-
The
a1(1260) resonance with quantum numbersJPC=1++ is a candidate for the chiral partner ofρ meson [1-3]. It is also described as aq¯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 ofa1(1260) has also been investigated using theτ decay spectrum into three pions [10-12], and multi-pion decays of light vector mesons [13, 14]. Recently, thea1(1260) resonance was studied in Ref. [15] in the decay ofτ→ντπ−a1(1260) through a triangle mechanism.The dynamically generated nature of
a1(1260) has been tested in the radiative decay process. The decay ofa1(1260) intoπγ in Ref. [16] was also studied in Refs. [17, 18] and found to be in agreement with the experimental data ifa1(1260) is associated with the dynamically generated picture. In Ref. [19], the lattice result for the coupling constant ofa1(1260) into theρπ channel is similar to the one obtained in Ref. [8]. Recently, the production ofa1(1260) in theπ−p→a1(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 elementaryqˉq component ofa1(1260) is comparable to the hadronic composite [21-23]. Using the chiral unitary approach, the largeNc behavior of thea1(1260) state was investigated in Ref. [22], and it was found thatqˉq is not the main component ofa1(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) isMa1(1260) = 1230± 40 MeV andΓa1(1260)=(250−600) MeV, respectively [26]①. The large uncertainties of the mass and width ofa1(1260) in the Particle Data Group (PDG) [26] show that the knowledge ofa1(1260) is very limited. Therefore, a study ofa1(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 theπγ radiative width ofa1(1260) very likely exceeds that ofa2(1320) [16, 34-36]. The absence of theJPC=1++ state in charge exchange photoproduction is puzzling. In this paper, by investigating theγp→a1(1260)+n process within theπ -exchange mechanism, we calculate its total cross-section. Theπ+π+π− mass distribution and the total cross-section ofγp→π+π+π−n are studied. In addition, we consider the non-resonance contributions to theγp→π+π+π−n resonance, which involve nucleon pole terms. Other contributions, which involveΔ(1232) and nucleon excited states, can be removed based on theπ+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 ofa1(1260) is discussed and theπ+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 thea1(1260) production in theγp→ a1(1260)+n reaction via theπ -exchange process.For the
πNN vertex, we adopt the commonly used effective LagrangianL=−igπNNˉNγ5(→τ⋅→π)N=−igπNN(ˉpγ5pπ0+√2ˉpγ5nπ++√2ˉnγ5pπ−−ˉnγ5nπ0),
(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 theπNN vertex, the form factor satisfies the relationFπNN(qπ)=Λ2π−m2πΛ2π−q2π,
(2) where
Λπ is a cut-off parameter [39, 40], which will be discussed in the following.qπ is the momentum of the exchangedπ meson.The vertex depicting the interaction of
a1(1260) andπγ is [17, 18]ta+1→π+γ=ga1πγ(gμν−pμγpνa1pγ⋅pa1)εμ(pa1)εν(pγ),
(3) where
εμ(pa1) andεν(pγ) are the polarization vectors corresponding toa1(1260) and photon, respectively.With the vertex above, we can easily get the partial decay width of
a1→πγ ,Γa1→πγ=g2a1πγ24πM3a1(M2a1−m2π),
(4) where
Ma1=1230 MeV is the nominal mass ofa1(1260) . Using the partial decay widthΓa1→πγ=640±246 keV ofa1(1260) as listed in PDG [26], we getga1πγ=244±94 MeV, where the error is from the uncertainties ofΓa1→πγ and the mass ofa1(1260) . In the following calculations, we take the average valuega1πγ=244 MeV.With the above integrants, one can get the scattering amplitude of the
γ(p1)p(p2)→a1(1260)+(p4)+n(p3) process asM=−√2igπNNga1πγq2π−m2πˉu(p3)γ5u(p2)×(gμν−pμ1pν4p1⋅p4)εμ(p4)εν(p1)FπNN(qπ).
(5) By defining
s=(p1+p2)2 , the corresponding unpolarized differential cross-section readsdσdcosθ=132πs|→p c.m.4||→p c.m.1|(14∑spins|M|2),
(6) where
θ is the scattering angle ofa+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 finala+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 viaπ exchange increases very rapidly close to the threshold, and the peak position of the total cross-section isEγ∼2.6 GeV. The total cross-section is proportional tog2a1πγ , which indicates that the cross-section is proportional to the partial decay widthΓa1→πγ . Since the exact value ofΓa1→πγ is not determined by theory or experiment, in this work we takeΓa1→πγ=640 keV. The result is comparable with the cross-section ofa2(1320) photoproduction [41]. -
Next, we consider the
γp→a1(1260)+n→ρ0π+n→ π+π+π−n andγp→ρ0p→π+π+π−n processes. Hereγp→ρ0p→π+π+π−n can occur via the nucleon pole term [42]. -
The
γp→a1(1260)+n→ρ0π+n→π+π+π−n reaction withπ exchange is shown in Fig. 3, where the relevant kinematic variables are shown. As discussed in the introduction, we take the coupling ofa1(1260) to theρπ channel as obtained in Ref. [8].The
a+1ρ0π+ vertex can be written as−it1=−iga1ρπ√2εμa1εμ,
(7) where
εa1 andε are the polarization vectors ofa1(1260) andρ , respectively.ga1ρπ is the coupling ofa1(1260) toρπ . We takega1ρπ=(−3795+i2330) MeV as obtained in Ref. [8], where only theS -wave interaction was considered. Note that there is also aD -wave contribution to thea1ρπ vertex as investigated in Ref. [43], where theD -wave contribution was found to be small.For the vertex of
a1(1260)+ interacting withρ0π+ , we also introduce a form factorFa1ρπ , which isFa1ρπ(qa1)=Λ4a1Λ4a1+(q2a1−M2a1)2,
(8) with a typical value of
Λa1=1.5 GeV as in Refs. [20, 44].The
a1(1260) propagator isGαβa1(qa1)=i−gαβ+qαa1qβa1/M2a1q2a1−M2a1+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 processa1(1260)→ ρπ→3π [44], andΓ0 is the decay width for the other processes. Following the experimental result in Ref. [24] for the total decay width ofa1(1260) , we takeΓ0=201 MeV forΓa1=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 inSU(3) , andg=mV2f , withmV=mρ , andf=93 MeV is the pion decay constant. Theρππ vertex can then be written as−it=−i√2g(p7−p6)λελ(p4).
(12) For the vertex of
ρ interacting withππ , we also introduce a form factorFρππ , which satisfies the relationFρππ(qρ)=Λ4ρΛ4ρ+(q2ρ−m2ρ)2,
(13) with a typical value of
Λρ=1.5 GeV as used in Ref. [44].The
ρ propagator isGσλρ(qρ)=i−gσλ+qσρqλρ/m2ρq2ρ−m2ρ+imρΓρ,
(14) where
Γρ is energy dependent. Because the dominant decay channel ofρ isππ , we takeΓρ(M2inv)=Γon(qoffqon)3mρMinv,
(15) with
Γon=149.1 MeV, andqon=√m2ρ−4m2π2,
(16) qoff=√M2inv−4m2π2,
(17) where
M2inv=q2ρ=(p6+p7)2 or(p5+p7)2 is the invariant mass squared of theπ+π− system. We takemρ=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 theρ into two pions.We finally obtain the scattering amplitude for the diagram shown in Fig. 3,
MI=√2igπNNga1πγq2π−m2πˉu(p3)γ5u(p2)(gμν−pμ1qνa1p1⋅qa1)×εν(p1)Ga1μσ(qa1)FπNN(qπ)Fa1ρπ(qa1)(gρπg)×(Gσλρ(p6+p7)(p7−p6)λFρππ(p6+p7)+(Gσλρ(p5+p7)(p7−p5)λFρππ(p5+p7)).
(18) -
Besides the resonance contribution of the
a1(1260) resonance, we study another kind of reaction mechanism for theγp→π+π+π−n reaction, which is depicted in Fig. 4, where we have considered the contribution fromγp→ρ0p→π+π−π+n . In Fig. 4, the relevant kinematic variables are also shown.To compute the contribution of Fig. 4, we take the interaction density for
ργπ as [49, 50],Lργπ=egργπmρϵμναβ∂μρν∂αAβπ,
(19) where
Aβ,π andρν denote the fields of the photon,π andρ , respectively. The coupling constantgργπ can be obtained from the experimental decay widthΓρ0→π0γ [26] , which leads togργπ=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=−√2gπNNgπNNq2π−m2πegργπmρgFπNN(qπ)ˉu(p3)γ5×((p3/+p5/ )+mp(p3+p5)2−m2pγ5u(p2)FN(p3+p5)ϵμναβ×(p6+p7)αp1βϵνGμσρ(p6+p7)(p7−p6)σFρππ(p6+p7)+(p3/+p6/ )+mp(p3+p6)2−m2pγ5u(p2)FN(p3+p6)ϵμναβ(p5+p7)α×p1βϵνGμσρ(p5+p7)(p7−p5)σFρππ(p5+p7)),(20)
with
FN(qp)=Λ4NΛ4N+(q2p−m2p)2,
(21) where
Λπ=0.6 GeV andΛN=0.5 GeV are taken from Refs. [49, 50, 51]. This choice of the cut-off leads to a satisfactory explanation of theρ0 photoproduction at low energies. Note that the value ofΛπ is different from the one we used for theγp→na1(1260)+ production. Other cut-off parameters are the same as given above. -
The total cross-section of the
γp→π+π+π−n reaction can be obtained by integrating the invariant amplitude in the four-body phase space:dσ(γp→π+π+π−n)=12!2mp⋅2mn4|p1⋅p2|(14∑spins|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)8√s|→p ∗a6||→p ∗b5||→p3|dΩ∗a6dΩ∗b5dΩ3dMπ+π−dMπ+π+π−,
(24) where
|→p ∗a6| andΩ∗a6 are the three-momentum and solid angle of the out-goingπ+ in the c.m. frame of the finalπ+π− system,|→p ∗b5| andΩ∗b5 are the three-momentum and solid angle of the out-goingπ+ in the c.m. frame of the finalπ+π+π− system, and|→p3| andΩ3 are the three-momentum and solid angle of the out-goingn in the c.m. frame of the initialγp system. In the above equation,Mπ+π− is the invariant mass of theπ+π− two body system, andMπ+π+π− is the invariant mass of theπ+π+π− three body system, ands=(p1+p2)2 is the invariant mass squared of the initialγp system.In Ref. [33], the
γp→π+π+π−n reaction was studied in the photon energy range4.8−5.4 GeV. The 3π mass distributions are measured from the1++(ρπ)S partial wave. In Fig. 5, we show the theoretical results,c1dσ/dMπ+π+π− , for theπ+π+π− invariant mass distributions for theγp→π+π+π−n reaction atEγ=5.1 GeV, compared with the experimental measurements of Ref. [33]. The theoretical results are obtained withc1=21.5 andc1=18 forMa1=1080 and1230 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 around1.4−1.6 GeV may account for the nuclear pole contribution. If we useMa1=1080 MeV, theπ+π+π− invariant mass distributions agree well with the experimental data. On the other hand, the theoretical results withMa1=1230 MeV can not describe the bump structure around 1.1 GeV.Figure 5. The 3
π invariant mass spectrum for theγp→π+π+π−n process compared with the data obtained by the CLAS collaboration from the1++(ρπ)S partial wave [33]. Left and right plots correspond toMa1=1080 and 1230 MeV , respectively.In addition to the differential cross-section, we also calculated the total cross-section for the
γp→π+π+π−n process as a function of the photon beam energyEγ . 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 atEγ=2.5 and 2.9 GeV corresponding toMa1=1080 and1230 MeV, respectively. The differential and total cross-sections could be checked in future experiments, such as those at CLAS. -
In recent years, it has been found that the
a1(1260) resonance, although long accepted as an ordinaryq¯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 thea1(1260) resonance in the photoproduction process. Sincea1(1260) was observed in the radiative decay ofa1(1260)+→π+γ , theγp→a1(1260)+n reaction byπ meson exchange is the main process for producinga1(1260) . Our numerical results show that the total cross-section ofγp→a1(1260)+n is of the order of 10 μb, which is comparable with the cross-section for photoproduction ofa2(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, theπ+π+π− invariant mass distributions from theγp→π+π+π−n reaction were studied. WithMa1=1080 MeV, we can describe the experimental data for theπ+π+π− invariant mass distributions fairly well. The total cross-section of theγp→π+π+π−n reaction was also studied using the model parameters determined from a comparison with the experimental data for theπ+π+π− invariant mass distributions. It is expected that our model calculations could be tested by future experiments with theγp→π+π+π−n reaction at the photon beam energyEγ around 2.5~2.9 GeV.One of us (Xu Zhang) would like to thank Yin Huang for helpful discussions.
