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Photo-production of lowest Σ1/2 state within the Regge-effective Lagrangian approach

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Yun-He Lyu, Han Zhang, Neng-Chang Wei, Bai-Cian Ke, En Wang and Ju-Jun Xie. Photo-production of lowest Σ1/2 state within the Regge-effective Lagrangian approach[J]. Chinese Physics C. doi: 10.1088/1674-1137/acc4ab
Yun-He Lyu, Han Zhang, Neng-Chang Wei, Bai-Cian Ke, En Wang and Ju-Jun Xie. Photo-production of lowest Σ1/2 state within the Regge-effective Lagrangian approach[J]. Chinese Physics C.  doi: 10.1088/1674-1137/acc4ab shu
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Received: 2023-01-20
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Photo-production of lowest Σ1/2 state within the Regge-effective Lagrangian approach

  • 1. School of Physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, China
  • 2. School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 101408, China
  • 3. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
  • 4. Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000, China

Abstract: Because the lowest Σ state with quantum numbers spin-parity JP=1/2 is far from being established experimentally and theoretically, we perform a theoretical study on the Σ1/2 photo-production within the Regge-effective Lagrangian approach. Considering that Σ1/2 couples to the ˉKN channel, we study the contributions from the t-channel K exchange diagram. Moreover, the contributions from the t-channel K exchange, s-channel nucleon pole, u-channel Σ exchange, and contact term are considered. The differential and total cross sections of the process γnK+Σ1/2 are predicted with our model parameters. The results should help in experimentally searching for the Σ1/2 state in the future.

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    I.   INTRODUCTION
    • The study of the low-lying excited Λ and Σ hyperon resonances is one of the most important issues in hadron physics. In particular, since Λ(1405) was experimentally discovered [1, 2], its nature has garnered significant attention [38], and one explanation for Λ(1405) is the ˉKN hadronic molecular state [915]. In addition, for the isospin I=1 partner of Λ(1405), the lowest Σ1/2 is crucial to understand light baryon spectra. At present, Σ(1620) with JP=1/2 is listed in the latest version of the Review of Particle Physics (RPP) [16]. It should be emphasized that the Σ(1620) state is a one-star baryon resonance. Many studies indicate that the lowest Σ1/2resonance is still far from being established, and its mass has been predicted to lie in the range 13801500 MeV [13, 1720]. Thus, searching for the lowest Σ1/2 is helpful for understanding low-lying excited baryons with JP=1/2 and light flavor baryon spectra.

      The analyses of relevant data on the process KpΛπ+π suggest that a Σ1/2 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 of KΣ photo-production also indicate that Σ1/2 is possibly buried under the Σ(1385) peak with a mass of 1380 MeV [22], and the search for Σ1/2 in the process Λ+cηπ+Λ has been proposed [23]. A more delicate analysis of CLAS data on the process γpKΣπ [24] suggests that the Σ1/2 peak should be around 1430 MeV [13]. In Refs. [25, 26], we suggest searching for such a state in the processes χc0(1P)ˉΣΣπ and χc0(1P)ˉΛΣπ. In addition, in Ref. [27], one Σ1/2 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 process γpK+Σ01/2.

      It is worth mentioning that a Σ(1480) resonance with JP=1/2 has been listed on the previous version of the RPP [29]. As early as 1970, the Σ(1480) resonance was reported in the Λπ+, Σπ, and pˉK0 channels of π+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 the K0Sp(ˉp) invariant mass spectrum of inclusive deep inelastic ep scattering by the ZEUS Collaboration [32]. Furthermore, a signal for a resonance at 1480±15 MeV with a width of 60±15 MeV was observed in the process ppK+pY0 [33]. Σ(1480) has been investigated theoretically within different models [3437]. In Ref. [37], S-wave meson-baryon interactions with strangeness S=1 were studied within the unitary chiral approach, and one narrow pole with a pole position of 1468i 13 MeV was found in the second Riemann sheet, which may be associated with the Σ(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 and Λ, and the LEPS [38] and CLAS [24] Collaborations have accumulated considerable relevant experimental data. For instance, with these data, we analyzed the process γpKΛ(1405) to deepen our understanding of the nature of Λ(1405) in Ref. [39]. To confirm the existence of Σ(1480), we propose to investigate the process γNKΣ(1480) within the Regge-effective Lagrangian approach.

      Considering the Σ(1480) signal was first observed in the π+Λ invariant mass distribution of the process π+pπ+K+Λ, and the significance is approximately 34σ [31], we search for charged Σ(1480) in the process γnK+Σ1/2, which may also avoid the contributions of possible excited Λ 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 process γnK+Σ1/2, which helps the experimental search for Σ1/2.

      This paper is organized as follows. In Sec. II, the theoretical formalism for studying the γnK+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.

    II.   FORMALISM
    • 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 and K exchange terms, s-channel nucleon pole term, u-channel Σ exchange term, and contact term.

      Figure 1.  Mechanisms of the γnK+Σ1/2 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, and p2 represent the four-momenta of the initial photon, kaon, neutron, and Σ(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, 4045],

      LγKK=ie[K(μK)(μK)K]Aμ,

      (1)

      LγKK=gγKKϵμναβμAναKβK,

      (2)

      LγNN=eˉN[γμˆeˆκN2MNσμνν]AμN,

      (3)

      LγΣΣ=eμΣΣ2MNˉΣγ5σμννAμΣ+h.c.,

      (4)

      LKNΣ=igKNΣˉNγ5ΣK+h.c.,

      (5)

      LKNΣ=igKNΣ3ˉKμˉΣγμγ5N+h.c.,

      (6)

      LKNΣ=gKNΣˉKˉΣN+h.c.,

      (7)

      where e(=4πα) is the elementary charge unit, Aμ is the photon field, and ˆe(1+τ3)/2 denotes the charge operator acting on the nucleon field. ˆκNκpˆe+κn(1ˆe) is the anomalous magnetic moment, and we take κn=1.913 for the neutron [16]. MN and MΣ denote the masses of the nucleon and the ground-state of the Σ hyperon, respectively. The strong coupling gKNΣ is taken to be 4.09 from Refs. [4648]. gγKK=0.254 GeV1 is determined from the experimental data of ΓKK+γ [16], and the value of gKNΣ=3.26i 0.06 is taken from Ref. [27]. In addition, the coupling gKNΣ=8.74 GeV is taken from Ref. [37], and the transition magnetic moment μΣΣ=1.28 is taken from Ref. [28].

      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 and uN 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 amplitudes Mμh are written as

      MμK=gγKKgKNΣ3(tM2K)ϵαβμνk1αk2βγνγ5,

      (10)

      MμK=2iegKNΣtM2Kkμ2,

      (11)

      MμΣ=ieμΣΣgKNΣ2Mn(uM2Σ)(uMΣ)σμνk1ν,

      (12)

      Mμn=κngKNΣ2Mn(sM2n)σμνk1ν(s+Mn).

      (13)

      To maintain gauge invariance in the full photoproduction amplitudes considered here, we adopt the amplitude of the contact term

      Mμc=iegKNΣpμ2p2k1,

      (14)

      for γnK+Σ1/2.

      It is known that the Reggeon exchange mechanism plays a crucial role at high energies and forward angles [5154]; 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 follows

      1tM2KFReggeK=(ssK0)αK(t)παKsin(παK(t))Γ(1+αK(t)),

      (15)

      1tM2KFReggeK=(ssK0)αK(t)1παKsin(παK(t))Γ(αK(t)),

      (16)

      where αK(t)=0.7GeV2×(tM2K) and αK(t)=1+0.83GeV2×(tM2K) are the linear Reggeon trajectories. The constants sK0 and sK0 are determined to be 3.0 GeV2 and 1.5 GeV2, respectively [55]. Here, αK and αK are the Regge-slopes.

      Then, the full photo-production amplitudes for the γnK+Σ1/2 reaction can be expressed as

      Mμ=(MμK+Mμc)(tM2K)FReggeK+MμΣfu+MμK(tM2K)FReggeK+Mμnfs,

      (17)

      where FReggeK and FReggeK represent the Regge propagators. The form factors fs and fu 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+(q2iM2i)2]2,i=s,u

      (18)

      where Mi and qi are the masses and four-momenta of the intermediate baryons, and Λi represents the cut-off values for the baryon exchange diagrams. In this study, we take Λs=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 γnKΣ1/2 reaction reads as

      dσdΩ=MNMΣ|kc.m.1||kc.m.2|8π2(sM2N)2λ,sp,sΣ|M|2,

      (19)

      where s denotes the invariant mass square of the center of mass (c.m.) frame for Σ1/2 photo-production, and dΩ=2πdcosθc.m., with θc.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|=sM2N2s,

      (20)

      |kc.m.2|=[s(MΣ+MK)2][s(MΣMK)2]2s.

      (21)
    III.   NUMERICAL RESULTS AND DISCUSSIONS
    • In this section, we present our numerical results on the differential and total cross sections for the γnK+Σ1/2 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 of Σ(1480) are M=1480±15 MeV and Γ=60±15 MeV, respectively [29].

      ParticleMass/MeV
      n939.565
      Σ1197.449
      K+493.677
      K493.677
      K891.66

      Table 1.  Particle masses used in this study.

      First, we show the angle dependence of the differential cross sections for the γnK+Σ1/2 reaction in Fig. 2, where the center-of-mass energies W=s varies from 2.0 to 2.8GeV. 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-channel K 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 process γnK+Σ1/2, mainly because of the Regge effects of t-channel K exchange. The K-Reggeon exchange exhibits a steadily increasing behavior with cosθc.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-channel K exchange term is small and can be safely neglected for the process γnK+Σ1/2, which is consistent with the results of Ref. [28].

      Figure 2.  (color online) γnK+Σ1/2 differential cross sections as a function of cosθc.m. are plotted for γn-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-channel K 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 for KNΣ does not affect our results significantly.

      Figure 3.  (color online) γnK+Σ1/2 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 γnK+Σ1/2 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-channel K exchange diagrams, respectively, whereas the cyan-dot-dashed curve represents the contribution of the contact term. For the γnK+Σ1/2 reaction, its total cross section attains a maximum value of approximately 4.2 μb at Eγ=2.4 GeV. It is expected that Σ(1480) can be observed by future experiments in the process γnK+Σ(1480)Σπ0/Σ0π/Σγ.

      Figure 4.  (color online) Total cross section for γnK+Σ1/2 is plotted as a function of the lab energy Eγ. 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-channel K exchange diagrams, respectively, whereas the cyan-dot-dashed curve represents the contribution of the contact term.

      Finally, we show the total cross section for γnK+Σ1/2 with the cut-off Λs/u=1.2, 1.5, and 1.8 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 of Σ(1480) are still unknown, future experiments would benefit from constraining these couplings if the state Σ(1480) is confirmed.

      Figure 5.  (color online) Total cross section for γnK+Σ1/2 with the cut-off Λs/u=1.2, 1.5, and 1.8 GeV.

    IV.   SUMMARY
    • The lowest Σ1/2 is far from established, and its existence is important to understand low-lying excited baryons with JP=1/2. There are many experimental hints of Σ(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 state Σ(1480) has a sizeable coupling to ˉKN according to the study of Ref. [37], we phenomenologically investigate the γnK+Σ1/2 reaction by considering the contributions from the t-channel K/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 of γnK+Σ1/2 is approximately 4.2 μb around Eγ=2.4 GeV. We encourage our experimental colleagues to further measure the γnK+Σ1/2 process.

    APPENDIX A: t-CHANNEL K EXCHANGE IN THE FORWARD/BACKWARD ANGLE
    • 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 are

      k02=s+m2KM2Σ2s,

      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 are

      ϵ(kc.m.1,λ=±1)=±12(01i0),

      ϵ(kc.m.1,λ=0)=0.

      We can easily find that kμ2ϵμ(kc.m.1,λ=0,±1)=0 for θ=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).

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