Narrow πNN resonance

I1 September 1997

PHYSICS LEl-fERS B Physics Letters B 408 (1997) 25-30

EISWiER

Narrow TNN resonance Department

A. Matsuyama ’ of Physics, Faculty of Science, Shizuoka University, 836 Ohya, Shizuoka 422, Japan Received 24 March 1997; revised manuscript received 9 June 1997 Editor: W. Haxton

Abstract We have studied a narrow ?rNN resonance ( Jp = O-, T = even) based on the three-body Faddeev equation. A ?rNN resonance pole is found close to the real energy axis, although its position largely depends on the two-body TN 53 off-shell property. @ 1997 Elsevier Science B.V. PAC.? 11.55.-m; 11.8O.J~; 13.75.G~; 14.2O.Pt Keywords: Dibaryon; Faddeev equation; 9NN system; Narrow resonance; S-matrix pole

The possible existence of a narrow rrNN resonance has recently revived the study of dibaryon resonances. So far the experimental evidences have been accumulated for pionic double charge exchange reactions [ 1,2] and pp -+ pp~-b reaction [ 31, and it is now expected aa Jp = O- T = even dibaryon resonance with the energy E II 2065 MeV and the width JYN 0.5 MeV. The theoretical study of the HN system has a long history based on the three-body unitary model

which takes into account all the related reactions of ?rNN and NN channels in a unified framework [ 41. In the ?TNNthree-body theory, T = 1 states which couple to NN channel have been actively studied in order to search dibaryon resonances [ 51. The poles of the S-matrix in the complex energy plane were calculated by Ueda [ 61, and Pearce and Afnan [ 71, and they have concluded that the pole position is far away from the real energy axis and it could not give rise to observable effects. Meanwhile Garcilazo predicted a rd,

’ Tel.: +81-54-238-4930; Fax: [email protected].

+81-54-237-9184;

E-mail:

~-nn bound state which has T = 2 and does not couple to NN channel [ 81. In spite of much experimental efforts, no evidence of a bound state has been found [ 9-121. In the later refined calculation, Garcilazo and Mathelitsch concluded that there would be no TNN bound states [ 131. On the other hand, there are several predictions of dibaryon states based on the quark model [ 14- 171, and some models predict bound states or resonances. However, quark model calculation is highly model-dependent and it can hardly make definite predictions. In these unsatisfactory situations, it is worth while studying the TNN system with T = even within the three-body unitary model and, especially, exploring whether or not it can predict a narrow resonance such as reported by experiments. The aim of this Letter is to investigate the possibility of a TNN resonance and point out there could exist such a narrow resonance, although the possibility largely depends on the so-called off-shell property of the two-body TN interactions. Let us start with a simple model in order to get some insights how to make a three-body resonance

0370.2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00822-8

26

A. Matsuyama / Physics Letters B 408 (I 997) 25-30

state. In Ref. [ 181, we have studied an identical threeboson system interacting with an attractive S-wave interaction and calculated the S-matrix pole trajectory. It is well known that the three-body system develops a bound state even if the two-body attractive interaction is not strong enough to make a two-body bound state. They are called Borromean according to the symbol of the Italian family Borromeo [ 191. The typical example is neutron-halo nucleus ’ 'Li which is made of 9Li-n -n. One of the conclusions of Ref. [ 181 is that, if the interaction strength is reduced, then a bound state pole of a three-boson Borromean goes to a resonance pole even in the J = 0 channel (both angular momenta of an interacting pair and a spectator are zero). This behaviour is quite different from the twobody system where an S-matrix pole goes to a virtual state (anti-bound state) for I= 0 channel and does not make a resonance since there is no centrifugal barrier. The pole of a Borromean in the complex energy plane behaves near the E = 0 threshold as [ 181 ImE 0: -(ReE)*

(1)

and it can generate a resonance state. This is due to the effect of the three-body phase-space which grows slower than the two-body phase-space near the threshold. This behaviour is formally the same as that in the two-body scattering with an effective angular momentum 1= 3/2. In the general case with the angular momentum of the interacting pair 11and the spectator angular momentum 12, the threshold behaviour is ImE cc -(ReE)2+‘1+12 .

(2)

In the case of Jp = O- ?rNN system, the dominant interacting NN pair is 1SOand ?TNpair is St 1 (Z33) for T = 0 (2). The NN ‘SO channel is very attractive and has a virtual state, while the ?TNP33 channel is also attractive to make A resonance. Therefore we may have a good chance to make a rNN three-body resonance. The three-body Faddeev equation with separable two-body interactions is written as [ 201 X,(E)

= Z,,(E)

+ xZ/s7(E)7,(E)X,,(E)

9

Y

sition matrix respectively, and rY( E) is the propagator of the interacting pair y. This is a matrix integral equation of Fredholm type with the kernel K(E) = Z ( E)T( E). The eigenvalue equation for the kernel K(E) = Z(E)T(E) is Z(E)r(E)I&(E))

= r),(E)]&(E))

9

(4)

and the solution of the Faddeev equation Eq. (3) can be written as X(E)

=

~GW#J~WIZ(E)

c n

l-vn(E)

’

Therefore S-matrix has a pole Ep where v,, ( Ep ) = 1. A pole with Ep < 0 corresponds to a three-body bound state, while a pole on the fourth quadrant of the unphysical Riemann sheet close to the real axis represents a resonance. In order to solve the eigenvalue equation Bq. (4) for a complex energy, we have adopted the contour deformation method of Ref. [ 211. In practice, the deformed contour of momentum variable starts from the origin 0 to A (zA = Poe-“, 0 < 0 < 7r/2) and then goes to an infinity (zoo = 00 - ipa sin 0) parallel to the real axis. Then the energy E can be analytically continued to the fourth quadrant of the Riemann sheet, which is roughly -26 < arg( E) 5 0, Im (mE) 2 -pz sin 28, with m being the nucleon mass. Let us solve the eigenvalue equation Eq. (4) for Jp = O-, T = 0,2 ?rNN system. The relevant attractive two-body channels are ‘So (NN) and St 1 (TN) for T = 0, while ‘SO (NN) and Ps3 (TN) for T = 2. We are interested in the qualitative feature whether or not a resonance can exist, and thus take only these attractive channels and neglect other channels which give minor effects. We will use separable potentials for the two-body interactions and non-relativistic kinematics in the following calculation. Since a three-body resonance should be close to the continuum threshold, these assumptions will be justified and numerical results would be qualitatively unchanged for another choice of potentials or kinematics. The relativistic effects will be discussed later. We will adopt rank-l one-term separable potential

(3) where the suffix specifies a channel of interacting pair with a spectator. Z,,(E) and Xp, (E) are the oneparticle-exchange transition matrix and the full tran-

K(P,P’) = AQ(P)Q(P’)

(6)

with a monopole form factor Q(P) = cxp’/(p2 + p2> for the channel of angular momentum 1 (1 = 0, 1) .

A. Matsuyamo

/ Physics Letters B 408 Cl 997) 25-30

For S-wave channels, we adjust the strength cyand the range p so that low energy scattering data are reproduced. (The sign factor is set to A = - 1.) The scattering length a = -23.47 fm and the effective range r = 2.401 fm are fitted for ‘So (NN) channel, while the scattering length a = -0.173 m;’ and the low energy phase shifts are fitted for Sr 1 (TN) channel. (A negative scattering length corresponds to an attractive interaction in our convention.) There are several ways of parametrizing the interaction of P33 ( ~TN) channel, i.e., so-called off-shell ambiguities. It will be seen that these ambiguities result in very different conclusions for the TNN three-body system. One of the motivations to study a three-body system is that one can distinguish several potentials which are on-shell equivalent but have different offshell behaviours, and thus determine the off-shell properties of the two-body interaction. We will employ two typical parametrizations and smoothly interpolate them. One is an energy-independent parametrization with A = -1 and the other is so-called isobar model parametrization, i.e., A cc (E - MO)-I where E is the center-of-mass energy and MO is the ‘bare mass difference’ which corresponds to the bare mass in the relativistic kinematics. In the energy-independent parametrization, two parameters cy,/3 are completely determined by fitting A resonance energy EA and width Ia. and the range turns out /l N 1100 MeV/c [ 221. This large range parameter in momentum space is inevitable since the range parameter /3 is determined by the pole position as j momentump=p, +ipi= the reduced-mass of the pion and the nucleon.) The calculated scattering volume with these parameters is a = -0.136 rn~~, which is about 60% of the experimental data a (exp) = -0.214 mi3. On the other hand, in the isobar model which takes an energydependent strength and introduces another parameter MO, it turns out MO = 425 MeV, p = 321 MeV/c in order to fit the scattering volume a as well as A resonance energy and width. The range /3 N 300 MeV/c is consistent with the isobar-hole calculation of the pion-nucleus scattering [ 231. Once A resonance energy, width and the scattering volume are fitted, phase shifts are well reproduced for the center-of-mass pion momentum 0 I pr 5 250 MeV/c. (Non-relativistic on-resonance momentum is pT N 195 MeV/c.) One of the measures which characterize the strength of

27

the attractive interaction is the ratio of strength parameters R = a*/~$where CQis the critical strength to make a two-body bound state. It turns out R = 0.975,0.566 for the energy-independent and the isobar model parametrizations respectively. Thus the energy-independent parametrization is effectively by far more attractive than the isobar model. The crucial observation is that the range parameters are very different, or in other words effective strengths of the attractive interactions are very different depending on the parametrization scheme. It will be shown that this ambiguity gives totally opposite conclusions for the n-NN three-body system. We have interpolated the energy-independent parametrization and the isobar model by employing the following formula

A(E)= &

= 0

1

E/MO-1 '

Although the parameter MO has a physical meaning of ‘bare mass difference’, we will take it as a free parameter (425 I MO I cc) and MO = cc corresponds to the energy-independent parametrization. In this scheme we cannot simultaneously reproduce the scattering volume a as well as A resonance energy Ep and width IA by adjusting two parameters CYand /?. Therefore we must enlarge the possibility of the form factor in order to fit three experimental data (a, EP,r~>. We have tried several possibilities within rank-l separable potentials, i.e., sum of two monopole form factors, sum of monopole and dipole form factors for example. However, we could not find reasonable form factors, since, either parameters are very uncertain to give almost the same form factor of one-term case, or solutions cease to exist as the parameter MO is increasing. In these situations, we will choose one-term separable potential to reproduce the experimental data as much as possible. In practice, we keep the original monopole form factor with two parameters LYand p, and try to fit either (A) ( EA,Ta) or (B) ( EA, a),which will be called parameter sets A and B respectively hereafter. It is important to reproduce the exact position of A resonance energy Ep , otherwise the calculated phase shifts are away from the experimental data. In the case of set A, the scattering volume (a] is not exactly fitted and it decreases as 0.214 rni3 2 la/ 2 0.136 mi3, while in the case of set B, the calculated resonance width I* increases

28

A. Matsuyama / Physics Letters B 408 (1997) 25-30

Table 1 S-matrix pole positions with the parameter MO varied MO

425 10” 2X 10s 5 x 103 104 2 x 104 5x 104 105

2x 10s 5x 10s 106 00

Set A (MeV)

Set B (MeV)

127.0-54.8i 130.1-44.81’ 132.3-38.91 131.2-26.51 121.1-14.31 97.1-4.4i 41.3-0.2i -6.4 -44.5 -75.0 -86.9 -99.9

127.0-54.81 118.2-47.71’ 117.3-40.81’ 114.6-28.4i 107.7-18.21 96.4- lO.Oi

Ei

I

79.2-4.Oi 68.9-2.2i 62.0-1.51’

57.l-l.li 55.4- 1.oi 53.5-0.91’

as 116 MeV 5 Ia I 190 MeV as the parameter Ma increases from MO = 425 to Ma = co. The ratio R = a’/(~; varies as 0.566 < R 5 0.975 (0.953) with Ma increasing for the set A (B) . Both parameter sets can reproduce the phase shifts fairly well for 0 < pR 5 200 MeV/c. One can reproduce the scattering volume a as well as ( Eh, rA ) by using an energy-independent separable potential with two-term form factor [ 241. However, our main concern is not to fit the two-body experimental data perfectly, but to study the two-body off-shell effects on the three-body system. Therefore these parameter sets would be satisfactory for our present objectives. Now let us discuss our numerical results. We have mainly studied Jp = O- T = 2 rrNN state which is a most promising candidate of the dibaryon resonance. Other states will be discussed later. There are two relevant channels with interacting pairs, i.e., ‘So (NN) +~r and Ps3 (TN) + N. Hereafter the channel should be understood to imply dynamical interaction and it is distinguished from the state which simply denotes the Fock space basis. The spectator angular momentum L and the total spin S making the total angular momentumJ=Oare(L,S)=(O,O)fortSa(NN)+?rstate and (L, S) = ( 1,l) for P33 (7rN) +N state. Therefore the Faddeev equation becomes 2 x 2 matrix equation. By appropriately choosing a deformed contour and mesh points, we have calculated the eigenvalue of the kernel K(E) = Z (E) T( E) and searched a complex energy EP at which an eigenvalue becomes unity. In Table 1 we show the pole positions of the .Smatrix for the parameter sets A and B, and the pole

I -100

t 0

I

100

El-

Fig. 1. S-matrix pole trajectories for the parameter sets A (solid line) and B (dashed line). A cut starting from the branch point Eh - irh/Z is shown by dot-dashed line.

trajectories are plotted in Fig. 1. There is a branch point at E = EA - iI’*/ coming from the pole of the A propagator. At Ma = 425, the pole position is EP = 127.0 - 54.81’ MeV on the fourth quadrant of the unphysical Riemann sheet. With these parameters, the Fredholm determinant of the Faddeev equation at the zero energy is DF( 0) = 0.77, which is less than unity and the three-body system has attractive nature. However, as we pointed out in Ref. [ 181, the value DF( 0) does not definitely tell us whether the system is attractive or repulsive in contrast to the case of a two-body system. Therefore we will mainly deal with the pole position in order to judge whether it is a resonance or not. As the parameter Ma increases, the pole position moves upward close to the real axis and then goes to the threshold E = 0. At the energy-independent limit Mu = 00, the pole position is EP = -99.9 MeV which is a mNN bound state (Borromean) for the parameter set A, while E,, = 53.5 - 0.9i MeV corresponding to a resonance for the set B. The range parameters at Ma = 00 are /? = 1158, 804 MeV/c for sets A and B respectively. The critical parameter to make a ?rNN bound state for the set A is Ma z 9.04 x 104. The pole position expected from the data is E,,(exp) 21 50 - 0.3i MeV. Therefore both parameter sets can re-

A. Matsuyama/ PhysicsLetters B 408 f 1997)25-30

produce the pole position reasonably well with an appropriate parameter MO, i.e., Ma = 4 x lo4 for set A and MO = 00 for set B. In fact, the parameter set A with MO = 4 x lo4 gives the pole position Ep = 56.7-0.51’MeV, and the calculated range and strength ratio are p = 1042 MeV/c and R = 0.967. Although the parameter MO looks changing very largely in Table 1, the strength ratio R = (Y*/LY~ increases rather slowly. For example, R = 0.769, 0.945 and 0.971 for MO = 103, lo4 and lo5 respectively in the case of set A. Therefore one should note that the pole position is very sensitive to the interaction strength. From these numerical results, one can understand that two different parametrizations of the Pss (rrN) interaction give completely opposite conclusions for the ?rNN three-body system. Namely, in the case of set A, the energy-independent parametrization (MO = co) gives a deeply bound lrNN system ( Ep = -99.9 MeV), while the isobar model (MO = 425) gives the pole position ( Ep = 127.0 - 54.81’ MeV) far from the real axis. With this large ambiguity, we cannot definitely conclude whether there exists a +nNN resonance, although reported experimental data suggest that they are consistent with the off-shell behaviour with the parameter MO N 4 x 104. We have also calculated the threshold behaviour for the parameter set A. One can expect the relation Ei 0: -E,” (E, = Re E, Ei = Im E) since the lowest angular momentum state is ‘SJ (NN) + r which has 1= 0 pair and L = 0 spectator angular momenta. However, we have found that it goes like Ei CC--@ for the region E, 2 50 MeV. In general, Ei is expressed as Ei = a2EF + a3E: + ah@ + . . * .

(8)

Thus E: and E,’ terms cease to contribute before & = 50 MeV, and Ei is approximately given by Ei N -5 x 10T8Z$ for 50 MeV 5 E, 5 100 MeV.

This means that the contribution of the decay channel ‘&(NN> + 7r is very small, or the coupling between the channels of ‘Sa(NN) + 7~and P33( vN) + N is very weak. Therefore the main mechanism to generate a dibaryon resonance is the multiple scattering between P33( ?TN)+N channels. (Note that there is no multiple scattering between ‘So (NN) + T channels since a particle must be exchanged.) In order to clarify the situation, we have calculated d’( dibaryon) +

29

the pole trajectory only by taking P33(7rN) + N channel. It turns out E,, = 127.2 - 55.Oi MeV at Ma = 425, and Ep = -90.4 MeV at MO = co. This result is almost the same as that of the calculation including ’ SC, (NN) + T channel. The critical parameter to make a lrNN bound state is MO li 1.0 x 105. In this case, the only decay channel is d’ + P33(7rN) + N and the threshold behaviour is expected to be Ei 0: -g since it has 1 = 1 interacting pair and L = 1 spectator angular momenta. Actually our numerical results show Ei 2~ -3 x lOegg. One should note that, although P33(?rN) +N state has certainly ‘So (NIV) +T stare by recoupling the angular momenta, the threshold behaviour of ‘&(NN> + 7r state is the same as P33(7rN) + N state since the final ?rNN state comes solely from the P33(?rN) + N channel. The weakness of the transition between the channels ‘So(NN) + n and Ps3( TN) + N is due to the fact that the overlap of the wave functions of the two-nucleon relative motion is very small. For example, if we make assumptions that the nucleon is static, i.e., rn=/rnN = 0, and the range of the ’ SO(NN) wave function is zero, then the transition is exactly forbidden. Therefore the effect of the ‘So(NN) channel is only to make the system slightly more attractive, and the main feature of the dibaryon resonance is completely determined by the multiple scattering between P33( ~TN) + N channels. For the case of set B which has no ?rNN bound state, the pole trajectory is approximately Ei N - 1 x 10e7g for 53.5 MeV < E, I 100 MeV. We have calculated the pole positions of other quantum numbers than Jp = O- T = 2. In the case of T = 2 system, all the other states Jp = O+, I*, 2* do not have resonance poles close to the real axis. In fact poles stay near the original A resonance branch point EA - X*/2 and we can not expect any observable effects. This is because the recoupling coefficients of the one-particle-exchange interaction 2~~ (E) are either small or negative. For the case of Jp = O- T = 0 state, we have carried out the same calculation by taking ‘So (NN) and Srr (~TN) channels. In this case, both channels have an interacting S-wave pair and a spectator with angular momentum L = 0. Although the overlap of the two-nucleon wave functions is large, the TN $1 attractive interaction itself is very weak (R= LY*/CX~21 0.3), and thus it cannot make a resonance. In fact, in order to make a &JN bound state, one must artificially increase the interaction strengths

A. Matsupama

30

/Physics

of both channels by ‘2= (1+ (l/RCY

l)S>C?

(9)

with 6 E 0.665. (6 = 0, ( 1) corresponds to LY’= a, (ao) .) Then, near the threshold, the pole trajectory goes like Ei N - IOE; with decreasing the strength LX’, which is consistent with the general formula E$. (2). We have also estimated the relativistic effects in a simple way, i.e., replacing the phase space factor d3p by the Lorentz invariant form d3p/2EP (EP = dm) and using the relativistic kinetic energy for the pion p2/2m, -+ dm - m,. The optimum parameter Ma which reproduces the scattering volume a, A resonance energy Eh and width I* turns out Ma = 310 MeV. With this parameter MO = 310, we have calculated the S-matrix pole of the ?rNN system ( Jp = O- , T = 2) only by taking Pss( TN) + N channel and found it at EP = 130.6 - 53.61’ MeV, which is close to that of the non-relativistic kinematics. Then we have increased the parameter MO while keeping ( EA, Ta ) fitted At the energy-independent limit MO = 03, the pole position is E,, = 100.7 - 2.3i MeV, which contrasts with the non-relativistic case where there exists a deeply bound state. It seems that the relativistic effects make the system less attractive, which is consistent with the observation of Ref. [ 71. However, one can see that the small width I/2 = 2.3 MeV is the indication of the narrow resonance. In fact, if the interaction strength is slightly increased by 6 = 0.2 using the formula Eq. (9)) i.e., a’ = 1.012~ with R = 0.894 in this case, the pole moves up to EP = 52.1 - 0.2i MeV which corresponds to the experimental data. This is again the indication of the sensitivity of the pole position to the interaction strength. In summary, we have studied the two-body offshell effects on the S-matrix pole position of the ?rNN (Jp = O- , T = even) three-body system. The energyindependent and isobar model parametrizations of the n-N F’s3 interaction give totally different pole positions for T = 2 state, i.e., a deeply bound state for the energy-independent parametrization, while a pole far away from the real energy axis for the isobar model. The pole trajectory interpolating these two models is approximately Ei 0: -e and the correct pole position of experiments can be reproduced by an appropriate parameter. The smallness of the resonance width is

Lefters

B 408 11997)

25-30

due to the angular momentum barrier of the three-body system. This behaviour indicates that the dibaryon resonance is generated mainly by the multiple scattering of Pss(rN) + N channels. We would thank Prof. K. Yazaki and theoretical nuclear physics group of the University of Tokyo for useful discussions. References [ I] R. Bilger et al., Z. Phys. A 343 ( 1992) 491. R. Bilger, H.A. Clement and M.G. Schepkin, Phys. Rev. Lett. 71 (1993) 42. 121 H. Clement, M. Schepkin, G.J. Wagner and 0. Zaboronsky, Phys. Lett. B 337 (1994) 43. [3] W. Brodowski et al., Z. Phys. A 355 (1996) 5. [4] H. Garcilazo and T. Mizutani, ?rNN systems (World Scientific, Singapore, 1990). [51 M.P. Lecher, M.E. Saino and A. Svarc, in: Advances in Nuclear Physics Vol. 17, eds. J.W. Negele and E. Vogt (Plenum Press, New York, 1986) Chap. 2. [61 T. Ueda, Phys. Lett. B 119 (1982) 281. [71 B.C. Pearce and I.R. Afnan, Phys. Rev. C 30 (1984) 2022. [81 H. Garcilazo, Phys. Rev. C 26 (1982) 2685; Nucl. Phys. A 408 ( 1983) 559; Phys. Rev. C 31 (1985) 257. 191 F.W.N. de Boer et al., Phys. Rev. Lett. 53 (1984) 423. 1101 E. Piasetzky et al., Phys. Rev. Lett. 53 ( 1984) 540. [Ill J. Lichtenstadt et al., Phys. Rev. C 33 (1986) 655. 1121 D. Ashery et al., Phys. Lett. B 215 (1988) 41. [I31 H. Garcilazo and L. Mathelitsch, Phys. Lett. B 234 ( 1990) 243. 1141 L.A. Kondratyuk, B.V. Martem’yanov and M.G. Shchepkin, Sov. J. Nucl. Phys. 45 ( 1987) 776. 1151 A. Valcarce, H. Garcilazo and E Femandez, Phys. Rev. C 52 (1995) 539. [I61 E. Moro, A. Valcarce, H. Garcilazo and E Femandez, Phys. Rev. C 54 (1996) 2085. [I71 K. Itonaga, A.J. Buchmann, G. Wagner and A. Faessler, Nucl. Phys. A 609 (1996) 422. [I81 A. Matsuyama and K. Yazaki, Nucl. Phys. A 534 (1991) 620. [I91 M.V. Zhukov, D.V. Fedorov, B.V. Danilin, J.S. Vaagen and J.M. Bang, Nucl. Phys. A 539 (1992) 177. 1201 I.R. Afnan and A.W. Thomas, in: Modem Three-Hadron Physics, ed. A.W. Thomas (Springer, Berlin, 1977). Ch. 1. [211 W. Glockle, Phys. Rev. C 18 (1978) 564. 1221 I.R. Afnan and A.W. Thomas, Phys. Rev. C 10 (1974) 109. [231 M. Hirata, E Lenz and K. Yazaki, Ann. Phys. 108 (1977) 116; M. Hi&a, J.H. Koch, F. Lenz and E.J. Moniz, Ann. Phys. 120 (1979) 205. ~241 A.W. Thomas, Nucl. Phys. A 258 (1976) 417.