Studies of mesic nuclei via decay reactions ^{†}^{†}thanks: Presented at International Symposium on ”Mesic Nuclei”, Kraków, Sept. 20, 2013
Abstract
Collisions in a system of two particles at energies close to a bound state in different channels are discussed. Next, the bound state decays into a third coupled channel. A phenomenological approach to He reaction is presented.
13.75.-n, 25.80.-e, 25.40.Ve
1 Introduction
The possibility of -nuclear quasi-bound states was first discussed by Haider and Liu [1, 2] a long time ago. The existence of such states has been elusive, however. At this moment the experimental evidence is rather indirect, getting the most clear indication from the measurements of He [3, 4] and of He [5, 6] , and from the realization [7] that the rapid slope of the cross section close to threshold may be a signal of a quasi-bound state. The same behavior of the total-cross section was also confirmed in the photon induced reaction He He [8]. The slope indicates large scattering length, but the final state He interaction does not allow to determine the sign of this length which would demonstrate that either a bound state or a virtual state is observed. Additional information is necessary. One possibility is to use the reaction on a three-nucleon target. Such an analysis indicates that the He system is not a bound but a virtual state [9].
Analogical enhancement close to the kinematic threshold was observed in the total cross-section of the He reaction [10, 11, 12]. Again, these results suggest a large scattering length, however, do not give a conclusive answer whether the bound state exists.
Having the scattering length one may extrapolate the scattering matrix
(1) |
where is the relative momentum, at some distance below the threshold. In this region becomes complex, . For Real one obtains the zero of the denominator and the singularity of the matrix on the physical sheet. That means a bound state for which Im becomes a quasi-bound state. In case of Real the zero of denominator may also happen but for , it lays on the second Riemann sheet of the complex energy plane. Such a state is called virtual and makes an analogy of nucleon-nucleon, spin 0, isospin 1 state known as anti-deuteron ( named so because of opposite sign of the pole position in the Im axis ). Going some distance below the threshold ( usually a short distance as depends on energy and equation 1 with constant looses its applicability)one may notice different behavior of in both cases. For a bound or quasi-bound state grows up until the energy of the bound state is reached. On the other hand, in case of a virtual state drops down immediately below the threshold.
Direct observation of the elastic scattering amplitude below the threshold is not feasible. However, one can observe a similar behavior in channels coupled to the channel where the bound state is suspected to exist. Thus in the case of reaction
(2) |
the channel of interest is He and the decay channel consists of three particles and .
As the He system seems almost bound, the He system is likely to be bound. The He might be expected to be the dominant decay channel. In such circumstances one could expect a subthreshold enhancement in the cross section for reaction (2). Surprisingly, there is no experimental confirmation of such an effect. Measurements [13, 14, 15, 16], offer a cross section of about 200 which is apparently due to a quasi-free reaction. An upper limit of the fraction that proceeds through a quasi-bound state is obtained at a level of 25 [14].
The aim of this work is to calculate/estimate the magnitude of
(3) |
reactions and to offer some speculations on the existence of the state.
2 Cross sections for the bound state formation and decays
2.1 Approximate amplitude for a two body process
Consider transition of two initial particles denoted by into two particles of a higher mass threshold. Particles are assumed to form an unstable, -wave, bound state of energy and width . There may be several modes of decay of this bound state and corresponding partial widths are denoted by .
The reaction of interest consists of three steps
Colliding particles generate unstable state ,
Unstable state lives for some time and
the unstable state decays into state .
In this section all these stages of reaction are described in a phenomenological way. The formulation used below is general but some approximations are made for a specific case : deuteron, He and .
It is assumed that the basic initial reaction
(4) |
has been studied experimentally in some region above the threshold. The relevant cross section in the threshold region may be presented in the form
(5) |
where the threshold behavior is described in part by - the relative momentum in the channel. The function is to be extracted from experiment. With deeply bound or broad states is a weakly energy dependent function, for weak binding it may indicate a sharp threshold peak.
This cross section is generated by an operator which in a standard way allows to calculate the related scattering amplitude
(6) |
is the reduced mass and , are the relative momenta in the corresponding channels. The cross section becomes
(7) |
A difficulty arises at this stage : from the scattering experiments one can extract which is the modulus of the on-shell transition amplitude for a given momentum while one needs the transition to the bound state . Formally
(8) |
where is the wave function of the bound state in the momentum space. Equation (8) involves integration over all momenta and not only over the momenta allowed by the energy conservation. To proceed without a specific model of we assume that the spacial range of this operator is characterized by the size of the final particle B ( that is He in the case of interest). This is the basic approximation of this calculation,
(9) |
where is the profile of single nucleon wave function in nucleus folded over nucleon interaction range and is a constant determined from the slope of the cross section. For an estimate we use a gaussian
(10) |
and to simplify the estimate we assume the bound state wave function in the same form
(11) |
The propagation of the bound state is described by
(12) |
where the complex part of the energy corresponds to the total decay rate.
Decay of the state into final state is given by an operator . The matrix element of this operator between the bound and the final state determines the partial width of the state. The Fermi formula gives
(13) |
from which we obtain
(14) |
In this calculation it is assumed that He is a spectator in the decay process and the final decay energy is carried by the meson and the proton. For simplicity the non-relativistic phase space is used. This may look suspicious in the meson case but the relevant reduced mass drops out from the final expression of the cross section. This calculation may be easily improved anyway. Here it serves also as a check of the normalization used.
2.2 Estimates of the cross section
The transition matrix element for the process in question is given by
(15) |
The related scattering amplitude is as in equation (6)
(16) |
and the cross section
(17) |
Collecting all factors and approximations the formula
(18) |
is obtained where the two radii were set equal .
The factor may be obtained from experimental cross sections measured and collected in ref.[17]. With the expected values , and one obtains at the peak. The result is most sensitive to the radius but the numbers obtained are below the experimental limit. However, the relation to the actual experimental limit is not that straightforward. It is the interference of reaction 3 with the quasi-free reaction that generates the experimental limit of 25 . This requires specific models and phase relations. At this moment the question :
3 Are there He bound states?
cannot be fully answered neither by experiment nor by theory. Simple, old calculations of the threshold behavior in He and He systems [18] indicated a He bound state. Since then two basic ingredients have changed :
better understanding of two nucleon decay mode.
a better knowledge of the subthreshold scattering amplitude.
The latter is represented by a best fit to multi-channel scattering data obtained in ref.[19] and plotted in figure 1. An average energy involved in the center of mass amounts to - 36 MeV ( 21 MeV binding and about 14 MeV of the residual nucleus recoil). So far below the threshold the absorptive part of the amplitude is fairly small and the rate of decay might be strongly reduced. That reduces the chance of observation via the reaction (3). This plot shows also that the attractive nuclear potential related to Re may be weaker than in the He case which involves about MeV subthreshold extrapolation.
On the other hand, the decay of the He bound state into two nucleon mode may be strongly enhanced. A phenomenological evaluation of the rates is possible as the cross sections for
(19) |
(20) |
(21) |
have been measured in the close to threshold region [22, 21]. The analysis based on the detailed balance corrected for final state interaction has been performed in ref. [20]. At central nuclear densities the related absorptive potential of the profile with a strength Im MeV was obtained. However, Helium nucleus is twice as dense and the corresponding absorptive potential rises to Im MeV. Such strong absorption may prevent binding or lead to much larger level width. To resolve some of the problems it would be useful to have also measurements of another
(22) |
decay process.
From the experimental field, the ongoing analysis of the reactions and from WASA-at-COSY, which will reach the sensitivity of several nb [16], should help to answer the question of the existence of the bound state.
Acknowledgements This work was supported by the the Polish National Science Center under grants No. 2011/03/B/ST2/00270 and No. 2011/01/B/ST2/00431
References
- [1] Q. Haider and L.C. Liu, Phys. Lett. B172, 257 (1986); Phys. Rev. C 34, 1845 (1986).
- [2] R.S. Bhalerao and L.C. Liu, Phys. Rev. Lett. 54, 865 (1985).
- [3] J. Berger, et al., Phys. Rev. Lett. 61, 919 (1988).
- [4] B. Mayer, et al., Phys. Rev. C53, 2068 (1996).
- [5] T. Mersmann et al., Phys. Rev. Lett. 98, 242301 (2007).
- [6] J. Smyrski et al., Phys. Lett. B 649, 258 (2007).
- [7] C. Wilkin, Phys. Rev. C 47, R938 (1993).
- [8] F. Pheron, et al. Phys. Lett. B709, 21-27 (2012).
- [9] A.M. Green and S. Wycech, Phys. Rev. C 68,061601(2003).
- [10] N. Willis et al., Phys.Lett. B406, 14 (1997).
- [11] R. Frascaria et al., Phys. Rev. C50, R537 (1994).
- [12] A. Wronska et al., Eur.Phys.J. A26, 421-428 (2005).
- [13] M. Skurzok et al., Prog. Part. Nucl. Phys. 67, 445 (2012).
- [14] P. Adlarson et al., Phys. Rev. C 87, 035204 (2013).
- [15] W. Krzemien et al., PoS Bormio 2013, 036 (2013).
- [16] W. Krzemien, Act. Phys. Pol.(2013) these proceedings
- [17] GEM Collaboration, A.Budzanowski et.al., Nucl.Phys. A 821, 193(2009)
- [18] S. Wycech, A.M. Green and J.A. Niskanen Phys. Rev. C 52,544(1995).
- [19] A.M. Green and S. Wycech, Phys. Rev. C 71,014001(2005)
- [20] J. Kulpa and S. Wycech, Acta. Phys. Pol. B 29,3077 (1998).
- [21] P. Moskal et al., Eur. Phys. J. A 43, 131-136 (2010).
- [22] R. Czyzykiewicz et al., Phys. Rev. Lett. 98, 122003 (2007).