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Study of the a1(1260) resonance in the γpπ+π+πn reaction

  • Within an effective Lagrangian approach and resonance model, we study the γpa1(1260)+n and γpπ+π+πn reactions via the π-exchange mechanism. For the γpπ+π+πn reaction, we perform a calculation of the differential and total cross-sections by considering the contributions of the a1(1260) intermediate resonance decaying into ρπ and then into π+π+π. Besides, the non-resonance process is also considered. With a lower mass of a1(1260), the experimental data for the invariant π+π+π mass distributions can be fairly well reproduced. For the γpa1(1260)+n reaction, with the model parameters, the total cross-section is of the order of 10 μb at the photon beam energy Eγ~2.5 GeV. It is expected that the model calculations in this work could be tested by future experiments.
  • The a1(1260) resonance with quantum numbers JPC=1++ is a candidate for the chiral partner of ρ meson [1-3]. It is also described as a q¯q state in the Numbu-Jona-Lasino model [4, 5]. Apart from the quark model, it is considered as a gauge boson of the local hidden symmetry [6, 7]. By using the chiral unitary approach, a1(1260) is a state arising from the interactions of pairs of hadrons in coupled channels [8, 9]. In addition, the nature of a1(1260) has also been investigated using the τ decay spectrum into three pions [10-12], and multi-pion decays of light vector mesons [13, 14]. Recently, the a1(1260) resonance was studied in Ref. [15] in the decay of τντπa1(1260) through a triangle mechanism.

    The dynamically generated nature of a1(1260) has been tested in the radiative decay process. The decay of a1(1260) into πγ in Ref. [16] was also studied in Refs. [17, 18] and found to be in agreement with the experimental data if a1(1260) is associated with the dynamically generated picture. In Ref. [19], the lattice result for the coupling constant of a1(1260) into the ρπ channel is similar to the one obtained in Ref. [8]. Recently, the production of a1(1260) in the πpa1(1260)p reaction within the effective Lagrangian approach was studied in Ref. [20] based on the results of [8]. Besides, it was found that the elementary qˉq component of a1(1260) is comparable to the hadronic composite [21-23]. Using the chiral unitary approach, the large Nc behavior of the a1(1260) state was investigated in Ref. [22], and it was found that qˉq is not the main component of a1(1260).

    Based on the values obtained by two different experimental groups [24, 25], it is estimated that the mass and Breit-Wigner width of a1(1260) is Ma1(1260) = 1230 ± 40 MeV and Γa1(1260)=(250600) MeV, respectively [26]. The large uncertainties of the mass and width of a1(1260) in the Particle Data Group (PDG) [26] show that the knowledge of a1(1260) is very limited. Therefore, a study of a1(1260) photoproduction could be helpful to determine the mass and width of this resonance.

    Meson photoproduction off a proton provides one of the most direct platforms to extract information about the hadronic structure [27, 28]. We should point out that in the experiments, no signal representing a1(1260)+n photoproduction [29-33] could be isolated even though the πγ radiative width of a1(1260) very likely exceeds that of a2(1320) [16, 34-36]. The absence of the JPC=1++ state in charge exchange photoproduction is puzzling. In this paper, by investigating the γpa1(1260)+n process within the π-exchange mechanism, we calculate its total cross-section. The π+π+π mass distribution and the total cross-section of γpπ+π+πn are studied. In addition, we consider the non-resonance contributions to the γpπ+π+πn resonance, which involve nucleon pole terms. Other contributions, which involve Δ(1232) and nucleon excited states, can be removed based on the π+n invariant mass spectrum from the experiments [33].

    The article is organized as follows. After the introduction, we present the reaction mechanism of a1(1260) photoproduction. The possible background relevant to the production of a1(1260) is discussed and the π+π+π mass distribution is presented in Sec. 3. This work ends with a discussion and conclusion.

    In this section, we discuss the a1(1260) production mechanism. Fig. 1 shows the basic tree-level Feynman diagram for the a1(1260) production in the γpa1(1260)+n reaction via the π-exchange process.

    Figure 1

    Figure 1.  Feynman diagram for the γpa1(1260)+n reaction via π-exchange.

    For the πNN vertex, we adopt the commonly used effective Lagrangian

    L=igπNNˉNγ5(τπ)N=igπNN(ˉpγ5pπ0+2ˉpγ5nπ++2ˉnγ5pπˉnγ5nπ0),

    (1)

    where the standard value, g2πNN/4π=14.4, is adopted as in Refs. [37, 38]. In addition, the form factor is introduced for suppressing the vertex coupling when one or two interacting particles go off-shell. For the πNN vertex, the form factor satisfies the relation

    FπNN(qπ)=Λ2πm2πΛ2πq2π,

    (2)

    where Λπ is a cut-off parameter [39, 40], which will be discussed in the following. qπ is the momentum of the exchanged π meson.

    The vertex depicting the interaction of a1(1260) and πγ is [17, 18]

    ta+1π+γ=ga1πγ(gμνpμγpνa1pγpa1)εμ(pa1)εν(pγ),

    (3)

    where εμ(pa1) and εν(pγ) are the polarization vectors corresponding to a1(1260) and photon, respectively.

    With the vertex above, we can easily get the partial decay width of a1πγ,

    Γa1πγ=g2a1πγ24πM3a1(M2a1m2π),

    (4)

    where Ma1=1230 MeV is the nominal mass of a1(1260). Using the partial decay width Γa1πγ=640±246 keV of a1(1260) as listed in PDG [26], we get ga1πγ=244±94 MeV, where the error is from the uncertainties of Γa1πγ and the mass of a1(1260). In the following calculations, we take the average value ga1πγ=244 MeV.

    With the above integrants, one can get the scattering amplitude of the γ(p1)p(p2)a1(1260)+(p4)+n(p3) process as

    M=2igπNNga1πγq2πm2πˉu(p3)γ5u(p2)×(gμνpμ1pν4p1p4)εμ(p4)εν(p1)FπNN(qπ).

    (5)

    By defining s=(p1+p2)2, the corresponding unpolarized differential cross-section reads

    dσdcosθ=132πs|p c.m.4||p c.m.1|(14spins|M|2),

    (6)

    where θ is the scattering angle of a+1 meson relative to the beam direction in the c.m. frame, while p c.m.1 and p c.m.4 are the three-momenta of the initial photon and the final a+1, respectively.

    In Fig. 2, the solid, dashed and dotted lines are obtained with Λπ=1.0, 1.3 and 1.6 GeV, respectively. From Fig. 2 one can see that the total cross-section via π exchange increases very rapidly close to the threshold, and the peak position of the total cross-section is Eγ2.6 GeV. The total cross-section is proportional to g2a1πγ, which indicates that the cross-section is proportional to the partial decay width Γa1πγ. Since the exact value of Γa1πγ is not determined by theory or experiment, in this work we take Γa1πγ=640 keV. The result is comparable with the cross-section of a2(1320) photoproduction [41].

    Figure 2

    Figure 2.  Dependence of the total cross-section of γpa1(1260)+n as a function of Eγ.

    Next, we consider the γpa1(1260)+nρ0π+nπ+π+πn and γpρ0pπ+π+πn processes. Here γpρ0pπ+π+πn can occur via the nucleon pole term [42].

    The γpa1(1260)+nρ0π+nπ+π+πn reaction with π exchange is shown in Fig. 3, where the relevant kinematic variables are shown. As discussed in the introduction, we take the coupling of a1(1260) to the ρπ channel as obtained in Ref. [8].

    Figure 3

    Figure 3.  Feynman diagram for the γpa1(1260)+nρ0π+nπ+π+πn reaction via π exchange.

    The a+1ρ0π+ vertex can be written as

    it1=iga1ρπ2εμa1εμ,

    (7)

    where εa1 and ε are the polarization vectors of a1(1260) and ρ, respectively. ga1ρπ is the coupling of a1(1260) to ρπ. We take ga1ρπ=(3795+i2330) MeV as obtained in Ref. [8], where only the S-wave interaction was considered. Note that there is also a D-wave contribution to the a1ρπ vertex as investigated in Ref. [43], where the D-wave contribution was found to be small.

    For the vertex of a1(1260)+ interacting with ρ0π+, we also introduce a form factor Fa1ρπ, which is

    Fa1ρπ(qa1)=Λ4a1Λ4a1+(q2a1M2a1)2,

    (8)

    with a typical value of Λa1=1.5 GeV as in Refs. [20, 44].

    The a1(1260) propagator is

    Gαβa1(qa1)=igαβ+qαa1qβa1/M2a1q2a1M2a1+iMa1Γa1,

    (9)

    where the width Γa1 is dependent on its four-momentum squared, and we can take the form as in Refs. [45, 46],

    Γa1=Γ0+Γ3π,

    (10)

    where Γ3π is the decay width for the process a1(1260)ρπ3π [44], and Γ0 is the decay width for the other processes. Following the experimental result in Ref. [24] for the total decay width of a1(1260), we take Γ0=201 MeV for Γa1=367 MeV at q2a1=1230 MeV.

    For the structure of the ρππ vertex, we use the general interaction as,

    LPPV=ig<Vμ[P,μP]>,

    (11)

    where <> stands for the trace in SU(3) , and g=mV2f, with mV=mρ , and f=93 MeV is the pion decay constant. The ρππ vertex can then be written as

    it=i2g(p7p6)λελ(p4).

    (12)

    For the vertex of ρ interacting with ππ, we also introduce a form factor Fρππ, which satisfies the relation

    Fρππ(qρ)=Λ4ρΛ4ρ+(q2ρm2ρ)2,

    (13)

    with a typical value of Λρ=1.5 GeV as used in Ref. [44].

    The ρ propagator is

    Gσλρ(qρ)=igσλ+qσρqλρ/m2ρq2ρm2ρ+imρΓρ,

    (14)

    where Γρ is energy dependent. Because the dominant decay channel of ρ is ππ, we take

    Γρ(M2inv)=Γon(qoffqon)3mρMinv,

    (15)

    with Γon=149.1 MeV, and

    qon=m2ρ4m2π2,

    (16)

    qoff=M2inv4m2π2,

    (17)

    where M2inv=q2ρ=(p6+p7)2 or (p5+p7)2 is the invariant mass squared of the π+π system. We take mρ=775.26 MeV in this work.

    It is worth to mention that the parametrization of the width of ρ meson shown in Eq. (15) is meant to take into account the phase space of each decay mode as a function of the energy [40, 47, 48]. In the present work we take explicitly the phase space for the P-wave decay of the ρ into two pions.

    We finally obtain the scattering amplitude for the diagram shown in Fig. 3,

    MI=2igπNNga1πγq2πm2πˉu(p3)γ5u(p2)(gμνpμ1qνa1p1qa1)×εν(p1)Ga1μσ(qa1)FπNN(qπ)Fa1ρπ(qa1)(gρπg)×(Gσλρ(p6+p7)(p7p6)λFρππ(p6+p7)+(Gσλρ(p5+p7)(p7p5)λFρππ(p5+p7)).

    (18)

    Besides the resonance contribution of the a1(1260) resonance, we study another kind of reaction mechanism for the γpπ+π+πn reaction, which is depicted in Fig. 4, where we have considered the contribution from γpρ0pπ+ππ+n. In Fig. 4, the relevant kinematic variables are also shown.

    Figure 4

    Figure 4.  Feynman diagram for the γpρ0π+nπ+π+πn reaction via π exchange.

    To compute the contribution of Fig. 4, we take the interaction density for ργπ as [49, 50],

    Lργπ=egργπmρϵμναβμρναAβπ,

    (19)

    where Aβ,π and ρν denote the fields of the photon, π and ρ, respectively. The coupling constant gργπ can be obtained from the experimental decay width Γρ0π0γ [26] , which leads to gργπ=0.76.

    Other vertexes are the same as given above. With the above preparation, we get the transition amplitude for the diagram shown in Fig. 4,

    MII=2gπNNgπNNq2πm2πegργπmρgFπNN(qπ)ˉu(p3)γ5×((p3/+p5/ )+mp(p3+p5)2m2pγ5u(p2)FN(p3+p5)ϵμναβ×(p6+p7)αp1βϵνGμσρ(p6+p7)(p7p6)σFρππ(p6+p7)+(p3/+p6/ )+mp(p3+p6)2m2pγ5u(p2)FN(p3+p6)ϵμναβ(p5+p7)α×p1βϵνGμσρ(p5+p7)(p7p5)σFρππ(p5+p7)),(20)

    with

    FN(qp)=Λ4NΛ4N+(q2pm2p)2,

    (21)

    where Λπ=0.6 GeV and ΛN=0.5 GeV are taken from Refs. [49, 50, 51]. This choice of the cut-off leads to a satisfactory explanation of the ρ0 photoproduction at low energies. Note that the value of Λπ is different from the one we used for the γpna1(1260)+ production. Other cut-off parameters are the same as given above.

    The total cross-section of the γpπ+π+πn reaction can be obtained by integrating the invariant amplitude in the four-body phase space:

    dσ(γpπ+π+πn)=12!2mp2mn4|p1p2|(14spins|M|2)×(2π)4dϕ4(p1+p2;p3;p5,p6,p7),

    (22)

    with

    M=MI+MII,

    (23)

    where 2! is a statistical factor for the final two π+ mesons, and the four-body phase space is defined as [26]

    dϕ4(p1+p2;p3;p5,p6,p7)=116(2π)8s|p a6||p b5||p3|dΩa6dΩb5dΩ3dMπ+πdMπ+π+π,

    (24)

    where |p a6| and Ωa6 are the three-momentum and solid angle of the out-going π+ in the c.m. frame of the final π+π system, |p b5| and Ωb5 are the three-momentum and solid angle of the out-going π+ in the c.m. frame of the final π+π+π system, and |p3| and Ω3 are the three-momentum and solid angle of the out-going n in the c.m. frame of the initial γp system. In the above equation, Mπ+π is the invariant mass of the π+π two body system, and Mπ+π+π is the invariant mass of the π+π+π three body system, and s=(p1+p2)2 is the invariant mass squared of the initial γp system.

    In Ref. [33], the γpπ+π+πn reaction was studied in the photon energy range 4.85.4 GeV. The 3π mass distributions are measured from the 1++(ρπ)S partial wave. In Fig. 5, we show the theoretical results, c1dσ/dMπ+π+π, for the π+π+π invariant mass distributions for the γpπ+π+πn reaction at Eγ=5.1 GeV, compared with the experimental measurements of Ref. [33]. The theoretical results are obtained with c1=21.5 and c1=18 for Ma1=1080 and 1230 MeV, respectively, which have been adjusted to the experimental data reported by the CLAS collaboration [33]. From Fig. 5, it is seen that the bump structure around 1.41.6 GeV may account for the nuclear pole contribution. If we use Ma1=1080 MeV, the π+π+π invariant mass distributions agree well with the experimental data. On the other hand, the theoretical results with Ma1=1230 MeV can not describe the bump structure around 1.1 GeV.

    Figure 5

    Figure 5.  The 3 π invariant mass spectrum for the γpπ+π+πn process compared with the data obtained by the CLAS collaboration from the 1++(ρπ)S partial wave [33]. Left and right plots correspond to Ma1=1080 and 1230 MeV , respectively.

    In addition to the differential cross-section, we also calculated the total cross-section for the γpπ+π+πn process as a function of the photon beam energy Eγ. The results are shown in Fig. 6, where one can see that the total cross-section increases rapidly near the threshold, and the peak of the total cross-section is at Eγ=2.5 and 2.9 GeV corresponding to Ma1=1080 and 1230 MeV, respectively. The differential and total cross-sections could be checked in future experiments, such as those at CLAS.

    Figure 6

    Figure 6.  Total cross-section for the γpπ+π+πn process. Left and right plots correspond to Ma1=1080 and 1230 MeV , respectively.

    In recent years, it has been found that the a1(1260) resonance, although long accepted as an ordinary q¯q state, can be dynamically generated from the pseudoscalar-meson-vector-meson interaction, and therefore qualify as a pseudoscalar-vector molecule. In this work, we have proposed to study the a1(1260) resonance in the photoproduction process. Since a1(1260) was observed in the radiative decay of a1(1260)+π+γ, the γpa1(1260)+n reaction by π meson exchange is the main process for producing a1(1260). Our numerical results show that the total cross-section of γpa1(1260)+n is of the order of 10 μb, which is comparable with the cross-section for photoproduction of a2(1320).

    In addition, taking the coupling constant obtained from the picture where the a1(1260) resonance is a dynamically generated state from pseudoscalar-meson-vector-meson interaction, the π+π+π invariant mass distributions from the γpπ+π+πn reaction were studied. With Ma1=1080 MeV, we can describe the experimental data for the π+π+π invariant mass distributions fairly well. The total cross-section of the γpπ+π+πn reaction was also studied using the model parameters determined from a comparison with the experimental data for the π+π+π invariant mass distributions. It is expected that our model calculations could be tested by future experiments with the γpπ+π+πn reaction at the photon beam energy Eγ around 2.5~2.9 GeV.

    One of us (Xu Zhang) would like to thank Yin Huang for helpful discussions.

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Xu Zhang and Ju-Jun Xie. Study of the a1(1260) resonance in the γpπ+π+πn reaction[J]. Chinese Physics C, 2019, 43(6): 064104-1-064104-6. doi: 10.1088/1674-1137/43/6/064104
Xu Zhang and Ju-Jun Xie. Study of the a1(1260) resonance in the γpπ+π+πn reaction[J]. Chinese Physics C, 2019, 43(6): 064104-1-064104-6.  doi: 10.1088/1674-1137/43/6/064104 shu
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Study of the a1(1260) resonance in the γpπ+π+πn reaction

    Corresponding author: Ju-Jun Xie, xiejujun@impcas.ac.cn
  • 1. Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
  • 2. University of Chinese Academy of Sciences, Beijing 101408, China

Abstract: Within an effective Lagrangian approach and resonance model, we study the γpa1(1260)+n and γpπ+π+πn reactions via the π-exchange mechanism. For the γpπ+π+πn reaction, we perform a calculation of the differential and total cross-sections by considering the contributions of the a1(1260) intermediate resonance decaying into ρπ and then into π+π+π. Besides, the non-resonance process is also considered. With a lower mass of a1(1260), the experimental data for the invariant π+π+π mass distributions can be fairly well reproduced. For the γpa1(1260)+n reaction, with the model parameters, the total cross-section is of the order of 10 μb at the photon beam energy Eγ~2.5 GeV. It is expected that the model calculations in this work could be tested by future experiments.

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    1.   Introduction
    • The a1(1260) resonance with quantum numbers JPC=1++ is a candidate for the chiral partner of ρ meson [1-3]. It is also described as a q¯q state in the Numbu-Jona-Lasino model [4, 5]. Apart from the quark model, it is considered as a gauge boson of the local hidden symmetry [6, 7]. By using the chiral unitary approach, a1(1260) is a state arising from the interactions of pairs of hadrons in coupled channels [8, 9]. In addition, the nature of a1(1260) has also been investigated using the τ decay spectrum into three pions [10-12], and multi-pion decays of light vector mesons [13, 14]. Recently, the a1(1260) resonance was studied in Ref. [15] in the decay of τντπa1(1260) through a triangle mechanism.

      The dynamically generated nature of a1(1260) has been tested in the radiative decay process. The decay of a1(1260) into πγ in Ref. [16] was also studied in Refs. [17, 18] and found to be in agreement with the experimental data if a1(1260) is associated with the dynamically generated picture. In Ref. [19], the lattice result for the coupling constant of a1(1260) into the ρπ channel is similar to the one obtained in Ref. [8]. Recently, the production of a1(1260) in the πpa1(1260)p reaction within the effective Lagrangian approach was studied in Ref. [20] based on the results of [8]. Besides, it was found that the elementary qˉq component of a1(1260) is comparable to the hadronic composite [21-23]. Using the chiral unitary approach, the large Nc behavior of the a1(1260) state was investigated in Ref. [22], and it was found that qˉq is not the main component of a1(1260).

      Based on the values obtained by two different experimental groups [24, 25], it is estimated that the mass and Breit-Wigner width of a1(1260) is Ma1(1260) = 1230 ± 40 MeV and Γa1(1260)=(250600) MeV, respectively [26]. The large uncertainties of the mass and width of a1(1260) in the Particle Data Group (PDG) [26] show that the knowledge of a1(1260) is very limited. Therefore, a study of a1(1260) photoproduction could be helpful to determine the mass and width of this resonance.

      Meson photoproduction off a proton provides one of the most direct platforms to extract information about the hadronic structure [27, 28]. We should point out that in the experiments, no signal representing a1(1260)+n photoproduction [29-33] could be isolated even though the πγ radiative width of a1(1260) very likely exceeds that of a2(1320) [16, 34-36]. The absence of the JPC=1++ state in charge exchange photoproduction is puzzling. In this paper, by investigating the γpa1(1260)+n process within the π-exchange mechanism, we calculate its total cross-section. The π+π+π mass distribution and the total cross-section of γpπ+π+πn are studied. In addition, we consider the non-resonance contributions to the γpπ+π+πn resonance, which involve nucleon pole terms. Other contributions, which involve Δ(1232) and nucleon excited states, can be removed based on the π+n invariant mass spectrum from the experiments [33].

      The article is organized as follows. After the introduction, we present the reaction mechanism of a1(1260) photoproduction. The possible background relevant to the production of a1(1260) is discussed and the π+π+π mass distribution is presented in Sec. 3. This work ends with a discussion and conclusion.

    2.   γpa1(1260)+n reaction
    • In this section, we discuss the a1(1260) production mechanism. Fig. 1 shows the basic tree-level Feynman diagram for the a1(1260) production in the γpa1(1260)+n reaction via the π-exchange process.

      Figure 1.  Feynman diagram for the γpa1(1260)+n reaction via π-exchange.

      For the πNN vertex, we adopt the commonly used effective Lagrangian

      L=igπNNˉNγ5(τπ)N=igπNN(ˉpγ5pπ0+2ˉpγ5nπ++2ˉnγ5pπˉnγ5nπ0),

      (1)

      where the standard value, g2πNN/4π=14.4, is adopted as in Refs. [37, 38]. In addition, the form factor is introduced for suppressing the vertex coupling when one or two interacting particles go off-shell. For the πNN vertex, the form factor satisfies the relation

      FπNN(qπ)=Λ2πm2πΛ2πq2π,

      (2)

      where Λπ is a cut-off parameter [39, 40], which will be discussed in the following. qπ is the momentum of the exchanged π meson.

      The vertex depicting the interaction of a1(1260) and πγ is [17, 18]

      ta+1π+γ=ga1πγ(gμνpμγpνa1pγpa1)εμ(pa1)εν(pγ),

      (3)

      where εμ(pa1) and εν(pγ) are the polarization vectors corresponding to a1(1260) and photon, respectively.

      With the vertex above, we can easily get the partial decay width of a1πγ,

      Γa1πγ=g2a1πγ24πM3a1(M2a1m2π),

      (4)

      where Ma1=1230 MeV is the nominal mass of a1(1260). Using the partial decay width Γa1πγ=640±246 keV of a1(1260) as listed in PDG [26], we get ga1πγ=244±94 MeV, where the error is from the uncertainties of Γa1πγ and the mass of a1(1260). In the following calculations, we take the average value ga1πγ=244 MeV.

      With the above integrants, one can get the scattering amplitude of the γ(p1)p(p2)a1(1260)+(p4)+n(p3) process as

      M=2igπNNga1πγq2πm2πˉu(p3)γ5u(p2)×(gμνpμ1pν4p1p4)εμ(p4)εν(p1)FπNN(qπ).

      (5)

      By defining s=(p1+p2)2, the corresponding unpolarized differential cross-section reads

      dσdcosθ=132πs|p c.m.4||p c.m.1|(14spins|M|2),

      (6)

      where θ is the scattering angle of a+1 meson relative to the beam direction in the c.m. frame, while p c.m.1 and p c.m.4 are the three-momenta of the initial photon and the final a+1, respectively.

      In Fig. 2, the solid, dashed and dotted lines are obtained with Λπ=1.0, 1.3 and 1.6 GeV, respectively. From Fig. 2 one can see that the total cross-section via π exchange increases very rapidly close to the threshold, and the peak position of the total cross-section is Eγ2.6 GeV. The total cross-section is proportional to g2a1πγ, which indicates that the cross-section is proportional to the partial decay width Γa1πγ. Since the exact value of Γa1πγ is not determined by theory or experiment, in this work we take Γa1πγ=640 keV. The result is comparable with the cross-section of a2(1320) photoproduction [41].

      Figure 2.  Dependence of the total cross-section of γpa1(1260)+n as a function of Eγ.

    3.   γpπ+π+πn reaction
    • Next, we consider the γpa1(1260)+nρ0π+nπ+π+πn and γpρ0pπ+π+πn processes. Here γpρ0pπ+π+πn can occur via the nucleon pole term [42].

    • 3.1.   γpa1(1260)+nρ0π+nπ+π+πn reaction

    • The γpa1(1260)+nρ0π+nπ+π+πn reaction with π exchange is shown in Fig. 3, where the relevant kinematic variables are shown. As discussed in the introduction, we take the coupling of a1(1260) to the ρπ channel as obtained in Ref. [8].

      Figure 3.  Feynman diagram for the γpa1(1260)+nρ0π+nπ+π+πn reaction via π exchange.

      The a+1ρ0π+ vertex can be written as

      it1=iga1ρπ2εμa1εμ,

      (7)

      where εa1 and ε are the polarization vectors of a1(1260) and ρ, respectively. ga1ρπ is the coupling of a1(1260) to ρπ. We take ga1ρπ=(3795+i2330) MeV as obtained in Ref. [8], where only the S-wave interaction was considered. Note that there is also a D-wave contribution to the a1ρπ vertex as investigated in Ref. [43], where the D-wave contribution was found to be small.

      For the vertex of a1(1260)+ interacting with ρ0π+, we also introduce a form factor Fa1ρπ, which is

      Fa1ρπ(qa1)=Λ4a1Λ4a1+(q2a1M2a1)2,

      (8)

      with a typical value of Λa1=1.5 GeV as in Refs. [20, 44].

      The a1(1260) propagator is

      Gαβa1(qa1)=igαβ+qαa1qβa1/M2a1q2a1M2a1+iMa1Γa1,

      (9)

      where the width Γa1 is dependent on its four-momentum squared, and we can take the form as in Refs. [45, 46],

      Γa1=Γ0+Γ3π,

      (10)

      where Γ3π is the decay width for the process a1(1260)ρπ3π [44], and Γ0 is the decay width for the other processes. Following the experimental result in Ref. [24] for the total decay width of a1(1260), we take Γ0=201 MeV for Γa1=367 MeV at q2a1=1230 MeV.

      For the structure of the ρππ vertex, we use the general interaction as,

      LPPV=ig<Vμ[P,μP]>,

      (11)

      where <> stands for the trace in SU(3) , and g=mV2f, with mV=mρ , and f=93 MeV is the pion decay constant. The ρππ vertex can then be written as

      it=i2g(p7p6)λελ(p4).

      (12)

      For the vertex of ρ interacting with ππ, we also introduce a form factor Fρππ, which satisfies the relation

      Fρππ(qρ)=Λ4ρΛ4ρ+(q2ρm2ρ)2,

      (13)

      with a typical value of Λρ=1.5 GeV as used in Ref. [44].

      The ρ propagator is

      Gσλρ(qρ)=igσλ+qσρqλρ/m2ρq2ρm2ρ+imρΓρ,

      (14)

      where Γρ is energy dependent. Because the dominant decay channel of ρ is ππ, we take

      Γρ(M2inv)=Γon(qoffqon)3mρMinv,

      (15)

      with Γon=149.1 MeV, and

      qon=m2ρ4m2π2,

      (16)

      qoff=M2inv4m2π2,

      (17)

      where M2inv=q2ρ=(p6+p7)2 or (p5+p7)2 is the invariant mass squared of the π+π system. We take mρ=775.26 MeV in this work.

      It is worth to mention that the parametrization of the width of ρ meson shown in Eq. (15) is meant to take into account the phase space of each decay mode as a function of the energy [40, 47, 48]. In the present work we take explicitly the phase space for the P-wave decay of the ρ into two pions.

      We finally obtain the scattering amplitude for the diagram shown in Fig. 3,

      MI=2igπNNga1πγq2πm2πˉu(p3)γ5u(p2)(gμνpμ1qνa1p1qa1)×εν(p1)Ga1μσ(qa1)FπNN(qπ)Fa1ρπ(qa1)(gρπg)×(Gσλρ(p6+p7)(p7p6)λFρππ(p6+p7)+(Gσλρ(p5+p7)(p7p5)λFρππ(p5+p7)).

      (18)
    • 3.2.   γpρ0pππ+pπ+π+πn reaction

    • Besides the resonance contribution of the a1(1260) resonance, we study another kind of reaction mechanism for the γpπ+π+πn reaction, which is depicted in Fig. 4, where we have considered the contribution from γpρ0pπ+ππ+n. In Fig. 4, the relevant kinematic variables are also shown.

      Figure 4.  Feynman diagram for the γpρ0π+nπ+π+πn reaction via π exchange.

      To compute the contribution of Fig. 4, we take the interaction density for ργπ as [49, 50],

      Lργπ=egργπmρϵμναβμρναAβπ,

      (19)

      where Aβ,π and ρν denote the fields of the photon, π and ρ, respectively. The coupling constant gργπ can be obtained from the experimental decay width Γρ0π0γ [26] , which leads to gργπ=0.76.

      Other vertexes are the same as given above. With the above preparation, we get the transition amplitude for the diagram shown in Fig. 4,

      MII=2gπNNgπNNq2πm2πegργπmρgFπNN(qπ)ˉu(p3)γ5×((p3/+p5/ )+mp(p3+p5)2m2pγ5u(p2)FN(p3+p5)ϵμναβ×(p6+p7)αp1βϵνGμσρ(p6+p7)(p7p6)σFρππ(p6+p7)+(p3/+p6/ )+mp(p3+p6)2m2pγ5u(p2)FN(p3+p6)ϵμναβ(p5+p7)α×p1βϵνGμσρ(p5+p7)(p7p5)σFρππ(p5+p7)),(20)

      with

      FN(qp)=Λ4NΛ4N+(q2pm2p)2,

      (21)

      where Λπ=0.6 GeV and ΛN=0.5 GeV are taken from Refs. [49, 50, 51]. This choice of the cut-off leads to a satisfactory explanation of the ρ0 photoproduction at low energies. Note that the value of Λπ is different from the one we used for the γpna1(1260)+ production. Other cut-off parameters are the same as given above.

    • 3.3.   Numerical results

    • The total cross-section of the γpπ+π+πn reaction can be obtained by integrating the invariant amplitude in the four-body phase space:

      dσ(γpπ+π+πn)=12!2mp2mn4|p1p2|(14spins|M|2)×(2π)4dϕ4(p1+p2;p3;p5,p6,p7),

      (22)

      with

      M=MI+MII,

      (23)

      where 2! is a statistical factor for the final two π+ mesons, and the four-body phase space is defined as [26]

      dϕ4(p1+p2;p3;p5,p6,p7)=116(2π)8s|p a6||p b5||p3|dΩa6dΩb5dΩ3dMπ+πdMπ+π+π,

      (24)

      where |p a6| and Ωa6 are the three-momentum and solid angle of the out-going π+ in the c.m. frame of the final π+π system, |p b5| and Ωb5 are the three-momentum and solid angle of the out-going π+ in the c.m. frame of the final π+π+π system, and |p3| and Ω3 are the three-momentum and solid angle of the out-going n in the c.m. frame of the initial γp system. In the above equation, Mπ+π is the invariant mass of the π+π two body system, and Mπ+π+π is the invariant mass of the π+π+π three body system, and s=(p1+p2)2 is the invariant mass squared of the initial γp system.

      In Ref. [33], the γpπ+π+πn reaction was studied in the photon energy range 4.85.4 GeV. The 3π mass distributions are measured from the 1++(ρπ)S partial wave. In Fig. 5, we show the theoretical results, c1dσ/dMπ+π+π, for the π+π+π invariant mass distributions for the γpπ+π+πn reaction at Eγ=5.1 GeV, compared with the experimental measurements of Ref. [33]. The theoretical results are obtained with c1=21.5 and c1=18 for Ma1=1080 and 1230 MeV, respectively, which have been adjusted to the experimental data reported by the CLAS collaboration [33]. From Fig. 5, it is seen that the bump structure around 1.41.6 GeV may account for the nuclear pole contribution. If we use Ma1=1080 MeV, the π+π+π invariant mass distributions agree well with the experimental data. On the other hand, the theoretical results with Ma1=1230 MeV can not describe the bump structure around 1.1 GeV.

      Figure 5.  The 3 π invariant mass spectrum for the γpπ+π+πn process compared with the data obtained by the CLAS collaboration from the 1++(ρπ)S partial wave [33]. Left and right plots correspond to Ma1=1080 and 1230 MeV , respectively.

      In addition to the differential cross-section, we also calculated the total cross-section for the γpπ+π+πn process as a function of the photon beam energy Eγ. The results are shown in Fig. 6, where one can see that the total cross-section increases rapidly near the threshold, and the peak of the total cross-section is at Eγ=2.5 and 2.9 GeV corresponding to Ma1=1080 and 1230 MeV, respectively. The differential and total cross-sections could be checked in future experiments, such as those at CLAS.

      Figure 6.  Total cross-section for the γpπ+π+πn process. Left and right plots correspond to Ma1=1080 and 1230 MeV , respectively.

    4.   Conclusion and discussion
    • In recent years, it has been found that the a1(1260) resonance, although long accepted as an ordinary q¯q state, can be dynamically generated from the pseudoscalar-meson-vector-meson interaction, and therefore qualify as a pseudoscalar-vector molecule. In this work, we have proposed to study the a1(1260) resonance in the photoproduction process. Since a1(1260) was observed in the radiative decay of a1(1260)+π+γ, the γpa1(1260)+n reaction by π meson exchange is the main process for producing a1(1260). Our numerical results show that the total cross-section of γpa1(1260)+n is of the order of 10 μb, which is comparable with the cross-section for photoproduction of a2(1320).

      In addition, taking the coupling constant obtained from the picture where the a1(1260) resonance is a dynamically generated state from pseudoscalar-meson-vector-meson interaction, the π+π+π invariant mass distributions from the γpπ+π+πn reaction were studied. With Ma1=1080 MeV, we can describe the experimental data for the π+π+π invariant mass distributions fairly well. The total cross-section of the γpπ+π+πn reaction was also studied using the model parameters determined from a comparison with the experimental data for the π+π+π invariant mass distributions. It is expected that our model calculations could be tested by future experiments with the γpπ+π+πn reaction at the photon beam energy Eγ around 2.5~2.9 GeV.

      One of us (Xu Zhang) would like to thank Yin Huang for helpful discussions.

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