Reaction of two pion production pd → pd ππ in the resonance region

. The ANKE@COSY data on the reaction of two-pion production in the GeV region are analyzed within the theoretical model by Platonova and Kukulin. The model includes excitation of the dibaryon resonance D IJ = D 03 (2380) with the spin J = 3 and isospin I = 0 observed by the WASA@COSY in the reaction pn → d π 0 π 0 , and its decay D 03 → D 12 + π → d + π + π , where D 12 (2150) is another dibaryon resonance. Distributions on the invariant masses of the ﬁnal d ππ and ππ systems are calculated.


Introduction
Search for dibaryon resonances in two-nucleon systems has a long history (for review see [1]). At present one of the most realistic candidate to dibaryon is the resonance D I J = D 03 observed by the WASA@COSY [2] in the total cross section of the reaction of two-pion production pn → dπ 0 π 0 , where I = 0 is the isospin and J = 3 is the total angular momentum of this resonance. The mass of the resonance 2.380 GeV is close to the ∆∆-threshold, but its width Γ = 70 MeV is twice lower as compared to the width of the free ∆-isobar. This narrow width is considered as the most serious indication to a non-hadronic, but most likely, quark content of the observed resonance state. Quark model calculations with presence of the hidden color allows one to explain the observed narrow width of this state [3]. On the other hand, in a pure hadronic picture with the nucleon, ∆(1232)-isobar and pion degrees of freedom, it is also possible to get a resonance state in the πN∆ system [4][5][6], with rather narrow width if one assumes excitation of the dibaryon resonance D 12 in the N∆-interaction. As it was shown recently by Niskanen [7], the bound ∆∆-system has the narrow width if one takes into account that a part of the energy of this system is assigned to internal motion of two ∆'s and, therefore, cannot be used for decay of the ∆-isobars.
Besides of the resonance behaviour of the total cross section of the reaction pn → dπ 0 π 0 as a function of the total energy √ s, there is also resonance behavior of the differential cross section of this reaction as a function of the invariant mass of the final two-pion system M ππ . This feature is known as the ABC effect first observed in the reaction pd → 3 Hππ in [8]. Now it is assumed that the ABC effect observed in pd-and dd-systems is caused just by excitation of the D 03 resonance in the intermediate state [9]. Different mechanisms of the  Figure 1. Mechanisms of the reaction pd → pdππ with the Roper (left) and ∆∆ (right) excitation studied in [12] ABC effect in the reaction pn → dπ 0 π 0 were discussed in [9]. One possible mechanism of the reaction pn → dπ 0 π 0 suggested by Platonova and Kukulin in [10] involves sequential excitation and decay of two dibaryon resonances, D 03 (2380) and D 12 (2150).
The spin-parity of this resonance J P = 3 + was established by polarized measurements, however, information about its production (decay) channels is still non-complete. Recently a resonance structure was observed by the ANKE@COSY in the differential cross section of the two-pion production reaction pd → pdππ at beam energies 0.8-2.0 GeV with high transferred momentum to the deuteron at small scattering angles of the final proton and deuteron [11]. In the distribution over the invariant mass M dππ of the final dππ system the resonance peak was observed at M dππ ≈ 2.380 GeV for beam energies 1.1 -1.4 GeV [11] that is the mass of the isoscalar two-baryon resonance D I J = D 03 , while the kinematic conditions differ considerably from that in [2]. Furthermore, the ABC-type effect was observed in [11] in the distribution of the invariant mass of two final pions M ππ . Two mechanisms of this reaction involving excitation of the Roper resonance N * (1440) and two ∆-isobars in one-loop diagrams depicted in Fig. 1 were studied in [12]. Both of these mechanisms predict too low cross section as compared to the data [11]. The reason is that these mechanisms contain the deuteron elastic form factor S (Q) (clearly visible in the limit of the impulse approximation) that leads to low cross section at high transferred momentum Q to the deuteron that is the case in the ANKE experiment [11]. The mechanism with the Roper resonance N * (1440) (Fig. 1, left) predicts proper position of the peak in the cross section of the reaction pd → pdππ, while its width is twice large, but underestimates the cross section by two orders of magnitude. A similar result but with shifted peak was found for the ∆∆-mechanism (Fig. 1, right). Another mechanism suggested in [10] for the reaction pn → dπ 0 π 0 does not use the deuteron form factor but involves two dibaryon resonances, D 03 (2380) and D 12 (2150). We modify this model by inclusion of the σ-meson exchange between the proton and deuteron and apply it to the process pd → pdππ (Fig. 2). As it was found in [10], the ABC effect in the reaction pn → dπ 0 π 0 can be explained if an additional decay mechanism is included, D 03 → dσ → dπ 0 π 0 , and partial restoration of the chiral symmetry in the D 03 dibaryon is assumed. Since this assumption is rather questionable and the contribution of this mechanism to the total cross section of the reaction pn → dπ 0 π 0 is rather small [10], we do not consider it here when analyzing the ANKE@COSY data [11].
Since not all required partial widths are known from the experiment [13] and theoretical analysis in hadronic picture [14] and quark model [3], we discuss mainly the shapes of the distributions over the invariant masses of the final dππ and ππ systems.

The model
The transition amplitude for the mechanism depicted in Fig. 2 has the following form: where p σ , m σ , Γ σ are the 4-momentum, mass, and the total width of the σ-meson, respectively; λ i (λ i ) is the spin projection of the initial (final) particle i. The amplitude of the virtual process p → pσ is based on the phenomenological σNN interaction [15] and its spin averaged form |M λ p λ p (p → p σ)| 2 is given in [12]. The amplitude of the subprocess σd → dππ in Fig. 2 is here the last term (π 1 ↔ π 2 ) takes into account symmetrization over the final identical pions; P D 03 , M D 03 and Γ D 03 (P D 12 , M D 12 and Γ D 12 ) are the 4-momentum, the mass and the total width of the D 03 (D 12 ), respectively; q is the 3-momentum of the initial deuteron in the c.m.s of the D 03 , k 1 is the 3-momentum of the pion π 1 in the c.m.s of D 03 , and k 2 is the 3-momentum of the pion π 2 in the c.m.s of the D 12 . We use in Eq. (2) standard notations for the Clebsch-Gordan coefficients ( j 1 m 1 j 2 m 2 |JM) and spherical functions Y lm (k) = k l Y lm (k). The orbital momenta in the vertices D 03 → πD 12 and D 12 → dπ are l 1 = l 2 = 1 and in the vertex d + σ → D 03 is l = 2. The vertex factors F in Eq. (2) are defined as in [10]: Here M 2 dππ = P 2 D 03 , M 2 dπ = P 2 D 12 , and P D 03 (P D 12 ) is the 4-momentum of the D 03 (D 12 ). The modules of 3-momenta q, k 1 , k 2 are determined by the invariant masses of the particles in the vertices σdD 03 , πD 12 D 03 and πdD 12 , respectively, via the triangle function λ(m 2 1 , m 2 2 , m 2 3 ): , where m π and m d are the masses of the π-meson and deuteron, respectively; k 10 (k 20 ) is the value of the k 1 (k 2 ) at the resonance point P 2 ). The invariant cross section can be written in the following form:  The results of the model calculations (see text) are normalized to the data ( ) [11] and shown by thick red lines for the real narrow interval of the deuteron scattering angle θ q in the c.m.s of the dππ system occurred in the experiment [11] and by thin blue lines for the hypothetical full interval θ q = 0 ÷ π Here s is the invariant mass of the initial pd system, p i (p f ) is the 3-momentum of the initial (final) proton in the total c.m.s of the reaction, k is the pion momentum in the c.m.s of the ππ system, and q is the momentum of the final deuteron in the c.m.s of the final dππ system. Eq. (4) determines distribution over the invariant mass M dππ (M ππ ) of the dππ (ππ) system. Integration intervals over variables M ππ , cosθ p f and cosθ q are determined by experimental conditions [11] assuming azimuthal symmetry. In the experiment [11] the final pions ππ were not detected, therefore, both the π 0 π 0 and π + π − isoscalar pairs have to be taken into account in the present model. According to calculations [3], the ratio of the decay width D 03 → dπ + π − /D 03 → dπ 0 π 0 is equal to 2 for unbroken isospin symmetry and 1.8 for the symmetry broken by the different masses of the π + and π 0 mesons. Therefore the isospin factor C T = 2.8 is introduced in calculation of the dσ in Eq. (4).

Numerical results and discussion
The  [10]. The partial widths Γ (l=2) D 03 →dσ and Γ (l=1) D 03 →D 12 π were not determined in [10] and are unknown at present. The results of our calculation based on Eq. (4) are shown in Figs. 3 and 4. One can see in Fig. 3 that maxima at low mass M ππ observed in [11] are reproduced by the model (thick red lines in Fig. 3). This is caused by the behavior of spherical functions Y 1m (θ, φ) in two vertices and the collinear kinematics in the ANKE experiment. If we go out of the collinear kinematics, when integrating over the full interval of the scattering angle θ q of the final deuteron, this effect disappears (thin blue lines in Fig. 3).
The absolute value of the cross section of the reaction pd → pdππ is proportional to the product of the partial widths D 03 → dσ, D 03 → D 12 π, D 12 → dπ. The experimental data on some partial widths are given in [13]. The partial widths Γ(D 12 → dπ) and Γ(D 12 → NN) were estimated in [16] from the analysis of the cross section of the reaction pp → dπ + . The partial width Γ(D 03 → dπ 0 π 0 ) is about 10 MeV [13]. If we assume that this width  Fig. 2 in comparison with the data (•) [11]. Theoretical curves are normalized to the data is completely determined by the decay D 03 → D 12 π → dπ 0 π 0 , which is the basis of the considered model, then in order to get agreement with the absolute value we should put Γ(D 03 → dσ) = 8.5 MeV. The contribution of the decay channel D 03 → dσ → dπ 0 π 0 to the total width of the D 03 and the cross section of the reaction pd → pdππ will be estimated in forthcoming paper.

Conclusion
The mechanism of the reaction pd → pdππ depicted in Fig. 2 was qualitatively considered in [11]. Here we present some numerical results on the basis of this mechanism. The shape of the calculated distributions over invariant mass M ππ is in a reasonable agreement with the data reflecting the typical ABC-effect behaviour. However, we found that this "ABC-type" shape is caused by the collinear kinematics of the performed experiment [11] and does not occur within the considered model for the case when scattering angle of the deuteron is in the full interval θ = 0 ÷ π. We found also that the shape of the distribution over invariant mass M dππ is in qualitative agreement with the data [11].