Threshold production of the $\Theta^+$ in a polarized proton reaction

We compute cross sections of $\Theta^+$ production near threshold for a polarized proton reaction, $\vec p \vec p \to \Sigma^+ \Theta^+$ which was recently proposed to determine unambiguously the parity of $\Theta^+$. Within theoretical uncertainties cross sections for the allowed spin configuration are estimated; it is of order of one microbarn for the positive parity $\Theta^+$ and about 1/10 microbarn for the negative parity $\Theta^+$ in the threshold energy region where the s-wave component dominates.

The discovery of the Θ + [1,2,3,4] has triggered a tremendous amount of research activities both from theories and experiments [5]. Hadron physics has now experienced a new stage of development with unexpected richness. The surprise came not only with its relatively low mass but also with a very narrow width, though for the latter only the upper limit is known so far. This feature is also shared by the recently observed Ξ states [6]. Quantum numbers such as spin and parity are not yet known neither. Since the parity reflects the internal dynamics of hadrons, it is crucially important to determine it by experiment and to understand by theory. The present theoretical situation, however, is not settled; chiral theories including the pioneering chiral soliton models [7,8,9], and the diquark model [10] predict positive parity, while recent lattice [11,12] and sum rule calculations [13,14] are on the other side.
Very recently, an unambiguous method was proposed in order to determine the parity of the Θ + using the reaction [15] p + p → Θ + + Σ + near threshold. ( This reaction has been previously considered for the production of Θ + [16], but it has turned out that it does more for the determination of the parity, in contrast with a number of recent attempts using other reactions which needed particular production mechanism [17]. In order to extract information of parity from (1), the only requirement is that the final state is dominated by the s-wave component. The s-wave dominance in the final state is then combined with the Fermi statistics of the initial two protons and conservations of the strong interaction, establishing the selection rule: If the parity of Θ + is positive, the reaction (1) is allowed at the threshold region only when the spin of the two protons is S = 0, while, if it is negative the reaction is allowed only when S = 1. This situation is similar to what was used in determining the parity of the pion. Experimentally, the pure S = 0 state may not be easy to set up. However, an appropriate combination of spin polarized quantities allows to extract information of S = 0 state. In this letter, we perform calculations for production cross sections of (1). Our purposes are: 1. To check that the production reaction is indeed dominated by the s-wave (in other word, there is no accidental vanishing of s-wave contributions to invalidate the above selection rule).
2. To estimate production cross sections within the present knowledge of theoretical models.
In order to estimate the production rate, we calculate the Born diagrams of pseudoscalar kaon (K(498)) and vector K * (K * (892)) exchanges, which are minimally needed for the present reaction (Fig. 1). Assuming that the parity of Θ + is positive, we can take effective interaction lagrangians as follows: with standard notations. If the parity of Θ + is negative, γ 5 matrix in (2) should be removed and in (4) γ 5 should be inserted. For the coupling terms of Σ + , we employ the values estimated from the previous analysis; g KN Σ = 3.54, g K * N Σ = −2.46 and g T K * N Σ = 1.15 [18]. Since the couplings to the Θ + is not known, we investigate several cases with different parameter values. For g KN Θ we mostly employ g KN Θ = 3.78, which is fixed by Γ Θ + →KN = 15 MeV. For each case, we employ for the unknown vector K * couplings, |g K * N Θ | = |g KN Θ |/2, as suggested by Ref. [19]. The tensor couplings are then varied within |g T K * N Θ | ≤ 2|g K * N Θ | = |g KN Θ | in order to see model dependence of this process. As for the form factor, we employ the following form of the monopole type: where q 2 is the four momentum square and m the mass of the exchanged particle (either K or K * ). The cut off parameter Λ is chosen to be Λ = 1 GeV. In Ref. [20] the authors employed a different type of form factor. However, the monopole type is more often used for meson-baryon vertices. In any events, the main points in the following discussions will not be changed by the use of different form factors. The calculation for the scattering amplitude is straightforward once having the interaction, Eqs. (right) panel is for the positive (negative) parity Θ + where the allowed initial spin is S = 0 (S = 1). For the allowed channels, five curves are shown using different coupling constants of g K * N Θ and g T K * N Θ ; zero and four different combinations of signs with the absolute values |g T K * N Θ | = 2|g K * N Θ | = |g KN Θ |, as indicated by the pair of labels in the figures, (sgn(g K * N Θ ), sgn(g T K * N Θ )). As shown in the figure, cross sections vary with about 50 % from the mean value for the vanishing K * exchanges. For the forbidden channels only the case of vanishing K * NΘ coupling constants is shown; cross sections using finite coupling constants vary within about 50 % just as for the allowed channels. In both figures, the s-wave threshold behavior is seen for the allowed channels as proportional to (s − s th ) 1/2 , while the forbidden channels exhibit the p-wave dependence of (s − s th ) 3/2 and with much smaller values than the allowed channel. The suppression factor is given roughly by [(wave number)·(interaction range)] 2 ∼ k/m K ∼ 0.1 (k = √ 2m K E), as consistent with the results shown in the figures. From these results, we conclude that the absolute value of the total cross section is of the order 1 [µb] for the positive parity Θ + and of the order 0.1 [µb] for the negative parity Θ + . The fact that the positive parity case has larger cross section is similar to what was observed in the photoproduction and hadron induced reaction also [17]. This is due to the p-wave nature of the KNΘ coupling with a relatively large momentum transfer for the Θ + production. When the smaller decay width of Θ + is used, the result simply scales as proportional to the width, if the K * NΘ couplings are scaled similarly.  In Fig. 3, we show the angular dependence in the center of mass system for several different energies above the threshold, √ s = 2730, 2740, 2750 and 2760 MeV. Here only K exchange is included but without K * exchanges. The angular dependence with the K * exchanges included is similar but with absolute values scaled as in the total cross sections. Once again, we can verify that the s-wave dominates the production reaction up to √ s < ∼

2750.
Recently, in Ref. [21], the authors discussed the experimental methods and obervables to determine the parity of the Θ + baryon with the polarized proton beam and target. They discussed the spin correlation parameter A xx as well as cross sections. It is computed by where σ 0 is the unpolarized total cross sections and the polarized cross section are denoted as 2S+1 σ Sz . In Fig. 4 we present A xx where we do not include K * exchange, but the results do not change very much by including K * . As shown in the figures A xx reflects very clearly the differences of the parity of Θ + . The cases with and without the form factor are similar and well fall into the region as indicated in Ref. [21]. In actual experiment, it is necessary to detect Σ also at the threshold region. Due to small energy (or velocity) of the final state particles in the center of system, produced Σ must be detected inside a very narrow cone forward peaked in the laboratory frame. Because of this fact, measurement at the existing facility of fixed target, such as COSY, would require an experimental challenge. Foundation (KRF-2003-070-C00015). The work of S.I.Nam has been supported by the scholarship endowed from the Ministry of Education, Science, Sports and Culture of Japan.