A model to explain the angular distribution of J/ψ and ψ(2S) decay into Λ¯Λ and Σ0¯Σ0

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M. Alekseev, A. Amoroso, R. Baldini Ferroli, I. Balossino, M. Bertani, D. Bettoni, F. Bianchi, J. Chai, G. Cibinetto, F. Cossio, F. De Mori, M. Destefanis, R. Farinelli, L. Fava, G. Felici, I. Garzia, M. Greco, L. Lavezzi, C. Leng, M. Maggiora, A. Mangoni, S. Marcello, G. Mezzadri, S. Pacetti, P. Patteri, A. Rivetti, M. Da Rocha Rolo, M. Savrié, S. Sosio, S. Spataro and L. Yan. A model to explain the angular distribution of J/ψ and ψ(2S) decay into Λ¯Λ and Σ0¯Σ0 [J]. Chinese Physics C, 2019, 43(2): 1-1. doi: 10.1088/1674-1137/43/2/023103
M. Alekseev, A. Amoroso, R. Baldini Ferroli, I. Balossino, M. Bertani, D. Bettoni, F. Bianchi, J. Chai, G. Cibinetto, F. Cossio, F. De Mori, M. Destefanis, R. Farinelli, L. Fava, G. Felici, I. Garzia, M. Greco, L. Lavezzi, C. Leng, M. Maggiora, A. Mangoni, S. Marcello, G. Mezzadri, S. Pacetti, P. Patteri, A. Rivetti, M. Da Rocha Rolo, M. Savrié, S. Sosio, S. Spataro and L. Yan. A model to explain the angular distribution of J/ψ and ψ(2S) decay into Λ¯Λ and Σ0¯Σ0 [J]. Chinese Physics C, 2019, 43(2): 1-1.  doi: 10.1088/1674-1137/43/2/023103 shu
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Received: 2018-10-23
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A model to explain the angular distribution of J/ψ and ψ(2S) decay into Λ¯Λ and Σ0¯Σ0

  • 1. Università di Torino, I-10125, Torino, Italy
  • 2. INFN Sezione di Torino, I-10125, Torino, Italy
  • 3. INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy
  • 4. INFN Sezione di Ferrara, I-44122, Ferrara, Italy
  • 5. Institute of High Energy Physics, Beijing 100049, People's Republic of China
  • 6. Università di Ferrara, I-44122, Ferrara, Italy
  • 7. Università del Piemonte Orientale, I-15121, Alessandria, Italy
  • 8. Università di Perugia, I-06100, Perugia, Italy
  • 9. INFN Sezione di Perugia, I-06100, Perugia, Italy

Abstract: BESIII data show a particular angular distribution for the decay of J/ψ and ψ(2S) mesons into Λ¯Λ and Σ0¯Σ0 hyperons: the angular distribution of the decay ψ(2S)Σ0¯Σ0 exhibits an opposite trend with respect to the other three channels: J/ψΛ¯Λ , J/ψΣ0¯Σ0 and ψ(2S)Λ¯Λ . We define a model to explain the origin of this phenomenon.

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    1.   Introduction
    • 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 and ψ(2S) mesons into baryon-antibaryon pairs B¯B=Λ¯Λ , Σ0¯Σ0 . The differential cross section of the process e+eψB¯B has the well known parabolic expression in cosθ [1]

      dNdcosθ1+αBcos2θ,

      where αB is the so-called polarization parameter and θ is the baryon scattering angle, i.e., the angle between the outgoing baryon and the beam direction in the e+e center-of-mass frame. As pointed out in Ref. [2], only the decay J/ψΣ0¯Σ0 has a negative polarization parameter αB . Figures 1 and 2 show the BESIII data [3] for the angular distribution of the four decays: J/ψΛ¯Λ , J/ψΣ0¯Σ0 , and ψ(2S)Λ¯Λ , ψ(2S)Σ0¯Σ0 .

      Figure 1.  (color online) Angular distribution of the baryon for the J/ψ decays into ΛˉΛ (upper panel) and Σ0ˉΣ0 (lower panel).

      Figure 2.  (color online) Angular distribution of the baryon for the ψ(2S) decays into ΛˉΛ (upper panel) and Σ0ˉΣ0 (lower panel).

    2.   Amplitudes and branching ratios
    • The Feynman amplitude for the decay ψB¯B can be written in terms of the strong magnetic and Dirac form factors as

      MψB¯B=iϵμψ¯u(p1)Γμv(p2)

      where the matrix Γμ is defined in Eq. (A2), ϵμψ is the polarization vector of the ψ 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 decay

      BψB¯B=1Γψ18π¯|MψB¯B|2|p1|M2ψ,

      where Γψ is the total width of the ψ meson. Using the mean value of the modulus squared of the amplitude, written in terms of the Sachs couplings,

      ¯|MψBˉB|2=43M2ψ(|gBM|2+2M2BM2ψ|gBE|2).

      we obtain the BR as

      BψB¯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

      BψB¯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=12|ABE|2/|ABM|21+2|ABE|2/|ABM|2.

    3.   Effective model
    • The SU(3) baryon octet states can be described in matrix notation as follows [4]

      OB=(Λ/6+Σ0/2Σ+pΣΛ/6Σ0/2nΞΞ02Λ/6),

      OˉB=(ˉΛ/6+ˉΣ0/2ˉΣ+ˉΞ+ˉΣˉΛ/6ˉΣ0/2ˉΞ0ˉpˉn2ˉΛ/6),

      where the first matrix is for baryons and the second for antibaryons. We can consider J/ψ and ψ(2S) mesons as SU(3) singlets. In view of the SU(3) symmetry, the zero level Lagrangian density should have the SU(3) invariant form L0 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(100010002),

      where gm is the effective coupling constant. This matrix describes the mass breaking effect due to the mass difference between s and u and d quarks, where the SU(2) isospin symmetry is assumed, so that mu=md . This SU(3) breaking is proportional to the 8th Gell-Mann matrix λ8 . The EM breaking effect is related to the fact that the photon coupling to quarks, described by the four-current

      12¯qγμ(λ3+λ8/3)q,

      is proportional to the electric charge. This effect can be parametrized using the following spurion matrix

      Se=ge3(200010001),

      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)+dTr({O,OˉB}Sm)+fTr([O,OˉB]Sm),

      where g , d , f , d and f are coupling constants. We can extract the Lagrangians describing the J/ψ and ψ(2S) decays into Λ¯Λ and Σ0¯Σ0

      LΣ0ˉΣ0=(G0+G1)Σ0ˉΣ0,LΛˉΛ=(G0G1)ΛˉΛ,

      (3)

      where G0 and G1 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

      BψΣ0ˉΣ0=|AΣE|2+|AΣM|2,BψΛˉΛ=|AΛE|2+|AΛM|2.

      Moreover, as obtained in Eq. (3), the amplitudes can be further decomposed as combinations of leading, E0 and M0 , and sub-leading terms, E1 and M1 , with opposite relative signs, i.e.,

      BψΣ0ˉΣ0=|E0+E1|2+|M0+M1|2=|E0|2+|E1|2+2|E0||E1|cos(ρE)+|M0|2+|M1|2+2|M0||M1|cos(ρM),BψΛˉΛ=|E0E1|2+|M0M1|2=|E0|2+|E1|22|E0||E1|cos(ρE)+|M0|2+|M1|22|M0||M1|cos(ρM),

      where ρE and ρM are the phases of the ratios E0/E1 and M0/M1 .

    4.   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 [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 and ρM . The values ρE=0 and ρM=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 for J/ψ and ψ(2S) are represented as functions of the phases with ρE[π/2,π/2] and ρM[π/2,3π/2] , the obtained results are quite stable, and the central values ρE=0 , ρM=π maximise the hierarchy between the moduli of leading, E0 and M0 , and sub-leading amplitudes, E1 and M1 . 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-leading J/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 for J/ψ . Bottom left: moduli of amplitudes E0 and E1 for ψ(2S) . Bottom right: moduli of amplitudes M0 and M1 for ψ(2S) .

      Decay BR Pol. par. αB
      J/ψΣ0¯Σ0 (11.64±0.04)×104 0.449±0.020
      J/ψΛ¯Λ (19.43±0.03)×104 0.461±0.009
      ψ(2S)Σ0¯Σ0 (2.44±0.03)×104 0.71±0.11
      ψ(2S)Λ¯Λ (3.97±0.03)×104 0.824±0.074

      Table 1.  Branching ratios and polarization parameters from Ref. [3]. The value of αB for the decay J/ψΛ¯Λ is from Ref. [6].

      Ampl. J/ψ ψ(2S)
      |E0| (2.16±0.02)×102 (0.42±0.07)×102
      |E1| (0.42±0.02)×102 (0.03±0.05)×102
      |M0| (3.15±0.02)×102 (1.72±0.02)×102
      |M1| (0.90±0.02)×102 (0.23±0.02)×102

      Table 2.  Moduli of the leading and sub-leading amplitudes.

      FFs J/ψ ψ(2S)
      |gΣE| (1.99±0.04)×103 (0.6±0.1)×103
      |gΣM| (0.94±0.02)×103 (0.94±0.02)×103
      |gΛE| (1.37±0.04)×103 (0.6±0.1)×103
      |gΛM| (1.64±0.03)×103 (1.20±0.02)×103

      Table 3.  Moduli of the strong Sachs form factors

    5.   Conclusions
    • The Λ and Σ0 angular distributions can be explained using an effective model with the SU(3)-driven Lagrangian

      LΣ0¯Σ0+Λ¯Λ=(G0+G1)Σ0¯Σ0+(G0G1)Λ¯Λ.

      The interplay between the leading G0 and sub-leading G1 contributions to the decay amplitude determines the sign and value of the polarization parameter αB .

      In particular, the different behavior of the J/ψΣ0¯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 the J/ψ and ψ(2S) decays.

      It is interesting to note that the angular distributions of Σ0(1385) and Σ±(1385) , measured by BESIII [12, 13], show the same Σ0 behavior.

      The process e+eJ/ψΣ+¯Σ is currently under investigation [14]. The behavior of its angular distribution could add important information to the knowledge of the J/ψ decay mechanism.

    Appendix A: Production cross section
    • We consider the decay of a charmonium state, a c¯c vector meson ψ , produced via e+e annihilation, into a baryon-antibaryon B¯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+eψBˉB , the red hexagon represents the ψBˉB coupling.

      Me+eψB¯B=ie2JμB Dψ(q2)¯v(k2)γμu(k1),

      where JμB=¯u(p1)Γμv(p2) is the baryonic four-current, Dψ(q2) is the ψ propagator, which includes the γ - ψ electromagnetic (EM) coupling, and ¯v(k2)γμu(k1) is the leptonic four-current. The four-momenta follow the labelling of Eq. (A1). The Γμ matrix can be written as [15]

      Γμ=γμfB1+iσμνqν2MBfB2,(A2)

      where MB is the baryon mass and fB1 and fB2 are constant form factors that we call “strong” Dirac and Pauli couplings; they weigh the vector and tensor parts of the ψB¯B vertex). 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 quantities fB1 , fB2 , gBE and gBM are in general complex numbers. The differential cross section of the process e+eψB¯B in the e+e center-of-mass frame, in terms of the two Sachs couplings, reads

      dσdcosθ=πα2β2M2ψ(|gBM|2+4M2BM2ψ|gBE|2)(1+αBcos2θ),

      where

      β=14M2BM2ψ

      is the velocity of the out-going baryon at the ψ mass, θ is the scattering angle, and the polarization parameter αB is given by

      αB=M2ψ|gBM|24M2B|gBE|2M2ψ|gBM|2+4M2B|gBE|2,(A3)

      αB[1,1] and depends only on the modulus of the ratio gBE/gBM , e.g., Fig. A2 shows the behavior of αΛ in the case of ψ=J/ψ as function of |gΛE|/|gΛM| . The strong Sachs and Dirac and Pauli couplings are related through αB . 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 for ψ=J/ψ as function of the ratio |gΛE|/|gΛM| . The masses are from Ref. [18].

      αB=1    gBE=0,  fB1=M2ψ4M2BfB2,gBM=fB1(14M2BM2ψ)=fB2(1M2ψ4M2B),

      the relative phase between fB1 and fB2 is iπ , and the ratio of the moduli is M2ψ/(4M2B) .

      With maximum negative polarization we have

      αB=1    gBM=0,  fB1=fB2,

      gBE=fB1(1M2ψ4M2B)=fB2(M2ψ4M2B1),

      so that in this case the strong magnetic Sachs coupling vanishes, the relative phase between fB1 and fB2 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 couplings

      αB=0    |gBE||gBM|=Mψ2MB.

Reference (18)

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