and substructures from , radiative and semi-leptonic decays
Using an improved “analytic -matrix model”, we reconsider the extraction of the and widths from scatterings data of Crystal Ball and Belle. Our main results are summarized in Tables 3 and 4. The averaged “direct width” to is 0.16(3) keV which confirms a previous result of MNO () and which does neither favour a large four-quark / molecule nor a pure components. The “direct width” of the of 0.28(2) keV is much larger than the four-quark expectation but can be compatible with a or a gluonium component. We also found that the rescattering part of each amplitude is relatively large indicating an important contribution of the meson loops in the determinations of the and total widths. This is mainly due to the large couplings of the and to and/or , which can also be due to a light scalar gluonium with large OZI violating couplings but not necessary to a four-quark or molecule state. Our average results for the total (direct+rescattering) widths: are comparable with the ones from dispersion relations and PDG values. Using the parameters from QCD spectral sum rules, we complete our analysis by showing that the production rates of unmixed scalar gluonia and G (1.5-1.6) agree with the data from , radiative and semi-leptonic decays.
keywords:and scatterings, radiative decays, light scalar mesons, gluonia and four-quark states, QCD spectral sum rules and low-energy theorems.
In previous series of papers MNO (); KMN (); MNW (), we have used an improved version of the K-matrix model originally proposed in MENES () for studying the hadronic and couplings of the meson 111Some other applications of the model have been discussed in PEAN (); LAYSSAC (). . We found that the “direct” coupling of the to is more compatible with a large gluon component in its wave function rather than with a (too large width) or four-quark (too small width). More recently, we have extended the analysis for studying the hadronic couplings of the and mesons KMN (); MNW (). We found an unexpected relatively large coupling of the to : 222Analogous values have been obtained in MENES (); OTHERS (); ZHENG09 () and in Fits 9 and 10 of ACHASOV06 () though not favoured by ACHASOV06 ()., which disfavours its large molecule and four-quark components, while the large coupling of to : , excludes its pure content. These phenomenological observations go in lines with the fact that, in the channel, the gluon component is expected to play an essential rôle through the scalar QCD anomaly (dilaton) VENEZIA (); NSVZ (); SNG0 (); SN06 (); LANIK (); SCHEC (); CHANO (); OCHS (); FRASCA (); ARRIOLA () which manifests through the trace of the QCD energy momentum tensor:
where is the quark mass anomalous dimension; is the quark field; is the Gell-Mann-Low QCD -function and is the gluon field strength.
In this paper, we pursue the test of the nature of these scalar mesons by studying their couplings to inside the energy region below 1.4 GeV, where new data on from BELLE BELLE () is available in addition to the old data of Crystal Ball CBALL () and MARK II MARK2 ().
2 The analytic K-matrix model for
In so doing, we shall work with a specific analytic K-matrix model originally introduced by the authors in Ref. MENES (), where one can separate the direct and rescattering couplings,
which is not always feasible using dispersion relations.
In this approach, the strong processes are described by a K-matrix model representing the amplitudes by a set of resonance poles and where the dispersion relations in the multi-channel case can be solved explicitly. The model can be reproduced by a set of Feynman diagrams, which are easily interpreted within the Effective Lagrangian approach, including resonance (bare) couplings to and and (in the original model MENES ()) 4-point and interaction vertices which we have omitted for simplicity in MNO () and here. A subclass of bubble pion loop diagrams including resonance poles in the s-channel are resummed (unitarized Born). In a previous work MNW (), we have discussed the approach for the case of : 1 channel 0 “bare” resonance (so-called model), 1 channel 1 “bare” resonance (K-matrix pole) and 2 channels 2 “bare” resonances and we have restricted to the symmetric shape function. We have introduced a real analytic form factor shape function, which takes explicitly into account left-handed cut singularities for the strong interaction amplitude, and which allows a more flexible parametrisation of the data. In our low energy approach, the shape function can be conveniently approximated by 333Here and in the following is negative and is opposite in sign with the one used in our previous works MNO (); KMN (); MNW ().:
which multiplies the scalar meson couplings to . In this form, the shape function allows for an Adler zero at and a pole for simulating the left hand cut.
1 channel 1 “bare” resonance
Let’s first illustrate the method in this simple case. The unitary amplitude is then written as:
where with ; are the bare coupling squared and :
with: below and above threshold . The “physical” couplings are defined from the residues, with the normalization:
The amplitude near the pole where and is:
The real part of is obtained from a dispersion relation with subtraction at and one obtains:
with: –, is the residue of at and: where: from MENES ().
Generalization to 2 channels 2 “bare” resonances
3 The process
Expression and normalization of the amplitudes
The amplitude of mass can be written in terms of the invariants 444We use the same normalization as MENES ().:
with . Helicity and amplitudes are denoted by F and G, which, in terms of partial wave amplitudes, read respectively:
and are the usual -functions normalized as in PDG PDG (), while is the scattering angle between and , which can be expressed in terms of s and t as:
For unpolarized photons, the cross section reads:
where the integration should be done from 0 to 1 for the neutral and from -1 to 1 for the charged cases. In the following analysis, we find convenient to express the charged and neutral amplitudes in terms of the and isospin ones 555We use the same convention as MENES () where is opposite in sign with OLLER08 ().:
corresponding to the following states:
The example of one pion exchange for
The expression of the Born term amplitude due to one pion exchange reads 666Here and in the following, we use the same normalization as in MENES () and we use the gauge conditions: .:
from which one can deduce the helicity amplitudes:
Amplitudes for 1 channel 1 resonance below 0.7 GeV
– The isospin I=0 channel: starting from the S wave amplitude in Eq. (3), we derive the amplitude for the electromagnetic process for isospin as:
Here the contribution from the Born term of is given by as defined in MENES (), a real analytic function in the plane with left cut . The function represents rescattering; it is regular for but has a right cut for with:
which vanishes at . With this definition the Watson theorem is fulfilled, i.e. the phase of is the same as the one of the elastic amplitude in Eq. (3). The real part is derived from a dispersion relation with subtraction at for to satisfy the Thomson limit, and has a representation similar to the one in Eq. (7), but by replacing by . The function is defined as below Eq. (7) but with replaced by everywhere. It vanishes at and is regular at this point. Finally, the polynomial reflects the ambiguity from the dispersion relations and is set here to . It represents the direct coupling of the resonance to . The residues at the pole of the rescattering and direct contributions to in Eq. (16), respectively, are obtained as:
from which one can deduce the branching ratio:
– The isospin I=2 channel: similarly, we parametrize the -wave amplitude by introducing the shape function :
where and: These amplitudes are again both subtracted at as in case of and one finds in analogy:
where: ; are the residues of at and is defined as in Eq. (22) but with replaced by . The cross sections for the and scattering processes are obtained from the previous expressions of the amplitudes.
Results of the analysis
4 Extension of the analysis below 1.09 GeV
In this paper, we extend the previous analysis by including and loops and work in the region just above the threshold (the minimal of our fit is obtained for GeV), where the contributions are dominant.
S-waves masses and hadronic couplings
In this case with 2 resonances 2 channels, the hadronic masses and couplings of the and and the corresponding values of the “bare parameters” of the model are given in Tables 2 and 3 of MNW () which we shall use in the extraction of their couplings. The values of the masses are (in units of MeV):
and the ratios of the hadronic couplings are:
D-wave mass and hadronic couplings
In the energy region where we shall work below 1.09 GeV, the -wave contribution can be also important. However, an accurate parametrization of the -wave contribution is not available. Assuming that it is dominated by the , we extract its complex pole position and hadronic couplings and the corresponding “bare parameters” of the model from the fit of the phase shift measured in FP () and HYAMS73 (). In so doing, we parametrize, as in MENES (), the D-wave scattering amplitudes:
where is a pole on the negative real axis to simulate left hand singularity, and:
The function satisfies:
with the phase-space function. The real part of can be obtained from dispersion relation:
where the last poles are adjusted to cancel the poles at and 777Note the extra factor 1/2 in compared to the one in MENES ().:
The resulting values of the bare parameters are given in Table 1, from which we derive the pole position in the 2nd sheet and the residues of 888There is also pole GeV in the 3rd sheet.. Compared with the PDG data 999Notice that the data of the inelasticity are not quite good which induces a relatively bad .:
one can notice that the pole position and the width are well reproduced.
The vector meson contributions
In the energy-region where we shall work, exchange of vector mesons in the t-channel can become important. We introduce their couplings to via the effective interaction:
where (resp. ) are the vector (resp. electromagnetic) field strengths; is the pion field. The coupling is normalized as:
Using the standard vector form of the vector propagator, the Born contribution to the amplitude due to the vector meson exchange is:
from which we deduce the helicity amplitudes:
The axial-vector contributions
The axial-vector contribution
We describe the in the same way as the meson 101010A tensor formulation of the axial-vector meson has been proposed in DERAFAEL () where the form of the propagator differs from the standard one. If we use this propagator, we reproduce the expression of the amplitude given in OLLER08 () where an extra contact term is added in Eq. (37). We shall see in our analysis that the presence of this term would decrease the strength of the direct coupling of the scalar resonance but the total = direct+rescattering contribution remains almost unchanged. This term might be absorbed by some other counter terms of the complete effective lagrangian. , where is simply replaced by , which can either be determined from the width or from the ChPT coupling constants DERAFAEL (); OLLER08 ():
where is the pion decay constant.
The size of the different radiative couplings
These can extracted from the data and using symmetry relations and read in units of GeV:
5 K-matrix model analysis of below 1.09 GeV
The Born and unitarized -wave amplitudes
The Born and unitarized terms can be calculated unambiguously using the effective lagrangians. Taking into account the t-channel exchange of pion, vector and axial-vector mesons discussed in the previous section and shown in Fig. 2, the Born and unitarized parts of the amplitude given in Eq. (16) generalize to (normalized to ):
where the values of are in Eq. (40), are Clebsh-Gordan coefficients for projecting exchanges on the s-channel amplitudes:
The partial -wave Born amplitudes read :