The $X(3872)\rightarrow J/\psi \pi \gamma$ and $X(3872)\rightarrow J/\psi \pi\pi \gamma$ decays

We study the $\rho$ and $\omega$ meson contribution to the radiative decays $X(3872)\rightarrow J/\psi \pi \gamma$ and $X(3872)\rightarrow J/\psi \pi\pi \gamma$. The $X(3872)\rightarrow J/\psi \pi \gamma$ is dominated by the $\omega$ meson. As for the $X(3872)\rightarrow J/\psi \pi\pi \gamma$, the contributions of the cascade decays through the $\rho$ and $\omega$ mesons are strongly suppressed with respect to the diagrams which proceed either through the $\psi(2S)$ or the three body decay of $\rho$. The branching ratios of $X(3872)\rightarrow J/\psi \pi \gamma$ and $X(3872)\rightarrow J/\psi \pi\pi \gamma$ are $(8.10^{+3.50}_{-2.88})\times10^{-3}$ and $(2.38\pm1.06)\%$, which may be accessible by the BESIII and LHCb Collaborations. Especailly, the $X(3872)\rightarrow J/\psi \pi \gamma$ and $X(3872)\rightarrow J/\psi \pi^+\pi^- \gamma$ decays can be employed to extract the couplings $g_{X\psi\omega}$ and $g_{X\psi\rho}$, which probe the isoscalar and isovector components of the X(3872) wave function respectively.

Since the discovery of the X(3872), there have been tremendous efforts to investigate its inner structure experimentally and theoretically. The exotic nature of the X(3872) was embodied in its mass and width, which are listed in Table I. One  of the most intriguing feature of the X(3872) is that its mass almost coincides with the D 0D * 0 threshold. Considering the narrow width of X(3872), it is natural to regard the X(3872) as a DD * hadronic molecule [21][22][23][24][25][26]. The molecule picture not only explains the coincidence of the mass of the X(3872) with the D 0D * 0 threshold naturally, but also explains its isospin violation in the J/ψρ decay mode [22,27,28]. However, some molecule models meet with difficulties when explaining the following phenomena: • The ratio Γ(B 0 → K 0 X)/Γ(B + → K + X) is about unity according to an estimation based on the molecule picture [29][30][31], which is about two times larger than mea- * Electronic address: wu qi@pku.edu.cn † Electronic address: wangjzh2022@pku.edu.cn ‡ Electronic address: zhusl@pku.edu.cn surements by the BaBar [17] and Belle [32] Collaborations.
• As a loosely bound hadronic molecule with a small binding energy, X(3872) was expected to be so fragile that it would be hard to explain the observed production rate in the high energy pp collisions at the Tevatron [33].
Actually, the above difficulties indicate that there should exist a significant cc component in the wave function of the X(3872) [34,35]. In other words, the coupled channel effect may play an important role in the formation of the X(3872).
To date, the inner structure of the X(3872) is still an open question and remains challenging. In addition to the mass spectrum, the decay patterns also encode important dynamical information and hence provide another perspective about its underlying structure. The ratio B[X → J/ψπ + π − π 0 ]/B[X → J/ψπ + π − ] has been measured by several experiments [36][37][38], which indicates a large isospin violation. This ratio is of great interest and has been investigated in different scenarios [23,28,34,[39][40][41][42][43][44]. Different components in the wave function of the X(3872) will affect the decays either in the long distance or the short distance. In other words, the decay patterns encode very important information on the underlying structure and can be used to test different theoretical explanations. For example, the X → D 0D0 π/γ, which proceeds through the decays of either D * 0 orD * 0 and thus belongs to the long-distance decays, can be used to study the longdistance structure of the X(3872) [21]. Pionic transitions from the X(3872) to χ cJ were investigated in Refs. [45][46][47]. The relative rates for these transitions to the final states with different J is very sensitive to the inner structure of the X(3872) as a pure charmonium state or a four-quark/molecular state [45]. The predictions of the ratio B[X → ψ ′ γ]/B[X → J/ψγ] from the DD * molecule [22,48], pure charmonium state [49] and molecule−charmonium mixture [50,51] turned out to be dramatically different from each other, which reflects the importance of the cc component in the X(3872).  [20].

Decay channels
Branching ratios In order to pin down the nature of the X(3872), searching for more decay modes is crucial. In Table II, we list the observed decays of the X(3872). The dominant decay channel is the open-charm decay, which is 37% for the D 0D * 0 and 49% for the D 0D0 π 0 . The branching ratios of the radiative decays J/ψγ and ψ ′ γ are of the same order as those of the hiddencharm decays. Are there other radiative decays of the X(3872) whose decay rates could be as large as those of the J/ψγ and ψ ′ γ?
Recently, the LHCb Collaboration observed a sizeable ω contribution to X(3872) → J/ψππ decay [52]. Inspired by the recent LHCb Collaboration measurements, we study the ρ and ω meson contributions to the radiative decay processes X(3872) → J/ψπγ and X(3872) → J/ψππγ in this work. In Ref. [53], the authors noted that the dominant contributions to X(3872) → J/ψπ + π − and X(3872) → J/ψπ + π − π 0 arise from the diagrams with the X(3872) coupling to the J/ψρ and J/ψω, respectively. One may wonder whether the same scenario still holds in the X(3872) → J/ψπγ and X(3872) → J/ψππγ.
Compared with X(3872) → J/ψππ, X(3872) → J/ψπγ has an advantage in exploring the isospin violation of the J/ψρ mode. The LHCb experiment has proved that there is a sizeable ω contribution to X(3872) → J/ψππ. In other words, the X(3872) → J/ψππ is not a clean process to study the isospin violation of the J/ψρ mode. In Fig. 1(a), the X(3872) → J/ψπγ decay occurs through the intermediate ρ or ω meson. The ω meson dominates this process because g Xψω and g ωπγ are both much larger than g Xψρ and g ρπγ respectively. Thus, the X(3872) → J/ψπγ should be a cleaner process to extract the coupling g Xψω . By the same token, the X(3872) → J/ψππγ is a cleaner process to study the isospin violation channel of J/ψρ. For this purpose, we will not only check the contribution of the ρ and ω mesons to the X(3872) → J/ψπγ process but also the contributions of diagrams with the X(3872) coupling to the J/ψρ or J/ψω in the X(3872) → J/ψππγ process. Besides the ρ and ω contributions, there are some nonresonant contributions which should be considered as the background contribution. We will predict the branching ratios of X(3872) → J/ψπγ and X(3872) → J/ψππγ, which could be tested by the BESIII and LHCb Collaborations. This paper is organized as follows. After the introduction, we present the theoretical framework in the calculation of X(3872) → J/ψπγ and X(3872) → J/ψππγ. We derive the in-variant decay amplitudes and invariant mass distributions using the effective Lagrangian method. In Sec. III, we present the invariant mass distribution of πγ and ππγ, and the branching ratios of X(3872) → J/ψπγ and X(3872) → J/ψππγ. Sec. IV is a short summary.

II. THEORETICAL FRAMEWORK
In this work, we utilize the effective Lagrangian method to study the radiative processes X(3872) → J/ψπγ and X(3872) → J/ψππγ. In the following subsections, we introduce the effective Lagrangian and invariant decay amplitudes and the formulas of the invariant mass distributions related to the radiative processes X(3872) → J/ψπγ and X(3872) → J/ψππγ.  Table II, the branching ratios of the decays X(3872) → π 0 χ c1 and X(3872) → γψ(2S ) are sizable. In addition, the branching ratios of ψ(2S ) → J/ψπ + π − and ψ(2S ) → J/ψπ 0 π 0 are (34.68 ± 0.30)% and (18.24 ± 0.31)% respectively [20]. The branching ratio of the χ c1 → γJ/ψ is also quite large. Thus, the diagrams Fig. 1(b) and Fig. 2(c) will also contribute to the background. In contrast, the πγ and ππγ invariant mass spectrum tend to peak around the ρ and ω mass for our concerned ρ and ω contributions. Besides, the QED gauge invariance requires the existence of Fig. 2(d). One notes that the ω may also contribute to Fig. 2(d). The branching ratio of ω → π 0 π 0 γ is (6.7 ± 1.1) × 10 −5 [20]. The branching ratio of ω → π + π − γ has not been measured yet. If one neglects the long range contributions and considers the isospin symmetry, the branching ratio of ω → π + π − γ is just twice the ω → π 0 π 0 γ. In contrast, the branching ratio of ρ → π + π − γ is around 10 −2 . In other words, the ω contribution to Fig. 2(d) is much smaller than the ρ contribution. Thus, we only consider the diagram in Fig. 2
In evaluating the decay amplitudes of X(3872) → J/ψπγ and X(3872) → J/ψππγ associated with the ρ and ω mesons in Figs. 1 and 2, we include the form factors for the ρ and ω mesons since they are not point-like particles [58]. In this work we adopt the following form factor: where we adopt Λ ρ = Λ ω = 598 MeV as a result of Γ ρ being a constant [53]. We have checked that our results barely depend on the form factor. In Ref. [53], the coupling constants g Xψρ and g Xψω are determined to be 0.09 ± 0.02 and 0.31 ± 0.06 by fitting to the LHCb data with Γ ρ being a constant. Other coupling constants can be determined from the corresponding experimental partial widths. With the effective Lagrangian in Eqs. (2)-(6), the decay widths of ρ → πγ, ω → πγ, X(3872) → ψ(2S )γ, X(3872) → χ c1 π and χ c1 → J/ψγ are where p f ρ , p f ω , p f X and p f χ c1 are the three momenta of the final mesons in the ρ, ω, X(3872) and χ c1 rest frame, respec- and |g PS γ | = 0.23GeV −1 . g ωρπ can be determined from the experimentally measured partial decay of ω → ρπ → πππ, which is |g ωρπ | = 50GeV −1 with Γ ρ being a constant [53]. Note that one can only obtain the absolute value of the coupling constant from the partial decay width. The phase can not be fixed. In this work, the default values of the above coupling constants are real and positive. The total invariant decay amplitudes of X(3872) → J/ψπγ and X(3872) → J/ψππγ are where φ ω stands for the relative phase between the ω and ρ terms, φ ψ ′ stands for the relative phase between ρ/ω and ψ ′ terms. We adopt the phase angle φ ω obtained by fitting the LHCb data in Ref. [53], which is 134.5 • .

C. Invariant mass distributions
The invariant π 0 γ mass distribution of the X(3872) → J/ψπ 0 γ decay is given by where p * 1 and (θ 1 , φ 1 ) are the three-momentum and decay angle of the outgoing π 0 /γ in the center-of-mass (c.m.) frame of the final π 0 γ system, p 4 is the three-momentum of the final J/ψ meson in the rest frame of X(3872), and M π 0 γ is the invariant mass of the final π 0 γ system.
with M ππγ the invariant mass of ππγ system. The p * 1 and (θ 1 , φ 1 ) are the three-momentum and decay angles of the outgoing π in the πγ center-of-mass (c.m.) frame. The p ′ 3 and (θ 2 , φ 2 ) are the three-momentum and decay angles of the outing π 0 in the ππγ c.m. frame. The p 4 is the three-momentum of the final J/ψ meson in the X(3872) rest frame. Definitions of these variables in the phase space integration of the X(3872) → J/ψππγ decay can be found in the Appendix of Ref. [53].

A. X(3872) → J/ψπγ
In this work, we assume that the phase angle φ ω in X(3872) → J/ψπγ is the same as in the X(3872) → J/ψππ. Unfortunately, due to the absence of the experimental data, the other phase φ χ c1 is unknown. We first investigate the φ χ c1 dependence of the interference term by setting the m πγ = 0.5 GeV, which is shown in Fig. 3. One can see that the interference term is not drastically dependent on the φ χ c1 . Thus it is reasonable to choose the phase angle φ χ c1 = 223 • to estimate the invariant mass distribution of πγ for the X(3872) → J/ψπγ, which corresponds to the central value of the interference term. In Fig. 4, we present the invariant mass distribution of πγ for the X(3872) → J/ψπγ when the φ ω and φ χ c1 are both fixed. Different from the X(3872) → J/ψππ in Ref. [53] which is dominated by the ρ meson, the decay of X(3872) → J/ψπγ is dominated by the ω meson. The line shape of the ω contribution and the total contribution are almost coincident in the high invariant mass region. The differential decay rate with respect to πγ from the ω contribution is two orders of magnitude larger than that from the ρ meson since g Xψω and g ωπγ are both three times larger than g Xψρ and g ρπγ respectively. Thus, the dominant resonance contribution of X(3872) → J/ψπγ is the ω meson. The χ c1 term provides the dominant the nonresonance contribution, which serves as the background. Due to the absolute dominance of the ω in X(3872) → J/ψπγ, X(3872) → J/ψπγ becomes a clean and ideal process to explore the isospin conservation channel J/ψω of X(3872). In the line shape of the total invariant mass distribution, there is a dip around 766 MeV, which results from the dip of the interference term. After integrating over the πγ invariant mass, the branching ratio of X(3872) → J/ψπγ is (8.10 +3.44 −2.84 ) × 10 −3 considering the ρ and ω contributions only.
The above branching ratio does not include the contribution from the χ c1 term. To gain the total branching ratio of X(3872) → J/ψπγ including the χ c1 term, the φ χ c1 dependence of the total branching ratio of X(3872) → J/ψπγ should be clarified.
In Fig. 5, we present the φ χ c1 dependence of the total branching ratio of X(3872) → J/ψπγ by fixing the φ ω to be 134.5 • and varying the φ χ c1 from 0 • to 360 • . The φ χ c1 dependence of the total branching ratio of X(3872) → J/ψπγ is fairly stable. Finally, the predicted branching ratio of X(3872) → J/ψπγ is (8.10 +3.59 −2.89 ) × 10 −3 . The central value is obtained by taking φ χ c1 = 180 • , the errors come from the variation of the φ χ c1 . Under the assumption that X(3872) is a DD * molecule and that its decay proceeds through the transitions to J/ψρ and J/ψω, the branching ratio of X(3872) → J/ψπγ was estimated to be 0.17×B[X → J/ψππ] [59], which is similar to our estimation. Our results indicate that the branching ratio of X(3872) → J/ψπγ is almost of the same order as those of the hidden-charm and radiative decays to ψ ′ /J/ψ of the X(3872), In the hidden charm decay of X(3872) → J/ψπππ, the coupling constants g Xψω and g ρππ are both larger than g Xψρ and g ωππ respectively. As a result, the diagram where the X(3872) couples to J/ψω is far more important than the diagram where the X(3872) couples to the J/ψρ [53].
For the radiative decay of X(3872) → J/ψππγ, g Xψω is larger than g Xψρ , while g ρπγ is smaller than g ωπγ as shown in Figs. 2(a)-(b). Thus, the contribution of Fig. 2(a) is probably comparable to that of Fig. 2(b). Here, it should be noted that Fig. 2(a) only contributes to the X(3872) → J/ψπ 0 π 0 γ process. In Fig. 6, we show the results of the ππγ invariant mass spectrum based on the contributions of Fig. 2(a) and 2(b), which are governed by the J/ψρ and J/ψω coupling, respectively. It can be seen that the contribution of the J/ψω channel is still larger than that of the J/ψρ channel. After integrating over the ππγ invariant mass, the branching ratios of X(3872) → J/ψπ 0 π 0 γ are (3.84 +1.90  [20], the branching ratios of X(3872) → γψ ′ → γJ/ψπ 0 π 0 is (0.82 ± 0.37)%.
In the present estimation, all the involved coupling constants are extracted from the corresponding experimental data. Thus, one should get the same results regardless of the molecular or other scenarios for the X(3872). On the other hand, the X(3872) → J/ψπγ and X(3872) → J/ψππγ decays are very helpful for constraining the coupling constants g Xψρ and g Xψω , Note that we have assumed that the interference of the diagrams in Fig. 2 is negligible. From B[X → J/ψπγ], the coefficient of g 2 Xψω is so large that we can easily extract the coupling of Xψω in X → J/ψω → J/ψπγ. The coefficients of g 2 Xψρ and g 2 Xψω in B[X → J/ψπ 0 π 0 γ] are pretty small and thus it is difficult to obtain any useful information about these couplings in X → J/ψπ 0 π 0 γ. In contrast, it is very interesting to see that the coefficient of g 2 Xψρ in B[X → J/ψπ + π − γ] is very large. Thus X → J/ψπ + π − γ is a very good process to extract the coupling Xψρ. We look forward to the measurement of the branching ratios of X(3872) → J/ψπγ and X(3872) → J/ψππγ in the near future. At that time, not only the predicted branching ratios can be tested, but also the coupling constants g Xψρ and g Xψω can be extracted.

IV. SUMMARY
As the first established charmonium-like state, X(3872) is one of the best studied exotic hadron states both experimentally and theoretically. Since its discovery, the mass spectrum, decay behaviors and production mechanism of the X(3872) have been studied extensively. The DD * hadronic molecule is the most popular explanation, with which most of the phe-nomena related to X(3872) could be best explained. However, the other interpretations can not be easily rule out.
In this work, we have studied the ρ and ω meson contribution to the radiative decays X(3872) → J/ψπγ and X(3872) → J/ψππγ using an effective Lagrangian method. We obtain the invariant decay amplitudes of the possible diagrams which contribute to X(3872) → J/ψπγ and X(3872) → J/ψππγ. We first investigate the φ χ c1 dependence of the interference term in X(3872) → J/ψπγ, which is not drastic. Thus, we choose a central value of φ χ c1 to analyse the invariant mass distribution of πγ for the X(3872) → J/ψπγ. The total branching ratio of X(3872) → J/ψπγ reaches (8.10 +3.59 −2.89 ) × 10 −3 , which barely depends on φ χ c1 .
Although the ρ meson contribution is dominant in X(3872) → J/ψππ, the ω contribution is also sizable as recently measured by the LHCb Collaboration [52]. Our numerical results strongly indicate that the X(3872) → J/ψπγ is dominated by the ω meson. Compared with X(3872) → J/ψππ, X(3872) → J/ψπγ is an ideal place to extract the coupling of X(3872) with J/ψω, which probes the isoscalar component of the X(3872).
As for the X(3872) → J/ψππγ cascade decays, the J/ψω contribution is much more important than that of the J/ψρ, which is similar to the case of X(3872) → J/ψπππ. The branching ratios of X(3872) → J/ψππγ with the ρ and ω contribution are in order of 10 −7 ∼ 10 −6 . However, the contributions of the above cascade decays through the ρ and ω mesons are strongly suppressed with respect to the diagrams which proceed either through the ψ(2S ) in Fig. 2(c) or the three body decay of the ρ meson in Fig. 2(d). The QED gauge invariance demands the existence of the seagull diagram Fig. 2(d). The branching ratio of X(3872) → J/ψρ → J/ψπ + π − γ may reach 10 −4 . The radiative transition of X(3872) → J/ψπ + π − γ seems to be a very clean process to precisely study the isospin violation property of X(3872) and extract the coupling of X(3872) with J/ψρ, which probes the isovector component of the X(3872).
The branching ratios of X(3872) → J/ψπγ and X(3872) → J/ψππγ are accessible for the BESIII and LHCb Collaborations. With the relationships between the branching ratios of X(3872) → J/ψπ(π)γ and the coupling constants g Xψρ/ω , we can extract g Xψρ and g Xψω if the branching ratios of X(3872) → J/ψπγ and X(3872) → J/ψππγ are measured in the near future. These couplings encode very important information on the inner structure of the X(3872).