Search for a D \bar{D} bound state in the Λ_b \rightarrow ΛD\bar{D} process
Le-Le Wei, Hong-Shen Li, En Wang, Ju-Jun Xie, De-Min Li, Yu-Xiao Li
SSearch for a D ¯ D bound state in the Λ b → Λ D ¯ D process Le-Le Wei, Hong-Shen Li, En Wang, ∗ Ju-Jun Xie,
2, 3, 1, † De-Min Li, and Yu-Xiao Li School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences, Beijing 101408, China
We have investigated the process of Λ b → Λ D ¯ D , by taking into account the contributions from the s -wave D ¯ D interaction within the coupled-channel unitary approach, and the intermediate ψ (3770)resonance. In addition to the peak of the ψ (3770), an enhancement near the D ¯ D mass thresholdis found in the D ¯ D invariant mass distributions, which should be the reflection of the D ¯ D boundstate. We would like to encourage our experimental colleagues to measure the D ¯ D invariant massdistribution of the Λ b → Λ D ¯ D process, which is crucial to search for the D ¯ D bound state and tounderstand the heavy-hadron heavy-hadron interactions. PACS numbers:
I. INTRODUCTION
Although the quark model was proposed by Gell-Mannand Zweig more than half century ago [1, 2], it is stillvalid in classifying all known hadrons by now. Sincethe X (3872) was observed by the Belle Collaboration in2003 [3], many charmonium-like states were reported ex-perimentally [4], and most of them cannot be explainedas the conventional mesons ( q ¯ q ) or baryons ( qqq ) [5, 6].There are many explanations about those states, such astetraquark states, molecular states, the conventional c ¯ c mesons, or the mixing between different components [7–11]. However, it is surprising that many resonant struc-tures are observed around thresholds of a pair of heavyhadrons, such as X (3872) and Z c (3900) ± around the D ¯ D ∗ threshold, Z cs (3985) around the ¯ D s D ∗ and ¯ D ∗ s D thresholds, and X (3930) around D s ¯ D s threshold. As dis-cussed in Ref. [12], such structures should appear at anythreshold of a pair of heavy-quark and heavy-antiquarkhadrons which have attractive interaction at threshold.Thus, the experimental information about the thresh-old structures is crucial to deeply understand the heavy-hadron heavy-hadron interactions, and the internal struc-tures of the hidden-charm states [13, 14].In Ref. [15], one new hidden charm resonance withmass around 3700 MeV (denoted as X (3700) in this ar-ticle) is predicted within the coupled channel unitaryapproach involving the D + D − , D ¯ D , D s ¯ D s , K + K − , K ¯ K , π + π − , π π , ηη , and π η channels. Later itwas suggested to search for this predicted D ¯ D boundstate in several processes, such as B → D ¯ DK [16], ψ (3770) → γX (3700) → γηη (cid:48) , ψ (4040) → γX (3700) → γηη (cid:48) , and e + e − → J/ψX (3700) → J/ψηη (cid:48) [17]. Ac-cording to the studies of Refs. [18, 19], the experimentaldata of e + e − → J/ψD ¯ D measured by the Belle Col-laboration [20, 21] are compatible with the existence ofsuch a D ¯ D bound state around 3700 MeV, though other ∗ Electronic address: [email protected] † Electronic address: [email protected] possibilities cannot be discarded due to the present qual-ity of the Belle data. In Ref. [24], we have performeda global fit to the data of γγ → D ¯ D [22, 23] and the e + e − → J/ψD ¯ D [21], by taking into account the s -wave D ¯ D final state interactions. Our results are con-sistent with the experimental data considering the un-certainties of the fitted parameters, and the modulussquared of the amplitude | t D ¯ D → D ¯ D | show peaks around3710 ∼ D ¯ D bound state withbinding energy B = 4 . +5 . − . MeV was also predicted ac-cording to the Lattice calculation in Ref. [25]. Thus, itis crucial to search for the signal of this predicted state.On the other hand, the decays of Λ b is one of the im-portant tool to study the hidden charm resonances [26],such as the processes of Λ b → J/ψ
Λ, Λ b → ψ (2 S )Λ [27–29]. The process Λ b → Λ X c ( X c ≡ c ¯ cu ¯ u ( d ¯ d ) , c ¯ cs ¯ s ) isalso proposed to search for the XY Z states in Ref. [30].In this work, we will propose to search for the signal ofthe D ¯ D bound state in the single-Cabibbo-suppressedprocess of Λ b → Λ D ¯ D , which has not been measured ex-perimentally up to our knowledge. It should be pointedout that the Λ b → Λ D ¯ D process is expected to havea larger branching fraction than the double-Cabibbo-Suppressed process Λ b → Λ K + K − with the branchingfraction B (Λ b → Λ K + K − ) = (15 . ± . ± . ± . × − measured by the LHCb Collaboration [31].Since the predicted mass of the D ¯ D bound state islower than the D ¯ D threshold, it will manifest itself as theenhancement near the D ¯ D threshold, the similar work isfound in Refs. [16, 32]. For instance, a peak observed inthe φω threshold in the J/ψ → γφω reaction [33] was in-terpreted as the manifestation of the f (1710) resonancebelow the φω threshold [34]. In Ref. [35] the BESIII Col-laboration has seen a bump structure close to thresholdin the K ∗ ¯ K ∗ mass distribution of the J/ψ → ηK ∗ ¯ K ∗ decay, which can be interpreted as a signal of the for-mation of an h resonance [34, 36]. We expect therewill be an enhancement near the threshold in the D ¯ D in-variant mass distribution. On the other hand, since the ψ (3770), with a mass close to the D ¯ D threshold, mainlydecays into D ¯ D in p -wave, we will take into account thecontribution from the ψ (3770). a r X i v : . [ h e p - ph ] F e b The paper is organized as follows. In Sect. II, we intro-duce our model for the process Λ b → Λ D ¯ D . Numericalresults for the D ¯ D invariant mass distribution and dis-cussions are given in Sect. III, and a short summary isgiven in the last section. II. FORMALISM
In analogy to Refs. [37–41], the mechanism of the decayΛ b → Λ D ¯ D ( D ¯ D ≡ D ¯ D , D + D − ) can happen via threesteps: the weak decay, hadronization, and the final stateinteraction. In the first step as depicted in Fig. 1, the b quark of the initial Λ b weakly decays into a c quark anda W − boson, followed by the W − boson decaying into a¯ cs quark pair, | Λ b (cid:105) = 1 √ b ( ud − du ) ⇒ V p c ¯ c √ s ( ud − du )= V p c ¯ c Λ , (1)where we take the flavor wave functions Λ b = b ( ud − du ) / √ s ( ud − du ) / √
2, and V p is thestrength of the production vertex that contains all dy-namical factors. bud sudc ¯ c ¯ uu + ¯ dd + ¯ ss W − FIG. 1: The quark level diagram for the weak decay Λ b → Λ c ¯ c . In order to give rise to the final state D ¯ D Λ (or D + D − Λ), the quark c and antiquark ¯ c need to hadronizetogether with the ¯ qq ( ≡ ¯ uu + ¯ dd + ¯ ss ) created from thevacuum with J P C = 0 ++ , which could be expressed asthe mechanisms of the internal W − emission and exter-nal W − emission, respectively shown in Figs. 2(a) and2(b). Thus, we have, | H (cid:105) in = V p (cid:12)(cid:12)(cid:12)(cid:12) c (cid:0) ¯ uu + ¯ dd + ¯ ss (cid:1) ¯ cs √ ud − du ) (cid:29) = V p (cid:0) D ¯ D + D + D − + D + s D − s (cid:1) Λ , (2)for the internal W − emission mechanism of Fig. 2(a) ,and | H (cid:105) ex = V p × C × D + s D − s Λ , (3) bud sudc ¯ c ¯ uu + ¯ dd + ¯ ss W − ( a ) bud cud ¯ ss ¯ csW − ( b ) FIG. 2: The mechanisms of (a) the internal W − emissionmechanism and (b) the external W − emission for the weakdecay Λ b and the hadronization of the c ¯ c through ¯ qq createdfrom the vacuum. for the external W − emission mechanism of Fig. 2(b).Here the color factor C accounts for the relative weightof the external W − emission with respect to the internal W − emission, and we take C = 3 in the case of colornumber N c = 3 [42–44].The final states can also undergo the interactions of the D ¯ D and Λ D , which may generate dynamically the reso-nances. The interaction of the coupled channels includingΛ D was studied within a unitary coupled-channel ap-proach which incorporates heavy-quark spin symmetry,and two resonances Ξ c (2790) and Ξ c (2815) are identifiedas the dynamically generated resonances [45]. Since theirmasses are about 150 ∼
200 MeV below the Λ D thresh-old, their contributions do not affect the structure closeto the D ¯ D threshold, which can be easily understoodfrom the Dalitz plot of Fig. 3. Thus, we neglect the Λ D interaction in this work, because only the D ¯ D invariantmass distribution near the threshold is relevant for the D ¯ D bound state.The next step is to consider the final state interac-tion of these channels to give D ¯ D (or D + D − ) at theend. We can have the final states of D ¯ D (or D + D − )through the direct production in the Λ b decay, or there-scattering of the primarily produced channels D ¯ D , D + D − , or D + s D − s , as shown in Figs. 4(a) and 4(b), re- M D Λ [ M e V ] M D ¯ D [MeV] Ξ c (2815) FIG. 3: The Dalitz plot for the Λ b → Λ D ¯ D . The green bandstands for the region of 3710 ∼ D ¯ D bound state lies in. spectively. Apart from the three coupled channels D ¯ D , D + D − , and D + s D − s , we only consider one light channel ηη to account for the width of the D ¯ D bound state, asin Refs. [16–18, 24]. Λ b Λ D ( D + )¯ D ( D − )( a )Λ b Λ D ( D + )¯ D ( D − )( b ) FIG. 4: The decays Λ b → Λ D ¯ D and Λ b → Λ D + D − , (a)direct production, (b) the re-scattering of the channels D ¯ D , D + D − , or D + s D − s . Then, the total amplitudes for the Λ b → Λ D ¯ D and Λ b → Λ D + D − can be expressed as, t s -waveΛ b → Λ D ¯ D = V p [1 + G D + D − t D + D − → D ¯ D + G D ¯ D t D ¯ D → D ¯ D +(1 + C ) G D + s D − s t D + s D − s → D ¯ D (cid:105) , (4) t s -waveΛ b → Λ D + D − = V p [1 + G D + D − t D + D − → D + D − + G D ¯ D t D ¯ D → D + D − +(1 + C ) G D + s D − s t D + s D − s → D + D − (cid:105) , (5)where G l is the loop function for the two-meson propa-gator in the l -th channel, G l = i (cid:90) d q (2 π ) q − m + i(cid:15) P − q ) − m + i(cid:15) = 116 π (cid:20) α l + ln m µ + m − m + s s ln m m + p √ s × (cid:18) ln s − m + m + 2 p √ s − s + m − m + 2 p √ s + ln s + m − m + 2 p √ s − s − m + m + 2 p √ s (cid:19)(cid:21) , (6)with the subtraction constant α l = − . l = 1 , , , D ¯ D , D + D − , D + s D − s , and ηη , respectively) and µ = 1500 MeV as Ref. [15]. P ≡√ s = M D ¯ D is the invariant mass of the two mesons inthe l -th channel. m and m are the masses of the twomesons in the l -th channel. p is the three-momentumof the meson in the center of mass frame of the meson-meson system, p = λ / ( s, m , m )2 √ s , (7)with the K¨allen function λ ( x, y, z ) = x + y + z − xy − yz − zx .With the isospin doublets ( D + , − D ), ( ¯ D , D − ), wehave, (cid:12)(cid:12) D + D − (cid:11) = 1 √ (cid:12)(cid:12) D ¯ D, I = 0 , I = 0 (cid:11) + 1 √ (cid:12)(cid:12) D ¯ D, I = 1 , I = 0 (cid:11) , (8) (cid:12)(cid:12) D ¯ D (cid:11) = 1 √ (cid:12)(cid:12) D ¯ D, I = 0 , I = 0 (cid:11) − √ (cid:12)(cid:12) D ¯ D, I = 1 , I = 0 (cid:11) . (9)Taking the averaged mass of D meson in Eqs.(4) and(5), it is easy to find that only the isospin I = 0 compo-nent of the D ¯ D has the contribution to the Λ b → Λ D ¯ D process, G D + D − t D + D − → D ¯ D + G D ¯ D t D ¯ D → D ¯ D = G D ¯ D t I =0 D ¯ D → D ¯ D , (10) G D + D − t D + D − → D + D − + G D ¯ D t D ¯ D → D + D − = G D ¯ D t I =0 D ¯ D → D ¯ D . (11)The scattering matrices t i → j in Eqs. (4) and (5) are ob-tained by solving the Bethe-Salpeter equation in coupledchannels, t = [1 − V G ] − V, (12)where the elements of the diagonal matrix G is theloop function of Eq. (6), and the matrix element V i,j are the transition potential of the i -th channel tothe j -channel. The transition potentials V i,j ( i, j = D ¯ D , D + D − , D + s D − s ) are tabulated in the Appendix Aof Ref. [15]. We introduce the potentials of ηη → D ¯ D and ηη → D + D − with a dimensionless strength a = 50to give the width of the D ¯ D bound state, and the tran-sition potentials of ηη → ηη and ηη → D + s D − s are notrelevant and are taken as zero [16–18, 24]. Both the G l and t i → j in Eqs. (4) and (5) are the functions of the D ¯ D invariant mass M D ¯ D .The obtained modulus squared of the transition am-plitude | t D + D − → D + D − | and | t D + D − → D + s D − s | are shownin Fig. 5, and one can find a peak around 3720 MeV,which could be associated to the D ¯ D bound state. Onthe other hand, from Fig. 5, the | t D + D − → D + D − | is twotimes larger than | t D + D − → D + s D − s | , which indicates thatthe X (3700) state coups mostly to D ¯ D channel. | t i → j | × − M D ¯ D [MeV] D + D − → D + D − D + D − → D + s D − s FIG. 5: The modulus squared of the transition ampli-tudes | t D + D − → D + D − | and | t D + D − → D + s D − s | calculated withEq. (12). In addition, we also take into account the decaysΛ b → Λ D ¯ D and Λ b → Λ D + D − via the intermedi-ate resonance ψ (3770), which is depicted in Fig. 6. Theamplitude can be written as, t p -wave = βV p × M ψ (3770) ˜ p D M D ¯ D − M ψ (3770) + iM ψ (3770) ˜Γ ψ (3770) , (13)where the normalization factor V p is the same as the onein Eqs. (4) and (5), and we introduce the parameter β toaccount for the relative weight of the ψ (3770) strength with respect to the s -wave contribution of Eqs. (4) and(5). ˜ p D is the momentum of the D (or D + ) in the restframe of the D ¯ D (or D + D − ) system,˜ p D = λ / (cid:0) M D ¯ D , M D , M D (cid:1) M D ¯ D . (14)We take the width for ψ (3770) energy dependent,which is given by,˜Γ ψ (3770) = Γ ψ (3770) × (cid:113) M D ¯ D − M D (cid:113) M ψ (3770) − M D , (15)with M ψ (3770) = 3773 . ψ (3770) = 27 . M D = ( M D + + M D ) / .
24 MeV [4]. Λ b Λ D ( D + )¯ D ( D − ) ψ (3770) FIG. 6: The microscopic diagram for the decays Λ b → Λ D ¯ D and Λ b → Λ D + D − . With the amplitudes of Eqs. (4), (5) and (13), we canwrite the differential decay width for the decays Λ b → Λ D ¯ D and Λ b → Λ D + D − ,dΓd M D ¯ D = ˜ p D p Λ M Λ M Λ b (2 π ) M b (cid:104) | t s -wave | + | t p -wave | (cid:105) , (16)with p Λ = λ / (cid:0) M Λ b , M Λ2 , M D ¯ D (cid:1) M Λ b . (17) III. NUMERICAL RESULTS AND DISCUSSION
In our model, we have three free parameters, the globalnormalization V p , the color factor C , and β . V p is aglobal factor and its value does not affect the shapes ofthe D ¯ D and D + D − invariant mass distributions. β represents the relative weight of the ψ (3770) strengthwith respect to the one of s -wave, and we take its value β = 0 .
15 to give the contributions from the s -wave D ¯ D interaction and the ψ (3770) with the same order of mag-nitude. Next, we first show the results with the colorfactor C = 3 and V p = 1, and will present the results fordifferent values of C and β .We show the D ¯ D and D + D − invariant mass dis-tributions in Fig. 7. One can find a clear enhance-ment near the D ¯ D threshold in the D ¯ D invariantmass distribution of the Λ b → Λ D ¯ D , due to the pres-ence of the X (3700) resonance below the D ¯ D threshold.The enhancement structure near the threshold is a littleweaker for the D + D − invariant mass distribution of theΛ b → Λ D + D − , because the D + D − threshold is higherthan the D ¯ D one and farther away from the peak of X (3700). d Γ / d M D ¯ D [ a r b . un i t s ] M D ¯ D [MeV] total p -wave s -wave01503004506007509003720 3750 3780 3810 3840 3870 3900(b) d Γ / d M D + D − [ a r b . un i t s ] M D + D − [MeV] total p -wave s -wave FIG. 7: The D ¯ D (a) and D + D − (b) invariant mass distribu-tions of the processes Λ b → Λ D ¯ D and Λ b → Λ D + D − . Theblue dashed curve shows the contribution from the meson-meson interaction in s -wave, the green dash-dotted curve cor-responds to the results for the intermediate meson ψ (3770),and the red solid curve shows the total contributions. In Fig. 8, we show the D ¯ D and D + D − invariant massdistributions with the different values of color factor C =3 . , . , .
0. One can find that both mass distributionsnear the threshold do not change too much, since thevalue of color factor C only affects the contribution fromthe D + s D − s loop of Fig. 4(b), which is smaller than thecontributions from the D + D − and D ¯ D .We also present our results for the different values of β = 0 . , . , .
10 in Fig. 9. One can see that the en-hancement near the threshold will be identified difficultlyfor the larger value of β . Indeed, the ψ (3770) would pro-vide the dominant contribution for the Λ b → Λ D ¯ D pro-cess, however, it is still expected to find an enhancementnear the D ¯ D threshold, especially the D ¯ D one, if the D ¯ D bound state do exist, as predicted in Refs. [15, 25]. d Γ / d M D ¯ D [ a r b . un i t s ] M D ¯ D [MeV] C =3.0 C =2.5 C =2.001503004506007509003720 3750 3780 3810 3840 3870 3900(b) d Γ / d M D + D − [ a r b . un i t s ] M D + D − [MeV] C =3.0 C =2.5 C =2.0 FIG. 8: The D ¯ D (a) and D + D − (b) invariant mass distri-butions of the processes Λ b → Λ D ¯ D and Λ b → Λ D + D − with different values of C = 3.0, 2.5, 2.0. Furthermore, since the ψ (3770) state couples to D ¯ D in p -wave, the partial wave analysis of this reaction wouldbe helpful to test the existence of the D ¯ D bound state.At present, the LHCb Collaboration has accumulateda large number of Λ b events, thus, we would like tocall the attention of the experimentalists to measure theΛ b → Λ D ¯ D decay, which should be useful to confirm theexistence of X (3700) and to understand its nature. IV. CONCLUSIONS
The study of the charmonium-like states is crucial tounderstand the heavy-hadron heavy-hadron interactions,and also the internal structures of the hidden-charmstates. One D ¯ D bound state around 3700 MeV was pre-dicted within the coupled channel unitary approach [15],and also the lattice investigation of the D ¯ D and D s ¯ D s scattering [24]. Although our previous studies on the e + e − → J/ψD ¯ D and γγ → D ¯ D data support the ex-istence of the D ¯ D bound state, the other possibilitiescannot be discarded due to the present quality of theexperimental data [18, 24]. Investigating the processesinvolving the s -wave D ¯ D system could provide the infor-mation about the existence of the D ¯ D bound state. d Γ / d M D ¯ D [ a r b . un i t s ] M D ¯ D [MeV] β =0.30 β =0.15 β =0.100500100015002000250030003720 3750 3780 3810 3840 3870 3900(b) d Γ / d M D + D − [ a r b . un i t s ] M D + D − [MeV] β =0.30 β =0.15 β =0.10 FIG. 9: The D ¯ D (a) and D + D − (b) invariant mass distri-butions of the processes Λ b → Λ D ¯ D and Λ b → Λ D + D − with different values of β =0.30, 0.15, 0.10. In this paper, we have investigated the processes Λ b → Λ D ¯ D and Λ b → Λ D + D − within the coupled chan-nel unitary approach, by taking into account the s -wavemeson-meson interactions and the contribution from theintermediate resonance ψ (3770). The D ¯ D and D + D − invariant mass distributions in the Λ b → Λ D ¯ D reactionare investigated, and our results show an enhancementstructure near the D ¯ D threshold, which should be thereflection of the D ¯ D bound state. Therefore, we stronglyencourage our experimental colleagues to measure theΛ b → Λ D ¯ D process, which would be crucial to confirmthe existence the X (3700) resonance, and to understandthe heavy-hadron heavy-hadron interactions. Acknowledgments
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