PAPER • OPEN ACCESS
A model to explain the angular distribution of
and
decay into
and
To cite this article: M. Alekseev et al 2019 Chinese Phys. C 43 023103
J
/ψ
ψ(2S)
ΛΛ
Σ
0Σ
0A model to explain the angular distribution of
and
decay into
and
M. Alekseev1,2 A. Amoroso1,2 R. Baldini Ferroli3 I. Balossino4,5 M. Bertani3 D. Bettoni4 F. Bianchi1,2
J. Chai2 G. Cibinetto4 F. Cossio2 F. De Mori1,2 M. Destefanis1,2 R. Farinelli4,6 L. Fava7,2 G. Felici3
I. Garzia4 M. Greco1,2 L. Lavezzi2,5 C. Leng2 M. Maggiora1,2 A. Mangoni8,9;1) S. Marcello1,2 G. Mezzadri4
S. Pacetti8,9 P. Patteri3 A. Rivetti1,2 M. Da Rocha Rolo1,2 M. Savrié6 S. Sosio1,2 S. Spataro1,2 L. Yan1,2
1Università di Torino, I-10125, Torino, Italy 2INFN Sezione di Torino, I-10125, Torino, Italy 3INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy 4INFN Sezione di Ferrara, I-44122, Ferrara, Italy 5Institute of High Energy Physics, Beijing 100049, People's Republic of China 6Università di Ferrara, I-44122, Ferrara, Italy 7Università del Piemonte Orientale, I-15121, Alessandria, Italy 8Università di Perugia, I-06100, Perugia, Italy 9INFN Sezione di Perugia, I-06100, Perugia, Italy J/ψ ψ(2S ) ΛΛ Σ0Σ0 ψ(2S ) → Σ0Σ0 J/ψ → ΛΛ J/ψ → Σ0Σ0 ψ(2S ) → ΛΛ
Abstract: 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 oth-er three channels: , and . We define a model to explain the origin of this phe-nomenon.
J/ψ ψ(2S )
Keywords: and hadronic decays, effective Lagrangian model, polarization parameters
PACS: 12.38.Aw, 13.25.Gv DOI: 10.1088/1674-1137/43/2/023103
1 Introduction
cc
Since their discovery, charmonia, i.e., mesons, have become unique tools for extending our knowledge of strong interaction dynamics at low and medium ener- gies. In the case of lightest charmonia, their decay mech-anisms 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 chromo-dynamics.
J/ψ ψ(2S ) BB= ΛΛ Σ0Σ0
e+e−→ ψ → BB
cosθ
We study the decays of and mesons into baryon-antibaryon pairs , . The differential cross section of the process has the well known parabolic expression in [1] dN d cosθ∝ 1 + αBcos 2θ, αB θ e+e−
where is the so-called polarization parameter and is the baryon scattering angle, i.e., the angle between the outgoing baryon and the beam direction in the cen-ter-of-mass frame. As pointed out in Ref. [2 ], only the de-J/ψ → Σ0Σ0 α B J/ψ → ΛΛ J/ψ → Σ0Σ0 ψ(2S ) → ΛΛ ψ(2S ) → Σ0Σ0 cay has a negative polarization parameter .
Figures 1 and 2 show the BESIII data [3] for the angular distribution of the four decays: , ,
and , .
2 Amplitudes and branching ratios
ψ → BB
The Feynman amplitude for the decay can be written in terms of the strong magnetic and Dirac form factors as
Mψ→BB= −iϵψµu(p1)Γµv(p2)
Γµ ϵψµ
ψ
where the matrix is defined in Eq. (A2), is the po-larization vector of the 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 de-cay Bψ→BB=Γ1 ψ 1 8π Mψ→BB 2 ⃗p1 Mψ2, Received 23 October 2018, Published online 1) E-mail: alessio.mangoni@pg.infn.it Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must main-tain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pub-lishing Ltd
Γψ ψ
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, Mψ→B ¯B 2= 4 3M 2 ψ |gBM|2+ 2M2B M2ψ |g B E|2 . we obtain the BR as Bψ→BB= Mψβ 12πΓψ |gBM|2+ 2M2B M2ψ |g B E|2 . (1) αB
Since it does not depend on , it cannot be used to de-termine the polarization parameter.
The above expression for BR can be written as the sum of the moduli squared of two amplitudes Bψ→BB= ABM 2+ ABE 2, (2) where, comparing with Eq. (1), ABM= √ Mψβ 12πΓψg B M, ABE= √ Mψβ 6πΓψ MB Mψg B E. It follows that the polarization parameter of Eq. (A3) can also be written as αB= 1− 2|ABE|2/|ABM|2 1+ 2|ABE|2/|AB M|2 .
3 Effective model
The SU(3) baryon octet states can be described in matrix notation as follows [4] OB= Λ/ √ 6+ Σ0/√2 Σ+ p Σ− Λ/√6− Σ0/√2 n Ξ− Ξ0 −2Λ/√6 , OB¯= ¯ Λ/√6+ ¯Σ0/√2 Σ¯+ Ξ¯+ ¯ Σ− Λ/¯ √6− ¯Σ0/√2 Ξ¯0 ¯p ¯n −2 ¯Λ/√6 , J/ψ ψ(2S ) L0∝ Tr(BB)
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 in-teraction. The first can be parametrized by introducing the spurion matrix [5] Sm= gm 3 1 0 0 0 1 0 0 0 −2 , gm s u d mu= md 8th λ 8
where is the effective coupling constant. This matrix describes the mass breaking effect due to the mass differ-ence between and and quarks, where the SU(2) isospin symmetry is assumed, so that . This SU(3) breaking is proportional to the Gell-Mann matrix . The EM breaking effect is related to the fact that the photon coupling to quarks, described by the four-current 0 0.5×105 1.0×105 1.5×105 2.0×105 Number of events 0 0 2.0×104 4.0×104 6.0×104 Number of events J/ψ→ΛΛ cos (θ) −1 −0.5 0.5 1 J/ψ→Σ0Σ0 J/ψ Λ ¯Λ Σ0Σ¯0 Fig. 1. (color online) Angular distribution of the baryon for the decays into (upper panel) and (lower panel).
0 0.5×104 1.0×104 Number of events −1 0 0.2×104 0.4×104 −0.5 0 0.5 1 Number of events ψ(2S)→ΛΛ ψ(2S)→Σ0Σ0 cos (θ) ψ(2S ) Λ ¯Λ Σ0Σ¯0 Fig. 2. (color online) Angular distribution of the baryon for the decays into (upper panel) and (lower panel).
1 2qγ µ(λ 3+ λ8/ √ 3)q,
is proportional to the electric charge. This effect can be parametrized using the following spurion matrix Se= ge 3 2 0 0 0 −1 0 0 0 −1 , ge where is the effective EM coupling constant. The most general SU(3) invariant effective Lagrangi-an density is given by [5]
L =gTr(OOB¯)+ dTr({O,OB¯}Se)+ f Tr([OOB¯]Se)
+ d′Tr({O,O ¯ B}Sm)+ f′Tr([O,OB¯]Sm), g d f d′ f′ J/ψ ψ(2S ) ΛΛ Σ0Σ0
where , , , and are coupling constants. We can extract the Lagrangians describing the and de-cays into and
LΣ0Σ¯0= (G0+G1)Σ0Σ¯0,LΛ ¯Λ= (G0−G1)Λ ¯Λ, (3) G0 G1
where and are combinations of coupling constants, i.e.,
G0= g, G1=
d
3(2gm+ ge).
Using the structure of Eq. (2), the BRs can be ex-pressed in terms of the electric and magnetic amplitudes as Bψ→Σ0Σ¯0= |AΣE|2+ |AΣM|2, Bψ→Λ ¯Λ= |AΛ E|2+ |AΛM|2. E0 M0 E1 M1
Moreover, as obtained in Eq. (3), the amplitudes can be further decomposed as combinations of leading, and , and sub-leading terms, and , with opposite relative signs, i.e.,
Bψ→Σ0Σ¯0= |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→Λ ¯Λ= |E0− E1|2+ |M0− M1|2
= |E0|2+ |E1|2− 2|E0||E1|cos(ρE)
+ |M0|2+ |M1|2− 2|M0||M1|cos(ρM),
ρE ρM E0/E1
M0/M1
where and are the phases of the ratios and .
4 Results
In this work, we have used the data from precise measurements [3, 6 ] of the branching ratios and polariza-tion 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 ex-periments [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 polar-ρE ρM ρE= 0 ρM= π |E0| |E1| |M0| |M1| J/ψ ψ(2S ) ρE∈ [−π/2,π/2] ρM∈ [π/2,3π/2] ρE= 0 ρM= π E0 M0 E1 M1 |E0| |E1| |M0| |M1| |gE| |gM| J/ψ |E1| |M1| ization parameters), we fix the relative phases and . The values and appear as phenomenologic-ally 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 rep-resented as functions of the phases with
and , the obtained results are quite stable, and the central values , maximise the hier-archy between the moduli of leading, and , and sub-leading 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 sub-leading amplitudes , (see Table 2 and Fig. 3
) are respons-10−4 10−3 10−2 10−1 4 Moduli of E0, 1 and M 0, 1 amplitudes |E0| |E1| |M0| |M1| M (GeV) 3 3.25 3.5 3.75 Fig. 3. (color online) Moduli of the parameters from Table 2 as function of the charmonium state mass M. 0 0.001 0.002 0.003 3
Moduli of strong form factors
3.25 3.5 3.75 4 M (GeV) |gΣ M| |gΣE| |gΛE| |gΛM| Fig. 4. (color online) Moduli of the parameters from Table 3 as function of the charmonium state mass M.
|gB E| |g
B M|
ible for the inversion of the , hierarchy (see Fig. 4
and Table 3).
5 Conclusions
Λ Σ0
The and angular distributions can be explained using an effective model with the SU(3)-driven Lagrangi-an
αB J/ψ → ΛΛ
Table 1. Branching ratios and polarization parameters from Ref. [3]. The value of for the decay is from Ref. [6].
Decay BR Pol. par. αB J/ψ → Σ0Σ0 (11.64 ± 0.04) × 10−4 −0.449 ± 0.020 J/ψ → ΛΛ (19.43 ± 0.03) × 10−4 0.461 ± 0.009 ψ(2S ) → Σ0Σ0 (2.44 ± 0.03) × 10−4 0.71 ± 0.11 ψ(2S ) → ΛΛ (3.97 ± 0.03) × 10−4 0.824 ± 0.074 Table 2. Moduli of the leading and sub-leading amplitudes. 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 3. Moduli of the strong Sachs form factors FFs J/ψ ψ(2S ) |gΣ E| (1.99 ± 0.04) × 10−3 (0.6 ± 0.1) × 10−3 |gΣ M| (0.94 ± 0.02) × 10−3 (0.94 ± 0.02) × 10−3 |gΛ E| (1.37 ± 0.04) × 10−3 (0.6 ± 0.1) × 10−3 |gΛ M| (1.64 ± 0.03) × 10−3 (1.20 ± 0.02) × 10−3 0 0.005 0.01 0.015 0.02 0.025 −0.2 0 0.01 0.02 0.03 0.04 0 0.002 0.004 0.006 0 0.005 0.01 0.015 0.02 0.8 |E 0 |, | E1 | of J/ ψ |M 0 |, | M 1 | of J/ ψ |E0 |, | E1 | of ψ (2 S) |M 0 |, | M 1 | of ψ (2 S) ρM/π ρE/π0 0.1 0.2 −0.1 −0.2 −0.1 0 0.1 0.2 ρE/π 0.9 ρM/π1 1.1 1.2 0.8 0.9 1 1.1 1.2 J/ψ J/ψ ψ(2S ) ψ(2S ) Fig. 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 . Top right: moduli of amplitudes M0 and M1 for . Bottom left:
moduli of amplitudes E0 and E1 for . Bottom right: moduli of amplitudes M0 and M1 for .
LΣ0Σ0+ΛΛ= (G0+G1)Σ 0Σ0+ (G 0−G1)ΛΛ. G0 G1 αB
The interplay between the leading and sub-lead-ing contributions to the decay amplitude determines the sign and value of the polarization parameter .
J/ψ → Σ0Σ0
|E1| |M1|
In particular, the different behavior of the
angular distribution is due to the large values of the sub-leading amplitudes and . This implies that the
SU(3) mass breaking and EM effects, which are respons-J/ψ ψ(2S )
ible for these amplitudes, play a different role in the dy-namics of the and decays.
Σ0(1385) Σ±(1385)
Σ0
It is interesting to note that the angular distributions of and , measured by BESIII [12, 13], show the same behavior.
e+e−→ J/ψ → Σ+Σ−
J/ψ
The process is currently under in-vestigation [14]. The behavior of its angular distribution could add important information to the knowledge of the
decay mechanism.
Appendix A: Production cross section
cc
ψ e+e−
BB
We consider the decay of a charmonium state, a vector meson , produced via annihilation, into a baryon-antibaryon
pair, i.e., the process
e−(k1)+ e+(k2)→ ψ(q) → B(p1)+ B(p2), (A1) where the 4-momenta are given in parentheses. The Feynman dia-gram is shown in Fig. A1 and the corresponding amplitude is
Me+e−→ψ→BB= −ie 2Jµ BDψ ( q2)v(k2)γµu(k1), JµB= u(p1)Γµv(p2) Dψ ( q2) ψ γ ψ v(k2)γµu(k1)
where is the baryonic four-current, is the propagator, which includes the - electromagnetic (EM) coupling, and is the leptonic four-current. The four-momenta fol-Γµ low the labelling of Eq. (A1). The matrix can be written as [15] Γµ= γµf1B+iσ µνqν 2MB f B 2, (A2) MB f1B f2B ψBB
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 vertex1). We introduce the
strong electric and magnetic Sachs couplings [16] gBE= f1B+ M2 ψ 4M2 B f2B, gBM= f1B+ f2B, Mψ fB 1 f2B gBE gB M e+e−→ ψ → BB e+e− 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 sec-tion of the process in the center-of-mass frame, in terms of the two Sachs couplings, reads dσ d cosθ= πα2β 2M2 ψ gBM 2 +4MB2 M2 ψ gBE 2 (1+ αBcos2θ), where β = v t 1−4M2B M2 ψ ψ θ αB
is the velocity of the out-going baryon at the mass, is the scat-tering angle, and the polarization parameter is given by αB=M 2 ψ gBM 2 − 4M2B|gB E|2 M2 ψ gBM 2 + 4M2 B gBE 2, (A3) αB∈ [−1,1] gBE/gB M and depends only on the modulus of the ratio ,
αΛ ψ = J/ψ
|gΛ E|/|gΛM|
αB
αB= 1
e.g., Fig. A2 shows the behavior of in the case of as function of . The strong Sachs and Dirac and Pauli coup-lings are related through . Let us consider three special cases. With maximum positive polarization, , the strong electric
e− e+ B B γ c c e+e−→ ψ→ B ¯B ψB ¯B
Fig. A1. (color online) Feynman diagram of the process , the red hexagon represents the coupling.
−1 −0.5 0 0.5 1 0 |gΛE|/|gΛM| αΛ |g Λ E|/| g Λ M| 1 2 3 4 5 αΛ ψ = J/ψ gΛE / gΛM Fig. A2. (color online) Polarization parameter for as function of the ratio . The masses are from Ref. [18].
Γµ γBB
Γµ ψBB
1) When the non-constant 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.
Sachs coupling vanishes, i.e., αB= 1 → gB E= 0, f1B= − M2 ψ 4M2 B fB 2, gBM= f1B 1−4M 2 B M2 ψ = fB 2 1− M 2 ψ 4M2 B , fB 1 f2B iπ M2 ψ/(4M2B)
the relative phase between and is , and the ratio of the moduli is . With maximum negative polarization we have αB= −1 → gB M= 0, f B 1 = − f B 2, gB E= f1B 1− M 2 ψ 4M2 B = fB 2 M 2 ψ 4M2 B − 1 , f1B f2B −iπ
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.
αB= 0
Finally, in the case with no polarization, , we obtain the modulus of the ratio between the Sachs couplings αB= 0 → |g B E| |gB M| =2MBMψ. References S. J. Brodsky and G. P. Lepage, Phys. Rev. D, 24: 2848 (1981) 1
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