A Refined Analysis on the Resonance
We study the property of the meson by analyzing the and decay processes. The competition between the rescattering mediated through a Breit–Wigner resonance and the rescattering generated from a local interaction is carefully studied through an effective lagrangian approach. Three different fits are performed: pure Breit-Wigner case, pure molecule case with only local rescattering vertices (generated by the loop chain), and the mixed case. It is found that data supports the picture where X(3872) is mainly a () Breit–Wigner resonance with a small contribution to the self–energy generated by final state interaction. For our optimal fit, the pole mass and width are found to be: MeV and MeV.
July 12, 2019
The is a narrow resonance close to the threshold, which was first observed in by the BELLE Collaboration BELLE1 (), and later confirmed by CDF CDF1 (), D0 D01 () and BABAR Collaborations BABAR1 (). It has also been observed at LHCb LHCb1 () and CMS CMS1 (). The new results of Belle show a mass MeV and width less than MeV BELLE3 (). A recent angular distribution analysis of the decay by LHCb has determined the quantum numbers to be LHCb2 ().
The decay of the X(3872) including and and final states are studied by BESIII, BABAR and BELLE BESIII (); BABAR2 (); BABAR3 (); BELLE4 (); BELLE2 (); BABAR4 (). Furthermore, other decay channels that have been observed experimentally are psigamma (), psi2sgamma () and psi2sgamma (), with relative branching ratios
The decay mode was further confirmed by the LHCb Collaboration recently LHCb:psi2sgamma (). As for the hadronic transition modes, the dipion spectrum in the is mainly given by resonance whereas the tripion spectrum in comes mainly from the meson. The ratio in Eq. (1) shows that these two processes are of the same order. One should note that the threshold of is about 8 MeV higher than , and the width of is only about 8 MeV PDG2014 (). Thus, the isospin symmetry breaking is not as serious as that shown in Eq. (1) since the phase space of decay mode is extremely suppressed compared with that of . Moreover, since the mass of is very close to the threshold of but not to that of , the rescattering effects through the loops can generate large isospin symmetry breaking at the amplitude level, and the number in Eq. (1) can be roughly accounted for even if the original decay particle has isospin ccbar3 ().
On the theory side, the nature of the is still a controversial issue, where different approaches have not reached yet a full agreement. The analysis of Ref. boundstate1 (); boundstate2 (); boundstate3 (); boundstate4 () favors a bound state, as the mass is very close to the threshold. Other works describe the as a virtual state virtualstate (), a tetraquark tetraquark () or a hybrid state hybrid (). On the other hand, it has also been considered as a mixture of a charmonium with a component ccbar1 (); ccbar2 (). The mixing can be induced by the coupled-channel effects, and the S-wave coupling can also explain the closeness of to the threshold of naturally LMC:coupledchannel (); Simonov:coupledchannel (). Furthermore, the existence of the substantial component in the state is supported by the analyses of the lager production rates of both in decays ccbar1 (); ccbar0 () and at hadron colliders hanhao ().
In Ref. ZhangO (), it is proposed to use the pole counting rule morgan92 () to study the nature of X(3872). A couple channel Breit–Wigner propagator is used to describe X(3872) and it is found that two nearby poles are needed in order to describe data. Based on this it is argued that the X(3872) is mainly of nature heavily renormalized by loop. However, Ref. ZhangO () did not consider the impact effect of the bubble chain generated by loops, which may generate a molecular type pole. Hence it might have been argued that the conclusion made in Ref. ZhangO () was not general. The purpose of this paper is to extend the analysis of Ref. ZhangO () by further including the effect of a contact . As we will see later, the major conclusions obtained in Ref. ZhangO () remain unchanged.
In this paper, we only focus on and final states. An effective lagrangian is constructed to calculate the decay into and . Three different scenarios are taken into consideration to fit experimental data: a single elementary particle propagating in the –channel; only bubble chains with contact rescattering; and the mixed situation, i.e., an elementary particle combined with the effect of bubble chain. In Sec. II, the effective lagrangian is introduced and the and amplitudes are calculated. In Sec. III, the numerical fits are performed and the resonance poles are analyzed. A brief summary is provided in Sec. IV. Minor technical details, such as the suppression of the longitudinal component of the amplitude with respect to the transverse one near threshold, are relegated to the appendixes.
Ii Theoretical analysis
ii.1 The Effective Lagrangian
The has been identified as a –wave resonance in the and final states with isospin , and the similar situation for the p-wave final states near the was studied in GYCheng (); Achasov () Likewise, as discussed in the introduction, we will assume to be an axial-vector resonance, with . To simplify the notations, from now on the channels are labeled just as , and the channels are labeled as (unless specifically stated otherwise). Likewise, when the two channels and appear together, they are labeled as in what follows.
In this section we construct the effective lagrangian of the interactions between X(3872) and other particles. The lagrangian of interactions has been constructed by Lag1 (); Lag2 (); Fleming (). However, operators such as , , are not considered previously and only constant form factors were used in these previous calculation missing information from decay vertex.
We will consider a model written in a relativistic form but intended for the description of and invariant energies close to the production threshold. We begin by constructing operators in our lagrangian with the lowest number of derivatives and fulfilling invariance under , and isospin symmetry. Hence, the interaction between the and the pair will occur through the combination of isospin, and ,
where the minus sign stems from the positive –parity and the usual assignment for and , with and , where indicates the type of or meson (e.g. ,etc) Bratten (). The vectors and gather the isospin doublets Doublet (), , , with the transposed conjugates , . Hence, following the previous symmetry prescriptions we consider the isospin, and invariant effective lagrangian given by the operators,
with the isospin doublets and . In the present model they are combined in such a way that the charge of the outgoing kaon always coincides with the charge of the incoming -meson, as we are interested in processes where the remaining decay product state is neutral and isoscalar (the quantum numbers of the ). In addition to Eq. (II.1), for the decay into we have the following operators,
with denoting or . In principle, one may also have a direct decay through the corresponding operator. However, since we are interested in the –resonant structure, we will not discuss this term in the lagrangian since it only contributes to the background term. The couplings have dimensions and while the other coupling constants are dimensionless.
The general structure of contains two coupling constants and , being consistent with heavy quark symmetry Lag1 (). They provide the contact rescattering. Operators with higher derivatives are regarded as corrections in this model and will be neglected. It is convenient to expand the operator together with the Lagrangian in the explicit form
However, in order to match the non-relativistic effective field theory near threshold (the minimal charm meson model) Lag2 (), one needs . Hence, under this condition the gets the simplified form:
Notice that this contact scattering matrix projects into the flavor structure of the transition. Hence, it accounts only for the local rescattering with the quantum number of the . From now on, we will use all along the article.
ii.2 Amplitude of
Based on the lagrangian in Eqs. (II.1) and (II.1), we extract the decay amplitude . Fig. 1 shows the interaction vertices and Fig. 2 describes Feynmann diagrams for and . In the first line of the Fig. 2, one can observe the final state interaction coming in part from a bubble chain of local scatterings through the operator. In general, every rescattering is produced by two kinds of interactions: contact interaction and exchanges in the –channel. The rescattering effective vertex denoted as (3) in Fig. 1 is given by
where is the momentum of the system, and the mass, provides the local scattering of the meson pairs and provides the precise structure for the various flavor scatterings. For a massive particle, like the X(3872), the Proca propagator has two components,
with and the transverse and longitudinal projection operators, respectively. The longitudinal part happens to be suppressed in the decay by an extra factor near the threshold and will produce a much smaller impact. Thus, no pole will be generated in the longitudinal channel in the neighbourhood of the threshold. Detailed discussion on this point can be found in Appendix A. The effective vertices in the blobs (1) and (4) in Fig. 1 also have contact interaction and exchanges. However, in the effective vertex (2) only exchanges have been taken into account in our model. Possible contact interactions will be treated as background to the spectrum in our later phenomenological analysis.
After taking into account the rescattering effect, the decay amplitude can be separated into transverse and longitudinal components,
where , and are the mass of X(3872), the momentum of K meson and the polarization, respectively, and . The total one-loop contributions are given by and . The factor 2 results from the identity between the and contributions. For instance, the one loop integral is given by
The contributions and have similar structure but with charged masses instead of neutral, having thus a different production threshold MeV. The threshold is placed at MeV, 8 MeV below the charged one. The expressions of and are given in Appendix B. Therein, and are proven to be, respectively, proportional to and near the threshold, with , being here the three-momentum in the center-of-mass rest frame.
There are also other decay channels with much lighter production thresholds, such as , etc, which affect the propagator. Since all such channel thresholds are far away from the one and the energy region under study is a narrow range around the latter, the contributions to the self energies from these channels can be fairly approximated as a constant. In order to account for these absorptive contributions, in the transverse part of the amplitude in Eq. (11) we make the replacement
where the effective parameters and will be determined by our fits to experimental data, and and are the partial widths of the X(3872) from and contribution. We denote the coupling as to distinguish it from the coupling, denoted as . The widths and are
where , , , and , are the mass of , , and the width of , respectively.
The invariant mass spectrum is provided by
where is the invariant mass of the system; is a normalization factor; is a constant, which multiplies the phase space () and models the background contribution.
In the situation, we assume that the meson firstly decays into , and then decays into . The vertices are presented in Fig.1. As explained before, no contact interaction is considered in the present study. It is assumed to be part of the constant background term below.
In the second line of Fig. 2 we show only the final state interaction. As the energy range under study is very close to the threshold, the rescattering dependence on the energy and other channels are accounted through the constant width introduced in the above section in Eq. (13) together with the and contributions therein.
The amplitude is now given by the much involved expression as the following:
where and are the momentum and the polarization of V meson, and is the polarization of . The other symbols are the same as in Eq. (11), and the also needs to be replaced by as before.
An adequate determination of the decay into amplitude can be extracted from the amplitude by inserting the propagator of the particle, the , as discussed in previous sections. The is studied for the decay. Considering the cascade decay , the spectrum is given by PDG2014 ()
where is the normalization constant, parametrizes the background, is the invariant mass and is the pion three-momentum in the rest-frame. The constants and are the mass and width of the vector meson, respectively.
Iii Numerical results and pole analysis
iii.1 Fits to the amplitudes
In above sections we have calculated the and invariant mass spectra, taking into account both the Breit–Wigner particle propagation (elementary X(3872)) and the bubble chain mechanisms. In this section we will carefully study the interfence and competition between the two mechanisms through a numerical fit to data. We anticipate here the prefers the elementary scenario than the molecular one, though the mixed situation (i.e., with both mechanisms involved) may not be excluded.
We perform the following three fits:
Case I: We assume that loops only renormalized the X(3872) self-energy through vertex with coupling constant . There is no contact interaction and in Eqs. (11) and (II.3). This situation implies that there is a pre-existent elementary X(3872), which is not a molecular bound state generated by intermediate state.
Case II: Among the interactions in Fig. 1 only the direct local interaction is taken into account and intermediate exchanges are discarded. That means setting in amplitudes (11) and (II.3), and corresponds to the situation where the bubble loop chains are responsible for the experimentally observed peak, i.e., since line-shape and pole are both related but what generates both is the bubble chain.
In such a situation, the structure of amplitudes takes then the form
where and denote the corresponding numerators in the amplitudes. When there is the propagation (Case I), the contributions from other channels, such as , are taken into account through the constant width in the propagator. In the present situation (Case II) there is no intermediate elementary particle, the contributions from other channels are taken into consideration by shifting the coupling constant to :
where is a real constant which accounts for lower thresholds contributions. The role of is to shift the pole from real axis to the complex plane (i.e., contributes a small width to the bound state). Now the amplitude takes the form:
As near the threshold, the pole in the transverse component is determined by the sign of , which will be discussed in the next subsection.
Case III: We also tried to incorporate both fit I and fit II features by switching on all the interactions in Figs. 1 and 2, allowing both direct contact interactions and intermediate state exchanges in the –channel. However, as we will see later, this does not improve the total with respect to Case I.
In next subsection we will try to examine which of the above scenario is favored by experimental data.
iii.2 Data fitting
Using Eqs. (16) and (18), we proceed now to fit two sets of data and two of data. The two sets of data are: from BELLE BELLE2 () and the from BABAR BABAR4 (). The and are reconstructed from and , respectively. We perform our fits from the threshold up to MeV for BELLE and MeV for BABAR. There are also two data samples (BELLE BELLE4 () and BABAR BABAR3 ()). We fit from MeV up to MeV for BELLE BELLE4 () and from MeV up to 3897.6 MeV for BABAR BABAR3 ().
As explained in the previous subsection, we consider three fit cases: Pure elementary particle (Fit I), where we set (and of course ); Pure molecule picture (Fit II), where and are set to zero; and a mixing of the elementary particle and molecule state (Fit III), where we have all the parameters except in Table 1, with real. Unfortunately, the Fit III was found to be unstable: Since too many parameters are involved, no convergent solution is found with positive error matrix. The total is not meaningfully improved in comparison to Fit I. Hence we will focus mainly on the first two fits and relegate the discussion in the following.
The most important parameters for the X(3872) pole position are the two coupling constants and . The fitting results are presented in Table 1. The (i=1,2,3) and (i=1,2) are normalization constants for the and processes, and the (i=1,2,3) and () parameterize the background contributions for the and data, respectively. Since each spectrum has a different normalization constant , in general it is not really possible to determine the and the couplings , and , independently.
In Table 1, the are obviously larger in Fit II than in Fit I. This is due to the contributions proportional and in Fit I, which are absent in Fit II and have to be compensated by large values of and .
|Fit I||Fit II|