Relativistic three-body calculations of a Y = 1,I = 3,JP = 2+ 2 πΛN − πΣN dibaryon H. Garcilazo1,∗ and A. Gal2,† 1Escuela Superior de F´ısica y Matem´aticas Instituto Polit´ecnico Nacional, Edificio 9, 07738 M´exico D.F., Mexico 2 2Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel 1 0 (Dated: November 7, 2012) 2 v Abstract o N TheπΛN−πΣN coupled-channelsystemwithquantumnumbers(Y,I,JP) = (1, 3,2+)isstudied 2 6 in a relativistic three-body model, using two-body separable interactions in the dominant p-wave ] h pion-baryon and 3S YN channels. Three-body equations are solved in the complex energy plane t 1 - l c to search for quasibound-state and resonance poles, producing a robust narrow πΛN resonance u n about 10–20 MeV below the πΣN threshold. Viewed as a dibaryon, it is a 5S quasibound state 2 [ consisting of Σ(1385)N and ∆(1232)Y components. Comparison is made between the present 2 v 2 relativistic model calculation and a previous, outdated nonrelativistic calculation which resulted 9 9 in a πΛN bound state. Effects of adding a K¯NN channel are studied and found insignificant. 3 . Possible production reactions of this (Y,I,JP) = (1, 3,2+) dibaryon are discussed. 9 2 0 2 1 PACS numbers: 14.20.Pt,13.75.Gx, 13.75.Ev,11.80.Jy : v i Keywords: dibaryons, pion-baryoninteractions, hyperon-nucleon interactions, Faddeev equations X r a ∗Electronic address: [email protected] †Electronic address: [email protected] 1 I. INTRODUCTION In recent work [1, 2] we have studied the πΛN −πΣN coupled channel system, in which the dominant two-body configurations are the pion-nucleon p-wave ∆(1232) resonance with s-wave hyperon spectator, the pion-hyperon p-wave Σ(1385) resonance with s-wave nucleon spectator, and the YN (Y ≡ Λ,Σ) 3S coupled channels with p-wave pion spectator. The 1 contributions of these two-body configurations obviously maximize in the three-body chan- nel with (I,JP) = (3,2+), where I,J,P denote the total isospin, total angular momentum 2 and parity, respectively. Substantial attraction in this three-body configuration was found in a nonrelativistic three-body calculation, resulting in a possible πΛN bound state. Having presented very recently a relativistic three-body Faddeev formalism appropriate for systems with p-wave two-body interactions [3], it is natural to apply it to the πΛN − πΣN cou- pled channels system with I = 3/2 and JP = 2+. The main consequence of adopting a relativistic formalism, as shown below, is that the πΛN bound state dissolves, becoming a πΛN resonance below the πΣN threshold. We note that a relativistic three-body formalism equivalent to that of Ref. [3] was already applied in the context of searching for a K¯NN (I = 1/2,JP = 0−) quasibound state for which the dominant two-body configurations are all s-waves [4]. We have also studied the effect of adding to the (3,2+) πYN channels a 2 K¯NN channel, induced through a Σ(1385)-mediated two-body p-wave K¯N −πY coupling, and found it to be relatively insignificant. This is to be expected, observing that none of the Pauli-allowed s-wave NN configurations fits into a (3,2+) K¯NN channel with a p-wave 2 meson spectator. For a recent overview of dibaryon candidates and related studies, see Refs. [5–7]. The paper is organized as follows: input two-body interactions are described in Sect. II and three-body equations are derived in Sect. III. Results are described in Sect. IV and discussed in Sect. V. Several production reactions by which to search for the present (Y,I,JP) = (1, 3,2+) dibaryon candidate are listed and briefly discussed in the Summary 2 Sect. VI. 2 II. TWO-BODY INTERACTIONS As discussed in Ref. [1], the dominant two-body interactions are in the p-wave πN (I,JP) = (3, 3+) ∆(1232) and πΛ − πΣ (I,JP) = (1, 3+) Σ(1385) channels, and in the 2 2 2 s-wave ΛN − ΣN (I = 1,3S ) channel. We note that these two-body interactions, taken 2 1 here in separable form, are independent of energy whereas the resulting two-body ampli- tudes are obviously energy dependent, and even resonate in the p-wave channels. Since the introduction of two-body energy-dependent interactions geared to simulate additional energy-dependent background amplitudes poses problems within a relativistic kinematics treatment (see Ref. [8] for a recent discussion) we limit the two-body interaction input in the present three-body relativistic calculation to energy-independent separable forms de- scribed below. Our notational convention is to assign particle indices 1,2,3 to hyperons, nucleon and pion, respectively. A. The πN subsystem The Lippmann-Schwinger equation for the pion-nucleon interaction is given by [3]: ∞ t (p ,p′;ω ) = V (p ,p′)+ p′′2dp′′ 1 1 1 0 1 1 1 1 1 Z 0 1 × V (p ,p′′) t (p′′,p′;ω ), (1) 1 1 1 ω − m2 +p′′2 − m2 +p′′2 +iǫ 1 1 1 0 0 N 1 π 1 p p so that using the separable potential V (p ,p′) = γ g (p )g (p′), (2) 1 1 1 1 1 1 1 1 one gets t (p ,p′;ω ) = g (p )τ (ω )g (p′), (3) 1 1 1 0 1 1 1 0 1 1 where 1 ∞ g2(p ) [τ (ω )]−1 = − p2dp 1 1 . (4) 1 0 γ1 Z0 1 1ω0 − m2N +p21 − m2π +p21 +iǫ p p A fit to the P phase shift and scattering volume using the form factor 33 g (p ) = p [exp(−p2/β2)+Cp2exp(−p2/α2)], (5) 1 1 1 1 1 1 1 1 andasetofparameterslistedinTableI,rowmarkedP , wasshownanddiscussed inRef.[3]. 33 This form factor and parameters are used in the present calculations. Listed in the same 3 row are also r.m.s. radii values of momentum-space and coordinate-space representations of the P form factor. These were discussed too in Ref. [3]; here we recall that g˜(r), the 33 1 coordinate-space Fourier transform of g (p), is not necessarily a nodeless function at finite 1 values of r, so that an appropriate measure of its spatial extension is provided by the value (πN) of its (single) zero r , given by the last entry. This does not appear to present a problem 0 in the case of the πN P form factor, where the difference between the listed values of 33 < r2 > and r(πN) is small, but it does present a problem in the case of the πY form g˜1 0 p factor where the squared radius < r2 > assumes occasionally negative values. Returning g˜1 to Table I, listed in the row marked P are parameters fitted to the P phase shifts which 13 13 are considerably smaller than the P resonating phase shifts. This πN P channel will act 33 13 in the three-body calculation only together with a spectator Σ hyperon, and its inclusion serves the purpose of estimating the role of πB channels other than the resonating ones. For notational simplicity, and since the πN P channel is excluded frommost of the calculations 13 reported here, it is suppressed in the derivation of the three-body equations below. TABLE I: Fitted parameters of the πN separable p-wave interaction (2) with form factor g (p) 1 (5). Values of the r.m.s. momentum < p2 > (fm−1), r.m.s. radius < r2 > and zero r(πN) g1 g˜1 0 p p (both in fm) of the Fourier transform g˜(r) are listed for the dominant P channel. 1 33 channel γ (fm4) α (fm−1) β (fm−1) C (fm2) < p2 > < r2 > r(πN) 1 1 1 g1 g˜1 0 p p P −0.075869 2.3668 1.04 0.23 4.07 1.47 1.36 33 P 0.033 – 1.325 0.0 13 The πN P amplitude in the three-body system can have either Λ or Σ hyperon as 33 spectator and is given by tY(p ,p′;W ,q ) = g (p )τY(W ,q )g (p′), (6) 1 1 1 0 1 1 1 1 0 1 1 1 where W is the invariant mass of the three-body system, q is the relative momentum 0 1 between the hyperon and the c.m. of the πN subsystem and 1 ∞ g2(p ) [τY(W ,q )]−1 = − p2dp 1 1 , 1 0 1 γ1 Z0 1 1 2 W − m2 +p2 + m2 +p2 +q2 − m2 +q2 +iǫ 0 r N 1 π 1 1 Y 1 (cid:16)p p (cid:17) p (7) where Y is either Λ or Σ. 4 B. The πΛ−πΣ subsystem Since we have in this case two coupled channels the corresponding Lippmann-Schwinger equation is ∞ tYY′(p ,p′;ω ) = VYY′(p ,p′)+ p′′2dp′′ 2 2 2 0 2 2 2 2 2 Z XY′′ 0 1 × VYY′′(p ,p′′) tY′′Y′(p′′,p′;ω ). (8) 2 2 2 ω − m2 +p′′2 − m2 +p′′2 +iǫ 2 2 2 0 0 π 2 Y′′ 2 p p Here we used the separable potential VYY′(p ,p′) = γ gY(p )gY′(p′), (9) 2 2 2 2 2 2 2 2 so that the solution of the Lippmann-Schwinger equation is tYY′(p ,p′;ω ) = gY(p )τ (ω )gY′(p′), (10) 2 2 2 0 2 2 2 0 2 2 with 1 ∞ [gY(p )]2 τ−1(ω ) = − p2dp 2 2 . (11) 2 0 γ2 XY Z0 2 2ω0 − m2π +p22 − m2Y +p22 +iǫ p p The two-body amplitude in the three-body system with a nucleon as spectator is given by expressions analogous to (6) and (7). Following Ref. [3] we used the form factors gΛ(p ) = p (1+Ap2)exp(−p2/β2), gΣ(p ) = BgΛ(p ), (12) 2 2 2 2 2 2 2 2 2 2 where the four parameters γ , β , A and B were fitted to the three pieces of data available, 2 2 namely, the position and width of the Σ(1385) resonance and the branching ratio for its two main decay modes. A family of such parameters is given in Table II, for a range of A (πY) values such that the spatial size (here r ) associated with the resulting πY form factors is 0 (πN) related physically to the spatial size r associated with the P πN form factor of Table I. 0 33 For more details and discussion, see Ref. [3].1 C. The YN subsystem In the case of isospin 1 which corresponds to the coupled ΛN −ΣN subsystem we have 2 two coupled channels so that applying Eq. (8) to the separable potential VYY′(p ,p′) = γYY′gY(p )gY′(p′) (13) 3 3 3 3 3 3 3 3 1 We note that the superscripts Λ and Σ are erroneously interchanged in Eq. (7) of the published journal version where they appear as subscripts. None of the results in Ref. [3] is affected by this typo. 5 TABLE II: Fitted parameters of the πΛ − πΣ p-wave separable interaction defined by Eqs. (9) and (12), for chosen values of the parameter A. Listed also are values of the r.m.s. momentum < p2 > (in fm−1), the r.m.s. radius < r2 > (whenever real) and zero r(πY) (both in fm) of g2 g˜2 0 p p the Fourier transform g˜(r). 2 A (fm2) γ (fm4) β (fm−1) B < p2 > < r2 > r(πY) 2 2 g2 g˜2 0 p p 0.25 −0.0091851 2.5810 0.93671 4.30 0.33 1.36 0.30 −0.0090934 2.4765 0.95132 4.13 0.23 1.41 0.35 −0.0089513 2.3919 0.96559 4.00 – 1.45 0.40 −0.0087763 2.3216 0.97949 3.89 – 1.48 0.45 −0.0085787 2.2619 0.99298 3.80 – 1.51 leads to tYY′(p ,p′;ω ) = gY(p )τYY′(ω )gY′(p′), (14) 3 3 3 0 3 3 3 0 3 3 where τYY′(ω ) are easily obtained. We used Yamaguchi form factors 3 0 1 gY(p ) = , (15) 3 3 1+(p /αY)2 3 3 so that there are five free parameters, three strengths and two ranges. These five parameters were fitted to the ΛN S = 1 scattering length a1 = 1.41 fm and effective range r1 = 3.36 1 1 2 2 fm, the real and imaginary parts of the ΣN S = 1 scattering length a′ = 2.74+i1.22 fm, 1 1 2 and the phase of the ΛN −ΣN S = 1 transition scattering length ψ = 23.8◦ obtained in the chiral quark model [9]. These parameters are given in Table III. TABLE III: Parameters of the spin-triplet YN separable potentials defined by Eqs. (13) and (15) for isospin values I = 1,3. YN 2 2 I γΛΛ (fm2) γΛΣ (fm2) γΣΣ (fm2) αΛ (fm−1) αΣ (fm−1) YN 3 3 3 3 3 1/2 −0.37704 −0.047865 −0.0059699 1.46 0.4 3/2 – – 0.36416 – 1.491 The spin-triplet hyperon-nucleon subsystem with isospin 3 corresponds to pure ΣN scat- 2 tering and it requires only two free parameters, one strength and one range. These two parameters were fitted to the ΣN S = 1 scattering length a′ = −0.44 fm and effective 3 1 2 6 range r′ = −2.09 fm obtained in the chiral quark model [9]. These parameters are also 3 1 2 given in Table III. D. Compact form of the two-body amplitudes The two-body amplitudes discussed above can be written in compact form as tY = |gπNiτYhgπN|, Y = Λ,Σ, (16) 1 1 1 1 |gπΛi t =  2 τ hgπΛ| hgπΣ| , (17) 2 |gπΣi 2(cid:16) 2 2 (cid:17) 2   |gΛNiτΛN→ΛNhgΛN| |gΛNiτΛN→ΣNhgΣN| 3 3 3 3 3 3 t =  . (18) 3 |gΣNiτΣN→ΛNhgΛN| |gΣNiτΣN→ΣNhgΣN| 3 3 3 3 3 3   Forapplicationswishingtoextendthesystemoftwo-bodyπY coupledchannelsintoasystem of πY −K¯N channels, coupled through the Σ(1385) isobar, Eq. (17) is to be replaced by |gπΛi 2   t = |gπΣi τ hgπΛ| hgπΣ| hgK¯N| . (19) 2 2 2 2 2 2   (cid:16) (cid:17) |gK¯Ni  2    III. THREE-BODY EQUATIONS Normally, the Faddeev amplitudes are labeled by the spectator particle which in general has the same label as the interacting pair. However, when there is particle conversion as in the present case one can have different interacting pairs for the same spectator or different spectators for the same interacting pair. For example, whereas πN is the interacting pair in the amplitude T and the spectator is either Λ or Σ, the interacting pair in the amplitude 1 T is either πΛ or πΣ and the spectator is a nucleon. Thus, we will label the corresponding 2 Faddeev amplitudes either by the spectator or by the interacting pair as helpful as to make the notation clear. In this way, considering all possible transitions, one obtains the Faddeev equations TY = tYG (πYN)TπY +tYG (πYN)TYN, (20) 1 1 0 2 1 0 3 7 TπY = tπY→πY′G (πY′N)TY′ + tπY→πY′G (πY′N)TY′N, (21) 2 2 0 1 2 0 3 XY′ XY′ TYN = tYN→Y′NG (πY′N)TπY′ + tYN→Y′NG (πY′N)TY′. (22) 3 3 0 2 3 0 1 XY′ XY′ For applications wishing to extend the two-body πY coupled channels into a system of ¯ πY −KN channels coupledthroughtheΣ(1385)isobar, theFaddeevamplitude(21)acquires the additional term tπY→K¯NG (K¯NN)TK¯N on the r.h.s., where 2 0 2 TK¯N = tK¯N→K¯NG (K¯NN)TK¯N 2 2 0 2 + tK¯N→πYG (πYN)TY + tK¯N→πYG (πYN)TYN. (23) 2 0 1 2 0 3 XY XY If we substitute Eq. (22) into Eqs. (20) and (21), using the expressions for the two-body amplitudes (16)–(18), we get that TY = |gπNiXY, TπY = |gπYiX , (24) 1 1 1 2 2 2 where the new amplitudes XY and X satisfy the equations 1 2 XY = τYhgπN|G (πYN)|gπYiX 1 1 1 0 2 2 + τYhgπN|G (πY′N)|gY′NiτY′N→Y′′NhgY′′N|G (πY′′N)|gπY′′iX 1 1 0 3 3 3 0 2 2 YX′Y′′ + τYhgπN|G (πY′N)|gY′NiτY′N→Y′′NhgY′′N|G (πY′′N)|gπNiXY′′, (25) 1 1 0 3 3 3 0 1 1 YX′Y′′ X = τ hgπY|G (πYN)|gπNiXY 2 2 2 0 1 1 XY + τ hgπY|G (πYN)|gYNiτYN→Y′NhgY′N|G (πY′N)|gπY′iX 2 2 0 3 3 3 0 2 2 XYY′ + τ hgπY|G (πYN)|gYNiτYN→Y′NhgY′N|G (πY′N)|gπNiXY′. (26) 2 2 0 3 3 3 0 1 1 XYY′ As shown in Ref. [3], the one-dimensional integral equations corresponding to the Faddeev equations for the πΛN − πΣN system can be read off from the AGS form Eqs. (25) and (26). For applications wishing to extend the description of the Σ(1385) isobar in terms of πY coupled channels into πY − K¯N coupled channels, the definition of X in Eq. (24) is 2 generalized to TπY |gπYi 2 2   =  X , (27) 2 TK¯N |gK¯Ni 2 2     with Eq. (26) modified by adding on its r.h.s. the term τ hgK¯N|G (K¯NN)|gK¯NiX . 2 2 0 2 2 8 IV. RESULTS We started by searching for (I = 3/2,JP = 2+) πΛN − πΣN bound-state poles, i.e. considering real values of W < m +m +m for which there are no three-body singulari- 0 π Λ N ties. The one-dimensional integral equations which follow from the coupled-amplitude AGS equations (25) and (26) were solved. Unlike the nonrelativistic cases studied in [1] and [2] we found no pole which would correspond to a bound state. In order to artificially generate such a pole we multiplied the strengths γ and γ by factors f > 1 and f > 1 which exactly 1 2 1 2 produce a bound state pole at the πΛN threshold W = m +m +m . We then rotated 0 π Λ N the integration contour into the complex plane as described in [3], i.e., q → q exp(−iφ) i i which allowed us to reduce slowly the factors f and follow the bound state pole into the i complex plane to its final position once f = f = 1. Finally, we checked that the position 1 2 of the pole is independent of the value of φ. TABLE IV: Energy position of the πΛN resonance pole, relative to the πΣN threshold, calculated for the g form factors of Table II, listed according to their A parameter and the zero of g˜. 2 2 A (fm2) r(πY) (fm) E (MeV) 0 0.25 1.36 −19.8−i2.6 0.30 1.41 −17.6−i2.9 0.35 1.45 −15.6−i3.2 0.40 1.48 −13.7−i3.5 0.45 1.51 −11.9−i3.8 In Table IV we list the energy eigenvalues, measured with respect to the πΣN threshold, as calculated using the P πN form factor from Table I and the family of πY form factors 33 recorded in Table II. The sensitivity of the calculated pole energy to the parametrization of the πY form factor amounts to less than 10 MeV. In all cases the eigenvalue lies above the πΛN threshold, but below the πΣN threshold. If we neglect the YN interaction, the real part of the pole energy rises approximately 10 MeV while the imaginary part remains almost the same. Finally, in order to check the effect of other non-resonating partial waves, we repeated the calculation of the first row in Table IV adding the πN P partial wave 13 from the second row of Table I. The energy changed then from E = −19.755−i2.611 MeV 9 to E = −19.734−i2.613 MeV, demonstrating that this effect is quite negligible. V. DISCUSSION In this section we discuss two aspects of the present relativistic three-body calculation, (i) relativistic vs nonrelativistic and (ii) the inclusion of a K¯NN channel. A. Relativistic vs Nonrelativistic As observed in the previous section the effects of a relativistic treatment are quite im- portant for the πΛN −πΣN system, removing the πΛN bound-state solution obtained in the nonrelativistic (NR) model [1, 2]. In order to understand the origin of the discrepancy between the relativistic and NR results we have repeated the calculation of the πΛN problem [1] for the simple case where there is no coupling to the πΣN channel and one neglects the YN interaction. In this case, the Faddeev equations of the πΛN bound-state problem are X = τ hg |G (πΛN)|g iX , (28) πN πN πN 0 πΛ πΛ X = τ hg |G (πΛN)|g iX , (29) πΛ πΛ πΛ 0 πN πN where τ with i=N,Λ are the isobar propagators of the πi subsystems and πi hg |G (πΛN)|g i are the one-pion-exchange diagrams. The πN and πΛ separable po- πi 0 πj tentials used in [1] are of the form V (p,p′) = γ g (p)g (p′), (30) πi πi πi πi with g (p) = p(1+p2)exp(−p2/α2 ), (31) πi πi where the parameters γ and α were fitted to the position and width of the resonances πi πi as given by the Particle Data Group [10]. We list these parameters in Table V as well as the corresponding ones obtained using the relativistic formulation in Ref. [3]. Using the parameters listed in the table, the NR model predicts a bound state at about −110 MeV while in the case of the relativistic model there is no bound state. If in the relativistic model we replace the one-pion-exchange diagrams by their NR versions we obtain almost the same 10