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The study of the low-lying excited
Λ∗ andΣ∗ hyperon resonances is one of the most important issues in hadron physics. In particular, sinceΛ(1405) was experimentally discovered [1, 2], its nature has garnered significant attention [3–8], and one explanation forΛ(1405) is theˉKN hadronic molecular state [9–15]. In addition, for the isospinI=1 partner ofΛ(1405) , the lowestΣ∗1/2− is crucial to understand light baryon spectra. At present,Σ∗(1620) withJP=1/2− is listed in the latest version of the Review of Particle Physics (RPP) [16]. It should be emphasized that theΣ∗(1620) state is a one-star baryon resonance. Many studies indicate that the lowestΣ∗1/2− resonance is still far from being established, and its mass has been predicted to lie in the range1380∼1500 MeV [13, 17–20]. Thus, searching for the lowestΣ∗1/2− is helpful for understanding low-lying excited baryons withJP=1/2− and light flavor baryon spectra.The analyses of relevant data on the process
K−p→Λπ+π− suggest that aΣ∗1/2− resonance may exist with a mass of approximately 1380 MeV [17, 18], which is consistent with the predictions of unquenched quark model [21]. The analyses ofK∗Σ photo-production also indicate thatΣ∗1/2− is possibly buried under theΣ∗(1385) peak with a mass of 1380 MeV [22], and the search forΣ∗1/2− in the processΛ+c→ηπ+Λ has been proposed [23]. A more delicate analysis of CLAS data on the processγp→KΣπ [24] suggests that theΣ∗1/2− peak should be around 1430 MeV [13]. In Refs. [25, 26], we suggest searching for such a state in the processesχc0(1P)→ˉΣΣπ andχc0(1P)→ˉΛΣπ . In addition, in Ref. [27], oneΣ∗1/2− state was found with a mass of approximately 1400 MeV by solving coupled channel scattering equations, and Ref. [28] suggests to search for this state in the photo-production processγp→K+Σ∗01/2− .It is worth mentioning that a
Σ∗(1480) resonance withJP=1/2− has been listed on the previous version of the RPP [29]. As early as 1970, theΣ∗(1480) resonance was reported in theΛπ+ ,Σπ , andpˉK0 channels ofπ+p scattering in the Princeton-Pennsylvania Accelerator 15-in. hydrogen bubble chamber [30, 31]. In 2004, a bump structure around 1480 MeV was observed in theK0Sp(ˉp) invariant mass spectrum of inclusive deep inelasticep scattering by the ZEUS Collaboration [32]. Furthermore, a signal for a resonance at1480±15 MeV with a width of60±15 MeV was observed in the processpp→K+pY∗0 [33].Σ∗(1480) has been investigated theoretically within different models [34–37]. In Ref. [37], S-wave meson-baryon interactions with strangenessS=−1 were studied within the unitary chiral approach, and one narrow pole with a pole position of1468−i 13 MeV was found in the second Riemann sheet, which may be associated with theΣ∗(1480) resonance. However,Σ∗(1480) signals are insignificant, and the existence of this state still needs to be confirmed within more precise experimental measurements.Photo-production reactions have been used to study the excited hyperon states
Σ∗ andΛ∗ , and the LEPS [38] and CLAS [24] Collaborations have accumulated considerable relevant experimental data. For instance, with these data, we analyzed the processγp→KΛ∗(1405) to deepen our understanding of the nature ofΛ∗(1405) in Ref. [39]. To confirm the existence ofΣ∗(1480) , we propose to investigate the processγN→KΣ∗(1480) ① within the Regge-effective Lagrangian approach.Considering the
Σ∗(1480) signal was first observed in theπ+Λ invariant mass distribution of the processπ+p→π+K+Λ , and the significance is approximately3∼4σ [31], we search for chargedΣ∗(1480) in the processγn→K+Σ∗−1/2− , which may also avoid the contributions of possible excitedΛ∗ states. We consider the t-, s-, and u-channel diagrams in the Born approximation by employing the effective Lagrangian approach, and the t-channel K/K∗ exchanges terms within the Regge model. Then, we calculate the differential and total cross sections of the processγn→K+Σ∗−1/2− , which helps the experimental search forΣ∗1/2− .This paper is organized as follows. In Sec. II, the theoretical formalism for studying the
γn→K+Σ∗−(1480) reaction is presented. The numerical results of the total and differential cross sections and discussion are shown in Sec. III. Finally, a brief summary is given in the last section. -
The reaction mechanisms of the
Σ∗(1480)(≡Σ∗) photo-production process are depicted in Fig. 1, where we consider the contributions from the t-channel K andK∗ exchange terms, s-channel nucleon pole term, u-channel Σ exchange term, and contact term.Figure 1. Mechanisms of the
γn→K+Σ∗−1/2− process. (a) t-channel K/K∗ exchange terms, (b) s-channel nuclear term, (c) u-channel Σ exchange term, and (d) contact term.k1 ,k2 ,p1 , andp2 represent the four-momenta of the initial photon, kaon, neutron, andΣ∗(1480) , respectively.To compute the scattering amplitudes of the Feynman diagrams shown in Fig. 1 within the effective Lagrangian approach, we use the Lagrangian densities for the electromagnetic and strong interaction vertices, as in Refs. [28, 40–45],
LγKK=−ie[K†(∂μK)−(∂μK†)K]Aμ,
(1) LγKK∗=gγKK∗ϵμναβ∂μAν∂αK∗βK,
(2) LγNN=−eˉN[γμˆe−ˆκN2MNσμν∂ν]AμN,
(3) LγΣΣ∗=eμΣΣ∗2MNˉΣγ5σμν∂νAμΣ∗+h.c.,
(4) LKNΣ=−igKNΣˉNγ5ΣK+h.c.,
(5) LK∗NΣ∗=igK∗NΣ∗√3ˉK∗μˉΣ∗γμγ5N+h.c.,
(6) LKNΣ∗=gKNΣ∗ˉKˉΣ∗N+h.c.,
(7) where
e(=√4πα) is the elementary charge unit,Aμ is the photon field, andˆe≡(1+τ3)/2 denotes the charge operator acting on the nucleon field.ˆκN≡κpˆe+κn(1−ˆe) is the anomalous magnetic moment, and we takeκn=−1.913 for the neutron [16].MN andMΣ denote the masses of the nucleon and the ground-state of the Σ hyperon, respectively. The strong couplinggKNΣ is taken to be 4.09 from Refs. [46–48].gγKK∗=0.254 GeV−1 is determined from the experimental data ofΓK∗→K+γ [16], and the value ofgK∗NΣ∗=−3.26−i 0.06 is taken from Ref. [27]. In addition, the couplinggKNΣ∗=8.74 GeV is taken from Ref. [37], and the transition magnetic momentμΣΣ∗=1.28 is taken from Ref. [28]②.In addition to the pseudoscalar coupling of Eq. (5), the vertex of
KNΣ may be described with the Lagrangian density of axial-vector coupling as follows [49, 50]LKNΣ=−gKNΣ2MNˉNγ5γμ(∂μK)Σ+h.c..
(8) We discuss the difference between the two schemes in the next section. In this work, we perform the calculations with the Lagrangian density of Eq. (5) for the vertex of
KNΣ in the following.With the effective interaction Lagrangian densities given above, the invariant scattering amplitudes are defined as
M=ˉuΣ∗(p2,sΣ∗)MμhuN(k2,sp)ϵμ(k1,λ),
(9) where
uΣ∗ anduN represent the Dirac spinors,ϵμ(k1,λ) is the photon polarization vector, and the sub-index h corresponds to the different diagrams of Fig. 1. The reduced amplitudesMμh are written asMμK∗=gγKK∗gK∗NΣ∗√3(t−M2K∗)ϵαβμνk1αk2βγνγ5,
(10) MμK−=−2iegKNΣ∗t−M2Kkμ2,
(11) MμΣ−=−ieμΣΣ∗gKNΣ2Mn(u−M2Σ)(q̸u−MΣ)σμνk1ν,
(12) Mμn=κngKNΣ∗2Mn(s−M2n)σμνk1ν(q̸s+Mn).
(13) To maintain gauge invariance in the full photoproduction amplitudes considered here, we adopt the amplitude of the contact term
Mμc=−iegKNΣ∗pμ2p2⋅k1,
(14) for
γn→K+Σ∗−1/2− .It is known that the Reggeon exchange mechanism plays a crucial role at high energies and forward angles [51–54]; thus, we adopt the Regge model when modeling the t-channel K and
K∗ contributions by replacing the usual pole-like Feynman propagator with the corresponding Regge propagators as follows1t−M2K→FReggeK=(ssK0)αK(t)πα′Ksin(παK(t))Γ(1+αK(t)),
(15) 1t−M2K∗→FReggeK∗=(ssK∗0)αK∗(t)−1πα′K∗sin(παK∗(t))Γ(αK∗(t)),
(16) where
αK(t)=0.7GeV−2×(t−M2K) andαK∗(t)=1+0.83GeV−2×(t−M2K∗) are the linear Reggeon trajectories. The constantssK0 andsK∗0 are determined to be 3.0 GeV2 and 1.5 GeV2 , respectively [55]. Here,α′K andα′K∗ are the Regge-slopes.Then, the full photo-production amplitudes for the
γn→K+Σ∗−1/2− reaction can be expressed asMμ=(MμK−+Mμc)(t−M2K)FReggeK+MμΣ−fu+MμK∗(t−M2K∗)FReggeK∗+Mμnfs,
(17) where
FReggeK andFReggeK∗ represent the Regge propagators. The form factorsfs andfu are included to suppress the large momentum transfer of the intermediate particles and describe their off-shell behavior because the intermediate hadrons are not point-like particles. For s-channel and u-channel baryon exchanges, we use the following form factors [40, 56]fi(q2i)=[Λ4iΛ4i+(q2i−M2i)2]2,i=s,u
(18) where
Mi andqi are the masses and four-momenta of the intermediate baryons, andΛi represents the cut-off values for the baryon exchange diagrams. In this study, we takeΛs=Λu=1.5 GeV and discuss the results with different cut-offs.Finally, the unpolarized differential cross section in the center of mass (c.m.) frame for the
γn→KΣ∗−1/2− reaction reads asdσdΩ=MNMΣ∗|→kc.m.1||→kc.m.2|8π2(s−M2N)2∑λ,sp,sΣ∗|M|2,
(19) where s denotes the invariant mass square of the center of mass (c.m.) frame for
Σ∗1/2− photo-production, anddΩ=2πdcosθc.m. , withθc.m. as the polar outgoing K scattering angle. Here,→kc.m.1 and→kc.m.2 are the three-momenta of the photon and K meson in the c.m. frame,|→kc.m.1|=s−M2N2√s,
(20) |→kc.m.2|=√[s−(MΣ∗+MK)2][s−(MΣ∗−MK)2]2√s.
(21) -
In this section, we present our numerical results on the differential and total cross sections for the
γn→K+Σ∗−1/2− reaction. The masses of the mesons and baryons are taken from the RPP [16], as given in Table 1. In addition, the mass and width ofΣ∗(1480) areM=1480±15 MeV andΓ=60±15 MeV, respectively [29].Particle Mass /MeV n 939.565 Σ− 1197.449 K+ 493.677 K− 493.677 K∗ 891.66 Table 1. Particle masses used in this study.
First, we show the angle dependence of the differential cross sections for the
γn→K+Σ∗−1/2− reaction in Fig. 2, where the center-of-mass energiesW=√s varies from2.0 to2.8 GeV. The black curves labeled as "Total" are the results of all the contributions from the t-, s-, and u-channels and the contact term. The blue-dot and red-dashed curves represent the contributions from the u-channel Σ exchange and t-channel K exchange mechanisms, respectively. The magenta-dot-dashed and green-dot curves correspond to the contributions from the s-channel and t-channelK∗ exchange diagrams, respectively, whereas the cyan-dot-dashed curves represent the contributions from the contact term. According to the differential cross sections, we find that the t-channel K meson exchange term plays an important role at forward angles for the processγn→K+Σ∗−1/2− , mainly because of the Regge effects of t-channel K exchange. The K-Reggeon exchange exhibits a steadily increasing behavior withcosθc.m. and falls off drastically at very forward angles, which is consistent with the results of Ref. [28]. In the appendix, we show that the contribution from the t-channel K exchange is zero in the forward angle (θc.m.=0 ) and the backward angle (θc.m.=π ). In addition, the u-channel Σ exchange term mainly contributes to the backward angles. It should be emphasized that the contribution from the t-channelK∗ exchange term is small and can be safely neglected for the processγn→K+Σ∗−1/2− , which is consistent with the results of Ref. [28].Figure 2. (color online)
γn→K+Σ∗−1/2− differential cross sections as a function ofcosθc.m. are plotted forγn -invariant mass intervals(in GeV units) . The black curve labeled as "Total" represent the results of all the contributions, including the t-, s-, and u-channels and the contact term. The blue-dot and red-dashed curves denote the contributions from the effective Lagrangian approach u-channel Σ exchange and t-channel K exchange mechanisms, respectively. The magenta-dot-dashed and green-dot-dashed curves represent the contributions of the s-channel and t-channelK∗ exchange diagrams, respectively, whereas the cyan-dot-dashed curves represent the contributions of the contact term.As mentioned in Sec. II, in addition to the pseudoscalar coupling of Eq. (5), the vertex of
KNΣ may also be described with the Lagrangian density of axial-vector coupling [Eq. (8)]. We also present the contribution from the u-channel Σ exchange with the Lagrangian densities of Eqs. (5) and (8) in Fig. 3; we find that both of them contribute to the backward angles. Because the contribution from the u-channel is small, it is expected that either pseudoscalar coupling or axial-vector coupling forKNΣ does not affect our results significantly.Figure 3. (color online)
γn→K+Σ∗−1/2− differential cross sections with only the contribution from u-channel Σ exchange. The magenta-dot-dashed curves represent the results obtained with the axial-vector coupling of Eq. (8), and the blue-dotted curves represent the results obtained with the pseudoscalar coupling of Eq. (5).In addition to the differential cross sections, we also calculate the total cross section of the
γn→K+Σ∗−1/2− reaction as a function of the initial photon energy. The results are shown in Fig. 4. The black curve labeled as "Total" represents the results of all the contributions, including the t-, s-, and u-channels and the contact term. The blue-dot and red-dashed curves represent the contributions from the u-channel Σ exchange and t-channel K exchange mechanisms, respectively. The magenta-dot-dashed and green-dot curves represent the contributions of the s-channel and t-channelK∗ exchange diagrams, respectively, whereas the cyan-dot-dashed curve represents the contribution of the contact term. For theγn→K+Σ∗−1/2− reaction, its total cross section attains a maximum value of approximately4.2 μ b atEγ=2.4 GeV . It is expected thatΣ∗(1480) can be observed by future experiments in the processγn→K+Σ∗−(1480)→Σ−π0/Σ0π−/Σ−γ .Figure 4. (color online) Total cross section for
γn→K+Σ∗1/2− is plotted as a function of the lab energyEγ . The black curve labeled as "Total" represents the results of all the contributions, including the t-, s-, and u-channels and the contact term. The blue-dot and red-dashed curves represent the contributions from the effective Lagrangian approach u-channel Σ exchange and t-channel K exchange mechanisms, respectively. The magenta-dot-dashed and green-dot curves represent the contributions of the s-channel nucleon term and t-channelK∗ exchange diagrams, respectively, whereas the cyan-dot-dashed curve represents the contribution of the contact term.Finally, we show the total cross section for
γn→K+Σ∗−1/2− with the cut-offΛs/u=1.2 ,1.5 , and1.8 GeV in Fig. 5. We find that the total cross sections are weakly dependent on the value of the cut-off. Because the precise couplings ofΣ∗(1480) are still unknown, future experiments would benefit from constraining these couplings if the stateΣ∗(1480) is confirmed. -
The lowest
Σ∗−1/2− is far from established, and its existence is important to understand low-lying excited baryons withJP=1/2− . There are many experimental hints ofΣ∗(1480) , as listed in the previous version of the RPP. We propose to search for this state in the photoproduction process to confirm its existence.Assuming that the
JP=1/2− low lying stateΣ∗(1480) has a sizeable coupling toˉKN according to the study of Ref. [37], we phenomenologically investigate theγn→K+Σ∗−1/2− reaction by considering the contributions from the t-channelK/K∗ exchange term, s-channel nucleon term, u-channel Σ exchange term, and contact term within the Regge-effective Lagrangian approach. The differential and total cross sections for these processes are calculated with our model parameters. The total cross section ofγn→K+Σ∗−1/2− is approximately4.2 μ b aroundEγ=2.4 GeV . We encourage our experimental colleagues to further measure theγn→K+Σ∗−1/2− process. -
In this appendix, we show that the contributions from the t-channel K exchange in the forward angle (
θ=0 ) and backward angle (θ=π ) are zero. In the c.m frame, the four-momenta of the outgoing K arek02=s+m2K−M2Σ∗2√s,
k12=|→kc.m.2|sinθ,
k22=0,
k32=|→kc.m.2|cosθ,
where
→kc.m.2 is the three-momentum of K [Eq. (21)]. The polarization vectors of the photon with momentum→kc.m.1 in the helicity basis areϵ(→kc.m.1,λ=±1)=±1√2(01∓i0),
ϵ(→kc.m.1,λ=0)=0.
We can easily find that
kμ2ϵμ(→kc.m.1,λ=0,±1)=0 forθ=0,π , which implies that the amplitude of Eq. (11) for t-channel K exchange will be zero in the forward angle (θ=0 ) and backward angle (θ=π ).
Photo-production of lowest Σ∗1/2− state within the Regge-effective Lagrangian approach
- Received Date: 2023-01-20
- Available Online: 2023-05-15
Abstract: Because the lowest