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Radiative decay of the Ξ(1620) in a hadronic molecule picture

  • Last year, the Ξ(1620) state, which is cataloged in the Particle Data Group (PDG) with only one star, was reported again in the Ξπ+ final state by the Belle Collaboration. Its properties, such as the spectroscopy and decay width, cannot be simply explained in the context of conventional constituent quark models. This inspires an active discussion on the structure of this resonance. In this paper, we study the radiative decays of the newly observed Ξ(1620) assuming that it is a meson-baryon molecular state of ΛˉK and ΣˉK with spin-parity JP=1/2 developed in our previous study. The partial decay widths of the ΛˉKΣˉK molecular state into Ξγ and Ξπγ final states through hadronic loops are evaluated using effective Lagrangians. The partial widths for Ξ(1620)0γΞ is evaluated to be approximately 118.76174.21 keV, which may be accessible for the LHCb experiment. If Ξ(1620) is a ΛˉKΣˉK molecule, the radiative transition strength Ξ(1620)0γˉKΛ is considerably small and the decay width is of the order of 0.01 eV. Future experimental measurements of these processes can be useful to test the molecule interpretations of the Ξ(1620).
  • Searching for hadrons beyond the constituent quark model is one of the most important topics in the hadron physics community. In the conventional quark model, a hadron is composed of qˉq as a meson or qqq as a baryon. It is natural to expect the existence of hadrons composed of more quarks, which are called exotic states. Because of the significant experimental progress over the past twenty years, more exotic states have been observed [1]. Last year, the Ξ(1620) state, which is cataloged in the Particle Data Group (PDG) [1] with only one star, was reported again in the Ξπ+ final state by the Belle Collaboration [2]. The observed resonance parameters of the structure are

    M=1610.4±6.0(stat)+5.93.5(syst)MeV,Γ=59.9±4.8(stat)+2.83.0(syst)MeV,

    (1)

    which are consistent with the earlier measured values [3, 4]. However, its spin-parity remains undetermined.

    From the observed decay model, Ξ(1620) is a conventional baryon composed of uss or dss. However, its properties, such as the spectroscopy and decay width, cannot be simply explained in the context of conventional constituent quark models [5-8]. As indicated in Refs. [9-16], Ξ(1620) can be understood as a molecular state in comparison with the Belle data [2]. A molecular state with a narrow width and a mass of approximately 1606 MeV was predicted in the unitarized coupled channels approach [9-12]. However, a Ξ bound state with a mass of approximately 1620 MeV and JP=1/2 was predicted by studying the ˉKΛ interaction in the framework of the one-boson-exchange (OBE) model [15]. Additionally, Ref. [15] considered Ξ(1620) as a pure molecular state composed of ˉKΛ component. A possible explanation for these results of Refs. [9-12, 15] is that Ξ(1620) has a larger ˉKΛ component [14, 16]. Moreover, the ˉKΣ component cannot be underestimated to reproduce the total decay width of Ξ(1620) and a study of only the spectroscopy does not provide a complete description of its nature [16].

    From the above discussion, Ξ(1620) may be a molecular state. However, currently, we cannot fully exclude other possible explanations such as a mixture of three-quark and five-quark components (as long as quantum numbers permit, this might be the reality). Further research is required to distinguish whether it is a molecular or compact multi-quark state. The photon coupling with a quark is significantly different from the coupling of the photon to the constituent ˉKΛ and ˉKΣ of Ξ(1620) [17]. Hence, a precise measurement of the radiative decays is useful to test different interpretations of Ξ(1620). In this paper, we study the radiative decay of Ξ(1620) using the hadronic molecule approach developed in our previous study [16].

    The remainder of the paper is organized as follows. The theoretical formalism is explained in Sec. II. The predicted partial decay widths are presented in Sec. III, followed by a short summary in the final section.

    In our previous study [16], the total decay width of Ξ(1620) was reproduced with the assumption that Ξ(1620) is an S-wave ˉKΛˉKΣ bound state with JP=1/2. Based on the molecular scenario, the radiative decay widths Ξ(1620)γΞ, Ξ(1620)γˉKΛ, and Ξ(1620)γπΞ are studied to elucidate the internal structure of the Ξ(1620) state. To calculate the radiation decay, we first employ the Weinberg compositeness rule to determine Ξ(1620) couplings to its constituents ˉKY (YΛ,Σ). Radiative decays occurs from the exchange of a suitable hadron between the ˉKY pair, which then transforms into Ξγ,γˉKΛ, and Ξπγ. The corresponding Feynman diagrams are shown in Fig. 1 and Fig. 2.

    Figure 1

    Figure 1.  (color online) Feynman diagrams of the Ξ0γΞ0 decay processes. We also indicate the definitions of the kinematics (p,k1,k2,p1,p2, and q) used in the calculation.

    Figure 2

    Figure 2.  (color online) Feynman diagrams of the Ξ0γΞ0π0, γΞπ+, and γˉKΛ decay processes. We also indicate the definitions of the kinematics (p,k1,k2,p1,p2,p3, and q) used in the calculation.

    To compute the diagrams shown in Figs. 1-2, we require the effective Lagrangian densities for the relevant interaction vertices. For the Ξ(1620)ˉKY vertices, the Lagrangian densities can be expressed as [16, 18, 19]

    LΞ(1620)(x)=gΞ(1620)ˉKYd4yΦ(y2)ˉK(x+ωYy)×Y(xωˉKy)ˉΞ(1620)(x),

    (2)

    where ωˉK=mˉK/(mˉK+mY) and ωY=mY/(mˉK+mY). For an isovector baryon Σ, Y should be replaced with Yτ, where τ is the isospin matrix. Φ(y2) is an effective correlation function that is introduced to describe the distribution of the constituents ˉK and Y in the hadronic molecular Ξ(1620) state, which is often selected to be of the following form [16, 18-26]:

    Φ(p2E)exp(p2E/β2),

    (3)

    where pE is the Euclidean Jacobi momentum and β is the size parameter that characterizes the distribution of the components inthe molecule. Currently, the value of β=1.0 is determined by experimental data [16, 18-26] (and references therein).

    The coupling constant gΞ(1620)ˉKY is determined by the compositeness condition [16, 18-26], which implies that the renormalization constant of the bound state wave function Ξ(1620) is set to zero

    ZΞ(1620)=xˉKΣ+xˉKΛdΣΞ(1620)dk0|k0=mΞ(1620)=0,

    (4)

    where xAB is the probability of the Ξ(1620) being in the hadronic state AB with normalization xˉKΣ+xˉKΛ=1.0. ΣΞ(1620) is the self-energy of the Ξ(1620) and can be computed using the Feynmann diagrams shown in Fig. 3.

    Figure 3

    Figure 3.  (color online) Self-energy of the Ξ(1620) state.

    ΣΞ(1620)(k0)=Y=Λ,Σ0,Σ+(CY)2g2Ξ(1620)ˉKY0dα0dη×(Δ2zk0+mY)16π2z2exp{1β2[(2ω2Yη+Δ24z)k20+αm2Y+ηm2ˉK]}.

    (5)

    where z=2+α+η and Δ=4ωY2η. k20=m2Ξ(1620) with k0,mΞ(1620) denoting the four-momenta and mass of the Ξ(1620), respectively; k1, mˉK, and mY are the four-momenta, mass of the ˉK meson, and mass of the Y baryon, respectively. Here, we set mΞ(1620)=mY+mˉKEb with Eb being the binding energy of Ξ(1620). Isospin symmetry implies that

    CY={1Y=Λ1/3Y=Σ02/3Y=Σ+  .

    (6)

    To estimate the radiative decays of the diagrams shown in Figs. 1-2, we require the effective Lagrangian densities related to the photon fields, which are [27, 28]

    LγΣΣ=ˉΣ[eΣAeκΣ2mNσμννAμ]Σ,

    (7)

    LγΛΛ=eκΛ2mNˉΛσμννAμΛ,

    (8)

    LγΣΛ=eμΣΛ2mNˉΣ0σμννAμΛ,

    (9)

    LKKγ=gK+K+γ4eϵμναβFμνK+αβK+gK0K0γ4eϵμναβFμνK0αβˉK0+h.c.,

    (10)

    LKKγ=ieAμKμK+,

    (11)

    where the strength tensors are defined as σμν= i2(γμγνγνγμ), Fμν=μAννAμ, and Kμν=μKννKμ.MN is the mass of p, and α=e2/4π=1/137 is the electromagnetic fine structure constant. The anomalous and transition magnetic moments of the baryons are provided by the PDG [1] and are shown in Table 1.

    Table 1

    Table 1.  Anomalous and transition magnetic moments.
    κΣ=0.16 κΣ0=0.65 κΣ+=1.46
    κΛ=0.61 μΣΛ=1.61
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    The coupling constants gK+K+γ and gK0K0γ are introduced to obtain consistent results with the experimental measurements of K+K+γ and K0K0γ. The theoretical decay widths of K+K+γ and K0K0γ are

    Γ(K+K+γ)=αg2K+K+γ24mK+(m2K+m2K+),

    (12)

    Γ(K0K0γ)=αg2K0K0γ24mK0(m2K0m2K0).

    (13)

    According to the experimental widths Γ(K+K+γ)=0.0503 keV [1] and Γ(K0K0γ)=0.125 keV [1], the coupling constant gKKγ is fixed as

    gK+K+γ=0.580GeV1,gK0K0γ=0.904GeV1,

    (14)

    and the signs of these coupling constants are fixed according to the quark model [29, 30].

    In addition to the Lagrangians above, the meson-baryon interactions are also required and can be obtained from the following chiral Lagrangians [31, 32]

    LVBB=g(ˉBγμ[Vμ,B]+ˉBγμBVμ),

    (15)

    LPBB=F2ˉBγμγ5[uμ,B]+D2ˉBγμγ5{uμ,B},

    (16)

    LPBPB=i4f2ˉBγμ[(PμPμPP)BB(PμPμPP)]

    (17)

    where g=4.64, F=0.51, D=0.75 [16, 31, 33]; at the lowest order, uμ=2μP/f with f=93 MeV;... denotes the trace in the flavor space; B, P, and Vμ are the SU(3) pseudoscalar meson, vector meson, and baryon octet matrices, respectively, which are

    B=(12Σ0+16ΛΣ+pΣ12Σ0+16ΛnΞΞ026Λ),

    (18)

    P=(π02+η6π+K+ππ02+η6K0KˉK026η),

    (19)

    Vμ=(12(ρ0+ω)ρ+K+ρ12(ρ0+ω)K0KˉK0ϕ)μ.

    (20)

    Putting all the pieces together, we obtain the decay amplitudes that are shown in Appendix A. The total amplitudes of the Ξ0Ξ0γ, Ξ0Ξ0π0γ, and Ξ0π+Ξγ are the sum of these individual amplitudes, respectively:

    MT(Ξ0Ξγ)=i=a,b,c,d,e,fMi(Ξ0Ξγ),

    (21)

    MT(Ξ0πΞγ)=i=a,b,c,dMi(Ξ0πΞγ),

    (22)

    MT(Ξ0ˉKΛγ)=i=e.fMi(Ξ0ˉKΛγ).

    (23)

    When the amplitudes are determined, the corresponding partial decay width can be easily obtained, which is expressed as

    dΓ(Ξ(1620)0γΞ)=12J+1132π2|p1|m2Ξ0¯|M|2dΩ,

    (24)

    dΓ(Ξ(1620)0γΞπ,γˉKΛ)=12J+11(2π)5116m2ׯ|M|2|p3||p2|dm13dΩp3dΩp2.

    (25)

    The complete explanation of calculating these two equations is available in Refs. [1, 34]. J is the total angular momentum of the Ξ(1620); |p1| is the three-momenta of the decay products in the center of mass frame, and the overline indicates the sum over the polarization vectors of the final hadrons; (p3,Ωp3) is the momentum and angle of the particle π in the rest frame of π and Ξ; Ωp2 is the angle of the photon in the rest frame of the decaying particle; m13 is the invariant mass for π and Ξ, and mπ+mΞm13m.

    Using the values obtained above, the radiative decay width of the Ξ(1620)0 into Ξ0γ, Ξγπ, and γˉKΛ that are shown in Fig. 1 and Fig. 2 can be calculated. However, the amplitudes of the Ξ(1620)0γΞ0 and Ξ(1620)0γΞπ cannot satisfy the gauge invariance of the photon field. To ensure the gauge invariance of the total amplitudes, the contact diagram must be included. The corresponding Feynman diagrams are shown in Fig. 4. For the this calculation, we adopt the following form to satisfy pμ2MTotalμ(MT+Mcom)=0

    Figure 4

    Figure 4.  (color online) Contact diagram for Ξ0Ξ0γ, Ξ0Ξ0π0γ, and Ξ0π+Ξγ. We also indicate definitions of the kinematics (p1,p2,p3,k1,k2, and p) used in the calculation.

    Macom(ΞγΞ0)=D+F2feCYgΞΣ+K0dα0dη0dζ116π2y2β2ˉu(p1)(Cμ1T1+Cμ2T2)γ5u(p)ϵμ(p2),

    (26)

    Mbcom(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞKΣ+4f2{1210dα0dη0dζ116π2y2β2×ˉu(p1)(Dμ1T1+Dμ2T2)u(p)ϵμ(p2),

    (27)

    where

    T1=exp{1β2[α(p21+m2K)+ηm2Σ++ζ(p22+m2Σ+)(p1ωΣ+p2ωK)2+14z(H2p2H1p1)2]},

    (28)

    T2=exp{1β2[α(p22+m2K)+ηm2K+ζ(p21+m2Σ+)(p2ωΣ+p1ωK)2+14z(H2p1H1p2)2]},

    (29)

    Cμ1=m21H318y3(2pμ1m1γμ)m31H214y2γμ+m21H212y2pμ1m1m2Σ+H12yγμ+m1m2Σ+γμ+m1mΣ+H212y2pμ1+((m+m1)mΣ+H1H22y2m1mΣ+H1y)pμ1mΣ+m1yγμ+H21H24y3p1p2(2pμ1m1γμ)+m1H1H22y2p1p2γμ+(m+m1)m1H1H22y2pμ1+3H12y2(2pμ1m1γμ),

    (30)

    Cμ2=m21H324y3pμ1m21H224y2pμ1(m+m1)mΣ+H1H22y2pμ1(m+m1)mΣ+H12ypμ1m1mΣ+H222y2pμ1m1mΣ+H22ypμ1+mΣ+yγμ+H1H222y3p1p2pμ1+H1H22y2p1p2pμ1m1(m+m1)H222y2pμ1m1(m+m1)H22ypμ1+H22y2(2pμ1m1γμ)+m1H22y2γμ+2H2y2pμ1+2ypμ1,

    (31)

    Dμ1=(m2Σ+H12y+m2Σ+)(2pμmγμ)mΣ+H212y2(mp2)pμ+mΣ+H1H22y2p2pμ+mΣ+H1y(mp2)pμ+mΣ+yγμmm213H318y3γμ+m(m2m213)H21H28y3γμm213H214y2(2pμmγμ)+H21m2132y2pμ+(H1H2(m2m213)4y2+1y)×(2pμmγμ)mH1H22y2p2pμ+3mH12y2γμ,

    (32)

    Dμ2=mΣ+H1H22y2p2pμ+mΣ+H222y2(mp2)pμmΣ+yγμ+(m2m213)H1H224y3pμmH1H22y2p2pμm213H324y3pμ+m213H222y2pμ+3H2y2pμmyγμ,

    (33)

    with y=1+α+η+ζ,H2=2(ζ+ωK),H1=2(α+ωΣ+), and m213=(p1+p3)2. m and m1 are the masses of the Ξ and Ξ, respectively.

    To compute the radiative decay widths of the considered processes, the coupling constants of Ξ(1620) to its components can be estimated using the compositeness conditions provided by Eq. (4). xΛˉK in Eq. (4) is the probability of Ξ(1620) being in the hadronic state ΛˉK. The value of the xΛˉK is in the a range of 0.0-1.0. If xΛˉK= 0.0 or 1.0, Ξ(1620) is a pure molecular state containing the components ΣˉK and ΛˉK, respectively. Otherwise Ξ(1620) is a ˉKΛ-ˉKΣ molecular state. Currently, the value of xΛˉK cannot be accurately determined from first principles; therefore, it should better be determined using experimental data. As the free parameter, xΛˉK=0.520.68 is fixed by fitting the experimental decay data on possible ˉKΛ-ˉKΣ bound states within the same theoretical framework adopted in Ref. [16]. In this region, the total decay width for this state is predicted to be approximately 50.3968.79 MeV, which is comparable with the experimental data [2].

    Considering Ξ(1620) loosely as an S-wave ˉKΛ-ˉKΣ hadronic molecule, the coupling constants gΞ(1620)ˉKΛ and gΞ(1620)ˉKΣ that are dependent on the parameter xˉKΛ are plotted in Fig. 5. We observe that the coupling constant gΞ(1620)ˉKΛ monotonously increases with increasing xˉKΛ in our considered xˉKΛ range, while the coupling constant gΞ(1620)ˉKΣ decreases with increasing xˉKΛ. The opposite trend can be easily understood, as the coupling constants gΞ(1620)ˉKΛ and gΞ(1620)ˉKΣ are directly proportional to the corresponding molecular compositions [23].

    Figure 5

    Figure 5.  (color online) xˉKΛ dependence of gΞ(1620)ˉKΛ and gΞ(1620)ˉKΣ.

    With the obtained total amplitude, the radiative decay width of Ξ(1620)0 into Ξ0γ can be calculated. The dependence of the corresponding radiative decay width on xˉKΛ is depicted in Fig. 6. For the xˉKΛ value within a reasonable range from 0.52 to 0.68, the radiative decay width of the Ξ(1620)0γΞ0 contribution from the ˉKΣ channel monotonously decreases. However, it increases for the ˉKΛ channel contribution to the Ξ(1620)0γΞ0. Moreover, the ˉKΣ component provides the dominant contribution to the partial decay width of the γΞ0 two-body channel. The ˉKΛ contribution to the γΞ0 two-body channel is very small. This is different from our results in Ref. [16] in that the ˉKΛ component provides the dominant contribution to the strong decay width of the Ξ(1620). A possible explanation for this may be that the interaction between the Σ baryon and photon is stronger than the γΛ interaction since the Σ0 decays completely to the final state containing the Λ baryon and γ [1].

    Figure 6

    Figure 6.  (color online) Decomposed contributions to the decay width of the Ξ(1620)0 into Ξ0γ.

    Figure 6 also indicates that the total radiative decay width decreases for Ξ(1620)0γΞ0 when xˉKΛ is changed from 0.52 to 0.68. We also observe that the interference between the ˉKΣ and ˉKΛ channels is considerably small, resulting in a total decay width of Ξ(1620)0γΞ0 being primarily contributed by the ˉKΣ channel. This does not alter the conclusion that the ˉKΛ channel strongly couples to the ˉKΣ channel [9, 16]. The main reason for this is that the radiative decay widths are often in the keV regime and are significantly lower than their strong counterparts. Indeed, the total radiative decay width for Ξ(1620)0γΞ0 is predicted to be 118.76174.21 KeV, which is significantly lower than the total decay width, which is predicted to be approximately 50.39-68.79 MeV [16].

    The individual contributions of K, ˉK, Σ, Λ exchanges, and contact term for the reaction Ξ(1620)0γΞ0 are shown in Fig. 7. The amplitudes corresponding to the K-exchange and Σ+-exchange are not gauge invariant, while the remainder are gauge invariant. We can observe that the contract term and K-exchange provide a dominant contribution to the total decay width, and is at last sixty orders of magnitude larger than those of the amplitudes corresponding to the ˉK, Σ, and Λ exchanges for the studied xˉKΛ range.

    Figure 7

    Figure 7.  (color online) Partial decay widths from K(red dash line), ˉK(cyan dash dot line), Σ (blue dot line), Λ(magenta dash dot dot line), and the remainder is the contact term exchange contribution for the Ξ(1620)0γΞ0 as a function of the parameter xˉKΛ.

    Next, we examine the three-body radiative decays Ξ(1620)0γπΞ and Ξ(1620)0γˉKΛ. The decay widths with xˉKΛ varying from 0.52 to 0.68 for such two transitions are depicted in Fig. 8. Since the phase space is small compared with the two-body radiative decay channel Ξ(1620)0γΞ0, the decay width should be the smallest for the Ξ(1620)0γˉKΛ channel, should be intermediate for the Ξ(1620)0γπΞ channel, and the largest for the Ξ(1620)0γΞ0 channel. Indeed, our study shows that the partial width of Ξ(1620)0γˉKΛ is rather small, weakly increasing as xˉKΛ increases. In particular, the partial width varies from 0.016 to 0.020 eV in the studied xˉKΛ range. However, the partial width of Ξ(1620)0γΞπ decreases as xˉKΛ increases, and the partial width of the Ξ(1620)0γΞπ is estimated to be 68.7558.19 eV.

    Figure 8

    Figure 8.  (color online) Partial decay widths of the Ξ(1620)0γΞπ and Ξ(1620)0γˉKΛ.

    In Fig. 2, the diagrams with pion emitted directly from the intermediated Λ and Σ should be included. Our estimations indicate that in the discussed parameter range, the radiative transition strength for these diagrams are considerably small and the decay width is of the order of approximately 0.1 eV. Moreover, the interferences among them are also rather small. Therefore, the contributions from these channels are not considered in this paper.

    We have studied the two-body and three-body radiative decays of the Ξ(1620) state assuming that it is a bound state of ˉKΛ-ˉKΣ. The coupling of Ξ(1620) to its components are fixed according to the Weinberg compositeness condition. The radiative decays for Ξ(1620)0γΞ0 and Ξ(1620)0γπΞ are obtained via triangle diagrams with exchanges of a pseudoscalar meson K, vector meson K, and baryons Σ and Λ. The three-body decay for the Ξ(1620)0γˉKΛ occur at the tree level. In the relevant parameter region, the partial widths are evaluated as

    Γ(Ξ(1620)0γΞ0)=118.76174.21keV,Γ(Ξ(1620)0γΞπ)=58.1968.75eV,Γ(Ξ(1620)0γˉKΛ)=0.0160.020eV.

    (34)

    Our calculation indicates the partial widths for Ξ(1620)0γπΞ and Ξ(1620)0γˉKΛ are too small to be observed, while for Ξ(1620)0γΞ0, the partial width can reach up to 174 keV at Ec.m.=1.620 GeV. Experimentally, with the current integrated luminosity that the LHCb experiment has accumulated at 1.620 GeV, the process Ξ(1620)0γΞ0 may be searched with only γΞ0 identified. Such research can also be conducted in the forthcoming Belle II experiment.

    Using the Lagrangians in Section II, the amplitudes for the diagrams shown in Figs. 1-2 can be easily obtained:

    Ma(Ξ0Ξ0γ)=i(i)3{(D3F)κΛ23,(F+D)μΣΛ2}egΞΛˉK4mNfd4q(2π)4Φ[(k1ωΛk2ωˉK0)2]×ˉu(p1)k1γ5q+mYq2m2Y(γμp2p2γμ)k2+mΛk22m2Λu(p)1k21m2ˉK0ϵμ(p2),

    (A1)

    Mb(Ξ0Ξ0γ)=i(i)3eD+F2fCYgΞΣ+ˉKd4q(2π)4Φ[(k1ωΣ+k2ωˉK)2]ˉu(p1)k1γ5×q+mYq2m2Y[γμ+κY4mN(γμp2p2γμ)]k2+mΣ+k22m2Σ+u(p)1k21m2Kϵμ(p2),

    (A2)

    Mc(Ξ0Ξ0γ)=i(i)3{(D3F)μΣΛ23,(F+D)κΣ02}egΞΛˉKCY4mNfd4q(2π)4Φ[(k1ωΣ0k2ωˉK0)2]×ˉu(p1)k1γ5q+mYq2m2Y(γμp2p2γμ)k2+mΣ0k22m2Σ0u(p)1k21m2ˉK0ϵμ(p2),

    (A3)

    MKd(Ξ0Ξ0γ)=i(i)3D+F2feCYgΞΣ+Kd4q(2π)4Φ[(k1ωΣ+k2ωK)2]ˉu(p1)qγ5×k2+mΣ+k22m2Σ+u(p)1k21m2K1q2m2K+(qk1)ϵ(p2),

    (A4)

    MKd(Ξ0Ξ0γ)=(i)3egCYgKKγ4d4q(2π)4Φ[(k1ωΣ+k2ωK)2]ˉu(p1)γρk2+mΣ+k22m2Σ+u(p)×1k21m2Kgρσ+qρqσ/m2Kq2m2Kϵμναβ(p2μgνηp2νgμη)(qαgβσqβgασ)ϵη(p2),

    (A5)

    Me(Ξ0Ξ0γ)=(i)36egCYgKKγ8d4q(2π)4Φ[(k1ωΛk2ωˉK0)2]ˉu(p1)γρk2+mΛk22m2Λu(p)×1k21m2ˉK0gρσ+qρqσ/m2ˉK0q2m2ˉK0ϵμναβ(p2μgνηp2νgμη)(qαgβσqβgασ)ϵη(p2),

    (A6)

    Mf(Ξ0Ξ0γ)=(i)3egCYgKKγ42d4q(2π)4Φ[(k1ωΣ0k2ωˉK0)2]ˉu(p1)γρk2+mΣ0k22m2Σ0u(p)×1k21m2ˉK0gρσ+qρqσ/m2ˉK0q2m2ˉK0ϵμναβ(p2μgνηp2νgμη)(qαgβσqβgασ)ϵη(p2),

    (A7)

    where {A, and B} are Λ and Σ baryon exchanges, respectively. The amplitudes of the Ξ0(1620)γπΞ,γˉKΛ can be also easily obtained:

    Ma(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞˉK0Λ32mNf2{{3κΛ,μΣΛ}{6κΛ,2μΣΛ}d4q(2π)4Φ[(k1ωΛk2ωˉK0)2]ˉu(p1)k1q+mYq2+m2Y×(γμp2p2γμ)k2+mΛk22m2Λu(p)ϵμ1k21m2ˉK0,

    (A8)

    Mb(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞKΣ+4f2{121d4q(2π)4Φ[(k1ωΣ+k2ωK)2]ˉu(p1)k1q+mYq2+m2Y×[γμ+κΣ+4mN(γμp2p2γμ)]k2+mΣ+k22m2Σ+u(p)ϵμ1k21m2K,

    (A9)

    Mc(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞKΣ+4f2{121d4q(2π)4Φ[(k1ωΣ+k2ωK)2]ˉu(p1)qk2+mΣ+k22+m2Σ+u(p)×(qμ+kμ1)1k21m2K1q2m2Kϵμ,

    (A10)

    Md(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞˉK0Σ032mNf2{{3μΣΛ,κΣ0}{6μΣΛ,2κΣ0}d4q(2π)4Φ[(k1ωΣ0k2ωˉK0)2]ˉu(p1)k1q+mYq2+m2Y×(γμp2p2γμ)k2+mΣ0k22m2Σ0u(p)ϵμ1k21m2ˉK0,

    (A11)

    Me(Ξ0ˉK0Λγ)=ieκΛgΞˉK0ΛCY4mNΦ[(k1ωΛk2ωˉK0)2]ˉu(p2)(γμp1p1γμ)k2+mΛk22m2Λu(p)ϵμ(p1),

    (A12)

    Mf(Ξ0ˉK0Λγ)=ieμΣΛgΞˉK0Σ0CY4mNΦ[(k1ωΣ0k2ωˉK0)2]ˉu(p2)(γμp1p1γμ)k2+mΣ0k22m2Σ0u(p)ϵμ(p1)

    (A13)

    where the expressions in the curly brackets, {AB, are for Ξ0π0Ξ0γ and Ξ0π+Ξγ, respectively.

    [1] P. A. Zyla et al. (Particle Data Group), Review of Particle Physics, PTEP 2020, 083C01 (2020)
    [2] M. Sumihama et al. (Belle Collaboration), Phys. Rev. Lett. 122, 072501 (2019) doi: 10.1103/PhysRevLett.122.072501
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    [4] E. Briefel et al., Phys. Rev. D 16, 2706 (1977) doi: 10.1103/PhysRevD.16.2706
    [5] S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986)
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    [7] W. H. Blask, U. Bohn, M. G. Huber et al., Z. Phys. A 337, 327 (1990)
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    [10] C. Garcia-Recio, M. F. M. Lutz, and J. Nieves, Phys. Lett. B 582, 49 (2004) doi: 10.1016/j.physletb.2003.11.073
    [11] D. Gamermann, C. Garcia-Recio, J. Nieves et al., Phys. Rev. D 84, 056017 (2011) doi: 10.1103/PhysRevD.84.056017
    [12] K. Miyahara, T. Hyodo, M. Oka et al., Phys. Rev. C 95, 035212 (2017) doi: 10.1103/PhysRevC.95.035212
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    [18] Y. Huang, C. j. Xiao, L. S. Geng et al., Phys. Rev. D 99, 014008 (2019) doi: 10.1103/PhysRevD.99.014008
    [19] Y. Dong, A. Faessler, T. Gutsche et al., Phys. Rev. D 81, 074011 (2010) doi: 10.1103/PhysRevD.81.074011
    [20] Y. Huang, M. Z. Liu, Y. W. Pan et al., Phys. Rev. D 101, 014022 (2020) doi: 10.1103/PhysRevD.101.014022
    [21] A. Faessler, T. Gutsche, V. E. Lyubovitskij et al., Phys. Rev. D 76, 114008 (2007) doi: 10.1103/PhysRevD.76.114008
    [22] Y. Dong, A. Faessler, T. Gutsche et al., Phys. Rev. D 79, 094013 (2009) doi: 10.1103/PhysRevD.79.094013
    [23] Y. Dong, A. Faessler, T. Gutsche et al., J. Phys. G 38, 015001 (2011) doi: 10.1088/0954-3899/38/1/015001
    [24] Y. Dong, A. Faessler, and V. E. Lyubovitskij, Prog. Part. Nucl. Phys. 94, 282 (2017) doi: 10.1016/j.ppnp.2017.01.002
    [25] C. J. Xiao, Y. Huang, Y. B. Dong et al., Phys. Rev. D 100, 014022 (2019) doi: 10.1103/PhysRevD.100.014022
    [26] Y. Huang, C. j. Xiao, Q. F. Lü et al., Phys. Rev. D 97, 094013 (2018) doi: 10.1103/PhysRevD.97.094013
    [27] S. H. Kim, S. i. Nam, A. Hosaka et al., Phys. Rev. D 88, 054012 (2013) doi: 10.1103/PhysRevD.88.054012
    [28] S. H. Kim and H. C. Kim, Phys. Lett. B 786, 156 (2018) doi: 10.1016/j.physletb.2018.09.041
    [29] F.E.Close, Academic Press/london 1979, 481p
    [30] H. Garcilazo and E. Moya de Guerra, Nucl. Phys. A 562, 521-568 (1993) doi: 10.1016/0375-9474(93)90129-L
    [31] E. J. Garzon and E. Oset, Eur. Phys. J. A 48, 5 (2012) doi: 10.1140/epja/i2012-12005-x
    [32] E. Oset and A. Ramos, Nucl. Phys. A 635, 99-120 (1998) doi: 10.1016/S0375-9474(98)00170-5
    [33] B. Borasoy, Baryon axial currents, Phys. Rev. D 59, 054021 (1999) doi: 10.1103/PhysRevD.59.054021
    [34] J.D. Jackson in High Energy Physics, Les Houches 1965 Summer School, Gordon and Breach Science Publishers (1965), p. 348
  • [1] P. A. Zyla et al. (Particle Data Group), Review of Particle Physics, PTEP 2020, 083C01 (2020)
    [2] M. Sumihama et al. (Belle Collaboration), Phys. Rev. Lett. 122, 072501 (2019) doi: 10.1103/PhysRevLett.122.072501
    [3] R. T. Ross, T. Buran, J. L. Lloyd et al., Phys. Lett. B 38, 177 (1972)
    [4] E. Briefel et al., Phys. Rev. D 16, 2706 (1977) doi: 10.1103/PhysRevD.16.2706
    [5] S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986)
    [6] S. Capstick and N. Isgur, AIP Conf. Proc. 132, 267 (1985) doi: 10.1063/1.35361
    [7] W. H. Blask, U. Bohn, M. G. Huber et al., Z. Phys. A 337, 327 (1990)
    [8] Y. I. Azimov, R. A. Arndt, I. I. Strakovsky et al., Phys. Rev. C 68, 045204 (2003) doi: 10.1103/PhysRevC.68.045204
    [9] A. Ramos, E. Oset, and C. Bennhold, Phys. Rev. Lett. 89, 252001 (2002) doi: 10.1103/PhysRevLett.89.252001
    [10] C. Garcia-Recio, M. F. M. Lutz, and J. Nieves, Phys. Lett. B 582, 49 (2004) doi: 10.1016/j.physletb.2003.11.073
    [11] D. Gamermann, C. Garcia-Recio, J. Nieves et al., Phys. Rev. D 84, 056017 (2011) doi: 10.1103/PhysRevD.84.056017
    [12] K. Miyahara, T. Hyodo, M. Oka et al., Phys. Rev. C 95, 035212 (2017) doi: 10.1103/PhysRevC.95.035212
    [13] Y. Oh, Phys. Rev. D 75, 074002 (2007) doi: 10.1103/PhysRevD.75.074002
    [14] Z. Y. Wang, J. J. Qi, J. Xu et al., Eur. Phys. J. C 79, 640 (2019) doi: 10.1140/epjc/s10052-019-7135-3
    [15] K. Chen, R. Chen, Z. F. Sun et al., Phys. Rev. D 100, 074006 (2019) doi: 10.1103/PhysRevD.100.074006
    [16] Y. Huang and L. Geng, Eur. Phys. J. C 80, 837 (2020) doi: 10.1140/epjc/s10052-020-8421-9
    [17] R. Koniuk and N. Isgur, Phys. Rev. D 21, 1868 (1980); Erratum: Phys. Rev. D 23, 818 (1981)
    [18] Y. Huang, C. j. Xiao, L. S. Geng et al., Phys. Rev. D 99, 014008 (2019) doi: 10.1103/PhysRevD.99.014008
    [19] Y. Dong, A. Faessler, T. Gutsche et al., Phys. Rev. D 81, 074011 (2010) doi: 10.1103/PhysRevD.81.074011
    [20] Y. Huang, M. Z. Liu, Y. W. Pan et al., Phys. Rev. D 101, 014022 (2020) doi: 10.1103/PhysRevD.101.014022
    [21] A. Faessler, T. Gutsche, V. E. Lyubovitskij et al., Phys. Rev. D 76, 114008 (2007) doi: 10.1103/PhysRevD.76.114008
    [22] Y. Dong, A. Faessler, T. Gutsche et al., Phys. Rev. D 79, 094013 (2009) doi: 10.1103/PhysRevD.79.094013
    [23] Y. Dong, A. Faessler, T. Gutsche et al., J. Phys. G 38, 015001 (2011) doi: 10.1088/0954-3899/38/1/015001
    [24] Y. Dong, A. Faessler, and V. E. Lyubovitskij, Prog. Part. Nucl. Phys. 94, 282 (2017) doi: 10.1016/j.ppnp.2017.01.002
    [25] C. J. Xiao, Y. Huang, Y. B. Dong et al., Phys. Rev. D 100, 014022 (2019) doi: 10.1103/PhysRevD.100.014022
    [26] Y. Huang, C. j. Xiao, Q. F. Lü et al., Phys. Rev. D 97, 094013 (2018) doi: 10.1103/PhysRevD.97.094013
    [27] S. H. Kim, S. i. Nam, A. Hosaka et al., Phys. Rev. D 88, 054012 (2013) doi: 10.1103/PhysRevD.88.054012
    [28] S. H. Kim and H. C. Kim, Phys. Lett. B 786, 156 (2018) doi: 10.1016/j.physletb.2018.09.041
    [29] F.E.Close, Academic Press/london 1979, 481p
    [30] H. Garcilazo and E. Moya de Guerra, Nucl. Phys. A 562, 521-568 (1993) doi: 10.1016/0375-9474(93)90129-L
    [31] E. J. Garzon and E. Oset, Eur. Phys. J. A 48, 5 (2012) doi: 10.1140/epja/i2012-12005-x
    [32] E. Oset and A. Ramos, Nucl. Phys. A 635, 99-120 (1998) doi: 10.1016/S0375-9474(98)00170-5
    [33] B. Borasoy, Baryon axial currents, Phys. Rev. D 59, 054021 (1999) doi: 10.1103/PhysRevD.59.054021
    [34] J.D. Jackson in High Energy Physics, Les Houches 1965 Summer School, Gordon and Breach Science Publishers (1965), p. 348
  • 加载中

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Yin Huang, Feng Yang and HongQiang Zhu. Radiative decay of the Ξ(1620) in a hadronic molecule picture[J]. Chinese Physics C. doi: 10.1088/1674-1137/abfd28
Yin Huang, Feng Yang and HongQiang Zhu. Radiative decay of the Ξ(1620) in a hadronic molecule picture[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abfd28 shu
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Radiative decay of the Ξ(1620) in a hadronic molecule picture

    Corresponding author: HongQiang Zhu, 20132013@cqnu.edu.cn
  • 1. School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China
  • 2. College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 401331, China

Abstract: Last year, the Ξ(1620) state, which is cataloged in the Particle Data Group (PDG) with only one star, was reported again in the Ξπ+ final state by the Belle Collaboration. Its properties, such as the spectroscopy and decay width, cannot be simply explained in the context of conventional constituent quark models. This inspires an active discussion on the structure of this resonance. In this paper, we study the radiative decays of the newly observed Ξ(1620) assuming that it is a meson-baryon molecular state of ΛˉK and ΣˉK with spin-parity JP=1/2 developed in our previous study. The partial decay widths of the ΛˉKΣˉK molecular state into Ξγ and Ξπγ final states through hadronic loops are evaluated using effective Lagrangians. The partial widths for Ξ(1620)0γΞ is evaluated to be approximately 118.76174.21 keV, which may be accessible for the LHCb experiment. If Ξ(1620) is a ΛˉKΣˉK molecule, the radiative transition strength Ξ(1620)0γˉKΛ is considerably small and the decay width is of the order of 0.01 eV. Future experimental measurements of these processes can be useful to test the molecule interpretations of the Ξ(1620).

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    I.   INTRODUCTION
    • Searching for hadrons beyond the constituent quark model is one of the most important topics in the hadron physics community. In the conventional quark model, a hadron is composed of qˉq as a meson or qqq as a baryon. It is natural to expect the existence of hadrons composed of more quarks, which are called exotic states. Because of the significant experimental progress over the past twenty years, more exotic states have been observed [1]. Last year, the Ξ(1620) state, which is cataloged in the Particle Data Group (PDG) [1] with only one star, was reported again in the Ξπ+ final state by the Belle Collaboration [2]. The observed resonance parameters of the structure are

      M=1610.4±6.0(stat)+5.93.5(syst)MeV,Γ=59.9±4.8(stat)+2.83.0(syst)MeV,

      (1)

      which are consistent with the earlier measured values [3, 4]. However, its spin-parity remains undetermined.

      From the observed decay model, Ξ(1620) is a conventional baryon composed of uss or dss. However, its properties, such as the spectroscopy and decay width, cannot be simply explained in the context of conventional constituent quark models [5-8]. As indicated in Refs. [9-16], Ξ(1620) can be understood as a molecular state in comparison with the Belle data [2]. A molecular state with a narrow width and a mass of approximately 1606 MeV was predicted in the unitarized coupled channels approach [9-12]. However, a Ξ bound state with a mass of approximately 1620 MeV and JP=1/2 was predicted by studying the ˉKΛ interaction in the framework of the one-boson-exchange (OBE) model [15]. Additionally, Ref. [15] considered Ξ(1620) as a pure molecular state composed of ˉKΛ component. A possible explanation for these results of Refs. [9-12, 15] is that Ξ(1620) has a larger ˉKΛ component [14, 16]. Moreover, the ˉKΣ component cannot be underestimated to reproduce the total decay width of Ξ(1620) and a study of only the spectroscopy does not provide a complete description of its nature [16].

      From the above discussion, Ξ(1620) may be a molecular state. However, currently, we cannot fully exclude other possible explanations such as a mixture of three-quark and five-quark components (as long as quantum numbers permit, this might be the reality). Further research is required to distinguish whether it is a molecular or compact multi-quark state. The photon coupling with a quark is significantly different from the coupling of the photon to the constituent ˉKΛ and ˉKΣ of Ξ(1620) [17]. Hence, a precise measurement of the radiative decays is useful to test different interpretations of Ξ(1620). In this paper, we study the radiative decay of Ξ(1620) using the hadronic molecule approach developed in our previous study [16].

      The remainder of the paper is organized as follows. The theoretical formalism is explained in Sec. II. The predicted partial decay widths are presented in Sec. III, followed by a short summary in the final section.

    II.   THEORETICAL FORMALISM
    • In our previous study [16], the total decay width of Ξ(1620) was reproduced with the assumption that Ξ(1620) is an S-wave ˉKΛˉKΣ bound state with JP=1/2. Based on the molecular scenario, the radiative decay widths Ξ(1620)γΞ, Ξ(1620)γˉKΛ, and Ξ(1620)γπΞ are studied to elucidate the internal structure of the Ξ(1620) state. To calculate the radiation decay, we first employ the Weinberg compositeness rule to determine Ξ(1620) couplings to its constituents ˉKY (YΛ,Σ). Radiative decays occurs from the exchange of a suitable hadron between the ˉKY pair, which then transforms into Ξγ,γˉKΛ, and Ξπγ. The corresponding Feynman diagrams are shown in Fig. 1 and Fig. 2.

      Figure 1.  (color online) Feynman diagrams of the Ξ0γΞ0 decay processes. We also indicate the definitions of the kinematics (p,k1,k2,p1,p2, and q) used in the calculation.

      Figure 2.  (color online) Feynman diagrams of the Ξ0γΞ0π0, γΞπ+, and γˉKΛ decay processes. We also indicate the definitions of the kinematics (p,k1,k2,p1,p2,p3, and q) used in the calculation.

      To compute the diagrams shown in Figs. 1-2, we require the effective Lagrangian densities for the relevant interaction vertices. For the Ξ(1620)ˉKY vertices, the Lagrangian densities can be expressed as [16, 18, 19]

      LΞ(1620)(x)=gΞ(1620)ˉKYd4yΦ(y2)ˉK(x+ωYy)×Y(xωˉKy)ˉΞ(1620)(x),

      (2)

      where ωˉK=mˉK/(mˉK+mY) and ωY=mY/(mˉK+mY). For an isovector baryon Σ, Y should be replaced with Yτ, where τ is the isospin matrix. Φ(y2) is an effective correlation function that is introduced to describe the distribution of the constituents ˉK and Y in the hadronic molecular Ξ(1620) state, which is often selected to be of the following form [16, 18-26]:

      Φ(p2E)exp(p2E/β2),

      (3)

      where pE is the Euclidean Jacobi momentum and β is the size parameter that characterizes the distribution of the components inthe molecule. Currently, the value of β=1.0 is determined by experimental data [16, 18-26] (and references therein).

      The coupling constant gΞ(1620)ˉKY is determined by the compositeness condition [16, 18-26], which implies that the renormalization constant of the bound state wave function Ξ(1620) is set to zero

      ZΞ(1620)=xˉKΣ+xˉKΛdΣΞ(1620)dk0|k0=mΞ(1620)=0,

      (4)

      where xAB is the probability of the Ξ(1620) being in the hadronic state AB with normalization xˉKΣ+xˉKΛ=1.0. ΣΞ(1620) is the self-energy of the Ξ(1620) and can be computed using the Feynmann diagrams shown in Fig. 3.

      Figure 3.  (color online) Self-energy of the Ξ(1620) state.

      ΣΞ(1620)(k0)=Y=Λ,Σ0,Σ+(CY)2g2Ξ(1620)ˉKY0dα0dη×(Δ2zk0+mY)16π2z2exp{1β2[(2ω2Yη+Δ24z)k20+αm2Y+ηm2ˉK]}.

      (5)

      where z=2+α+η and Δ=4ωY2η. k20=m2Ξ(1620) with k0,mΞ(1620) denoting the four-momenta and mass of the Ξ(1620), respectively; k1, mˉK, and mY are the four-momenta, mass of the ˉK meson, and mass of the Y baryon, respectively. Here, we set mΞ(1620)=mY+mˉKEb with Eb being the binding energy of Ξ(1620). Isospin symmetry implies that

      CY={1Y=Λ1/3Y=Σ02/3Y=Σ+  .

      (6)

      To estimate the radiative decays of the diagrams shown in Figs. 1-2, we require the effective Lagrangian densities related to the photon fields, which are [27, 28]

      LγΣΣ=ˉΣ[eΣAeκΣ2mNσμννAμ]Σ,

      (7)

      LγΛΛ=eκΛ2mNˉΛσμννAμΛ,

      (8)

      LγΣΛ=eμΣΛ2mNˉΣ0σμννAμΛ,

      (9)

      LKKγ=gK+K+γ4eϵμναβFμνK+αβK+gK0K0γ4eϵμναβFμνK0αβˉK0+h.c.,

      (10)

      LKKγ=ieAμKμK+,

      (11)

      where the strength tensors are defined as σμν= i2(γμγνγνγμ), Fμν=μAννAμ, and Kμν=μKννKμ.MN is the mass of p, and α=e2/4π=1/137 is the electromagnetic fine structure constant. The anomalous and transition magnetic moments of the baryons are provided by the PDG [1] and are shown in Table 1.

      κΣ=0.16 κΣ0=0.65 κΣ+=1.46
      κΛ=0.61 μΣΛ=1.61

      Table 1.  Anomalous and transition magnetic moments.

      The coupling constants gK+K+γ and gK0K0γ are introduced to obtain consistent results with the experimental measurements of K+K+γ and K0K0γ. The theoretical decay widths of K+K+γ and K0K0γ are

      Γ(K+K+γ)=αg2K+K+γ24mK+(m2K+m2K+),

      (12)

      Γ(K0K0γ)=αg2K0K0γ24mK0(m2K0m2K0).

      (13)

      According to the experimental widths Γ(K+K+γ)=0.0503 keV [1] and Γ(K0K0γ)=0.125 keV [1], the coupling constant gKKγ is fixed as

      gK+K+γ=0.580GeV1,gK0K0γ=0.904GeV1,

      (14)

      and the signs of these coupling constants are fixed according to the quark model [29, 30].

      In addition to the Lagrangians above, the meson-baryon interactions are also required and can be obtained from the following chiral Lagrangians [31, 32]

      LVBB=g(ˉBγμ[Vμ,B]+ˉBγμBVμ),

      (15)

      LPBB=F2ˉBγμγ5[uμ,B]+D2ˉBγμγ5{uμ,B},

      (16)

      LPBPB=i4f2ˉBγμ[(PμPμPP)BB(PμPμPP)]

      (17)

      where g=4.64, F=0.51, D=0.75 [16, 31, 33]; at the lowest order, uμ=2μP/f with f=93 MeV;... denotes the trace in the flavor space; B, P, and Vμ are the SU(3) pseudoscalar meson, vector meson, and baryon octet matrices, respectively, which are

      B=(12Σ0+16ΛΣ+pΣ12Σ0+16ΛnΞΞ026Λ),

      (18)

      P=(π02+η6π+K+ππ02+η6K0KˉK026η),

      (19)

      Vμ=(12(ρ0+ω)ρ+K+ρ12(ρ0+ω)K0KˉK0ϕ)μ.

      (20)

      Putting all the pieces together, we obtain the decay amplitudes that are shown in Appendix A. The total amplitudes of the Ξ0Ξ0γ, Ξ0Ξ0π0γ, and Ξ0π+Ξγ are the sum of these individual amplitudes, respectively:

      MT(Ξ0Ξγ)=i=a,b,c,d,e,fMi(Ξ0Ξγ),

      (21)

      MT(Ξ0πΞγ)=i=a,b,c,dMi(Ξ0πΞγ),

      (22)

      MT(Ξ0ˉKΛγ)=i=e.fMi(Ξ0ˉKΛγ).

      (23)

      When the amplitudes are determined, the corresponding partial decay width can be easily obtained, which is expressed as

      dΓ(Ξ(1620)0γΞ)=12J+1132π2|p1|m2Ξ0¯|M|2dΩ,

      (24)

      dΓ(Ξ(1620)0γΞπ,γˉKΛ)=12J+11(2π)5116m2ׯ|M|2|p3||p2|dm13dΩp3dΩp2.

      (25)

      The complete explanation of calculating these two equations is available in Refs. [1, 34]. J is the total angular momentum of the Ξ(1620); |p1| is the three-momenta of the decay products in the center of mass frame, and the overline indicates the sum over the polarization vectors of the final hadrons; (p3,Ωp3) is the momentum and angle of the particle π in the rest frame of π and Ξ; Ωp2 is the angle of the photon in the rest frame of the decaying particle; m13 is the invariant mass for π and Ξ, and mπ+mΞm13m.

      Using the values obtained above, the radiative decay width of the Ξ(1620)0 into Ξ0γ, Ξγπ, and γˉKΛ that are shown in Fig. 1 and Fig. 2 can be calculated. However, the amplitudes of the Ξ(1620)0γΞ0 and Ξ(1620)0γΞπ cannot satisfy the gauge invariance of the photon field. To ensure the gauge invariance of the total amplitudes, the contact diagram must be included. The corresponding Feynman diagrams are shown in Fig. 4. For the this calculation, we adopt the following form to satisfy pμ2MTotalμ(MT+Mcom)=0

      Figure 4.  (color online) Contact diagram for Ξ0Ξ0γ, Ξ0Ξ0π0γ, and Ξ0π+Ξγ. We also indicate definitions of the kinematics (p1,p2,p3,k1,k2, and p) used in the calculation.

      Macom(ΞγΞ0)=D+F2feCYgΞΣ+K0dα0dη0dζ116π2y2β2ˉu(p1)(Cμ1T1+Cμ2T2)γ5u(p)ϵμ(p2),

      (26)

      Mbcom(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞKΣ+4f2{1210dα0dη0dζ116π2y2β2×ˉu(p1)(Dμ1T1+Dμ2T2)u(p)ϵμ(p2),

      (27)

      where

      T1=exp{1β2[α(p21+m2K)+ηm2Σ++ζ(p22+m2Σ+)(p1ωΣ+p2ωK)2+14z(H2p2H1p1)2]},

      (28)

      T2=exp{1β2[α(p22+m2K)+ηm2K+ζ(p21+m2Σ+)(p2ωΣ+p1ωK)2+14z(H2p1H1p2)2]},

      (29)

      Cμ1=m21H318y3(2pμ1m1γμ)m31H214y2γμ+m21H212y2pμ1m1m2Σ+H12yγμ+m1m2Σ+γμ+m1mΣ+H212y2pμ1+((m+m1)mΣ+H1H22y2m1mΣ+H1y)pμ1mΣ+m1yγμ+H21H24y3p1p2(2pμ1m1γμ)+m1H1H22y2p1p2γμ+(m+m1)m1H1H22y2pμ1+3H12y2(2pμ1m1γμ),

      (30)

      Cμ2=m21H324y3pμ1m21H224y2pμ1(m+m1)mΣ+H1H22y2pμ1(m+m1)mΣ+H12ypμ1m1mΣ+H222y2pμ1m1mΣ+H22ypμ1+mΣ+yγμ+H1H222y3p1p2pμ1+H1H22y2p1p2pμ1m1(m+m1)H222y2pμ1m1(m+m1)H22ypμ1+H22y2(2pμ1m1γμ)+m1H22y2γμ+2H2y2pμ1+2ypμ1,

      (31)

      Dμ1=(m2Σ+H12y+m2Σ+)(2pμmγμ)mΣ+H212y2(mp2)pμ+mΣ+H1H22y2p2pμ+mΣ+H1y(mp2)pμ+mΣ+yγμmm213H318y3γμ+m(m2m213)H21H28y3γμm213H214y2(2pμmγμ)+H21m2132y2pμ+(H1H2(m2m213)4y2+1y)×(2pμmγμ)mH1H22y2p2pμ+3mH12y2γμ,

      (32)

      Dμ2=mΣ+H1H22y2p2pμ+mΣ+H222y2(mp2)pμmΣ+yγμ+(m2m213)H1H224y3pμmH1H22y2p2pμm213H324y3pμ+m213H222y2pμ+3H2y2pμmyγμ,

      (33)

      with y=1+α+η+ζ,H2=2(ζ+ωK),H1=2(α+ωΣ+), and m213=(p1+p3)2. m and m1 are the masses of the Ξ and Ξ, respectively.

    III.   NUMERICAL RESULTS AND DISCUSSIONS
    • To compute the radiative decay widths of the considered processes, the coupling constants of Ξ(1620) to its components can be estimated using the compositeness conditions provided by Eq. (4). xΛˉK in Eq. (4) is the probability of Ξ(1620) being in the hadronic state ΛˉK. The value of the xΛˉK is in the a range of 0.0-1.0. If xΛˉK= 0.0 or 1.0, Ξ(1620) is a pure molecular state containing the components ΣˉK and ΛˉK, respectively. Otherwise Ξ(1620) is a ˉKΛ-ˉKΣ molecular state. Currently, the value of xΛˉK cannot be accurately determined from first principles; therefore, it should better be determined using experimental data. As the free parameter, xΛˉK=0.520.68 is fixed by fitting the experimental decay data on possible ˉKΛ-ˉKΣ bound states within the same theoretical framework adopted in Ref. [16]. In this region, the total decay width for this state is predicted to be approximately 50.3968.79 MeV, which is comparable with the experimental data [2].

      Considering Ξ(1620) loosely as an S-wave ˉKΛ-ˉKΣ hadronic molecule, the coupling constants gΞ(1620)ˉKΛ and gΞ(1620)ˉKΣ that are dependent on the parameter xˉKΛ are plotted in Fig. 5. We observe that the coupling constant gΞ(1620)ˉKΛ monotonously increases with increasing xˉKΛ in our considered xˉKΛ range, while the coupling constant gΞ(1620)ˉKΣ decreases with increasing xˉKΛ. The opposite trend can be easily understood, as the coupling constants gΞ(1620)ˉKΛ and gΞ(1620)ˉKΣ are directly proportional to the corresponding molecular compositions [23].

      Figure 5.  (color online) xˉKΛ dependence of gΞ(1620)ˉKΛ and gΞ(1620)ˉKΣ.

      With the obtained total amplitude, the radiative decay width of Ξ(1620)0 into Ξ0γ can be calculated. The dependence of the corresponding radiative decay width on xˉKΛ is depicted in Fig. 6. For the xˉKΛ value within a reasonable range from 0.52 to 0.68, the radiative decay width of the Ξ(1620)0γΞ0 contribution from the ˉKΣ channel monotonously decreases. However, it increases for the ˉKΛ channel contribution to the Ξ(1620)0γΞ0. Moreover, the ˉKΣ component provides the dominant contribution to the partial decay width of the γΞ0 two-body channel. The ˉKΛ contribution to the γΞ0 two-body channel is very small. This is different from our results in Ref. [16] in that the ˉKΛ component provides the dominant contribution to the strong decay width of the Ξ(1620). A possible explanation for this may be that the interaction between the Σ baryon and photon is stronger than the γΛ interaction since the Σ0 decays completely to the final state containing the Λ baryon and γ [1].

      Figure 6.  (color online) Decomposed contributions to the decay width of the Ξ(1620)0 into Ξ0γ.

      Figure 6 also indicates that the total radiative decay width decreases for Ξ(1620)0γΞ0 when xˉKΛ is changed from 0.52 to 0.68. We also observe that the interference between the ˉKΣ and ˉKΛ channels is considerably small, resulting in a total decay width of Ξ(1620)0γΞ0 being primarily contributed by the ˉKΣ channel. This does not alter the conclusion that the ˉKΛ channel strongly couples to the ˉKΣ channel [9, 16]. The main reason for this is that the radiative decay widths are often in the keV regime and are significantly lower than their strong counterparts. Indeed, the total radiative decay width for Ξ(1620)0γΞ0 is predicted to be 118.76174.21 KeV, which is significantly lower than the total decay width, which is predicted to be approximately 50.39-68.79 MeV [16].

      The individual contributions of K, ˉK, Σ, Λ exchanges, and contact term for the reaction Ξ(1620)0γΞ0 are shown in Fig. 7. The amplitudes corresponding to the K-exchange and Σ+-exchange are not gauge invariant, while the remainder are gauge invariant. We can observe that the contract term and K-exchange provide a dominant contribution to the total decay width, and is at last sixty orders of magnitude larger than those of the amplitudes corresponding to the ˉK, Σ, and Λ exchanges for the studied xˉKΛ range.

      Figure 7.  (color online) Partial decay widths from K(red dash line), ˉK(cyan dash dot line), Σ (blue dot line), Λ(magenta dash dot dot line), and the remainder is the contact term exchange contribution for the Ξ(1620)0γΞ0 as a function of the parameter xˉKΛ.

      Next, we examine the three-body radiative decays Ξ(1620)0γπΞ and Ξ(1620)0γˉKΛ. The decay widths with xˉKΛ varying from 0.52 to 0.68 for such two transitions are depicted in Fig. 8. Since the phase space is small compared with the two-body radiative decay channel Ξ(1620)0γΞ0, the decay width should be the smallest for the Ξ(1620)0γˉKΛ channel, should be intermediate for the Ξ(1620)0γπΞ channel, and the largest for the Ξ(1620)0γΞ0 channel. Indeed, our study shows that the partial width of Ξ(1620)0γˉKΛ is rather small, weakly increasing as xˉKΛ increases. In particular, the partial width varies from 0.016 to 0.020 eV in the studied xˉKΛ range. However, the partial width of Ξ(1620)0γΞπ decreases as xˉKΛ increases, and the partial width of the Ξ(1620)0γΞπ is estimated to be 68.7558.19 eV.

      Figure 8.  (color online) Partial decay widths of the Ξ(1620)0γΞπ and Ξ(1620)0γˉKΛ.

      In Fig. 2, the diagrams with pion emitted directly from the intermediated Λ and Σ should be included. Our estimations indicate that in the discussed parameter range, the radiative transition strength for these diagrams are considerably small and the decay width is of the order of approximately 0.1 eV. Moreover, the interferences among them are also rather small. Therefore, the contributions from these channels are not considered in this paper.

    IV.   SUMMARY
    • We have studied the two-body and three-body radiative decays of the Ξ(1620) state assuming that it is a bound state of ˉKΛ-ˉKΣ. The coupling of Ξ(1620) to its components are fixed according to the Weinberg compositeness condition. The radiative decays for Ξ(1620)0γΞ0 and Ξ(1620)0γπΞ are obtained via triangle diagrams with exchanges of a pseudoscalar meson K, vector meson K, and baryons Σ and Λ. The three-body decay for the Ξ(1620)0γˉKΛ occur at the tree level. In the relevant parameter region, the partial widths are evaluated as

      Γ(Ξ(1620)0γΞ0)=118.76174.21keV,Γ(Ξ(1620)0γΞπ)=58.1968.75eV,Γ(Ξ(1620)0γˉKΛ)=0.0160.020eV.

      (34)

      Our calculation indicates the partial widths for Ξ(1620)0γπΞ and Ξ(1620)0γˉKΛ are too small to be observed, while for Ξ(1620)0γΞ0, the partial width can reach up to 174 keV at Ec.m.=1.620 GeV. Experimentally, with the current integrated luminosity that the LHCb experiment has accumulated at 1.620 GeV, the process Ξ(1620)0γΞ0 may be searched with only γΞ0 identified. Such research can also be conducted in the forthcoming Belle II experiment.

    APPENDIX A
    • Using the Lagrangians in Section II, the amplitudes for the diagrams shown in Figs. 1-2 can be easily obtained:

      Ma(Ξ0Ξ0γ)=i(i)3{(D3F)κΛ23,(F+D)μΣΛ2}egΞΛˉK4mNfd4q(2π)4Φ[(k1ωΛk2ωˉK0)2]×ˉu(p1)k1γ5q+mYq2m2Y(γμp2p2γμ)k2+mΛk22m2Λu(p)1k21m2ˉK0ϵμ(p2),

      (A1)

      Mb(Ξ0Ξ0γ)=i(i)3eD+F2fCYgΞΣ+ˉKd4q(2π)4Φ[(k1ωΣ+k2ωˉK)2]ˉu(p1)k1γ5×q+mYq2m2Y[γμ+κY4mN(γμp2p2γμ)]k2+mΣ+k22m2Σ+u(p)1k21m2Kϵμ(p2),

      (A2)

      Mc(Ξ0Ξ0γ)=i(i)3{(D3F)μΣΛ23,(F+D)κΣ02}egΞΛˉKCY4mNfd4q(2π)4Φ[(k1ωΣ0k2ωˉK0)2]×ˉu(p1)k1γ5q+mYq2m2Y(γμp2p2γμ)k2+mΣ0k22m2Σ0u(p)1k21m2ˉK0ϵμ(p2),

      (A3)

      MKd(Ξ0Ξ0γ)=i(i)3D+F2feCYgΞΣ+Kd4q(2π)4Φ[(k1ωΣ+k2ωK)2]ˉu(p1)qγ5×k2+mΣ+k22m2Σ+u(p)1k21m2K1q2m2K+(qk1)ϵ(p2),

      (A4)

      MKd(Ξ0Ξ0γ)=(i)3egCYgKKγ4d4q(2π)4Φ[(k1ωΣ+k2ωK)2]ˉu(p1)γρk2+mΣ+k22m2Σ+u(p)×1k21m2Kgρσ+qρqσ/m2Kq2m2Kϵμναβ(p2μgνηp2νgμη)(qαgβσqβgασ)ϵη(p2),

      (A5)

      Me(Ξ0Ξ0γ)=(i)36egCYgKKγ8d4q(2π)4Φ[(k1ωΛk2ωˉK0)2]ˉu(p1)γρk2+mΛk22m2Λu(p)×1k21m2ˉK0gρσ+qρqσ/m2ˉK0q2m2ˉK0ϵμναβ(p2μgνηp2νgμη)(qαgβσqβgασ)ϵη(p2),

      (A6)

      Mf(Ξ0Ξ0γ)=(i)3egCYgKKγ42d4q(2π)4Φ[(k1ωΣ0k2ωˉK0)2]ˉu(p1)γρk2+mΣ0k22m2Σ0u(p)×1k21m2ˉK0gρσ+qρqσ/m2ˉK0q2m2ˉK0ϵμναβ(p2μgνηp2νgμη)(qαgβσqβgασ)ϵη(p2),

      (A7)

      where {A, and B} are Λ and Σ baryon exchanges, respectively. The amplitudes of the Ξ0(1620)γπΞ,γˉKΛ can be also easily obtained:

      Ma(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞˉK0Λ32mNf2{{3κΛ,μΣΛ}{6κΛ,2μΣΛ}d4q(2π)4Φ[(k1ωΛk2ωˉK0)2]ˉu(p1)k1q+mYq2+m2Y×(γμp2p2γμ)k2+mΛk22m2Λu(p)ϵμ1k21m2ˉK0,

      (A8)

      Mb(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞKΣ+4f2{121d4q(2π)4Φ[(k1ωΣ+k2ωK)2]ˉu(p1)k1q+mYq2+m2Y×[γμ+κΣ+4mN(γμp2p2γμ)]k2+mΣ+k22m2Σ+u(p)ϵμ1k21m2K,

      (A9)

      Mc(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞKΣ+4f2{121d4q(2π)4Φ[(k1ωΣ+k2ωK)2]ˉu(p1)qk2+mΣ+k22+m2Σ+u(p)×(qμ+kμ1)1k21m2K1q2m2Kϵμ,

      (A10)

      Md(Ξ0π0Ξ0γ,π+Ξγ)=i3eCYgΞˉK0Σ032mNf2{{3μΣΛ,κΣ0}{6μΣΛ,2κΣ0}d4q(2π)4Φ[(k1ωΣ0k2ωˉK0)2]ˉu(p1)k1q+mYq2+m2Y×(γμp2p2γμ)k2+mΣ0k22m2Σ0u(p)ϵμ1k21m2ˉK0,

      (A11)

      Me(Ξ0ˉK0Λγ)=ieκΛgΞˉK0ΛCY4mNΦ[(k1ωΛk2ωˉK0)2]ˉu(p2)(γμp1p1γμ)k2+mΛk22m2Λu(p)ϵμ(p1),

      (A12)

      Mf(Ξ0ˉK0Λγ)=ieμΣΛgΞˉK0Σ0CY4mNΦ[(k1ωΣ0k2ωˉK0)2]ˉu(p2)(γμp1p1γμ)k2+mΣ0k22m2Σ0u(p)ϵμ(p1)

      (A13)

      where the expressions in the curly brackets, {AB, are for Ξ0π0Ξ0γ and Ξ0π+Ξγ, respectively.

Reference (34)

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