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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 orqqq 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, theΞ(1620) state, which is cataloged in the Particle Data Group (PDG) [1] with only one star, was reported again in theΞ−π+ final state by the Belle Collaboration [2]. The observed resonance parameters of the structure areM=1610.4±6.0(stat)+5.9−3.5(syst)MeV,Γ=59.9±4.8(stat)+2.8−3.0(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 ofuss ordss . 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, aΞ bound state with a mass of approximately 1620 MeV andJP=1/2− was predicted by studying theˉKΛ interaction in the framework of the one-boson-exchange (OBE) model [15]. Additionally, Ref. [15] consideredΞ(1620) as a pure molecular state composed ofˉKΛ component. A possible explanation for these results of Refs. [9-12, 15] is thatΞ (1620) has a largerˉKΛ component [14, 16]. Moreover, theˉKΣ component cannot be underestimated to reproduce the total decay width ofΞ(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 constituentˉKΛ andˉKΣ ofΞ(1620) [17]. Hence, a precise measurement of the radiative decays is useful to test different interpretations ofΞ(1620) . In this paper, we study the radiative decay ofΞ(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.
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In our previous study [16], the total decay width of
Ξ(1620) was reproduced with the assumption thatΞ(1620) is anS -waveˉKΛ−ˉKΣ bound state withJP=1/2− . Based on the molecular scenario, the radiative decay widthsΞ(1620)→γΞ ,Ξ(1620)→γˉKΛ , andΞ(1620)→γπΞ are studied to elucidate the internal structure of theΞ(1620) state. To calculate the radiation decay, we first employ the Weinberg compositeness rule to determineΞ(1620) couplings to its constituentsˉKY (Y≡Λ,Σ ). Radiative decays occurs from the exchange of a suitable hadron between theˉKY pair, which then transforms intoΞγ ,γˉKΛ , andΞπγ . The corresponding Feynman diagrams are shown in Fig. 1 and Fig. 2.Figure 1. (color online) Feynman diagrams of the
Ξ∗0→γΞ0 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
Ξ∗0→ γΞ0π0 ,γΞ−π+ , andγˉKΛ 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)ˉKY∫d4yΦ(y2)ˉK(x+ωYy)×Y(x−ωˉKy)ˉΞ(1620)(x),
(2) where
ωˉK=mˉK/(mˉK+mY) andωY=mY/(mˉK+mY) . For an isovector baryonΣ , Y should be replaced with→Y⋅→τ , whereτ is the isospin matrix.Φ(y2) is an effective correlation function that is introduced to describe the distribution of the constituentsˉK and Y in the hadronic molecularΞ(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 andβ is the size parameter that characterizes the distribution of the components inthe molecule. Currently, the value ofβ=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 functionΞ(1620) is set to zeroZΞ(1620)=xˉKΣ+xˉKΛ−dΣΞ(1620)d⧸k0|⧸k0=mΞ(1620)=0,
(4) where
xAB is the probability of theΞ(1620) being in the hadronic stateAB with normalizationxˉKΣ+xˉKΛ= 1.0 .ΣΞ(1620) is the self-energy of theΞ(1620) and can be computed using the Feynmann diagrams shown in Fig. 3.ΣΞ(1620)(k0)=∑Y=Λ,Σ0,Σ+(CY)2g2Ξ(1620)ˉKY∫∞0dα∫∞0dη×(−Δ2z⧸k0+mY)16π2z2exp{−1β2[(−2ω2Y−η+Δ24z)k20+αm2Y+ηm2ˉK]}.
(5) where
z=2+α+η andΔ=−4ωY−2η .k20= m2Ξ(1620) withk0,mΞ(1620) denoting the four-momenta and mass of theΞ(1620) , respectively;k1 ,mˉK , andmY are the four-momenta, mass of theˉK meson, and mass of the Y baryon, respectively. Here, we setmΞ(1620)=mY+mˉK−Eb withEb being the binding energy ofΞ(1620) . Isospin symmetry implies thatCY={1Y=Λ√1/3Y=Σ0−√2/3Y=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[eΣ⧸A−eκΣ2mNσμν∂νAμ]Σ,
(7) LγΛΛ=eκΛ2mNˉΛσμν∂νAμΛ,
(8) LγΣΛ=eμΣΛ2mNˉΣ0σμν∂νAμΛ,
(9) LK∗Kγ=gK∗+K+γ4eϵμναβFμνK∗+αβK−+gK∗0K0γ4eϵμναβFμνK∗0αβˉK0+h.c.,
(10) LKKγ=ieAμK−↔∂μK+,
(11) where the strength tensors are defined as
σμν= i2(γμγν−γνγμ) ,Fμν=∂μAν−∂νAμ , andK∗μν=∂μK∗ν− ∂νK∗μ .MN is the mass of p, andα=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 andgK∗0K0γ are introduced to obtain consistent results with the experimental measurements ofK∗+→K+γ andK∗0→K0γ . The theoretical decay widths ofK∗+→K+γ andK∗0→K0γ areΓ(K∗+→K+γ)=αg2K∗+K+γ24mK∗+(m2K∗+−m2K+),
(12) Γ(K∗0→K0γ)=αg2K∗0K0γ24mK∗0(m2K∗0−m2K0).
(13) According to the experimental widths
Γ(K∗+→K+γ)= 0.0503 keV [1] andΓ(K∗0→K0γ)=0.125 keV [1], the coupling constantgK∗Kγ is fixed asgK∗+K+γ=0.580GeV−1,gK∗0K0γ=−0.904GeV−1,
(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]⟩+⟨ˉBγμB⟩⟨Vμ⟩),
(15) LPBB=F2⟨ˉBγμγ5[uμ,B]⟩+D2⟨ˉBγμγ5{uμ,B}⟩,
(16) LPBPB=i4f2⟨ˉBγμ[(P∂μP−∂μPP)B−B(P∂μP−∂μPP)]⟩
(17) where
g=4.64 ,F=0.51 ,D=0.75 [16, 31, 33]; at the lowest order,uμ=−√2∂μP/f withf=93 MeV;⟨...⟩ denotes the trace in the flavor space; B, P, andVμ are theSU(3) pseudoscalar meson, vector meson, and baryon octet matrices, respectively, which areB=(1√2Σ0+1√6ΛΣ+pΣ−−1√2Σ0+1√6ΛnΞ−Ξ0−2√6Λ),
(18) P=(π0√2+η√6π+K+π−−π0√2+η√6K0K−ˉK0−2√6η),
(19) Vμ=(1√2(ρ0+ω)ρ+K∗+ρ−1√2(−ρ0+ω)K∗0K∗−ˉK∗0ϕ)μ.
(20) Putting all the pieces together, we obtain the decay amplitudes that are shown in Appendix A. The total amplitudes of the
Ξ∗0→Ξ0γ ,Ξ∗0→Ξ0π0γ , andΞ∗0→π+Ξ−γ are the sum of these individual amplitudes, respectively:MT(Ξ∗0→Ξγ)=∑i=a,b,c,d,e,fMi(Ξ∗0→Ξγ),
(21) MT(Ξ∗0→πΞγ)=∑i=a,b,c,dMi(Ξ∗0→πΞγ),
(22) MT(Ξ∗0→ˉKΛγ)=∑i=e.fMi(Ξ∗0→ˉKΛγ).
(23) When the amplitudes are determined, the corresponding partial decay width can be easily obtained, which is expressed as
dΓ(Ξ(1620)0→γΞ)=12J+1132π2|→p1|m2Ξ∗0¯|M|2dΩ,
(24) dΓ(Ξ(1620)0→γΞπ,γˉKΛ)=12J+11(2π)5116m2ׯ|M|2|→p∗3||→p2|dm13dΩ∗p3dΩp2.
(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; (→p∗3,Ω∗p3 ) is the momentum and angle of the particleπ in the rest frame ofπ andΞ ;Ωp2 is the angle of the photon in the rest frame of the decaying particle;m13 is the invariant mass forπ andΞ , andmπ+mΞ⩽ m13⩽m .Using the values obtained above, the radiative decay width of the
Ξ(1620)0 intoΞ0γ ,Ξγπ , andγˉKΛ that are shown in Fig. 1 and Fig. 2 can be calculated. However, the amplitudes of theΞ(1620)0→γΞ0 andΞ(1620)0→ γΞπ 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 satisfypμ2MTotalμ(≡MT+Mcom)=0 Figure 4. (color online) Contact diagram for
Ξ∗0→Ξ0γ ,Ξ∗0→Ξ0π0γ , andΞ∗0→π+Ξ−γ . We also indicate definitions of the kinematics (p1,p2,p3,k1,k2 , and p) used in the calculation.Macom(Ξ∗→γΞ0)=D+F√2feCYgΞ∗Σ+K−∫∞0dα∫∞0dη∫∞0dζ116π2y2β2ˉu(p1)(Cμ1T1+Cμ2T2)γ5u(p)ϵ∗μ(p2),
(26) Mbcom(Ξ∗0→π0Ξ0γ,π+Ξ−γ)=−i3eCYgΞ∗K−Σ+4f2{1√21∫∞0dα∫∞0dη∫∞0dζ116π2y2β2×ˉu(p1)(Dμ1T1+Dμ2T2)u(p)ϵ∗μ(p2),
(27) where
T1=exp{−1β2[α(−p21+m2K−)+ηm2Σ++ζ(−p22+m2Σ+)−(p1ωΣ+−p2ωK−)2+14z(H2p2−H1p1)2]},
(28) T2=exp{−1β2[α(−p22+m2K−)+ηm2K−+ζ(−p21+m2Σ+)−(p2ωΣ+−p1ωK−)2+14z(H2p1−H1p2)2]},
(29) Cμ1=−m21H318y3(2pμ1−m1γμ)−m31H214y2γμ+m21H212y2pμ1−m1m2Σ+H12yγμ+m1m2Σ+γμ+m1mΣ+H212y2pμ1+((m+m1)mΣ+H1H22y2−m1mΣ+H1y)pμ1−mΣ+−m1yγμ+H21H24y3p1⋅p2(2pμ1−m1γμ)+m1H1H22y2p1⋅p2γμ+(m+m1)m1H1H22y2pμ1+3H12y2(2pμ1−m1γμ),
(30) Cμ2=−m21H324y3pμ1−m21H224y2pμ1−(m+m1)mΣ+H1H22y2pμ1−(m+m1)mΣ+H12ypμ1−m1mΣ+H222y2pμ1−m1mΣ+H22ypμ1+mΣ+yγμ+H1H222y3p1⋅p2pμ1+H1H22y2p1⋅p2pμ1−m1(m+m1)H222y2pμ1−m1(m+m1)H22ypμ1+H22y2(2pμ1−m1γμ)+m1H22y2γμ+2H2y2pμ1+2ypμ1,
(31) Dμ1=(−m2Σ+H12y+m2Σ+)(2pμ−mγμ)−mΣ+H212y2(m−⧸p2)pμ+mΣ+H1H22y2⧸p2pμ+mΣ+H1y(m−⧸p2)pμ+mΣ+yγμ−mm213H318y3γμ+m(m2−m213)H21H28y3γμ−m213H214y2(2pμ−mγμ)+H21m2132y2pμ+(H1H2(m2−m213)4y2+1y)×(2pμ−mγμ)−mH1H22y2⧸p2pμ+3mH12y2γμ,
(32) Dμ2=−mΣ+H1H22y2⧸p2pμ+mΣ+H222y2(m−⧸p2)pμ−mΣ+yγμ+(m2−m213)H1H224y3pμ−mH1H22y2⧸p2pμ−m213H324y3pμ+m213H222y2pμ+3H2y2pμ−myγμ,
(33) with
y=1+α+η+ζ ,H2=2(ζ+ωK−) ,H1=2(α+ωΣ+) , andm213=(p1+p3)2 . m andm1 are the masses of theΞ∗ andΞ , 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 ofΞ(1620) being in the hadronic stateΛˉK . The value of thexΛˉK is in the a range of 0.0-1.0. IfxΛˉK = 0.0 or 1.0,Ξ(1620) is a pure molecular state containing the componentsΣˉK andΛˉK , respectively. OtherwiseΞ(1620) is aˉKΛ -ˉKΣ molecular state. Currently, the value ofxΛˉK cannot be accurately determined from first principles; therefore, it should better be determined using experimental data. As the free parameter,xΛˉK=0.52−0.68 is fixed by fitting the experimental decay data on possibleˉKΛ -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 approximately50.39−68.79 MeV, which is comparable with the experimental data [2].Considering
Ξ∗(1620) loosely as an S-waveˉKΛ -ˉKΣ hadronic molecule, the coupling constantsgΞ∗(1620)ˉKΛ andgΞ∗(1620)ˉKΣ that are dependent on the parameterxˉKΛ are plotted in Fig. 5. We observe that the coupling constantgΞ∗(1620)ˉKΛ monotonously increases with increasingxˉKΛ in our consideredxˉKΛ range, while the coupling constantgΞ∗(1620)ˉKΣ decreases with increasingxˉKΛ . The opposite trend can be easily understood, as the coupling constantsgΞ∗(1620)ˉKΛ andgΞ∗(1620)ˉKΣ are directly proportional to the corresponding molecular compositions [23].With the obtained total amplitude, the radiative decay width of
Ξ(1620)0 intoΞ0γ can be calculated. The dependence of the corresponding radiative decay width onxˉKΛ is depicted in Fig. 6. For thexˉKΛ value within a reasonable range from 0.52 to 0.68, the radiative decay width of theΞ(1620)0→γΞ0 contribution from theˉKΣ channel monotonously decreases. However, it increases for theˉKΛ channel contribution to theΞ(1620)0→γΞ0 . Moreover, theˉKΣ component provides the dominant contribution to the partial decay width of theγΞ0 two-body channel. TheˉKΛ contribution to theγΞ0 two-body channel is very small. This is different from our results in Ref. [16] in that theˉKΛ component provides the dominant contribution to the strong decay width of theΞ(1620) . A possible explanation for this may be that the interaction between theΣ baryon and photon is stronger than theγ−Λ interaction since theΣ0 decays completely to the final state containing theΛ baryon andγ [1].Figure 6 also indicates that the total radiative decay width decreases for
Ξ(1620)0→γΞ0 whenxˉKΛ is changed from 0.52 to 0.68. We also observe that the interference between theˉKΣ andˉKΛ channels is considerably small, resulting in a total decay width ofΞ(1620)0→γΞ0 being primarily contributed by theˉKΣ channel. This does not alter the conclusion that theˉKΛ channel strongly couples to theˉKΣ 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 forΞ(1620)0→γΞ0 is predicted to be118.76−174.21 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 reactionΞ(1620)0→ γΞ0 are shown in Fig. 7. The amplitudes corresponding to theK− -exchange andΣ+ -exchange are not gauge invariant, while the remainder are gauge invariant. We can observe that the contract term andK− -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 theˉK∗ ,Σ , andΛ exchanges for the studiedxˉKΛ 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 theΞ(1620)0→γΞ0 as a function of the parameterxˉKΛ .Next, we examine the three-body radiative decays
Ξ(1620)0→γπΞ andΞ(1620)0→γˉKΛ . The decay widths withxˉKΛ 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 channelΞ(1620)0→γΞ0 , the decay width should be the smallest for theΞ(1620)0→γˉKΛ channel, should be intermediate for theΞ(1620)0→γπΞ channel, and the largest for theΞ(1620)0→γΞ0 channel. Indeed, our study shows that the partial width ofΞ(1620)0→γˉKΛ is rather small, weakly increasing asxˉKΛ increases. In particular, the partial width varies from 0.016 to 0.020 eV in the studiedxˉKΛ range. However, the partial width ofΞ(1620)0→γΞπ decreases asxˉKΛ increases, and the partial width of theΞ(1620)0→γΞπ is estimated to be68.75−58.19 eV.In Fig. 2, the diagrams with pion emitted directly from the intermediated
Λ andΣ 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 ofˉKΛ -ˉKΣ . The coupling ofΞ(1620) to its components are fixed according to the Weinberg compositeness condition. The radiative decays forΞ(1620)0→γΞ0 andΞ(1620)0→γπΞ are obtained via triangle diagrams with exchanges of a pseudoscalar meson K, vector mesonK∗ , and baryonsΣ andΛ . The three-body decay for theΞ(1620)0→γˉKΛ occur at the tree level. In the relevant parameter region, the partial widths are evaluated asΓ(Ξ(1620)0→γΞ0)=118.76−174.21keV,Γ(Ξ(1620)0→γΞπ)=58.19−68.75eV,Γ(Ξ(1620)0→γˉKΛ)=0.016−0.020eV.
(34) Our calculation indicates the partial widths for
Ξ(1620)0→γπΞ andΞ(1620)0→γˉKΛ are too small to be observed, while forΞ(1620)0→γΞ0 , the partial width can reach up to 174 keV atEc.m.=1.620 GeV. Experimentally, with the current integrated luminosity that the LHCb experiment has accumulated at1.620 GeV, the processΞ(1620)0→ γΞ0 may be searched with onlyγΞ0 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(Ξ∗0→Ξ0γ)=i(i)3{(D−3F)κΛ2√3,(F+D)μΣΛ2}egΞ∗ΛˉK4mNf∫d4q(2π)4Φ[(k1ωΛ−k2ωˉK0)2]×ˉu(p1)⧸k1γ5⧸q+mYq2−m2Y(γμ⧸p2−⧸p2γμ)⧸k2+mΛk22−m2Λu(p)1k21−m2ˉK0ϵ∗μ(p2), (A1) Mb(Ξ∗0→Ξ0γ)=−i(i)3eD+F√2fCYgΞ∗Σ+ˉK−∫d4q(2π)4Φ[(k1ωΣ+−k2ωˉK−)2]ˉu(p1)⧸k1γ5×⧸q+mYq2−m2Y[γμ+κY4mN(γμ⧸p2−⧸p2γμ)]⧸k2+mΣ+k22−m2Σ+u(p)1k21−m2K−ϵ∗μ(p2),
(A2) Mc(Ξ∗0→Ξ0γ)=i(i)3{(D−3F)μΣΛ2√3,(F+D)κΣ02}egΞ∗ΛˉKCY4mNf∫d4q(2π)4Φ[(k1ωΣ0−k2ωˉK0)2]×ˉu(p1)⧸k1γ5⧸q+mYq2−m2Y(γμ⧸p2−⧸p2γμ)⧸k2+mΣ0k22−m2Σ0u(p)1k21−m2ˉK0ϵ∗μ(p2),
(A3) MKd(Ξ∗0→Ξ0γ)=−i(i)3D+F√2feCYgΞ∗Σ+K−∫d4q(2π)4Φ[(k1ωΣ+−k2ωK−)2]ˉu(p1)⧸qγ5×⧸k2+mΣ+k22−m2Σ+u(p)1k21−m2K−1q2−m2K+(q−k1)⋅ϵ∗(p2),
(A4) MK∗d(Ξ∗0→Ξ0γ)=(i)3egCYgK∗Kγ4∫d4q(2π)4Φ[(k1ωΣ+−k2ωK−)2]ˉu(p1)γρ⧸k2+mΣ+k22−m2Σ+u(p)×1k21−m2K−−gρσ+qρqσ/m2K∗−q2−m2K∗−ϵμναβ(p2μgνη−p2νgμη)(qαgβσ−qβgασ)ϵ∗η(p2),
(A5) Me(Ξ∗0→Ξ0γ)=(i)3√6egCYgK∗Kγ8∫d4q(2π)4Φ[(k1ωΛ−k2ωˉK0)2]ˉu(p1)γρ⧸k2+mΛk22−m2Λu(p)×1k21−m2ˉK0−gρσ+qρqσ/m2ˉK∗0q2−m2ˉK∗0ϵμναβ(p2μgνη−p2νgμη)(qαgβσ−qβgασ)ϵ∗η(p2),
(A6) Mf(Ξ∗0→Ξ0γ)=−(i)3egCYgK∗Kγ4√2∫d4q(2π)4Φ[(k1ωΣ0−k2ωˉK0)2]ˉu(p1)γρ⧸k2+mΣ0k22−m2Σ0u(p)×1k21−m2ˉK0−gρσ+qρqσ/m2ˉK∗0q2−m2ˉK∗0ϵμναβ(p2μgνη−p2νgμη)(qαgβσ−qβgασ)ϵ∗η(p2),
(A7) where
{A , andB} areΛ andΣ baryon exchanges, respectively. The amplitudes of theΞ0(1620)→γπΞ,γˉKΛ can be also easily obtained:Ma(Ξ∗0→π0Ξ0γ,π+Ξ−γ)=i3eCYgΞ∗ˉK0Λ32mNf2{{−√3κΛ,μΣΛ}{√6κΛ,√2μΣΛ}∫d4q(2π)4Φ[(k1ωΛ−k2ωˉK0)2]ˉu(p1)⧸k1⧸q+mYq2+m2Y×(γμ⧸p2−⧸p2γμ)⧸k2+mΛk22−m2Λu(p)ϵ∗μ1k21−m2ˉK0,
(A8) Mb(Ξ∗0→π0Ξ0γ,π+Ξ−γ)=i3eCYgΞ∗K−Σ+4f2{1√21∫d4q(2π)4Φ[(k1ωΣ+−k2ωK−)2]ˉu(p1)⧸k1⧸q+mYq2+m2Y×[γμ+κΣ+4mN(γμ⧸p2−⧸p2γμ)]⧸k2+mΣ+k22−m2Σ+u(p)ϵ∗μ1k21−m2K−,
(A9) Mc(Ξ∗0→π0Ξ0γ,π+Ξ−γ)=i3eCYgΞ∗K−Σ+4f2{1√21∫d4q(2π)4Φ[(k1ωΣ+−k2ωK−)2]ˉu(p1)⧸q⧸k2+mΣ+k22+m2Σ+u(p)×(qμ+kμ1)1k21−m2K−1q2−m2K−ϵ∗μ,
(A10) Md(Ξ∗0→π0Ξ0γ,π+Ξ−γ)=i3eCYgΞ∗ˉK0Σ032mNf2{{−√3μΣΛ,κΣ0}{√6μΣΛ,√2κΣ0}∫d4q(2π)4Φ[(k1ωΣ0−k2ωˉK0)2]ˉu(p1)⧸k1⧸q+mYq2+m2Y×(γμ⧸p2−⧸p2γμ)⧸k2+mΣ0k22−m2Σ0u(p)ϵ∗μ1k21−m2ˉK0,
(A11) Me(Ξ∗0→ˉK0Λγ)=−ieκΛgΞ∗ˉK0ΛCY4mNΦ[(k1ωΛ−k2ωˉK0)2]ˉu(p2)(γμ⧸p1−⧸p1γμ)⧸k2+mΛk22−m2Λu(p)ϵ∗μ(p1),
(A12) Mf(Ξ∗0→ˉK0Λγ)=−ieμΣΛgΞ∗ˉK0Σ0CY4mNΦ[(k1ωΣ0−k2ωˉK0)2]ˉu(p2)(γμ⧸p1−⧸p1γμ)⧸k2+mΣ0k22−m2Σ0u(p)ϵ∗μ(p1)
(A13) where the expressions in the curly brackets,
{AB , are forΞ∗0→π0Ξ0γ andΞ∗0→π+Ξ−γ , respectively.
