Photoproduction of $K^*$ for the study of $\Lambda(1405)$

The photo-induced $K^*$ vector meson production is investigated for the study of the $\Lambda(1405)$ resonance. This reaction is particularly suited to the isolation of the second pole in the $\Lambda(1405)$ region which couples dominantly to the $\bar K N$ channel. We obtain the mass distribution of the $\Lambda(1405)$ which peaks at 1420 MeV, and differs from the nominal one. Combined with several other reactions, like the $\pi^- p \to K^0 \pi \Sigma$ which favours the first pole, this detailed study will reveal a novel structure of the $\Lambda(1405)$ state.

In recent years, we have been observing a remarkable development in hadron physics, especially in baryon resonances. Chiral models which implement strong s-wave meson-baryon interactions have been showing that some of the 1/2 − resonances, such as the Λ(1405), are strongly dominated by quasi-bound states of coupled meson-baryon channels [1,2,3,4]. The case of the Λ(1405) as a quasibound state is not a merit of the chiral Lagrangians since it was previously obtained in a unitary coupled channel approach in Refs. [5,6,7]. The use of chiral Lagrangians has allowed a systematic approach to face the meson-baryon interaction.
In these pictures, the resonances generated may be regarded as another realization of 5quark dominated states, although the lowest Fock space to generate the quantum numbers of Λ starts from 3-quark states. The s-wave meson-baryon interaction at lowest order in the chiral Lagrangians is given by the Weinberg-Tomozawa term which contains an attractive interaction in theKN channel as well as its couplings to other meson-baryon channels.
Therefore, confirmation of this picture is important in order to understand better the nonperturbative dynamics of QCD.
One interesting finding concerns the structure of the Λ(1405) resonance; several groups have reported that there are two poles in the region of Λ(1405) in analyses based on the chiral unitary models [8,9,10,11,12,13,14,15]. The existence of two poles was first found in the context of the cloudy bag model [16], and recent studies of chiral dynamics reveal the detailed structure of these poles. For instance, in Ref. [17], they found two poles at z 1 = 1390 − 66i MeV and z 2 = 1426 − 16i; the former at lower energy and with a wider width couples dominantly to πΣ channels, while the latter at higher energy with a narrower width couples dominantly toKN channels. If this is the case, the form of the invariant mass distribution of πΣ, where the Λ(1405) is seen, depends on the particular reaction used to generate the Λ(1405) [17]. In fact, different shapes of mass distributions were observed in previous theoretical studies [18,19] and in Ref. [20] it was found that the π − p → K 0 πΣ reaction was particularly selective of the first Λ pole. It is therefore desirable to study the nature of the Λ(1405) focusing on whether such two poles really exist in the nominated resonance region.
In this paper, we propose another, hopefully better, reaction induced by photons for the extraction of the second pole around the Λ(1405) resonance: γp → K * Λ(1405) → πKπΣ.
A great advantage of this reaction is the use of a linearly polarized photon beam and the observation of the angular distribution of πK decaying from K * , which is correlated with the linear polarization of the photon.
As we shall see below, if we produce a πK system in a plane perpendicular to the photon polarization, the t-channel exchanged particle is dominated by the kaon; heavier strange mesons contributions should be suppressed due to their larger masses. Ignoring (hopefully small) background contributions from, for instance, unknown higher nucleon resonances also, it is sufficient to consider only the processes with kaon exchange, as shown in Fig. 1. The exchanged kaon rescatters in isospin I = 0 and 1 channels. Then the former couples strongly to Λ(1405), especially to the higher pole, while the latter does it to Σ(1385). We utilize the swave meson-baryon scattering amplitude calculated by the chiral unitary model [9,12]. The amplitude generates the Λ(1405) resonance dynamically, while the Σ(1385) is not generated because it is a p-wave resonance. In order to perform a realistic calculation, we introduce the Σ(1385) field explicitly.
The scattering amplitude as described by the diagram of Fig. 1 can be divided into two The former part (−it γ→K − Kπ ), as shown in Fig. 2, is derived from the following effective Lagrangians [21,22] where K, A µ , V µ and P are the kaon, photon, octet vector meson and octet pseudoscalar meson fields, respectively. The coupling constants are determined from the empirical partial decay width of K * : Γ K * ± →K ± γ = 0.05 MeV and Γ K * ± →Kπ = 51 MeV. The resulting values are |g γK * ± K ± | = 0.252 [GeV −1 ] and g V P P = −6.05. The latter g V P P is the universal vector meson coupling constant. We note that the effective Lagrangian (2) is consistent with a vector meson dominance model [23]. Using the above interaction Lagrangians, the where p and k are the momenta of the particle in parentheses, ǫ µ (γ) is the polarization vector of photon, and Γ K * is the total decay width of K * , for which we use the energy dependent one for a virtual K * , Γ K * = Ap 3 CM , where p CM is the two-body relative momenta of the final state, and A = 2.05 × 10 −6 [MeV −2 ] such that Γ K * ∼ 51 MeV at the resonance position.
Eq. (4) is instructive to show the correlations between the photon polarization and the K 0 and π + momenta. In order to maximize the contribution of the t-channel we select the K * in the direction of the photon. Then, it is easy to see that the amplitude is proportional to sin φ where φ is the angle between the plane defined by the K 0 and π + momenta and the photon polarization (in the Coulomb gauge, ǫ 0 = 0). Hence, the maximum strength of the amplitude occurs when this plane is perpendicular to the photon polarization.
In addition one needs not to worry about symmetrization in the case where there are two equal charge pions in the final state. In this case the interference term is zero and one can omit the symmetrization and the 1/2 factor in the cross section.
The amplitude (−it K − p→M B ) consists of two parts, as shown in Fig. 3 − where −it ChU is the meson-baryon scattering amplitude derived from the chiral unitary model, and −it Σ * is the Σ(1385) pole term. M I is the invariant mass for K − p system, which 1 For the final state K + π 0 , the amplitude is reduced by factor 1/ √ 2, and therefore, the resulting cross section becomes one half. In the rest of this paper, we show the result for K 0 π + .
In the chiral unitary model [9,12], the coupled channel amplitudes are obtained by where G is the meson-baryon loop function and V is the kernel interaction derived from the Weinberg-Tomozawa term of the chiral Lagrangian. This amplitude reproduces well the total cross sections for several channels. It also leads to dynamically generated resonances in good agreement with experiment. Since the Σ(1385) is not generated in this resummation because it is a p-wave resonance, we introduce it explicitly with its coupling to channel i (Σ(1385) → M B) which is deduced from the πN ∆ using SU (6) symmetry in [24,25] and given by where g A = 1.26, and we use the meson decay constant f = 93 × 1.123 MeV [9]. This is a nonrelativistic form for the transition between spin 1/2 and 3/2 particles, where S is a spin transition operator [26] and the coefficients c i are given in Table 1. Note that these couplings reproduce well the observed branching ratio of Σ(1385) decay into πΛ and πΣ. Then we have the amplitude where we have introduced a strong form factor F f (k 1 ) for the vertex K − pΣ * in order to account for the finite size structure of the baryons. We adopt a monopole type F f (q) = In the present reaction around the region of Λ(1405), the effect of the form factor is not very large.
The cross section is then given by the squared amplitude of Eq. (1) integrated over the four-body phase space. After eliminating four momenta variables from the twelve momenta Table 1: (of four particles), we can write a total cross section as a function of the incident energy √ s: where p 1 (2) and ω 1(2) are the momenta and energy of the final K(π) from K * , andP 3 is the relative three momentum of M B (∼ πΣ or πΛ) in their center of mass frame. The angle θ denotes the relative angle of M B in the CM frame of the total system. We have performed this integration by the Monte-Carlo method. As we have mentioned before, the advantage of this reaction is that the identification of the K * production is cleanly done in experiments.
Observation of the three pions in the process K * + → π + + K 0 → π + + (π + π − ) can be made with high efficiency and with all three momenta measured.  Table 2, two charged and two neutral channels. In the present case, since we have the K − p channel initially, the I = 2 component of πΣ channel is not allowed.
It is worth showing the isospin decomposition of the distributions of charged states [18] dσ(π ± Σ ∓ ) where T (I) is the amplitude with isospin I. The factor 1/2 in front of |T (1) | 2 and the ratio of the couplings g 2 Σ * π ± Σ ∓ /g 2 Σ * π 0 Λ = 1/3 (see Table. 1) explain why the Σ(1385) does not affect the charged πΣ channels very much, as compared with the πΛ final state. The difference between π + Σ − and π − Σ + comes from the crossed term Re(T (0) T (1) * ), and when we sum up the two distributions this term vanishes. We also show the result for the sum of the charged πΣ channels in Fig. 5 (thin solid line). The feature that the initial K − p couples dominantly to the second pole of the Λ(1405) is well preserved in the total mass distribution, although the width of this distribution is slightly larger than that of the I=0 resonance because it contains some contribution from the Σ(1385). This is a nice feature and suggests that by observing the mass distributions of the charged state from the intermediate baryon, it would be possible to study the nature of the second pole of the Λ(1405) resonance.
It is also interesting to see the I = 1 s-wave amplitude in this energy region, where the existence of another pole is discussed [8,17]. It was shown in Ref. [17] that in the SU (3) decomposition of the meson baryon states the interaction was attractive in a singlet and two octets, hence it is natural to expect the existence of another s-wave I = 1 resonance in addition to the Σ(1620) already reported in Ref. [9]. Indeed, a pole is found at 1410 − 40i MeV in the model of Ref. [8]. However, the properties of this I = 1 pole are very sensitive to the details of the model since in different models or approximations it appears in different Riemann sheets, but there is still some reflection on the amplitudes in all cases. Therefore, investigation of the I = 1 s-wave amplitude would bring further information of resonance properties.
We could have the I = 1 amplitude by combining the three πΣ channels (see Eq. (10)).
However, the |T (1) | 2 term will contain contributions both from s and p-wave, although the contribution of the p-wave to the πΣ channels is small. In order to extract the I = 1 s-wave amplitude we separate the T-matrix into partial waves as Since we are looking at the cross sections where the angle variable among M B is integrated, the product of s-and p-wave amplitude vanishes. Then, the difference of the distributions for the two charged states contains only the T We plot this magnitude in Fig. 6 with a dashed line. In principle, it is possible to extract The results are shown in Fig. 6 (Solid line) and a small peak is seen as a reflection of the approximate resonant structure predicted in Refs. [8,17]. Finally we show the results for the sum of all πΣ and πΛ channels in Fig. 7. This corresponds to the most feasible case in experiment in which the three pions decaying from K * are identified. In the total spectrum as a function of M I (right panel), we find a twobump structure reflecting both the Λ(1405) and the Σ(1385). In the actual case, there would be a further contribution from theKN channel, raising at around 1430 MeV which we do not include in the calculation. This contribution starts where the mass distribution in Fig.   7 has already dropped down and therefore will not blur the shape of the distribution. This is the case in a related reaction studied in Ref. [18]. This figure is also illustrating because it reveals a large strength in the region of 1420 MeV, which makes this shape clearly distinct from the one observed experimentally in the π − p → K 0 πΣ reaction [28] with a neat peak around 1400 MeV. Hence, this measurement is valuable by itself. Yet, to get the individual contributions one should measure the channels shown in Fig. 5. It is interesting to recall that in the chiral model of Ref. [20] it was shown that the π − p → K 0 πΣ reaction favoured the lower mass Λ pole.
In this paper, we have proposed a reaction γp → π + K 0 M B for the study of the second pole possibly existing in the Λ(1405) region. This second resonance has been shown to couple more strongly toKN than to πΣ in several chiral models, the present reaction is suitable for the isolation of this pole. Although the coupling to Σ(1385) might contaminate the pure πΣ mass distribution from the second Λ(1405) pole, the resulting total mass distribution still   [20]. A similar mass distribution to the present one was observed in the former study of K − p → γπΣ [19], where the photon is emitted from the initial state and hence the Λ(1405) production is also induced by a K − .
Experimental evidence on the existence of such two Λ * states would provide more information on the nature of the current Λ(1405) and thus new clues to understand non-perturbative dynamics of QCD.