A Phenomenological analysis on isospin-violating decay of X(3872)
aa r X i v : . [ h e p - ph ] F e b A Phenomenological analysis on isospin-violating decay of X (3872) Qi Wu , ∗ Dian-Yong Chen , † and Takayuki Matsuki ‡ School of Physics, Southeast University, Nanjing 210094, China Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, Japan (Dated: February 18, 2021)In a molecular scenario, we investigate the isospin-breaking hidden charm decay processes of X (3872), i.e., X (3872) → π + π − J /ψ , X (3872) → π + π − π J /ψ , and X (3872) → π χ cJ . We assume that the source of thestrong isospin violation comes from the di ff erent coupling strengths of X (3872) to its charged components D ∗ + D − and neutral components D ∗ ¯ D as well as the interference between the charged meson loops and neutralmeson loops. The former e ff ect could fix our parameters by using the measurement of the ratio Γ [ X (3872) → π + π − π J /ψ ] / Γ [ X (3872) → π + π − J /ψ ]. With the determined parameter range, we find that the estimated ratio Γ [ X (3872) → π χ c / Γ [ X (3872) → π + π − J /ψ ] is well consistent with the experimental measurement from theBESIII collaboration. Moreover, the partial width ratio of π χ cJ for J = , , . ∼ .
65 :1 : 1 . ∼ .
43, which could be tested by further precise measurements of BESIII and Belle II.
PACS numbers: 13.25.GV, 13.75.Lb, 14.40.Pq
I. INTRODUCTION
As the first observed charmonium-like state, X (3872) hasbeen comprehensively investigated both from experimentaland theoretical sides. It was first observed in π + π − J /ψ in-variant mass spectrum of B → K π + π − J /ψ process by theBelle Collaboration in 2003 [1]. The mass and width weremeasured to be m = . ± . . ) ± . . ) MeVand Γ < . . L . ), respectively. The mea-sured mass is very close to the thresholds of D ∗ ¯ D , whichare 3871 .
63 MeV and 3879 .
91 MeV for charged and neutralcharmed meson pairs, respectively. Another interesting prop-erty of X (3872) is its narrow width. These particular prop-erties stimulated physicists with great interests in the natureof X (3872). After the observation by the Belle Collabora-tion, the charmonium-like state X (3872) has been confirmedby the BaBar[2–11], CDF[12–15], D0[16], CMS[17–22],LHCb[23–36], and BESIII[37–40] Collaborations in the in-variant mass spectra of π + π − J /ψ [1, 3–5, 9, 12–27, 32–37, 41], π + π − π J /ψ [11, 39, 42], D ∗ ¯ D [8, 40, 43], D ¯ D π [6, 44], χ cJ π [38], γ J /ψ [7, 10, 40, 42], and γψ (2 S )[10, 28, 40].The quantum number of X (3872) has been determined to be I G ( J PC ) = + (1 ++ ) by the LHCb collaboration [26]. In Table I,we have collected the experimental observations of X (3872).Beside the resonance parameters, the experimental mea-surements indicated that the π + π − invariant mass for X (3872) → π + π − J /ψ concentrates near the upper kinematicboundary, which corresponds to the ρ meson mass [1]. As for X (3872) → π + π − π J /ψ , the π + π − π invariant mass distribu-tion has a strong peak between 750 MeV and the kinematiclimit of 775 MeV, suggesting that the process is dominated bythe sub-threshold decay X (3872) → ω J /ψ . The ratio of thebranching fractions of π + π − J /ψ and π + π − π J /ψ is determined ∗ Electronic address: [email protected] † Electronic address: [email protected] (corresponding author) ‡ Electronic address: [email protected] to be [11, 39, 42], B [ X → J /ψπ + π − π ] B [ X → J /ψπ + π − ] = . ± . ± . . + . − . BESIII0 . ± . ω J /ψ and ρ J /ψ channels furthermakes the nature of X (3872) complicated and confusing. Tounderstand the particular properties of X (3872), a large num-ber of attempts were made to reveal its nature. Since the massof X (3872) is very close to the threshold of D ¯ D ∗ , it is nat-ural to consider it as an analogue of the deuteron, i.e., D ¯ D ∗ loosely bound state, which was supported by the spectrum es-timation in potential model [45–47, 49–55, 57, 58, 137, 138],by the QCD sum rule [59], the decay and production propertyinvestigations [60–90, 92–94, 136]. However, the estimationin a chiral quark model disfavors the S − wave D ¯ D ∗ molecularinterpretation even if all the possible meson exchanges weretaken into account [95]. In Refs. [96, 97], the authors indi-cated that the interaction of D ¯ D ∗ may not be strong enough toform a bound state, but their interaction can still lead to a cuspstructure near the threshold.It should be noticed that in all the observed decay modesof X (3872), the final states contain charm and anti-charmquarks. Thus, the quark components of X (3872) include atleast c ¯ c , which implies that X (3872) could be a good char-monium candidate. Considering the J PC quantum numberand the mass of X (3872), one can assign it only as χ c (2 P )state [98–100, 102–105, 126]. However, the resonance pa-rameters and decay behaviors make us hardly interpret it inthe charmonium scenario. Then, the tetraquark interpretationwith constituent c ¯ cq ¯ q [106–116] and hybrid with constituent c ¯ cg [117–119] have been proposed.To date, the nature of X (3872) still remains unclear. Be-sides the mass spectrum, the investigations of decay behav-iors are also crucial to understand the properties of X (3872).The isospin breaking e ff ects of X (3872) were studied inRef. [120], where X (3872) was considered as a dynamicallygenerated state and the coupling strengths of X (3872) D ∗ + D − TABLE I:
The experimental measurements of X (3872) from di ff erent experiments, where ✓ denotes that the decay channel wasobserved but the mass of X (3872) was not reported and ✗ indicates that channel was not observed. Experiments Channel Mass (MeV) Channel Mass (MeV)Belle π + π − J /ψ [1] 3872 . ± . D ¯ D π [44] 3875 . + . − . π + π − J /ψ [41] 3871 . ± . D ∗ ¯ D [43] 3872 . + . − . γ J /ψπ + π − π J /ψ [42] ✓ BaBar η J /ψ [2] ✗ π + π − J /ψ [3] 3873 . ± . π + π − J /ψ [4] ✓ π + π − J /ψ [5] 3871 . ± . B − )3868 . ± . B ) D ¯ D π [6] ✓ γ J /ψ [7] ✓ D ∗ ¯ D [8] 3875 . + . − . π + π − J /ψ [9] 3871 . ± . B + )3868 . ± . B ) γ J /ψγ J /ψ ′ [10] ✓ π + π − π J /ψ [11] 3873 . + . − . CDF π + π − J /ψ [12] 3871 . ± . π + π − J /ψ [13] ✓ π + π − J /ψ [14] ✓ π + π − J /ψ [15] 3871 . ± . π + π − J /ψ [16] 3871 . ± . π + π − J /ψ [17] 3870 . ± . π + π − J /ψ [18] ✓ π + π − J /ψ [19] ✓ π + π − J /ψ [20] ✓ π + π − J /ψ [21] ✓ π + π − J /ψ [22] ✓ LHCb π + π − J /ψ [23] 3871 . ± . π + π − J /ψ [24] ✓ π + π − J /ψ [25] 3871 . ± . π + π − J /ψ [26] ✓ π + π − J /ψ [27] ✓ γψ ′ [28] 3873 . ± . J /ψ )3869 . ± . ψ ′ ) ρ J /ψ [29] ✓ p ¯ p [30] ✗ φφ [31] ✗ π + π − J /ψ [32] ✓ π + π − J /ψ [33] ✓ π + π − J /ψ [34] 3871 . ± . π + π − J /ψ [35] 3871 . ± . π + π − J /ψ [36] ✓ BESIII π + π − J /ψ [37] 3871 . ± . π χ c (1 P )[38] ✓ ω J /ψ [39] 3873 . ± . D ∗ ¯ D + c . c .γ J /ψγψ (2 S ) γ D + D − [40] ✓✓✗✗ and X (3872) D ∗ ¯ D were assumed to be the same. Under2 P assignment, the decay channel X (3872) → ρ/ω + J /ψ was estimated via intermediate charmed meson loops [126].Using a phenomenological Lagrangian approach, the authorsstudied radiative decays to J /ψ/ψ (2 S ) with the X (3872) be-ing a composite state containing both D ¯ D ∗ molecule and a c ¯ c component [78, 121], and hidden charm and radiative de-cays of X (3872) were investigated with the X (3872) being acomposite state comprised of the dominant molecular D ¯ D ∗ component and other hadronic pairs, which could be D ± D ∗∓ and J /ψω/ρ [122]. The final state interaction e ff ects of thehidden charm decay of X (3872) were examined in Ref. [75],they found that the FSI contribution to X (3872) → J /ψρ is tiny. Assuming the decays of X (3872) through ρ J /ψ and ω J /ψ , the authors in Ref. [68] calculated the decay rates of X (3872) → π + π − J /ψ and X (3872) → π + π − π J /ψ .Besides π + π − J /ψ and π + π − π J /ψ , the pionic transitionfrom X (3872) to χ cJ was predicted in Refs. [77, 123], in whichit is found that the ratio of di ff erent transitions with di ff er-ent angular momentum J was sensitive to the inner structureof X (3872). In 2019, the BESIII Collaboration searched forthe process e + e − → γ X (3872) by using the collision datawith center-of-mass energies between 4.15 and 4.30 GeVand a new decay mode, χ c π , of X (3872) was observedwith a statistical significance of more than 5 σ but no signifi-cant X (3872) signal was observed in the invariant mass dis-tributions of π χ c , . The ratios of the branching ratios of X (3872) → π χ cJ for J = , , X (3872) → π + π − J /ψ were measured to be [38], B [ X → π χ cJ ] B [ X → π + π − J /ψ ] = . + . − . ± . J = . + . − . ± . J = . + . − . ± .
04 (1 . J = J = X (3872) in B + → χ c π K + decay, and the ratiowas measured to be B [ X → π χ c ] / B [ X → π + π − J /ψ ] < . X (3872).In the present work, we attempt to hunt for the source ofthe isospin violation in the molecular scenario by assum-ing that X (3872) is an S -wave molecule with J PC = ++ given by the superposition of D ¯ D ∗ and D ± D ∗∓ hadronicconfigurations. The fundamental source of the isospin vio-lation is the mass di ff erence of up and down quarks. Spe-cific to the present discussed issue, the concrete manifesta-tion is the mass di ff erence of charged and neutral charmedmesons, which leads to the di ff erent coupling strengths of X (3872) D ∗ ¯ D and X (3872) D ∗ + D − . This coupling strengthdi ff erence in part provides the source of the isospin viola-tion in the decays of X (3872). As a molecular state, thehidden charm decays of X (3872) occur via the charmed me-son loops, where the interferences between the charged andneutral meson loops provide another important source of theisospin violation. In the present work, we consider these twosources of isospin violation, the uncertainties of the formerone, i.e. the di ff erent coupling strengths, can be determinedby the ratio B [ X → π + π − π J /ψ ] / B [ X → π + π − J /ψ ], and thenwith the fixed parameters, we can further estimate the ratios B [ X → π χ cJ ] / B [ X → π + π − J /ψ ]. Comparing the present es-timation for J = J = J =
2, thepresent estimations can narrow down the ratios’ range, whichcould be tested by further measurements.This paper organized as follows: After introduction,we present the model used in the present estimations of X (3872) → ρ/ω J /ψ and X (3872) → χ cJ π . The numericalresults and discussions are presented in Sec. III, and Sec. IVis devoted to a short summary. II. HIDDEN CHARM DECAY OF X (3872) As discussed above, we assume that the coupling strengthsof X (3872) D ∗ ¯ D and X (3872) D ∗ + D − are di ff erent. The ef-fective coupling of X (3872) with its components can be, L X (3872) = g X √ X † µ h sin θ (cid:16) D ∗ µ ¯ D + D ¯ D ∗ µ (cid:17) + cos θ (cid:16) D ∗ + µ D − + D + D ∗− µ (cid:17)i (3) X (3872) J/ψρD ¯ D ∗ D X (3872) J/ψρD ¯ D ∗ D ∗ ( a ) ( b ) X (3872) J/ψρ ¯ D ∗ D ¯ D X (3872) J/ψρ ¯ D ∗ D ¯ D ∗ ( c ) ( d )FIG. 1: Diagrams contributing to X (3872) → J /ψρ . The chargeconjugate diagrams are not shown but included in the calculations. X (3872) χ c πD ¯ D ∗ D X (3872) χ c π ¯ D ∗ D ¯ D ∗ ( a ) ( b ) X (3872) χ c πD ¯ D ∗ D ∗ X (3872) χ c π ¯ D ∗ D ¯ D ∗ ( c ) ( d )FIG. 2: Diagrams contributing to X (3872) → χ cJ π , the charge con-jugate diagrams are not shown but included in the calculations. where g X is the coupling constant, θ is a phase angle describ-ing the proportion of neutral and charged constituents.It should be mention that from a more fundamental quarklevel point of view, the di ff erence in coupling strength shouldbe dynamically generated from u , d quark-antiquark pair gen-erating processes and also from the wave function di ff erenceof the charged and neutral D ¯ D ∗ . Furthermore, from the dis-persion discussion in Refs. [125, 126], the di ff erent couplingstrength comes also from the dispersion integrals where thethresholds are di ff erent for the charged and neutral channels.From the phenomenological point of view, we can parame-terize the coupling strength and the isospin breaking e ff ectsinto a common factor g X and a phase angle θ in the e ff ectiveLagrangian as shown in Eq. (3).Moreover, the distributions of the components, i.e., D ¯ D ∗ ,in the molecular state could be described by a wave function,which would then be integrated in the Feynman diagram cal-culations and a ff ect the magnitude of partial widths. In thepresent work, we mainly focus on the ratios of the partialwidths as given in Eqs. (1)-(2), the form factor appears in boththe numerators and denominators. As discussed in Ref. [78],the estimated ratio in the nonlocal case may not be too muchdi ff erent from the local one. Then the simple parameterizationin Eq. (3) could be a reasonable approximation in estimatingthe order of magnitude of the ratio.In the present work, the hidden charm decay processes of X (3872) occur via charmed meson loops, i.e., the charmo-nium and light meson in the final state couple to the com-ponents of X (3872) by exchanging a proper charmed meson.The diagrams contributing to X (3872) → ρ J /ψ are presentedin Fig. 1. The diagrams contributing to X (3872) → ω J /ψ could be obtained by replacing the ρ meson by the ω meson,while the diagrams contributing to X (3872) → π χ cJ with J = , , D ∗ DJ /ψρ , which isequivalent to the DD ∗ coupling to J /ψρ through an exchangeof an excited charmed meson. In the intermediate meson loopmodel, the dominant contributions are usually supposed tocome from the ground states of mesons which have strongcoupling with the final states, while the contribution from theexcited mesons is suppressed.In the present work, all these diagrams are estimated inhadronic level and all the involved interactions are depictedby e ff ective Lagrangians. In heavy quark limit, one can con-struct the e ff ective Lagrangian for charmonium and charmedmesons, which are [127–129] L ψ D ( ∗ ) D ( ∗ ) = − ig ψ DD ψ µ ( ∂ µ DD † − D ∂ µ D † ) + g ψ D ∗ D ε µναβ ∂ µ ψ ν ( D ∗ α ↔ ∂ β D † − D ↔ ∂ β D ∗† α ) + ig ψ D ∗ D ∗ ψ µ ( D ∗ ν ↔ ∂ ν D ∗† µ + D ∗ µ ↔ ∂ ν D ∗† ν − D ∗ ν ↔ ∂ µ D ∗ ν † ) , L χ cJ D ( ∗ ) D ( ∗ ) = g χ c DD χ c D i D † i + g χ c D ∗ D ∗ χ c D ∗ i µ D ∗ µ † i + ig χ c D ∗ D χ µ c ( D ∗ i µ D † i − D i D ∗† i µ ) + g χ c D ∗ D ∗ χ µν c D ∗ i µ D ∗† i ν , (4)The coupling between light meson and charmed mesonscould be obtained based on the heavy quark limit and chiralsymmetry, which are [129–131] L = − ig D ∗ D P (cid:16) D i ∂ µ P i j D ∗ j † µ − D ∗ i µ ∂ µ P i j D j † (cid:17) + g D ∗ D ∗ P ε µναβ D ∗ µ i ∂ ν P i j ↔ ∂ α D ∗ β † j − ig DD V D † i ↔ ∂ µ D j ( V µ ) ij − f D ∗ D V ǫ µναβ ( ∂ µ V ν ) ij ( D † i ↔ ∂ α D ∗ β j − D ∗ β † i ↔ ∂ α D j ) + ig D ∗ D ∗ V D ∗ ν † i ↔ ∂ µ D ∗ j ν ( V µ ) ij + i f D ∗ D ∗ V D ∗† i µ ( ∂ µ V ν − ∂ ν V µ ) ij D ∗ j ν + H . c ., (5)where the D ( ∗ ) † = ( ¯ D ( ∗ )0 , D ( ∗ ) − , D ( ∗ ) − s ) is the charmed mesontriplet, P and V µ are 3 × P = π √ + αη + βη ′ π + K + π − − π √ + αη + βη ′ K K − ¯ K γη + δη ′ , V = ρ √ + ω √ ρ + K ∗ + ρ − − ρ √ + ω √ K ∗ K ∗− ¯ K ∗ φ . (6)With the Lagrangian listed above, we can obtain the de-cay amplitude corresponding to X (3872) → ρ J /ψ, ω J /ψ , and π χ cJ with J = , ,
2. For brevity, we collect all the am-plitudes corresponding to diagrams in Figs. 1 and 2 in Ap-pendix. A and leave the coupling constants to be discussed inthe following section.In the present estimation, since the threshold of J /ψρ is veryclose to the mass of X (3872), the width of ρ meson should beincluded, and then, the width of X (3872) → ρ J /ψ should be, Γ X (3872) → J /ψρ = W ρ Z ( m X − m J /ψ ) (2 m π ) ds f ( s , m ρ , Γ ρ ) × | ~ p | π m X | M tot X (3872) → J /ψρ ( m ρ → √ s ) | (7)where W ρ = R ( m X − m J /ψ ) (2 m π ) ds f ( s , m ρ , Γ ρ ), f ( s , m ρ , Γ ρ ) is a rela-tivistic form of the Breit-Wigner distribution, which reads f ( s , m ρ , Γ ρ ) = π m ρ Γ ρ ( s − m ρ ) + m ρ Γ ρ , (8)and the amplitude M tot X (3872) → J /ψρ ( m ρ → √ s ) can be obtainedby replacing the mass of ρ meson by √ s in the amplitudeslisted in the appendix, in the same way the momentum of thefinal state becomes, | ~ p | = q [ m X − ( √ s − m J /ψ ) ][ m X − ( √ s + m J /ψ ) ]2 m X (9)As for X (3872) → J /ψπ + π − π , it is a sub-threshold de-cay process, hence, the Breit-Wigner distributions of ω me-son should be also considered in a similar way. However, thelower limit of integral in Eq. (7) and W ρ should be replacedwith (3 m π ) . With the partial widths of X (3872) → ω/ρ J /ψ ,one can obtain the partial widths of X (3872) → π + π − J /ψ and X (3872) → π + π − π J /ψ , which are, Γ [ X (3872) → J /ψπ + π − ] =Γ [ X (3872) → ρ J /ψ ] B [ ρ → π + π − ] Γ [ X (3872) → J /ψπ + π − π ] =Γ [ X (3872) → ω J /ψ ] B [ ω → π + π − π ]where B [ ρ → π + π − ] ≃ B [ ω → π + π − π ] = (89 . ± . ρ → π + π − and ω → π + π − π , respectively. III. NUMERICAL RESULTS AND DISCUSSION
Since mass di ff erence of X (3872) and D ∗ ¯ D is very tiny,the coupling constants g X are very sensitive to the mass of X (3872). Thus, in the present work, we mainly focus on theratios of the hidden charm decay channels, which are inde-pendent on the coupling constants g X . Moreover, the involvedcharmonia in the present estimation are J /ψ and χ cJ . In theheavy quark limit, the coupling constants of the involved char-monia and charmed mesons can be related to the gauge cou-plings g and g by, g ψ DD = g √ m ψ m D , g ψ D ∗ D = g q m ψ m D ∗ / m D , g ψ D ∗ D ∗ = g √ m ψ m D ∗ , g χ c DD = − √ g √ m χ c m D , g χ c D ∗ D ∗ = − √ g √ m χ c m D ∗ g χ c D ∗ D = √ g q m χ c m D m D ∗ g χ c D ∗ D ∗ = g √ m χ c m D ∗ where g = √ m ψ / (2 m D f ψ ), g = − p m χ c / / f χ c , and f ψ = f χ c = J /ψ and χ c decay con-stants [128], respectively.In the heavy quark and chiral limits, the charmed mesoncouplings to the light vector and pesudoscalar mesons havethe following relationship [129, 131], g DDV = g D ∗ D ∗ V = β g V √ , f D ∗ DV = f D ∗ D ∗ V m D ∗ = λ g V √ , (10) g D ∗ D P = gf π √ m D m D ∗ , g D ∗ D ∗ P = g D ∗ D P √ m D m D ∗ (11)where the parameter g V = m ρ / f π with f π =
132 MeV beingthe pion decay constant and β = . λ = .
56 GeV − and g = .
59 [132].In the amplitudes, the form factors should be considered todepict the inner structures and o ff shell e ff ects of the charmedmesons in the loop. However, the mass of X (3872) is veryclose to the thresholds of D ∗ D , which indicates that the com-ponents of X (3872), i.e., the charmed mesons connected to X (3872) in Figs. 1-2, are almost on shell. Therefore, we in-troduce only one form factor in a monopole form to depictthe inner structure and the o ff -shell e ff ects of the exchangedcharmed meson, which is [47, 131, 133–135], F (cid:16) q (cid:17) = m − Λ q − Λ (12)where the parameter Λ can be further reparameterized as Λ D ( ∗ ) = m D ( ∗ ) + α Λ QCD with Λ QCD = .
22 GeV and m D ( ∗ ) is themass of the exchanged meson. The model parameter α shouldbe of order of unity [47, 133–135], but its concrete value can-not be estimated by the first principle. In practice, the value of α is usually determined by comparing theoretical estimateswith the corresponding experimental measurements. α θ ( D e g r ee ) FIG. 3: The ratio Γ [ X → J /ψπ + π − π ] / Γ [ X → J /ψπ + π − ] dependingon the parameter θ and α . The black solid lines are the upper andlower limit from the BESIII measurements, while the dashed linesare the upper and lower limits determined by BABAR and Belle col-laborations. In Fig. 3, we present the ratio Γ [ X → J /ψπ + π − π ] / Γ [ X → J /ψπ + π − ] depending on the parameter θ and α . For compari-son, we also present the upper and lower limits measured bythe BESIII, BABAR, and Belle collaborations [11, 39, 42].From the figure, one can see the ratio is almost independenton the parameter α due to the similarity of ρ J /ψ and ω J /ψ decay modes. Taking the latest BESIII data [39] as a scale,the determined θ range is 66 ◦ ∼ ◦ , which indicates that inthe X (3872), the weight of D ¯ D ∗ component is (83 ∼ X (3872) [120].As we have discussed in the introduction, there are twosources of the isospin violation in X (3872) decays. Whenone takes θ = ◦ , the coupling strengths of X (3872) D ∗ ¯ D and X (3872) D ∗ + D − are the same. Hence, the isospin vi-olation comes from the interference between the chargedand neutral meson loops. In this case, the ratio Γ [ X → π + π − π J /ψ ] / Γ [ X → π + π − J /ψ ] is estimated to be of the or-der of 10 , which indicates that the contributions from theinterference between the charged and neutral meson loopsare rather small and the dominant source of isospin viola-tion in X (3872) decays should come from the di ff erent cou-pling strengths of X (3872) D ∗ ¯ D and X (3872) D ∗ + D − . InRefs. [125, 126], the authors assigned X (3872) as a 2 P char-moninum, and the large isospin violating came from thedispersion integrals where the mass of charged and neu-tral in the loops are di ff erent. In a c ¯ c -two-meson hybridmodel, the ratio Γ [ X (3872) → π + π − π J /ψ ] / Γ [ X (3872) → π + π − J /ψ ] was estimated to be 1 . ∼ .
24 [136]. Inthe D ∗ ¯ D molecular scenario, considering the S − D mix- α θ ( D e g r ee ) FIG. 4: The ratio Γ [ X → χ c π ] / Γ [ X → J /ψπ + π − ] depending onthe parameter α and θ . The red dashed curve indicates the centervalues of the ratio reported by the BESIII collaboration [38], whilethe black solid lines indicate the range determined by the ratio Γ [ X → J /ψπ + π − π ] / Γ [ X → J /ψπ + π − ]. α θ ( D e g r ee ) FIG. 5: The same as Fig. 4 but for Γ [ X → χ c π ] / Γ [ X → J /ψπ + π − ]. ing and the isospin mass splitting [137, 138], the ratio Γ [ X (3872) → π + π − π J /ψ ] / Γ [ X (3872) → π + π − J /ψ ] was eval-uated as 0 .
42 [137], which was close to the lower limit ofthe experimental data. In Ref. [139], X (3872) was consideredas a tetraquark state and the isospin violating decay process X (3872) → π + π − J /ψ was assumed to occur through ρ me-son pole which is caused by the ω − ρ mixing.In Fig. 4, we present the ratio Γ [ X → χ c π ] / Γ [ X → J /ψπ + π − ] depending on the parameter α and θ . In the θ range determined by Γ [ X → π + π − π J /ψ ] / Γ [ X → π + π − J /ψ ],we find the ratio Γ [ X → χ c π ] / Γ [ X → π + π − J /ψ ] is deter-mined to be 0 . ∼ .
17, which is well consistent with themeasurement of the BESIII collaboration, which is Γ [ X → χ c π ] / Γ [ X → π + π − J /ψ ] = . + . − . ± .
10 [38]. α θ ( D e g r ee ) FIG. 6: The same as Fig. 4 but for Γ [ X → χ c π ] / Γ [ X → J /ψπ + π − ]. α θ ( D e g r ee ) R R FIG. 7: The ratios R = Γ [ X → π χ c ] / Γ [ X → π χ c ] (left panel)and R = Γ [ X → π χ c ] / Γ [ X → π χ c ] (right pannel) depending onthe parameter α and θ . The success in reproducing the ratio Γ [ X → χ cJ π ] / Γ [ X → π + π − J /ψ ] with J = J = J =
2, whichare presented in Figs. 5 and 6. Within the determined θ range, the ratio Γ [ X → χ cJ π ] / Γ [ X → π + π − J /ψ ] are es-timated to be 1 . ∼ .
07 and 1 . ∼ .
28 for J = J =
2, respectively, which indicate the partial widthsof X (3872) → π χ cJ are very similar to each other. Fur-thermore, from Figs. 4- 6, one can find that the α and θ de-pendences of these ratios are also very similar, which are re-sulted form the similarity of χ cJ , J = { , , } . In Fig. 7, wepresent the ratios R = Γ [ X → π χ c ] / Γ [ X → π χ c ] and R = Γ [ X → π χ c ] / Γ [ X → π χ c ] depending on the param-eter α and θ . In the determined θ range, the ratios are deter-mined to be Γ [ X → χ c π ] : Γ [ X → χ c π ] : Γ [ X → χ c π ] = (1 . ∼ .
65) : 1 : (1 . ∼ . . . . IV. SUMMARY
In the present work, we have investigated the decay be-haviors of X (3872) → π + π − J /ψ , π + π − π J /ψ , and π χ cJ in amolecular scenario and tried to understand isospin violationsin X (3872) hidden charm decays. The fundamental sourceof the isospin violation has been shown in two di ff erent as-pects. The mass di ff erence of the charged and neutral charmedmesons leads to di ff erent coupling strengths of X (3872) D ∗ ¯ D and X (3872) D ∗ + D − , which in part provides the source ofisospin violation in the decay of X (3872). Another impor-tant source of isospin violation is the interference between thecharged and neutral meson loops.By comparing our estimate of the ratio Γ [ X → π + π − π J /ψ ] / Γ [ X → π + π − J /ψ ] with the experimental data,we have determined the relative coupling strengths of X (3872) D ∗ ¯ D and X (3872) D ∗ + D − . Then, with the fixedparameters, we have further estimated the ratios Γ [ X → π χ cJ ] / Γ [ X → π + π − J /ψ ] with J = , ,
2. We have furthertested our estimate by comparing our results of J = J = J =
2, whichare Γ [ X → π χ c ] / Γ [ X → π + π − J /ψ ] = . ∼ .
07 and Γ [ X → π χ c ] / Γ [ X → π + π − J /ψ ] = . ∼ .
28, respec-tively. Moreover, in the determined parameter range, the par-tial width ratio of π χ cJ for J = , , . ∼ .
65) : 1 : (1 . ∼ . ACKNOWLEDGMENTS
This project is partially supported by the National Natu-ral Science Foundation of China under Grant No. 11775050.Qi Wu was supported by the Scientific Research Founda-tion of Graduate School of Southeast University (Grants no.YBPY2028).
Appendix A: Decay amplitudes
The amplitudes for X (3872) → J /ψρ corresponding to dia-grams in Fig. 1 are M a = i Z d q (2 π ) h g X ǫ X µ ih − ig ψ DD ǫ J /ψν ( − i )( p ν + q ν ) i × h − f D ∗ DV ε ρσαβ ip ρ ǫ σρ/ω i ( q α − p α ) i p − m p − m − g µβ + p µ p β / m q − m q F ( q , m q ) M b = i Z d q (2 π ) h g X ǫ X µ ih g ψ D ∗ D ε ρσαβ ip ρ ǫ σ J /ψ ( − )( iq β + ip β ) i × h ig D ∗ D ∗ V g ντ g θν ( − i )( q κ − p κ ) ǫ κρ/ω + i f D ∗ D ∗ V g τκ g θν i ( p κ ǫ νρ/ω − p ν ǫ κρ/ω ) i p − m − g µτ + p µ p τ / m p − m − g αθ + q α q θ / m q q − m q F ( q , m q ) M c = i Z d q (2 π ) h g X ǫ X µ ih g ψ D ∗ D ε ρσαβ ip ρ ǫ σ J /ψ ( − )( − ip β − iq β ) ih − ig DDV ( − i )( p ν − q ν ) ǫ νρ/ω i − g µα + p µ p α / m p − m p − m q − m q F ( q , m q ) , M d = i Z d q (2 π ) h g X ǫ X µ ih ig ψ D ∗ D ∗ ǫ τ J /ψ ( − i ) (cid:16) g ξθ g λτ ( p θ + q θ ) + g ξτ g λθ ( p θ + q θ ) − g ξθ g θλ ( p τ + q τ ) (cid:17)ih f D ∗ DV ε ρσαβ ip ρ ǫ σρ/ω ( − i )( p α − q α ) i − g µλ + p µ p λ / m p − m p − m − g ξβ + q ξ q β / m q q − m q F ( q , m q ) , (A1)The amplitudes for X (3872) → χ cJ π corresponding to dia-grams in Fig. 2 are M a = i Z d q (2 π ) h g X ǫ X µ ih g χ c DD ih − ig D ∗ DP ip ν i p − m − g µν + p µ p ν / m p − m q − m q F ( q , m q ) M b = i Z d q (2 π ) h g X ǫ X µ ih g χ c D ∗ D ∗ ih − ig D ∗ DP ip α i − g µν + p µ p ν / m p − m p − m − g να + q ν q α / m q q − m q F ( q , m q ) M c = i Z d q (2 π ) h g X ǫ X µ ih − ig χ c D ∗ D ǫ χ c ν i × h g D ∗ D ∗ P ε ρσαβ ip σ ( − i )( p α − q α ) i p − m − g µβ + p µ p β / m p − m − g νρ + q ν q ρ / m q q − m q F ( q , m q ) , M d = i Z d q (2 π ) h g X ǫ X µ ih g χ c D ∗ D ∗ ǫ χ c αβ ih − ig D ∗ DP ip ν i − g µα + p µ p α / m p − m p − m − g βν + q β q ν / m q q − m q F ( q , m q ) , (A2) [1] S. 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