Production of the X(3872) in charmonia radiative decays
Feng-Kun Guo, Christoph Hanhart, Ulf-G. Meißner, Qian Wang, Qiang Zhao
PProduction of the X (3872) in charmonia radiative decays Feng-Kun Guo a, ∗ , Christoph Hanhart b, † , Ulf-G. Meißner a,b, ‡ , Qian Wang b, § , Qiang Zhao c, ¶ a Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,Universit¨at Bonn, D-53115 Bonn, Germany b Institut f¨ur Kernphysik, Institute for Advanced Simulation, and J¨ulich Center for Hadron Physics,D-52425 J¨ulich, Germany c Institute of High Energy Physics and Theoretical Physics Center for Science Facilities,Chinese Academy of Sciences, Beijing 100049, China
September 10, 2018
Abstract
We discuss the possibilities of producing the X (3872) , which is assumed to be a D ¯ D ∗ boundstate, in radiative decays of charmonia. We argue that the ideal energy regions to observe the X (3872) associated with a photon in e + e − –annihilations are around the Y (4260) mass andaround 4.45 GeV, due to the presence of the S -wave D ¯ D (2420) and D ∗ ¯ D (2420) threshold,respectively. Especially, if the Y (4260) is dominantly a D ¯ D molecule and the X (3872) a D ¯ D ∗ molecule, the radiative transition strength will be quite large. ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail address: [email protected] ¶ E-mail address: [email protected] a r X i v : . [ h e p - ph ] J un Introduction
Since its discovery by the Belle Collaboration [1], the X (3872) , which is extremely close to the D ¯ D ∗ threshold, has stimulated a lot of efforts, both experimental and theoretical. It is regardedas one of the most promising candidates for a hadronic molecule, which are formed of two or morehadrons — analogous to the deuteron, the shallow bound state made of a proton and a neutron. Thequantum numbers of the X (3872) have been determined to be J P C = 1 ++ [2], in accordance with thehadronic molecular interpretations which can be either an S -wave bound state [3, 4, 5, 6] or a virtualstate in the D ¯ D ∗ system [7]. Another puzzling new charmonium state is the Y (4260) with quantumnumbers J P C = 1 −− , which was observed by the BaBar Collaboration [8]. It is difficult to be put inthe vector family of the c ¯ c in potential models. Various interpretations were proposed. One intriguingpossibility is that the main component of the Y (4260) is a D ¯ D (2420) bound state [9, 10, 11]. Fora comprehensive review of the X (3872) , Y (4260) and other XY Z states observed in the last decade,we refer to Ref. [14].So far the X (3872) has been observed in several different processes. The discovery was madein B -meson decays in the processes B ± → K ± J/ψπ + π − by the Belle Collaboration [1] and laterconfirmed by the BaBar Collaboration [15]. It was also observed in the proton–antiproton annihila-tions p ¯ p → J/ψπ + π − X by both the CDF [16] and D0 [17] Collaborations, and in proton–protoncollisions by the LHCb Collaboration [2, 18]. It is quite natural to search for the X (3872) also in thedecays of higher charmonia, especially the −− states, which can be easily and copiously producedin electron-positron collisions at, e.g., the Beijing Electron-Positron Collider II (BEPC-II). However,so far no evidence of the X (3872) in the radiative charmonium decays has been reported. In this pa-per, we will investigate the production of the X (3872) in the radiative decays of charmonium states,which include the ψ (4040) , the ψ (4160) , the Y (4260) and the ψ (4415) , which are all in the energyrange of the BESIII experiment [19] at the BEPC-II. As will be shown later on, among the vectorcharmonium(-like) states, the Y (4260) is the most promising one for producing the X (3872) , if thelong-distance part of its wave function is dominated by the D ¯ D hadronic molecule component —note that the mass of the Y (4260) is located close to the S -wave D ¯ D threshold.Our paper is organized as follows: In Sec. 2, based on a nonrelativistic effective field theory(NREFT), we will identify the most important mechanism for the X (3872) production, namely thetriangle loops with the coupling of the initial charmonium(-like) state with charmed mesons being S -wave. Using the effective Lagrangians given in Sec. 3, we will calculate the partial decay widths ofthe radiative transitions of the charmonia, especially parameter-free predictions for the Y (4260) → X (3872) γ and will be made, and the results will be given in Sec. 4. A brief summary will be given inthe last section. In general, a hadronic molecule is not a pure two-meson state since it can couple to other components,such as a q ¯ q or a compact multiquark state, when these have the same quantum numbers. Thus, sucha hadronic molecule can be produced through either the compact quark component or the hadronicconstituents. It is a process-dependent question and in some cases one of those two mechanisms is Notice that there are two D states of similar masses, and the one in question should be the narrower one, i.e. the D (2420) , because it is not sensible to discuss a constituent with a width comparable or even larger than the range offorces [12, 13]. ∗ ¯ D ∗ D γX (3872)1 −− ( c ) D ¯ D D ∗ γX (3872)1 −− ( d ) D ¯ D ∗ D γX (3872)1 −− ( e ) D ¯ D D ∗ γX (3872)1 −− ( a ) D ∗ ¯ D D ∗ γX (3872)1 −− ( b ) Figure 1: Relevant triangle loops for the production of the X (3872) in the vector charmonium radia-tive decays. The charge-conjugated diagrams are not shown.more important than the other. Let us take the X (3872) as an example, which may be decomposed as | X (3872) (cid:105) = α | c ¯ c (cid:105) + α √ (cid:12)(cid:12) D ¯ D ∗ + c.c. (cid:11) . (1)Then the production amplitude is composed of two parts, P X (3872) = α P c ¯ c + α P D ¯ D ∗ , where P c ¯ c and P D ¯ D ∗ represents the production of the c ¯ c and D ¯ D ∗ + c.c. , respectively (in the following,the charge conjugated channel will not be shown for simplicity but will be included in the numericalcalculation). We assume that the X (3872) is mainly a D ¯ D ∗ molecule, i.e. | α | (cid:29) | α | . In this case,if P D ¯ D ∗ is not heavily suppressed, then the X (3872) will dominantly be produced through the longdistance D ¯ D ∗ component — see Refs. [20, 21] for a more detailed discussion.Both the short and long distance production of the X (3872) in the radiative decays of the ψ (4040) and ψ (4160) are considered in Refs. [22, 23] in the framework of the so-called X-EFT [24]. Here, wewill focus on the contribution from intermediate charmed meson loops, i.e. the quantity P D ¯ D ∗ definedabove. The mechanism is shown in Fig. 1. Both the initial charmonium and the X (3872) couple to apair of charmed and anticharmed mesons. The X (3872) couples to the D ¯ D ∗ pair in an S -wave. Withthe quantum numbers being −− , the initial charmonium can couple to either two S -wave charmedmesons in a P -wave, or one P -wave and one S -wave charmed mesons in an S - or D -wave. As wewill show in the following, the mechanism with an S -wave coupling to the initial charmonium willgreatly facilitate the production processes.The charmed meson channels which could have significant effects are those close to the mass ofthe considered charmonium. In this work we will consider the s P(cid:96) = − ( S -wave), and s P(cid:96) =
32 + ( P -wave) charmed mesons, where s (cid:96) is the total angular momentum of the light quark system whichincludes the light quark spin and orbital angular momentum. The pertinent triangle loops are shownin Fig. 1. There is no D analogue of diagrams (d,e) because, different from the D case, its quantumnumbers does not allow that all vertices are in S -wave. One should notice that although the X (3872) can have a sizable D + D ∗− component [27, 28], because the magnetic coupling to the neutral charmedmesons is much larger than that to the charged ones, see, e.g. Ref. [25], we only consider the neutralcharmed mesons in the loops. Because all the charmonia considered are close to the open charm thresholds in question, theintermediate charmed and anticharmed mesons are nonrelativistic. We are thus allowed to use a non-relativistic power counting, the framework of which was introduced for studying the intermediate In fact, there can be photonic coupling to the charged charmed mesons from gauging the ψD ( ∗ ) ¯ D ( ∗ ) vertex and thekinetic energy of the charmed mesons, see e.g. [26]. However, they are of order O ( v ) , thus less important, in the powercounting scheme to be detailed in the following. Furthermore, the loops involving such vertices are divergent and henceneed unkown counterterms. In contrast, Ref. [29] states that including the charged charmed mesons would largely increasethe partial decay width of the X (3872) → γJ/ψ based on a vector meson dominance model in a flavor SU(4) formalism. v is much smaller than 1. Thus, the loop diagramsas shown in Fig. 1 can be organized through a velocity counting, where the three-momentum scalesas v , the kinetic energy scales as v , and each of the nonrelativistic propagators scales as v − . Inleading order, the S -wave coupling is momentum independent and does not contribute any power tothe velocity counting. The P -wave coupling scales as v [30] or as the external momentum [31, 32]depending on the process in question.Let us focus on the last two diagrams of Fig. 1 first. The D meson has s (cid:96) = 1 / , and D has s (cid:96) = 3 / . Thus, they can couple to L = 2 but not to L = 0 , where L is the orbital angular momentum,in the heavy quark limit. As a result, only the D -wave charmonia can couple to the D ¯ D in an S -wave,and for the S -wave charmonia the coupling must be D -wave. Thus, if the initial state is a D -wavecharmonium or has a significant D ¯ D molecular component (as might be the case for the Y (4260) ),the loop integral scales as v ( v ) E γ = E γ v , (2)where E γ is the external photon energy. The decay amplitude is the product of the loop integral andthe coupling constants for the three vertices. One sees that the amplitude is greatly enhanced for smallvelocity. It was shown in Ref. [33] that the value of the velocity should be understood as the averageof two velocities which correspond to the two cuts in the triangle diagram. These two velocitiesmay be estimated as (cid:112) | m + m − M i | / ¯ m and (cid:112) | m + m − M f | / ¯ m , where m is the massof the charmed meson between the two charmonia, m is the mass of the meson between the initial(final) charmonium and the photon, ¯ m ij = ( m i + m j ) / , and M i ( f ) is the mass of the initial (final)charmonium. Therefore, the amplitude is most enhanced when both the initial and final charmoniaare close to the corresponding thresholds.For diagrams (a), (b) and (c) of Fig. 1, the vertex involving the initial charmonium is in a P -wave.The momentum in that vertex has to be contracted with the external photon momentum q , and thusshould be counted as q . The decay amplitude through this type of loops scales as v ( v ) q m = E γ m v , (3)where m is a quantity of the dimension mass, and the factor of m − is introduced to make the aboveexpression have the same dimension as that obtained in Eq. (2). This factor in fact accounts for thedifferent dimensions of the coupling constants for the P -wave and S -wave vertices in diagrams (a, b,c) and (d, e), respectively, i.e. m = | g /g | where g and g are the coupling constants to be definedin Eq. (12) below. If all the coupling constants are of natural size, that is m ∼ GeV, then this loopshould be suppressed relative to the one in Eq. (2) for a soft photon. This is supported by the numericalresults in Sec. 4. Notice that only neutral charmed mesons are involved so that the P -wave vertex,although it contains a derivative, will not get gauged and the triangle diagrams are gauge invariant.If the initial charmonium is the ψ (4040) or the ψ (4415) , which are the radial exceptions of J/ψ and thus S -wave charmonia, the coupling to the D ¯ D is in a D -wave in the heavy quark limit, asoutlined above. In this case, the ψD ¯ D vertex should be counted as v . Using the same powercounting, the loops in Fig. 1 (d, e) should scale as v E γ , and thus are suppressed rather than enhancedfor small values of v .In the above discussions, we have neglected the width of the D (2420) , which presents a newscale. One concern is whether it would break the power counting established above. The width of the D (2420) is . ± . MeV [34], thus Γ (cid:46) | b | , where b = m + m − M i . From Eq. (A.4),4hich is the nonrelativistic scalar loop function where one of the intermediate mesons carries a finiteconstant width, one can conclude that the power counting scheme will not be modified by the presenceof the finite width of the D (2420) (as long as the width is sufficiently small). Because the charmed mesons do not have definite charge parity, it is necessary to clarify the phaseconvention under charge conjugation to be used in our paper, which is C D C − = ¯ D, C D ∗ C − = ¯ D ∗ , C D C − = ¯ D . (4)The X (3872) has a positive C -parity, and the Y (4260) as well as all the other vector charmoniumstates have negative C -parity. Thus, the flavor wave functions of the X (3872) and Y (4260) in termsof the charmed mesons are convention dependent. With the convention specified above, the D ¯ D ∗ and D ¯ D components of the X (3872) and Y (4260) can be written as | X (3872) (cid:105) = 1 √ (cid:12)(cid:12) D ¯ D ∗ + D ∗ ¯ D (cid:11) , | Y (4260) (cid:105) = 1 √ (cid:12)(cid:12) D ¯ D − D ¯ D (cid:11) . (5)Because we work with nonrelativistic kinematics for the charmed mesons and charmonia through-out this work, the two-component notation introduced in Ref. [25] is very convenient. In this simpli-fied notation, the field for the ground state charmed mesons is H a = (cid:126)V a · (cid:126)σ + P a , where (cid:126)σ are thePauli matrices, P a and V a annihilates the pseudoscalar and vector charmed mesons, respectively, and a is the flavor label for the light quarks. The quantum numbers of the light quark system in these twomesons are s P(cid:96) = − . Under the convention specified in Eq. (4), the field annihilating the ground statemesons containing an anticharm quark is [36] ¯ H a = σ (cid:16) (cid:126) ¯ V a · (cid:126)σ T + ¯ P a (cid:17) σ = − (cid:126) ¯ V a · (cid:126)σ + ¯ P a . (6)The field for the s P(cid:96) = 3 / + charmed mesons can be written as T ia = P ij a σ j + (cid:114) P i a + i (cid:114) (cid:15) ijk P j a σ k , (7)where P a and P a annihilate the charmed mesons D (2420) and D (2460) , respectively. Thecharmed antimesons are collected in ¯ T ia = − ¯ P ij a σ j + (cid:112) / P i a − i (cid:112) / (cid:15) ijk ¯ P j a σ k , where theconvention C D C − = ¯ D is adopted. Under parity and charge conjugation and with the conventionspecified above, these fields transform as H a P → − H a , H a C → σ ¯ H Ta σ , ¯ H a P → − ¯ H a , ¯ H a C → σ H Ta σ , (8) T ia P → T ia , T ia C → σ ¯ T i Ta σ , ¯ T ia P → ¯ T ia , ¯ T ia C → σ T i Ta σ . (9)Analogously, we can construct the field for the S -wave charmonia, which is J = (cid:126)ψ · (cid:126)σ + η c , where ψ and η c annihilate the vector and pseudoscalar charmonia, respectively. The leading coupling of the S -wave charmonium with the charmed and anticharmed mesons reads as L S = i g (cid:68) ¯ H † a (cid:126)σ · ←→ ∂ H † a J (cid:69) + H.c. , (10) In the literature, some authors write the wave function of the X (3872) with a different relative sign of the two terms, | X (3872) (cid:105) = √ (cid:12)(cid:12) D ¯ D ∗ − D ∗ ¯ D (cid:11) . This corresponds to a different convention for the C -parity transformation for the D ∗ , C D ∗ C − = − ¯ D ∗ . Notice that only the flavor neutral mesons are eigenstates of the C -parity, the physical observablesshould be independent of the convention. For a detailed discussion in the case of the X (3872) , see Ref. [35]. A ←→ ∂ B ≡ A ( (cid:126)∂B ) − ( (cid:126)∂A ) B and (cid:104) . . . (cid:105) denotes the trace in flavor space. Notice that all thecharmed meson and charmonium fields in the above Lagrangian and the ones in the following arenonrelativistic and have dimension mass / .Some of the −− charmonia in question are D -wave states. For instance, the ψ (4160) is widelyconsidered as the D state [37, 38]. The field for the D -wave charmonia in two-component notationcan be written as [23] J ij = 12 (cid:114) (cid:0) ψ i σ j + ψ j σ i (cid:1) − √ δ ij (cid:126)ψ · (cid:126)σ, (11)where only the −− state relevant for our discussions is included. Considering parity, C -parity, spinsymmetry and Galilean invariance, the leading order Lagrangian for the coupling of the D -wavecharmonia to the charmed and anticharmed mesons can be written as L D = i g (cid:68) ¯ H † a σ i ←→ ∂ j H † a J ij (cid:69) + g (cid:68)(cid:16) ¯ T j † σ i H † − ¯ H † σ i T j † (cid:17) J ij (cid:69) + H.c. , (12)where the first term has already been introduced in Ref. [23] ( g is denoted by g in that paper).In order to calculate the triangle diagrams depicted in Fig. 1, we need to know the photoniccoupling to the charmed mesons. The magnetic coupling of the photon to the S -wave heavy mesonsis described by the Lagrangian [39, 25] L HHγ = e β Tr (cid:104) H † a H b (cid:126)σ · (cid:126)B Q ab (cid:105) + e Q (cid:48) m Q Tr (cid:104) H † a (cid:126)σ · (cid:126)B H a (cid:105) , (13)where B k = (cid:15) ijk ∂ i A j is the magnetic field, Q is the light quark charge matrix, and Q (cid:48) is the heavyquark electric charge (in units of the proton charge e ). These two terms describe the magnetic couplingdue to the light and heavy quarks, respectively. The E1 transition of the
32 + charmed mesons to the − states may be parameterized in terms of a simple Lagrangian L T Hγ = (cid:88) a c a Tr (cid:104) T ia H † a (cid:105) E i + H.c. (14)Note that here the coefficients are light-flavor-dependent.At last, assuming that the X (3872) and Y (4260) are hadronic molecules, we parameterize theircoupling to the charmed mesons in terms of the following Lagrangian L XY = y √ Y i † (cid:0) D i a ¯ D a − D a ¯ D i a (cid:1) + x √ X i † (cid:0) D ∗ i ¯ D + D ¯ D ∗ i (cid:1) + H.c. , (15)where we assume that the Y (4260) couples to the D ¯ D in an isospin symmetric manner so that thelight flavor index a runs through u and d , and neglect all the other components except for the D ¯ D ∗ for the X (3872) . Considering a state slightly below an S -wave two-hadron threshold, the effective coupling of thisstate to the two-body channel is related to the probability of finding the two-hadron component in thephysical wave function of the bound state, λ , and the binding energy, (cid:15) = m + m − M [40, 41] g NR = λ πµ (cid:114) (cid:15)µ (cid:104) O (cid:16)(cid:112) µ(cid:15) r (cid:17)(cid:105) , (16)6here µ = m m / ( m + m ) is the reduced mass and r is the range of forces, and the nonrelativisticnormalization is used. After proper renormalization (see Ref. [42]), the coupling constants in Eq. (15)are given by the one in the above equation. Notice that the coupling constant gets maximized for apure bound state, which has λ = 1 by definition.The threshold of the D and D ∗ using the PDG fit values for the masses [34] is . ± . MeV. The mass of the X (3872) is . ± . MeV [34]. With M Y = 4263 +8 − MeV, andthe isospin averaged masses of the D and D mesons, we obtain the mass differences between the X (3872) and Y (4260) and their corresponding thresholds, respectively, M D + M D ∗ − M X = 0 . ± . MeV , M D + M D (2420) − M Y = 27 +9 − MeV . (17)Assuming that the X (3872) and Y (4260) are pure hadronic molecules, which corresponds to theprobability of finding the physical states in the two-hadron states λ = 1 , we obtain | x | = 0 . +0 . − . ± . GeV − / , | y | = 3 . +0 . − . ± . GeV − / , (18)where the first errors are from the uncertainties of the binding energies, and the second ones are dueto the approximate nature of Eq. (16). The range of forces is estimated by r − ∼ √ µ ∆ th where µ is the reduced mass and ∆ th is the difference between the threshold of the components and the nextclose one, which is M D ∗ + + M D + − M D ∗ − M D for the X (3872) and M D + M D ∗ − M D − M D for the Y (4260) , respectively.The value of β in the magnetic coupling of the S -wave charmed mesons is not precisely known.We will use the value β − = 276 MeV determined with m c = 1 . GeV in Ref. [25]. There is noexperimental measurement on the radiative decays of the P -wave charmed mesons. However, therehave been a few calculations using various quark models. Taking the predictions of Γ( D → D ( ∗ )0 γ ) in Refs. [43, 44, 45] as a guidance, the value for the coupling constant for the neutral charmed mesons c is in the range [0 . , . . ψ (4040) → γX (3872) and ψ (4415) → γX (3872) The ψ (4040) and ψ (4415) were widely accepted as the S and S vector charmonium states, respec-tively [37]. In the heavy quark limit, spin symmetry requires that the S -wave charmonium couples tothe D ( ∗ ) ¯ D in a D -wave. As shown in Sec. 2, such a D -wave vertex will cause the charmed mesonloops to be suppressed. Thus, we will neglect these loops, and consider only the loops involving the S -wave charmed mesons D and D ∗ , which correspond to the diagrams shown in Fig. 1 (a), (b) and (c).Assuming that the two-body S -wave charmed mesons saturate the decay width of the ψ (4040) and90% of width of the ψ (4415) — the only relatively well measured branching fraction is the sequentialdecay into the D D − π + + c.c. through the D ¯ D (2460) which is (10 ± , we may obtain an upperlimit for the coupling constant g for both the S and S charmonium states, (cid:12)(cid:12) g S ] (cid:12)(cid:12) < . GeV − / , (cid:12)(cid:12) g S ] (cid:12)(cid:12) < . GeV − / . (19)As a result, the upper limits for the production of the X (3872) are Γ( ψ (4040) → γX (3872)) (a,b,c) < . keV , Γ( ψ (4415) → γX (3872)) (a,b,c) < . keV , (20)which correspond to tiny branching fractions of order − .However, even a small D -wave c ¯ c mixture would greatly enhance the decay width of the ψ (4415) .This is because the ψ (4415) is only 10 MeV below the D ∗ ¯ D threshold, and the velocity, the relevant7arameter for the power counting, is as small as 0.04. Considering such an admixture, we obtain fromthe last two diagrams in Fig. 1 Γ( ψ (4415) → γX (3872)) (d,e) = 287 sin θ ( g x GeV ) c keV (cid:46)
89 sin θ (cid:0) g GeV (cid:1) keV , (21)where c (cid:39) . is used, and sin θ is the mixture of the D -wave component in the ψ (4415) wavefunction. In Ref. [46], θ ≈ ◦ is suggested from an analysis of the e + e − decay widths of thevector charmonia. We have assumed spin symmetry for the coupling of the initial charmonium to thecharmed mesons. ψ (4160) → γX (3872) As discussed before, being the D charmonium state, the ψ (4160) couples to a pair of S -wavecharmed mesons in a P -wave, and to one S -wave and one P -wave charmed mesons in an S -wave.Thus all the diagrams shown in Fig. 1 contribute to its radiative decay into the X (3872) . For thediagrams (a), (b) and (c), we can derive an upper limit for their contributions. The upper limit for thecoupling g for the ψ (4160) may be obtained by saturating its total decay width by two-body decaysinto a pair of S -wave charmed mesons. We obtain (cid:12)(cid:12) g D ] (cid:12)(cid:12) < . GeV − / . Using this value, thecontribution of the S -wave charmed mesons to the width of the ψ (4160) → γX (3872) is less than0.20 keV. We should mention that our numerical result for the width of the ψ (4160) → γX (3872) issmaller than the estimate in Ref. [23] using a different method and using the BaBar measurement ofthe X (3872) → γψ (cid:48) [47], which was not confirmed by the Belle Collaboration [48], as input. M Ψ (cid:64) GeV (cid:68) (cid:71) (cid:72) Ψ (cid:72) n D (cid:76) (cid:174) Γ X (cid:76) (cid:144) (cid:72) g x G e V (cid:76) k e V Figure 2: Dependence of the partial decay width of a D -wave charmonium into γX (3872) on the massof the charmonium. The solid and dotted curves are obtained with and without taking into account thewidth of the D (2420) , respectively. Here, only the contributions from Fig. 1 (d) and (e) are included,and c = 0 . is used.The value of g , which is needed for evaluating the diagrams (d) and (e), is unknown. Thus, weexpress the result from these two diagrams in terms of g Γ( ψ (4160) → γX (3872)) (d,e) = 19 . g x GeV ) c keV (cid:46) . (cid:0) g GeV (cid:1) keV , (22)8here we have taken c (cid:39) . . Expressing g by g = g m , if m ∼ GeV, then the approximateupper limit obtained from diagrams (d) and (e), 3 keV, is one order of magnitude larger than thatfrom diagrams (a), (b) and (c). This can be understood from the power counting. The momentumof the photon in this decay is 280 MeV. Thus, the factor q/m presents a suppression of the firstthree diagrams relative to the last two at the amplitude level. With the total width of the ψ (4160) being ± ‘MeV [34], a width of a few keV only amounts to a branching fraction of the order of − . Although larger than the 0.2 keV arising from the first three diagrams, it is still small so that anexperimental observation will be difficult.However, notice that the ψ (4160) is far off the optimized region for the observation of the X (3872) .This can be seen from Fig. 2, which shows the dependence of the radiative decay width of a D -wavecharmonium into the γX (3872) on the charmonium mass, where the solid and dotted curves representthe results with and without taking into account the finite width of the D (2420) , respectively. Onesees pronounced peaks slightly above the D ¯ D and D ∗ ¯ D thresholds in the dashed curve. This isdue to the closeness of the X (3872) to the D ¯ D ∗ threshold, which makes the kinematics so specialthat ( c (cid:48) − c ) / (2 √− a c ) — a, c and c (cid:48) are defined in Eq. (A.3) — is close to 1, and thus produces themaxima (recall that the imaginary part of arctan( i ) is infinite, c.f. Eq. (A.2)). The pronounced peaksget smeared by the finite width of the D (2420) , as can be seen from the solid curve. Still, the widthdivided by g peaks around 4.29 GeV and 4.45 GeV. Thus, as stated in Sec. 4.1, one might be able tomake an observation through a D -wave admixture in the ψ (4415) . Y (4260) → γX (3872) We assume that the Y (4260) is a D ¯ D molecule according to the suggestions of Refs. [9, 10, 11]. Theproduction of the recently observed charged charmonium Z c (3900) [49, 50, 51] can be understood inthis interpretation [11, 52] if it is a D ¯ D ∗ hadronic molecule [11, 53, 54, 55, 56]. Radiative decays ofthe Y (4260) into a pair of charmed mesons was studied based on this assumption very recently [57].In this picture, the radiative decay of the Y (4260) into the X (3872) will be a long-distance process,and the dominant decay mechanism is shown in Fig. 1 (d). With the the loop function given in theAppendix, we obtain the width Γ( ψ (4260) → γX (3872)) (d) = 141 +136 − (cid:0) x GeV (cid:1) c keV , (23)where the uncertainty is dominated by the use of Eq. (16), which is mainly due to neglecting thecoupled channel D ∗ ¯ D in this case. The velocity counting is well controlled since v (cid:39) . . UsingEq. (A.4), we have checked that including a finite constant width for the D only causes a minorchange of about 3%. The value of c is in the range of [0 . , . using the width predictions inthree different quark models [43, 44, 45]. Taking c = 0 . , we plot the dependence of the width of the Y (4260) → γX (3872) on the binding energy of the X (3872) in Fig. 3, where the value of x is relatedto the binding energy via Eq. (16). Therefore, depending on the precise location of the X (3872) , thebranching fraction can reach the order of − . We have argued that the triangle loops with all the intermediate states being the S -wave charmedmesons are suppressed relative to the ones with one S -wave and one P -wave charmed mesons whenthe initial charmonium is a D -wave state. This is based on the assumption that the coupling constantsare of natural size so that m = | g /g | ∼ GeV. If g is unnaturally small, then these two kinds of9 .0 0.1 0.2 0.3 0.4010203040506070 Ε X (cid:64) MeV (cid:68) (cid:71) (cid:72) Y (cid:72) (cid:76) (cid:174) Γ X (cid:72) (cid:76)(cid:76) k e V Figure 3: Dependence of the width of the Y (4260) → X (3872) γ in terms of the binding energy ofthe X (3872) , (cid:15) X = M D + M D ∗ − M X . Here the D D ∗ γ coupling constant is taken as c = 0 . .mechanisms might be comparable. One can check which one is dominant by measuring certain angu-lar distribution. This is because the two different types of loops have a different angular dependence— the one with two S -wave vertices does not depend on any angle with respect to the photon threemomentum while the other does, as can be seen from the expressions A (d,e) = A ( (cid:126)(cid:15) ψ × (cid:126)(cid:15) γ ) · (cid:126)(cid:15) X , A (a,b,c) = B ˆ q · (cid:126)(cid:15) ψ (ˆ q × (cid:126)(cid:15) X ) · (cid:126)(cid:15) γ + C ˆ q · (cid:126)(cid:15) X (ˆ q × (cid:126)(cid:15) ψ ) · (cid:126)(cid:15) γ , (24)where ˆ q is the unit vector along the three momentum of the photon, and (cid:126)(cid:15) ψ , (cid:126)(cid:15) γ and (cid:126)(cid:15) X are the corre-sponding polarization vectors. The expressions for A , B and C in terms of loop functions are given inAppendix B. Because the vector charmonium produced in e + e − collisions is transversely polarized,the angle between the photon momentum and the ψ polarization vector can be related to that withrespect to the beam axis. The relation follows from (cid:88) λ =1 , (cid:12)(cid:12)(cid:12) ˆ q · (cid:126)(cid:15) ( λ ) ψ (cid:12)(cid:12)(cid:12) = 12 sin θ q , (cid:88) λ =1 , (cid:12)(cid:12)(cid:12) ˆ q × (cid:126)(cid:15) ( λ ) ψ (cid:12)(cid:12)(cid:12) = 12 (1 + cos θ q ) , (25)where θ q is the angle between the photon momentum and the beam axis. Thus, we have the angulardistribution from diagrams (a,b,c) d Γ (a,b,c) d cos θ q ∝ (cid:88) λ =1 , (cid:18) | B | (cid:12)(cid:12)(cid:12) ˆ q · (cid:126)(cid:15) ( λ ) ψ (cid:12)(cid:12)(cid:12) + | C | (cid:12)(cid:12)(cid:12) ˆ q × (cid:126)(cid:15) ( λ ) ψ (cid:12)(cid:12)(cid:12) (cid:19) ∝ ρ cos θ q , (26)where ρ = (cid:0) | C | − | B | (cid:1) / (cid:0) | B | + | C | (cid:1) . For the ψ (4160) → γX (3872) , the value is ρ = − . so that the angular distribution is almost ∼ sin θ q . Thus, when the long-distance part dominates theproduction of the X (3872) , one may use the angular distribution to distinguish the P -wave D ( ∗ ) ¯ D ( ∗ ) threshold and S -wave D ¯ D ( ∗ ) threshold effects. A similar idea of using angular distributions to probethe structure of the X (3872) was already proposed in Refs. [22, 23]. The value of ρ shows that | B | (cid:29) | C | , which is due to a strong cancellation between different loops in C . Summary
In this paper, we have investigated the production of the X (3872) in the radiative decays of excitedcharmonia. These states include the ψ (4040) , ψ (4160) , ψ (4415) and the Y (4260) , which are the S , D , S charmonium and a conjectured D ¯ D molecule, respectively. Assuming the X (3872) isa D ¯ D ∗ bound state, we considered its production through the mechanism with intermediate charmedmesons. Using a NREFT, we argue that the meson loops with all the vertices being in an S -waveshould provide the most prominent contributions. We present a power counting that is confirmed byour numerical studies. It predicts that the closer to the threshold of the open charm intermediate statesthe initial charmonium is located, the more important the loops are. In this context, the production ratein the decays of the S -wave charmonia ψ (4040 , , contrary to that for the D -wave charmonium ψ (4160) , should be small since they couple to the D ( ∗ ) ¯ D in a D -wave and D ( ∗ ) ¯ D ( ∗ ) in a P -wave.The production in the Y (4260) decays will be strongly enhanced compared to all the other transitionsstudied in this work, if the Y (4260) is a D ¯ D molecule, as suggested in Refs. [9, 10, 11], since the S -wave coupling constant is maximized in such a case. Especially, if the mechanism for the productionof Z c (3900) in Y (4260) → πZ c proposed in Ref. [11] is correct, the X (3872) must be copiouslyproduced in Y (4260) → X (3872) γ .We also show that the measurement of the angular distribution of the radiated photon in e + e − → Y (4160) → γX (3872) should be sensitive to the underlying transition mechanisms.In this study, the ψ (4415) was assumed to be an S -wave charmonium. However, if it has a sizablemixing with a D -wave c ¯ c component or an S -wave D ∗ ¯ D component (notice that it is only 10 MeVbelow the D ∗ ¯ D threshold), then it can also decay into the X (3872) γ through the enhanced loops with S -wave couplings. Based on our calculation, we strongly suggest to search for the X (3872) associatedwith a photon in the energy region around the Y (4260) and 4.45 GeV in the e + e − collisions. Acknowledgments
We are appreciate to Thomas Mehen, Eulogio Oset and Roxanne Springer for useful discussionsand comments. This work is supported in part by the DFG and the NSFC through funds pro-vided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD”, the EUI3HP “Study of Strongly Interacting Matter” under the Seventh Framework Program of the EU, theNSFC (Grant No. 11165005 and 11035006), and the Ministry of Science and Technology of China(2009CB825200).
A Loop functions
When we neglect the widths of all the intermediate mesons, the decay amplitudes can be expressed inthe scalar three-point loop function I ( m , m , m , (cid:126)q ) = i (cid:90) d l (2 π ) (cid:0) l − m + i(cid:15) (cid:1) (cid:2) ( P − l ) − m + i(cid:15) (cid:3) (cid:2) ( l − q ) − m + i(cid:15) (cid:3) , (A.1)where m i ( i = 1 , , are the masses of the particles in the loop. This loop integral is convergent.Since all the intermediate mesons in the present case are highly nonrelativistic, the explicit expression11s derived as I ( m , m , m , (cid:126)q )= − i m m m (cid:90) d d l (2 π ) d (cid:16) l − (cid:126)l m + i(cid:15) (cid:17) (cid:16) l + b + (cid:126)l m − i(cid:15) (cid:17) (cid:104) l + b − b − ( (cid:126)l − (cid:126)q ) m + i(cid:15) (cid:105) = µ µ π m m m √ a (cid:34) arctan (cid:32) c (cid:48) − c (cid:112) a ( c − i(cid:15) ) (cid:33) + arctan (cid:32) a + c − c (cid:48) (cid:112) a ( c (cid:48) − a − i(cid:15) ) (cid:33)(cid:35) , (A.2)where µ ij = m i m j / ( m i + m j ) are the reduced masses, b = m + m − M , b = m + m + q − M with M the mass of the initial particle, and a = (cid:18) µ m (cid:19) (cid:126)q , c = 2 µ b , c (cid:48) = 2 µ b + µ m (cid:126)q . (A.3)For more information about the loop function, we refer to Refs. [32, 42]. The two arctangent functionscorrespond to the two cuts in the triangle diagram [33].In the following, we give the expression for the loop with one of the mesons having a finite width.By assigning a constant width Γ to the meson with a mass m , the first propagator in Eq. (A.2) ismodified to l − (cid:126)l / (2 m ) + i Γ / . Thus, the first cut of the triangle diagram involving m will be influenced, and the scalar loop integralbecomes I ( m , m , m , (cid:126)q )= µ µ π m m m √ a (cid:34) arctan (cid:32) c (cid:48) − c (cid:112) a ( c − iµ Γ ) (cid:33) + arctan (cid:32) a + c − c (cid:48) (cid:112) a ( c (cid:48) − a − i(cid:15) ) (cid:33)(cid:35) . (A.4) B Coefficients in the decay amplitudes A = (cid:114) N g x c E γ (cid:104) I (cid:16) m D , m D , m D *0 , (cid:126)q (cid:17) + I (cid:16) m D , m D *0 , m D , (cid:126)q (cid:17)(cid:105) B = 43 (cid:114) i N e g x (cid:126)q (cid:18) β + 1 m c (cid:19) (cid:104) I (1) ( m D , m D , m D *0 , (cid:126)q ) + 2 I (1) ( m D ∗ , m D ∗ , m D , (cid:126)q ) (cid:105) C = 23 (cid:114) i N e g x (cid:126)q (cid:20) (cid:18) β − m c (cid:19) I (1) ( m D *0 , m D , m D ∗ , (cid:126)q ) − (cid:18) β + 1 m c (cid:19) I (1) ( m D *0 , m D *0 , m D , (cid:126)q ) (cid:21) (B.5)where N = (cid:112) M X M ψ accounts for the nonrelativistic normalization, and the expression for thevector loop integral I (1) ( m , m , m , (cid:126)q ) can be found in Ref. [32].12 eferences [1] S. K. Choi et al. [Belle Collaboration], Phys. Rev. Lett. (2003) 262001 [hep-ex/0309032].[2] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. (2013) 222001 [arXiv:1302.6269 [hep-ex]].[3] A. De Rujula, H. Georgi, S.L. 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