the 4-momenta of the D + meson, and ǫ n (λ * ) is the polarization The two-body decay width in the X 2 rest-frame in this case reads [5]: The sum over theD * and X 2 polarizations can be easily done, and we get The above tensor structure should be contracted with I im ( q ) I rl ( q ). We see that the sum over polarizations of Eq. (20) is orthogonal to δ im and δ rl , which guarantees also here that only the UV finite I 0 term of the three-point loop function is relevant. The contraction leads to a q 4 factor, 12 and thus the width scales like | q | 5 , as expected for a d-wave decay.
Results for the X 2 → DD * decay widths are also compiled in Table I. We only show predictions for the X 2 (4013) → D + D * − (D 0D * 0 ) decays, because being the X 2 an even C-parity state, its decay modes into the charge conjugated final states have the same decay widths. In what respects to the effect of the form factors, the discussion runs in parallel to that in the Sect. III A 1, though the effect of the pion-exchange form factor is significantly smaller here (a factor 4 or 5 at most). As 12 In the DD mode, studied in Sect. III A 1 a factor of 2 q 4 /3 is obtained instead. Thus, neglecting the D − D * mass difference, because the loop integrals are the same, we would find This extra factor 3/2 due to the spin-1 polarization vector produces an enhancement of the DD * decay mode with respect to the DD one, which partially compensates the smaller available phase space.
expected, the widths are comparable to those found for the X 2 → DD decays. Finally, we estimate the partial X 2 → DD * (D * D ) width (including both the charged and neutral channels as well as the charge-conjugated modes) to be where the errors have been estimated as in Eq. (16). The above result, together with that obtained previously for the DD channel, leads to a total X 2 width of the order of 2-8 MeV, assuming its existence.

B. Bottom decays
Thanks to the heavy flavor symmetry, the results of the previous subsection can be trivially extended to the bottom sector. There, we have a robust prediction, even when HQSS uncertainties (20%) are taken into account, for the X b2 resonance (see Fig. 2). Moreover, all sort of non-relativistic approximations adopted in the current scheme are now more suited, since the range of variation of the internal pion momentum in the loops is similar to that shown in Fig. 4 for the charm sector.
On the other hand, as discussed in Sect. II C, we do not expect any significant isospin breaking effects and the X b2 resonance would be a pure I = 0 state, with equal coupling to its neutral and charged components. For simplicity, we will also neglect the tiny difference between B 0 and B ± masses, and we will use a common mass m B = (m B 0 + m B ± )/2 = 5279. 42 MeV. Yet, for the pion mass that appears in the loop integral, we take the isospin averaged value m π = (2m π ± + m π 0 )/3.
Note that the relevant internal pion momentum is around 750 MeV. With all these approximations, we find for any charge channel (B + B − , B 0B0 , B + B * − , B 0B * 0 ) or charge conjugation mode (B * B ). Our results for these decay widths are presented in Table II. We notice in passing that following heavy flavor symmetry we use the same value of g = 0.570 ± 0.006 in the charm and bottom decays.
It agrees very well with a recent lattice calculation with relativistic bottom quarks which gives  Table I, but now considering the X b2 mass histograms displayed in Fig. 2 and the coupling given in Eq. (10). The decay width of the X b2 →BB * mode is the same because of charge conjugation symmetry.
of g ∞ = 0.492 ± 0.029 [48]. Thus we expect that the decay widths of Table II slightly overestimate the real ones.
For the BB mode we find a pronounced dependence on the Gaussian cutoff Λ employed in the dynamical generation of the resonance. This is inherited from the strong dependence of the X b2 mass on this UV cutoff, as discussed in Eq. (10), which affects the available phase space for the decay. With all these shortcomings, we expect a partial width in the 1-10 MeV range, when both charge modes are considered.
In the BB * decay mode, the impact of the Gaussian regulator is even larger, because it turns out that for Λ = 1 GeV, the central value of the resonance mass M X b2 = 10594 +22 −26 MeV is located below the threshold m B + m B * ∼ 10604 MeV. Thus, in that case, the decay will be forbidden. For Λ = 0.5 GeV, we estimate a width also in the 4-12 MeV range, when the four possible decay modes IV. THE X 2 AND X b2 RADIATIVE DECAYS In this section, we will study the X 2 → DD * γ and X b2 →BB * γ decays. The interaction of the photon with the s-wave heavy mesons contains two contributions that correspond to the magnetic couplings to the light and heavy quarks [49] (see also Appendix A 3). Both terms are needed to understand the observed electromagnetic branching fractions of the D * + and D * 0 because a cancellation between the two terms accounts for the very small width of the D * + relative to the D * 0 [5]. Actually, one finds [49,50]: where E γ is the photon energy, m c the charm quark mass and α ∼ 1/137.036 the fine-structure constant. In the non-relativistic constituent quark model β = 1/m q ∼ 1/330 MeV −1 , where m q is the light constituent quark mass. Heavy meson chiral perturbation theory allows one to improve upon this approximation by including corrections from loops with light Goldstone bosons, which give O( √ m q ) corrections [49].
Using isospin symmetry to relate Γ(D * 0 → D 0 π 0 ) and Γ(D * + → D 0 π + ), correcting by the slightly different available p-wave phase space in each of the two decays, and taking into account the experimental D * 0 and D * + widths and radiative branching fractions quoted by the PDG [5], we find: These values differ from those used in Ref. [50] because of the recent accurate BABAR measurement of the D * + decay width, mentioned in Sect. III, which is around 10% smaller than the previous CLEO one used in Ref. [50]. Fixing the charm quark mass to m c = 1.5 GeV, we fit the parameter β to the above experimental values and find β −1 = (293 ± 11) MeV.
In what follows, we will study decays of the type X 2 → PP * γ, being P and P * pseudoscalar and vector heavy-light mesons, respectively. Let us define p µ 1 , p µ 2 and p µ 3 as the four vectors of the final photon, pseudoscalar and vector mesons, respectively. Besides, let us define the invariant masses Since, as we will see, the Feynman amplitudes depend only on the invariant masses m 2 12 and m 2 23 of the final γP and PP * pairs, respectively, we can use the standard form for the Dalitz plot [5] with |T | 2 the absolute value squared of the decay amplitude with the initial and final polarizations being averaged and summed, respectively. Thus, we readily obtain where for a given value of m 2 23 , the range of m 2 12 is determined by its values when p P is parallel or anti-parallel to p γ [5]: Because of parity conservation, this is a p-wave decay and hence the photon momentum appears always in the amplitudes. In the X 2 rest frame, it is given by A. X 2 (4013) → DD * γ We will first consider the X 2 (4013) → D 0D * 0 γ decay, which proceeds according to the Feynman diagrams depicted in Fig. 6. This decay can take place directly through the radiative transition of the constituent D * 0 as shown in Fig. 6(a), which is the tree level approximation. However, there are other mechanisms driven by the DD * FSI. After emitting the photon, the vector meson D * 0 transits into the D 0 , and it can interact with the other constituent in the X 2 as shown in Fig. 6(b). There is a third (c) mechanism in which the photon is emitted from theD * 0 meson, and the virtual D * 0D0 rescatter into D 0D * 0 . Finally, Fig. 6

Tree Level Approximation
The Feynman amplitude of the mechanism depicted in Fig. 6(a) reads (as in the previous sections, we treat the charm mesons non-relativistically) with m 12 the invariant mass of the final γD 0 pair. Besides, ǫ i (λ γ ) is the polarization vector of the final photon with helicity λ γ , p γ is its three momentum and for the normalization of the heavy meson fields and the X 2 D * 0D * 0 coupling. Finally, The Gaussian form factor is inherited from the D ( * )D( * ) EFT UV renormalization scheme.
We have neglected the D * 0 width in the above propagator because, since it is quite small, its inclusion only leads to small numerical variations in the decay rate which are certainly smaller than uncertainties induced by the errors in the coupling g X 2 0 and the mass of the X 2 (4013) resonance.
Similarly, the use of the non-relativistic D * 0 propagator instead of m 2 12 − m 2 D * 0 + iε −1 leads also to numerically negligible differences, as compared to the HQSS corrections. The sum over theD * 0 , γ and X 2 polarizations can be easily done, and we get The amplitude for D +D * − γ decay is readily obtained from Eq. (32) making the obvious replacements: g X 2 0 → g X 2 c , β 1 → β 2 and (m D 0 , m D * 0 ) → (m D + , m D * + ). Performing the phase space integration, we find at tree level (assuming the existence of the X 2 state) Γ(X 2 (4013) → D + D * − γ) tree = 0.10 +0.10 −0.05 0.09 +0.06 −0.03 keV, where the values outside and inside the parentheses are obtained with Λ = 0.5 and 1 GeV, respectively. The errors account for the uncertainty, both in the mass of the X 2 state and in its couplings g X 2 0,c , derived from the X(3872) input (M X(3872) and the ratio R X(3872) of the decay amplitudes for the X(3872) → J/ψρ and X(3872) → J/ψω decays) and the HQSS corrections, as explained in the the caption of Table I. We have neglected additional uncertainties stemming from the error on β (≃ 3%), because it is totally uncorrelated to those discussed above, and it is much smaller than those affecting for instance the g X 2 0,c couplings. The neutral mode width is much larger that the charged one, thanks to the bigger magnetic D * Dγ coupling and a larger available phase space.
In analogy with the discussion of Eqs. (20) and (21) in Ref. [33] for the X(3872) → D 0D0 π 0 decay, in the X 2 radiative processes the relative distance of the D * D * pair can be as large as allowed by the size of the X 2 (4013) resonance, since the final state is produced by the one body decay of theD * meson instead of by a strong two body transition. Thus, the radiative DD * γ decays might provide details on the long-distance part of the resonance wave function. For instance, the dΓ/d| pD * 0| [dΓ/d| p D * − |] distribution is related to the X 2 (4013) wave-function Ψ( pD * 0) [Ψ( p D * − )] [33]. This is in sharp contrast to the DD and DD * decay modes studied in the previous sections, which turned out to be strongly sensitive to short distance dynamics of the resonance, as revealed by the notorious dependence on the UV form factors.
However, all these considerations are affected by the DD * FSI effects to be discussed next.

DD * FSI Effects
To account for the FSI effects, we include in the analysis the DD * → DD * and D * D → DD * T - The X 2 → D 0D * 0 γ decay amplitude for the mechanisms depicted in Fig. 6(b) and (c) reads where m 23 is the invariant mass of the final D 0D * 0 pair, and the three-point loop integral function J (M 1 , M 2 , M 3 , p γ ) is given in the Appendix C 2. The integral is convergent, however, for consistency it is evaluated using the same UV regularization scheme as that employed in the D ( * )D( * ) EFT. In sharp contrast with the hadronic decays studied above, the momenta involved in these integrals are rather low.
On the other hand, we see that in the (b)+(c) contribution there appears the combination T D 0D * 0 →D 0D * 0 (m 23 ) + T D * 0D0 →D 0D * 0 (m 23 ). In the isospin limit, when the mass differences between the neutral (D 0D * 0 ) and charged (D + D * − ) channels are neglected, we will find T 00→00 = (T I=0 C=−1 + T I=1 C=−1 )/2. From Eqs. (A8) and (A10), we find the C-parity odd isospin amplitudes, 13 with G DD * ≃ G D 0D * 0 ≃ G D +D * − a common loop function. Note that the kernel of this LSE is fixed by the isoscalar (C 0Z ) and isovector (C 1Z ) C(charge conjugation) = −1 terms of V D ( * )D( * ) . This is a trivial consequence of the conservation of this symmetry, taking into account that the X 2 and the photon are even and odd C-parity states, respectively.
The (d) and (e) FSI contributions of Fig. 6 are similar, with obvious replacements. We find where and in the isospin limit, we would have with F Λ (E) = Diag f neu Λ (E), f ch Λ (E) , where the Gaussian form factors are defined after Eq. (A13). The Z b (10610) observed in Ref. [51] carries electric charge, and its neutral partner was also reported by the Belle Collaboration [52]. It lies within a few MeV of the BB * threshold and it is tempting to speculate about it as a hadronic molecule. Belle also reported the discovery of a second exotic electrically charged bottomonium state [51], Z b (10650) in the vicinity of the B * B * threshold.
For both the Z b (10610) and Z b (10650) states, J P = 1 + are favored from angular analyses.
Within our scheme, we assume that the Z b (10610) resonance is an isovector BB * + B * B / √ 2 s-wave bound state with J P C = 1 +− [15]. Note, HQSS predicts the interaction of the B * B * system with I = 1, J P C = 1 + quantum numbers to be identical to that of the BB * pair in the Z b (10610) sector. Thus, HQSS naturally explains [53] the approximate degeneracy of the Z b (10610) and Riemann sheet [15], which probably correspond to the recently observed charged charmonium-like Z c (3900) and Z c (4025) states [55][56][57][58][59]. These resonances lie close to the DD * and D * D * thresholds, respectively, while J P = 1 + quantum numbers are favored from some angular analyses.
Due to the presence of the Z c (3900) close to threshold, one should expect the loop (FSI) mechanisms depicted in Fig. 6 to be important since T I=1 C=−1 must have a pole. However, the value of C 0Z = (C 0A − C 0B ) is still unknown. It is not determined by the inputs deduced from the X(3872) and Z b (10610) states used in the present analysis. Depending on the value of C 0Z , there can be an isoscalar J P C = 1 +− DD * s-wave bound state or not. For instance, considering the case for Λ = 0.5 GeV and taking the central value for C 0Z ∼ −2.5 fm 2 one finds a bound state pole in the DD * system bound by around 10 MeV; if C 0Z ∼ −1.5 fm 2 , there will be a DD * bound state almost at threshold; if the value of C 0Z is larger, there will be no bound state pole any more. Therefore, the information of C 0Z will be crucial in understanding the DD * system and other exotic systems related to it through heavy quark symmetries [15,35]. Conversely, as we will see, the X 2 radiative decay width could be used to extract information on the fourth LEC, C 0Z , thanks to the FSI effects.
To investigate the impact of the FSI, in Fig. 7 we show the dependence of the partial X 2 (4013) → DD * γ decay widths on C 0Z . For comparison, the tree-level results are also shown in the same plots.
As expected, for the decay into the D 0D * 0 γ channel, the FSI effects turn out to be important, and for some values of C 0Z , they dominate the decay width. The maximum effects of the FSI mechanisms approximately occur for values of C 0Z which give rise to an isoscalar 1 +− DD * bound or virtual state close to threshold. One can see an apparent deviation from the tree-level results in this region. When C 0Z takes smaller values, the binding energy of the bound state increases and moves apart from threshold; when C 0Z takes larger values, the pole moves deeper into a non-physical RS and becomes a virtual state further from the threshold. In both situations, the FSI corrections turn out to be less important. On the other hand, the FSI corrections are always important in the D + D * − γ channel. This is because the tree level amplitude involves only the D * ± D ± γ magnetic coupling, while FSI brings in the neutral magnetic coupling, which is much larger than the former one. This is also the reason, besides phase space, why the tree level width is much larger in the neutral mode than in the charged one, as commented above.
Notice that in the above calculations, we did not include the contribution from the coupledchannel FSI D * D * → DD * which can come from replacing the D * Dγ vertices in Fig. 6 by the D * D * γ ones 14 . We have checked that this is a good approximation when the re-summation of the charmed meson scattering is switched off. This is partly because the loop integral defined in Eq. (C8) takes a much larger absolute value for the considered diagrams than for those with the D * D * γ vertices in most regions of the phase space. The only exception is when the photon has a very low energy so that the D * D * are almost on shell. However, after integrating over the phase space, this region provides a small contribution to the partial decay width because of the p 3 γ suppression due to relative p-wave between the photon and the 1 +− DD * system. The re-summation would introduce a further complication because of the presence of the Z c (4025) which couples dominantly to the D * D * in the 1 +− channel. It contributes mainly to the region close to the pole position of the Z c (4025). Again, this corresponds to the region with the very low-energy photon and thus it is suppressed due to phase space. Therefore, we expect that the neglected contribution discussed here has little impact, and its numerical effects should be safely covered by the sizable HQSS uncertainty exhibited in the results.
To better understand the dependence of the X 2 (4013) → DD * γ decay widths on C 0Z in the bottom panel of Fig. 8 we show the pole positions of the odd C-parity DD * → DD * T -matrix as functions of this LEC and for an UV cutoff of 0.5 GeV. There are two coupled channels, the neutral one which has the lowest threshold, and the charged channel. As a consequence, there are three 14 The electric part of the D * D * γ vertex does not contribute to the FSI X2 decay width amplitude when the quantum numbers of the final D * D * pair are 1 +− , with the two heavy mesons in relative s−wave. Thus, we are left with the contribution from the magnetic coupling, as in the D * Dγ case. both decay modes as a function of C 0Z , which were already presented in Fig. 7. As can be seen, it The red solid curve shows the evolution of the bound state with C 0Z , while the dashed and the dash-dotted curves show that of the virtual ones. The vertical black line marks the value of C 0Z for which a DD * bound state is generated at the D 0D * 0 threshold. is around this critical value C 0Z = −1.5 fm 2 , when the FSI effects are larger for both decays, due to the vicinity of the pole to the threshold. This happens regardless of whether it is a bound or a virtual state, since the presence of the pole in both situations greatly enhances the odd C-parity DD * → DD * T -matrix near threshold. This can be appreciated in the top panel of Fig. 8, where the dependence of the T 00→00 and T +−→+− scattering lengths on C 0Z is shown.
Thus we have understood why in the region of values of C 0Z around −1.5 fm 2 , FSI corrections strongly affect the D 0D * 0 γ decay width: this channel has the lowest threshold and the bound or virtual state is located on or nearby it. For the charged decay mode, the width exhibits a maximum for values of C 0Z also in this region, followed by a clear minimum placed now in the vicinity of C 0Z ∼ C 1Z . Notice that when C 0Z = C 1Z , T 00→+− vanishes (see Eq. (45)) and therefore the contribution due to the neutral mesons, driven by the largest magnetic coupling (β 1 ), in the FSI loops disappears. The exact position of the minimum is modulated by the further interference between the tree level and the FSI charged loops, which are comparable.
In the bottom panel, we also observe a virtual state pole, the position of which is rather insensitive 15 to C 0Z . It is originated by the interaction in the I = 1 sector, C 1Z , and it should be related to the Z c (3900) exotic charmonium-like state reported by the BESIII and Belle collaborations. Within our LO EFT scheme, we do not find a DD * bound state, but instead a pole located near threshold in a non-physical RS [15]. 15 The situation is more complicated, as can be seen in the plot. There is a narrow region of values of C0Z around  25) and (26) can be used to predict the widths for the B * radiative decays [49], where we have taken the value m b = 4.8 GeV for the bottom quark mass.
As in the study of its hadronic decays, we assume the X b2 be a pure I = 0 state, with equal coupling to its neutral and charged components, g X b2 The isospin breaking effects for the B * mesons are expected to be small and the tiny difference between the B 0 and B ± masses can safely be neglected as well. In this limit we find at tree level where β a = β 1 (β 2 ) for the where the values have been obtained with Λ = 0.5 GeV. We remind that for Λ = 1 GeV, the central value of the resonance mass M X b2 is located below the threshold (m B + m B * ) ∼ 10604 MeV and the decay is forbidden. The errors reflect the uncertainty in the inputs from the X(3872) and the HQSS breaking corrections, as outlined in the caption of Table II;  The amplitude for the FSI mechanisms is readily evaluated and we find FSI corrections turn out to be important, as can be appreciated in Fig. 9. This is because we are generating in the T I=1 C=−1 amplitude a bound state [Z b (10610)], almost at threshold (binding energy (2.0±2.0) MeV [54]), that enhances the loop mechanisms, as we discussed in the charm sector. If we pay attention for instance to the charged B − B * + γ mode, we could appreciate a distinctive feature: there appears a destructive interference pattern between the tree level and the FSI amplitudes.
Thanks to our MC procedure where correlations are consistently propagated, we also observe a reduction of the size in the uncertainties. Besides the uncertainties on the mass and the couplings of the X b2 resonance, the errors on C 1Z quoted in Eq. (46) are also accounted for in the 68% CL bands displayed in the panels. Actually, these latter uncertainties should have also an important impact on the total CL bands. This is because variations of C 1Z allow for situations where the pole is located precisely at threshold (zero binding energy) or bound by about 4 MeV. In the first case the FSI contribution should be larger than that obtained with the central value of C 1Z , which correspond to a binding energy of 2 MeV. These big 68% CL bands makes hard to disentangle any further dependence on C 0Z , which in this case turns out to be quite mild.

V. CONCLUSIONS
In this work we have studied the hadronic and radiative decays of a molecular P * P * state with quantum numbers J P C = 2 ++ in the charm (X 2 ) and bottom (X b2 ) sectors using an EFT approach.
We have considered the X(3872) resonance as a J P C = 1 ++ DD * hadronic molecule. The X 2 and the X b2 states will be HQSFS partners of the X(3872) with masses and couplings to the P * P * heavy meson pair determined by the properties of the X(3872) resonance.
The hadronic d-wave X 2 → DD and X 2 → DD * two-body decays are driven via one pion exchange. We observed that as a result of the contribution from highly virtual pions, which is out of control in the low-energy EFT, these hadronic decay widths (hence the total width of the X 2 as well) bear a large systematic uncertainty. Even though the momenta involved in these decays probably lie outside the range of applicability of EFT the calculations are still valuable as a way to find reasonable estimates of these partial decay widths, which we expect to almost saturate the X 2 decay width. To this end and in analogy to the Bonn potential, we have included a monopole pion-exchange form factor, with a cutoff around 1 GeV, in each of the D * Dπ and D * D * π vertices to suppress the contribution of large momenta. We finally estimate the partial widths of both processes to be of the order of a few MeV. The analysis runs in parallel in the bottom sector with the assumption that the bare contact terms in the Lagrangian are independent of the heavy flavor.
In this sector, we also find widths of the order of a few MeV.
We discussed the radiative X 2 → DD * γ and X b2 →BB * γ decays as well. The widths are small, of the order of keV's (eV's) in the charm (bottom) sectors. Furthermore, they are affected by the DD * or BB * FSI mechanisms. FSI effects are large because they are enhanced by the presence of the isovector Z c (3900) and Z b (10610) resonances located near the D 0D * 0 andBB * thresholds, respectively. In the charm sector, FSI corrections turn out to be also sensitive to the negative Cparity isoscalar DD * interaction (C 0Z ). Thus, future precise measurements of these radiative decay widths might provide valuable information on this LEC, which cannot be in principle determined from the properties of the X(3872), Z b (10610) and Z b (10650) resonances. Constraints on this latter LEC are important in order to understand the dynamics of the P ( * )P ( * ) system.
The definition for H (Q) a also specifies our convention for charge conjugation, which is CP with c the Dirac space charge conjugation matrix satisfying cγ µ c −1 = −γ T µ .

Quadruple-heavy-meson contact interaction
At very low energies, the interaction between a heavy and anti-heavy meson can be accurately described just in terms of a contact-range potential. The LO Lagrangian respecting HQSS reads [60] L with τ the Pauli matrices in isospin space, and C

(τ )
A,B light flavor independent LECs, which are also assumed to be heavy flavor independent. Note that in our normalization the heavy or anti-heavy meson fields, H (Q) or H (Q) , have dimensions of E 3/2 (see [61] for details). This is because we use a non-relativistic normalization for the heavy mesons, which differs from the traditional relativistic one by a factor √ M H . For later use, the four LECs that appear above are rewritten into C 0A , C 0B and C 1A , C 1B which stand for the LECs in the isospin I = 0 and I = 1 channels, respectively. The relations read The LO Lagrangian determines the contact interaction potential V = −L/4, which is then used as kernel of the two body elastic LSE (see Eq. (4) and the related discussion).
The LECs that appear in the J P C = 1 ++ and 2 ++ sectors [Eqs.
(3) and (7)] turn out to be C 0X ≡ C 0A + C 0B and C 1X ≡ C 1A + C 1B . The contact interaction in the Z b (10610) sector (I = 1, On the other hand, the interaction in the D 0D * 0 , D * 0D0 , D + D * − , D * + D − space reads: 16 with C 0Z = C 0A − C 0B and the orthogonal matrix A given by: Equation (A8) trivially follows from the fact that the L 4H interaction of Eq. (A6) is diagonal in the isospin basis and the charge conjugation is well defined. 17 The interaction given in Eq. (A8) can be used as the kernel of an UV finite LSE to obtain the T -matrix that we use to account for the FSI in the radiative decays studied in Section IV, with the two particle regularized matrix propagator defined as where trivially G D 0D * 0 = G D * 0D0 and G D + D * − = G D * + D − . In addition, the on-shell UV Gaussian form factor matrix reads , with a = (neu), (ch).

P ( * )P ( * ) π interactions
The relevant term in the LO Lagrangian of the heavy meson chiral perturbation theory [40][41][42][43] that provides the D * Dπ coupling is 16 In the bottom sector, the corresponding basis is: (1 ↔ +, 2 ↔ −). In our convention, the C-parity of these states is independent of the isospin and it is equal to ∓1.
with φ a relativistic field that describes the pion, 18 g is the heavy flavor independent P P * π coupling and f π = 92.2 MeV the pion decay constant. Note that in our normalization, the pion field has a dimension of energy, while the heavy meson or antimeson fields H (Q) or H (Q) have dimensions of E 3/2 , as we already mentioned.

HHγ interactions
The magnetic coupling of the photon to the s-wave heavy mesons is described by the Lagrangian [49,50] is the light quark charge matrix, and Q ′ is the heavy quark electric charge (in units of the proton Besides, m Q is the heavy quark mass and β is the parameter introduced in Ref. [49]. These two terms describe the magnetic coupling due to the light (preserves HQSS) and heavy quarks (suppressed by 1/m Q ), respectively.
Both terms are needed to understand the observed electromagnetic branching fractions of the D * + and D * 0 because a cancellation between the two terms accounts for the very small width of the D * + relative to the D * 0 [5].
In the non-relativistic constituent quark model β = 1/m q ∼ 1/330 MeV −1 , where m q is the light constituent quark mass. Heavy meson chiral perturbation theory provides contributions from Goldstone boson loops, which give O( √ m q ) corrections to the decay rates [49]. If these loop corrections are evaluated in an approximation where heavy hadron mass differences are neglected, the correction to the above formulas can be incorporated by making the following replacements [49] with f K ∼ 1.2f π . 18 We use a convention such that φ = φx−iφy √ 2 creates a π − from the vacuum or annihilates a π + , and the φz field creates or annihilates a π 0 . We adopt the usual convention C( τ · φ)C −1 = ( τ · φ) T . In this appendix, we will argue that the d-wave DD may be treated perturbatively in the 2 ++ system. Even though this was already discussed in Ref. [7], we have included here a new argument grounded on a different EFT to make a more compelling case on the smallness of this contribution to the X 2 mass. We will compare the power counting of the self-energy diagrams of the X 2 from the d-wave DD and the s-wave D * D * two-point loops, see Fig. 10. If the DD loop is suppressed in comparison with the D * D * one, it will validate the perturbative treatment of the DD. Because in our case the heavy mesons are non-relativistic, we can apply a velocity counting for the loops analogous to the power counting of the heavy meson loops in heavy quarkonium transitions [62,63].
For the D * D * loop, the velocity counting of the self-energy reads as where g S denotes the value of the s-wave coupling of the X 2 to the D * D * , v denotes the velocity of the D * meson, v 5 is for the loop integral measure since the non-relativistic energy is counted as O(v 2 ), and 1/(v 2 ) 2 accounts for the two non-relativistic propagators.
Similarly, for the DD loop, denoting the velocity of the D meson by w, the velocity counting is given by where g D is the d-wave coupling constant normalized to have the same dimension as g S , and the factor of w 4 in the denominator comes from the two d-wave vertices.
Therefore, we obtain the ratio The question is now how g D compares with g S . We can estimate g D by considering the one-pion exchange diagram considered in this work as illustrated in Fig. 11. Because the X 2 is very close to the D * D * threshold, we should count each of the D * (D * ) propagators as 1/v 2 . This is equivalent to affirm that the cut due to the D * D * in the triangle diagram in Fig. 11 is the same as that in the D * D * bubble diagram of the X 2 self-energy. Thus, we can count the D * in both diagrams in the same way. But the pion propagator should be counted differently. The reason is that because the X 2 couples to the DD in a d-wave, the momenta in the D * Dπ vertices of the one-pion exchange diagram should become the momenta of the D andD, q D = m D w, and the pion momentum is of the same order as we discussed in Sect. III A 1. This is to say that the pion propagator should be counted as 1/w 2 rather than 1/v 2 . Thus, expressing the content of Fig. 11 in terms of power counting gives where g is the axial coupling constant in Eq. (A14), and Λ χ = 4πf π is the hard scale for the chiral expansion.
Numerically, for the case of the X 2 , we have w ≃ (M X 2 − 2m D )/m D ≃ 0.38, and v ≃ (M X 2 − 2m D * )/m D * ∼ 0.06 if we take 7 MeV as the binding energy (recall that we have the charged D * D * channel explicitly whose threshold is around 7 MeV above the neutral one). With these values, we use Eqs. (B3) and (B4), which leads to g D ∼ 0.6 g S , to obtain an estimate of the contribution of the d-wave DD to the X 2 self-energy relative to the s-wave D * D * , r D/S ∼ 0.05.
The above value suggests a high suppression of the d-wave DD in comparison with the s-wave D * D * . We notice that the power counting of Ref. [7] indicates that the size of the DD loop is N 4 LO (next-to-next-to-next-to-next-to-leading order), in line with the velocity power counting arguments.
the integration over x gives zero. The convergence of the integral is greatly enhanced because P 2 (x) is orthogonal to x as well. Moreover, the same type of arguments guarantees that I 0 (M, m; M X 2 , q 2 ) ∼ const. in the q 2 → 0 limit. Numerically, we use non-relativistic kinematics to compute | q | in the evaluation of I 0 (M, m; M X 2 , q 2 ) in Eq. (C5), i.e., q 2 ≃ 2µ F 1 F 2 (M X 2 − m F 1 − m F 2 ). However, to guarantee the appropriate d-wave phase space, we use relativistic kinematics to evaluate q i q j in Eq. (C4) and the | q | phase-space factor that appears in Eqs. (13) and (19).
In addition, the exchanged pion is highly virtual, and one might include a vertex form factor of the form in each of the two πP ( * ) P ( * ) vertices.

Radiative decays
In the computation of the FSI effects on the radiative decays of the X 2 and X b2 resonances in the Sect. IV, the following three-point loop function appeared with P µ = (M X 2 , 0) in the rest frame of the X 2 and µ being finite, it should be evaluated using the same UV renormalization scheme as that employed in the D ( * )D( * ) EFT. This is accomplished by including in the integrand of Eq. (C8) a Gaussian form factor, F Λ ( q ) defined as F Λ ( q ) = e −( q 2 −γ 2 )/Λ 2 e −( q 2 cm − q 2 on shell )/Λ 2 . (C10) Here γ 2 = 2µ 12 (M X 2 − M 1 − M 2 ), q 2 on shell = 2µ 23 (m 23 − M 2 − M 3 ), with m 2 23 = (P − p γ ) 2 = M 2 X 2 − 2M X 2 E γ , and Note that the first exponential factor accounts for the off-shellness in the X 2 D * 0D * 0 coupling, as in Eq. (33), while the second one accounts for the virtuality of the incoming mesons in the DD * → DD * and D * D → DD * T -matrices. Note that, after the inclusion of this factors, an analytical expression for the integral cannot be easily obtained, and it needs to be computed numerically.