diff --git "a/0tFRT4oBgHgl3EQfkzf9/content/tmp_files/load_file.txt" "b/0tFRT4oBgHgl3EQfkzf9/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/0tFRT4oBgHgl3EQfkzf9/content/tmp_files/load_file.txt" @@ -0,0 +1,750 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf,len=749 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='13597v1 [hep-ph] 31 Jan 2023 The scalar exotic resonances X(3915), X(3960), X(4140) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='Badalian and Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='Simonov NRC “Kurchatov Institute” Moscow, Russia February 1, 2023 Abstract The scalar resonances X(3915), X(3960), X(4140) are considered as exotic four-quark states: cq¯c¯q, cs¯c¯s, cs¯c¯s, while the X(3863) is proved to be the c¯c, 2 3P0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The masses and the widths of these reso- nances are calculated in the framework of the Extended Recoupling Model, where a four-quark system is formed inside the bag and has relatively small size (<∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='0 fm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Then the resonance X(3915) ap- pears due to the transitions: J/ψω into D∗+D∗− (or D∗0 ¯D∗0) and back, while the X(3960) is created due to the transitions D+ s D− s into J/ψφ and back, and the X(4140) is formed in the transitions J/ψφ into D∗+ s D∗− s and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The characteristic feature of the recoupling mechanism is that this type of resonances can be predominantly in the S-wave decay channels and has JP = 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In two-channel case the reso- nance occurs to be just near the lower threshold, while due to coupling to third channel (like the c¯c channel) it is shifted up and lies by (20– 30) MeV above the lower threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The following masses and widths are calculated: M(X(3915)) = 3920 MeV, Γ(X(3915)) = 20 MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' M(X(3960)) = 3970 MeV, Γ(X(3960) = 45(5) MeV, M(X(4140)) = 4120(20) MeV, Γ(X(4140)) = 100 MeV, which are in good agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 1 1 Introduction In the region (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='9–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2) GeV there are now three scalar resonances and the X(3915) was the first, observed by the Belle in the e+e− → J/ψωK process [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Later this resonance was confirmed by the BaBar [2] and in several other experiments [3]), in particular, in two-photon collisions [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' For some years this resonance was assumed to be the conventional c¯c meson – χco(2P), although this interpretation has called out some doubts [6, 7] (see discussion in the reviews [8, 9]) and does not agree with predictions in different relativistic potential models (RPM) [10]-[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The experimental masses of the X(3915) and χc2(2P) were found to be almost equal, while in the RPMs a smaller mass, M(2 3P0) ∼= 3870 ± 30 MeV, and much larger mass difference, δ20(2P) = M(χc2(2P) − M(χc0(2P) ∼= (70 − 100) MeV, were predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that large mass difference δ20 is kept even if the coupling of the χc0(2P) to open channels is taken into account [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Such theoretical expectations were supported by the Belle observation of the wide scalar X(3860) resonance [16], both in e+e− → J/ψD+D− and e+e− → J/ψD0 ¯D0 decays, which has the mass M = 3862+26 −32 +40 −82 MeV and large width Γ ∼= 200 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The existence of the scalar X(3860) resonance is confirmed by the analysis of two-photon production, γγ → D ¯D in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Very recently the LHCb [18] has observed two more scalar resonances X(3960), X(4140) in the D+ s D− s mass spectrum in the B+ → D+ s D− s K+ de- cays with the parameters: M(X(3960)) = (3956±5±10) MeV, Γ(X(3960)) = (43 ± 13 ± 8) MeV, M(X(4140)) = (4133 ± 6 ± 6) MeV, Γ(X(4140)) = (67 ± 17 ± 7) MeV, both with JP C = 0++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' These new scalar resonances evi- dently look as exotic states and the X(3960) was interpreted as the molecular D+ s D− s state within the QCD sum rules approach [19, 20] and in a coupled- channel model [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' in [22] it appears due to the triangle singularity, while in [23] the parameters of the X(3960), as a diquark-antidiquark state, were obtained in a good agreement with experiment, using the QCD sum rules approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that the masses of the X(3960) and X(4140) resonances lie by ∼ 20 MeV above the thresholds: D+ s D− s and J/ψφ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In our paper we assume that the X(3915) and both the X(3960), X(4140) belong to exotic four-quark states cq¯c¯q and cs¯c¯s and to define their parame- ters we will use the Extended Recoupling Model (ERM), recently suggested in [24], which develops the Recouplimg Model, presented earlier [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The ERM allows to calculate the mass and width of a scalar four-quark states, however, within suggested mechanism such resonances cannot exist in the 2 systems with two identical mesons, like D+ s D+ s , D∗+ s , D∗+ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' This theoretical prediction is supported by the Belle experiment [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the ERM the system of two mesons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (J/ψ + φ), can transfer into another pair of the mesons (D+ s , D− s ) by rearranging confining strings and back in the infinite chain of transformations, like J/ψφ → (D+ s ¯D− s ) → J/ψφ → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='. Note that such se- quences can also be treated, for example, in the standard OBE approximation with the meson exchanges, which, however, does not produce the singulari- ties near the thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the coupled-channel models (CCM) [27, 28] the interaction between hadrons, like D+ s D− s , J/ψφ, is usually neglected, while in the ERM such interaction is taken into account, introducing the four- quark bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' It is important that all hadrons involved have rather small sizes, ∼= (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='40 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='55) fm and only ω(1S) has a bit larger r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='7 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' We would like to underline the characteristic features of the ERM [24]: first, due to the string rearrangement of a four-quark system the singularity lies close to the lower threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' second, this mechanism produces the resonance in the S-wave hadron-hadron system and therefore, the quantum numbers of these resonances JP C = 0++, 1++, 2++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' third, a resonance does not appear, if hadrons are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the literature there are still a controversy, concerning the X(3915), and different interpretations were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' This resonance was considered in tetraquark model within the Born–Oppenheimer approach in [29, 30, 31, 32], due to the triangle singularity [22] and the threshold effects [33], as the molecular Ds ¯Ds bound state [34] or the lightest cs¯c¯s state [35] and as the diquark-antidiquark state, using the QCD sum rule method [23, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In con- trast to a molecular structure of four-quark states in the ERM these systems are assumed to be compact systems, similar to the diquark-antidiquark states studied in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In such compact systems their wave functions at the origin are not small and therefore they can be produced in the γγ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In our paper we will shortly discuss the higher scalars, X(4500), X(4700), observed by the LHCb [38], which admit different interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In next section we shortly remind the basic formulas in two-channel case and give the values of the parame- ters, needed to define the masses and widths of the recoupled four-quark resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In section 3 more general matrix representation of the ERM is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In section 4 we calculate the transition amplitudes and give the masses and widths of the scalar resonances, and compare them with exper- imental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In section 4 the masses of high X(4500), X(4700) resonances, as the c¯c states, are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Our conclusions are presented in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 3 2 The two-channel approach in the Extended Recoupling Model We study the experimental process where, among other products, two hadrons are produced and one pair of hadrons (the pair 1) can transfer into another pair of hadrons (the pair 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In [24] the probability amplitude of this tran- sition was denoted as V12(p1, p2), with p1, p2 – relative momenta of the hadrons, referring to the pair 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' If an infinite set of the transfor- mations was supposed and the total production amplitude A2 of the pair 2 was written as a product of the slowly varying function F(E) and the singular factor f12(E) = 1 1−N , then the amplitude A2 = F(E)f12(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' This definition of the transition amplitude V12 = V21 differs of that in other ap- proaches, where one or more the OBE diagrams with meson exchanges are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the ERM [24] the process occurs through the intermediate stage of the Quark Compound Bag (QCB) [39, 40], where all quarks and antiquarks of two hadrons are participating in the string recoupling and, possibly, the spin recoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Denoting the QCB wave functions as Φ(qi) (i = 1, 2, 3, 4) and the two-hadron wave functions as Ψi(h1, h2), the amplitude V12 can be written as, V12 = (Ψ1(ha1hb1)Φ(qi))(Φ(qi)Ψ2(ha2hb2) = V1(p1)V2(p2), (1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the amplitude V12 = 1 1−N acquires the factorized form: V12(p1, p2) = v1(p1)v2(p2) with the factor N, written as N = z(E)I1(E)I2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (2) Here z = z(E) can be called the transition probability, while I1(E), I2(E) are the following integrals (see [24]): Ii(E) = viGivi = � d3pi (2π)3 v2 i (pi) E′(pi) + E ′′(pi) − E , (3) where the hadron energies E′(pi), E ′′(pi) in the i-th pair near thresholds, E′(p) = p2 2m′ + m′, include corresponding thresholds Eth i and the reduced masses µi, namely, Eth i = m′(i) + m ′′(i), µi = m′(i)m ′′(i) m′(i) + m ′′(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (4) 4 The result of the integration in Ii(E) can be approximated by the form: Ii = consti 1 νi − i � 2µi(E − Eth i ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (5) with µi, defined in (4), while νi is expressed via the parameters of the hadron wave functions, which were calculated explicitly in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Here we would like to underline that the transition probability z(E) appears to be the only fitting parameter in the ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The whole series of the transitions from the pair 1 to 2 and back is summed up to the amplitude f12, f12(E) = 1 1 − zI1I2 , Ii = 1 νi − i � 2µi(E − Eth i ) , (6) where νi are found from the four-quark wave functions, as in [37, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (6) takes place for the energies E > E1, E2, while for E < E1, E2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' below thresholds, the amplitude f1 = � 1 ν1+√ 2µ1(|E−E1|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' It is important that in the ERM the process proceeds with the zero relative angular momentum between two mesons, L = 0, otherwise the transition probability z12(E) is much smaller and a resonance may not appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Note also that if the recoupling mechanism is instantaneous, or the tran- sition from one pair of the mesons to another proceeds instantaneously, then the transition amplitude V (12) does not factorize into V (1)V (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' such an assumption was used in the original Recoupling Model [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' However, in this approximation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' for the Tcc resonance agreement with experiment was not reached [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' On the contrary, in the ERM [24] the recoupling mecha- nism proceeds in two stages: at first stage the hadrons h1, h2 collapse into common “compound bag” [39, 40], where the four quarks are kept together by the confining interaction between all possible quark pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' This compound bag has its own wave function Φi(q1, q2, q3, q4) and the probability amplitude of the h1, h2 → Φ transition, which defines the factor V1(p1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In a similar way the transition from the Bag state to the final hadrons h3, h4 defines the factor V2(p2) and we obtain the relation: v1(pi) = � d3q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='d3q4ψh1ψh2Φi(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='.q4), (7) and similar equation for v2(p2), replacing h1, h2 by h3, h4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' From vi(pi) the function Ii (3) is defined and using (6), one obtains νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 5 Now we give experimental data and corresponding the ERM parame- ters, referring to the four-quark systems, cq¯c¯q for X(3915) and cs¯c¯s for the X(3960), X(4140).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' We give also the threshold energies E1, E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The parameters of the four-quark resonances 1) X(3915), JP = 0+, Γ(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 20(5) MeV [1, 3], J/ψω → D∗ ¯D∗, E1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='880, E2 = 4020, µ1 = M(J/ψ)M(ω) M(J/ψ)+M(ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='624, µ2 = M(D∗)M( ¯D∗) M(D∗)+M( ¯D∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='050 (all in GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' From [24] ν1(J/ψω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='21 GeV, ν2(D∗ ¯D∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='44 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 2) X(3960), JP = 0+, Γ(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 43(21) MeV [18], [J/ψφ] → [D− s D+ s ], E1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='936, E2 = 4116, µ1 = MJ/ψMφ MJ/ψ+Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='767, µ2 = M(D+ s )M(D− s ) M(D+ s +M(D−) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' ν1(J/ψφ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='265, ν2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='424 (all in GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 3) X(4140), JP = 0+, Γ(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 67(24) MeV[18], [J/ψφ] → [D∗− s D∗+ s ], E1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='116, E2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='224, µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='767, µ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='056, ν1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='265, ν2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='410 (all in GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Here q can be u, d quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' To define the structure of the cross sections we start with the value of the recoupling probability z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2 GeV2 and the parameters from the item 1) to obtain the distribution |f12(E)|2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the values of |f12(E)|2 will be given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the amplitude f12(E) the resulting singularity can be found in the form of (6) and for equal threshold masses it produces a pole nearby thresholds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' however, real distance between the thresholds is large, ∼ 100 MeV and the actual singularity structure can be more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 3 The matrix approach in the ERM In previous Section we have presented the ERM equations in the case of two channels, which are convenient to define the mass of a resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' However, they do not allow to study some details of the process, or to consider a larger number of channels, which can have a influence at the properties of a four- quark system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Therefore here we present a more general representation of the amplitude using the unitarity relation, when the standard form of the transition amplitudes fij(E) (for L = 0) is fij − f ∗ ji = � n 2iknfinf ∗ jn, (8) 6 or the unitarity relation can be realized through the M-matrix representation, ˆfM = 1 ˆ M − iˆk , (9) where ˆf, ˆ M, ˆk are the matrices in the channel numbers [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In some cases instead of the ˆ M it is more convenient to use the ˆK matrix, ˆ M = − ˆK−1, where the matrix elements (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') Mik(E) are the real analytic functions of E with the dynamical cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' For two-channel system ˆfM can be written as ˆfM = 1 ˆ M − iˆk = ˆN D(E), (10) with ˆN = � M22 − ik2 −M21 −M12 M11 − ik1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (11) Here D(E) = (M11 − ik1)(M22 − ik2) − M12M21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (12) One can easily establish the relation between the equations (10)- (12) and the amplitude f12(ERM) (6) in two-channel case, which is a partial case of these equations: f12(ERM) = N11N22 D(E) , D(E) = (ν1 − ik1)(ν2 − ik2) − z, (13) and z = M12M21, νi ≡ Mii(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (14) One can see that for z > 0 the values νi = Mii are real analytic functions of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the ERM [24] νi were positive constants (defined via the parameters of the compound bag model), while in general case Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (12)-(14) include other transition m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='s fik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Later in our analysis we will be interested only in the denominator D(E) (12) and the factors in (13), (14), which fully define the position of a resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The value of z, in principle, can be calculated within the ERM, however, it can depend on many unknown parameters, and at the present stage we prefer to keep z as a single fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' It can be shown that z depends on the width of a resonance, but weakly depends on the resonance position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Now we consider three channels case to study more realistic case and choose the situation, when a resonance lies above the threshold 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Here we do 7 not need to specify the channel 3, which for example, may be a conventional c¯c state with JP C = 0++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' We introduce the 3 × 3 amplitude ˆfM(E) with three thresholds Ei (i = 1, 2, 3) and the momenta ki = � 2µi(E − Ei), µi = m1im2i m1i+m2i, and Ei = m1i + m2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Here m1i, m2i are the masses of two hadrons in the channel i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In this case the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (9) is kept, ˆf3(E) = ˆN3 D3(E), D3(E) = ((M11−ik1)(M22−ik2)−M12M21))(M23−ik3)+∆M, (15) where ∆M is ∆M = M31M12M23+M32M21M13−M13M31(M22−ik2)−M32M23(M11−ik1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (16) For the energy E below the thresholds, 1 and 2, −ik1 = |k1|, −ik2 = |k2|, and the factor ∆M is a real function of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' For the threshold 3 below thresholds of 1 and 2 one can define the poles of the amplitude ˆf3, or the zeroes of D3(E), and rewrite the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (15) as, D3 = (M11 − ik1)(M22 − ik2) − ˜z(E), (17) where the transition probability ˜z(E) ˜z(E) = M12M21 − ∆M(M33 + ik3) M2 33 + k2 3 (18) One can see that ˜z(E) acquires imaginary part, which can be of both signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Therefore the influence of the third (or more) open channels, lying below the thresholds E1, E2 in the 2 × 2 matrix f12(E), may be important in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The channel 3 can be taken into account, introducing complex values of z(E), which can depend on the energy as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 4 The masses and widths of the scalar reso- nances We start with the X(3915) resonance and consider the following recoupling process: J/ψω → D∗ ¯D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' At first we look at two-channel situation and choose the recoupling parameter z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='18 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' For the X(3915) structure – cq¯c¯q the parameters µi, νi, Ei are given in the item 1) of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Then inserting 8 all parameters to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (13), one obtains the distribution |f12(E)|2 (f2 ≡ f12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Its values for different E are given in Table 1, which show that the maximum takes place at E = 3880 MeV, just near the lower threshold, and Γ2 = Γ(2 − channels) ∼= 15 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In experiment for this resonance, observed by the Belle group in the process e+e− → e+e−J/ψω [1], the larger mass M(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = (3918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='9) MeV and Γ(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = (20 ± 5) MeV [3] were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the case of 3-channels, when e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the coupling to the c¯c channel is taken into account, the factor z3(E) acquires an imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In this case we calculate the amplitude f3(E), taking z3 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='18−i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20) GeV2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the values of |f3(E)|2 are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Table 1: The values of the |f12(E)|2 for X(3915) E(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='915 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='93 |f2(E)|2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='68 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='08 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='72 |f3(E)|2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30 348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='4 360 243 From Table 1 one can see that in the 3-channel case the peak is shifted up by ∼ 35 MeV and corresponds the mass ER ∼= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='915 GeV and the width Γ3 ∼= 20 MeV, which are in good agreement with the experimental mass and Γ(exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 20(5) MeV [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The scalar resonance X(3960) with JP C = 0++ was recently observed by the LHCb in the B+ → J/ψφK+ [18] and within the ERM it can be explained due to the infinite chain of the transitions: J/ψφ → D+ s D− s and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In two-channel approximation the X(3960) parameters (νi, µi, Ei, (i = 1, 2) are given in the item 2) (Section 2), which are used to define the amplitude (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' First, we choose z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30 GeV2 and calculate the transition amplitudes |f12(E)|2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' their values are given in the Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the two-channel approximation the numbers from Table 2 show the peak at E = 3940 MeV, near D+ s D− s threshold, and Γ(2 − ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') ∼= 15 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the 3-channel case the mass of the X(3960) resonance is shifted up to the position M(3 − ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 3970 MeV and the width increases to the value Γ(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') ∼= 45(5) MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' these values are in agreement with the experimental numbers: M(X(3960)) = 3956(15) MeV, Γ(X(3960)) = (43 ± 21) MeV [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In [18] the LHCb has reported about another, the X(4140) resonance, with JP C = 0++, in the B+ → D+ s D− s K+ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Its mass M(X(4140) = 9 Table 2: The transition probability |f12|2 as a function of the energy E for the X(3960) resonance E(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='05 |f12|2(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='93 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='50 |f3|2(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='02 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='7 198 500 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2 4133(12) MeV is close to the J/ψφ threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' We consider this resonance as the cs¯c¯s system and first calculate the squared amplitudes |f12(E)|2 in two- channel case, taking the parameters µi, νi, Ei from the item 3) of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In this 2-channel case: J/ψφ and D∗+ s D∗− s the transition probability z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='35 is taken and the calculated values of |f12|2 are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In three-channel case the channel D+ s D− s is added as the third one, then the values |f3|2 are calculated for z3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20 and given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Table 3: The values of the |f12(E)|2 and |f3(E)|2 for the X(4140) E(GeV) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='07 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='22 |f12(E)|2(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='35) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='40 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='45 |f3|2(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='54 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='87 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='12 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='66 From Table 3 one can see the peak at ER = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='01) GeV, Γ(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 60 MeV in two-channel approximation and the peak at ER = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='02) GeV with the width Γ(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') ∼= 100 MeV in tree-channel case, which are in good agreement with the experimental mass M(X(4140)) = (4133 ± 12) MeV and Γ(X(4140)) = (67 ± 24) MeV [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Our numbers in Tables 1–3 show that in two-channel case the resonance always lies just near the lower threshold, however, if the coupling to the third channel is taken into account, then it is shifted up and its position occurs to be close to the experimental number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The masses and widths of the exotic resonances, X(3915), X(3960), X(4140), defined in the ERM, are given in the Table 4 together with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' From Table 4 one can see that in the ERM the predicted masses and the widths of the scalar four-quark resonances are in good agreement with 10 Table 4: The ERM predictions for the masses and widths (in MeV) of exotic resonances with JP C = 0++ Resonance M(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') M(exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') Γ(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') Γ(exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') X(3915) 3920 3918 (2) 20 20(5) [3] X(3960) 3970 3956(15) 45(5) 43(21) [18] X(4140) 4120(20) 4133(12) 100 67(24) [18] experiment, if besides two channels, which creates the resonance, the coupling of the resonance to third channel is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Comparing our results with those in literature, one can notice that our conclusions on the four-quark structure of the X(3915), X(3960, X(4140)) also agree with the analysis in the paper [33], based on the coupled channel model of the c¯c and meson-meson systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that the general structure of the channel-coupling matrix elements in both approaches is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 5 The scalar X(4500), X(4700) resonances High scalar resonances X(4500), X(4700), or χc0(4500), χc0(4700), [38], were studied in many papers and for them two interpretations were suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' First, the X(4500) and X(4700) are considered as the c¯c states – 4 3P0 and 5 3P0 and their masses were calculated in relativistic quark models, where coupling to open channels was taken into account [14, 15, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In [41] the influence of open channels is studied using the so-called screened potential [11], while in [13] the spectrum was calculated using the relativistic string Hamiltonian [42] with the flattened confining potential [43];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' this flattening effect arises due to creation of virtual q¯q pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that the flattened confining potential appears to be universal for all types of the mesons and it produces the hadronic shifts down ∼ (100 − 130) MeV for the 4P, 5P char- monium states and gives the masses of the 4 3P0, 5 3P0 states in a reasonable agreement with experiment [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' On the contrary, in [44], within the 3P0 model, much smaller shifts due to the coupled-channel effects, <∼ 30 MeV , were obtained for the 4 3P0, 5 3P0 states, while in [41] these states acquire too large mass shifts for the chosen screened potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Model-independent analysis of the c¯c spectrum can also be done by means 11 of the Regge trajectories, if they are defined not for the meson mass M(nL) but for the excitation energy: E(nL) = M(nL) − 2 ¯mQ [45], where ¯mQ is the current heavy quark mass [13]: (M(n 3P0)−2 ¯mc)2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='06+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='08nr, (inGeV2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' n = nr +1, ¯mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (19) This Regge trajectory gives M(4 3P0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='474 GeV and M(5 3P0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='719 GeV, in good agreement with the LHCb data [38] (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Table 5: The Regge trajectory predictions for the masses of the charmonium n 3P0 states (in MeV) state M(nP) exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' mass 1 3P0 3429 3414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='8(3)) 2 3P0 3863 3862+26 −32 [16] 3 3P0 4194 abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 4 3P0 4473 4474 ± 6 [38] 5 3P0 4719 4694 ± 4+16 −3 [38] 6 3P0 4941 abs In Table 5 the masses M(2 3P0) = 3863 MeV, M(4 3P0) = 4473 MeV and M(5 3P0) = 4719 MeV, show very good agreement with those of χc0(3862) [16], X(4500) and X(4700) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' At present other high excitations with JP = 1+, 2+ (n = 4, 5) are not yet found and their observation would be very important to understand the fine-structure effects of high charmonium, in particular, the fine-structure splitting have to decrease for a screened GE potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that the resonance X(4700) lies very close to the ψ(2S)φ threshold and this fact indicates a possible connection between the c¯c and the cs¯c¯s states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The four-quark interpretation of the X(4500), X(4700) was discussed in different models [19],[46]-[49], where in the mass region (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='4–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='8) GeV the radial or orbital excitations of a diquark-antidiquark systems can exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 12 6 Conclusions In our paper the scalar resonances X(3915), X(3960), X(4140) are assumed to be the four-quark states, produced due to recoupling mechanism, when one pair of mesons can transform into another pair of mesons infinitely many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' These resonances do not exist in the c¯c spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' As the four-quark states they have several specific features: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The resonance appears only in the S-wave decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Within the ERM it lies rather close to the lower threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The scalar four-quark resonance can be created in two channel case due to transitions between channels, but it can also be coupled to another channel 3, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the c¯c channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' These resonances have no large sizes, being the compact systems, and this fact may be important for their observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the case of the X(3915) this statement is confirmed by the Belle analysis of the Q2 distribution of the X(3915) → J/ψω decays in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The masses and widths of the X(3915), X(3960), X(4140), presented in Ta- ble 4, are obtained in a good agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The authors are grateful to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Igumnova for collaboration.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Galkin, EPJC 58, 399 (2008) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' arXiv: 0808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='3912;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' [32] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Braaten, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} 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