Probing the existence of \eta^3He mesic nucleus with a few-body approach

Motivated by the two recent observations in the WASA-at-COSY detector, we investigate the $\eta^3$He nucleus with the $\eta NNN$ few body method. We construct the effective $s$-wave energy dependent $\eta N$ potential which reproduce the $\eta N$ subthreshold scattering amplitude in the 2005 Green-Wycech model. It gives the $\eta$ separation energy and decay width of 0.19 MeV and 1.71 MeV, respectively. We also construct various sets of effective $s$-wave energy independent $\eta N$ potentials where the corresponding complex scattering lengths (a) are within the range given in most theoretical models. We obtain the bound $\eta^3$He nucleus with decay width of about 5 MeV when a is (1.0 fm, 0.3 fm), and of about 10 MeV when a is (1.0 fm, 0.5 fm).


Introduction
The  * (1535) resonance, which is close to the nucleon () threshold ( th =1487 MeV), results in a strong attractive force between the  meson and the nucleon.It is first examined by Bhalerao and Liu in the pioneering work [1], using the - coupled channels method.Soon it's verified in the dynamical calculation of the  * (1535)  11 resonance [4].Since then, the  interaction has been studied with several coupled-channel models which generate out a wide range of the real part of the  scattering length from 0.2 to 1.0 fm [2][3][4][5][6][7].At these works, the imaginary  scattering lengths are found to have a narrower range from 0.2 to 0.5 fm.
Due to the attractive  interaction, it is possible to form  mesic bound states in nuclei.The evidences that the  mesic quasibound states may exist are given in Refs.[8][9][10][11].Since then, various optical model calculations have been used for searching the  nuclear bound state [12][13][14][15][16].In Ref. [17], Xie et al. obtained a 0.3 MeV's binding energy of the  3 He and a decay width around 3 MeV by evaluating the  →  3 He near-threshold reaction.A recent interpretation of  quasibound states constrained data in the photon and hadron induced reactions implies that  is unbound,  3 He might be bound while  4 He is bound [18].
Besides the optical potential model calculations, there are also several few-body calculations concerning the ,  or  systems [19,20].With precise fewbody stochastic variational method (SVM) and several energy dependent -wave  interactions derived from several coupled channel models of the  * (1535) resonance, Barnea et al. [20] reported a bound  3 He nucleus in the condition that the real part of the  scattering length should exceed 1.0 fm approximately which yields a few MeVs binding energy between  and 3 He.For the  4 He, they found that the real  scattering length should be at least 0.7 fm to form a bound nuclei.Similar conclusions were given with qwu@nju.edu.cn(Q.Wu) ORCID(s): applying SVM and a pionless effective field theory (EFT) [21].However, in Ref. [22], with solving the four and five body Alt-Grassberger-Sandhas equations, it said neither  3 He and  4 He are bound when the  scattering length is 0.97 f m + 0.27 fm.
On the experimental side, various experiments using photon, pion, proton or deuteron beams have given signals of bound  mesic nuclei [23][24][25][26][27][28] but none of them can conclude clear existence [29].A very recent experimental campaign of searching the  mesic nuclei was conducted by the WASAat-COSY collaboration.The search for the  3 He bound state in the WASA-at-COSY detector were realizd through the  → ( 3 He, ) → ( 3 He, 2) and  → ( 3 He, ) → ( 3 He, 6) reactions.They reported a  3 He bound nucleus with decay width Γ above 20 MeV and binding energy   between 0 and 15 MeV [30].But this observation is within the range of the systematic error which doesn't allow one to make a definite conclusion.The search in the  → ( 3 He, ) → (,  0 ) reaction gives 13 to 24 nb for the bound  3 He giving the decay width between 5 and 50 MeV and the   between 0 and 40 MeV [31].
Considering the uncertainty (large error bar) in the observation of the binding energy and decay width of the  3 He nucleus in the WASA-at-COSY detector, we investigate the  3 He system under various sets of effective -wave  potentials with the few-body method, which allows us to have a certain range of the binding energy and the decay width.We construct several sets of energy independent  potentials where the  scattering lengths are contained within the range of the values given in most theoretical literature.Besides, since the scattering amplitude given in most theoretical models declines rapidly as going deeper into the subthreshold energy region, a energy dependent  potential might be necessary.Thus in this work, we also apply the energy dependent  potential in the study of the  3 He nucleus.
Based on the above calculations, we will show the correspondences between the theoretical sets of  potentials and the binding energies (decay widths) of the  3 He nucleus within the framework of few-body system.Then, if people have more definite conclusions or more certain values from the experimental side in the future, we will filter out the suitable  potential if it reproduces the experimental values.This might help us have a better understanding of the  hadronic interaction and the  mesic nuclei.
The paper is organized as follows: In Sec.2.1, we introduce the method we used to solve the four body system.Then in Sec.2.2, we introduce the effective energy dependent wave  potentials which reproduce the scattering amplitude given by the GW model [5].The energy independent -wave  potentials giving the relations with the  scattering lengths are also given.In Sec.3.1 and Sec.3.2, the binding energy and the decay width of the  3 He nucleus are investigated respectively by using the energy dependent and the energy independent  potentials.Sec. 4 is devoted to summary.In this work, we investigate the  3 He system via solving the four body  Schrödinger equation.The Hamiltonian is written as follows:

Gaussian expansion method
where   and    are the kinetic operators of  meson and nucleons, respectively.     represents the interactions between nucleons and    denotes  potential.In order to solve the  four body Schrödinger equations, We apply the Gaussian expansion method (GEM) [32,33] and write the four body  wave function with the   = 1∕2 + and isospin ( ,   ) = (1∕2, 1∕2) as: where the sets of Jacobi coordinates ( = 1 − 4) are shown in Fig. 1.
and  represent the spin and isospin wave function of the nucleons, respectively.It should be noted that both the intrinsic spin and isospin of the  meson is 0 thus we neglect its spin and isospin wave function.The  is the anti-symmetrization operator between nucleons.The relative spatial wave functions between the nucleons and eta meson in , corresponding to the three Jacobi coordinates,   1  1 (),   2  2 () and   3  3 (), are expanded by using the following Gaussian basis functions, applying the GEM, The Gaussian variational parameters are chosen to have geometric progression below, Then, the eigen energies and the coefficients   and   are obtained with applying the Rayleigh-Ritz variational method.We use the AV8'  interaction which is a modified version of AV18 interaction [34] where a tensor  interaction is included.The calculated binding energy of deuteron, 3 He and 3 H with AV8'  interaction are 2.24, 7.11 and 7.82 MeV, respectively.

Construction of effective 𝜂𝑁 potential
We construct the effective  potential through the following equation which gives its relation with the scattering amplitude  or the scattering length : where  and  0 represent for the scattering phase shift and effective range, respectively. is the wave number with  = √ 2   where  = √  − √  ℎ .For later convenience, we replace  with  √  for distinguishing with  3 He total binding energy .  is the reduced mass of the  meson and nucleon.Note that we have  =  when  → 0. For the further convenience, we only focus on the absolute value of the  ( ). 1  We use Gaussian-type  potential as: where the range parameter  and potential strength  0 need to be determined.In the case of energy dependent  potential, the  0 shall has the formula as  0 ( √ ) which is dependent on the  center-of-mass energy  √ .We then solve the two body -wave  Schrödinger equation: under the boundary condition: As mentioned in the first section, in this work, we both construct the energy dependent and energy independent  potentials.First, we construct the energy dependent  potential via reproducing the scattering amplitude  ( √ ) given in the GW model in Ref. [5] (shown in Fig. 2).The reason we choose the GW model is as follows: As will be discussed in Sec.3.2, the smallest scattering length to form a bound  3 He nucleus is around 0.7 ∼ 0.8 fm.And as will be shown in Fig. 3, the strength of the  potential drops significantly as the energy goes deeper into the subthreshold energy region.Thus, we need a relative larger value of the Real scattering length ( ( √  = 0)) to form a bound  3 He nucleus when the  √  goes deeper.Among several models, the GW model gives a very large real scattering length at 0.96 fm.For instance, in Ref. [35], it gives the real scattering length at 0.67 fm which not large enough to form a bound  3 He nucleus.
The calculated  0 ( √ ) (real and imaginary parts) are expressed in Fig. 3. Two different Gaussian range parameters  (1.0 and 4.0 fm −2 ) are used and the reasons for adopting these two values will be given in the end of this subsection.As we can see, both the real and imaginary parts of the  0 ( √ ) experience a slight increase and then drops rapidly as the energy goes smaller.
Second, we build several sets of energy independent wave  potentials which give the  scattering length within the following range as mentioned in the Sec.1: Here   and   represent the real and imaginary  scattering length for short, respectively.In Fig. 4, we depict the  0 ( Real 0 for the real part and  Imag 0 for the imaginary part) by 1 Obviously, we only consider the attractive  potential while it is not strong enough to form a bound  two body bound state.Thus, the scattering length always keeps in the same sign.separating them with different scattering length.Namely, in Fig. 4, each point has a corresponding scattering length (  , Table 1 The momentum scale parameter Λ used in several EFT models.
Refs [4,36] [7] Λ (fm −1 ) 3.9 3.2   ).In each folding line, the   is fixed and the six points represent the six different   , which are 0 ∼ 0.5 fm (step = 0.1 fm) from down to the top.) of the energy independent  potentials under different complex scattering lengths in the case of potential range parameter  = 1.0 fm −2 (up panel) and 4.0 fm −2 (down panel).In each line, the real scattering length (  ) is fixed and the six points in each line are corresponding to different imaginary scattering lengths, 0, 0.1, 0.2, 0.3, 0.4, and 0.5 fm, respectively.
One interesting thing in Fig. 4 is that, the  Real 0 has no change when the   grows from 0 to 0.1 fm but it has more and more significant enhancement when the   is around 0.5 fm.Namely, each folding line turns flatter as the   grows, specially in the case of  = 4.0 fm −2 .
It is obvious that our potentials strongly depend on the range parameter .It is often identified with the momentum cutoff Λ (Λ = 2 √ ) which is used to treat the divergent loop integrals in on-shell EFT  * (1535) models [37,38].In Table 1, we give the Λ values used in several different EFT  * (1535) models.It gives a range of  around 2 ∼ 4 fm −2 .It should be noted that in Ref. [35], Λ = 6.6 fm −1 is used which gives the potential range 1∕ √  = 0.3 fm.But according to the Ref. [19], choosing a potential range smaller than 0.47 fm ( ∼ 4 fm −2 ) would be inconsistent with staying within a purely hadronic basis.Therefore, we won't consider any  values which are larger than 4 fm −2 and use  = 1.0 and 4.0 fm −2 as a benchmark in this work.

Energy dependent 𝜂𝑁 potential
In this subsection, we show the numerical results of the  3 He nucleus using the energy dependent  potential in Fig. 3. Similar as Ref. [19], the  center-of-mass energy  √  in the environment of the  3 He nucleus can be given with: where  () =  () ∕(  +   ),   and   are the nuclear and  kinetic energy operators evaluated in center-of-mass frame. is the total binding energy of the  3 He nucleus and   is the  separation energy with respect to the 3 He threshold.Then, the binding energy of the  3 He nucleus is obtained with a self-consistent procedure.In Fig. 3  The  separation energy   (MeV), the decay width Γ (MeV) and the ⟨ √ ⟩ (MeV) obtained in the case of  = 4.0 fm −2 are in the second row.Three similar works taken from Refs.[19,21] are shown in row 3 to 5, which also use the energy dependent  potential but with a different few-body method.Note that here we temporarily only consider the real part of the  potential and treat the imaginary  potential perturbatively.

⟨𝛿
In the case of  = 1.0 fm −2 , it gives no self-consistent bound state of  3 He.As shown in Fig. 5, the intersection point of the line ( √ ) and the line ⟨ √ ⟩ =  √  is above the 3 He threshold.
As for  = 4.0 fm −2 case, the intersection point is below the 3 He threshold giving a bound state.The  binding energy (  ), ⟨ √ ⟩ and decay width Γ are shown in Table 2.For comparison, three similar calculations in Refs.[19,21] are shown in Table 2, which also employ the energy dependent  potential reproducing the GW scattering amplitude in Fig. 2 but with a different few-body method and different  potentials.All of these works are consistent with each other giving a weakly bound  3 He nucleus and the decay width around 1.5 MeV 2 .

Energy independent 𝜂𝑁 potential
As shown in Table 2, the decay width given by several few body calculations including the present work which reproduce the GW scattering amplitude are all between 1 ∼ 2 MeV, which are much less than the observed values in Ref. [30] of >20 MeV and in Ref. [31] of 5∼50 MeV.As we mentioned before, the strength of the energy dependent  potential decreases as it goes deeper into the subthreshold energy region.Thus, in order to obtain a larger decay width, we exclude this suppression and employ the  3 He system with the energy independent  potential.
First, we neglect the imaginary part of the  potential.In Fig. 6, we show the   under different   .We find that to form a  3 He bound system, the smallest scattering length for range parameter  = 4.0 fm −2 is around 0.7 fm.And the smallest scattering length for  = 1.0 fm −2 is 0.83 fm.The corresponding binding energy with respect to the 3 He threshold is between 0 ∼ 2 MeV.Then, we include the imaginary  potential and solve the four body complex Schrödinger equation.Note that here we do not treat the imaginary  potential perturbatively but fully diagonalize the Hamiltonian.We search the  3 He bound state of each set of the  potentials in Fig. 4. As we mentioned before, the smallest scattering lengths are 0.7 and 0.83 fm in order to form a  3 He bound state, respectively for  = 4.0 and  = 1.0 fm −2 cases when we only consider the real  potential.And considering the fact that the imaginary  potential have negative contribution to form the bound nuclei, it is not necessary to consider the cases when   < 0.83 fm of  = 1.0 fm −2 case and   < 0.70 fm of  = 4.0 fm −2 case.
In Fig. 7, we show whether there exists a bound  3 He nucleus under different scattering lengths, (  ,   ).The black circles mean there exists a bound  3 He nucleus while the crosses mean there doesn't exist any.It illustrates how the imaginary scattering length gives a negative contribution in forming a bound  3 He nucleus.When   reduces from 1.0 to 0.83 fm, the cases which have a bound state also reduce.Similar behaviors occur in the case of  = 4.0 fm −2 .The difference is that when   is around 0.9 to 1.0 fm, the  3 He nucleus is always bound due to the relative larger binding energy when   = 0. Thus, it keeps bound when   increases.
Besides, in Table 3 and Table 4, we show the   and Γ values of some of the black circles in Fig. 7, in which the  3 He nucleus is bound.It should be noted that the decay widths grow significantly as the imaginary scattering length is around 0.5 fm, which are around 10 MeV and much larger than the   value.Thus, with the energy independent  potential, we may obtain a much larger decay width.
As shown in Table 4, the decay width is around 10 MeV when   ∼ 0.5 fm.So we think the decay width might exceed 20 MeV when   ∼ 1.0 fm if there still exist a bound  3 He nucleus with   ∼1.0 fm.Although there has been no theoretical model which gives the imaginary

Table 3
The   (MeV) and decay width Γ (MeV) of  scattering length up to 1 fm, we still want to have a glimpse of the binding energy and the decay width in the case of   = 1.0 fm.In Table 5, the binding energy and the decay width in the case of (  ,   ) = (1.0,1.0) (in unit of fm) are shown together with the (  ,   ) = (1.0,0.3) case ( = 4.0 fm −2 ).It is surprise that the binding energy, 2.71 MeV, is even larger than the (1.0, 0.3) and (1.0, 0.5) cases.This may be due to the behavior we mentioned in the second last paragraph in Sec.2.2: The  Real 0 are enhanced more and more significantly as   is larger than 0.5 fm.Therefore, The binding energies (  ) and the decay widths (Γ) of the  3 He nucleus obtained in the present work in the case of scattering length (  ,   ) = (1.0,0.3) and (1.0, 1.0) (in unit of fm).The range parameter  = 4.0 fm −2 .The last two columns are the experimental values taken from Ref. [30] and Ref. [31], respectively.The   and Γ are in unit of MeV.
(1.0, 0.3) (1.0, 1.0) Ref. [30] Ref. [ the binding energy might have a inverse behavior and grows larger as the   increases from 0.5 fm.Shown in Table 5, the decay width in the (  ,   ) = (1.0,1.0) case is around 20 MeV, just as we expect.Together with two calculated results, we also put the experimental values in Refs [30,31].If the observed Γ > 20 MeV in Ref. [30] is positive, we may give a explanation here.But anyway, we need more definite experimental conclusions and values.

Summary
We examine the possible existence of the  3 He mesic nucleus via solving the four body Hamiltonian with the Gaussian expansion method.We construct the effective wave energy dependent  potential, which reproduces the subthreshold  scattering amplitude given by the GW model [5].The  separation energy and the decay width are then obtained by the self-consistent procedure [21], with 0.19 and 1.71 MeV, respectively.These values are consistent with the previous self-consistent few-body calculations with the same energy dependent  potential derived from the GW model [19,21].
We also construct several sets of the energy independent -wave  potentials and each set has a corresponding complex  scattering length (  ,   ).Then, we give whether or not the  3 He is bound under specific scattering length.The possibility of a large decay width is then investigated though these sets of  potential.We find that when the complex scattering length (  ,   ) is (1.0 fm, 0.3 fm), the  3 He is bound by 1.65 MeV with the decay width of about 5 MeV.And it's bound by 1.46 MeV with the width of about 10 MeV at (1.0 fm, 0.5 fm).

Figure 1 :
Figure 1: Jacobi coordinates of  four body system.

F 6 Figure 2 :
Figure 2: The scattering amplitude  ( √ ) (real part and imaginary part) at the subthreshold energy region obtained in the GW model [5].

Figure 3 :
Figure 3: The strength of the  energy dependent potential,  Real 0

Figure 5 :
Figure 5: The relation between the  center-of-mass energy  √  and total binding energy  in the  3 He system.The relation between the calculated ⟨ √ ⟩ (Eq.10) and the  √  is also given.The  = 1.0 fm −2 case and the  = 4.0 fm −2 case are shown with red hollow circle and black solid circle, respectively.The blue dashed line is drawn with ⟨ √ ⟩ =  √ .

Figure 6 :
Figure 6: The relations between the   values in the  3 He system and the  scattering lengths (only real part).The black solid line represents the  = 4.0 fm −2 and the red dashed line represents the  = 1.0 fm −2 .

Figure 7 :
Figure7: Existences of the bound 3 He system under different complex scattering lengths (  ,   ) when the range parameter  = 1.0 fm −2 (up panel) and 4.0 fm −2 (down panel).The circles mean there exist a bound state while the crosses mean there don't.
, we give the relation between the  potential and the  √ .With applying the  potential into the four body calculation, we obtain the relation between the binding energy  ( ) and the  √ .Then, based on Eq. 10, we obtain the relation between ⟨ √ ⟩ and  √ .The relations between the  and

Table 2
certainly give the self consistent value of the  3 He binding energy .

3
He system under different complex scattering length (  ,   ) (in unit of fm).The potential range parameter  is 1.0 fm −2 .

Table 4
The   (MeV) and decay width Γ (MeV) of 3He system under different complex scattering length (  ,   ) (in unit of fm).The potential range parameter  is 4.0 fm −2 .