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Since their discovery, charmonia, i.e.,
c¯c 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 low-energy regime, these processes are beyond the perturbative description of quantum chromodynamics.We study the decays of
J/ψ andψ(2S) mesons into baryon-antibaryon pairsB¯B=Λ¯Λ ,Σ0¯Σ0 . The differential cross section of the processe+e−→ψ→B¯B has the well known parabolic expression incosθ [1]dNdcosθ∝1+αBcos2θ,
where
αB is the so-called polarization parameter andθ is the baryon scattering angle, i.e., the angle between the outgoing baryon and the beam direction in thee+e− center-of-mass frame. As pointed out in Ref. [2], only the decayJ/ψ→Σ0¯Σ0 has a negative polarization parameterαB . Figures 1 and 2 show the BESIII data [3] for the angular distribution of the four decays:J/ψ→Λ¯Λ ,J/ψ→Σ0¯Σ0 , andψ(2S)→Λ¯Λ ,ψ(2S)→Σ0¯Σ0 . -
The Feynman amplitude for the decay
ψ→B¯B can be written in terms of the strong magnetic and Dirac form factors asMψ→B¯B=−iϵμψ¯u(p1)Γμv(p2)
where the matrix
Γμ is defined in Eq. (A2),ϵμψ is the polarization vector of theψ meson, and the four-momenta follow the labelling of Eq. (A1). The branching ratio (BR) is given by the standard form for the two-body decayBψ→B¯B=1Γψ18π¯|Mψ→B¯B|2|→p1|M2ψ,
where
Γψ is the total width of theψ meson. Using the mean value of the modulus squared of the amplitude, written in terms of the Sachs couplings,¯|Mψ→BˉB|2=43M2ψ(|gBM|2+2M2BM2ψ|gBE|2).
we obtain the BR as
Bψ→B¯B=Mψβ12πΓψ(|gBM|2+2M2BM2ψ|gBE|2).
(1) Since it does not depend on
αB , 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
Bψ→B¯B=|ABM|2+|ABE|2,
(2) where, comparing with Eq. (1),
ABM=√Mψβ12πΓψgBM, ABE=√Mψβ6πΓψMBMψgBE.
It follows that the polarization parameter of Eq. (A3) can also be written as
αB=1−2|ABE|2/|ABM|21+2|ABE|2/|ABM|2.
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The SU(3) baryon octet states can be described in matrix notation as follows [4]
OB=(Λ/√6+Σ0/√2Σ+pΣ−Λ/√6−Σ0/√2nΞ−Ξ0−2Λ/√6),
OˉB=(ˉΛ/√6+ˉΣ0/√2ˉΣ+ˉΞ+ˉΣ−ˉΛ/√6−ˉΣ0/√2ˉΞ0ˉpˉn−2ˉΛ/√6),
where the first matrix is for baryons and the second for antibaryons. We can consider
J/ψ andψ(2S) mesons as SU(3) singlets. In view of the SU(3) symmetry, the zero level Lagrangian density should have the SU(3) invariant formL0∝ Tr(B¯B) . 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]Sm=gm3(10001000−2),
where
gm is the effective coupling constant. This matrix describes the mass breaking effect due to the mass difference betweens andu andd quarks, where the SU(2) isospin symmetry is assumed, so thatmu=md . This SU(3) breaking is proportional to the8th Gell-Mann matrixλ8 . The EM breaking effect is related to the fact that the photon coupling to quarks, described by the four-current12¯qγμ(λ3+λ8/√3)q,
is proportional to the electric charge. This effect can be parametrized using the following spurion matrix
Se=ge3(2000−1000−1),
where
ge is the effective EM coupling constant.The most general SU(3) invariant effective Lagrangian density is given by [5]
L=gTr(OOˉB)+dTr({O,OˉB}Se)+fTr([OOˉB]Se)+d′Tr({O,OˉB}Sm)+f′Tr([O,OˉB]Sm),
where
g ,d ,f ,d′ andf′ are coupling constants. We can extract the Lagrangians describing theJ/ψ andψ(2S) decays intoΛ¯Λ andΣ0¯Σ0 LΣ0ˉΣ0=(G0+G1)Σ0ˉΣ0,LΛˉΛ=(G0−G1)ΛˉΛ,
(3) where
G0 andG1 are combinations of coupling constants, i.e.,G0=g,G1=d3(2gm+ge).
Using the structure of Eq. (2), the BRs can be expressed in terms of the electric and magnetic amplitudes as
Bψ→Σ0ˉΣ0=|AΣE|2+|AΣM|2,Bψ→ΛˉΛ=|AΛE|2+|AΛM|2.
Moreover, as obtained in Eq. (3), the amplitudes can be further decomposed as combinations of leading,
E0 andM0 , and sub-leading terms,E1 andM1 , with opposite relative signs, i.e.,Bψ→Σ0ˉΣ0=|E0+E1|2+|M0+M1|2=|E0|2+|E1|2+2|E0||E1|cos(ρE)+|M0|2+|M1|2+2|M0||M1|cos(ρM),Bψ→ΛˉΛ=|E0−E1|2+|M0−M1|2=|E0|2+|E1|2−2|E0||E1|cos(ρE)+|M0|2+|M1|2−2|M0||M1|cos(ρM),
where
ρE andρM are the phases of the ratiosE0/E1 andM0/M1 . -
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 [7-11]. 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
ρE andρM . The valuesρE=0 andρM=π appear as phenomenologically favored by the data. Indeed, (largely) different phases would give negative, and hence unphysical, values for the moduli|E0| ,|E1| ,|M0| and|M1| . Moreover, as shown in Fig. 5, where the four moduli forJ/ψ andψ(2S) are represented as functions of the phases withρE∈[−π/2,π/2] andρM∈[π/2,3π/2] , the obtained results are quite stable, and the central valuesρE=0 ,ρM=π maximise the hierarchy between the moduli of leading,E0 andM0 , and sub-leading amplitudes,E1 andM1 . These values for|E0| ,|E1| ,|M0| and|M1| are reported in Table 2 and shown in Fig. 3. The corresponding values of|gE| ,|gM| are reported in Table 3 and shown in Fig. 4. The large sub-leadingJ/ψ amplitudes|E1| ,|M1| (see Table 2 and Fig. 3) are responsible for the inversion of the|gBE| ,|gBM| hierarchy (see Fig. 4 and Table 3).Figure 3. (color online) Moduli of the parameters from Table 2 as function of the charmonium state mass M.
Figure 4. (color online) Moduli of the parameters from Table 3 as function of the charmonium state mass M.
Figure 5. (color online) Red and black bands represent moduli of leading and sub-leading amplitudes, respectively. The vertical width indicates the error. Top left: moduli of amplitudes E0 and E1 for
J/ψ . Top right: moduli of amplitudes M0 and M1 forJ/ψ . Bottom left: moduli of amplitudes E0 and E1 forψ(2S) . Bottom right: moduli of amplitudes M0 and M1 forψ(2S) .Decay BR Pol. par. αB J/ψ→Σ0¯Σ0 (11.64±0.04)×10−4 −0.449±0.020 J/ψ→Λ¯Λ (19.43±0.03)×10−4 0.461±0.009 ψ(2S)→Σ0¯Σ0 (2.44±0.03)×10−4 0.71±0.11 ψ(2S)→Λ¯Λ (3.97±0.03)×10−4 0.824±0.074 Ampl. J/ψ ψ(2S) |E0| (2.16±0.02)×10−2 (0.42±0.07)×10−2 |E1| (0.42±0.02)×10−2 (0.03±0.05)×10−2 |M0| (3.15±0.02)×10−2 (1.72±0.02)×10−2 |M1| (0.90±0.02)×10−2 (0.23±0.02)×10−2 Table 2. Moduli of the leading and sub-leading amplitudes.
FFs J/ψ ψ(2S) |gΣE| (1.99±0.04)×10−3 (0.6±0.1)×10−3 |gΣM| (0.94±0.02)×10−3 (0.94±0.02)×10−3 |gΛE| (1.37±0.04)×10−3 (0.6±0.1)×10−3 |gΛM| (1.64±0.03)×10−3 (1.20±0.02)×10−3 Table 3. Moduli of the strong Sachs form factors
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The
Λ andΣ0 angular distributions can be explained using an effective model with the SU(3)-driven LagrangianLΣ0¯Σ0+Λ¯Λ=(G0+G1)Σ0¯Σ0+(G0−G1)Λ¯Λ.
The interplay between the leading
G0 and sub-leadingG1 contributions to the decay amplitude determines the sign and value of the polarization parameterαB .In particular, the different behavior of the
J/ψ→Σ0¯Σ0 angular distribution is due to the large values of the sub-leading amplitudes|E1| and|M1| . 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 theJ/ψ andψ(2S) decays.It is interesting to note that the angular distributions of
Σ0(1385) andΣ±(1385) , measured by BESIII [12, 13], show the sameΣ0 behavior.The process
e+e−→J/ψ→Σ+¯Σ− is currently under investigation [14]. The behavior of its angular distribution could add important information to the knowledge of theJ/ψ decay mechanism. -
We consider the decay of a charmonium state, a
c¯c vector mesonψ , produced viae+e− annihilation, into a baryon-antibaryonB¯B pair, i.e., the process\scriptsizee−(k1)+e+(k2)→ψ(q)→B(p1)+¯B(p2),
where the 4-momenta are given in parentheses. The Feynman diagram is shown in Fig. A1 and the corresponding amplitude is
Figure A1. (color online) Feynman diagram of the process
e+e−→ψ→BˉB , the red hexagon represents theψBˉB coupling.Me+e−→ψ→B¯B=−ie2JμB Dψ(q2)¯v(k2)γμu(k1),
where
JμB=¯u(p1)Γμv(p2) is the baryonic four-current,Dψ(q2) is theψ propagator, which includes theγ -ψ electromagnetic (EM) coupling, and¯v(k2)γμu(k1) is the leptonic four-current. The four-momenta follow the labelling of Eq. (A1). TheΓμ matrix can be written as [15]Γμ=γμfB1+iσμνqν2MBfB2,(A2)
where
MB is the baryon mass andfB1 andfB2 are constant form factors that we call “strong” Dirac and Pauli couplings; they weigh the vector and tensor parts of theψB¯B vertex①). We introduce the strong electric and magnetic Sachs couplings [16]gBE=fB1+M2ψ4M2BfB2,gBM=fB1+fB2,
that have the structure of the EM Sachs form factors [17].
Mψ is the mass of the charmonium state. The four quantitiesfB1 ,fB2 ,gBE andgBM are in general complex numbers. The differential cross section of the processe+e−→ψ→B¯B in thee+e− center-of-mass frame, in terms of the two Sachs couplings, readsdσdcosθ=πα2β2M2ψ(|gBM|2+4M2BM2ψ|gBE|2)(1+αBcos2θ),
where
β=√1−4M2BM2ψ
is the velocity of the out-going baryon at the
ψ mass,θ is the scattering angle, and the polarization parameterαB is given byαB=M2ψ|gBM|2−4M2B|gBE|2M2ψ|gBM|2+4M2B|gBE|2,(A3)
αB∈[−1,1] and depends only on the modulus of the ratiogBE/gBM , e.g., Fig. A2 shows the behavior ofαΛ in the case ofψ=J/ψ as function of|gΛE|/|gΛM| . The strong Sachs and Dirac and Pauli couplings are related throughαB . Let us consider three special cases. With maximum positive polarization,αB=1 , the strong electric Sachs coupling vanishes, i.e.,Figure A2. (color online) Polarization parameter
αΛ forψ=J/ψ as function of the ratio|gΛE|/|gΛM| . The masses are from Ref. [18].αB=1 → gBE=0, fB1=−M2ψ4M2BfB2,gBM=fB1(1−4M2BM2ψ)=fB2(1−M2ψ4M2B),
the relative phase between
fB1 andfB2 isiπ , and the ratio of the moduli isM2ψ/(4M2B) .With maximum negative polarization we have
αB=−1 → gBM=0, fB1=−fB2,
gBE=fB1(1−M2ψ4M2B)=fB2(M2ψ4M2B−1),
so that in this case the strong magnetic Sachs coupling vanishes, the relative phase between
fB1 andfB2 is−iπ and the ratio of the moduli is one.Finally, in the case with no polarization,
αB=0 , we obtain the modulus of the ratio between the Sachs couplingsαB=0 → |gBE||gBM|=Mψ2MB.
A model to explain the angular distribution of J/ψ and ψ(2S) decay into Λ¯Λ and Σ0¯Σ0
- Received Date: 2018-10-23
- Available Online: 2019-02-01
Abstract: BESIII data show a particular angular distribution for the decay of