BESIII data show a particular angular distribution for the decay of
and
mesons into
and
hyperons: the angular distribution of the decay
exhibits an opposite trend with respect to the other three channels:
,
and
. We define a model to explain the origin of this phenomenon.
J/ψ and ψ(2S) hadronic decays
effective Lagrangian modelpolarization parameters12.38.Aw13.25.Gv
Article funded by SCOAP^{3}
arxivppt1809.04273Introduction
Since their discovery, charmonia, i.e.,
mesons, have become unique tools for extending our knowledge of strong interaction dynamics at low and medium energies. In the case of lightest charmonia, their decay mechanisms can only be studied by means of effective models, since, due to their lowenergy regime, these processes are beyond the perturbative description of quantum chromodynamics.
We study the decays of
and
mesons into baryonantibaryon pairs
,
. The differential cross section of the process
has the well known parabolic expression in
[1]
where
is the socalled polarization parameter and
is the baryon scattering angle, i.e., the angle between the outgoing baryon and the beam direction in the
centerofmass frame. As pointed out in Ref. [2], only the decay
has a negative polarization parameter
. Figures 1 and 2 show the BESIII data [3] for the angular distribution of the four decays:
,
, and
,
.
(color online) Angular distribution of the baryon for the
decays into
(upper panel) and
(lower panel).
(color online) Angular distribution of the baryon for the
decays into
(upper panel) and
(lower panel).
Amplitudes and branching ratios
The Feynman amplitude for the decay
can be written in terms of the strong magnetic and Dirac form factors as
where the matrix
is defined in Eq. (A2),
is the polarization vector of the
meson, and the fourmomenta follow the labelling of Eq. (A1). The branching ratio (BR) is given by the standard form for the twobody decay
where
is the total width of the
meson. Using the mean value of the modulus squared of the amplitude, written in terms of the Sachs couplings,
we obtain the BR as
Since it does not depend on
, it cannot be used to determine the polarization parameter.
The above expression for BR can be written as the sum of the moduli squared of two amplitudes
where, comparing with Eq. (1),
It follows that the polarization parameter of Eq. (A3) can also be written as
Effective model
The SU(3) baryon octet states can be described in matrix notation as follows [4]
where the first matrix is for baryons and the second for antibaryons. We can consider
and
mesons as SU(3) singlets. In view of the SU(3) symmetry, the zero level Lagrangian density should have the SU(3) invariant form
. Moreover, we consider two sources of SU(3) symmetry breaking: the quark mass and the EM interaction. The first can be parametrized by introducing the spurion matrix [5]
where
is the effective coupling constant. This matrix describes the mass breaking effect due to the mass difference between
and
and
quarks, where the SU(2) isospin symmetry is assumed, so that
. This SU(3) breaking is proportional to the
GellMann matrix
. The EM breaking effect is related to the fact that the photon coupling to quarks, described by the fourcurrent
is proportional to the electric charge. This effect can be parametrized using the following spurion matrix
where
is the effective EM coupling constant.
The most general SU(3) invariant effective Lagrangian density is given by [5]
where
,
,
,
and
are coupling constants. We can extract the Lagrangians describing the
and
decays into
and
where
and
are combinations of coupling constants, i.e.,
Using the structure of Eq. (2), the BRs can be expressed in terms of the electric and magnetic amplitudes as
Moreover, as obtained in Eq. (3), the amplitudes can be further decomposed as combinations of leading,
and
, and subleading terms,
and
, with opposite relative signs, i.e.,
where
and
are the phases of the ratios
and
.
Results
In this work, we have used the data from precise measurements [3, 6] of the branching ratios and polarization parameters, reported in Table 1, based on the events collected with the BESIII detector at the BEPCII collider. These data are in agreement with the results of other experiments [711]. Since for each charmonium state we have six free parameters (four moduli and two relative phases) and only four constrains (two BRs and two polarization parameters), we fix the relative phases
and
. The values
and
appear as phenomenologically favored by the data. Indeed, (largely) different phases would give negative, and hence unphysical, values for the moduli
,
,
and
. Moreover, as shown in Fig. 5, where the four moduli for
and
are represented as functions of the phases with
and
, the obtained results are quite stable, and the central values
,
maximise the hierarchy between the moduli of leading,
and
, and subleading amplitudes,
and
. These values for
,
,
and
are reported in Table 2 and shown in Fig. 3. The corresponding values of
,
are reported in Table 3 and shown in Fig. 4. The large subleading
amplitudes
,
(see Table 2 and Fig. 3) are responsible for the inversion of the
,
hierarchy (see Fig. 4 and Table 3).
(color online) Moduli of the parameters from Table 2 as function of the charmonium state mass M.
(color online) Moduli of the parameters from Table 3 as function of the charmonium state mass M.
(color online) Red and black bands represent moduli of leading and subleading amplitudes, respectively. The vertical width indicates the error. Top left: moduli of amplitudes E_{0} and E_{1} for
. Top right: moduli of amplitudes M_{0} and M_{1} for
. Bottom left: moduli of amplitudes E_{0} and E_{1} for
. Bottom right: moduli of amplitudes M_{0} and M_{1} for
.
Branching ratios and polarization parameters from Ref. [3]. The value of
for the decay
is from Ref. [6].
Decay
BR
Pol. par.
Moduli of the leading and subleading amplitudes.
Ampl.
Moduli of the strong Sachs form factors
FFs
Conclusions
The
and
angular distributions can be explained using an effective model with the SU(3)driven Lagrangian
The interplay between the leading
and subleading
contributions to the decay amplitude determines the sign and value of the polarization parameter
.
In particular, the different behavior of the
angular distribution is due to the large values of the subleading amplitudes
and
. This implies that the SU(3) mass breaking and EM effects, which are responsible for these amplitudes, play a different role in the dynamics of the
and
decays.
It is interesting to note that the angular distributions of
and
, measured by BESIII [12, 13], show the same
behavior.
The process
is currently under investigation [14]. The behavior of its angular distribution could add important information to the knowledge of the
decay mechanism.
Appendix A: Production cross section
We consider the decay of a charmonium state, a
vector meson
, produced via
annihilation, into a baryonantibaryon
pair, i.e., the process
where the 4momenta are given in parentheses. The Feynman diagram is shown in Fig. A1 and the corresponding amplitude is
(color online) Feynman diagram of the process
, the red hexagon represents the
coupling.
where
is the baryonic fourcurrent,
is the
propagator, which includes the

electromagnetic (EM) coupling, and
is the leptonic fourcurrent. The fourmomenta follow the labelling of Eq. (A1). The
matrix can be written as [15]
where
is the baryon mass and
and
are constant form factors that we call “strong” Dirac and Pauli couplings; they weigh the vector and tensor parts of the
vertex^{①)}
When the nonconstant matrix is introduced to describe the EM coupling , the tensor term contains also the anomalous magnetic moment, that, in this case where parametrises the strong vertex , has been embodied in the strong Pauli coupling.
. We introduce the strong electric and magnetic Sachs couplings [16]
that have the structure of the EM Sachs form factors [17].
is the mass of the charmonium state. The four quantities
,
,
and
are in general complex numbers. The differential cross section of the process
in the
centerofmass frame, in terms of the two Sachs couplings, reads
where
is the velocity of the outgoing baryon at the
mass,
is the scattering angle, and the polarization parameter
is given by
and depends only on the modulus of the ratio
, e.g., Fig. A2 shows the behavior of
in the case of
as function of
. The strong Sachs and Dirac and Pauli couplings are related through
. Let us consider three special cases. With maximum positive polarization,
, the strong electric Sachs coupling vanishes, i.e.,
(color online) Polarization parameter
for
as function of the ratio
. The masses are from Ref. [18].
the relative phase between
and
is
, and the ratio of the moduli is
.
With maximum negative polarization we have
so that in this case the strong magnetic Sachs coupling vanishes, the relative phase between
and
is
and the ratio of the moduli is one.
Finally, in the case with no polarization,
, we obtain the modulus of the ratio between the Sachs couplings