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Photoproduction of Theta baryon from the neutron

We study photoproduction reactions for Theta^+, which was recently observed as a candidate of an exotic five quark state uudd(bar s). We employ effective lagrangian interactions in the linear and non-linear formalism of chiral symmetry and investigate two cases of spin and parity of Theta^+, J^P = 1/2^+ and 1/2^-. We calculate cross sections and discuss possibilities to discriminate one from the other.


I. INTRODUCTION
Recent discovery of the pentaquark baryon Θ + with strangeness S = +1 has triggered a diverse interest in a new type of hadron containing more than three quarks. Experimental evidence was first reported by LEPS group at SPring-8 [1] in a reaction γn → K + K − n. This result was subsequently confirmed by DIANA (ITEP) [2], CLAS (J-Lab) and SAPHIR [3] collaborations. So far the mass and width of Θ + were observed with a relatively small mass M Θ ∼ 1540 MeV and a very narrow width Γ < ∼ 25 MeV. Experimentally, other properties are not yet known. On the contrary, there are several theoretical studies for some time using various effective models [4,5,6]. Also, there are numbers of preprints that attempt to explain the nature of Θ + [7,8,9,10,11,12,13,14,15,16,17,18]. Perhaps the most attractive predictions were made by Diakonov et al. in the chiral soliton model [5]. Using a few input parameters, the masses and decay widths of various channels were calculated with remarkable agreement with experimental data. In this model, Θ + was identified with a member of flavor antidecuplet10 with S = +1, isospin I = 0 and spin-parity J P = 1/2 + .
A peculiarity of these predictions is the positive parity of Θ + , since in a naive valence quark model, the ground state of the five quark state must have negative parity. There are several suggestions which brings down the 1/2 + state lower than the 1/2 − state [8,12,13,18].
For instance, in Ref. [12], the role of the hedgehog pion was emphasized, while in Ref. [13], diquark correlations were investigated. Definitely, more theoretical and experimental studies are needed in order to further understand the nature of Θ + .
The purpose of this paper is to provide a theoretical estimate for photo-induced Θ + production reaction γn → K − Θ + , and present various cross sections which will be utilized in further experiments. In actual experiments, the final state Θ + decays into KN eventually.
Here, we do not consider this part for the following reasons. First, the fact that the observed peak of the resonance is very sharp with a narrow width implies that at least in the vicinity of the resonance position the amplitude is dominated by Θ + terms with negligible interference effects with background terms. Second, the theoretical situation for computing the background process is not well under control. Hence our estimation given here is a crude one, but will provide a guide line for the study of the relevant reactions.
Here we concentrate on the process of the neutron target exclusively. In this particular process we investigate in rather detail theoretical aspects of this reaction, since so far there are ambiguities that we can not settle, associated with reaction mechanism, interactions and Θ + structure. Nevertheless, we attempt to draw some conclusions within acceptable approximations and ranges of unknown parameters. Furthermore, we consider a possibility to get information on some properties of Θ + from this reaction. In particular, we consider the Θ + production for two possible cases of positive and negative parities of Θ + . The spin and isospin are assumed to be J = 1/2 and I = 0. Motivated by theoretical interests, we perform calculations using effective lagrangians of sigma models of linear (with a pseudoscalar coupling, PS) and non-linear (with a pseudovector coupling, PV) types. Equivalence of the two methods in a certain case is used as a consistency check of our calculations. For quantitative discussions, we employ a hybrid model (HM) combining the PS and PV schemes with a suitable weight depending on the energy. We do not include heavier mesons such as scalar and vector mesons, since we do not know their interactions with Θ + . They may become important in actual processes. In this sense, our model lagrangians are minimal.
In a recent preprint, Liu and Ko studied various reactions producing Θ + , including photon and hadron induced processes [19]. For a photon induced process, they studied the case of a proton target, where they have assumed that Θ + has positive parity. This paper is organized as follow. In section II, we formulate effective lagrangians of sigma model type including the kaon and Θ + . Scattering amplitudes are then computed at the tree level. Results for numerical calculations are presented in section III, where various cross sections are shown for cases of positive and negative parity of Θ + . In the final section IV, we discuss results and summarize the present work.

II. EFFECTIVE LAGRANGIANS AND AMPLITUDES
We consider interaction terms for the calculation of the reaction γn → K − Θ + at the tree level (in literatures, these terms are often referred to as the Born terms). In many cases, we show the formulae for the positive parity Θ + . The interactions with the negative parity Θ + may be obtained by removing γ 5 in the following equations such as (1) and (2). The relevant interaction lagrangians (for the positive parity Θ + ) satisfying gauge and chiral symmetries are given by In these equations, Θ, n and K denote the fields of Θ + , the neutron and the kaon, respectively. In Eqs. (1) and (2), g, g * A and f π are the coupling constant of nΘ + K in PS, that in PV and the pion decay constant being set 93.2 MeV. The strengths of g and g * A will be determined later.
In Eqs. (3)-(5), e is a unit charge. In Eq. (4), κ n is the anomalous magnetic moment of the neutron (in units of nucleon magneton) and k µ the photon momentum. In Eq. (5), we have introduced the anomalous magnetic moment of Θ + , κ Θ (in units of Θ magneton). We do not know its value, and therefore, treat it as a parameter in the following calculations.
whereū and u are the Dirac spinors of Θ + and the neutron. p, p ′ , k and k ′ are the fourmomenta of the neutron, Θ + , photon and K − , respectively. Subscripts s, u and t indicate the channels corresponding to Figs. 1 (a), (b) and (c).
In the nonlinear framework of PV, the four diagrams of Figs. 1 (a) -(d) must be computed. The first three terms are given by the substitution in the result of PS, γ 5 → γ 5 / k ′ in Eqs. (6) and (7), Now we determine the coupling constants, g * A and g from the decay width of Θ + . We use a formula where P f , E cm and M i are the relative three momentum of the final state, the total energy in the center of mass (CM) energy frame and the mass of the initial particle, respectively. We use Eqs. (1), (2) and Γ = 15 MeV from chiral soliton model calculations. Then, we obtain g * A = 0.306 for PV and g = 4.05 for PS. If Θ + has negative parity, then we find g * A = 0.171 and g = 0.542. It is interesting that these coupling constants satisfy the (extended) Goldberger- Because of these GT relations, the amplitudes computed in the PS and PV become equivalent, when κ n = κ Θ = 0. This provides a good consistency check of our numerical calculations.
In reactions such as pion and kaon photoproductions, it is known that only the tree (or Born) diagrams are not sufficient to reproduce experimental data. In fact, as the energy is increased the cross sections of the Born terms rise too rapidly. In the hybrid model (HM) the linear and non-linear models are combined under the assumption that at low energies the non-linear model works well while at higher energies, the linear one does better. As proposed in Refs. [20,21], we consider the following HM amplitude where F HM = Λ 2 HM /(Λ 2 HM + | k ′ | 2 ) and Λ HM is the mixing parameter depending on the kaon momentum k ′ . Obviously the amplitude of HM satisfies the gauge invariance. For the mixing parameter, we employ Λ HM = 450 MeV [20]. Now we find for the difference between the PS and PV amplitudes Since κ n = −1.91, for a positive value of κ Θ the difference ∆M is suppressed, while for a negative value enhanced. As discussed in the next section, the effect of the mixing of the PS and PV in the HM is to make the energy dependence of the cross sections milder.
There is another factor we need to consider, which is the effect of the form factor. The status of theoretical treatment is still controversial due to subtleties of maintaining gauge invariance. However, the necessity of the form factor seems obvious from physics point of view as it should reflect the size (or internal structure) of hadrons. For instance, the Born terms in the K + Λ photoproduction [22] produces too large total cross sections as shown in Fig. 2. From the comparison in the theoretical and experimental cross sections of KΛ photoproduction, we may naturally expect a suppression also for the present reaction of K − Θ + production. More details will be discussed in the next section.

III. NUMERICAL RESULTS
We present the total and differential cross sections for the cases of positive and negative increase. Especially, the rate of the increase in the PV seems very large. However, such a behavior may not be realistic if we recall the result of KΛ production as seen in Fig. 2 In fact, this has motivated us to employ the scheme of the hybrid model (HM). As shown in the figures, the energy dependence of the HM result is much milder, and the total cross section is suppressed at large E CM . Because of this, in the following discussions we often refer to the results of the HM when we address actual numbers. The total cross section in HM averaged over the energy range E CM < dominates. The t-channel contribution is relatively small. Also from the figures, we observe that the threshold behavior of the s, u and KR terms are different from that of the t-channel meson exchange term. The former produces the E 0 + amplitude in S-wave of the final state KΘ + , while the latter is for higher multipoles.
In Fig.5 differential cross sections at a typical value E CM = 2.25 GeV are shown for the same κ Θ as for the total cross sections. The angular dependence also vary significantly depending on the lagrangian scheme and on κ Θ . However, except for the PS case, the cross sections are forward peaking especially when a smaller κ Θ is employed. At this point we note that when κ Θ ∼ 0, the angular dependence has a forward peak for the cases of HM and PV.
Next, in Figs. 6 and 7 we show the results for a negative parity Θ + . The total cross sections are much smaller than the case of the positive parity by about a factor 1/10 ∼ 1/5.
This can be understood as follows. For instance, as we have seen in the PS scheme the Born diagrams of s and u-channel processes are dominant. In this case, the KNΘ coupling for positive parity Θ + is γ 5 (for negative parity it is 1 as in the non-relativistic case). The Born diagram contains also the electromagnetic coupling proportional to ǫ · γ, where ǫ is a spatial part of the photon polarization vector. From these structures, one can see that the amplitude of the direct Born term is of order 1 for the positive parity Θ + , while it is of order p/M for the negative parity Θ + . The propagation of the negative energy neutron in the Z-graph type contributes to the term of order 1 for the positive parity Θ + while there is no such a leading order term for the negative parity Θ + . Similar suppression is verified for the PV scheme in the KR term as it should be. We can also verify that t-channel diagram is also suppressed for a negative parity Θ + . In the non-relativistic limit and for on-shell particles, the KNΘ vertex has a structure of σ · q for the positive parity Θ + , while 1 for the negative parity Θ + , where q is an exchanged momentum. The strength of the coupling is fixed from the decay of Θ + in which the momentum is q ∼ 256 MeV. In the t-channel diagram, however, the relevant momentum is about 1 GeV, which suppresses the amplitude for the negative parity Θ + by the factor 1/4.
Once again, the total cross section depends on the lagrangian scheme and on κ Θ . The largest total cross section for the negative parity Θ + is obtained in the PV scheme for κ Θ = +1.79. But if we take the result of the HM and including the suppression factor of 18 % for a reasonable estimate, we obtain: 20 nb (for κ Θ = +1.79), 16 [nb] (for κ Θ = 0), 14 [nb] (for κ Θ = −1.79).
We have also shown in Fig. 7 differential cross sections for the case of a negative parity Θ + . In this case, it is interesting to see that the cross sections are suppressed at the forward angle in all cases (more precisely, the results for PS with κ Θ = 1.79 and 0 show only a local minimum at the forward angle.). This is a sharp contrast to the result of the positive parity Finally, we comment on the possible range of the decay width of Θ + . In DIANA collabora-tion [2], Γ(Θ → KN) was measured to be < 9 MeV. This result is consistent with the remark in Ref. [5] that Γ could be smaller than 15 MeV. In fact, by employing g (0) A ∼ 0.37 [24], Γ becomes about 10 MeV. If we use a smaller coupling constant corresponding to this narrower width, cross sections are reduced.

IV. SUMMARY AND DISCUSSIONS
In  we may use the two features to determine the parity of Θ + ; one is the total cross sections which differ by about a factor ten for the positive and negative parity cases, and the other is angular dependence which shows a forward peak structure for the positive parity Θ + but with a suppressed one for the negative parity Θ + especially near the threshold region. Both observables need more careful analysis when compared with data, since they depend very much on the lagrangian schemes and the unknown parameters. In particular, the total cross sections are very sensitive to the form factor as it should be. On the other hand, the qualitative structure of the angular dependence seems rather stable as long as the HM scheme is adopted.
As reported by the DIANA group, if the decay width of Θ + would smaller than 9 MeV, say for instance 5 MeV, the total cross section is reduced by the factor three and the estimated σ(γn → K − Θ + ) would be of order 10 nb. If we assume that the parity of Θ + is negative, the total cross section would be even smaller which would be of order one nb. Definitely, we need to study more both theoretically and experimentally in order to understand the nature of the pentaquark state Θ + .